Datasets:

---

ashishkgpian

/

adspapers_5_10

like 0

[ { "title": "ABSTRACT", "content": "Using multi-epoch observations of the Stripe 82 region done by Sloan Digital Sky Survey, we measure precise statistical proper motions of the stars in the Sagittarius stellar stream. The multi-band photometry and SDSS radial velocities allow us to efficiently select Sgr members and thus enhance the proper motion precision to ∼ 0.1 mas yr -1 . We measure separately the proper motion of a photometrically selected sample of the main sequence turn-off stars, as well as of a spectroscopically selected Sgr giants. The data allow us to determine the proper motion separately for the two Sgr streams in the South found in Koposov et al. (2012). Together with the precise velocities from SDSS, our proper motion provide exquisite constraints of the 3-D motions of the stars in the Sgr streams. Keywords: Galaxy: halo, stars: kinematics, methods: statistical, surveys", "pages": [ 1 ] }, { "title": "SAGITTARIUS STREAM 3-D KINEMATICS FROM SDSS STRIPE 82", "content": "Sergey E. Koposov 1,2 , Vasily Belokurov 1 , N. Wyn Evans 1 Draft version July 6, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The disintegrating Sagittarius (Sgr) dwarf galaxy remains a riddle, wrapped in a mystery, inside an enigma. Large scale photometric surveys, such as the Two Micron All-Sky Survey (2MASS) and the Sloan Digital Sky Survey (SDSS) have now revealed the structure of the tidal tails of the Sgr over more than 2 π radians on the Sky (Majewski et al. 2003; Belokurov et al. 2006). By tallying all the stellar debris in the streams and remnant, we now know that the progenitor galaxy had a luminosity of ∼ 10 8 L /circledot , comparable to the present day Small Magellanic Cloud (Niederste-Ostholt et al. 2010, 2012). The ingestation of such a large progenitor, together with its dismantling under the actions of the Galactic tides, can provide us with a wealth of information about both the Galaxy and the Sgr, if we can only decode it. Radial velocities, and sometimes metallicities and chemical abundances, are now known for many hundreds of stars in the Sgr tails (e.g., Majewski et al. 2004; Monaco et al. 2007; Chou et al. 2007, 2010). There are also ∼ 10 globular clusters associated with the Sgr tails (Law & Majewski 2010b). This rich mosaic of positions and velocities of Sgr tracers has proved surprisingly difficult to understand. Although there is no shortage of Sgr disruption models in the literature (see e.g., Helmi 2004; Law et al. 2005; Johnston et al. 2005; Fellhauer et al. 2006), they all have significant shortcomings, and fail to reproduce a substantial portion of the datasets. The most successful recent attempt is by Law & Majewski (2010a), though they do not explain the striking two stream morphology seen in the SDSS data (Belokurov et al. 2006; Koposov et al. 2012; Slater et al. 2013). Additionally they advocate the use of a triaxial halo for the Galaxy with minor axis contained with the Galactic plane, which is unattractive on other grounds (e.g. Kuijken & Tremaine 1994). Given the impasse, it is natural to look to proper motions of the Sgr stream as so as to obtain a clearer picture of its space motion. 1 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK Carlin et al. (2012) have provided the first measurements of the proper motion of the Sgr trailing tail. They took advantage of archival photographic plate data in some of Kapteyn's Selected Areas which provides a 90 year baseline. They derive proper motions for four 40 ' × 40 ' fields covering locations on the trailing tail between 70 · and 130 · from the Sgr core. However, the number of stars in each field remains modest ( ∼ 15 -55), and so the precision of the proper motion measurement is still quite low ( ∼ 0 . 2 -0 . 7 mas yr -1 ). Here, we will pursue a different tack to obtain proper motions of the trailing stream in roughly the same area of sky. As part of a project to detect supernovae, the Sloan Digital Sky Survey scanned a ∼ 290 square degree region on the Celestial Equator, known as Stripe 82 (e.g., Abazajian et al. 2009). Proper motions can be derived by matching objects between the ∼ 80 epochs (Bramich et al. 2008) obtained over time period of ∼ 7 years, whilst the co-added optical data is roughly 2 magnitudes deeper than a single epoch SDSS measurement. Although the baseline is small so the precision of a measurement of proper motion of a single star is still low, we can take advantage of the large number of Sgr tracers to get a high precision ( ∼ 0 . 1masyr -1 ) measurement for the proper motion of the ensemble. The paper is arranged as follows. Section 2 describes the extraction of proper motions for stars from the Stripe 82 data using background quasars to provide an absolute reference frame. Section 3 shows how to identify the Sgr stars in Stripe 82, where they occupy a distinctive niche in magnitude and radial velocity space. Section 4 discusses our modelling of the proper motions using both photometric and spectroscopic samples. We extract the proper motion for both the bright and faint Sgr streams identified by Koposov et al. (2012). In section 5, we compare our proper motions both with the earlier work of Carlin et al. (2012) and with the simulation data.", "pages": [ 1 ] }, { "title": "2. STRIPE 82 PROPER MOTION DETERMINATION", "content": "Stripe 82 has already been the subject of numerous studies. Its multi-epoch and multi-band imaging allows study of the variable sky and identification of many kinds of transient phenomena (see e.g., Sesar et al. 2007; Becker et al. 2008; Kowalski et al. 2009; Watkins et al. 2009). The Stripe 82 dataset has also been used to derive proper motions by Bramich et al. (2008). This lightmotion catalogue was subsequently exploited to build reduced proper motion diagrams (Vidrih et al. 2007) and analyse kinematical properties of Galactic disk and halo populations (Smith et al. 2009a,b, 2012). Even so, the proper motion measurements pioneered by Bramich et al. (2008) can be improved. For the bulk proper motion of Sgr, we are interested in the statistical properties of a large ensemble of faint tracers, so proper motions with the smallest possible systematic errors are highly desirable. The original catalogue by Bramich does not provide proper motions for stars fainter than r ∼ 20.5 and is known to have some noticeable systematics. Since the Sgr stream has a very large number of tracers in Stripe 82 (Watkins et al. 2009), we do not require a proper motion measurement for every star, but rather need small systematic errors and well understood errorbars for an ensemble. For this purpose, it makes sense to measure the proper motions relative to quasars (QSOs). Stripe 82 has a number of both spectroscopically and photometrically identified QSOs. In this work, we have used the catalogue of spectroscopic QSOs from Schneider et al. (2010) and the sample of photometrically identified QSOs from Richards et al. (2009) to extract denizens of Stripe 82. The purity of the spectroscopic catalogue of QSOs is guaranteed - all the objects are QSOs and must have zero proper motion. However, the photometric catalogue is known to have some contamination by stars. In order to minimize contaminants, we use the cut good ≥ 1, as recommended by Richards et al. (2009). This guarantees a small stellar contamination, certainly < 5%.", "pages": [ 1, 2 ] }, { "title": "2.1. Relative Proper Motions", "content": "Given a sample of QSOs each with zero proper motion, then for each star in the vicinity of the QSO, we may determine proper motion relative to the quasar. As an input catalog for the stars, we took the Stripe 82 co-add dataset (Annis et al. 2011), from which we select primary objects, classified by the SDSS pipeline as stars. The individual source detections are taken from the Stripe 82 portion of the SDSS DR7 database (O'Mullane et al. 2005) using only those fields having acceptable and good data quality flags. Matching co-added sources to detections at individual epochs is done with the 0.5 arcsec radius using the Q3C module for the PostgreSQL database (Koposov & Bartunov 2006). This procedure makes the catalogue incomplete for high proper motion objects (with proper motions /greaterorsimilar 100 mas yr -1 ), but we are not interested in such objects in our current study. Then, for each pair (star, QSO) observed multiple times by SDSS within one field, we analyse the positional offsets and errors. For the right ascension, these are defined via with similar equations for the declination. The model for the positional offsets is where µ is the proper motion of the star, t is the date of the observation, f is the fraction of outliers, and σ is the scatter around the linear relation, whilst R (∆) is the rectangular function to account for the outliers. The resulting likelihood is then minimized with respect to the 4 parameters µ , ∆ 0 , f , σ with the error-bars determined from the Hessian at the minimum. Repeating this procedure for the offsets in declination gives us the proper motions and their errors, µ α , σ µ,α , µ δ , σ µ,δ for all the sources with a spectroscopic or a photometric QSO nearby.", "pages": [ 2 ] }, { "title": "2.2. Systematic Errors in the Proper Motions", "content": "After performing the computation of the proper motion for individual stars, and before trying to measure the statistical proper motions for ensembles of stars, it is important to check for the presence of possible systematics, as well as to examine the accuracy of the error bars. Fig. 1 presents such an assessment. In this paper, we are focusing on the particular part of Stripe 82, which intersects with the Sgr stream. As systematic effects may depend on the right ascension, Fig. 1 uses only the proper motions in the right ascension range 20 · < α < 50 · . The left panel of Fig. 1 shows the histogram of proper motions of the spectroscopic QSOs measured relative to the photometric QSOs and normalized by the error bar provided by our fitting procedure. The red overplotted curve shows a Gaussian with the center at zero and unity dispersion. The excellent match of the histogram with the Gaussian curve shows that that the proper motions do not possess noticeable systematic offsets, and that the error bars on the proper motions are a faithful description of the precision. It is also quite clear from the plot that this is not true for proper motion measurements by Bramich et al. (2008), which have noticeable systematics and error underestimation. The middle panel of Fig. 1 shows the the proper motion of the spectroscopic QSOs relative to the photometric QSOs versus the g -r color difference. The points with error bars show the median proper motion in bins of g -r , while the error bars are 1.48 times the median absolute deviation of proper motions within a corresponding g -r bin. Since all the points lie within 1 σ of zero, we conclude that the color-dependent terms in the proper motions are negligible down to the precision 0.1 -0.2 mas yr -1 . This holds true at least within the color range -0 . 2 /lessorsimilar g -r /lessorsimilar 1, which is the color-range applicable to 99% percent of the photometric QSOs. Later we will see that the proper motion of the Sgr stream determined from the photometric sample with g -r ∼ 0 . 3, g -i ∼ 0 . 3 and the spectroscopic sample with g -r ∼ 0 . 5, g -i ∼ 0 . 75 agree each other, further confirming the small level of color-related systematic errors. And last, the right panel of Fig. 1 shows the precision of our proper motion measurements as a function of r -band magnitude. There is a considerable scatter, but the median proper motion precision of ∼ 2mas yr -1 is significantly better than that of Bramich et al. (2008) for - - - large range of magnitudes.", "pages": [ 2, 3 ] }, { "title": "3. THE SAGITTARIUS STREAM", "content": "The Sgr stream in the southern Galactic hemisphere is known to have a complex structure. Koposov et al. (2012) used main-sequence turn-off (MSTO) stars extracted from SDSS Data Release 8 (DR8) to demonstrate the existence of two streams - a thicker, brighter stream and a thinner, fainter stream displaced by ∼ 10 · . The brighter stream has multiple turn-offs as well as a prominent red clump, whereas the fainter stream does not. Koposov et al. (2012) argued that the brighter stream was composed of more than one stellar population, including some metal-rich stars, whereas the fainter stream is dominated by a single metal-poor population. Here, we are primarily interested in the intersection of the Sgr stream with the SDSS Stripe 82. Fig. 2 shows the density of MSTO stars extracted via the cuts 0 . 25 < g -i < 0 . 35 and 19 . 8 < r < 22 . 5 in the southern hemisphere, together with Stripe 82 demarcated by the red lines. Again, two distinct streams structures are clearly visible - a brighter one crossing the Equator and Stripe 82 at a right ascension of ∼ 35 · and a dimmer one crossing the Equator at a right ascension of ∼ 15 · . The magnitude distribution of MSTO stars along Stripe 82 is shown in the middle panel of Fig. 2. The Sgr streams are clearly evident, with the brighter stream visible at α ∼ 35 · and fainter one at α ∼ 15 · . Since the distances to the streams are not constant with α , and since Stripe 82 crosses the stream at an angle, we observe a clear distance gradient with right ascension. The right panel of Fig. 2 shows the radial velocity as a function of right ascension for giant stars, extracted by the cut log( g ) < 4 using the SDSS spectroscopic measurements. This time not only are stars within Stripe 82 included, but also those satisfying | δ | < 20 · and lying near the stream ( -15 · < B < 5 · ). As already found by Watkins et al. (2009), the Sgr stream is visible at the V GSR ∼ -150kms -1 3 . The figure also shows the variation of radial velocities as the stream stars return from the apocenter of the Sgr orbit. As they will be needed later, we extract the radial velocities along the stream, together with the velocity dispersion, by fitting a Gaussian to the Sgr signal. Although Figure 2 shows the radial velocities as a function of right ascension, it is important to realise that we perform these fits in bins along Λ, which is the angle along the stream from the remnant, measured positive in the trailing direction, as defined in Majewski et al. (2003)). The resulting measurements as a function of Λ are shown in Table 1. Importantly, Fig. 2 demonstrates that the Sgr stream stars are visible as a distinct bright feature in magnitude and velocity space. Therefore, this information can be used to select Sgr member stars and statistically measure their properties such as proper motion.", "pages": [ 3 ] }, { "title": "4. MODELING PROPER MOTIONS", "content": "As demonstrated above, in the Southern Galactic hemisphere, it is possible to achieve clean separation of the Sgr trailing tail stars and the Galactic foreground in R GS both apparent magnitude and the radial velocity space. However, even for high-confidence stream members, the uncertainty of individual proper motion measurements ( ∼ 2 -3mas yr -1 ; see Section 2) is comparable or higher than the expected tangential velocity of the stream ( ∼ 1 -3mas yr -1 ; Law & Majewski 2010a). It is therefore crucial to combine the signal from as many stream members as possible to beat down the noise. Koposov et al. (2010) have shown that, for the regions of apparent magnitude and radial velocity space dominated by the stream, an accurate measurement of the systemic proper motion of an ensemble of stars belonging to the stream can be obtained through simple background subtraction. Alternatively, the overall stellar distribution in the space of observables can be modeled, yielding the direct contributions of the Galaxy and the stream. If such models can be cast in the 3D space of proper motion and magnitude or radial velocity, it is feasible that a superior measurement of the proper motion can be achieved as the contribution of the background to both systematic and random noise will be reduced. Unfortunately, adequate analytical models of the Sgr stream and the Note . -α 1 , α 2 columns denote the edges of the boxes in right ascension in Stripe 82 used to perform the proper motion measurements Note . - Λ, B, X, Y, Z correspond to the centers of the fields where the proper motions are measured. a Spectroscopic Sample Galaxy are not readily available. Therefore, we choose to approximate these distributions by a sum of multidimensional Gaussians. This so-called Gaussian mixture is a well known semi-parametric technique widely used to model multi-dimensional datasets (McLachlan & Peel 2000). Gaussian mixtures have several key properties that make the model-fitting straightforward and fast. First, the uncertainties associated with the measure- ments (as well as the missing data) are naturally incorporated into the model. Secondly, there exists a guaranteed fast convergence procedure - the Expectation Minimisation (EM) algorithm (Dempster et al. 1977). In this work, we have used the extreme-deconvolution package, the open-source Gaussian mixture implementation by Bovy et al. (2011).", "pages": [ 3, 4, 5 ] }, { "title": "4.1. Photometric sample", "content": "As evident from the dissection of the SDSS dataset shown in Fig. 2, main sequence turn-off stars (MSTO) are the most numerous Sgr stream specimens available to carry out the proper motion analysis. The typical density of the MSTO stars in the stream is 150 stars per square degree, compared to the foreground density of around 50. Stripe 82 slices through both the bright and the faint streams at an angle, resulting in a slightly tilted bi-modal distribution in the plane of right ascension and r band magnitude. A faint Eastern wing to the bright Stream can also be discerned at higher right ascension (see middle panel of Fig. 2). Motivated by the evolution of the distance and the structure of the stream along the Stripe, we split the MSTO sample into three parts: the faint stream at 5 · < α < 22 · , the bright stream at 25 · < α < 40 · , and the Eastern wing of the bright stream at 42 · < α < 52 · (we label those fields as FP1, FP2, FP3 respectively; they are shown by red error-bars on Fig. 2). In each of the three right ascension bins, the overall 3D distribution of MSTO stars in the space of ( µ α , µ δ , r) was modeled with a mixture of 5 Gaussians, The initial guess for the free parameters was obtained by running the K-means algorithm (MacQueen 1967). The choice of number of Gaussians ( N gau ) was motivated by looking at the cross-validated log-likelihood (Arlot & Celisse 2010) as a function of N gau . This initially rises as a function of N gau , peaks at N gau ∼ 5 and stays roughly constant for N gau > 5, demonstrating the goodness of fit and the absence of overfitting. We have also made sure that the objects in the Sgr stream are represented by a single Gaussian. When applying the extreme deconvolution, we force the covariance matrix for the Sgr component to be diagonal, e.g. assuming zero covariance between the proper motion and the apparent magnitude. The proper motions for the stream stars were not corrected for the solar reflex motion. The errors in the proper motion measurement, i.e. the uncertainties in determining the centre of the Gaussian representing the Sgr Stream, were determined either from bootstrap procedure or from the Hessian matrix of the likelihood function. Fig. 3 shows the data analysed as well as the best fit Gaussian mixture model of the main Sgr stream 25 · < α < 40 · . It is reassuring to see that the Gaussian mixture model was able to describe the data distribution adequately. The resulting measurements of the proper motion together with the uncertainties are given in Table 2. Among ∼ 7000 stars analyzed, according to the model, ∼ 4000 stars belong to Sgr, while ∼ 3000 stars belong to the background/foreground population.", "pages": [ 6 ] }, { "title": "4.2. Spectroscopic sample", "content": "The number of members of the Sgr stream with SDSS spectroscopic measurements is significantly smaller than the number of MSTO stars used in the previous section. Despite this, knowledge of the radial velocities and surface gravities allows us to have much purer samples of the Sgr stream members. In this section, we perform a complementary measurement of the stream's proper motion using spectroscopic members only. To this end, we select all stars with spectra in Stripe 82 lying at 14 · < α < 50 · ( we label this area FS4 and show it with the blue error-bar on Fig. 2) and classified by the SDSS spectroscopic pipeline as giants log( g ) < 4. Despite very wide range of selected right ascensions, most of the Sgr members with spectroscopy in the Stripe 82 region located at the center of the bright stream, at α ∼ 30 · . According to the right panel of Fig. 2, the Sgr stream's radial velocity changes along the Stripe. Therefore, to ease the modeling, we subtract the variation of the radial velocity centroid from the data using the measurements presented in Table 1. The resulting radial velocity is approximately constant as a function of RA, and the measurements of proper motions and radial velocities µ α , µ δ , ˜ V GSR = V GSR -V model , GSR (Λ) can now be represented by a mixture of Gaussians. We run the extreme deconvolution on the sample of ∼ 1500 stars and find that 3 Gaussian components are sufficient to describe the dataset. We follow the strategy outlined in Section 4.1, with the difference that the covariance matrix of the Gaussian representing the stream component is now set free. Fig. 4 shows the density distribution of the data, together with the best-fit Gaussian mixture model for the Stripe 82 stars with spectra. By comparing the grey-scale density with the red contours, we can confirm that the model has captured the properties of the dataset reasonably well. The last line of Table 2 reports the values of the proper motion for the spectroscopic sample, and confirms that the measurements for the two independent stream samples agree within 2 σ . It is also particularly reassuring since Sgr members in the spectroscopic sample have very different colors (g -i ∼ 0.75) from the Sgr members in the photometric sample (g -i ∼ 0.3) of the Sgr giants. Therefore the agreement of the proper motions from two samples is a proof of a small level of color-related systematic effects. The proper motion signals we have measured have a very large contribution of the solar reflex motion. Since we have distance estimates to the Sgr stream in the South measured elsewhere (e.g. Koposov et al. 2012), we can correct for this and check whether the proper motions are actually aligned with the streams. Figure 5 shows the proper motion vectors after applying the solar reflex corrections. As we can clearly see, the proper motions are indeed properly aligned with the streams and are consistent with each other. For this calculation we have still used the distance to the faint stream as given by Koposov et al. (2012, 2013), although, there is evidence in Slater et al. (2013) that the stream the faint stream is 3 -5 kpc closer.", "pages": [ 6, 7 ] }, { "title": "5. COMPARISONS TO EARLIER WORK", "content": "There are many models of the Sgr stream in the literature, all purporting to provide the distances, velocities and proper motions as a function of position on the sky (e.g. Fellhauer et al. 2006; Pe˜narrubia et al. 2010; Law & Majewski 2010a). Although quite detailed, all the models fail to reproduce at least some of the features that we see on the sky (e.g., Niederste-Ostholt et al. 2010; Koposov et al. 2012). But, it still instructive to see where our data measurements lie relative to the existing models. We have chosen the Law & Majewski (2010a) model as a comparison benchmark, because it is arguably the most comprehensive and up-to-date. There are several caveats to be borne in mind. The first is related to the fact that while our central measurement in Stripe 82 at α ∼ 35 · or (Λ , B ) ∼ (100 · , 0 · ) corresponds directly to the center of the trailing tail in the simulation, the fainter stream which crosses Stripe 82 at α ∼ 15 · or Λ , B ∼ (90 · , 10 · ) doesn't have a counterpart in the simulation by Law & Majewski (2010a). The second is related to the choice of rotation velocity of the Local Standard of Rest. As shown in Carlin et al. (2012), the observed proper motion signal is sensitive to the adopted V LSR . The models of Law & Majewski (2010a) have been computed and fitted under the assumption of the IAU standard value of the V LSR =220kms -1 , while the current best estimates are slightly higher at 230 -250kms -1 (Bovy et al. 2009; Reid et al. 2009; Deason et al. 2011). Throughout the paper, we adopted the value of 235 km s -1 . Fig. 6 shows our data points overplotted onto the distribution of tracers from Law & Majewski (2010a) model, where we have selected tracers from the trailing tail only ( Lmflag = -1 and Pcol < = 7), and lying within 20 degrees of Stripe 82 ( | δ | < 20). The Λ values of our measurements correspond to the centers of the rectangular areas used to perform the measurements, and the error-bars indicate the extent of those areas. The agreement for the central field (Λ ∼ 105 · ), as well as for the edge of the bright stream (Λ ∼ 120 · ), is quite good. At the location of the fainter Sgr stream (empty circle on the plot), the model doesn't have many particles, as expected. In order to facilitate further comparisons, Table 3 gives the positions and velocities corresponding to our measurements. Here, ( X,Y,Z ) and ( U, V, W ) are in a right handed Galactocentric coordinate system with the Sun located at X = -8.5 kpc, and V LSR = 235kms -1 . The distances taken from the work by Koposov et al. (2012), and the error on the velocities does not take into account any possible systematic error distance of ∼ 10%. Readers wishing to compare Sgr disruption models with our observations should use Table 3 only for rough checks. A proper comparison entails measuring the average proper motions within the same area of the sky as we do and comparing these numbers directly with our Table 2. We also compare our measurements with the data from Carlin et al. (2012). Out of 6 fields analysed by Carlin et al. (2012), 4 have measurable Sgr signal and only 2 of those (SA 93 and SA 94) lie near the Stripe 82. The other 2 fields with detectable Sgr signal are ∼ 15 degrees away from the Stripe 82. As Figure 6 shows, the error-bars of our measurements are much tighter than Carlin et al. (2012), and the proper motions themselves agree approximately within the combined errors.", "pages": [ 7 ] }, { "title": "6. CONCLUSIONS", "content": "We have measured the proper motion of the Sgr stream using Stripe 82 data. By tying the astrometry to the known QSOs and combining the measurements for large samples of stars, we have been able to achieve high precision proper motions. Our measurements have been performed for two distinct groups - spectroscopically selected red giants/subgiants with SDSS radial velocities and photometrically selected MSTO stars. The proper motions of those agree very well and after correcting for the solar reflex motion are tightly aligned with the Sgr stream. Our results are in agreement with earlier measurements by Carlin et al. (2012), but, by virtue of the large numbers of stars in our samples, our statistical error bars are substantially smaller, typically about 0 . 1 mas yr -1 . There are three fields in Stripe 82 for which the proper motion of the photometric sample has been measured, and one field for the spectroscopic sample. Combining this information with distances from Koposov et al. (2012) gives us the full six dimensional phase space coordinates of the Sgr trailing stream at four locations along Stripe 82. We provide a table of three dimensional positions and velocities of Sgr stream stars, in which the contributions of the motion of the Sun and LSR have been removed. To complement the work on the Sgr streams in the South, it would be particularly useful to carry out a corresponding kinematical analysis for the streams in the North. This is a subject to which we plan to return in a later contribution. The combination of the proper motions, radial velocities and distances in both Galactic hemispheres should allow us to make further progress in the understanding of the complicated structure of the Sgr streams and solve some of the riddles posed by their existence. The authors would like to thank Jo Bovy for making his extreme deconvolution code available and supporting it. The extreme deconvolution code version used in this paper was r112. Most of the results presented in this paper have been done using open source software numpy/scipy/matplotlib and scikitslearn (Pedregosa et al 2011). An anonymous referee helped us remove a number of obscurities from the paper. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the MaxPlanck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.", "pages": [ 7, 8 ] }, { "title": "REFERENCES", "content": "Pe˜narrubia, J. 2010, ApJ, 712, 516 Smith, M. C., Whiteoak, S. H., & Evans, N. W. 2012, ApJ, 746, 181 Vidrih, S., Bramich, D. M., Hewett, P. C., et al. 2007, MNRAS, 382, 515 Watkins, L. L., Evans, N. W., Belokurov, V., et al. 2009, MNRAS, 398, 1757", "pages": [ 8, 9 ] } ]

[ { "title": "ABSTRACT", "content": "The rich statistics of galaxy rotations as captured by the velocity function (VF) provides invaluable constraints on galactic baryon physics and the nature of dark matter (DM). However, the comparison of observed galaxy rotations against cosmological models is prone to subtle caveats that can easily lead to misinterpretations. Our analysis reveals full statistical consistency between ∼ 5000 galaxy rotations, observed in line-of-sight projection, and predictions based on the standard cosmological model (ΛCDM) at the mass-resolution of the Millennium simulation (H i line-based circular velocities above ∼ 50 km s -1 ). Explicitly, the H i linewidths in the H i Parkes All Sky Survey (HIPASS) are found consistent with those in S 3 -SAX, a post-processed semi-analytic model for the Millennium simulation. Previously found anomalies in the VF can be plausibly attributed to (1) the mass-limit of the Millennium simulation, (2) confused sources in HIPASS, (3) inaccurate inclination measurements for optically faint sources, and (4) the non-detectability of gas-poor early-type galaxies. These issues can be bypassed by comparing observations and models using linewidth source counts rather than VFs. We investigate if and how well such source counts can constrain the temperature of DM.", "pages": [ 1 ] }, { "title": "CONFRONTING COLD DARK MATTER PREDICTIONS WITH OBSERVED GALAXY ROTATIONS", "content": "Danail Obreschkow 1 , Xiangcheng Ma 2 , Martin Meyer 1 , 3 , Chris Power 1 , 3 , Martin Zwaan 4 , Lister Staveley-Smith 1 , 3 , Michael J. Drinkwater 5 1 International Centre for Radio Astronomy Research (ICRAR), M468, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia 2 The University of Sciences and Technology of China, Centre for Astrophysics, Hefei, Anhui 230026, China 3 ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) 4 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching b. Munchen, Germany and 5 School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia ApJ, accepted 15/02/2013", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Mass and angular momentum are crucial galaxy properties, since their global conservation laws constrain the history and future of galaxy evolution (Bullock et al. 2001b,a). Moreover, measurements of mass and angular momentum uncover hidden dark matter and potentially constrain its nature (Zavala et al. 2009; Obreschkow et al. 2013). In recent decades, the mass statistics have been studied in detail via the mass function (MF, Li & White 2009), the luminosity function (LF, Loveday et al. 2012), and the auto-correlation function (Blake et al. 2011). By contrast, angular momentum remains a side-topic, normally addressed indirectly via the Tully-Fisher relation (TFR, McGaugh 2012) or used as a means of recovering the mass distribution in individual galaxies (de Blok et al. 2008). Spatial statistics of angular momentum and the related circular velocity function (VF, Gonzalez et al. 2000; Desai et al. 2004; Zwaan et al. 2010; Papastergis et al. 2011) remain relatively unexplored. This is despite the fact that the VF offers a tremendous potential with regard to comparing LFs obtained in different wave-bands (Gonzalez et al.), measuring various mechanisms of feedback in the evolution of galaxies (Sawala et al. 2012), and constraining the temperature of dark matter (Zavala et al. 2009). In fact, measuring a galaxy's rotational velocity is challenging, since it requires both a measurement of the galaxy inclination, typically drawn from a spatially resolved optical image, as well as a measurement of the line-of-sight rotational velocity, typically obtained from the Doppler-broadening of the 21 cm emission line of neutral hydrogen (H i ). Today, only two H i surveys offer reasonably large samples to construct VFs, the H i Parkes All Sky Survey (HIPASS, Barnes et al. 2001) and the ongoing Arecibo Legacy Fast ALFA (ALFALFA, Giovanelli et al. 2005a,b). They are the largest surveys by the cosmic volume and by the number of galaxies, respectively. The VFs derived from HIPASS (Zwaan et al. 2010) and the 40%-release of ALFALFA (Papastergis et al. 2011) were both compared against theoretical models, including predictions by the S 3 -SAX- model (Obreschkow et al. 2009a), the only current model of frequency-resolved H i -emission lines in a cosmological simulation. These comparisons uncovered statistically significant differences, some of which could be attributed to gas-poor massive early-type galaxies (Zwaan et al.), but the physical implications remained unclear. Differences in the faint-end of the velocity function (Fig. 9 in Papastergis et al.), near the resolution limit of the S 3 -SAX-model, seemed to hint a possible breakdown of the current cosmological model. In a new attempt to understand and exploit these differences, we successively found them to be subtle artifacts of the comparison itself, hence motivating a more detailed analysis. This paper presents a revised comparison between the H i line profiles in HIPASS and S 3 -SAX. We deliberately focus on HIPASS, while reserving a similar analysis of the ongoing ALFALFA survey for the future, because HIPASS already has optical inclinations available, exhibits a detailed completeness function, and contains less cosmic variance than 40%-ALFALFA in terms of the redshift-distribution of the galaxies (see Fig. 4a by Martin et al. 2010 versus Fig. 2 bottom by Zwaan et al. 2005). The HIPASS data is compared against the S 3 -SAX-model in various ways. A key result, worth highlighting early, is the full consistency between the 50percentile H i linewidth W 50 in HIPASS and S 3 -SAX, as illustrated by the counts in Fig. 1. In this work we compare both apparent H i linewidths and inclinationcorrected circular velocities using source counts, as well as space density functions. The different aspects uncovered by these functions are discussed in detail, as well as their reliability as statistical estimators. Based on the results, we finally conjecture that linewidth counts might be a useful tool for measuring the temperature of dark matter, and discuss how well HIPASS can, in principle, constrain this temperature. The manuscript is organized as follows. Section 2 first explains the observed dataset (HIPASS with optical imaging) and its simulated counterpart (S 3 -SAX). Five statistically independent simulations are generated specifically to assess the effects of cosmic variance. The observed and simulated datasets are then truncated to congruent subsamples suitable for their comparison. This comparison is presented in detail in Section 3. In Section 4, the consistency between HIPASS and S 3 -SAX is interpreted and discussed with respect to the TFR and alternative models of dark matter. Section 5 summarizes the results in a list of key messages.", "pages": [ 1, 2 ] }, { "title": "2.1. Observed data: HIPASS", "content": "HIPASS is a blind search for H i emission at declinations dec < +25 · in the velocity range -1 , 280 km s -1 < cz < 12 , 700 km s -1 , where c is the speed of light and z is the redshift. This survey resulted in 5317 identified galaxies, gathered in two catalogs: the HIPASS galaxy catalogue (HICAT, Meyer et al. 2004; Zwaan et al. 2004) containing 4315 sources with dec < +2 · , and its northern extension (NHICAT, Wong et al. 2006) containing 1002 sources with +2 · < dec < +25 · . The H i lines of these 5317 sources have been parameterized in various ways. In this work, we will use the luminosity distance D L , given in Mpc, the velocity-integrated line flux S int , given in Jy km s -1 , the corresponding H i mass M HI = 2 . 36 · 10 5 S int D 2 L (1 + z ) -1 , given in M /circledot , the peak-flux density S p , given in mJy, and the linewidth W 50 (' W max 50 ' in HICAT), given in km s -1 and measured at 50% of the peak flux density. HIPASS uses a channel width of 13 . 2 km s -1 , but parameterization was carried out after two stages of smoothing (Tukey and Hanning), resulting in a full-width-half-max resolution of 26 . 4 km s -1 for W 50 . Doyle et al. (2005) presented optical counterparts for HICAT, identified in the b J -band plates of the SuperCOSMOS Sky Survey (Hambly et al. 2001). To each of these galaxies they fitted an ellipse to measure the semimajor axis a , the semi-minor axis b , and the position angle. There are 3618 sources in HOPCAT with identified values a and b . From those values the galaxy inclinations i can be estimated using the spheroid assumption, where q ≡ b/a and q 0 denotes the intrinsic axis ratio, here taken to be q 0 = 0 . 2 to remain consistent with Zwaan et al. (2010). As in the latter work, we here define the circular velocity V c of a galaxy as although the actual asymptotic rotational velocity may slightly differ from V c . This section summarizes S 3 -SAX (Obreschkow et al. 2009a), the first cosmological model of resolved H i -emission lines of galaxies. S 3 -SAX builds on model-galaxies generated by a semianalytic model (SAM, De Lucia & Blaizot 2007). The latter relies on the Millennium simulation (Springel et al. 2005) that tackles the evolution of cold dark matter (CDM) in a comoving box measuring (500 h -1 Mpc) 3 , where h is defined via the local Hubble constant H 0 ≡ 100 h kms -1 Mpc -1 . This simulation uses the standard cosmological model (ΛCDM) with parameters h = 0 . 73, Ω m = 0 . 25, Ω b = 0 . 045, Ω Λ = 0 . 75, and σ 8 = 0 . 9. From this simulation CDM halos and their merging histories are extracted. The SAM then assigns galaxies to the centers of these halos using a cooling model and evolves global galaxy properties, such as stellar mass, gas mass, and morphology according to physical rules allowing for feedback from black holes and supernovae. The free parameters in this SAM were tuned to the locally observed color-magnitude distribution, but there is no explicit fit to galaxy sizes, rotations, and gas properties. The number of model-galaxies at a time of 13 . 7 · 10 9 yr, i. e., today, is to about 3 · 10 7 . Although the cosmological parameters of the Millennium simulation are slightly inconsistent with the newest estimates (Komatsu et al. 2011), the present-day galaxy properties remain nearly unaffected according to calculations by ? . Given the evolving model-galaxies of the SAM, Obreschkow et al. (2009a) assigned refined cold gas properties to each galaxy. Their method, sketched out in Fig. 2, can be summarized as follows. The scale radius of galactic disks is calculated directly from the spin of the dark matter halo. To do so, a variable ratio ξ ∈ [0 . 5 -1] between the specific angular momentum of baryons and dark matter was adopted. This ratio is a function of the Hubble-type and the stellar mass, adjusted such that the resulting disk scale radii optimally reproduce those of the real galaxies in The HI Nearby Galaxy Survey (THINGS, Walter et al. 2008). Given the disk scale radius and the total cold gas mass from the SAM, the radial H i surface density Σ HI ( r ) is calculated using a pressure-based model for the ratio between molecular (H 2 ) and atomic (H i ) hydrogen (Obreschkow & Rawlings 2009), derived from the THINGS sample (Leroy et al. 2008). In parallel, circular velocity profiles V total c ( r ) = ( V halo c 2 ( r ) + V disk c 2 ( r )+ V bulge c 2 ( r )) 1 / 2 are calculated from the circular velocities implied by the gravitational potentials of the dark matter halo, the galactic disk, and the central bulge, respectively. Convolving V total c ( r ) with Σ HI ( r ) then results in a model for the frequency-resolved H i emission line (dashed line in Fig. 2c), which, when convolved with a Gaussian Kernel for dispersion, turns into a smooth profile (solid line). Departing from the cubic box of the Millennium simulation populated with model-galaxies with resolved H i emission lines, Obreschkow et al. (2009b) produced a sky-model with apparent extra-galactic H i emission as seen by a fixed observer. To do so, the Cartesian coordinates ( x, y, z ) of the model-galaxies were mapped onto apparent positions, i.e., right-ascension (RA), declination (dec), and redshift z , using the method of Blaizot et al. (2005). This method explicitly accounts for the fact that galaxies more distant from the observer are seen at an earlier stage in their cosmic evolution. Along with this mapping, the intrinsic H i luminosities are transformed into observable fluxes. Moreover, the H i emission line of each galaxy is corrected for the inclination of the galaxy, respecting, however, the isotropy of turbulent/thermal dispersion. The linewidth W 50 at the 50%-level of the peak flux density is then measured from the apparent H i line of the inclined model-galaxy. To allow a clean comparison with observations, we then calculate the circular velocity V c of a model-galaxy via eq. (2). This is an important step, since V c can differ from the asymptotic value of V total c ( r →∞ ) by up to 30% for some galaxies with relatively compact H i distributions. For the purpose of this paper we realized five different virtual skies by placing the cosmic volume probed by HIPASS five times inside the simulation box of the Millennium simulation as shown in Fig. 3. There is no overlap between these five sub-volumes, making them (almost) statistically independent. These five virtual sky volumes will be used to quantify the effects of cosmic variance, that is the random effects of the locally inhomogeneous large scale structure. In the aim of comparing the S 3 -SAX-model against HIPASS it is crucial to note that the gridded beam of the HIPASS data measures 15.5' at full-width-half-max. Using the S 3 -SAX sky-model, we find that this limited spatial resolution implies a non-negligible probability for two or more H i disks to be confused, i.e., to fall inside the same beam and simultaneously overlap in frequency. This confusion must hence be accounted for when comparing observations against simulations. We do so by merging simulated galaxies, whose centroids are separated less than 15.5' and whose H i lines overlap in frequency. The common H i mass is then taken as the sum of the individual components and W 50 is measured from the combined line as shown in Fig. 4. We further define the circular velocity V c of the merged object as the H i mass-weighted average of the circular velocities of the components. This procedure reduces the number of simulated sources by about 2%.", "pages": [ 2, 3, 4 ] }, { "title": "2.3. Sample selection", "content": "Let us now construct subsamples of sources in HIPASS and S 3 -SAX using identical selection criteria. Two types of samples will be considered depending on whether optical counterparts are required for HIPASS sources. These counterparts are needed when considering circular velocities V c , since those require estimates of the galaxy inclinations. However, masses M HI and linewidths W 50 do not require optical data. We shall call the larger galaxy sample, in which optical counterparts are irrelevant, the 'reference sample'. A subsample of this reference sample, in which all galaxies have optical inclinations and thus estimates of V c , is then called the ' V c -sample'. The precise selection criteria of these two samples are listed in Tab. 1 and explained in the following.", "pages": [ 4 ] }, { "title": "2.3.1. Reference sample", "content": "Volume truncation: Since the volume of S 3 -SAX exceeds that of HIPASS, the former must be truncated to the field-of-view (FOV) and redshift range of HIPASS, i.e., dec ≤ +25 · and cz ≤ 12 , 700 kms -1 . These criteria truncate S 3 -SAX to sub-volumes matching the colored regions in Fig. 3. Mass limit: The limiting H i mass, above which S 3 -SAX can be considered complete, is about 10 8 h -1 70 M /circledot (Obreschkow et al. 2009a), where h 70 ≡ h/ 0 . 7, i.e., h 70 = 1 if H 0 = 70kms -1 Mpc -1 , as consistent with current observations (Jarosik et al. 2011). The simulated H i MF The upper list shows the criteria applied to all data presented in this paper. The lower list shows the additional selection criteria applied when comparing circular velocities V c , which require optical inclination measurements. Here, R ∈ [0 , 1] denotes a random number drawn separately for every galaxy and equation. drops rapidly below this limiting mass due to the limited mass resolution of the Millennium simulation. We must therefore limit the comparison between simulated and observed sources to the mass range M HI ≥ 10 8 h -1 70 M /circledot . This criterion removes 71 nearby galaxies from HIPASS, that is about 1 . 3% of the 5317 identified sources. Limiting linewidth: HIPASS does not resolve sources with linewidths smaller than 25 km s -1 (see Section 3.1 in Meyer et al. 2004). For correctness we therefore apply the selection criterion W 50 ≥ 25 km s -1 to S 3 -SAX, although this only reduces the number of simulated sources by about 0 . 1%. Completeness limit: In HIPASS, real sources are detected with a probability approximated by where E(x) ≡ max { 0 , erf(x) } . The parameters are p = (0 . 036 , 19 , 0 . 36 , 1 . 1) for dec ≤ +2 · (Zwaan et al. 2004) and p = (0 . 02 , 5 , 0 . 14 , 1) for dec > +2 · (Wong et al. 2006). The same completeness function must be applied to S 3 -SAX. This is done by drawing a random number R uniformly from the interval [0 , 1] for every simulated galaxy, and retaining the galaxy only if C ( S p , S int ) ≥ R . In addition, we must account for the fact that the completeness function itself is very uncertain for C ( S p , S int ) < 0 . 5 (e.g., Fig. 6 in Zwaan et al. 2004). As in Zwaan et al. (2010), we therefore only retain galaxies with C ( S p , S int ) ≥ 0 . 5. These completeness cuts reduce the total number of simulated sources in the HIPASS volume to about 8%, while reducing the number of observed sources by 514, i.e., by an additional 9 . 8% after the 1 . 3% mass cut. This concludes the construction of the reference samples.", "pages": [ 4 ] }, { "title": "2.3.2. V c -sample", "content": "To compare observed and simulated values of V c , the reference sample must be further reduced to a HOPCAT equivalent sample, i.e., a subsample with optically measured inclinations. Volume truncation: We must exclude the galaxies with dec > +2 · , for which optical inclinations are not readily available in HIPASS. In doing so, the number of observed and modeled objects shrinks by roughly 20%. HOPCAT completeness: Out of all galaxies in the reference sample with dec ≤ +2 · only 86% yield optical inclinations. Most of the remaining objects lie too close to the galactic plane, where the stellar foreground deteriorates extragalactic optical imaging. To account for this incompleteness, we reduce the number of simulated galaxies to 86% by only retaining the objects satisfying 0 . 86 ≤ R , where R ∈ [0 , 1] is a random number. Inclination selection: Galaxies with inclinations close to face-on exhibit poor inclination measurements, which, given their small values of i , result in highly uncertain values of V c when using eq. (2). As in Zwaan et al. (2010) we therefore only retain galaxies with i ≥ 45 · , hence reducing the sample sizes by an additional 29%.", "pages": [ 4, 5 ] }, { "title": "3. COMPARISON BETWEEN HIPASS AND S 3 -SAX", "content": "Given the identically selected samples of observed and simulated galaxies, we shall now compare the statistics of the H i line profiles. This comparison will be carried out both at the level of direct source counts (Section 3.2) and at the level of space density functions (Section 3.3).", "pages": [ 5 ] }, { "title": "3.1. Sample size", "content": "Let us first consider the raw size of the observed and simulated samples given in Tab. 2. The mean number of sources in the five simulated reference samples is about 4926 with a standard deviation of 482. This standard deviation is significantly higher than the Poisson shot noise of ∼ √ 4926 ≈ 70, demonstrating the non-negligible effect of large scale structure in HIPASS. The number of observed sources in the reference sample is clearly consistent with the simulation. We therefore expect the normalization of corresponding source count statistics and space density functions to be consistent between observation and simulation. By contrast, the mean number of sources in the simulated V c -samples (about 2013) undershoots the number of observed sources by about 339 or 14%. This difference is slightly larger than the characteristic value of cosmic variance of 267, estimated from the standard deviation of the object-numbers in the five simulated V c -samples. As argued in Section 3.2, this moderately significant difference between the sizes of the observed and simulated V c -samples is at least partially explainable by a small fraction of inaccurate inclination measurements in HOPCAT. Those tend to assign high inclinations ( i ≥ 45 · ) to objects, which in actual fact have low inclinations ( i < 45 · ) and should hence be removed from the observed V c -sample.", "pages": [ 5 ] }, { "title": "3.2. Source counts", "content": "A refined statistical analysis consists of counting the number of galaxies, binned by specific galaxy properties. The properties of particular interest are the H i linewidth W 50 and the circular velocity V c defined by eq. (2). For completeness we also analyze the statistics of the H i mass M HI . The source counts of M HI and W 50 are derived from the reference samples. In turn, the source counts of V c , which require inclination measurements, must be performed using the smaller V c -samples. Number of sources in each sample. The selection criteria are listed in Tab. 1 and the simulated volumes are shown in Fig. 3. Figs. 5a-c show the observed (bars) and simulated (lines) counts of M HI , W 50 , and V c , respectively. The grey solid lines correspond to the five individual simulations, while the black lines represent the geometric means of these functions. Variations between the five models are due to cosmic variance. The observed source counts exhibit several error bars, representing the uncertainties described in Tab. 3. Some of these uncertainties are statistical, while others are systematic and thus correlated across different bins. The observed and simulated M HI counts in Fig. 5a are moderately consistent. Four of the five models and the mean model show a slight bump around M HI ≈ 4 · 10 8 M /circledot . This seems to be a feature of the particular SAM chosen here, since it is also present in the b J -band LF of the same SAM (see Fig. 8 right of Croton et al. 2006), but absent in other SAMs building on the Millennium simulation (e.g. Baugh et al. 2005). Fig. 5b is the central plot of this paper and extends on Fig. 1. It demonstrates that the simulated linewidths W 50 are fully consistent with the observed ones. We emphasize that this consistency requires that the simulated and observed samples are constructed according to identical selection criteria (see Tab. 1). Experimenting with different completeness functions C further revealed the importance of using the smooth completeness function C ( S p , S int ) provided for HICAT and NHICAT. A hard sensitivity limit, i.e., C ( S p , S int ) as a step-function, is not sufficient in that it induces variations larger than the error bars. Moreover, accounting for the confusion of sources turns out to be vital. If instead all individual galaxies in the simulated sky were considered distinguishable, then the mean source counts are given by the dashed line in Fig. 5b. The difference is most pronounced at the largest linewidths of W 50 /greaterorsimilar 500 km s -1 (artifact '2'). Thus, the largest values of W 50 in the observed data are mostly due to confused sources, i.e., galaxies within the same telescope beam and with H i line profiles overlapping in frequency space. In constructing the original HICAT dataset (Meyer et al. 2004) an effort was made to flag and separate sources exhibiting confused H i line profiles. About 9% of the sources with W 50 ≥ 500 km s -1 in the reference-sample have been flagged as confused (as opposed to 7% in the whole reference sample). By contrast, our modelling revealed that most sources with W 50 ≥ 500km s -1 are confused. This means that it may be impossible to identify most instances of confusion by relying exclusively on the information in the HIPASS data. An example of a confused source is shown in Fig. 6. The counts of circular velocities V c are shown in Fig. 5c. The models are consistent with the observations for V c > 50 km s -1 , but drastically differ for smaller velocities (artifact '3'). The only major difference between Fig. 5b and Fig. 5c is the inclination-correction [see eq. (2)]; therefore the excess of observed sources with V c < 50 km s -1 suggests an issue with their inclinations. A systematic visual inspection of the b J -band images of the SuperCOSMOSSky Survey used in HOPCAT uncovered that a vast majority ( > 90%) of the galaxies with V c < 50 km s -1 (about 11% of the 2352 objects in the V c -sample or 6% of all 4315 galaxies in HICAT/HOPCAT) are problematic. They are either too faint or too irregular for an optical estimation of the inclination, or they simply exhibit erroneous shape parameterisations. Fig. 7 displays three representative examples of the latter case. The ellipses in Fig. 7 represent the original parameterization in terms of minor axis, major axis, and position angle. The axis ratios of these ellipses imply inclinations i > 45 · [via eq. (1)]. To the naked eye, however, these three galaxies are nearly face-on spiral disks ( i < 45 · ), especially in the multi-color image of the source HIPASSJ1200-00, which is about two magnitudes deeper than SuperCOSMOS. Using the 'correct' inclination for this source rather than that suggested by HOPCAT, increases V c roughly by a factor two. Since the correct inclination is then below 45 · , this source would be rejected from the V c -sample and thus disappear from Fig. 5c. In conclusion, there is a small fraction of incorrect shape identifications in HOPCAT, which happens to dominate the low-end of the V c counts. Incidentally, this also explains the asymmetric scatter skewed towards low rotational velocities in the HOPCAT-based TFR (upper panels in Fig. 3 in Meyer et al. 2008).", "pages": [ 5, 7 ] }, { "title": "3.3. Space densities", "content": "The source counts presented in the previous section depend on the selection criteria of the survey listed in Tab. 1. Survey-independent and thus more fundamental statistical measures are the space density functions φ ≡ d N/ d V . These functions represent the absolute number of sources, detected or not, per unit of cosmic volume and per unit of galaxy properties, such as M HI (H i MF) or V c (VF). Evaluating these functions from empirical data requires inverting the com- teness function, as well as removing the effects of cosmic variance. This is achieved by the two-dimensional stepwise maximum likelihood (2DSWML) method developed by Zwaan et al. (2003) and applied by Zwaan et al. (2005) and Zwaan et al. (2010) to recover the observed space density functions of M HI , W 50 , and V c , shown in Figs. 5d-f. Note that the data shown here include all Hubble-types. Figs. 5d-f also display the simulated counterparts (solid lines), obtained simply by binning all galaxies contained in the redshift z = 0 box of the Millennium simulation. This box is large enough for cosmic variance to be neglected. However, the observed space density functions still obey the same cosmic variance as the respective source counts. Therefore the cosmic variance uncertainty is plotted with the observed data, although we derive its value from the variations between the five simulated source counts. Fig. 5d reveals that the simulated and observed H i MFs are only marginally consistent in the sense that the simulation falls within the error bars for about 50% of the data points rather than 67%. The fact that the agreement was slightly better in source count statistics of Fig. 5a might indicate a minor artifact in the reconstruction of the observed H i MF. For example, as suggested by Zwaan et al. (2004), the 'true' completeness function C exhibits a slight dependence on the shape of the H i line profile (single-peaked, double-peaked, flat-top) in addition to the main dependence on S p and S int . This small higher-order effect could be captured by extending the 2DSWML method to 3D using C ( S p , S int , shape ). Explanation of the different error bars shown in Fig. 5. The error 'type' refers to whole sample. For example, distance errors due to peculiar velocities are systematic for an individual source, but statistical at the level of a sample of sources with random peculiar motions. In combining multiple errors into a single error bar, statistical errors are added in quadrature, while systematic errors are added linearly. the models and observations. In the small velocity range, these inconsistencies (artifact '1') directly relate to the mass resolution limit of the Millennium simulation. This limit implies a significant incompleteness of simulated objects with W 50 /lessorsimilar 80 km s -1 and V c /lessorsimilar 50 km s -1 (and M HI < 10 8 M /circledot to the left of Fig. 5d). In turn, this masslimit is probably linked to the spurious bumps around W 50 ≈ 120 km s -1 and V c ≈ 70 km s -1 . A more subtle feature in Figs. 5e and 5f are the significant deviations at W 50 /greaterorsimilar 500 km s -1 and V c /greaterorsimilar 200 km s -1 (artifact '4'). Those deviations are absent in the corresponding source counts of Figs. 5b and 5c. A systematic investigation of the simulated galaxies in this high-velocity regime reveals them to be dominated by early-type galaxies of numerical Hubble-type T ≤ 0 (E, S0) hosting low-mass, but fast-rotating H i disks. Excluding those objects from the simulation modifies the predicted functions in Figs. 5e and 5f to the dot-dashed lines, which are in much better agreement with the observed data, as already noted by Zwaan et al. (2010). In other words, the model predicts that the high-end of the VF is dominated by gas-poor early-type galaxies, but it also predicts that HIPASS is unlikely to detect these galaxies; hence the consistent source counts. To show this explicitly, let us calculate the maximal comoving distance D max (in Mpc) out to which a galaxy { M HI , V c } can be detected in the sense that the completeness function C drops to 50% at that distance. Substituting S p for 10 3 S int V -1 c (approximation for i > 45 · ) and S int for 4 . 2 · 10 -6 M HI D -2 max (approximation for z /lessmuch 1), C ( S p , S int ) = 0 . 5 [using eq. (3)] numerically solves to D 2 max ≈ 7 · 10 -6 M HI exp( -0 . 4 V 0 . 34 c ) for HICAT and NHICAT. The cosmic volume V max (in Mpc 3 ), in which HIPASS can detect a galaxy specified by { M HI , V c } then becomes V max ≈ 0 . 63 · (4 π/ 3) D 3 max , where 0.63 is the sky-coverage of HIPASS, i.e., On the other hand, the S 3 -SAX model allows us to predict the space-density ρ ( M HI , V c ) of a source { M HI , V c } , defined as the average number of sources per Mpc 3 within a pixel { log 10 M HI ± ∆ / 2 , log 10 V c ± ∆ / 2 } (here using ∆ = 0 . 1). The product then approximates the predicted number of HIPASS detections per pixel in the { M HI , V c } -plane. Fig. 8 displays ρ ( M HI , V c ) colored by galaxy type with isolines of n ( M HI , V c ). The region n ( M HI , V c ) < 1 contains less than one detection per pixel and thus represents a 'blind zone' of HIPASS. This blind zone contains the gas-poor ( M HI /lessorsimilar 10 9 M /circledot ), fast-rotating ( V c /greaterorsimilar 200 km s -1 ) earlytype galaxies predicted by the model. Since HIPASS is very insensitive to these galaxies, it is simply unable to recover the predicted high-end of the VF. Surveys deeper than HIPASS are needed to verify whether the predicted amount of massive gas-poor early-type galaxies is correct. For now, it seems safe to conclude that the HIPASS VF approximates the VF of late-types, even if no Hubbletype cut is applied to the dataset. On a side-note, the deeper ALFALFA survey does indeed find significant differences in the high-velocity end of the velocity function (e.g. Fig. 4 in Papastergis et al. 2011). In principle, the artifacts '2' and '3' of Figs. 5b and 5c are still present in Figs. 5e and 5f, but they are occluded by the even stronger artifacts '1' and '4'. This shows that the comparison between models and observations is less prone to spurious artifacts, when performed using source counts. Furthermore, within the source counts, W 50 is a less problematic quantity than V c due to artifact '3'.", "pages": [ 7, 8, 9 ] }, { "title": "4. DISCUSSION", "content": "This section discusses the physical implications of the excellent consistency between observed and simulated H i linewidths, as well as potential applications.", "pages": [ 9 ] }, { "title": "4.1. Interpretation of the consistency of W 50", "content": "What does the consistency between the observed and modeled W 50 -counts (Fig. 5b) tell us? Does it strengthen the case of the ΛCDM model or does it merely manifest the empirical tuning of the free parameters in the galaxymodel? There is, as argued here, a bit of both. The local galaxy stellar MF in the model has been adjusted indirectly by tuning the feedback from star formation and black holes on the interstellar medium to reproduce the observed b J -band and K -band LFs (Croton et al. 2006). Moreover, the radii of galaxies match the locally observed mean stellar mass-to-scale radius relation (Obreschkow et al. 2009a). One might therefore expect the galaxy rotations, which depend roughly on mass and radius, to align with local observations. In this argument, it should nonetheless be emphasized that the free modelparameters (feedback coefficients and the spin ratio of baryonic matter to dark matter) have only been varied within the restricted ranges consistent with current highresolution observations and high-resolution simulations. Therefore, we can at least conclude that the consistency of the W 50 -counts in Fig. 5b confirms ΛCDM within the current uncertainties of galaxy-modelling. Moreover, it is worth emphasizing that the relation between stellar mass and scale radius is subject to very large scatter, both observationally and in the model (Fig. 2 in Obreschkow et al. 2009a). Therefore, even if the mean relation between stellar mass and scale radius is fixed to observations, this merely corresponds to an overall normalization of the VF and the corresponding W 50 -counts. The details of these functions depend on the shape of the multi-dimensional probability-distribution of halo mass, stellar mass and disk scale radius. This shape has not been constrained by empirical fits. Instead, it depends directly on the masses, spins, and merging histories of the dark halos in the Millennium simulation. This argument increases the support of ΛCDM.", "pages": [ 9 ] }, { "title": "4.2. Constraints on the dark matter type", "content": "Quantifying the degree to which the W 50 -counts support ΛCDM is of course a more delicate affair. For example, what is the actual range of allowed dark matter particle masses m DM , assumed infinite in CDM but finite in Warm Dark Matter (WDM) models? Answering this question would require a large array of different WDM models, similar to the Millennium simulation, equipped with SAMs, where all the uncertainties associated with every free parameter are tackled down to the W 50 -counts. The mammoth numerical requirements of this task lie at the edge of current super-computing capacities. Here, we limit the analysis to a first order approximation of the variation of the W 50 -counts as function of m DM , keeping the free parameters of the galaxy-model fixed to their best values in ΛCDM. This approximation is obtained by rescaling the number density of each galaxy in S 3 -SAX, one-by-one, by φ WDM ( M halo ) /φ CDM ( M halo ), where M halo is the mass of the halo containing the galaxy and φ WDM ( M halo ) and φ CDM ( M halo ) are the local halo MFs of WDM and CDM halos, respectively. These MFs are modeled analytically by evolving the initial density field using the formulation of Sheth & Tormen (1999). WDM models are obtained by subjecting the initial CDM power spectrum to a transfer function following Bode et al. (2001). For consistency, these calculations were performed using the cosmological parameters of the Millennium simulation. Fig. 9 shows the W 50 -counts for CDM and two WDM scenarios with particle masses m DM = 1 keV c -2 and m DM = 0 . 5 keV c -2 , respectively. Although the observed W 50 -counts are only marginally consistent with m DM = 1 keV c -2 and inconsistent with m DM = 0 . 5 keV c -2 , those WDM cosmologies need not to be incompatible with the observed W 50 -counts. In fact, we cannot exclude that varying the free parameters of the SAM within the currently allowed ranges can bring the WDM models in line with the observed data. However, Fig. 9 conveys that if all free parameters in the galaxy-model can be replaced by independently determined precise values, then the W 50 -counts from HIPASS can indeed discriminate between CDM and WDM with 1 keV c -2 particles.", "pages": [ 9, 10 ] }, { "title": "4.3. Tully-Fisher relation", "content": "So far, we have shown that the H i masses and circular velocities of the galaxies in the SAM (as modeled via S 3 -SAX) are consistent with observations; and Croton et al. (2006) showed that the stellar masses are consistent with local observations as well. However, the fact that circular velocities and masses are independently consistent with observations does not, in fact, imply that their twodimensional distribution is correct, too. Therefore, we shall finally discuss the two-dimensional distribution of circular velocities and baryon masses, i.e., the baryonic TFR. To remain consistent with observational standards the circular velocity is here approximated as V 20 , defined as half the apparent H i linewidth W 20 (measured at the 20% peak flux level), corrected for inclinations. The observational data is drawn from McGaugh et al. (2000) and corrected for h = 0 . 73. These data include galaxy types from dwarfs to giant spirals, whose values of V 20 have been recovered from H i line measurements, corrected for inclinations drawn from optical imaging. Only inclinations above 45 · were retained to restrict the uncertainties of sin i . The comparison of these data against S 3 -SAX in Fig. 10 reveals a good consistency, although the observational scatter is 50% larger than that of S 3 -SAX. This difference is explainable by measurement uncertainties, especially regarding the inclination corrections in the low-mass end of Fig. 10a according to McGaugh et al.. Additionally, the S 3 -SAX-model probably underestimates the scatter in V 20 by ignoring the detailed substructure of H i , such as turbulent mixing in mergers, high-velocity clouds, warps, and gas-rich satellites. Unlike the baryonic TFR (Fig. 10b), the stellar mass TFR (Fig. 10a) clearly departs from a power-law relation for galaxies with V 20 < 200 km s -1 . As emphasized before (e.g. McGaugh et al.), this reflects the trend for high gas-fractions in low-mass galaxies and confirms that the TFR is fundamentally a relation between circular velocity and total mass, which is a function of the baryon mass (Papastergis et al. 2012).", "pages": [ 10 ] }, { "title": "5. CONCLUSION", "content": "This paper presented a detailed comparison between the H i lines is HIPASS and those in S 3 -SAX, a cosmological model of galaxies with resolved H i lines. The results can be condensed into a list of key messages. different masses seen at different inclinations. However, the W 50 counts of HIPASS are nonetheless sensitive to the temperature of dark matter. These cosmological tests and prospects promise to become particularly fruitful when applied to future H i surveys, such as the full ALFALFA survey and ultimately the ASKAP HI All-Sky Survey (WALLABY) with the Australian Square Kilometer Array Pathfinder (ASKAP). Those future surveys should be paralleled by equally sophisticated simulated counterparts, namely mock-skies produced from galaxy-models extending to considerably smaller masses and circular velocities than those based on the Millennium simulation.", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "Barnes D. G., et al., 2001, MNRAS, 322, 486 Baugh C. M., Lacey C. G., Frenk C. S., Granato G. L., Silva L., Bressan A., Benson A. J., Cole S., 2005, MNRAS, 356, 1191 Blaizot J., Wadadekar Y., Guiderdoni B., Colombi S. T., Bertin E., Bouchet F. R., Devriendt J. E. G., Hatton S., 2005, MNRAS, 360, 159 Blake C., et al., 2011, MNRAS, 415, 2892 Bode P., Ostriker J. P., Turok N., 2001, ApJ, 556, 93 Bullock J. S., Dekel A., Kolatt T. S., Kravtsov A. V., Klypin A. A., Porciani C., Primack J. R., 2001a, ApJ, 555, 240 Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S., Kravtsov A. V., Klypin A. A., Primack J. R., Dekel A., 2001b, MNRAS, 321, 559 Croton D. J., et al., 2006, MNRAS, 365, 11 de Blok W. J. G., Walter F., Brinks E., Trachternach C., Oh S.-H., Kennicutt Jr. R. C., 2008, AJ, 136, 2648 De Lucia G., Blaizot J., 2007, MNRAS, 375, 2 Desai V., Dalcanton J. J., Mayer L., Reed D., Quinn T., Governato F., 2004, MNRAS, 351, 265 Doyle M. T., et al., 2005, MNRAS, 361, 34 Driver S. P., et al., 2011, MNRAS, 413, 971 Freedman W. L., Madore B. F., Scowcroft V., Burns C., Monson A., Persson S. E., Seibert M., Rigby J., 2012, ApJ, 758, 24 Giovanelli R., et al., 2005a, AJ, 130, 2613 -, 2005b, AJ, 130, 2598 Gonzalez A. H., Williams K. A., Bullock J. S., Kolatt T. S., Primack J. R., 2000, ApJ, 528, 145 Guo Q., White S., Angulo R. E., Henriques B., Lemson G., Boylan-Kolchin M., Thomas P., Short C., 2012, ArXiv e-prints Hambly N. C., Irwin M. J., MacGillivray H. T., 2001, MNRAS, 326, 1295 Jarosik N., et al., 2011, ApJS, 192, 14 Komatsu E., et al., 2011, ApJS, 192, 18 Leroy A. K., Walter F., Brinks E., Bigiel F., de Blok W. J. G., Madore B., Thornley M. D., 2008, AJ, 136, 2782 Li C., White S. D. M., 2009, MNRAS, 398, 2177 Loveday J., et al., 2012, MNRAS, 420, 1239", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "We present results from a study to determine whether relations - established in the local Universe - between the mass of supermassive black holes (SMBHs) and their host galaxies are in place at higher redshifts. We identify a well-constructed sample of 18 X-ray-selected, broad-line Active Galactic Nuclei (AGN) in the Extended Chandra Deep Field South - Survey with 0 . 5 < z < 1 . 2. This redshift range is chosen to ensure that HST imaging is available with at least two filters that bracket the 4000 Å break thus providing reliable stellar mass estimates of the host galaxy by accounting for both young and old stellar populations. We compute single-epoch, virial black hole masses from optical spectra using the broad MgII emission line. For essentially all galaxies in our sample, their total stellar mass content agrees remarkably well, given their BH masses, with local relations of inactive galaxies and active SMBHs. We further decompose the total stellar mass into bulge and disk components separately with full knowledge of the HST point-spread-function. We find that ∼ 80% of the sample is consistent with the local M BH -M ∗ , Bulge relation even with 72% of the host galaxies showing the presence of a disk. In particular, bulge dominated hosts are more aligned with the local relation than those with prominent disks. We further discuss the possible physical mechanisms that are capable building up the stellar mass of the bulge from an extended disk of stars over the subsequent eight Gyrs. Subject headings: galaxies: evolution - galaxies: active", "pages": [ 1 ] }, { "title": "THE BLACK HOLE - BULGE MASS RELATION OF ACTIVE GALACTIC NUCLEI IN THE EXTENDED CHANDRA DEEP FIELD - SOUTH SURVEY", "content": "MALTE SCHRAMM 1 AND JOHN D. SILVERMAN Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa, Japan 277-8583 (Kavli IPMU, WPI) Accepted for publication in The Astrophysical Journal on December 6, 2012", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Adetermination of the physical mechanisms through which supermassive black holes are built up at the centers of galaxies have been one of the key issues in astrophysics (see Kormendy & Richstone 1995). Such processes are thought to further provide a link black hole growth and the formation of the bulges of their host galaxies based on both observations and theory. Correlations between the mass of the central black hole and absolute magnitude (Magorrian et al. 1998; Marconi & Hunt 2003; Häring & Rix 2004), and/or stellar velocity dispersion (Gebhardt et al. 2000; Merritt & Ferrarese 2001) of the spheroidal component indicate that the mass ratio between a SMBH and its bulge is constant over a wide dynamic range in mass (e.g. MBH / MBulge = 0.0014; Häring & Rix (2004); hereafter MBH -MBulge relation). We will refer to this relation as the local relation. Over the past years, several studies have addressed whether there is an evolution of the mass relations between the central black hole and its host galaxy. Such studies must rely upon galaxies with accreting SMBHs (i.e., Active Galactic Nuclei; AGN) since the region of influence surrounding black holes cannot be resolved at higher redshifts. While those hidden by obscuration (i.e., type 2 AGNs) give a rather clean view of their host galaxy, unobscured (type 1) AGN are the only systems for which black hole masses can be measured. Although, an estimate of the mass of the host bulge is challenging due to the glare of a luminous AGN that only gets more difficult at high redshift. Fortunately, optical imaging from space with HST can be used to disentangle the light between an AGN and its host galaxy (e.g., Sánchez et al. 2004; Jahnke et al. 2004, 2009; Bennert et al. 2011b; Cisternas et al. 2011) due to the high spatial resolution and well understood point spread func- ion. Alternatively, it is also possible to measure the stellar velocity dispersion from optical spectra for less luminous AGNs (Woo et al. 2008); this method requires high signal-to-noise spectra that limits its application to high redshift AGNs. Even if the host galaxy is resolved only limited spectral coverage is usually available to estimate stellar masses. Single-band studies are therefore restricted to the black hole mass - luminosity relation or have to make assumptions on the mass-to-light ratio of the host galaxy (see Peng et al. 2006a,b; Decarli et al. 2010a,b). Merloni et al. (2010) implemented a new approach to measure the stellar mass content of AGNhost galaxies through template fitting of the broad-band photometric spectral energy distribution (Brusa et al. 2009; Xue et al. 2010). With this approach, Merloni et al. (2010) estimate the total stellar mass content which provides only an upper limit to the bulge mass. Bennert et al. (2011) take a significant step forward by using the multi-band HST data available in the GOODS (Giavalisco et al. 2004) fields to decompose the AGN and host galaxy light including a bulge component tractable through multiple filter bandpasses. Unfortunately, the sample is selected to be a redshifts (1 < z < 2) for which the optical imaging falls below the rest-frame 4000 Å break. Surprisingly, the aforementioned studies find elevated black hole masses as compared to either the bulge component (Woo et al. 2008; Bennert et al. 2011b) or total (Merloni et al. 2010) stellar mass of their host galaxy. Recently, Jahnke et al. (2009) and Cisternas et al. (2011) report that the mass ratio between the black hole and the total stellar mass of its host galaxy is similar to local values possibly indication of an undermassive bulge. Even with the considerable effort achieved to date, there are several challenges that need to be met in order to accurately determine the evolution of the M BH -M ∗ , Bulge at higher redshift. First, the decomposition of optical light is more dif- cult due to the strong surface brightness dimming of the host galaxy as compared to the AGN. To mitigate this effect, high resolution imaging with high signal-to- noise is needed to adequately resolve the host galaxy especially for bright AGN. Equally important, at least, one rest-frame optical color and a luminosity is needed to constrain the stellar mass content of the host galaxy (Bell et al. 2003). A color that covers the 4000 Å break provides a good estimator on the underlying stellar mass-to-light ratio. It is worth highlighting that the 4000 Å break moves out of the optical filter bands at z > 1 . 2 thus requiring deep high-resolution NIR imaging. Furthermore, due to the limited physical resolution at high redshift and the fact that galaxies become more compact, it may be challenging to classify galaxies morphologically such as distinguishing between disturbed and undisturbed hosts. In this study, we determine the M BH -M ∗ , Total and M BH -M ∗ , Bulge relations at 0 . 5 < z < 1 . 2 using a sample of 18 X-ray selected broad-line AGN (BLAGN) from the Extended Chandra Deep Field - South Survey. Based on HST/ACS imaging from GEMS (Rix et al. 2004) and GOODS(Giavalisco et al. 2004), we measure the stellar mass content of their host galaxies including the bulge component. We specifically focus on this redshift range so that there is at least one HST band above and below the 4000 Å break thus providing a rest-frame color required for accurate conversion of light to mass. Black hole masses are determined using single epoch virial mass estimation based on the MgII emission line. In Section 2, we describe our sample. In Section 3, we describe our analysis of the HST/ACS data that involves the image decomposition of the total light into AGN, bulge and disk decomposition, and stellar mass estimation. Black hole masses are fully detailed in Section 4. Section 5 and 6 presents the results including a discussion of the relations between the mass of the SMBH and their total/bulge stellar mass. Finally, in Section 7 we give a summary of the results. Throughout this paper we assume a flat cosmology with H 0 = 70 kms -1 Mpc -1 , Ω M = 0 . 3 and Ω Λ = 0 . 7. A determination of the M BH -M ∗ , Bulge relation using AGN samples, also requires an assessment of the possible biases originating from selection of AGN (see Salviander et al. 2007; Lauer et al. 2007). While quiescent galaxies are selected by their magnitude or luminosity, active galaxies (e.g., unobscured, broad-line AGN) are often selected by their optical nuclear luminosity or magnitude. The bias introduced by the luminosity (i.e., mass) limit will have a stronger effect at the high mass end of the black hole mass function which is strongly decreasing. Offsets from the local M BH -M ∗ , Bulge relation seen in samples of luminous AGN with massive BHs (e.g., Merloni et al. 2010; Bennert et al. 2011b; Peng et al. 2006a,b) may be explained by such a bias. Therefore, a sample selected at lower luminosities that fall well below the knee of the black hole mass function should be less impacted by such a bias.", "pages": [ 1, 2 ] }, { "title": "2. AGN SAMPLE", "content": "Currently, broad-line (type 1) AGNs provide the only means to establish the relation between BH mass and galaxy mass beyond the local universe. This is due to the fact that BH mass measurements rely upon a determination of the velocity widths of gas in the vicinity of the BH as provided by broad emission lines (e.g., Kaspi et al. 2000; Vestergaard & Peterson 2006). High-resolution imaging (best if taken from space) can then be used to detect the extended emission from the underlying host galaxy. There have been numerous studies of the host galaxies of type 1 AGNs using such techniques (e.g., Jahnke et al. 2004, 2009; Sánchez et al. 2004; Bennert et al. 2011b; Cisternas et al. 2011). We aim to take advantage of type 1 AGNs that are found in X-ray surveys, such as the Chandra Deep Field South - Survey, that reach faint depths. These X-ray sources are likely to have a wide range in their optical properties that includes those of lower luminosity both missed in opticallyselected samples such as SDSS, and more favorable for the study of their host galaxy. There are numerous papers on the host galaxies of X-ray selected AGN that may be of interest to the reader (e.g., Grogin et al. 2005; Pierce et al. 2007; Ammons et al. 2009; Silverman et al. 2008). Here, we specifically select type 1 AGNs from the compilation of Silverman et al. (2010) that provide spectroscopic redshifts and classification of X-ray sources in the the Extended Chandra Deep Field South - Survey (Lehmer et al. 2005). These are objects with at least one broad emission line having a FWHM greater than 2000 km s -1 . We further require that an available spectrum has a good enough quality to perform our emission line fitting procedure to estimate virial black hole masses. We then demand that each type 1 AGN has been observed by HST. The ECDFS is covered by the GEMS (Rix et al. 2004) and GOODS (Giavalisco et al. 2004) surveys in the central area. GEMS consists of imaging in two optical HST filters (ACS F606W and F850LP) while GOODS has four filters (ACS F435W, F606W, F775W and F850LP). Unfortunately, some sources are located on the outskirts of the ECDFS and therefore no HST coverage is available. Even though, extensive ground based data is available of the full ECDFS area from various observing campaigns (e.g., MUSYC survey; Cardamone et al. 2010), we choose to avoid any biases that may appear due to the inclusion low resolution data. We further stress that is essential to have at least two filters that bracket the 4000 Å break in the rest-frame of the host galaxy for accurate estimation of the mass-to-light ratio and the stellar mass (see the following section). To do so, we elect to restrict the type 1 AGN sample to 0 . 5 < z < 1 . 2 that allows us to determine accurate rest-frame B-V colors for the entire sample. In addition, we apply the same selection to the deeper 2Msec catalog (Luo et al. 2008) and identify one additional source. Our final sample consists of 18 type 1 AGN with half falling in the GOODS area and the other half within the GEMS field. In Figure 1, we show the distribution of Xray flux and R-band optical magnitude of Chandra sources and highlight those within our type 1 AGN sample. It is apparent that the sample spans about two dex in both X-ray flux or luminosity, and optical brightness. The final sample covers the full region of the fX -R plane as the overall BLAGN sample.", "pages": [ 2, 3 ] }, { "title": "3.1. AGN-host decomposition and bulge correction", "content": "The first step to obtain information on the host galaxy is to remove the contribution of the AGN component from the broad-band HST images. The separation of galaxy light from that of the nuclear point source in luminous AGN is challenging even at lower redshifts since the AGN can outshine the host galaxy by several magnitudes. Objects in our study have lower luminosities (due to their X-ray selection with deep observations) thus the contamination from the point source is substantially weaker compared to similar studies using optically-seleted quasars. Knowledge of the point spread function (PSF) at the position of the AGN is crucial for our further analysis. The ACS PSF in the GEMS survey is known to vary across the field (Jahnke et al. 2004). Therefore, we create a local PSF for each AGN by averaging all stars within a radius of 60 arcsec around the target. Each high S/N PSF consists of about 30-40 stars. The remaining uncertainties between individual stars are included in the variance frame as an additional contribution from the rms image of each PSF. In the GOODS fields, the estimation of a proper PSF is more difficult. Each tile has only a limited number of unsaturated stars (5- 20) with strong spatial variations, in some cases, between stars located in the center and at the edges of each tile. For most of our objects, we used the same strategy as for GEMS by creating a local mean PSF for each AGN position. In three cases, the AGN was close to the edge of the tile thus we used the nearest star as our PSF reference. We use GALFIT (Peng et al. 2002, 2010) to fit the twodimensional light distribution of each AGN with a point source model represented by an empirical PSF plus a Sersic model for the host galaxy. The nucleus component is either an average PSF created from various field stars around the target, or a single PSF star as described above. The decomposition of the images is done in several steps. First, we conservatively subtract a PSF scaled to the flux contained in a small aperture (typically 2 pixels) around the central pixel. For the second step, we perform a full decomposition by adding a Sersic model as a second component. We then optimize the fit to minimize the residuals. This step requires several iterations of the fit using different starting values to ensure convergence to a global minimum in the parameter space. If necessary, we add further components to fit asymmetries in the host (e.g., arm structures). Neighboring galaxies are either masked out or fitted simultaneously (see ID-333 for such an example in the bottom panel of Figure 2) to avoid flux spilling over from one object to the other. If the host galaxy flux is below 2%, we decide that the host galaxy is unresolved. In Figure 2, we show three representative examples of the image decomposition procedure for objects with different nuclear-to-host (N/H) 2 ratio and bulge-to-total (B/T) stellar light ratio. As shown, these cases demonstrate the effectiveness of both the short (F606W) and long (F850LP) wavelength HST imaging. The top panel shows ID-158 that exhibits a nuclear component attributed to the AGN and a clearly extended component characterized by a Sersic index of 4.2. With no discernable disk, the morphology is determined to be that of an early-type galaxy. In the middle panels, the host galaxy of AGN (ID-417) is well resolved above our detection limit even though it has a high N/H ratio. The Sersic index is found to be n=2.1 that does not favor either a simple early or late type morphology. Only through a decomposition of the bulge and disk components (as described below) can we determine whether this object is truly bulge or disk dominated. The third example (ID-333) shown in the bottom panels has a galaxy contribution with a Sersic index of 1.25 indicating a strong disk contribution to the overall morphology. In Table 1, we list the sample properties and the results of our fitting routine. In Figure 3, we show the PSF-subtracted host images of the entire sample. We estimate the uncertainties of our measurements through a series of simulations. We create artificial AGN images using empirical PSFs and host galaxy models superimposed with artificial noise to match the flux levels measured in the real images. We estimate statistical errors on the host galaxy and nuclear magnitude by comparing the input and output values for our fit parameters. Host galaxy apparent magnitude, radius and morphology are extracted from the Sersic model fits. Since the Sersic index can be underestimated, in some cases, using GALFIT, especially when the nuclear-to-host (N/H) ratio is high (see Sánchez et al. 2004; Kim et al. 2008a,b), we use the simulations to correct the Sersic index. Here, we also want to point out the importance of the HST data again specially in cases such as ID-333 (See Fig. 2) where the AGN is strongly contaminated by a nearby companion that is hardly resolved in ground-based data. A direct bulge/disk decomposition is only possible for objects which have low N/H ratios (typically N / H < 2). We find 5/18 host galaxies to have nsersic , corrected > 3; therefore, we classify them as truly bulge dominated. If the single Sersic fit of the host galaxy indicates the possible presence of a disk component with nsersic , corrected < 3 (13/18 objects), we refit the galaxy with two Sersic models each representing the disk and the bulge. We put limits on the Sersic Index (0 . 5 < n < 1 . 5 for the disk and 3 < n < 5 for the bulge) of each component to achieve an effectively reduced chi square of the residuals as compared to using single values typical for a disk (n=1) and bulge (n=4) components. In the end, we are able to directly decompose 14/18 host galaxies in the F850LP filter into either a purely bulge or bulge+disk component. Some bulge+disk component fits failed due to the disturbed morphology of the host galaxy (i.e. ID-271 or ID-516) even though the AGN was weak ( N / H < 1). Since the F850LP filter provides the best contrast between nuclear and host component, we can use the best fit parameters (i.e. disk/bulge radius, position angle, axis ratios) as constraint in the bluer filter bands with typically higher N / H ratios. The best fit radii for the bulge components range from 0.6-6 kpc and are consistent with the typical sizes 2 The nuclear-to-host ratio (N/H) is simply the flux attributed to the AGN (N) divided by that of the host galaxy (H). Both determined through decomposition of the HST images. Sample: TARGET LIST AND RESULTS FROM THE ANALYSIS. THE DIFFERENT COLUMNS SHOW (1) OBJECT ID TAKEN FROM THE LEHMER ET AL., (2),(3) RA DEC COORDINATES, (4) REDSHIFT, (5) ABSOLUTE V-BBAND MAGNITUDE,(6) REST-FRAME U-V COLOR OF THE HOST GALAXY ,(7) SERSIC INDEX FROM SINGLE SERSIC FIT,(8) HALF-LIGHT RADIUS OF THE BULGE AND DISK IN ARCSEC IN F850LP (9) TOTAL STELLAR MASS, (10) BULGE-TO-TOTAL LUMINOSITY RATIO, (11) FWHM OF THE MGII EMISSION LINE, (12) CONTINUUM LUMINOSITY AT 3000Å, (13) BH MASS, (14) EDDINGTON RATIO, (15) NUCLEAR-TO-HOST RATIO IN F850LP, (16) SURVEY FIELD:GEMS [1] & GOODS [2] of elliptical galaxies at similar redshifts (Trujillo et al. 2007). In four cases, the radius of the bulge component is less than 3.5 pixel and the fit can be treated as an upper limit. But we also want to caution that the radii are more sensitive as a free parameter of the fit than the fluxes of the components. We can further check whether the inclusion of a pseudobulge component provides a reasonable fit to the surface brightness profile. To do so, we allow the bulge component to have a lower Sersic index. While it is challenging to distinguish between a pseudo-bulge and a classical bulge with the data in hand, we do find stronger residuals in the nuclear region if we fit with a pseudo-bulge component. These new fits lead to only small changes in the fluxes of each component (<0.1 mag) and bulge-to-disk ratio (<13%), hence only a small impact on the stellar mass estimates. The poorer fits, seen when incorporating a pseudo-bulge component, may indicate that SMBHs are more directly related to classical bulges than pseudo-bulges (Kormendy et al. 2011). For N/H > 3, we estimate a bulge correction through extensive simulations rather than a direct measurement. For each filter band starting with the longest wavelength due to lower contrast between AGN and host galaxy, we create a set of artificial AGN+host images using the best fit parameters for the nucleus component. The host galaxy consists of two Sersic models, one for the disk and one for the bulge component. We vary the free parameters to mimic a broad range of bulges and disks. We add appropriate noise measured from the HST images and fit the artificial images with a single Sersic model plus PSF model. We compare the fit solutions of the simulations with our single Sersic fit of the real data and select the B/T ratios of all model fits that recover the original fit parameters within their uncertainties. We fit all filter bands separately to account for possible color gradients between the disk and the bulge. Due to higher N/H ratios at shorter wavelength and typically lower S/N (fainter objects) we constrain some of the free parameters such as size, ellipticity and centroid position to the range of solutions found in the F850LP filter band. In Figure 4, we show the B/T distributions recovered for two of our host galaxies (ID-417 and ID-375) together with an image of the host galaxy. Figure 4 also shows the good agreement between the recovered B/T ratio and the host morphology. For example ID-375 has a low B/T ratio of about 0.1 indicating the presence of a dominating disk and the image of the host galaxy shows some spiral structures with very little light concentration in the center of the galaxy. The mean value of the B/T distribution and its uncertainty is then used to estimate the bulge mass. The error bars on the photometry are typically larger by 0.15-0.3 mag.", "pages": [ 3, 5, 6 ] }, { "title": "3.2. Total/Bulge Stellar Mass Estimates", "content": "Our main goal is the estimation of the stellar mass content of each host galaxy and its bulge component in the sample by converting the rest frame optical colors into mass-to-light ratios. For targets in the GEMS area, we have only a single optical color while for GOODS we have multiple colors based on four filter bands. This method has been success- fully employed in several studies on AGN host galaxies (e.g., Schramm et al. 2008; Jahnke et al. 2004, 2009; Sánchez et al. 2004). As demonstrated by Bell et al. (2003) for a variety of star formation histories, the stellar mass-to-light ratio (M/L) can be robustly predicted from the B-V color. We adopt their formula based on a Chabrier IMF: where M / LV is given in solar units. The choice of an IMF has a systematic effect on the final mass estimation. Using a Salpeter IMF would typically increase our mass-to-light ratios by factor of 1.4. The ECDFS area is well covered by broad-band photometry from various instruments ranging from the ultraviolet to the infrared. The available photometry provides another approach to estimate the total stellar mass content of the host galaxies through direct SED fitting (as shown by Merloni et al. 2010). Although this would be a powerful alternative, the data suffers from additional uncertainties such as source confusion in the ground based data or variability of the sources due to multi-epoch data. In any case, we decided to implement SED fitting only as a consistency check on our total stellar mass estimates based on the HST data. For the procedure, we use our own algorithm based on a Levenberg-Marquart χ 2 minimization. We use a set of SED model templates (Maraston 2005) with declining star formation histories based on a Kroupa IMF (which gives similar results as a Chabrier IMF), solar metallicity and a dust extinction law following Calzetti et al. (2000). We make use of the broad-band image decomposition (AGN+host) based on the HST results to constrain the template models that includes the photometric errors in each filter band. First, we fit a template AGN model (Richards et al. 2006) to the photometry of the nucleus obtained from the decomposition and subtract this from the total (ground-and/or space-based) photometry. Next, we fit the residual fluxes with either a single template or a two component template. Strong contamination from unresolved sources in the ground based data (i.e. in ID-333) are taken into account and subtracted separately using the HST photometry as an additional constraint for the companion template model. To estimate errors on the stellar mass, we use a Monte Carlo approach. We vary the observed flux in each bandpass by a random number which is Gaussian distributed with a sigma defined by the flux error. We generate 100 simulated SEDs and recompute the fit. Masses from both approaches typically agree within 0.13 dex. In Figure 6, we show four examples (ID-339, ID-170, ID-465 and ID-250) of the SED decomposition compared to the single epoch spectrum used for the BH mass estimation. These four AGN represent different level of AGN and host stellar continuum throughout our sample. In addition, these objects also have different B/T ratios: 1.0 (ID-339, ID-250), 0.5 (ID170) and 0.24 (ID-465). These examples illustrate the clear advantage gained from having the HST photometry. Only with HST resolution, we can constrain the flux of the AGN and the host galaxy including the bulge and disk components separately. The mass estimates from Bell et al. (2003) and SED fitting typically agree within 0.15 dex. In Figure 6, we plot the U-V rest-frame color versus the total stellar mass. Nearly all hosts concentrate at the high mass end and below the red sequence (i.e., the green valley). The masses of our host galaxies are comparable to typical red sequence galaxies but the colors of the host indicate a population of recently formed stars. ✑ ✒ ✔ ✕ ✗ ✘ ✚ ✛", "pages": [ 6, 7 ] }, { "title": "4. BLACK HOLE MASSES", "content": "We measure black hole masses for our entire type 1 AGN sample using single-epoch spectra that provide both a velocity width of a broad emission line and the monochromatic luminosity of the continuum. We use optical spectra acquired mainly from the followup of X-ray sources (Szokoly et al. 2004; Silverman et al. 2010). We supplement these with spectra taken with FORS2 on the VLT but not yet publicly available. Several prescriptions to estimate black hole masses are available from the literature using various emission lines such as H β , MgII or CIV (Kaspi et al. 2000; Vestergaard & Peterson 2006; Collin et al. 2006; McLure & Jarvis 2002). Due to the redshift range of our sample and optical spectroscopic coverage, we use the MgII emission line to estimate virial black hole masses in all cases. Although most of the black hole mass calibrations are based on reverberation mapping data of H β several studies have shown that there is good agreement between the mass estimates based on MgII and the Balmer lines ( H β , H α ) out to high redshifts (Shen & Liu 2012; Matsuoka et al. 2012) by combining optical and NIR spectroscopy. The prescription for estimating black hole mass as given in McLure & Jarvis (2002) is implemented; although, we recognize that similar recipes are available elsewhere (Kong et al. 2006, McGill et al. 2008, Wang et al. 2009) with each of these agreeing essentially to within 0.2-0.3 dex. Weperforman iterative least-squares minimization to fit the MgII line for each AGN to measure its line width. Our procedure is a modified version of the one used in Gavignaud et al. (2008). The number of components to fit the line depends on the characteristics of the objects and quality of the data. We fit the region around the emission line using a model that includes a pseudo-continuum and one or two Gaussian components to characterize the line profile. We find that for the local continuum a powerlaw+broadened Fe-template (provided by M. Vestergaard: see Vestergaard & Wilkes 2001) gives the best results. Specially the strength of the Fe-emission in the wings of the MgII line can vary strongly (see ID-250 for strong Fe and ID-170 for very weak Fe) and affects the outcome of the fit. We try to both minimize the number of model components and optimize the residuals around the emission line. We either interpolate over absorption features or mask them out. A FWHM of the line profile is determined using either a one or two component Gaussian model. We have tested the same algorithm on the sample from Merloni et al. (2010). Even tough we find some scatter for the individual fits, there is no systematic offset in the final black hole mass estimates. Two of our objects overlap with the study from Bennert et al. (2011b); our mass estimates agree within 0.1 dex using the same recipe. In the next step, we measure the continuum luminosity at 3000 Å required to estimate a radius to the BLR. For luminous AGN ( L bol > 45) the continuum luminosity can be directly measured from the spectrum due to the typically low impact of the host galaxy. For our sample, we find that in several cases, there is a significant host galaxy contribution that must be taken into account (see Fig. 5). Therefore, we decided to measure the monochromatic luminosity at 3000 Å by decomposing the HST/ACS images. The procedure enables us to isolate the AGN (i.e., nuclear) emission from its host galaxy most effectively. We then fit an average quasar SED template (Richards et al. 2006), accounting for dust attenuation to estimate the intrinsic continuum luminosity at 3000 Å. We find that the continuum luminosity based on HST imaging agrees with that determined from the decomposition of the broad-band SED to within 5%. Monte Carlo realizations using the uncertainties of the FWHM and L 3000 measurements enable us to estimate the uncertainties on the black hole mass in addition to 0.4 dex uncertainty inherent in the scaling relations. In Figure 7, we present examples of the fits to the broad emission lines in six AGN with different quality of data. A summary of the results of our line fits are shown in Table 1.", "pages": [ 8 ] }, { "title": "5.1. The BH Mass-Total Stellar Mass Relation", "content": "Wefirst present the relation between black hole mass ( M BH) and total stellar mass ( M ∗ , Total) in Figure 8 ( le f t panel). From the distribution of data points, it is apparent that our sample does not have the dynamic range in either stellar mass or black hole mass to establish both a slope and normalization simultaneously of a linear fit. Fortunately, we can compare with the local relation established using inactive galaxies (mainly ellipticals or S0) as done by Häring & Rix (2004) and determine whether an offset exists. We find that essentially all of our AGN fall along the local M BH -M ∗ , Bulge relation. It is important to highlight that the bulge mass is equivalent to the total stellar mass for the local comparison sample. To be more specific, we find that 17/18 objects, considering their 1 σ errors, are consistent with the typical region of 0.3 dex scatter around the best fit local relation having a slope of 1.12 (Häring & Rix 2004). Given our limitations in mass coverage as mentioned above, we fit a linear regression model to our data while fixing the slope to the value given above thus determining only the normalization. We find the best-fit normalization to be 8.31 by using FITEXY (Press et al. 1993), which estimates the parameters of a linear fit while considering errors on both variables. The fit is affected by the single target offset from the relation. If excluded for no obvious reason, the constant would be 8.24. With a simple Monte Carlo test, we can reject the null hypothesis that the two samples are significantly different. While the local inactive sample is established using dynamical masses, we do not expect these to differ substantially from the stellar masses; this is in fact the ✤ case as demonstrated in Bennert et al. (2011a). Ideally, we would like to compare our sample with a local sample of active SMBHs with stellar mass measurements of their hosts. The work of Bennert et al. (2011a) allows such a direct comparison. We show these data in Figure 8 as marked by small black circles. Carrying out the same fit as for the ECDFS AGNs, we find the best fit constant to be 8.30 for the local AGNs. We use the total stellar mass for the regression fit of both active samples (Bennert et al. (2011a), ECDFS AGNs) and find no significant deviation between them in the M BH -M ∗ , Total relation. Our result agrees well with findings of recent studies of the M BH -M ∗ , Total relation at high redshift. In particular, Jahnke et al. (2009) use a similar technique of decomposing HSTimages and converting rest-frame optical colors into stel- r mass-to-light ratios based on a sample of AGNs at z > 1 in COSMOS with NICMOS coverage. Their sample consists of ten objects with seven for which they achieve a decomposition in multiple bands and find no offset in black hole mass given their total stellar masses. Our study effectively improves the statistics by a factor of 2.5 and fills in a gap in redshift coverage (see Figure 8, right panel). In addition, these results are supported by the findings of Cisternas et al. (2011) who explored the same relation on a sample of BLAGN at 0 . 3 < z < 0 . 9 from the COSMOS survey; although, only one HST band is available to constrain the stellar mass content of the host galaxy. Taken together, these studies (Jahnke et al. 2009; Cisternas et al. 2011), including our own, clearly contrast with other works at high redshift that claim an increasing offset in black hole mass for a given stellar mass. In the right panel of Figure 8, we show the redshift evolution of M BHM ∗ , Total ratio compared to various other studies probing the same relation. Some studies are using different mass estimators for the black hole masses or stellar masses. For example, Merloni et al. (2010) use the prescription of McGill et al. (2008) for their black hole mass estimations and assume a Salpeter IMF for their stellar mass estimates. When necessary, we convert the masses of different studies to the prescription based on the formula from McLure & Jarvis (2004) and a Chabrier IMF for the stellar mass estimates. In case of Merloni et al. (2010), the corrections have only a marginal effect on the M BH -M ∗ , Total relation. Based on our results, we cannot confirm or rule out a stronger evolution at higher redshift ( z > 1 . 5). In particular, our mean bolometric luminosity is log Lbol = 44 . 7 while the higher redshift sample of Merloni et al. (2010) is at log Lbol = 45 . 5. As a consequence, the mean BH mass is shifted to higher masses and therefore a direct comparison with these objects and any trend implied by the data might be biased by the differences in the sample properties. It is worth highlighting that our results are likely to be less biased due to selection since our BH masses are typically below 10 9 M /circledot , the knee in the black hole mass function; we may be effectively avoiding the problems fully presented in Lauer et al. (2007).", "pages": [ 8, 9, 10 ] }, { "title": "5.2. The BH Mass-Bulge Stellar Mass Relation", "content": "While the total stellar mass is well-determined using different methods (Schramm et al. 2008; Jahnke et al. 2009; Merloni et al. 2010; Cisternas et al. 2011; Bennert et al. 2011b), we usually do not know how much of the total mass is present in the bulge. As stated above, only 5/18 of our AGN hosts have a Sersic Index n > 3 indicating a purely bulgedominated host galaxy. We make the assumption that for these objects the total mass is the same as the bulge mass. For the remainder, we estimate the bulge contribution to the total mass by corrections to the total mass by accounting for the contribution of the disk. Applying the same cut at n < 3, we find that ∼ 72% of the host galaxies show a disk component. Although the fraction is in good agreement with the results presented by Schawinski et al. (2011) on a sample of X-ray selected AGN in the Chandra Deep Field South at 2 < z < 3. Although, we draw a different conclusion on the importance of the disk component, in terms of the mass contribution to the total mass. Our bulge/disk decomposition shows that, even though a disk is present, the mass of the central bulge can still dominate the total mass of the host galaxy. The different redshift regimes might play an important role since there is about 3-5 Gyr of galaxy evolution between our study and that of Schawinski et al. (2011). Using the B/T ratio to divide our sample into bulge and disk dominated systems, we find that ∼ 50% of the sample has a significant bulge component with B / T > 0 . 5; this can even be true for objects with a surface brightness profile of the host galaxy described by a fit with a Sersic index of ∼ 2. We can now establish the M BH -M ∗ , Bulge relation at 0 . 5 < z < 1 . 2. In Figure 9, we plot the M BH -M ∗ , Bulge relation and compare our results with the sample of inactive galaxies from Häring & Rix (2004) and local AGN from Bennert et al. (2011a). The stellar mass measurements for the local AGN allow a more direct comparison with our sample than the dynamical masses of Häring & Rix (2004). We find that the mass distributions for all three samples are very similar with each other. This can be clearly seen in a histogram of the mass ratio ( log MBH / M ∗ , Bulge ) shown in the top panel of Figure 10, where there is no significant difference in the median value. Overall, we find that 78% of the AGNs are consistent with the local relation. If we consider the single object undergoing a clear major merger (ID-333), there are only three objects that are significantly offset from the local relation. If we artificially move this object onto the relation, then 83% of AGNs in our sample are consistent with the local relation. We interpret this as evidence for a black hole-bulge relation, at these redshifts, to be similar to the local relation. Interestingly, we do find additional scatter in our sample compared to that in the local distributions. We further note that there are no objects well below the M BH -M ∗ , Bulge relation. We can further investigate where high-z AGN lie in respect to the local relation as a function of their bulge-to-total ratio. All objects with B / T > 0 . 5 fall nicely onto the local relation (see Figure 9 and the bottom panel of 10). They are also the most massive objects in the sample in terms of their bulge mass. Objects with a B / T < 0 . 5 are clearly separated in bulge mass (from bulge-dominated objects) and the majority are still in good agreement with the local relation. Only four objects have under massive bulges considering their 1 σ error bars including ID-333 which has a massive companion that might move the whole system onto the relation after the merger.", "pages": [ 10 ] }, { "title": "6. DISCUSSION", "content": "An important question for SMBHs and their host galaxies is their subsequent evolution in the black hole - bulge mass plane. As previously mentioned, 83% of the bulges in our sample are already massive enough that their M BH -M ∗ , Bulge ratio agrees well with that seen in inactive galaxies today (see Figure 9). We illustrate this further in Figure 10 (top panel), by comparing the distribution of the M BH -M ∗ , Bulge ratio between various samples. Interestingly, there are some outliers with undermassive bulges, relative to their BH mass, that are preferentially disk dominated galaxies. In the bottom panel of Figure 10, we compare the distributions of this ratio for the bulge and disk dominated subsamples separately to the distribution of the local AGNs. Even though the number statistics are small, we find no difference for the bulge dominated subsample by looking at their median ratios. The situation is different for objects in the diskdominated subsample. While some objects overlap with the distribution M BH -M ∗ , Bulge ratios of the local AGN, the median ratio of the disk dominated subsample is shifted by 0.5 dex towards a higher ratio. When comparing bulges of similar mass (log M ∗ , Bulge < 10 . 5), local AGN host galaxies have a smaller offset ( ∼ 0.25 dex). On the other hand for the same mass matched subsample which includes 22/25 objects in the local AGN sample and 12/18 from our sample, we find that the local AGN sample contains only ∼ 30% disk dominated systems while our subsample contains ∼ 80%disk dominated systems. Within the AGN population, we may be witnessing both a migration onto the local relation and a morphological transformation with cosmic time. We recognize that selection effects may impact such comparisons. Ideally, we want to have an AGN sample spanning a wide baseline in redshift with equivalent selection, BH mass indicators and sufficient statistics. This leads to the question how these host galaxies can grow their stellar bulge mass to match the bulge masses seen today. One possible track could be the event of a major merger that leads ultimately to a significant increase in stellar bulge mass. Mergers are seen to play a role in black hole growth for ✦ similar X-ray selected samples (Silverman et al. 2011). Out of our 18 AGN, only one (ID-333) shows signs of an ongoing major merger. Even though other host galaxies do show some signs of minor merger activity, we conclude that the growth of the bulge through a major merger event in the near future is not certain. On the other hand, the good agreement of the AGNhost galaxy M BHM ∗ , Total relation with the local relation clearly shows that all the mass needed to put our host galaxies onto the local M BH -M ∗ , Bulge relation is already in place within these galaxies at redshift z ∼ 1 (Jahnke et al. 2009). Therefore, mass transfer from the disk to the bulge is neccessary to grow their bulges. Any bulge growth through internal processes has to overcome the mass growth of the black hole otherwise the galaxy would just move on a diagonal track in the M BH -M ∗ , Bulge relation. While the BHs in their active phase are growing, we can also investigate how the host galaxy is growing in stellar mass by looking at their individual growth rates (i.e., SFR) and compare these to the BH growth rates. We estimate star-formation rates based on the UV continuum from our best fit SED models and converted these into growth rates (SFR/ M stell). In Figure 11, we compare the growth rates of the host galaxies with the growth rates of the BHs as determined by ˙ M / M BH. ˙ M is determined from Lbol = /epsilon1 ˙ Mc 2 . To estimate ★ the bolometric luminosities Lbol and Eddington ratios, we use the luminosity dependent corrections from Hopkins et al. (2007) applied to our derived continuum luminosities at 3000 Å. We find that apparently the BHs gain mass much stronger than the host galaxies by a factor of ∼ 30. These relative growth rates are broadly consistent with that seen in obscured AGN(Netzer 2009; Silverman et al. 2009). Such an offset implies that the typical duty cycle of an AGN (see Martini (2004) for an overview) during which it can grow its BH mass efficiently must be short enough (typically 10 7 -10 8 yr) to prevent a significant vertical movement in the black hole mass - bulge mass plane. If the growth rates are extrapolated over a period of 1Gyr, the host galaxies do not gain much stellar mass from the present level of star formation. As previously mentioned, only one object (ID-333) shows a possible major merger due to the presence of a more massive but inactive companion. While some objects show signs of minor merging activity (i.e. ID-712,ID-271), and we cannot exclude that we miss further minor merger events due to their low surface brightness, the stellar mass gain is expected to be low. Assuming the current growth rates and ignoring a possible major merger, all galaxies except one would need more than > ∼ 1Gyr to move more than 0.3 dex in the M BHM ∗ , Total. Therefore, we do not expect much evolution over the next 1Gyr for the majority of our sample.", "pages": [ 10, 11, 12 ] }, { "title": "7. SUMMARY", "content": "We have performed a detailed analysis of a sample of 18 type 1 AGN host galaxies at 0 . 5 < z < 1 . 2 to estimate their stellar mass content and explore the relation between the mass of the central BH and the mass of the host galaxy. Our sample is of moderate-luminosity due to a selection based initially on their X-ray emission as detected with the Extended Chandra Deep Field - South Survey. This results in a sample having black hole masses below the knee of the black hole mass function thus mitigating biases (Lauer et al. 2007) seen in other samples to date. For the chosen redshift range, HST imaging is available with at least two filters that bracket the 4000 Å break thus providing reliable stellar mass estimates of the host galaxy by accounting for both young and old stellar populations. We have estimated bulge masses for all galaxies through either direct decomposition of the imaging data into a bulge or bulge plus disk component, or through simulations where artificial host galaxies with different B/T ratios are compared to single Sersic fits of the host galaxy. We are now able to look separately into their relation of the BH mass with either total stellar mass content or bulge mass after the contribution from the disk is removed. Wefindthat the relation between M BH and M ∗ , Total is in very good agreement with the local M BH -M ∗ , Bulge which has been reported by several studies so far. From our morphological analysis and decomposition of bulge and disk components, we can quantify the fraction of bulge dominated objects with B / T > 0 . 5 to be 50% while 72% of the sample shows the presence of a disk component which is a significantly higher fraction than for a stellar mass matched local AGN sample. Even though the bulge mass is shifted towards lower masses given their BH mass in some cases, we find that ∼ 80% of the sample is in agreement with the local M BH and M ∗ , Bulge relation given their 1 σ error bars. We further compare the growth rates of the host galaxy and their BHs and find that assuming the present SFR and accretion rates (while ignoring possible major merger events), only one AGN in our sample would move more than 0.3 dex over the next 1Gyr. We highlight that bulge dominated galaxies are well in place at z ∼ 1 on the local M BH -M ∗ , Bulge relation. There is a significant fraction (20%) of our sample that is disk dominated and above the local relation which is not seen in either local inactive or active galaxy samples. For these galaxies to grow their bulges and align themselves on the local relation a physical mechanism is likely needed to redistribute their stars. While mergers may play a role, it is not yet clear whether this is the dominant process. The authors fully appreciate the comments given by an anonymousreferee that improved the paper and useful discussions with Tommaso Treu and Charles Steinhardt. This work was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.", "pages": [ 12, 13 ] }, { "title": "REFERENCES", "content": "Hopkins, P. F., Richards, G. T., & Hernquist, L. 2007, ApJ, 654, 731 Jahnke, K., Bongiorno, A., Brusa, M., Capak, P., Cappelluti, N., Cisternas, Silverman, J. D., Kampczyk, P., Jahnke, K., Andrae, R., Lilly, S. J., Elvis, M., Civano, F., Mainieri, V., Vignali, C., Zamorani, G., Nair, P., Le Fèvre, O., de Ravel, L., Bardelli, S., Bongiorno, A., Bolzonella, M., Cappi, A., Caputi, K., Carollo, C. M., Contini, T., Coppa, G., Cucciati, O., de la Torre, S., Franzetti, P., Garilli, B., Halliday, C., Hasinger, G., Iovino, A., Knobel, C., Koekemoer, A. M., Kovaˇc, K., Lamareille, F., Le Borgne, J.-F., Le Brun, V., Maier, C., Mignoli, M., Pello, R., Pérez-Montero, E., Ricciardelli, E., Peng, Y., Scodeggio, M., Tanaka, M., Tasca, L., Tresse, L., Vergani, D., Zucca, E., Brusa, M., Cappelluti, N., Comastri, A., Finoguenov, A., Fu, H., Gilli, R., Hao, H., Ho, L. C., & Salvato, M. 2011, ApJ, 743, 2 Silverman, J. D., Lamareille, F., Maier, C., Lilly, S. J., Mainieri, V., Brusa, M., Cappelluti, N., Hasinger, G., Zamorani, G., Scodeggio, M., Bolzonella, M., Contini, T., Carollo, C. M., Jahnke, K., Kneib, J.-P., Le Fèvre, O., Merloni, A., Bardelli, S., Bongiorno, A., Brunner, H., Caputi, K., Civano, F., Comastri, A., Coppa, G., Cucciati, O., de la Torre, S., de Ravel, L., Elvis, M., Finoguenov, A., Fiore, F., Franzetti, P., Garilli, B., Gilli, R., Iovino, A., Kampczyk, P., Knobel, C., Kovaˇc, K., Le Borgne, J.-F., Le Brun, V., Mignoli, M., Pello, R., Peng, Y., Perez Montero, E., Ricciardelli, E., Tanaka, M., Tasca, L., Tresse, L., Vergani, D., Vignali, C., Zucca, E., Bottini, D., Cappi, A., Cassata, P., Fumana, M., Griffiths, R., Kartaltepe, J., Koekemoer, A., Marinoni, C., McCracken, H. J., Memeo, P., Meneux, B., Oesch, P., Porciani, C., & Salvato, M. 2009, ApJ, 696, 396 Silverman, J. D., Mainieri, V., Lehmer, B. D., Alexander, D. M., Bauer, F. E., Bergeron, J., Brandt, W. N., Gilli, R., Hasinger, G., Schneider, D. P., Tozzi, P., Vignali, C., Koekemoer, A. M., Miyaji, T., Popesso, P., Rosati, P., & Szokoly, G. 2008, ApJ, 675, 1025 Silverman, J. D., Mainieri, V., Salvato, M., Hasinger, G., Bergeron, J., Capak, P., Szokoly, G., Finoguenov, A., Gilli, R., Rosati, P., Tozzi, P., Vignali, C., Alexander, D. M., Brandt, W. N., Lehmer, B. D., Luo, B., Rafferty, D., Xue, Y. Q., Balestra, I., Bauer, F. E., Brusa, M., Comastri, A., Kartaltepe, J., Koekemoer, A. M., Miyaji, T., Schneider, D. P., Treister, E., Wisotski, L., & Schramm, M. 2010, ApJS, 191, 124 Szokoly, G. P., Bergeron, J., Hasinger, G., Lehmann, I., Kewley, L., Mainieri, V., Nonino, M., Rosati, P., Giacconi, R., Gilli, R., Gilmozzi, R., Norman, C., Romaniello, M., Schreier, E., Tozzi, P., Wang, J. X., Zheng, W., & Zirm, A. 2004, ApJS, 155, 271 Trujillo, I., Conselice, C. J., Bundy, K., Cooper, M. C., Eisenhardt, P., & Ellis, R. S. 2007, MNRAS, 382, 109 Vestergaard, M. & Peterson, B. M. 2006, ApJ, 641, 689 Vestergaard, M. & Wilkes, B. J. 2001, ApJS, 134, 1 Woo, J., Treu, T., Malkan, M. A., & Blandford, R. D. 2008, ApJ, 681, 925 Xue, Y. Q., Brandt, W. N., Luo, B., Rafferty, D. A., Alexander, D. M., Bauer, F. E., Lehmer, B. D., Schneider, D. P., & Silverman, J. D. 2010, ApJ, 720, 368", "pages": [ 13, 14 ] } ]

[ { "title": "ABSTRACT", "content": "Off-center stellar tidal disruption flares have been suggested to be a powerful probe of recoiling supermassive black holes (SMBHs) out of galactic centers due to anisotropic gravitational wave radiations. However, off-center tidal flares can also be produced by SMBHs in merging galaxies. In this paper, we computed the tidal flare rates by dual SMBHs in two merging galaxies before the SMBHs become self-gravitationally bounded. We employ an analytical model to calculate the tidal loss-cone feeding rates for both SMBHs, taking into account two-body relaxation of stars, tidal perturbations by the companion galaxy, and chaotic stellar orbits in triaxial gravitational potential. We show that for typical SMBHs with masses 10 7 M /circledot , the loss-cone feeding rates are enhanced by mergers up to Γ ∼ 10 -2 yr -1 , about two order of magnitude higher than those by single SMBHs in isolated galaxies and about four orders of magnitude higher than those by recoiling SMBHs. The enhancements are mainly due to tidal perturbations by the companion galaxy. We suggest that off-center tidal flares are overwhelmed by those from merging galaxies, making the identification of recoiling SMBHs challenging. Based on the calculated rates, we estimate the relative contributions of tidal flare events by single, binary, and dual SMBH systems during cosmic time. Our calculations show that the off-center tidal disruption flares by un-bound SMBHs in merging galaxies contribute a fraction comparable to that by single SMBHs in isolated galaxies. We conclude that off-center tidal disruptions are powerful tracers of the merging history of galaxies and SMBHs. Subject headings: black hole physics - galaxies: active - galaxies: kinematics and dynamics - galaxies: nuclei - gravitational waves", "pages": [ 1 ] }, { "title": "Enhanced off-center stellar tidal disruptions by supermassive black holes in merging galaxies", "content": "F.K. Liu 1 and Xian Chen 2 , 3", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In the Λ-cold dark matter cosmology, both dark matter halos and galaxies form due to frequent mergers. In this paradigm, hierarchical galaxy mergers would incorporate multiple supermassive black holes (SMBHs) into a galaxy (Volonteri et al. 2003). When two SMBHs, initially embedded in the two cores of the merging galaxies, sink to the common center of the system due to dynamical friction and become gravita- ionally bound, a supermassive black hole binary (SMBHB) would form (Begelman et al. 1980). During the interaction between the SMBHB and the stellar and gaseous environments, if the two SMBHs could successfully evolve to a separation of hundreds of Schwarzschild radius, then gravitational wave (GW) radiation could lead to the coalescence of the SMBHs within a Hubble time, and the asymmetry of GW radiation is predicted to impart a recoiling velocity on the post-merger SMBH (Hughes 2009; Centrella et al. 2010). Detection of the GW radiation from coalescing SMBHB would be a vital test of the theory of general relativity (GR), and is the major goal of the ongoing Pulsar Timing Array (PTA) project and any future space-based GW mission. Despite many efforts to detect GW radia- tion from coalescing SMBHBs, theoretical studies found large uncertainties for the dynamical evolution of SMBHB in normal galaxies: in the absence of gas and efficient stellar relaxation, the evolution of SMBHB would stall at sub-parsec (pc) scale and not enter the GW radiation regime (Merritt & Milosavljevi'c 2008; Colpi & Dotti 2011), while recent N -body simulations suggest that efficient repopulation of stars to the galaxy core may be norm in real mergers (Preto et al. 2011; Khan et al. 2011). Observationally, it is difficult to test the dynamical evolution of SMBHB in stellar systems, because of the lack of electromagnetic (EM) radiation from the vicinity of the dormant SMBHs. Recently, 'tidal flares', the EM outbursts produced due to tidal disruption of stellar objects by SMBHs, have been identified as powerful probes of the mass and spin of the otherwise dormant SMBHs (Rees 1988; Komossa 2002; Donley et al. 2002; van Velzen et al. 2011; Gezari et al. 2009; Bloom et al. 2011; Burrows et al. 2011; Cenko et al. 2012). The flaring rate for a single SMBH in an isolated galaxy is estimated to be 10 -5 to 10 -4 yr -1 (Magorrian & Tremaine 1999; Syer & Ulmer 1999; Wang & Merritt 2004; Brockamp et al. 2011). It is predicted that the formation and evolution of bound SMBHB at galaxy center would significantly change the event rate and affect the light curves of tidal flares. Shortly after the formation of SMBHB, the three-body interaction between the binary and a bound stellar cusp will enhance the flaring rate to as high as 1 yr -1 (Ivanov et al. 2005; Chen et al. 2009, 2011; Wegg & Nate Bode 2011). After the stellar cusp is disrupted, mainly due to slingshot ejection, if stellar relaxation is inefficient in galaxy center, the flaring rate will become one order of magnitude lower than that in the single black hole system (Chen et al. 2008). When the SMBHB enters the GW-radiation regime, the interruption and recurrence of tidal-flare light curve by the perturbing secondary black hole occur on an observable timescale (Liu et al 2009), and the stars resonantly trapped by the inspiralling SMBHB may produce a tidal flare around the coalescence of the binary (Schnittman 2010; Seto & Muto 2011). After the coalescence of the binary, the launch of a recoiling SMBH might be also accompanied by a brief burst of tidal flares. As the recoil- ing SMBH travels outside the galaxy core, tidal disruption of the stars gravitationally bound to the hole may produce a flare apparently displaced from the galaxy center (Komossa & Merritt 2008; Merritt et al. 2009; Stone & Loeb 2012; Li et al. 2012). Because of the many differences between the flaring rates in single and binary SMBH systems, it is suggested that tidal flares can be utilized to constrain the fraction and dynamical evolution of SMBHBs in galaxy centers (Chen et al. 2008). Off-nuclear tidal disruption flares and ultra compact star clusters with peculiar properties are suggested to be the key features of gravitational recoiling SMBHs in galaxies. However, off-center tidal disruption flares can also be produced by SMBHs in merging galaxies, and star clusters with the proposed peculiar characters may also form by tidal truncation of secondary galaxies in minor mergers. In particular, tidal flaring rates would be enhanced at a stage when the SMBHs are still isolated in the cores of merging galaxies because of the mutual perturbation between merging galaxies, much earlier than the formation of SMBHBs. Roos (1981) pioneered the discussions on the stellar tidal disruptions by assuming that merging galaxies harbor Sgr A*-like SMBHs and by taking into account perturbations by companion galaxies. However, it is unclear by how much the tidal disruption rates can be boosted in physical galaxy models combining the correlations of central SMBHs and galactic bulges, how many tidal flares in the universe are contributed by this merger phase, and how they would affect the constraint on the merger history of SMBHs. As a first step toward addressing these issues, in this paper we calculate the stellar-disruption rates during galaxy mergers and investigate the prospect of using tidal flares to probe multiple SMBHs in merging galaxies. The outline of the paper is as follows. In § 2, we introduce the basic loss-cone theory and the stellar-disruption process in single SMBH systems. In § 3, we describe the stellar relaxation process in merging systems and generalize the losscone theory to calculate the corresponding stellardisruption rate. We also discuss our results for different merger parameters. Based on the calculated rates, we investigate the contribution of tidal fares by merging galaxies in § 4 and discuss our results and their implications in § 5.", "pages": [ 1, 2, 3 ] }, { "title": "2. Loss-cone feeding in single SMBH system", "content": "We first calculate stellar-disruption rates for isolated galaxies with single SMBHs, to prepare the basics for more complicated calculations for merging galaxies. A star with mass m ∗ and radius r ∗ would be tidally disrupted when it passes by an SMBH as close as the tidal radius (Hills 1975; Rees 1988), where M · is the black hole mass, M 7 = M · / 10 7 M /circledot , and R /circledot and M /circledot are, respectively, the solar radius and mass. In the following, we assume r ∗ = R /circledot and m ∗ = M /circledot unless mentioned otherwise. For these solar-type stars, when M · /lessmuch 4 × 10 7 M /circledot , tidal disruption happens outside the marginally bound orbit of the black hole, and collisions between the bound stellar debris, as well as the subsequent accretion on to the black hole, could produce an EM flare known as the 'tidal flare' (Rees 1988). The criterion for stellar disruption is then J ≤ J td /similarequal (2 GM · r t ) 1 / 2 , where G is Newtonian gravitational constant, J is the specific angular momentum, and J td is the specific angular momentum corresponding to a pericenter distance of r t . Here, the latter approximation accounts for the fact that most stars are disrupted along parabolic orbits, i.e., their specific binding energy E /lessmuch GM · /r t . When M · > 4 × 10 7 M /circledot , the marginally bound orbit of Schwarzschild black hole becomes greater than the tidal radius, then the criterion for stellar depletion becomes J < J mb , where J mb denotes the specific angular momentum for marginally bound geodesic. In general, J mb is a function of black-hole spin and inclination relative to the equatorial plane, but for simplicity we adopt the orientation-averaged value J mb = 4 GM · /c (Kesden 2012) in the following calculation, where c is the speed of light. For even greater black-hole mass M · /greaterorsimilar 10 9 M /circledot , tidal disruption occurs inside the event horizon of the central SMBH even when the black hole is maximally spinning, so no tidal flare could be produced by disrupting solar-type stars (Ivanov & Chernyakova 2006; Kesden 2012). As a result of tidal disruption and direct capture, a small fraction of stars are lost from the system during their pericenter passages. In a spherical system, the disruption rate of stars from distance r to r + dr from the central SMBH is approximately (Frank & Rees 1976; Syer & Ulmer 1999), where ρ ( r ) is the stellar mass density at r , t d ( r ) is the dynamical timescale, and θ 2 ( r ) estimates the fraction of stars subjected to lose from the system. The loss fraction θ 2 is dimensionless and can be interpreted geometrically as a solid angle, because at r the lost stars have velocity vectors pointing toward the SMBH within an angle of θ lc ( r ) = J lc /J c ( r ) and in an isotropic system their fraction is θ 2 = θ 2 lc . Here J c denotes the angular momentum for circular orbit and is of order rσ ( r ) given the stellar velocity dispersion σ ( r ). The cone-like region with half-opening angle θ lc toward the SMBH is therefore called 'loss cone'. The isotropy of stellar distribution breaks down at the edge of loss cone when the orbital-averaged rms velocity deflection angle θ d ( r ) is much smaller than θ lc (Lightman & Shapiro 1977; Cohn & Kulsrud 1978). Taking this effect into account, careful analysis of the loss-cone structure suggests that (Young 1977). Therefore, when θ d /greatermuch θ lc ('pinhole regime'), Equation (4) recovers θ 2 = θ 2 lc , because the stars act as if the loss cone does not exist and the system remains isotropic. On the other hand when θ d /lessmuch θ lc ('diffusive regime'), the loss cone becomes empty within one dynamical timescale, so afterwards only a fraction θ 2 d / | ln θ lc | of stars residing at the boundary layer θ lc ∼ θ lc + θ d of the loss cone will be depleted during one t d . The total stellar disruption rate Γ is an integration of Equation (3) over both pinhole and diffusive regimes. To calculate ρ ( r ), t d ( r ), and θ 2 ( r ), a physical model describing the stellar distribution in the host galaxy needs to be specified. We consider only the bulge component of a galaxy because it is the major source for stellar disruption. We model a bulge with a spherical model with double power laws, i.e., where r b is the break radius, ρ b is the stellar mass density at r b , γ and β are, respectively, the inner and outer power-law indices, and r max is the cut off radius to prevent divergence of the total stellar mass. The five model parameters, ( r b , r max , ρ b , γ, β ), are determined by the following five physically motivated conditions According to Jeans's equation in the isotropic limit the velocity dispersion σ ∝ r -1 / 2 when r /lessmuch r b and σ ∝ r 1 -β/ 2 when r /greatermuch r b ; therefore, we calculate σ with where σ b is the velocity dispersion at r b . By applying Equations (6) and (7) at r e , we first derive where A e ≡ M ∗ ( r e ) /M · and σ e is computed with M · -σ 4 e relation. Then the model parameters ( r b , r max , ρ b ) are calibrated according to their definitions, and the results are r b /similarequal r e [(6 -2 γ ) / (3 A e -βA e )] 1 / (3 -β ) , r max /similarequal ( A/A e ) 1 / (3 -β ) r e , and ρ b = (3 -γ ) M · / (2 πr 3 b ). For example, our fiducial galaxy model with M · = 10 7 M /circledot , γ = 1 . 75, and β = 2 corresponds to r b /similarequal 4 . 5 pc, r e /similarequal 260 pc, and r max /similarequal 820 pc. Having specified the galaxy model, we now calculate the deflection angle θ d which determines θ 2 in Equation (4). Two-body scattering is an inherent relaxation mechanism in stellar system and it gives a lower limit of θ 2 = J 2 /J c to θ d , where J 2 is the cumulative change of J due to two-body scattering during one dynamical timescale. Because successive two-body scatterings are uncorrelated (incoherent), we have J 2 = ( t d /t r ) 1 / 2 J c , where is the two-body relaxation timescale, ln Λ is the Coulomb logarithm (we assumed a fiducial value of 5), and is a correction factor of order unity. When twobody scattering dominates the relaxation process, J 2 is an increasing function of r , with the transition between pinhole and diffusive regimes ( J 2 ∼ J lc ) being situated at r ∼ r b . The differential loss rate d Γ /dr (eq. (3)) scales as r 9 / 2 -2 γ in the diffusive regime ( r /lessmuch r b ) and as r -1 -β/ 2 in the pinhole one ( r /greatermuch r b ); therefore, the stellar disruption rate peaks at the transition regime at r ∼ r b . Take our fiducial model with M · = 10 7 M /circledot , γ = 1 . 75, and β = 2 for example. The critical radius where θ 2 2 = θ 2 lc is r cri /similarequal 2 . 3 r b , and the total disruption rate due to two-body relaxation is Γ /similarequal 2 . 3 × 10 -5 yr -1 , consistent with previous calculations (e.g., Magorrian & Tremaine 1999; Syer & Ulmer 1999; Wang & Merritt 2004; Brockamp et al. 2011). If M · increases, r cri /r b will also increase, given the fact that θ 2 lc is a decreasing function of r/r b , and that θ 2 2 ∝ M -1 · and θ 2 lc ∝ M 1 / 3 · at any r/r b . On the other hand, the integrated stellar-disruption rate will decrease, mainly because the diffusive regime of loss cone becomes larger. A more accurate calculation of Γ could be carried out by solving the diffusion equation in the 2-D E J space (e.g., Lightman & Shapiro 1977; Cohn & Kulsrud 1978; Magorrian & Tremaine 1999; Wang & Merritt 2004), but it is considerably time-consuming and out of the scope of this paper. Nevertheless, the present scheme gives good approximation to the two-body disruption rate, and is sufficient to provide references for the sake of investigating the effects of galaxy mergers on the stellar-disruption rate.", "pages": [ 3, 4, 5 ] }, { "title": "3. Enhanced loss-cone feeding during galaxy merger", "content": "Because the loss cone is already 'full' in the pinhole regime, enhancing relaxation efficiency in this regime does not increase the fraction of losscone stars, therefore would not increase stellardisruption rate. On the other hand, the loss cone in the diffusive regime is largely empty, so the disruption rate can be enhanced if stellar relaxation in this regime becomes more efficient. Enhancement of stellar relaxation in the diffusive regime can be achieved by galaxy merger due to at least two processes. First, perturbation by the companion galaxy would secularly change the stellar angular momenta (Roos 1981). Second, the triaxial gravitational potential built up during merger (Preto et al. 2011; Khan et al. 2011) would drive stars to galaxy center in a chaotic manner (Poon & Merritt 2001). In this section we calculate the stellar-disruption rates due to the above two processes, and we show the rate for each of the two SMBHs in the merging system.", "pages": [ 5 ] }, { "title": "3.1. Basic Theory", "content": "A companion galaxy would tidally torque the stellar orbits in the central galaxy, secularly changing the orbital elements. Given mass M p of the perturber and its distance d from central galaxy, one can derive GM p r/d 3 for the tidal force exerted by M p across a stellar orbit of radius r /lessmuch d in the central galaxy. The corresponding tidal torque on the stellar orbit is of magnitude T p ∼ GM p r 2 /d 3 . Because of the tidal torque, the angular momentum of star changes coherently, i.e., ∆ J ∝ t , up to a timescale t ω , where t ω is determined by the shorter one between the dynamical timescale of the perturber and the apsidal precession timescale of the stellar orbit (Binney & Tremaine 2008). For t > t ω , the torque on stellar orbit adds up stochastically and in this case ∆ J 2 ∝ t . Therefore, averaged over a timescale much longer than t ω , the tidal torque changes J 2 by an amount of J 2 p = T 2 p t ω t d ( r ) during each stellar dynamical timescale. As a result, the deflection angle θ 2 d in Equation (4) increases by an amount of θ 2 p = ( J p /J c ) 2 . We note that the calculation of J p is analogous to the calculation of angular-momentum change due to resonant relaxation where the resonance torque is induced by the grainy gravitational potential (Rauch & Tremaine 1996; Hopman & Alexander 2006). Galaxy merger also increases the triaxiality of the gravitational potential (Preto et al. 2011; Khan et al. 2011). Poon & Merritt (2001) showed that when the triaxiality is large, a consistent fraction of stars are fed to the loss cone in a chaotic manner and the loss cone remains full. Suppose f c is the fraction of stars on chaotic orbits, the extra contribution to stellar-disruption rate can be calculated by replacing θ 2 d with θ 2 c = f c θ 2 lc ln θ lc (Merritt & Poon 2004). It has been shown that f c approaches unity when the triaxiality becomes greater than 0 . 25, but will rapidly decrease to 0 inside the influence radius of the central SMBH where the gravitational potential is largely spherical (Poon & Merritt 2004). Because of tidal perturbation and triaxiality during galaxy merger, the effective deflection angle θ 2 d increases to and in the diffusive regime the loss-cone-limited deflection angle ( θ 2 in Equation (4)) also becomes larger. Consequently, an enhancement of stellardisruption rate is anticipated. Now we have prepared Equations (3), (4), and (12) to calculate the stellar-disruption rate in merging galaxies. However, the equations are valid only in the adiabatic approximation, i.e., the gravitational potential varies on a timescale much longer than the typical timescale for stellar orbital evolution. If the adiabatic condition is violated, the galaxy core will be subject to significant heating and expansion on the dynamical timescale (Ostriker 1972; Merritt & Cruz 2001; Boylan-Kolchin & Ma 2007). For the stars at r ∼ r b which predominate the loss-rate enhancement, the maximum timescale of coherent angular-momentum change, t ω , is limited by the apsidal precession timescale, which is of order t d ( r b ). The timescale for chaotic orbital evolution is also of order t d ( r b ). The adiabatic limit therefore requires the orbital period of the merging galaxies to be longer than t d ( r b ). For this reason, the following calculations are restricted to d > 2 r b .", "pages": [ 5, 6 ] }, { "title": "3.2. Stellar-disruption Rates", "content": "We now calculate the stellar disruption rate for both galaxies in a merger. The black-hole and bulge components are modeled with the parameters ( M · , γ, β ), as is described in § 2. The mass ratio of the galaxies, by construction, equals the mass ratio of the SMBHs, q ≡ M · ,s /M · ,m ≤ 1, where the subscript m denotes the quantity for the bigger main galaxy and s for the smaller satellite galaxy. As we have shown that the contribution to stellar tidal disruptions is dominated by the stars at the break radius of galaxy, we can approximately construct a merger system of galaxies without loss of generality as follows. Given the distance d between the two galaxy centers, the total stellar density at any location is approximated by summing the densities of the two unperturbed bulges. In this density field, each galaxy approximately preserves its initial structure out to a radius min( r max , r tr ), where r tr is the truncation ra- utual tidal interaction, defined by the condition that the mean densities within r tr are the same for the two truncated galaxies. Figure 1 shows the density contours (upper panel), as well as the density distribution along the line connecting the two black holes (lower panel), for a merging system with M · = 10 7 M /circledot , q = 0 . 3, and d = 50 r b . Given the configuration of the merging system, we calculated θ 2 and Γ due to two-body relaxation for each of the two galaxies. To calculate θ p and Γ due to tidal perturbation, the perturber mass M p is derived by integrating the stellar and blackhole masses in the perturber galaxy enclosed by r tr . Note that the perturber is the satellite galaxy when calculating Γ for the main galaxy, but can also be the main galaxy when calculating Γ for the satellite. To calculate θ c and Γ due to chaotic loss-cone feeding, the triaxiality of galaxy needs to be determined. But our model is axisymmetric by construction, so the triaxiality cannot be derived self-consistently. We circumvent this inconsistency by assuming that at any radius where the density increment induced by the perturber excesses δ = 20% of the initially unperturbed density, a fraction of f c = 50% of stellar orbits are chaotic. Otherwise f c = 0 if δ < 20%. The radial range where f c = 50% is insensitive to the choice of δ because of the steep density profiles we adopted in the following calculations. Figure 2 shows the stellar disruption rates as a function of d for both main (upper panel) and satellite (lower panel) galaxies. The parameters are ( M 7 , q, γ, β ) = (1 , 0 . 3 , 1 . 75 , 2) by default. When d /greatermuch 100 r b ≈ 450 pc, the loss-cone filling in both galaxies is dominated by two-body relaxation (dotted lines) and the disruption rate is identical to that for isolated single SMBH. As the distance shrinks to d ∼ 100 r b , about 2 r e of the central galaxy, the disruption rates induced by companion galaxies start to exceed those due to two-body relaxation. This is because θ p ( r b ) becomes greater than θ 2 ( r b ). As d further decreases to d /lessorsimilar 10 r b ≈ 45 pc, θ c ( r b ) becomes greater than θ 2 ( r b ), so the contribution to Γ due to triaxial potential starts to exceed that due to two-body relaxation. When the two galaxy cores are as close as the break radius of the main galaxy, Γ in both galaxies have been enhanced by two orders of magnitude. In the subsequent evolution with d /lessorsimilar 2 r b for which our simple scheme cannot be applied, the three-body interactions between the two gravitationally bound SMBHs and the surrounding stars are expected to play an important role and to further enhance the disruption rates (Ivanov et al. 2005; Chen et al. 2009, 2011; Wegg & Nate Bode 2011). In a real merger, because galaxy orbitals are eccentric (Jiang et al. 2008), the distance d will not decrease monotonically, but oscillate between the apocenter distance r apo and pericenter distance r per , both distances decreasing with time due to dynamical friction. In this case, one can average the stellar-disruption rate over one orbital period according to ¯ Γ = ∫ r apo r per Γ( r ) v -1 r dr/ ∫ r apo r per v -1 r dr , where v r denotes the radial velocity of galaxy at distance r . Since in our model, where r/v r ∝ r β/ 2 and Γ( r ) ∝ r -η , both β and η are of order unity (see Section 3.3), we find that ¯ Γ differs from Γ( d ) by a factor of also order unity if we define d ≡ ( r per + r apo ) / 2. In this sense, the rates in Figure 2 can be used as the orbital-averaged stellardisruption rates for galaxy mergers with eccentric orbits. We also note that we may have underestimated the contribution from triaxial potential, because in our model by construction f c vanishes inside about the influence radius of black hole, as the stellar-density variation δ inside the sphere of radius r b is small (e.g. lower panel of Figure 1). In real galaxies, however, chaotic orbits may partially exist inside the influence radius of black hole (Poon & Merritt 2001).", "pages": [ 6, 7 ] }, { "title": "3.3. Dependence of disruption rate on model parameters", "content": "In § 3.2, we have shown that tidal perturbation by the companion galaxy dominates the enhancement of Γ in a merger. The enhancement occurs when θ p in the diffusive regime exceeds θ 2 . As a result, the critical radius r cri ,p that separates the pinhole and diffusive regimes is now determined by θ p ( r cri ,p ) = θ lc , and enhancement of stellar-disruption rate requires that r cri ,p < r cri . Now we investigate in what mergers the condition r cri ,p < r cri would be satisfied. According to J p ( r cri , p ) = J lc and the relation where we first derive the following scaling relation in the limit r cri , p /lessorsimilar r b and J lc = J td for the central galaxy: Since 1 < β < 3 for the majority of galaxies (Lauer et al. 2005), Equation (15) suggests that in general enhancement of Γ would occur when the perturbing galaxy is larger or the galaxy distance is smaller. When tidal perturbation dominates the loss-cone filling, according to Equation (3) and M · ∝ σ 4 e , the rate Γ in the limit r cri , p /lessorsimilar r b scales as For our fiducial model with γ = 7 / 4 and β = 2, we can derive Γ ∝ ( d/r b ) -5 / 7 , which is consistent with the numerical results given by the dashed lines at d < 30 r b in Figure 2. Figure 3 shows the dependence of Γ on black hole mass when q = 0 . 3. Equations (15) and (16) suggest that Γ ∝ M (24+ γ ) / 42 · when q and d/r b are fixed. The enhanced stellar-disruption rates in Figure 3 generally agree with this scaling when M · /lessorsimilar 4 × 10 7 M /circledot . When M · > 4 × 10 7 M /circledot , the dependence of Γ on M · steepens because direct capture of stars by SMBH (GR effect) becomes important, such that the scaling of loss-cone size changes from J 2 lc ∝ M 4 / 3 · to J 2 lc ∝ M 2 · . When M · /greaterorsimilar 10 8 M /circledot , the loss-cone stars will be directly captured by the central SMBH without producing tidal flares if the SMBH is non-rotating or rotates slowly (Ivanov & Chernyakova 2006; Kesden 2012), and the corresponding curves are shown in dashed lines. Note that even when the SMBH in the main galaxy is more massive than 10 8 M /circledot , the merging system could still produce tidal flares, due to the existence of a smaller SMBH in the satellite galaxy. We found that when 1 < M 7 /lessorsimilar 10, the event rates of tidal flares can be as high as ∼ 10 -2 yr -1 as d shrinks to about r b . Figure 5 shows the variation of stellar disruption rate when the density profile of the main galaxy changes. For the main galaxy, when the inner power-law index γ decreases from 1 . 75 to 1, the stellar disruption rates due to two-body relaxation and tidal perturbation both drop by a factor of a few, because of the slight decrement of the stellar density at r ∼ r b . Meanwhile, the dependence of Γ on d/r b at d /lessorsimilar 10 r b changes from ( d/r b ) -5 / 7 to ( d/r b ) -2 / 7 , resulting in an even smaller rate at d ∼ r b . When the outer power law index β decreases from 2 to 1 . 5, the stellar disruption rates in the main galaxies drop approximately by a factor of 20. This is because the galaxy with shallower outer density profile is more spatially extended and has lower central density. For the satellite, when γ or β of the main galaxy decrease, the enhancement of stellar disruption rate occurs at smaller d/r b and becomes weaker for a Figure 4 shows the dependence of Γ on the mass ratio q = M · ,s /M · ,m ≤ 1 of the two black holes, while M · ,m is fixed. For both main and satellite galaxies, the enhancement of Γ becomes more significant as q increases. It is worth noting that even q is as small as 0 . 01, the stellar disruption rate in the main galaxy can still be enhanced by two orders of magnitude when d shrinks to about r b . We also find that the enhanced stellar disruption rate in the satellite is more sensitive to q than that in the main galaxy. This is because the baseline stellar-disruption rate, i.e., the rate for single black hole in isolated galaxy, changes with q for satellite galaxy, but does not vary for the main galaxy since in the calculation M · ,m is fixed (e.g. see Equation [16]). Quantitatively speaking, according to Equations (15) and (16), when varying q while keeping M · ,m fixed, the enhanced stellar disruption rate for the main galaxy scales as q 3(1 -β s )(1 -2 γ ) / (14 β s ) , while the rate for the satellite scales as q (24+ γ s ) / 42 -(1 -2 γ s )(3+ ββ s -3 β s ) / (14 β ) . For example, given ( γ, β, γ s , β s ) = (1 . 75 , 2 , 1 . 75 , 2), one can derive Γ ∝ q 15 / 56 for the main galaxy and Γ ∝ q 59 / 84 for the satellite. fixed d/r b . This is because r b of the main galaxy becomes greater as γ or β decrease, so that for the satellite the physical distance of the perturber increases if d/r b is fixed. We notice that when β = 1 . 5, the stellar disruption rate in the main galaxy remains lower than that in the satellite as d decrease. This result implies that in mergers where the main galaxies have low surface brightness, the tidal flares are mostly contributed by the satellite galaxies. When the density profile of the satellite galaxy is varying, the resulting stellar disruption rates are shown in Figure 6. In general, the dependence of Γ on the density profile can be understood in the light of the analysis for Figure 5, except that now the role between the main and satellite galaxies switches. However, one difference is that when d shrinks to about r b , the disruption rate in the main galaxy is not sensitive to the density profile of the satellite. This is because when d ∼ r b the stellar cusp surrounding the SMBH in the satellite is almost completely striped off by the tidal filed of the main galaxy, so for the main galaxy the perturbing mass is approximately M · ,s . Figures 2-6 showed that galaxy merger starts enhancing stellar-disruption rate when the two galactic nuclei are still widely apart, well before the two SMBHs become gravitationally bound. The boost factor for each SMBH incorporated is about 10 2 ( M · / 10 7 M /circledot )( d/r b ) µ ( q/ 0 . 3) ν , where M · is the mass of the subject black hole, r b refers to the break radius of the more massive galaxy, and µ and ν are indices depending on the density profiles of the two galaxies. Less massive black holes have smaller boost factors because prior to merger they already have higher stellar-disruption rates. The exact boost factor depends on the stellar-disruption rate prior to galaxy merger, which deserves some discussion. When calculating Γ for isolated galaxies, we considered only two-body relaxation but not more efficient relaxation processes, such as resonant relaxation, perturbation by massive objects, or relaxation processes in triaxial gravitational potential (e.g. Rauch & Tremaine 1996; Perets et al. 2007; Merritt & Poon 2004). Resonant relaxation enhances stellar-disruption rate only mildly, less than a factor of a few in typical galaxies (Rauch & Ingalls 1998). Massive perturbs, such as molecular clouds and stellar-mass black holes, if highly concentrate inside the influence radius of an SMBH, in principle could enhance the stellar-disruption rate by orders of magnitude (Perets et al. 2007). But such galactic nuclei could only be transient, because large concentration of massive perturbs normally corresponds to short relaxation timescale. On the other hand, weak triaxiality seems intrinsic to galaxies, suggesting that chaotic loss-cone feeding may be important prior to galaxy mergers. If we use Equation (119) in Merritt & Vasiliev (2011) 2 to estimate the stellar-disruption rate induced by triaxial potential inside the black-hole influence radius, meanwhile use formulae derived in Sections 2 and 3.1 with f c = 0 . 1 to calculate the rate due to chaotic orbits outside the black-hole influence radius, then the total disruption rates for isolated fiducial galaxies become Γ /similarequal (5 . 8 , 8 . 9 , 47) × 10 -5 yr -1 when M · = (10 6 , 10 7 , 10 8 ) M /circledot . For comparison, the rates due to two-body relaxation only are (4 . 0 , 2 . 3 , 1 . 4) × 10 -5 yr -1 . The difference is the greatest in the case of M · = 10 8 M /circledot , because the 'gap' between r cri and r b is the largest. These results suggest that only in the most massive galaxies with M · /greaterorsimilar 10 8 M /circledot could intrinsic triaxiality make the enhancement of stellardisruption rate less significant.", "pages": [ 7, 8, 9, 10 ] }, { "title": "4. Contributions of tidal flares by merging galaxies", "content": "In a synoptic sky survey, the probability of catching tidal flares in merging galaxies does not depend only on the stellar-disruption rate, but also on the duration of galaxy mergers. In other words, the fractions of tidal flares in merging and in normal galaxies are proportional to the numbers of stellar-disruption events produced during, respectively, the merger and the quiescent phases. Since the duration of a galaxy merger is determined by the dynamical friction timescale, t df , the fraction of tidal flares in merging galaxies is proportional to the typical number of tidal stellar disruptions, n = t df Γ. Given the distance d between two merging galaxies, we calculate the dynamical friction timescale as (see eq. [8.13] in Binney & Tremaine 2008), where M g ( d ) here refers to the stellar mass enclosed by the radius d in the main galaxy and M s is the total mass of the truncated satellite. When the two galaxies are distant and Γ is not enhanced, the total number of disrupted stars is proportion to the dynamical friction timescale, which is We refer to this early evolutionary stage as phase I. During phase I, the dependence of t df Γ on d β/ 2 implies that the majority of tidal flares are contributed by wide galaxy pairs. When loss-cone feeding is enhanced due to tidal perturbation by the companion galaxy, the stellar-disruption rate Γ increases with decreasing d . We refer to this later evolutionary stage as phase II, and we note that main and satellite galaxies enter phase II at different times. During phase II, the dependence of t df Γ on d flattens compared to that in phase I, implying a enhanced detection rate of tidal flares in close galaxy pairs. Figure 7 gives the typical number of disrupted stars ( t df Γ) as a function of d in our fiducial model. In the calculation, we did not consider the decrease of stellar density due to tidal disruption, because the total mass of disrupted stars is negligible with respect to the stellar mass in the initial condition. In general, when the merger is in phase I, t df Γ scales as d , as predicted. During this phase, t df Γ is not sensitive to the total mass of the system as long as q is fixed, because massive systems where Γ is larger have shorter t df . Note that before the main galaxy (solid curve) enters phase II, the satellite galaxy (dashed curve) contributes comparable number of, if not more, tidal flares. This is because when two-body relaxation predominates the loss-cone filling, smaller galaxies have smaller diffusive loss cones, therefore will have higher stellar-disruption rates, as is explained in the end of Section 2. When the galaxy mergers enter phase II, which is marked by the dots, the t df Γ curve flattens, indicating an enhanced contribution of tidal flares by closer galaxy pairs. During this phase, the contribution of tidal flares from main galaxy is typically greater than that from satellite. Figure 8 shows the dependence of t df Γ on different parameters of galaxy merger, which are summarized as follows. the satellite is more susceptible to tidal stripping and t df is much longer than that in the fiducial case. Figures 7 and 8 suggest that during phase I the total number of disrupted stars scales roughly as insensitive to the total mass of the system or the stellar density profiles of the merging galaxies. During phase II, when the curve of t df Γ is nearly independent of distance d , the total number becomes about where m /similarequal 3(1 -β s )(1 -2 γ ) / (14 β s ) -1 and n = (45+ γ ) / 42 are power-law indices derived from Equations (15), (16) and (18). It is worth noting that each merger investigated above involves only two galaxies. However, mergers of group galaxies are also common and the tidal disruption rates are expected to be even more heavily enhanced because of stronger perturbations and larger triaxiality.", "pages": [ 10, 11, 12, 13 ] }, { "title": "5. Discussions", "content": "Formation of SMBHBs at galaxy centers is anticipated in the paradigm of hierarchical galaxy formation (Begelman et al. 1980), and coalescence of the binaries is predicted to induce recoiling velocities on the post-merger SMBHs (Centrella et al. 2010). In our previous works (Chen et al. 2008, 2009, 2011; Liu et al 2009), we investigated the possibility of using tidaldisruption flares to identify gravitationally bound SMBHBs of sub-pc separations in galactic nuclei. Recently, off-nuclear tidal flares have also been suggested in the literature to be probes of recoiling SMBHs (Komossa & Merritt 2008; Merritt et al. 2009; Stone & Loeb 2011, 2012; Li et al. 2012). However, an off-center tidal flare can also be produced by SMBHs embedded in merging galaxies. In this paper, we calculated the tidal flare rates produced by dual SMBHs in a particular evolutionary stage when the two SMBHs are still unbounded to each other and isolated in the cores of merging galaxies. We considered three major processes responsible for the loss-cone feeding in the merger system, namely, two-body stellar relaxation, tidal perturbation by the companion galaxy, and chaotic stellar orbits in triaxial gravitational potential. By employing an analytical model to calculate the stellar disruption rates for both SMBHs in the two merging galaxies, we found that prior to the formation of SMBHB the stellar disruption rate would be enhanced by as large as two orders of magnitude in both galaxies. The enhancement is dominated by tidal perturbation and occurs when the two galaxies are so close that the stars inside the influence radius of the central SMBH are significantly perturbed. We have shown that the enhanced stellar disruption rate depends on the masses, mass ratio, and density profiles of the two galaxies, as well as the distance d between the two galaxy cores. In the fiducial model with ( M 7 , q, γ, β ) = (1 , 0 . 3 , 1 . 75 , 2), the enhancement starts when the perturber galaxy approaches approximately twice the effective radius of the central galaxy ( d /similarequal 2 r e ). In more massive systems with M 7 > 10, where the stellar disruption rates due to two-body relaxation are generally lower, the enhancement starts as soon as d shrinks to 10 r e . As a result, the phase with enhanced stellar-disruption rate extends to an evolutionary stage much earlier than the formation of bound SMBHB, which considerably increases the detection rate of wide SMBH pairs in tidalflare surveys. When d shrinks to about the influence radius of the central SMBH ( d ∼ 2 r b ), the stellar disruption rate in the fiducial model increases to 3 × 10 -3 yr -1 in the main galaxy and to 2 × 10 -3 yr -1 in the satellite. Compared to the peak rates in the later evolutionary stages with gravitationally-bound binary SMBHs (e.g. Chen et al. 2009, 2011; Wegg & Nate Bode 2011), the total stellar-disruption rates before SMBHs become bounded are smaller by only a factor of a few. In more massive or equal-mass ( q > 0 . 3) mergers, the stellar disruption rates could be even higher. The above results showed that the tidal disruption rates by off-center SMBH pairs in merging galaxies are several order magnitudes higher than those by recoiling off-nuclear SMBHs (Komossa & Merritt 2008; Li et al. 2012; Stone & Loeb 2012), implying that off-center tidal disruption flares would be overwhelmed by the SMBH pairs in merging galaxies. Therefore, it would be challenging to distinguish recoiling SMBHs in off-center tidal disruption flares. One possible way to distinguish the two kinds of off-center tidal disruptions may be to identify the evolutionary stages of galaxies. Recoiling SMBHs are in galaxies at late stages of mergers, while un-bounded SMBH pairs are in galaxies at early or middle stages of mergers. Early stages of major mergers when galaxies are widely separated may be identified with the disturbed morphology of host galaxies. However, morphological signatures of galaxy merger are weak during the middle or late stages of major mergers, as well as during the whole stages of minor mergers, therefore it would be also a challenge to identify these merger stages. Another difference may be among the properties of star clusters around the off-center SMBHs. A recoiling SMBH is expected to reside in an ultra-compact bound star cluster of mass much smaller than the black-hole mass, of size much smaller than the black-hole influence radius, and of stellar-velocity dispersion much larger than that of host galaxy (Merritt et al. 2009; Li et al. 2012). It may also associate with a massive cloud of unbound stars, whose mass is comparable to the black-hole mass, size comparable to the black-hole influence radius, and stellar-velocity dispersion comparable to or greater than that of the host galactic nuclei (Li et al. 2012). While the star clusters hosting the the secondary black holes in minor mergers are the remnants of the tidally truncated satellite galaxies. These clusters are orders of magnitude heavier than the secondary SMBHs, their sizes are much larger than the influence radii of the secondary or the primary SMBHs, and their stellarvelocity dispersions are comparable to those of typical dwarf galaxies but significantly smaller than those of the primary galactic nuclei. Therefore, the two types of star clusters should differ significantly in their sizes, stellar-velocity dispersions, and the mass ratios between SMBHs and star clusters, which could be identified with deep photometrical and spectroscopical observations. When a pair of SMBHs evolve to about the influence radius, d ∼ r b , the enhanced stellardisruption rates can be as high as 10 -2 yr -1 . For such a high tidal disruption rate, multiple tidal flares may occur in the same galaxy within a time span of decades. Unlike the recurring tidal flares in binary or recoiling SMBH systems, the flares in merging galaxies are contributed by wide SMBH pairs separated by r b ∼ 1 -10 pc (depending on black hole mass and galaxy density profile). Note that a separation of 10 pc at redshift z = 0 . 1 (1) corresponds to an angular size of 5 (1) milliarcsec (mas). As a result, spatial offsets between successive tidal flares in such a merging system may be detected by instruments such as Gaia and LSST 3 . Figures 3 and 4 imply that such flipflop flares could occur in the galaxy mergers with 10 7 M /circledot < M · < 10 8 M /circledot and q > 0 . 1. The mergers with M · < 10 7 could not produce recurring flares because the stellar-disruption rate is too low. When 10 7 M /circledot < M · < 10 8 M /circledot but q < 0 . 1, most flares are produced in the main galaxy; therefore, the recurring flares are unlikely to display spatial offset, and would be indistinguishable from those in binary or newly-formed recoiling SMBH systems. When M · > 10 8 M /circledot , the SMBH in the main galaxy would directly capture stars, mostly without producing flares, while the SMBH in the satellite could still produce tidal flares if M · ,s < 10 8 M /circledot . In the last case, although the recurring flares occur at the same sky position, they should be displaced from the center of the minor merger by an amount of r b /greaterorsimilar 10 pc. Sources with such high flaring rate and large off-center displacement cannot be produced by binary or recoiling SMBHs. The above discussions suggest that with the aid of telescopes with high spatial resolution, the cause of the recurring tidal flares can be distinguished. In the universe, the fraction of tidal flares contributed by galaxy mergers is proportional to the total number of the disrupted stars during merger. During each merger, the number of tidal flares contributed by phase I, when the separations of galaxies are about r e /lessorsimilar d /lessorsimilar 10 r e , is about n I ∼ 10 3 q -1 ( d/ 10 3 r b ) β/ 2 [Figures (7) and (8)]. The scaling n I ∝ q -1 implies that n I is determined mainly by minor mergers. Suppose a galaxy experiences N mergers during a Hubble time ( ∼ 10 10 yr), then during one duty cycle, the number of tidal flares contributed by the isolation phase is about n s ∼ 2 × 10 5 /N , if two-body scattering is the dominant relaxation process. Since a galaxy with M 7 = 1 (10) at redshift z = 0 has experienced typically N ∼ 10 (100) mergers and most mergers have q ∼ 0 . 1 (Hopkins et al. 2010), according to the ratio n I : n s , we find that about ∼ 5% (50%) tidal flares are contributed by phaseI galaxy mergers ( d ∼ 10 3 r b ). For typical mergers with q /lessmuch 1, according to Figure 8, the majority of the tidal flares are produced in satellite galaxies during phase I, unless the satellite galaxies have low surface brightness. This result implies that a large fraction of genuine tidal flares would be displaced by several r e from the centers of the merging systems. Given that an offset of 2 r e ∼ 500 pc corresponds to 250 (60) mas at z = 0 . 1 (1), these offset tidal flares could be misidentified as supernovae or gammaray bursts by careless classification schemes. They may also be mistaken as ' naked ' recoiling quasars (e.g., Komossa & Merritt 2008) or ' orphan transients ' (X-/ γ -ray transients either uncorrelated with bursts in low-energy bands or without detection of optical counterparts, e.g., Horan et al. 2009) because of the relative dimness of the satellites. The mis-identification could be very common in massive galaxies, because the physical scale of r e is larger. During phase II when the stellar disruption rates are enhanced by galaxy mergers, the main galaxies would contribution typically more than half of the tidal flares, unless q ∼ 1 or the surface brightnesses of the main galaxies are low. This result indicates that in an advanced merger, where the separation between the two galaxy cores is less than the effective radius of the main galaxy, the tidal flares preferentially reside in the massive nucleus of the system. According to Figures 7 and 8, the number of tidal flares contributed by such advanced merger phase does not depend on d/r b , and scales as n II ∼ 200 q m M n 7 , where m < 0 and n > 0 are analytical indices derived in § 4. Therefore, the biggest contribution is expected to come from minor mergers in massive systems. Because n II is typically less than n I , the contribution of tidal flares from phase II is typically smaller than that from phase I. However, for the most massive systems with M 7 /greaterorsimilar 10 in which the main SMBHs mostly swallow the stars without producing tidal flares, one major merger ( q > 0 . 3), or one minor merger ( q /lessorsimilar 0 . 3) between galaxies of low surface brightness, would produce more tidal flares in phase II than in phase I. In these particular systems, a greater fraction of tidal flares would be contributed by close SMBH pairs with separations 10 /lessorsimilar d/r b /lessorsimilar 100. It is important to know the relative contributions of tidal flares by single ( n s ), binary ( n b ), and merging SMBH systems ( n I and n II ). The total number of flares produced by a recoiling black holes is typically smaller than 10 3 (Komossa & Merritt 2008; Stone & Loeb 2011; Li et al. 2012), therefore negligible in the comparison. According to Chen et al. (2011), during the lifetime of an SMBHB with q /lessmuch 1, the interaction between the binary and the surrounding dense stellar cusp will produce a number of n b /similarequal 7 × 10 4 q (2 -γ ) / (6 -2 γ ) M 11 / 12 7 of tidal flares. Suppose a galaxy on average experiences N mergers and M ( M ≤ N ) of them result in the formation of SMBHBs. Then being averaged by one duty cycle of galaxy merger, n s : n b : ( n I + n II ) is about 20 : 5 M : N , where we used q = 0 . 1 because minor mergers are the most common (Volonteri et al. 2003; Hopkins et al. 2010; McWilliams et al. 2013b). For galaxies of total masses (10 9 , 10 10 , 10 11 ) M /circledot , typical N are (1 , 10 , 10 2 ) (Hopkins et al. 2010) or significantly higher (McWilliams et al. 2013a,b; B'edorf & Portegies Zwart 2013), while M are predicted to be greater than 1 (Volonteri et al. 2003). These numbers highlight the significant contribution of tidal flares from merging systems with multiple SMBHs. To give more accurate calculations, one has to combine the cosmic merger history of galaxies, as well as the formation rate of SMBHBs of different masses and mass ratios. Such calculations and the assessment of their uncertainties are beyond the scope of the current paper and will be addressed in a future paper. We are grateful to Shuo Li, Zuhui Fan, Rainer Spurzem, and Thijs Kouwenhoven for helpful comments. We also thank Alberto Sesana for earlier discussions on this topic. This work is supported by the National Natural Science Foundation of China (NSFC11073002). F.K.L. also thanks the support from the Research Fund for the Doctoral Program of Higher Education (RFDP), and X.C. acknowledges the support from China Postdoc Science Foundation (2011M500001).", "pages": [ 13, 14, 15 ] }, { "title": "REFERENCES", "content": "Merritt, D. & Poon, M. Y., 2004, ApJ, 606, 788 Merritt, D., Schnittman, J. D., & Komossa, S. 2009, ApJ, 699,1690 Merritt, D., & Vasiliev, E. 2011, ApJ, 726, 61 Ostriker, J. P., Spitzer, L., & Chavalier, R. A., 1972, ApJ, 176, L47 Perets, H. B., Hopman, C., & Alexander, T., 2007, ApJ, 656, 709 Poon, M. Y., & Merritt, D., 2001, ApJ, 549, 192 Poon, M. Y., & Merritt, D., 2004, ApJ, 606, 774 Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26 Rauch, K. P., & Ingalls, B., 1998, MNRAS, 299, 1231 Rauch, K. P., & Tremaine, S. 1996, NewA, 1, 149 Rees, M. J. 1988, Nature, 333, 523 Roos, N., 1981, A&A, 104, 218 Schnittman, J. D. 2010, ApJ, 724, 39 Seto, N., & Muto, T. 2011, MNRAS, 415, 3824 Stone, N., & Loeb, A. 2011, MNRAS, 412, 75 Stone, N., & Loeb, A. 2012, MNRAS, 422, 1933 Syer, D. & Ulmer, A., 1999, MNRAS, 306, 35 Tremaine, S., et al. 2002, ApJ, 574, 740 van Velzen, S., Farrar, G. R., Gezari, S., et al. 2011, ApJ, 741, 73 Volonteri, M., Haardt, F., & Madau, P. 2003, ApJ, 582, 559 Wang, J.-X. & Merritt, D., 2004, ApJ, 600, 149 Wegg, C., & Nate Bode, J. 2011, ApJ, 738, L8 Young, P. J. 1977, ApJ, 215, 36", "pages": [ 17 ] } ]

[ { "title": "ABSTRACT", "content": "Explaining variability observed in the accretion flows of black hole X-ray binary systems remains challenging, especially concerning timescales less than, or comparable to, the viscous timescale but much larger than the inner orbital period despite decades of research identifying numerous relevant physical mechanisms. We take a simplified but broad approach to study several mechanisms likely relevant to patterns of variability observed in the persistently high-soft Roche-lobe overflow system LMC X-3. Based on simple estimates and upper bounds, we find that physics beyond varying disk/corona bifurcation at the disk edge, Compton-heated winds, modulation of total supply rate via irradiation of the companion, and the likely extent of the partial hydrogen ionization instability is needed to explain the degree, and especially the pattern, of variability in LMC X-3 largely due to viscous dampening. We then show how evaporation-condensation may resolve or compound the problem given the uncertainties associated with this complex mechanism and our current implementation. We briefly mention our plans to resolve the question, refine and extend our model, and alternatives we have not yet explored.", "pages": [ 1 ] }, { "title": "PROTOTYPING NON-EQUILIBRIUM VISCOUS-TIMESCALE ACCRETION THEORY USING LMC X-3", "content": "Hal J. Cambier Physics Department, University of California, Santa Cruz, CA 95064 and", "pages": [ 1 ] }, { "title": "David M. Smith", "content": "Physics Department, University of California, Santa Cruz, CA 95064, USA and Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA Draft version April 21, 2022", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The X-ray spectrum of black hole X-ray binaries (BHXRBs) often shows variability in intensity and hardness on timescales of order the viscous timescale, but much larger than the innermost orbital period, including the well-known 'q'-diagram hysteresis patterns traced by transient BHXRBs (Fender et al. 1999; Homan & Belloni 2005; Done et al. 2007, and see fig.1). The properties of such variability may also evolve over multiple viscous-timescale cycles. The high-soft (but subEddington) and quiescent intensity-hardness limits are understood as manifestations of the thin-disk (Shakura & Sunyaev 1973) and radiatively-inefficient advectiondominated accretion flow (ADAF; Narayan & Yi 1995a) accretion limits while states between are typically understood as some evolving combination of disk and ADAF flows (Chakrabarti & Titarchuk 1995; Esin et al. 1997; Nandi et al. 2012). Theoretical work has begun in this regime, but still cannot fully explain observations, especially regarding viscous timescales where detailed simulations require prohibitively many time steps and steadystate assumptions lose validity. This motivates us to develop theory in more detail starting with a system like LMC X-3, whose behavior is more constrained than that of the transients, but still exhibits substantial variability that is quantitatively challenging to explain. In the current paradigm, transient BHXRBs are those systems where the outer disk reaches temperatures low enough to trigger the partial hydrogen ionization instability (PHII) in which accretion proceeds via cycles as viscosity alternately concentrates mass into rings and diffuses it inward (see Cannizzo 1998, and Lasota 2001 for a review). The part of the cycle where viscosity concentrates mass leads to the quiescent phase where any emis- sion is presumably powered by some fraction of the flow that escapes the viscosity trap. Eventually, density increases enough to raise disk temperature above the transition point in viscosity, and the sudden jump in accretion rate leads to a rise in luminosity (often several orders of magnitude) while still in the hard state, followed by a softening of the spectrum at this same, peak luminosity (the vertical and horizontal shifts by the dashed line in fig.1). Jet emission is seen to shut off as the system enters the extreme high-soft state (Fender et al. 1999). Afterward, the system typically makes partial transitions in hardness (with small, erratic shifts in luminosity) on sub-viscous timescales and associated with intermittent jet emission on top of a secular decline in luminosity. Eventually the system transitions completely back to the hard state and then finishes fading back into quiescence. There are a handful of persistent BHXRBs and blackhole-candidate X-ray binaries that do not execute such extreme quiescence-flaring cycles, and consistent with the transient paradigm they appear to avoid the PHII through an appropriate combination of disk size, luminosity (for disk irradiation), and mass supply rate (Coriat et al. 2012). Despite the label, these systems may still show significant variability and a variety of behaviors. Cygnus X-1 appears to slide between canonical high-soft and low-hard states with some scatter but no clear hysteresis (Smith et al. 2002, hereafter SHS02). The black hole candidates GRS 1758-258 and 1E 1740.7-2942 exhibit transient-like hysteresis but seldom reach the soft or quiescent limits (SHS02). LMC X-3 is typically bright in soft X-rays, but shows some hysteresis that circulates in the opposite direction to transients (see fig.1), and does occasionally transition completely to the hard state (Wilms et al. 2001; Smale & Boyd 2012). Accounting for irradiation of the outer disk, LMC X-3 accretes at rates high enough to usually avoid the ionization instability completely (Coriat et al. 2012), though the outer disk likely becomes susceptible for the deeper drops in disk luminosity, and almost certainly for rare, complete state transitions. Such high accretion rates are explained via Roche-lobe overflow (RLO); optical measurements of the companion's spectral type that take disk irradiation into account indicate B5IV (Soria et al. 2001) or B5V (Val-Baker et al. 2007) spectral type. Such a star will fill the Roche lobe for a 1.7 day orbit around a black hole with mass equal to the recent 9.5 M glyph[circledot] lower bound (Val-Baker et al. 2007). Furthermore, Soria et al. (2001) have argued that feeding the observed X-ray luminosities via winds would lead to column densities far higher than measured. Although its distance precludes direct measurement or exclusion of radio jets, the jet quenching observed in transient BHXRBs as they approach the highsoft state also suggests that LMC X-3 does not usually possess strong jets (Fender et al. 1998). Thus, if LMC X3 shares similar variability mechanisms with transients, while being far less prone to the ionization instability and jet outflows, then LMC X-3 provides a more controlled setting to study such mechanisms. Below we list the major mechanisms we have considered so far in modeling LMC X-3, also summarized in fig.2. For RLO systems like LMC X-3, hard X-rays from the inner accretion disk can lead to supply-rate modulation (SRM) from the companion by inflating the companion's atmosphere to increase the density of gas at the L1 Lagrange point, as well as the area and pressure behind the nozzle (Lubow & Shu 1975; Meyer & Meyer-Hofmeister 1983). We elaborate in § 3.1. As the stream of gas from the companion dissipates energy and spirals onto the circularization radius, it may encounter the edge of the viscously spreading disk and from this point the flow can undergo disk corona bifurcation (DCB) as some fraction can efficiently shock, cool, and join the disk while some fraction may stream past the thin disk edge and maintain its virial temperature (Hessman 1999; Armitage & Livio 1998). Warping of the outer disk, whether driven by irradiation (Pringle 1992; Ogilvie & Dubus 2001; Foulkes et al. 2010) or a lift force (Montgomery & Martin 2010) can affect the accretion flow by changing the density profile that the disk presents to the RLO stream ( § 3.2), and by varying how the companion is exposed to or shadowed from inner disk X-rays. For LMC X-3 specifically, Ogilvie &Dubus (2001) and Foulkes et al. (2010) predict that the system is potentially unstable to irradiation-driven warping. Because the dominant long-term effects of warping are through bifurcation, and because we will lump them together as a boundary condition in our simulations, we combine discussion of them into § 3.2. X-rays from the inner disk can Compton heat gas in the disk atmosphere and any corona at large radii above the local virial temperature thus driving a Compton-heated wind (CHW), discussed in § 3.3, which is not only important for removing gas, but for removing hot corona that might otherwise help evaporate the disk or condense further inward (item (EC) below). As noted, for large enough disks and insufficient irradiation, a finite strip in the outer disk becomes susceptible to the PHII. For now, we do not treat it in any detail, but discuss the likely extent and manner of its effects in LMC X-3 in § 3.4. If the disk and corona are coupled thermally, then the disk and corona may exchange mass through evaporation and condensation (EC). Mayer & Pringle ((2007), hereafter MP07) provide a thorough introduction and numerical treatment, and Liu et al. (Liu et al., hereafter LTMHM07) and Meyer-Hofmeister et al. (2009) discuss more applications and provide the steady-state prescription for our modified method. Through EC, extant disks will tend to preserve the soft state down to lower luminosities via Compton-cooling-driven condensation, providing a natural explanation of why BHXRBs return to the hard state at lower luminosity and thus show hysteresis (Meyer-Hofmeister et al. 2009 focus on this aspect). We discuss other interesting effects possible in § 3.5. We will restrict our focus to an alpha-viscosity prescription for the disk. For the present work describing long-timescale variability over accretion rates typical to LMC X-3 where uncertainties regarding conventional mechanisms still loom large, we consider this perfectly adequate, but acknowledge the possibility of more intrinsic variability mechanisms ( § 5). In § 2 we review key features of LMC X-3's accretion behavior, summarize how we infer the innermost disk and corona/ADAF accretion rates from the X-ray data (additional details are provided in the Appendix), and critically examine the qualitatively simple bifurcationonly model in Smith et al. ( ? , hereafter SDS07). The latter motivates § 3, in which we furnish additional detail on the mechanisms as listed above, including estimated constraints on their effects and their current level of implementation in our modeling. In § 4.1 we argue that a model including mechanisms besides EC cannot explain the data, but does best when given an unreasonably small disk radius and very large variations in corona-disk ratio at this boundary. We then show in § 4.2 how EC may ef- ively recreate such seemingly ad-hoc conditions, but how it may also imply behavior inconsistent with observations, including an extremely easily triggered 'sympathetic' mode where the innermost disk and corona accretion rates rise and fall simultaneously. We briefly review the results and caveats of the current model, and state our current plans to resolve the question in § 5.", "pages": [ 1, 2, 3 ] }, { "title": "2. LMC X-3 AS PROTOTYPE", "content": "Besides simplifying initial modeling as discussed above, LMC X-3 offers additional practical advantages. The Rossi X-ray Timing Explorer ( RXTE ) monitored LMC X-3 for over 16 years, and at least five of those include observations each about a kilosecond long taken roughly twice a week, thus providing a long, uninterrupted history of accretion with sufficient resolution at the timescales we seek to study. Also, the X-ray blackbody component, when present, tracks the StefanBoltzmann law fairly well (see fig.3) indicating that inner disk geometry (i.e. truncation, warping) changes fairly little, and that the corona optical depth, τ c , is small, simplifying estimates of the inner corona accretion rate, ˙ M c . Unless specified otherwise, we will use symbols ˙ M d ( ˙ M c ) as shorthand for disk (corona) accretion rates at the inner disk radius, R id , and reserve the italic-face for general, local accretion rates ˙ M d = ˙ M d ( R,t ), ˙ M c = ˙ M c ( R,t ). Also, unless otherwise indicated, we will use the following system parameters: black hole mass, M bh =10 M glyph[circledot] , companion mass M ∗ =5 M glyph[circledot] , orbital period P sys =1 . 705d, inclination i =67 o and system distance, d sys =48 kpc (van der Klis et al. 1985; Val-Baker et al. 2007), which also imply a circularization radius of R circ = 2 . 7 × 10 11 cm. We first fit individual RXTE spectra with a disk blackbody and a power law of fixed photon index, Γ pli = 2 . 34 (as in SDS07) with total absorption of fixed column density n H = 3 . 8 × 10 20 cm -2 (Page et al. 2003), using the wabs*simpl*diskbb models in XSPEC (Arnaud 1996). To systematically identify transitions to the low-hard state we looked for cases where the first fitting gave reduced χ 2 > 1 . 1, and refit these with a wabs*(plaw) model where the power-law index is not frozen. Reassuringly, spectra identified this way were fit better with fewer parameters, and are also typically preceded by obvious declines in the blackbody component (fig.4). For low τ c one can describe the flows qualitatively by taking ˙ M d ( t ) ∼ T 4 bb and ˙ M c ( t ) proportional to the ratio of power-law to blackbody count fluxes (as in SDS07, though there the disk central temperature was confused with the effective temperature giving ˙ M d ( t ) ∼ T 20 / 6 bb ). We obtain absolute normalization for ˙ M d by fixing R id and comparing observed and predicted fluxes in the high state where agreement should be best, while for ˙ M c , we obtain an estimate based on the simple τ c and a typical ADAF solution, and check this against a more detailed calculation. We relegate the details to the Appendix to focus on a general description of accretion behavior (fig.5). From fig.5, one can see that the ˙ M c 'turns on' in pulses (referring to the secular month-long features and not the jagged week-long sub-pulses) roughly a viscous timescale apart and slightly shorter in duration, and that these pulses tend to anticipate drops in ˙ M d . This trend was already noted in (Smith et al. 2007) based on inferred qualitative accretion rates, and led the authors to posit a 'bifurcation-only' model where a fairly-constant total supply rate ( ˙ M s ) is split far from the black hole between non-interacting quickly-drainingcorona and slowly-draining-disk components. Our normalization estimates for ˙ M c ( t ) suggest that for any given episode there is generally insufficient total mass in a ˙ M c ( t ) pulse to explain the associated ˙ M d drop. Even if our overall normalization is off, we still found that scaling ˙ M c to conserve mass for one episode does not work very well for other episodes. This mass-conservation problem motivated considering mechanisms that can adjust the total supply rate, remove mass, and/or exchange it between disk and corona flows. The secular evolution of the episodes on super-viscous timescales also lends itself to interpretation as multiple mechanisms acting on similar timescales effectively generating 'beat-frequency' behaviors. The simple alternative of some mechanism(s) acting on super-viscous timescales coupled to viscous-timescale variability mechanisms lacks good candidates for the former. Nuclear evolution is too slow, we do not expect significant magnetic cycles from a companion with a radiative outer envelope (but keep the possibility in mind regarding other systems), and the inferred mass ratio in LMC X-3 is too high for slowly-growing tidal resonances to be significant (Frank et al. 2002). Furthermore, based on the observed inclination, the warps would have to reach heights of 30 o relative to the orbital plane, and survive the severe drops in ˙ M d , to exhibit precession effects if irradiation-driven, which poses difficulties if LMC X-3 is only marginally unstable to irradiation-driven warping as Ogilvie & Dubus (2001) suggest. The disk component in LMC X-3 tends to fall and recover more rapidly for larger drops than for shallow drops, a trend quantified in SDS07 and recently over an expanded data set in Smale & Boyd (2012). This aspect is qualitatively consistent with a bifurcation-only model given sufficient variation in the amplitude and duration of a drop at the outer edge - sensitivity to duration for a single input amplitude can be seen in figures 7&8 of ? . However, using their analytical machinery, with and without crude representation of SRM and outflow effects, we will later show that rough quantitative agreement with observations of LMC X-3 requires inputs that are extremely unlikely without additional physics ( § 4.1). An interesting exception to the usual of ˙ M c pulses heralding steep ˙ M d drops is the small drop in ˙ M d at 1100d into fig.5 not associated with any ˙ M c pulse above the typical noise level. Inferring accretion rates in the absence of the disk component introduces additional parameters and uncertainties, but we wish to make a few relevant observations while we work on a more definitive analysis of the hard state. The disk component drops and recovers on timescales of days in transitions into and out of the hard state, and tends to return more quickly than it decays when LMC X-3 is at its 'hardest' in our data, circa the 1500d mark in bottom of fig.5, consistent with the notion of an extant inner disk preserving itself through condensation. Also, the power-law component tends to increase before failed and successful disk restarts, which may physically correspond to the inner edge of a truncated disk moving inward to provide more and hotter seed photons, and/or rapid condensation.", "pages": [ 3, 4, 5 ] }, { "title": "3. VARIABILITY MECHANISMS CONSIDERED", "content": "Though the basic physics of companion irradiation and streaming are simple, the dynamics are potentially complicated to initialize and implement in detail, especially if the outer disk warps. However, we can estimate bounds on both mechanisms individually, and because they sit at the edge of the accretion flows, we can lump them into a manual boundary condition for now and still derive meaningful results. Compton-heated winds can be launched a bit further inward, but can be described fairly well by simple analytical functions of radius and X-ray luminosity assuming that the corona is easily replenished, and thus we can quickly obtain upper bounds on CHW losses. Evaporation-condensation can depend sensitively on disk and corona conditions at all radii making it the least amenable to simple estimates, and as noted earlier this same strong dependence on the system's state can naturally engender hysteresis. EC also allows the disk component to vary more substantially and more rapidly by evaporating disk material interior to the circularization radius, but this evaporated disk material can also condense further inward much faster than inner disk conditions change, potentially to the point that ˙ M d rises and falls simultaneously with ˙ M c . This 'sympathetic' accretion mode can be seen in the more detailed simulations of Mayer and Pringle (their fig.8) and in many cases we simulated (e.g. figures 8,11,12), but is effectively absent (or negligible) in our observations of LMC X-3, and thus primarily poses a challenge to our basic EC model.", "pages": [ 5 ] }, { "title": "3.1. SRM Estimates and Remarks", "content": "The total supply rate of mass through the L1 nozzle ˙ M s , will scale with the product of local gas density ρ L1 , speed at which gas streams through the nozzle (roughly the local sound speed c s ), and area of the nozzle A n where the latter has width and height roughly equal to the isothermal scale height in the local tidal field, H 2 L1 ≈ c 2 s / Ω 2 orb (Lubow & Shu 1975). Under X-ray irradiation, each layer of the atmosphere will tend to heat up until it emits the intrinsic stellar flux plus the incident X-ray flux at that altitude. Due to the very steep transition in density at the photosphere, we find most of the X-ray energy is deposited in a thin layer there, which we will take to be infinitesimally thin for now. Thus, the modulation with respect to a given reference state as a function of stellar temperature T glyph[star] , effective incident X-ray luminosity L x, eff , gravity-darkened stellar luminosity L glyph[star], eff , and distance between L1 and the photosphere d L 1 -d ph is given by (e.g. Meyer & Meyer-Hofmeister 1983): where Short of solving the structure of the stellar envelope in the Roche-lobe potential under time-varying irradiation, we can estimate the extent of SRM by computing the ratio of effective incident-to-intrinsic luminosity. One can make a simple estimate by computing the effective gravity at a point sitting about halfway between the nozzle and the pole of the companion giving L ∗ , eff /L ∗ ≈ 0 . 68, and also use the inclination of this point relative to the inner disk to get the fraction of X-rays emitted into this latitude, cos β x = 0 . 28, yielding The companion's effective stellar luminosity falls within ∼ 500 -1000 L glyph[circledot] based on the reported bolometric stellar luminosity 800 -1600 L glyph[circledot] (Soria et al. 2001). More carefully integrating the incident-to-intrinsic ratio over the irradiated face (again, with gravity darkening) agrees closely with this simple estimate as the projected area and fraction of disk flux fall concurrently with (and faster than) the effective gravity toward L1. For irradiation operating alone, choosing the maximum observed luminosity as the reference point in eqn.1 would permit drops to ≈ 50% of the observed maximum and only if the X-ray source were turned off completely, but this estimate is still fairly sensitive to companion temperature. Harder and more isotropic X-ray flux from a hot corona may enhance modulation, but for LMC X-3 the maximum observed power-law flux is barely a fifth that of the disk, roughly equal to the projection factor reducing inner-disk flux onto the companion. However, even if much deeper drops are possible, and irradiation-driven warping or some other mechanism were included to prevent the system from settling into a permanent steady high-soft state, the fact that SRM affects the flow at the very boundary means that any changes it introduces will suffer severe viscous dampening ( § 4.1). Altogether, this suggests that SRM is significant, but certainly cannot explain the steep ˙ M d declines by itself. Furthermore, we consider this simple model's predictions of the SRM magnitude an upper bound in light of as detailed two-dimensional hydrodynamic simulations of the envelope by Viallet & Hameury (2007). They find that irradiation will still drive gas toward the nozzle, but the gas will also have ample time to cool down as it crosses the disk's shadow. They note that because they do not solve for perpendicular velocity it may exceed their estimates near the nozzle, and we remark that warping of the outer disk might reveal more of the companion's equator and nozzle and negate the effects of cooling. For our disk/corona simulations, we ignored the delay between irradiation and changes in ˙ M s since we estimated the sound-crossing time of the envelope near L1 to be ≈ 16 hr, far less than the viscous timescale. However, in the case Viallet & Hameury (2007) studied they found that some of the gas may take longer, up to several system orbital periods, to reach the nozzle.", "pages": [ 5, 6 ] }, { "title": "3.2. Bifurcation (DCB) and warping estimates", "content": "Matter streaming from the L1 point typically collides with the edge of the disk, which usually sits outside the circularization radius due to viscous spreading. Because the disk is relatively cold at this radius, the collision is highly ballistic (Armitage & Livio 1998). The fraction of matter streaming around the disk instead of immediately joining it can then be estimated simply by finding the altitudes at which the vertical disk and stream (both roughly Gaussian) density profiles match, and supposing (Hessman 1999) that all the stream within this range immediately joins the disk while matter outside may stream further in. This yields a streaming fraction, where ρ d 0 and ρ s 0 are disk and stream densities at z = 0 and the stream scale height H s will not differ much from H L1 -we also refer to Hessman (1999) for fits to the results of Lubow & Shu (1975). Irradiation-driven warping of the outer disk may also affect the streaming fraction. Again, Ogilvie & Dubus (2001) and Foulkes et al. (2010) suggest warping is possible in LMC X-3, and the latter work specifically finds a disk tilt of 10 o likely for LMC X-3. However, both use an isotropic central luminosity, and the latter use an Eddington ratio in luminosity for LMC X-3 comparable to our derived maximum Eddington ratio in ˙ M d , so we consider their results an upper bound on warping. We generalize the f s ( t ) estimate to a stream that scans the edge of a disk tilted by an angle ϑ d ( t ) above the orbital plane. Here, the vertical density centroid follows z 0 = R d sin( ϑ d ( t ) cos(Ω syn t )) where R d is the radius of the disk edge, and Ω syn = Ω K ( R d ) -Ω sys is the beat frequency between the Keplerian frequency at the disk edge and the system orbital frequency. The finite travel time of the stream should add a roughly constant delay of order the local free-fall time, and for now we ignore this effect. Assuming dϑ d /dt glyph[lessmuch] Ω syn , the altitudes of equal density are Wewill see that EC can depend very non-linearly on f s ( t ) at the boundary, but for now we use the orbit-averaged f s ( t ) as a gauge of plausible DCB strength: We plot 〈 f s 〉 at the outer boundary for relevant ranges of total supply rate, ˙ M tot , and R d , and for ϑ d of 0 o and 10 o in fig.6. For an untilted disk, the contours are explained by the drop in disk scale height with radius and much slower drop with accretion rate, while for a tilted disk, the scanning greatly washes out the R d dependence leaving accretion rate as the dominant factor. Our simple estimate also does not resolve the fate of the surviving stream beyond the edge (Foulkes et al. 2010 do, but unfortunately not for LMC X-3 in particular), but should bound the fraction of mass diverted. In Begelman et al. (1983), the authors considered an optically thin corona subject to Compton heating/cooling (ignoring bremsstrahlung and other heating/cooling mechanisms) and pointed out that accretion X-rays can heat the corona at all radii up to a temperature, T iC , at which inverse-Compton heating and cooling equilibrate. Whether a wind is launched at a given radius then depends mostly on whether this T iC is greater or smaller than the local virial temperature, T vir , and the authors define a radius R iC by where the temperatures are equal, as well as a critical luminosity, L cr ≈ L Edd / 33 at which the gas can be Compton-heated to the virial temperature within the sound-crossing time of the local corona's scale height. Because the tidal gravitational field falls off faster than the source luminosity, gas flows out most easily at large radii. Though primarily a function of source X-ray luminosity and radius, the shape of the source spectrum can affect the mass-loss rate slightly but we ignore this effect. Begelman et al. (1983) computed mass-loss rates for total ˙ M w glyph[lessorsimilar] ˙ M tot , while later work addresses dynamics and wind limit cycles (Shields et al. 1986). Woods et al. (1996) performed simulations to test the previous analytical prescription and amend it slightlymostly by noting a shift in the location of R iC and providing corrections for low luminosities that do not immediately concern us. We take their fitting formula for wind losses per unit area where the normalization ˙ m ch is the ratio of corona pressure to sound speed at R iC , ξ = R/R iC ≈ 2 R/R circ , and their η = L/L cr . We then also introduce a factor f xh in η ≡ f xh L/L cr for how well X-ray luminosity from an inner disk Compton heats the outer corona compared to the point source considered in the references. Although the outflow geometry may permit parts of the outflow to eventually reach low inclinations relative to the inner disk, the chief hurdle is heating the gas when it is sitting deepest in the tidal gravity field. Integrating cos i over the solid angle subtended by the outer corona versus half the disk's sky gives f xh ≈ 0 . 025. This factor suppresses CHWconsiderably, while f xh ≈ 1 implies CHW will have significant effects at the maximum observed luminosities (fig.7). Furthermore, the f xh for depleting disk flow is likely different and smaller than the corona as the X-rays will have to reach higher inclination, and heat conduction from a transition layer will be competing with advection by the wind. Winds may also be driven by other means, i.e., magneto-centrifugal and line driving, but extensive simulations by Proga (2003) with parameters relevant to LMC X-3 indicate that these losses in LMC X-3 will be at most a few percent of the total accretion rate. To gauge CHW self-screening, or screening the companion, consider a wind carrying away 10 19 g s -1 (total, half this per disk face) at the local sound speed at R iC . If the density did not fall off with radius, the Thomson optical depth would be: where a is orbital separation. That this extremely generous upper bound gives marginal absorption indicates the Compton wind will not screen the companion. Instead, CHWand SRM will likely dampen each other's contribution to ˙ M tot -variability seen at inner radii as additional X-ray luminosity simultaneously increases ˙ M s supplied by the companion and ˙ M w lost to space. However, their interaction could enhance the scaling of ˙ M d / ˙ M c with L x at large radii.", "pages": [ 6, 7 ] }, { "title": "3.4. PHII limits and discussion", "content": "The PHII is fundamental to the picture of transient BHXRBs and thus to future extension of our work, but the physics itself is not trivial to implement let alone fully understood as the (60 page) review by Lasota (2001) attests. However, for LMC X-3, the strong, persistent disk emission should generally stabilize the disk within at least 1 R circ , and we will later show ( § 4.1) that even drastic disk variability outside R circ / 25 is still too viscously dampened to explain observations, though the PHII may still contribute to the magnitude of disk variability, and likely plays an important role during complete state transitions. Taking either the mean or median of ˙ M d ( t ) over our data set as a suitable proxy for supply rate gives 〈 ˙ M s 〉 ≈ 0 . 1 ˙ M Edd , and while the disk beyond ∼ R circ / 3 will be cool enough to experience the PHII absent irradiation at 0 . 1 ˙ M Edd (e.g. fig.1 of Janiuk & Czerny 2011), irradiation can stabilize more and possibly all of the outer disk (Coriat et al. 2012). Work by Dubus et al. (1999) indicates that LMC X-3's disk would become susceptible to instability just beyond R circ at 10 18 g s -1 for typical values of α (0.1) and an overall accretion to irradiation efficiency factor C , originally fit to light curves of the BHXRB A0620-00 and roughly consistent with simple calculations based on an annulus-to-annulus irradiation geometry (see discussion in Kim et al. 1999 and comparison at the end of Dubus et al. 1999 to King et al. 1997). Because we do not see ˙ M d ( t ) decay on R circ -viscous timescales in LMC X-3, it appears that the PHII would also lack a large span of starved inner disk for a heating front to propagate through. After the long, complete state transition of fig.5 however, the disk recovery is flarelike, consistent with the notion that the PHII can play a significant role in LMC X-3 at low enough disk blackbody flux.", "pages": [ 7, 8 ] }, { "title": "3.5. EC Background and Implementation", "content": "As stated in § 1, EC may occur if the disk and corona are thermally coupled-if the disk cannot efficiently radiate away corona heat conducted onto it, nor sufficiently cool the corona via inverse-Compton cooling, then it will experience net heating and evaporate, but otherwise it cools the corona which then condenses onto it. Thus the mass-exchange, or 'EC' rate ˙ M z , is sensitively dependent on the balance of heating and cooling, and the very different scalings of heating and cooling mechanisms involved make possible a wide variety of behaviors. At present, several EC models incorporate viscous and compressive heating, bremsstrahlung, and inverse-Compton cooling in the accretion flow including LTMHM07 and MP07. Besides separating the thresholds for disk formation/destruction normally degenerate under a bremsstrahlung-only density criterion via inverseCompton cooling, and thus engender hysteresis (MeyerHofmeister et al. 2009), it is also possible to evaporate the outer disk but condense it back onto the inner disk rapidly enough to drive correlated rises (and falls) of ˙ M d with ˙ M c (again, fig.8 of MP07 and prominently in the left panel of our fig.11). It is also possible to preferentially evaporate the middle of a disk to the point of destroying it as visible in Meyer et al. (2007), MP07, and several of our simulations. For our initial EC implementation, we do the following. We assume azimuthal symmetry for the accretion flow and divide it into 45 logarithmically-spaced radial zones with a single virtual corona zone associated with each disk zone (i.e. the code is 1.5D). We evolve the disk by solving mass fluxes with the standard viscousdisk equations (Frank et al. 2002) and a simple donorcell scheme. Meanwhile we assume that the local corona properties and EC rates match those of the steady-state corona and ˙ M z solutions from LTMHM07 for the same local accretion rate and evolve the corona working inward from the outer boundary condition. More specifically, for each disk zone we compute zoneboundary ( j ± 1 / 2) velocities with a viscosity based on a standard thermal equilibrium thin disk solution assuming Kramer's opacity as in Frank et al. (2002) : The overall disk and corona evolution is then governed by: and where ∆ A j is the zone area. The fluxes are also limited so as not to draw mass from a disk zone or the corona flow than is physically available, and the 'Rx' emphasizes that we are plugging in the EC rates of LTMHM07 as a function of radius, and local ˙ M c and effective disk temperature. If ˙ M c and ˙ M z are anywhere comparable (of the same order of magnitude) to the viscous disk fluxes, then the computations are performed only once. Otherwise, the latter two steps are relaxed further, and allowed to change Σ d but not ˙ M d , until their iterations converge within a given tolerance (or exceed an iteration limit). We note that large | ˙ M z | can lead to oscillatory behavior with this simple scheme, which can be subdued but not fundamentally fixed with smaller time steps and tolerances. This can be seen in the results of our simulations (fig. 11-13) as small sudden jumps in ˙ M c (and somewhat in ˙ M d when EC is strong at small radii), but by changing time step and tolerance we have found that this does not impact the general, longer timescale features that immediately concern us. We also explored small changes in the number of radial zones, obtaining similar results with 40 or more grid points, but our results diverged very quickly for coarser grids. For steady input conditions, we confirm that our method does well reproducing cases considered by LTMHM07 (treating the disk fully lets it spread viscously to larger sizes mildly enhancing condensation). To test our method's treatment of dynamic behavior, we looked at the same case MP07 studied: a disk spanning 3000 Schwarzschild radii around a 10 M glyph[circledot] black hole with fixed boundary corona fraction f s of 0 . 1 and mass supply rate of 10 -3 times Eddington ˙ M Edd . Their time-dependent method evolves the corona self-consistently on its dynamical timescale making it more physically realistic, but this also requires many more time steps. We found that with default parameters and physics our code never evaporates any part of the disk, while MP07 predict the formation of a gap that eats its way inward. However, we also found that if we scaled up the heat-conduction fluxes predicted by LTMHM07 for zones where inverse Compton cooling does and does not set the electron temperature by a factor of three and five respectively (adjusting the formula identifying the zones accordingly) then we do obtain good agreement with MP07 (see fig.8, and their fig.8). By preferentially scaling up heat fluxes in the zones where inverse-Compton cooling limits electron temperature we obtained more rapid evaporation starting further inward while doing the same for heat fluxes in non-Compton zones led to increased stability. In this particular case, we saw little change when lowering the magnetic-to-gas pressure ratio, β c , a global constant in LTMHM07, from 0.8 to 0.1. Both the models of LTMHM07 and MP07 necessarily neglect, or precede, some additional physical effects which may be relevant to our early results so we discuss them here (and summarize in fig.9) to motivate our more exploratory simulations ( § 4.2 and fig.13). In both models, condensation is a smooth, unresolved flow, but applying the results of Wang et al. (2012) shows that the corona is liable to clump at radii greater than roughly 100 R id under typical conditions for LMC X-3. Such clumping potentially increases cooling (thus condensation) efficiency. Both LTMHM07 and MP07 use Spitzer electron conduction throughout the problem domain, and both suspect that the effective thermal conduction coefficient κ may be significantly smaller. Although the degree of tangling in the magnetic fields of the transition zone is far harder to constrain, it is amenable to parameterization. Meanwhile, Cao (2011) provides a recent calculation for how much the ordered component of field shifts from predominantly poloidal outside ∼ 10 R id to predominantly toroidal inside. Lastly, neither model includes mechanisms to spontaneously produce corona, a point MP07 especially emphasize. Indeed, since Galeev et al. (1979) derived that within a certain radius, the buoyancy of magnetic loops formed within the disk can outpace their reconnection leading to a carpet of buoyant loops, this solar-like corona has often been invoked as a partial or complete source of corona. For LMC X-3, the condition on radius in Galeev et al. (1979) gives R glyph[lessorsimilar] 300( ˙ M Edd / ˙ M d ) R S . It is hard to imagine this mechanism alone generating ˙ M c pulses that anticipate ˙ M d drops lasting substantially longer than the viscous timescale at ∼ 100 R S , but it may play an important role by reheating and replenishing the corona, and affecting the local magnetic field geometry. An important caveat in applying the LTMHM07 model came to our attention after running our simulations. For ˙ M c glyph[greaterorsimilar] 0.1 Eddington near R ∼ 100 R S , the conduction or Compton-cooling (at high ˙ M d ) limiting temperatures may cross the coupling temperature (e.g. Bradley & Frank 2009). Under these conditions, the model will tend to overpredict condensation and thus exaggerate the amplitude and duration of the sympathetic mode, but triggering the effect requires strong, correlated ˙ M c , ˙ M d to already be underway.", "pages": [ 8, 9 ] }, { "title": "4.1. Models without EC", "content": "To illustrate how a combination of non-EC mechanisms has difficulty explaining the magnitude of ˙ M d drops and especially the pattern of rapid decline versus slow recovery, we employ a very simple model where disk accretion is computed using the analytical machinery of ? . The latter is derived assuming a time-independent viscosity with power-law dependence on radius, and we referred to the thin-disk solutions on pg.93 of Frank et al. (2002) for all relevant viscosity parameters. This method prevents incorporating radial dependence of ˙ M w , but because wind losses fall rapidly with decreasing radius, and since we predict they are fairly weak anyway, assuming that they take place near the boundary does not invalidate the main results of this toy model. This method also precludes incorporating he PHII, but as discussed in § 3.4 the main effects of the PHII should typically be limited to radii beyond ∼ R circ in LMC X-3. The left panel in fig.10 shows a calculation with this reduced model geared toward reproducing the first disk drop in fig.5. We set f s manually, and SRM and wind losses were also computed beforehand as functions of the observed ˙ M d . To reproduce the depth of the first drop, we first allowed sustained f s of 100% and when this proved insufficient we moved the outer disk radius inward to a mere 0.04 R circ for the runs shown, still employing f s of 100%, and leading to massive ˙ M c pulses compared to our estimates. This can be corrected somewhat by invoking a wind stronger than our estimates. Besides requiring unrealistic values with respect to our estimates, and an extreme ad-hoc disk truncation, the toy model resists efforts to simultaneously improve agreement with other major features of the data. To improve model-data agreement for the second disk drop in the left panel of fig.10 without increasing disk radius (which would obviously undo agreement with the first ˙ M d drop) requires either increased SRM or increasing the height and duration of the second coronal pulse. The latter will generate obvious disagreement with the second ˙ M c pulse by attaching a tail that is very clearly not observed; the former significantly reduces the amplitude of the first coronal pulse relative to the second so that one must invoke a stronger and more complicated wind mechanism. Again, the chief problem is that a disk flow will be viscously smeared too much to match observations unless variability is driven at a relatively small radius where even maximally efficient CHW should be insubstantial, and the PHII should not operate nor regularly drive heating fronts. In the next section, we will show how EC may introduce a evaporation-to-condensation transition or gap at radii comparable to the outer edge of the arbitrarily truncated disk of the toy model, but that it also tends to overpredict condensation at inner radii, gener- ating correlated ˙ M d -˙ M c rises and falls inconsistent with observations.", "pages": [ 9, 10 ] }, { "title": "4.2. Models including EC", "content": "Except where specifically noted otherwise, for these simulations we again use α d = 0 . 1, but a corona α c = 0 . 2, the standard Spitzer coefficient for electron thermal conduction, a β c = 0 . 8 for the LTMHM07 EC prescription, the observationally favored R circ = 2 . 7 × 10 11 cm, assume R id is 3 R S , and set f xh = 0 . 03. We note here that the SRM results from § 3.1 specifically correspond to a value of L ref x, eff /L ref glyph[star], eff = 0 . 4 for ˙ M d = 10 18 g s -1 , and to 6 . 5 scaleheights between the nozzle and the stellar surface for T ref glyph[star] = 16500K. For the EC-inclusive model, we first simulated a disk with standard density profile extending to the circularization radius reacting to a mild Gaussian f s ( t ) pulse, and show the results in the left panel of fig.11. Two immediately remarkable features include the sympathetic rise and fall of ˙ M d with ˙ M c , and the general saturation of ˙ M c response when the outer disk is most intensely siphoning onto the inner disk. This saturation physically arises from the steep transition between evaporation and condensation. In the mass exchange model of LTMHM07 that we employ, the local evaporation/condensation rate ˙ M z , varies with the local corona accretion rate in terms of Eddington ratio, ˙ m c , like where a and b are functions of many other parameters, local variables, and radius itself. If these other variables vary weakly with radius, then a very dense corona at some radius will lead to efficient condensation slightly further inward, and subsequent changes in ˙ M z about zero will be driven by the weaker variations in the critical value of ˙ m c . The issue of sympathetic accretion prompted us to consider a scenario in which the outer disk mass most vulnerable to being siphoned has already been evaporated away, such that there is a gap in the outer disk. We first studied the effects of simply truncating the disk, and show an example with radius 3 × 10 10 cm and default LTMHM07 EC parameters in the right panel of fig.11. Truncating the disk to this radius prevents triggering sympathetic accretion while generating appreciable ˙ M d variability and diminishing, but not eliminating, EC's role in amplifying variability over the simulation domain. We next attempted to generate this gap selfconsistently, while preserving the interior aspect of the accretion flow that works fairly well. To this end, we first subjected a full disk to a constant f s at the boundary. The run confirmed that a small seed corona can rapidly evaporate the outer disk, that this corona is immediately condensed only slight inward, but also that the process of forming a complete gap would take on the order of years in our standard EC implementation. To pursue this idea further, we ran simulations with a gap already inserted of which fig.12 is representative. Besides the initial relaxation of ˙ M d due to viscous spreading of the inner disk exceeding and resupply via condensation, one can see that sympathetic accretion is still an issue, and ˙ M c variability inward of the gap is severely suppressed again, as the high corona fraction of the gap triggers the saturation effect described above. Since our standard model and implementation of EC faces fundamental problems in reproducing major features of the data, we studied the effects of introducing physically motivated, if not yet rigorously justified modifications. Thus far, it appears that the most successful modifications follow a fairly strict pattern of effectively raising the coronal heating and critical evaporation-tocondensation ˙ m c over the innermost two decades in R/R id . The latter is nearly inversely proportional to b in eqn.13 which in the model of LTMHM07 scales with the corona viscosity parameter α c , radius, electron thermal conduction coefficient κ , and gas-to-total pressure ratio β c as though we should point out that β c enters their model strictly via a prescription for compressive heating. In the simulations producing fig.13, we raised α c linearly from 0.2 to 0.4 inwards over 100 R id , and independently adjusted b over radial zones spanning 10 0 -10 2 . 7 , 10 2 . 7 -10 3 . 4 , and 10 3 . 4 -10 4 . 6 in R/R id . For run A in fig.13, we scaled b in these zones by 0.1, 0.2, and 0.4 respectively, and for run B by 0.4, 0.6, and 0.8. Although the simulated ˙ M c pulses are far more massive than observations indicate, these modifications very clearly control the degree of hysteresis versus sympathetic accretion. If the overall magnitude of EC is smaller, then for higher f s , and/or SRM moderately larger than expected, this modified model could reproduce the observed hysteresis. The most conceivably adjustable parameters in eqn.14 are α c , especially if understood to include other heating mechanisms, and κ . Since the current contrast between simulation and observation still favors weaker EC, the requirement on additional heating would not likely be as extreme as implied by our modifications, but this then requires even greater deviation in κ which affects b rather weakly, and these parameters are not necessarily independent in reality.", "pages": [ 10, 11 ] }, { "title": "5. DISCUSSION AND CONCLUSIONS", "content": "Thus far, we still cannot offer a very definitive solution for LMC X-3's behavior, but we have more rigorously examined several physical mechanisms popularly invoked to explain variability in LMC X-3 and additionally considered evaporation and condensation. We have found that if condensation is suppressed at inner radii (or over-predicted in the current model), then EC may naturally reconcile the observational evidence for RLO accretion and associated circularization radius, as well as the large amplitude, long duration, and rapid declines in hysteresis episodes with our estimates of other major viscous timescale variability mechanisms and the viscous dampening that they would undergo. However, we wish to emphasize that our current EC implementation and the steady-state theory informing it by default led to excessive condensation when compared as accurately as possible to observations, and it led to significantly different predictions for the particular low-˙ M s case studied by MP07. As discussed in § 3.5, both the LTMHM07 and MP07 models necessarily neglected some physics, some of which might enhance heating and/or suppress conduction closer to the black hole, and thus help explain the discrepancy between observations and the predictions of our code with the default EC prescription. To this end, we have reproduced and are testing a code that follows MP07 and evolves the disk and corona mass and energy equations self-consistently. An additional advantage is that we can naturally include physics behind the PHII by incorporating detailed results for disk cooling as a function of density and central temperature from previous work - crucial to studying transient systems. However, this explicit method suffers from advancing by a very small time step. We have started building a parallelized implicit method which we hope to develop further during simulations with the explicit method, but we also hope to find ways to save on excessive computation through better physical understanding of the problem. Our estimates for the reduced efficiency of CHW, and cursory examination of theory results for MHD winds in similar systems. Proga (2003) suggest that winds in general will have little affect on accretion dynamics in LMC X-3. However, we will continue to consider how efficiency of CHW might be increased, or that MHD winds may be stronger than anticipated (e.g. King et al. 2012). Other assumptions in our current modeling that bear repeated mention include the alpha-prescription disk, and how we infer and interpret the disk and coronal accretion rates. Regarding the former, it is at least expected that under conditions associated with jet flow, angular momentum transport via the magnetically driven outflow may become substantial or dominant compared to viscous transport (i.e. the magneto-rotational instability) at least within the inner flow (e.g. Zanni et al. 2007; Casse & Ferreira 2000). Evolution of the magnetic fields in the disk may also lead to intrinsic variability out to ∼ 100 R id , as in de Guiran & Ferreira (2011). The potential for corona flow to stall centrifugally as a function of external flow conditions and local viscosity (e.g., Chakrabarti & Titarchuk 1995; Garain et al. 2012) might be realized in LMC X-3, but is also usually expected to occur well within the innermost 100 R id of the flow and to destroy the disk interior to the centrifugal shock, so it may be most relevant to state transitions. If more frequently prevalent though, the latter could alter our picture of spectral production, enhance ˙ M c variability especially on shorter timescales, and change the dynamics of EC. Based on the pattern of the power-law component to anticipate declines in blackbody flux and the viscous recovery timescale of the blackbody component, we are still naturally inclined to favor a picture where variability is driven outside-in so that mechanisms like these and EC would predominantly accelerate and enhance ˙ M d declines driven by known mechanisms operat- /OverDot /OverDot ing in the outer flow. Our immediate focus will be resolving the outstanding questions regarding the current mechanisms considered, especially evaporation and condensation. After un- derstanding and constraining these better, we hope to extend our investigations to additional physics, systems, and phenomena, especially the transients and jet launching.", "pages": [ 11, 12 ] }, { "title": "6. ACKNOWLEDGEMENTS", "content": "The authors acknowledge support through the NASA ADP program grant NNX09AC86G. The authors would also like to thank the referee for suggestions that improved the content and clarity of the paper.", "pages": [ 13 ] }, { "title": "RELATING ACCRETION RATES TO SPECTRA PRODUCTION", "content": "For the blackbody spectrum, Zimmerman et al. (2005) note that XSPEC fits the maximum temperature and normalization constant to a temperature profile of the form T ( R ) = T max ( R id /R ) 3 / 4 . The fit to the peak temperature using this profile is only ∼ 5% smaller than the peak temperature found fitting a temperature profile based on a zero torque boundary condition and color-correction factor f col (Ebisuzaki et al. 1984): where Because there is little difference in the fitted peak temperatures, because we plan to fix the black hole mass and R id , and because we prefer the physically-motivated temperature profile we will use it instead. Integrating over disk annuli then gives the familiar formula for flux: Because Shimura & Takahara (1995) predict that f col depends weakly on radius, accretion rate, and other parameters, we fix f col = 1 . 7. This then implies that the maximum disk accretion rate ranges from 0.07-0.29 ˙ M Edd for R id spanning 1 to 3 Schwarzschild radii. We scale the ˙ M d of fig.5 by matching the observation with the highest blackbody flux and temperature to the maximum 0.29 ˙ M Edd . We note that if dissipation interior to the last stable circular orbit is significant, this will also put the actual ˙ M d below our estimate (Beckwith et al. 2008; Shafee et al. 2008). Inferring ˙ M c ( t ) requires additional assumptions but many are well constrained within ADAF theory (Narayan & Yi 1995b). Specifically, theory predicts that corona ions are very effectively virialized at inner radii and much hotter than the electrons, whose exact temperature depends on many conditions, but is generally flat over the innermost 100 R id and of order 100keV in the cases considered by Narayan & Yi (1995b). The former means that we can confidently predict scale height given corona α c while the latter provides some justification for choosing a constant electron temperature in corona emission calculations. Taking R id = 3 R S , corona alpha parameter α c = 0 . 2, and gas-to-total pressure ratio β c = 0 . 8, we obtain (Narayan & Yi 1995b) the following estimates for corona density n c , corona height H c , and coronal optical depth τ in terms of ˙ m c = ˙ M c / ˙ M Edd , However, we remind the reader that the latter is fairly sensitive to α c , scaling roughly like α -1 c . If we take the scattering fraction to be P τ = 1 -e -τ then based on the scattering fractions returned by XSPEC for the mixed states, inversion gives us the simple ˙ M c estimate shown as the solid line in fig.5. We compare this simple estimate with a more detailed power-law flux calculation. Hua & Titarchuk (1995) find a Green's function for the output energy spectrum given the seed photon energy spectrum per scattered seed photon (specifically, their eqn.9), G ν ( x, x s , T e , P τ ). The latter depends again on scattering fraction, as well as corona electron temperature T e , and the output ( x ) and seed ( x s ) dimensionless photon energies ( x ∗ = hν ∗ /k B T e ). Using the result of a more detailed calculation for the scattering fraction in a slab geometry from Zdziarski et al. (1994), we convolve G ν ( x, x s , T e , P τ ) with the flux of seed photons that can scatter (overall P τ factor) out of the modified blackbody spectrum. In terms of r = R/R id , and T ∗ the power-law energy spectrum, F E is given by: . Note that we have assumed the corona emission is largely isotropic, but we have included the projection factor for blackbody emission from each annulus to half the height of the corona at r = 1, and we confirmed that r = 40 is a numerically acceptable cutoff. Fixing Γ pli = 2 . 34 and T e = 150keV ( T e dependence is relatively weak for T e glyph[greatermuch] max( T ∗ , 25 keV /k B )) we tabulated integrated flux for a range of T max and ˙ M c to be inverted numerically, ultimately obtaining the ˙ M c points in the upper panel of fig.5. For relatively high T bb and low τ they agree fairly well with the simpler method, with the main difference due to the less step-like ˙ M c -τ relationship in the more detailed P τ . However, to explain observations with higher F pl and lower T bb , the formula quickly requires excessively high ˙ M c , and at these implied optical depths the formula itself becomes unreliable.", "pages": [ 13, 14 ] }, { "title": "REFERENCES", "content": "Armitage, P. J., & Livio, M. 1998, ApJ, 493, 898 Meyer, F., & Meyer-Hofmeister, E. 1983, A&A, 121, 29 Meyer-Hofmeister, E., Liu, B. F., & Meyer, F. 2009, A&A, 508, 329 Montgomery, M., & Martin, E. 2010, ApJ, 722, 989 Nandi, A., Debnath, D., Mandal, S., & Chakrabarti, S. K. 2012, A&A, 542, A56 Narayan, R., & Yi, I. 1995a, ApJ, 444, 231 -. 1995b, ApJ, 452, 710 Ogilvie, G. I., & Dubus, G. 2001, MNRAS, 320, 485 Page, M. J., Soria, R., Wu, K., et al. 2003, MNRAS, 345, 639 Pringle, J. E. 1992, MNRAS, 258, 811 Proga, D. 2003, ApJ, 585, 406 Shafee, R., Narayan, R., & McClintock, J. E. 2008, ApJ, 676, 549 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Shields, G. A., McKee, C. F., Lin, D. N. C., & Begelman, M. C. 1986, ApJ, 306, 90 Shimura, T., & Takahara, F. 1995, ApJ, 445, 780 Smale, A. P., & Boyd, P. T. 2012, ApJ, 756, 146 Smith, D. M., Dawson, D. M., & Swank, J. H. 2007, ApJ, 669, 1138 Smith, D. M., Heindl, W. A., & Swank, J. H. 2002, ApJ, 569, 362 Soria, R., Wu, K., Page, M. J., & Sakelliou, I. 2001, A&A, 365, L273 Val-Baker, A. K. F., Norton, A. J., & Negueruela, I. 2007, in American Institute of Physics Conference Series, Vol. 924, The Multicolored Landscape of Compact Objects and Their Explosive Origins, ed. T. di Salvo, G. L. Israel, L. Piersant, L. Burderi, G. Matt, A. Tornambe, & M. T. Menna, 530-533 van der Klis, M., Clausen, J. V., Jensen, K., Tjemkes, S., & van Paradijs, J. 1985, A&A, 151, 322 Viallet, M., & Hameury, J.-M. 2007, A& A, 475, 597 Wang, J.-M., Cheng, C., & Li, Y.-R. 2012, ApJ, 748, 147 Wilms, J., Nowak, M. A., Pottschmidt, K., et al. 2001, MNRAS, 320, 327 Woods, D. T., Klein, R. I., Castor, J. I., McKee, C. F., & Bell, J. B. 1996, ApJ, 461, 767 Zanni, C., Ferrari, A., Rosner, R., Bodo, G., & Massaglia, S. 2007, A&A, 469, 811 Zdziarski, A. A., Fabian, A. C., Nandra, K., et al. 1994, MNRAS, 269, L55 Zimmerman, E. R., Narayan, R., McClintock, J. E., & Miller, J. M. 2005, ApJ, 618, 832", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "Rapid spectral changes in the hard X-ray on a time scale down to ∼ 0 . 1 s are studied by applying 'shot analysis' technique to the Suzaku observations of the black hole binary Cygnus X-1, performed on 2008 April 18 during the low/hard state. We successfully obtained the shot profiles covering 10200 keV with the Suzaku HXD-PIN and HXD-GSO detector. It is notable that the 100-200 keV shot profile is acquired for the first time owing to the HXD-GSO detector. The intensity changes in a time-symmetric way, though the hardness does in a time-asymmetric way. When the shot-phaseresolved spectra are quantified with the Compton model, the Compton y -parameter and the electron temperature are found to decrease gradually through the rising phase of the shot, while the optical depth appears to increase. All the parameters return to their time-averaged values immediately within 0.1 s past the shot peak. We have not only confirmed this feature previously found in energies below ∼ 60 keV, but also found that the spectral change is more prominent in energies above ∼ 100 keV, implying the existence of some instant mechanism for direct entropy production. We discuss possible interpretations on the rapid spectral changes in the hard X-ray band. Subject headings: accretion, accretion disks - X-rays: binaries - X-rays: individual (Cyg X-1)", "pages": [ 1 ] }, { "title": "RAPID SPECTRAL CHANGES OF CYGNUS X-1 IN THE LOW/HARD STATE WITH SUZAKU", "content": "S. Yamada 1 , H. Negoro 2 , S. Torii 3 , H. Noda 3 , S. Mineshige 5 , and K. Makishima 3,1 Accepted to ApJL: February 28, 2013", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Starting with the first identification of the black hole (BH) binary Cygnus X-1 (hereafter Cyg X-1) in the early 1970's (e.g., Oda et al. 1971; Tananbaum et al. 1972; Thorne and Price 1975), X-ray observations have been playing an important role to reveal spectral and temporal properties of BH binaries, which are largely classified into two distinct states: the high/soft state and the low/hard state (e.g., Remillard and McClintock 2006; Done et al. 2007). In contrast to the high/soft state characterized by the dominant disk emission (Mitsuda et al. 1984; Makishima et al, 1986) from the standard disk (Shakura & Sunyaev 1973), the spectrum in the low/hard state is expressed by a powerlaw with a photon index of ∼ 1.5 with an exponential cutoff at ∼ 100 keV (e.g., Sunyaev & Trumper 1979) from a hot 'corona' (e.g., Ichimaru 1977; Narayan & Yi 1995; chapter 8 in Kato, Fukue, and Mineshige 2008). Rapid time variabilities on a time scale of ∼ ms (e.g., Miyamoto et al. 1991) only seen in the low/hard state have been studied in many ways (e.g., Nowak et al. 1999; Poutanen 2001; Pottschmidt et al. 2003; Uttely et al. 2011; Torii et al. 2011), though the origin is still missing piece of puzzle, presumably due to observational difficulties of realizing both high sensitivity and large effective area. with Ginga . This method is the time-domain stacking analysis to obtain universal properties behind nonperiodic variability. It is in time-domain analysis that we can combine spectral information in a straightforward way. They found three main features: (1) the intensity changes time symmetrically, (2) both of the rise and decay curves are well represented by the superpositions of two exponential functions with time constants of ∼ 0 . 1 s and ∼ 1 . 0 s, and (3) the spectral variation is, by contrast, time asymmetric in the sense that it gradually softens toward the peak and instantly hardens across the peak (see Figure 2 in Negoro et al. 1994). These properties are further investigated with RXTE (Focke et al. 2005). The time constant of ∼ 1 s far exceeds the local (dynamical or thermal) timescale of the innermost region, and should thus reflect accreting motion of gas element. Manmoto et al. (1996) proposed an interesting explanation that inward-forwarding accreting blob, causing an increase in X-ray flux, are reflected as sonic wave when it reaches the BH (Kato, Fukue, and Mineshige 2008), though further observational constraints have been awaited. The extension of this approach towards higher higher energies, ∼ 200 keV, should be crucial because it may provide a hint to the physics causing the rapid spectral variation. Thus, we observed Cyg X-1 in the low/hard state with Suzaku (Mitsuda et al. 2007), by utilizing both the XIS (Koyama et al. 2007) located on the focus of the X-ray mirror (Serlemitsos et al. 2006) and the Hard Xray Detector (HXD: Takahashi et al. 2007; Kokubun et al. 2007; Yamada et al. 2011) (Takahashi et al. 2007; Kokubun et al. 2007; Yamada et al. 2011). The distance, the mass, and the inclination of Cyg X-1 are 1.86 +0 . 12 -0 . 11 kpc (Reid et al. 2011; Xiang et al. 2011), 14 . 8 ± 1 . 0 M /circledot , and 27 . 1 ± 0 . 8 · (Orosz et al. 2011), respectively. It has an O9.7 Iab supergiant, HD 226868 (Gies & Bolton 1986) with an orbital period of 5 . 599829 days (Brocksopp et al. 1999). Unless otherwise stated, errors refer to 90% confidence limits.", "pages": [ 1 ] }, { "title": "2. OBSERVATION AND DATA REDUCTION", "content": "Cyg X-1 data taken with Suzaku on 2008 April 18 (ObsID=403065010) are used in this letter, which is one of the 25 observations in its low/hard state (see Yamada 2011 in details). The XIS0 was operated in the timing mode, or the Parallel-Sum (P-sum) mode, which is one of the clocking modes in the XIS. A timing resolution of the P-sum mode is ∼ 7.8 ms. Data reduction of the timing mode are different from the standard one in the Grade selection criteria 7 : Grade 0 (single event), Grade 1, and 2 (double events) are used. The XIS background is not subtracted because it is less than 0.01 % of the signal events. The HXD data consisting of the PIN (10-60 keV) and GSO (50-300 keV) events are processed in the same manner as Torii et al. (2011). The events are selected by the criteria of elevation angle ≥ 5 · , cutoff rigidity ≥ 6 GV, and 500 s after and 180 s before the South Atlantic Anomaly. The non X-ray background (NXB) of the HXD modeled by Fukazawa et al. (2009) are subtracted from the HXD data. The NXB model reproduces the blacksky data with an accuracy of ∼ 1% when the exposure is longer than ∼ 40 ks. The cosmic X-ray background can be ignored in our analysis, since its contribution is less than ∼ 0.1%. The simultaneous exposure for the XIS0, PIN and GSO is 33.9 ks.", "pages": [ 2 ] }, { "title": "3.1. Definition and preparation", "content": "We formulate the definition of the 'shot'. Here t and C ( t ) is the event arriving time and the count rate at t . T is an interval of time over which C ( t ) is averaged, and C ( t ) T denotes the average count rate over an interval of t -T < t < t + T . t a and t b refer to any time after and before t , satisfying the conditions of t < t a < t + T and t -T < t b < t , and f is the dimensionless parameter of order unity for the threshold. The peak time t p of each 'shot' is a local maximum in C ( t ) defined as, bustly since no iteration is incorporated in this process. The only caveat is that when accidentally more than two adjacent bins of C ( t ) have the same value at the peak, the first of them is selected as the peak; e.g., when C ( t 1 ) < C ( t 2 ) = C ( t 3 ) > C ( t 4 ) is realized, t 2 is to be the peak. It is possible to impose optionally another constraint on the separation of time between the two successive peaks ∆ t ; e.g., ∆ t ≥ gT works to avoid accumulating the small peaks or fake events caused by the Poisson statistics, where g is the dimensionless parameter of order unity. Since we aimed at accumulating as many photons as possible to quantify spectral change, we adopted g = 1, which means that a minimum of ∆ t equals T . An example of the light curve segment of XIS0 is shown in Figure 1(a). The events with 7.8 ms time resolution were binned into 0.1 s bins, resulting in 20-60 counts per bin. Intensity changes by a factor of ∼ 2, so that Poisson fluctuation ( ∼ 15% at 1 σ ) is sufficiently smaller than the intrinsic variation. Employing f = 1 . 0 and T = 1 . 5 s, we actually applied the procedure of Equation (1) to the entire P-sum light curve, and identified 7524 shots in total. The distribution of ∆ t becomes a grossly exponential distribution in agreement with the previous reports (Focke et al. 2005). Figure 1(b) shows the shot profile obtained by stacking the 7524 shots with reference to each peak. We can see two exponential slopes in the shot profile; shorter and longer decay time constants of ∼ 0 . 1 s and 1-2 s. Our primary focus is to extend this analysis to the HXD band, so we do not further investigate the shot profile and the XIS0 spectra due to incomplete calibration of the P-sum mode.", "pages": [ 2 ] }, { "title": "3.2. Shot profiles of the HXD data", "content": "According to the peak time determined with the XIS0 light curve, we have accumulated the PIN and GSO events and their NXB events 8 . To estimate pileup effects in a phenomenological way, we purposely tried to use either 1 ' inside or outside image core of XIS0 to obtain the shot profiles based on Yamada et al. (2012), though the shot profiles were not significantly changed. To investigate the energy dependence of the shots, we have utilized four energy bands: 10-20 keV and 20-60 keV from PIN, and 50-100 keV and 100-200 keV from GSO. After subtracting the NXB events from the data and correcting them for dead time, we have obtained the 10-200 keV shot profiles with the HXD data. The stacked shot profiles are divided by the count rates averaged over -4 to -2 s and 2 to 4 s to approximately correct them for the differences in the efficiency. The normalized shot profiles are shown in Figure 2(a)(d). The derived profiles appear all very similar in shape to the one of XIS0. However, energy dependencies are certainly found when their widths of the peaks are carefully inspected. In Figure 2(a) and 2(d), the peak value in 10-20 keV is ∼ 1.7 while ∼ 1.5 in 100-200 keV, in- ting that the general trend that the higher photon energy is, the lower becomes the peak. This is neither due to incorrect background subtraction nor decrease in the sensitivity of the HXD, because the systematic uncertainty in the NXB subtraction is at most ∼ 3% of the signal intensity even in the 100-200 keV band, and because we are referring to relative changes, instead of absolute values. To clarify the differences among these profiles, we divided the normalized shot profiles in the higher three bands by that in the 10-20 keV. As shown in Figure 2(e)-(g), the hardness ratios (relative to 10-20 keV) gradually decrease towards the peak, but suddenly return to their average values immediately after (within 0.1 s) the peak. Although this feature has been found in energies below ∼ 60 keV in Negoro et al. (1994), we have not only confirmed the same trend up to ∼ 200 keV, but also found that the spectral change is more prominent in higher energy of E /greaterorsimilar 100 keV.", "pages": [ 2, 3 ] }, { "title": "3.3. Quantification of the shot-phase-resolved spectra", "content": "We then quantified its spectral change by accumulating the HXD spectra according to the shot phase. The NXB events were accumulated in the same ways and subtracted. Figure 3 shows three examples of the derived shot-phase-resolved HXD spectra, corresponding to 0.15 s before, right on, and 0.15 s after the peak. The exposure at the peak is 752.4 s (7524 shots × 0.1s). To grasp their characteristics in a model-independent way, we superposed the time-averaged spectrum, and show the ratio of the shot spectra to it in Figure 3. Aa shown evidently in Figure 3(b) by a clear turnover of the ratio above ∼ 100 keV, a spectral cutoff at the peak is lower than the averaged one. Furthermore, the spectral ratio before the peak shown in Figure 3(a) appears downward, while that after the peak is almost flat, which is consistent with the gradual softening before the peak and instant hardening at the peak as seen in Figure 2. To consider physics underlying this spectral evolution we have fitted the 13 shot-phase-resolved HXD spectra with a typical model of Comptonization, compps (Poutanen & Svensson 1996) in the same manner as that in Torii et al. (2011). The seed photon is assumed to be a disk black body emission (Mitsuda et al. 1984; Makishima et al. 1986; Makishima et al. 2008) with a temperature of 0.2 keV. The free parameters in the fits are the electron temperature T e , the optical depth τ or the Compton y parameter, and the normalization N dbb . Note that if τ is fixed, T e is affected more by a spectral slope than a spectral cutoff. To avoid such a misunderstanding, we kept both τ and T e left free. As the shot-phase-resolved spectra do not have sufficient photon statistics, we fixed the reflection fraction Ω at the value of 0.235, because the obtained value from the time-averaged spectrum is 0 . 235 +0 . 021 -0 . 020 . This implies that we assumed that the reflection follows the primary continuum within ∼ 0.1 s. The fits to all the spectra have been successful, resulting in the best-fit parameters in Table 1. Even when considering the systematic error of the NXB in the GSO spectra, its contributions to the resultant values are less than ∼ 1%. As the count rate increases on ∼ s time scale, T e and y decrease while τ increases; when the count rate starts to decrease, all the parameters appear to return to the averaged values. To visualize this, we plot in Figure 4 the de- ed parameters in Table 1, as well as the time-averaged ones. Since our composite shot profile comprises a large number of relatively small individual shots, the averaged parameters are close to those at ∼ 1 s from the peak. The gradual decrease in the y -parameter before the peak is consistent with the hardness decrease as seen in Figure 2. The decrease in T e around the peak clearly reflect the trend that the high-energy cutoff appears lowered at the peak as seen in Figure 3(b). Thus, the fitting results are consistent with the hardness ratios in Figure 2 and the spectral ratios in Figure 3. Note that N dbb also increases along the shot profile, though we could not confidently measure the inner radius without using the soft X-ray data (cf. Makishima et al. 2008).", "pages": [ 3, 4 ] }, { "title": "4. DISCUSSION AND SUMMARY", "content": "We performed the shot analysis to extract important information on understanding rapid hard X-ray variability, which can not be obtained by the Fourier transform (FT) methods (cf. Negoro et a. 2001, Legg et al. 2012, Torii et al. 2011). In general, FT methods are less arbi- trary than the stacking analysis, but phase information is lost in the FT analysis. Further FT methods require more photons than a stacking method. Thus, we chose to used the stacking method and successfully extended the higher energy limit of the shot analysis up to ∼ 200 keV, by utilizing the HXD data as well as the P-sum mode of the XIS. What we found are summarized as follows: (1) the shot feature is found at least up to ∼ 200 keV with high statistical significance, (2) the shot profiles are approximately symmetric, though the hardness changes progressively more asymmetrically toward higher energies of E /greaterorsimilar 100 keV, and (3) the 10-200 keV spectrum at the peak shows lower energy cutoff than the timeaveraged spectrum. By quantifing this feature in terms of the single-zone Comptonization, we found that (4) as a shot develops toward the peak, y and T e decrease, while τ and the flux increases, and immediately past the shot peak, T e and τ (and hence y ) suddenly return to their The fitting results of the shot-phase-resolved spectra. time-averaged values. Let us consider a possible physical mechanism to explain the new findings and the previously-known features as well. The shot profile does not show any plateau at least down to ∼ ms (Focke et al. 2005), which means that most of the luminosity is released almost timesymetrically within ∼ 1 s or much shorter. Meanwhile, the hardness changes instantly within 0.1 s as shown in Figure 2, or much shorter as shown in Negoro et al. (1994). These features could be explained by some physical impulse or a discrete phenomenon, which can change properties of the radiation source in a short time. When accreting matter is assumed to be an ideal and non-relativistic gas, the entropy of the accreting gas, s , with a temperature T and the density ρ is proportional to ln( P/ρ γ ) = ln( T/ρ γ -1 ), where γ is the ratio of specific heat capacities (5/3 for monatomic gas). It can be interpreted that the entropy decreases in some way as the flux increases, but instantly increases at the peak, and returns to the mean value after the peak. This suggests the existence of some instant mechanism for direct entropy production (or heating). One of the possible ideas on the rapid intensity change have been considered as magnetic flares analogous to the solar corona (Galeev et al. 1979), and recently more sophisticated (cf., Poutanen & Fabian 1999; Zycki 2002). The magnetic fields are amplified by the differential rotation of the disk, and rise up into the corona where they reconnect and finally liberate their energy in flare, making electrons accelerate. A typical magnetic model assumes so-called 'avalanche magnetic flare', in which each flare has a certain probability of triggering a neighboring one, producing long avalanches (Mineshige et al. 1994; Lyubarski, et al. 1997). An election temperature is expected to increase if magnetic reconnection accelerate electrons or protons. It contradicts with the gradual decrease in T e before the peak, while agrees with the instant increase at the peak and an expected time scale of ∼ 1-100 ms when the energy dissipation occurs within r /lessorsimilar 50 R g ( R g is the gravitational radius). However, it is still unknown how much magnetic power is stored in the corona. Magnetic reconnection in a plunge region may be likely based on the global three-dimensional MHD simu- tion (Machida and Matsumoto 2003). The mass propagating model (Manmoto et al. 1996, Negoro et al. 1995) gives an alternative explanation, because the viscous time scale of the corona is ∼ 1 s at 100 R g , and the only ∼ 20% initial perturbation of mass accretion rate at ∼ 100 R g can change the luminosity by ∼ 60%. Furthermore, the mass accretion reflected as a sonic ware can create the latter half of the peak; in the theoretical view point, the flow passing a Bonditype (not a disk-type) critical point does not always fall to the BH due to the strong outward centrifugal force unless angular momentum is very small (Kato, Fukue, and Mineshige 2008). The perturbation would start from overlapping region between the disk and corona, presumably ∼ 50-100 R g (Makishima et al. 2008), where intense turbulence is expected due to a large gap of the pressures and temperatures. Since the surface density of the corona is about four orders of the magnitude smaller than that of the disk, a little mass transfer from the disk to the corona in the overlapping region could increase mass accretion into the coronae. Shock phenomena might be possibly working for some reason because there are two or more sonic points around a BH (Nagakura and Yamada 2009). A rapid heating, such as magnetic reconnection, can explain the short ( ∼ 0.1s) timescale of the shots, though the long ( ∼ 1s) timescale would be related to mass accretion time scale. Further observational studies are needed to completely understand the physics causing the rapid variability. For instance, shot profiles with distinct features, such as polarization (Laurent et al. 2011) or a γ -ray emission (Ling et al. 1987), could provide a new hint, which would be precisely measured by new missions like GEMS (Black et al. 2010) and ASTRO-H SGD (Takahashi et al. 2010). We would like to express our thanks to Suzaku team members, and also to Ryoji Matsumoto, Hiroshi Oda, and Hiroki Nagakura for valuable discussions. The research has been financed by the Special Postdoctoral Researchers Program in RIKEN and JSPS KAKENHI Grant Number 24740129. S.T. and H.N are supported by Grant-in-Aid for JSPS Fellows.", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Black, J. K., Deines-Jones, P., Hill, J. E., et al. 2010, Proc. SPIE, Poutanen, J. 2001, X-ray Astronomy: Stellar Endpoints, AGN, and the Diffuse X-ray Background, 599, 310 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Remillard, R. A., & McClintock, J. E. 2006, ARA&A, 44, 49 Serlemitsos, P. J., et al. 2007, PASJ, 59, 9 Torii et al. 2011, PASJ, 63, 771 Thorne, K.S., Price, R.H., 1975, ApJL, 195, L101 Uttley, P., Wilkinson, T., Cassatella, P., Wilms, J., Pottschmidt, K., Hanke, M., and Bock, M., 2011, MNRAS, 414, L60-64 Yamada, S. et al., 2011, PASJ, SP, 63, S645 Yamada, S., Ph.D. Thesis, University of Tokyo, 2011 Yamada, S. et al., 2012, PASJ, 64, 53 ˙ Zycki, P. T. 2002, MNRAS, 333, 800", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "We have examined the relationship between supermassive black hole mass (M BH ) and the stellar mass of the host spheroid (M sph , ∗ ) for a sample of 75 nearby galaxies. To derive the spheroid stellar masses we used improved 2MASS K s -band photometry from the archangel photometry pipeline. Dividing our sample into core-S'ersic and S'ersic galaxies, we find that they are described by very different M BH -M sph , ∗ relations. For core-S'ersic galaxies - which are typically massive and luminous, with M BH /greaterorsimilar 2 × 10 8 M /circledot - we find M BH ∝ M 0 . 97 ± 0 . 14 sph , ∗ , consistent with other literature relations. However, for the S'ersic galaxies - with typically lower masses, M sph , ∗ /lessorsimilar 3 × 10 10 M /circledot - we find M BH ∝ M 2 . 22 ± 0 . 58 sph , ∗ , a dramatically steeper slope that differs by more than 2 standard deviations. This relation confirms that, for S'ersic galaxies, M BH is not a constant fraction of M sph , ∗ . S'ersic galaxies can grow via the accretion of gas which fuels both star formation and the central black hole, as well as through merging. Their black hole grows significantly more rapidly than their host spheroid, prior to growth by dry merging events that produce core-S'ersic galaxies, where the black hole and spheroid grow in lock step. We have additionally compared our S'ersic M BH -M sph , ∗ relation with the corresponding relation for nuclear star clusters, confirming that the two classes of central massive object follow significantly different scaling relations. Subject headings: black hole physics - galaxies: bulges - galaxies: nuclei - galaxies: fundamental parameters", "pages": [ 1 ] }, { "title": "THE SUPERMASSIVE BLACK HOLE MASS - SPHEROID STELLAR MASS RELATION FOR S ' ERSIC AND CORE-S ' ERSIC GALAXIES", "content": "Nicholas Scott, Alister W Graham Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Vic, 3122, Australia and James Schombert Department of Physics, University of Oregon, Eugene, OR 97403, USA Draft version July 29, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Supermassive black hole masses, M BH , are well known to scale with a number of properties of their host galaxy. This was first reported by Kormendy & Richstone (1995), who found a linear correlation between supermassive black hole mass and host spheroid luminosity. Later studies found log-linear correlations between supermassive black hole mass and: stellar velocity dispersion, σ (Ferrarese & Merritt 2000; Gebhardt et al. 2000), stellar concentration (Graham et al. 2001) and dynamical mass, M dyn ∝ σ 2 R (Magorrian et al. 1998; Marconi & Hunt 2003; Haring & Rix 2004). These initial studies reported strong log-linear correlations. Recent work using larger galaxy samples, with accurate measurements of their black hole masses and host galaxy properties, has indicated that these simple log-linear scaling relations are not always sufficient descriptions of the observed distribution. In particular, Graham & Driver (2007) showed that the black hole mass - S'ersic index ( n ) relation was not log-linear, and Graham (2012a) showed that the M BH -M sph , dyn relation requires two separate log-linear relations: with a slope of ∼ 2 for S'ersic galaxies (whose bulge surface brightness profiles are well-represented by a single S'ersic 1968, model 1 ) and with a slope ∼ 1 for core-S'ersic galaxies (whose bulge surface brightness profiles deviate from a single S'ersic function by having a partially depleted core: Graham et al. 2003; Graham & Guzm'an 2003; Trujillo et al. 2004; Ferrarese et al. 2006a). In addition Graham & Scott (2013, hereafter GS13) demonstrated that the M BH -L sph relation is also better described by two log-linear relations, with K s -band slopes ∼ 2 . 73 ± 0 . 55 and ∼ 1 . 10 ± 0 . 20 for the S'ersic and coreS'ersic galaxies respectively. In contrast the M BH -σ relation is well described by a single log-linear relation with slope ∼ 5 (Ferrarese & Merritt 2000; Graham et al. 2011; McConnell & Ma 2013; Park et al. 2012, GS13) over this same black hole mass range, once the offset barred and pseudobulge galaxies are properly taken into account (Graham 2008a; Hu 2008; Graham & Li 2009). These revisions to the supermassive black hole scaling relations are in fact expected, given other observed galaxy scaling relations. As observational samples have explored a broader range in galaxy luminosity, the Lσ relation has been revised from L ∝ σ 4 (Faber & Jackson 1976) to exhibit two different slopes at high and low luminosity. The relation for luminous galaxies is given by L ∝ σ 5 (Schechter 1980; Liu et al. 2008) but for M B > -20 . 5 mag L ∝ σ 2 (Davies et al. 1983; Matkovi'c & Guzm'an 2005). Given this form of the Lσ relation, and the loglinear M BH -σ relation, the M BH -L relation cannot be log-linear - it must exhibit the same bend as the Lσ relation. From these correlations, the expected form of the M BH -L sph relation is: M BH ∝ L 1 . 0 sph for luminous core-S'ersic galaxies and M BH ∝ L 2 . 5 sph for the typically less-luminous S'ersic galaxies, consistent with the findings of GS13. The steep M BH -L sph relation for S'ersic galaxies implies that their black hole must grow much more rapidly than their host spheroid. In these intermediate-mass galaxies, the accretion of gas plays a significant role in the growth of the spheroid and in fuelling the central supermassive black hole. This is seen at high redshift through the coexistence of Active Galactic Nuclei (AGN) with rapidly star-forming sub-millimetre galaxies (e.g., Blain et al. 1999; Page et al. 2001; Alexander et al. 2005; Pope et al. 2008; Page et al. 2012; Simpson et al. 2012) and ultraluminous FIR-detected galaxies (e.g., Norman & Scoville 1988; Alonso-Herrero et al. 2006; Daddi et al. 2007; Murphy et al. 2009), and through large spectroscopic surveys of AGN hosts (e.g., Silverman et al. 2009; Chen et al. 2013). At low redshift this is evident from the coincidence of ongoing star formation or young stellar populations and AGN activity (e.g., Kauffmann et al. 2003b; Netzer 2009; Rosario et al. 2013). These observations are all consistent with a model in which there is a strong physical link between star formation and AGN activity (Hopkins et al. 2006). In a large sample of lowredshift AGN, LaMassa et al. (2013) find that the star formation rate is related to the black hole accretion rate by: ˙ M ∝ SFR 2 . 78 , indicating that gas accretion contributes much more significantly to black hole growth than to star formation. An accurate determination of the true form of the supermassive black hole scaling relations is critical in a number of areas of extragalactic astrophysics. Supermassive black holes play a critical role in semi-analytic models of galaxy formation, through their ability to regulate star formation via AGN feedback (Silk & Rees 1998; Kauffmann & Haehnelt 2000). This feedback is vital in matching the predicted galaxy/bulge luminosity functions of such models to observations. Modern semi-analytic models use the observed local supermassive black hole scaling relations as a key constraint on the rate of black hole growth (e.g., Springel et al. 2005; Bower et al. 2006; Croton et al. 2006; Di Matteo et al. 2008; Booth & Schaye 2009; Dubois et al. 2012). Using an incorrect form of the local scaling relations can significantly alter the degree of AGN feedback in these simulations, resulting in an inaccurate determination of the efficacy of that feedback, or in incorrect rates of star formation and build-up of stellar mass. Another important application of the local supermassive black hole scaling relations is to the study of the evolution of the black hole - host spheroid connection with redshift. The M BH -L sph and M BH -M sph , ∗ relations have been determined over a range of redshifts, including at z /lessorsimilar 0 . 5 (Woo et al. 2006; Salviander et al. 2007; Shen et al. 2008; Canalizo et al. 2012; Hiner et al. 2012), z ∼ 1 (Peng et al. 2006; Salviander et al. 2007; Merloni et al. 2010; Bennert et al. 2011; Bluck et al. 2011; Zhang et al. 2012) and z > 2 (Borys et al. 2005; Kuhlbrodt et al. 2005; Peng et al. 2006; Shields et al. 2006). These studies typically search for changes in the high-redshift scaling relations with respect to the local scaling relations in an effort to identify evolution in the relationship between supermassive black holes and their host spheroids. From any apparent evolution, they then attempt to determine whether the onset of spheroid or black hole growth occurred first, and therefore which is the driving mechanism for the local correlations. Correctly determining the local scaling relations is therefore critical in identifying any evolution with redshift as the local scaling relations are much more accurately known than those at high redshift much of the constraint on evolution comes from the local scaling relations. Using an incorrect local scaling relation can hide (or incorrectly identify) evolution with redshift in the black hole - host spheroid connection. In this work we extend the investigation of the bent supermassive black hole scaling relations to include the stellar mass of the host spheroid; the M BH -M sph , ∗ relation. In Section 2 we present our sample of galaxies containing supermassive black holes and describe the derivation of their associated spheroid's stellar luminosity and mass. The luminosities used here differ slightly from those in GS13 due to our use of updated photometry, which we describe below. In Section 3 we use a linear regression to examine the bent nature of the M BH -M sph , ∗ relation. In Section 4 we discuss the implications of bent supermassive black hole scaling relations on the intrinsic scatter of supermassive black hole scaling relations, the coevolution of supermassive black holes and their host spheroids and the alleged common origins of nuclear star clusters and supermassive black holes and . We present our conclusions in Section 5.", "pages": [ 1, 2 ] }, { "title": "2. SAMPLE AND DATA", "content": "We make use of the sample of 78 supermassive black hole masses and host spheroid magnitudes presented in GS13. Following GS13, we continue to exclude M32 (due to an unknown amount of stellar stripping), the Milky Way (due to its uncertain bulge magnitude due to dust extinction) and NGC 1316 (due to its uncertain core type), giving a final sample of 75 galaxies. All galaxies in the sample have directly measured supermassive black hole masses, with typical uncertainties ∼ 50% (for detailed references see GS13, their section 2). GS13 provide apparent Band K s -band magnitude data and a prescription to determine absolute spheroid magnitudes. Although we make use of their Bband magnitudes, we use different initial K s -band apparent total galaxy magnitudes as described below. GS13 used K s -band magnitudes from the Two Micron All-Sky Survey (2MASS) Extended Source Catalogue (Jarrett et al. 2000), however they noted that some errors have recently been reported for the 2MASS catalogue photometry (Schombert & Smith 2012). Given these concerns we derive new total galaxy magnitudes from the same 2MASS K s -band images using the archangel photometry pipeline (Schombert & Smith 2012). The pipeline takes a galaxy's name as input, parsing the name through the NASA Extragalactic Database (NED) to resolve its position on the sky. The pipeline then extracts the four sky strip images surrounding the galaxy's coordinates from the 2MASS Atlas Image Server, stitching the images together to form a single raw image. The 2MASS project provides calibrated, flattened, kernel-smoothed, sky-subtracted images so these steps are not duplicated by the pipeline. An accurate sky determination is then made using user-defined sky boxes clear of obvious stellar or galaxy sources, which are then summed and averaged. The final step is the extraction of isophotal values as a function of radius using an ellipse fitting routine. Elliptical apertures are used to determine curves of growth for determination of photometric values. Our new K s magnitudes improve on the 2MASS Extended Source Catalogue values in three ways. First, as described in Schombert & Smith (2012), the sky background is over-subtracted by the 2MASS data reduction pipeline, resulting in an underestimation of the total galaxy magnitude. This over-subtraction truncates the surface brightness profile of luminous elliptical galaxies at large radii, causing the 2MASS pipeline to underestimate the physical size of each object. This results in the total magnitude being measured from an aperture that significantly underestimates the true physical size of an object, thus resulting in a total magnitude that is 10-40 % lower than the total magnitudes derived by archangel . Finally, our archangel derived magnitudes are determined from pipeline fits to the surface brightness profiles, extending out to radii where the uncertainty in the surface brightness of the object exceeds 1 mag arcsec -2 , which is typically further than the fourhalf-light-radii apertures used for 2MASS total magnitudes. For low S'ersic index profiles there is little difference in the total magnitudes derived from these two apertures, however, as shown in Figure 1, for S'ersic n /greaterorsimilar 3 the flux missed by the 2MASS aperture can be significant. In Figure 2 we show a comparison between the 2MASS catalogue magnitudes and our new archangel magnitudes for 67 galaxies. Eight galaxies of our 75 were too large on the sky to model readily with archangel . For these eight remaining galaxies we derived a correction to their 2MASS magnitudes using the following procedure. We divided the sample of 67 galaxies into three morphological types (ellipticals, lenticulars and spirals) and fit linear relations to the 2MASS vs. archangel magnitudes for each subset (indicated by the lines in Figure 2). Based on these relations and the 2MASS catalogue magnitude we derived corrected magnitudes for the 2/8 remaining elliptical galaxies and used the 2MASS magnitudes for the 6/8 S0 and Sa disk galaxies. The relation between the 2MASS and archangel K s -band magnitudes for elliptical galaxies in our sample is: The two galaxies for which this procedure was used are indicated with a † in Table 1, along with the six disk galaxies for which we used 2MASS photometry. As tabulated in GS13, total apparent Bband magnitudes were drawn from the Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et al. 1991). Following GS13, apparent magnitudes were converted to absolute magnitudes using distance moduli primarily from the surface brightness fluctuation based measurements of Tonry et al. (2001), after applying the 0.06 magnitude correction of Blakeslee et al. (2002, see GS13 for a full list of references for the distance determinations). All magnitudes were corrected for Galactic extinction following Schlegel et al. (1998), cosmological redshift dimming and K -corrections. In addition to the standard corrections mentioned above, two further 'corrections' were applied to the absolute magnitude of disk galaxies to derive spheroid magnitudes. Given the large sample size, individual spheroid fractions were not derived for each object, but instead a mean statistical correction was applied based on each object's morphological type and disk inclination. The relationship between the applied correction (for both dust and bulge fraction) is based on the galaxy's morphological type and inclination and is given by equation (5) in GS13. The dust correction follows the method of Driver et al. (2008) and depends on the galaxy inclination and passband. The correction for the bulge-todisk flux ratio was derived from the observed bulge-todisk flux ratios presented in Graham & Worley (2008), as adapted slightly by GS13. The bulge-to-disk correction depends on morphological type and passband, and is given in table 2 of GS13 for the Band K s -bands. While the above prescription results in significant uncertainties for the spheroid magnitudes of individual bulges, the ensemble average correction is considerably more accurate, scaling with √ N . Such a statistical correction is only now viable given the sufficiently large sample of supermassive black hole masses in disk galaxies that are available in the literature. Our updated inclination and dust corrected K s -band spheroid magnitudes for the full sample are given in Table 1, the Bband values are given in table 2 of GS13. We use the absolute galaxy magnitudes of the elliptical galaxies and the inclination and dust corrected bulge magnitudes of the disk galaxies to derive stellar masses for all spheroids in our final sample of 75 galaxies. Using the optical-NIR ( B -K s ) color of the spheroids, we derive stellar mass-to-light ratios (M/L) using the relations presented in Bell & de Jong (2001). For the K s -band stellar mass-to-light ratio, M/L K s and the ( B -K s ) color, the relation is: We derive stellar mass-to-light ratios in the K s -band as this ratio shows the smallest sensitivity to color in this band. For the range of B -K s colors found in our sample, M/L K s varies by a factor of 2, compared to a factor of 7 for the Bband mass-to-light ratio. K s -band magnitudes also suffer the least from dust extinction, though as noted by Bell & de Jong (2001) the conversion to stellar mass based on an optical-NIR color is not significantly affected by dust. The final stellar masses for all 75 spheroids in our sample are given in Table 1. Lastly, we note that for all 75 galaxies, GS13 identified them as either S'ersic or core-S'ersic galaxies. For the majority of galaxies this classification was based upon examination of their surface brightness profiles taken from Properties of our sample of 75 nearby galaxies hosting a † : Galaxies for which we did not obtain new archangel photometry. Column (1): Galaxy identifier. Column (2): Morphological type. Column (3): Core type. y indicates the galaxy contains a core, n indicates the galaxy does not have a depleted core. ? indicates that the classification is based on the velocity dispersion. Column (4): Supermassive black hole mass. References are provided in GS13. Column (5): Inclination and dust corrected K s absolute spheroid magnitude. For elliptical galaxies the typical uncertainty on K s , sph is 0.25 mag. For disk galaxies this increases to 0.75 mag, due to the additional dust and bulge-to-total corrections. Column (6): ( B -K s ) spheroid color. Column (7): Spheroid stellar mass. For elliptical galaxies the typical uncertainty on M sph , ∗ is 0.2 dex, for disk galaxies this increases to 0.36 dex. Hubble Space Telescope (HST) imaging. For 19 galaxies without HST imaging GS13 assigned a core type based on the galaxy's velocity dispersion (see GS13 for further details, including a list of those objects with velocity dispersion based core type assignments). The most massive galaxies are typically core-S'ersic galaxies and the least massive galaxies are exclusively S'ersic galaxies, however there is a significant region of overlap, with spheroids in the 10 10 -10 11 M /circledot region showing both profile types. For all of these galaxies in the overlap region the classification was based on their observed light profile.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2.1. Sources of uncertainty on M BH and M sph , ∗", "content": "The uncertainty on M BH for each individual object is given in Table 1; these are drawn from the same sources as the black hole mass measurements themselves and have been adjusted to the distances tabulated in GS13 - detailed references are given in GS13. The typical uncertainty on M BH is 50%. The total uncertainty on M sph , ∗ has contributions from two or three separate components. For elliptical galaxies these are the uncertainty on the K s magnitudes measured by archangel and the uncertainty in converting this magnitude into a stellar mass. The uncertainty in the magnitudes derived from the 2MASS photometry are determined by the archangel pipeline, and are typically ∼ 0 . 25 mags (or 0.1 dex). The uncertainty in the M/L used to convert these magnitudes to stellar masses depends on the uncertainty in the ( B -K s ) color, and on the uncertain star formation history of each object. Bell & de Jong (2001) give M/L K s for a range of star formation histories, allowing the uncertainty in M/L due to the uncertain star formation history to be estimated. For our sample, the uncertainty on M/L K s (due to both the uncertainty on the observed color and the uncertain star formation history) is typically 0.17 dex. In elliptical galaxies the typical total uncertainty on M sph , ∗ for disk-less systems is 0.2 dex. In systems with a stellar disk there is the additional source of uncertainty in converting the total magnitude into a spheroid magnitude by applying both a dust correction and a correction for the bulge-to-disk flux ratio. The uncertainty due to the dust correction is estimated by Driver et al. (2008) to be 5% in both the Band Kbands. The uncertainty in the bulge-to-disk flux ratio can be estimated from the data presented in Graham & Worley (2008, their table 4), and for the galaxies in our sample is typically 0.3 dex. For systems with a disk this is the dominant source of uncertainty and the total typical uncertainty in M sph , ∗ is 0.36 dex.", "pages": [ 5 ] }, { "title": "3. ANALYSIS", "content": "In Figure 3 we show the spheroid stellar mass plotted against the supermassive black hole mass for all galaxies in Table 1. We separate galaxies into S'ersic (filled blue symbols) and core-S'ersic (open red symbols). We fit separate linear regressions to the S'ersic and core-S'ersic galaxy subsamples, using the bces bisector regression of Akritas & Bershady (1996). This technique takes into account the measurement uncertainties in both black hole mass and stellar spheroid mass and accounts for (though does not determine) the intrinsic scatter. For the core-S'ersic galaxies the best-fitting symmetrical regression is: and for S'ersic galaxies the best-fitting symmetrical regression is: with rms residuals of 0.47 and 0.90 dex respectively in the log M BH direction. These linear relations are shown in Fig. 3 as the solid red and blue lines for the core-S'ersic and S'ersic galaxies respectively. For comparison, the linear regression to the combined sample is shown as the black dashed line, which has a slope of 1 . 50 ± 0 . 12 (c.f. Laor 2001)and an rms residual of 0.67 dex. The bestfitting regressions for the two different types of galaxy have significantly different slopes that are not consistent with each other given the confidence intervals on the slopes. S'ersic galaxies follow an approximately quadratic relation, whereas core-S'ersic galaxies follow an approximately linear relation. This is in agreement with the analysis and conclusions of Graham (2012a) who studied ⊙ the M BH -M sph , dyn relation. The bend or break in the M BH -M sph , ∗ distribution occurs where the core-S'ersic and S'ersic relations overlap. This is at a spheroid stellar mass M sph , ∗ ∼ 3 × 10 10 M /circledot , corresponding to a black hole mass M BH ∼ 2 × 10 8 M /circledot . The significance of this 'break mass' will be discussed in Section 4. Here we simply note that very few S'ersic galaxies have M BH greater than this break mass; equally no core-S'ersic galaxy has a black hole mass below this value. Park et al. (2012) examined the effect of using different linear regression techniques on the M BH -σ relation but their results are applicable to supermassive black hole scaling relations in general. They reported that three popular regression techniques, bces (as used in this work) a modified version of fitexy (Press et al. 1992; Tremaine et al. 2002) and a Bayesian technique developed by Kelly (2007), linmix err return consistent results. However, if the measurement uncertainties are larger than ∼ 15% in the ordinate when using the 'forward' regression, they find that the bces routine may be biased to higher slopes (as was noted by Tremaine et al. 2002). Because of the significant uncertainties on many of our low-mass spheroid stellar masses (due to our statistical bulge-disk separation), we redetermined our coreS'ersic and S'ersic relations using the modified fitexy and the linmix err linear regression methods to test the robustness of our result. Following GS13, we determine a symmetrical bisector regression from the linmix err routine by determining the 'forward' and 'inverse' regressions, then determining the line that bisects those two regressions. Using the linmix err method we find bisector slopes for the core-S'ersic and S'ersic samples of 1 . 10 ± 0 . 07 and 2 . 11 ± 0 . 46 respectively. We adopt a similar approach with fitexy ; determining the 'forward' and 'inverse' regressions then determining the bisector line. This method yields bisector regressions for the coreS'ersic and S'ersic samples with slopes of 1.08 and 2.48 respectively. With all three symmetric regression methods we find consistent best-fitting relations, and in all cases the slope for the S'erisc galaxies is greater than 2 σ steeper than that for core-S'erisc galaxies. We conclude that our principal result - that core-S'ersic and S'ersic galaxies follow different M BH -M sph , ∗ relations is robust against the choice of linear regression method. The core-S'ersic/S'ersic classification is similar to the slow rotator/fast rotator (SR/FR) classification of Emsellem et al. (2007), though as discussed in Emsellem et al. (2011) the overlap between the two systems is not perfect. Of our objects, 33/75 are part of the ATLAS 3D survey (Cappellari et al. 2011) and have FR/SR classifications from Emsellem et al. (2011). We also examined the spheroid stellar mass vs. supermassive black hole mass relation for these galaxies. We again fit linear regressions to the two samples of 9 SR and 24 FR galaxies. With this smaller sample we do not find significantly different slopes for the FR and SR samples; the slopes for the FR and SR samples are 1 . 71 ± 0 . 27 and 1 . 32 ± 0 . 44 respectively. While the slope for FRs is somewhat steeper, the difference is not significant given the formal uncertainty on the derived slopes. However, this may be caused by the galaxies with the most massive black holes and the least massive galaxies not having an FR/SR classification and therefore not being included in these regressions. This same sampling effect, where galaxies at the extremes of the supermassive black hole mass scaling relations were not well-sampled, led to a single log-linear relation being sufficient to describe the data in the past. It is only recently, where increased numbers of objects at the extremes of the relations have been added, has the bend in the supermassive black hole mass scaling relations become evident. Alternatively, the coreS'ersic/S'ersic classification may be more closely linked to the mechanism(s) responsible for black hole growth than the FR/SR classification. Indeed, FR lenticular galaxies with depleted cores are known to exist (Dullo & Graham 2013; Krajnovi'c et al. 2013). A larger sample of galaxies with both measured supermassive black hole masses and kinematic classifications would be desirable. From Eqn. (4), the expected variation of M BH /M sph , ∗ with M sph , ∗ is close to linear for the S'ersic galaxies, with the black holes of more massive spheroids representing a larger fraction of their host spheroid's mass. In Figure 4 we show the ratio of supermassive black ⊙ hole mass to spheroid stellar mass as a function of spheroid stellar mass. In previous studies which found M BH ∝ M ∼ 1 sph , ∗ , this ratio was thought to be constant, with the supermassive black hole having a mass ∼ 0 . 150 . 20% of its host spheroid's mass (Merritt & Ferrarese 2001; Marconi & Hunt 2003). Here we find that this remains approximately true for the core-S'ersic galaxies, albeit with M BH ∼ 0 . 55% of M sph , ∗ (the relation for core-S'ersic galaxies is consistent with a slope of 1 hence a constant mass fraction). However, for S'ersic galaxies, we find that the average M BH /M sph , ∗ is offset to a lower mean value of 0.3% for our particular sample's mass range and it also displays a large range, from ∼ 0 . 02-2%.", "pages": [ 5, 6, 7 ] }, { "title": "4.1. Comparison to previous studies", "content": "In this work we have identified a break in the M BH -M sph , ∗ diagram, caused by two separate log-linear relations with significantly different slopes for core-S'ersic and S'ersic galaxies. While this result is a significant change from the commonly accepted view of single loglinear scaling relations describing the relationship between black holes and their host spheroids, this work is not the first to identify this change. As noted in Section 1, Graham (2012a) and GS13 have both previously reported the need for separate log-linear relations for coreS'ersic and S'ersic galaxies to describe the trend of black hole mass with host galaxy dynamical mass and host spheroid luminosity respectively. In addition, Graham (2007) had previously noted that both the M BH -σ and M BH -L relations cannot both be log-linear due to the non-linear Lσ relation for early-type galaxies. While in this paper we have quantified the first bent relation between black hole mass and host spheroid stellar mass, we can compare our results with past work pertaining to the host's dynamical mass. GS13 reported an M BH -L K S relation for the S'ersic galaxies with a slope of 2 . 73 ± 0 . 55, which they equated to a slope of 2 . 34 ± 0 . 47 in the M BH -M dyn diagram using M/L K s ∝ L 1 / 6 K s (e.g., La Barbera et al. 2010; Magoulas et al. 2012). Prior to that, Graham (2012) had reported a slope of 1 . 92 ± 0 . 38 for the M BH -M dyn relation based on independent data. These two steep slopes compare well with our slope of 2 . 22 ± 0 . 58 for the S'ersic galaxy M BH -M sph , ∗ relation. For the core-Sersic galaxies, GS13 had reported a slope of 1 . 10 ± 0 . 20 in the M BH -L K S diagram. If core-S'ersic galaxies are predominantly built by dry mergers of galaxies near or above the high-mass end of the S'ersic distribution, then the slope in the M BH -M sph , ∗ diagram should be the same as the slope in the M BH -L diagram. Graham (2012) had additionally reported a slope of 1 . 01 ± 0 . 52 for the core-S'ersic galaxies in the M BH -M dyn , sph diagram (with the large uncertainty reflecting their small sample size). These two shallower slopes are consistent with our slope of 0 . 97 ± 0 . 14 in the M BH -M sph , ∗ diagram. Other authors have noted that low luminosity (or mass) galaxies are consistently offset from single loglinear black hole scaling relations derived from samples dominated by massive systems. Greene et al. (2008) identified a population of low mass galaxies (stellar masses in the range 10 9 -10 10 M /circledot ) whose black hole masses were offset below the M BH -M bulge relation of Haring & Rix (2004) by an order of magnitude. More recently, Mathur et al. (2012) determined black hole masses for a sample of 10 narrow-line Seyfert galaxies with host spheroid luminosities in the range 3 × 10 9 -3 × 10 10 L /circledot (corresponding to a range in stellar mass of approximately 10 10 -10 11 M /circledot ) and again found their galaxies to be offset below the Haring & Rix (2004) relation. While Mathur et al. (2012) attribute much of this offset to 5 of their objects being 'pseudobulges', the remaining 5 are 'classical' bulges and are still substantially offset from the M BH - L bulge relation of Gultekin et al. (2009). Greene et al. (2008) also identify many of their objects which are significantly offset below the classical log-linear M BH -M bulge relation as being well-fit with a de Vaucouleurs ( n = 4) profile, suggesting they are not pseudobulges. The offset objects found by both these studies are consistent with the log-linear scaling relations for S'ersic galaxies reported in this work, Graham (2012a) and in GS13. While we do not identify pseudobulges in our sample, we note that both NGC4486a and NGC821, both 'classical' elliptical galaxies with no indication of a pseudobulge, are consistent with our S'ersic scaling relation and offset from the core-S'ersic relation, arguing against the offset nature of the low-mass systems being a pseudobulge phenomena. Sani et al. (2011), Vika et al. (2012) and Beifiori et al. (2012) have all recently constructed M BH -M sph , ∗ relations (or in the case of Vika et al. 2012, the closelyrelated M BH -L K, sph relation) for large samples ( ∼ 50 galaxies). All three studies only considered single loglinear fits to their data and are dominated by massive galaxies with M sph , ∗ > 3 × 10 10 M /circledot . As expected from their high-mass-dominated samples, all three studies find M BH -M sph , ∗ relations with slopes ∼ 1, consistent with our finding for core-S'ersic galaxies. However, in all three studies a number of galaxies are offset below those authors' single log-linear relations. With host spheroid masses around 3 × 10 10 M /circledot , they are consistent with the high-mass regime of our S'ersic galaxy relation. These studies highlight the need to extend the range of supermassive black hole masses and host spheroid masses used to examine the black hole scaling relations.", "pages": [ 7, 8 ] }, { "title": "4.2. Scatter about the M BH -M sph , ∗ relation", "content": "As well as determining the slope of the supermassive black hole scaling relations, many studies also examine the scatter about the relations. In particular, this is done to argue that one of the relations has reduced intrinsic scatter compared to the other common scaling relations, and therefore is the 'fundamental' driving relation. At low masses the scatter in our Figure 3 is dominated by the large measurement uncertainties due to our statistical dust and bulge correction, making such a comparison difficult for this study. Instead, we will make a few simple observations on the revised expectations for the intrinsic scatter as a result of our bent scaling relation. As noted in Section 3, the observed scatter (the sum of the intrinsic scatter and measurement uncertainty) in the black hole mass direction is significantly larger for the S'ersic galaxies than for the core-S'ersic galaxies: 0.90 dex compared to 0.47 dex. An increase is expected given the typically larger measurement uncertainties for the S'ersic galaxies due to the statistical dust and bulge correction. However, we also expect the scatter in the vertical direction to increase because of the increased slope of the relation for the S'ersic galaxies. This is not the case for the intrinsic scatter orthogonal to the relation - we would expect this to be reduced relative to a single log-linear relation fit to the entire data. The orthogonal scatter we find for the core-S'ersic galaxies is 0.26 dex, and for the S'erisc galaxies it is 0.29 dex, an improvement over the 0.31 dex orthogonal scatter we find for the single log-linear relation. Finally, a number of theoretical studies have examined the idea of the supermassive black hole scaling relations being the product of the repeated merging of a randomly seeded initial population of black holes and host galaxies (Peng 2007; Jahnke & Macci'o 2011). These studies predict a decrease in the intrinsic scatter as host galaxy mass increases (though Jahnke & Macci'o 2011, note that the addition of star formation to their model reduces the rate of this decrease in scatter). Given the large measurement uncertainties for many of our S'ersic galaxies it is difficult to quantify any variation of the intrinsic scatter with host mass in our sample. Moreover, we expect that only for the core-S'ersic galaxies is the binary merging of both galaxies and black holes the dominant mechanism of growth. As these galaxies only span a relatively narrow range in host spheroid mass, any systematic variation of the intrinsic scatter within this subsample is uncertain.", "pages": [ 8 ] }, { "title": "4.3. Comparing supermassive black hole and nuclear star cluster scaling relations", "content": "Ferrarese et al. (2006b) and Wehner & Harris (2006) have argued that nuclear star clusters and super- ⊙ massive black holes form a single class of central massive object (CMO) based on their allegedly common mass scaling relations, though recent studies (Balcells et al. 2007; Graham & Spitler 2009; Graham 2012b; Erwin & Gadotti 2012; Leigh et al. 2012; Scott & Graham 2013) have since argued against this scenario. We briefly re-examine the connection between nuclear star cluster and supermassive black hole scaling relations in the light of the bent M BH -M sph , ∗ relation. In Figure 5 we show M CMO vs. M sph , ∗ for the S'ersic galaxies presented in this study and the nuclear star clusters presented in Scott & Graham (2013). The two lines show the best-fitting linear relations to the two samples. The M NC -M sph , ∗ line is taken from Scott & Graham (2013) and has a slope of 0 . 88 ± 0 . 19. The relation for the nuclear star clusters is 2 . 3 σ shallower than that for the supermassive black holes. This difference is more pronounced than that reported by Scott & Graham (2013), due to our improved bent supermassive black hole scaling relation, strongly suggesting that supermassive black holes and nuclear star clusters do not follow a common scaling relation and therefore do not have a common formation mechanism.", "pages": [ 8 ] }, { "title": "4.4. The growth of supermassive black holes and stellar spheroids", "content": "As noted in Section 3, the break or bend in the M BH -M sph , ∗ relation occurs at M sph , ∗ /similarequal 3 × 10 10 M /circledot . This mass scale has been identified by a number of authors as marking a change in early-type galaxy properties. Below this mass galaxies are typically young, lowsurface density objects (Kauffmann et al. 2003a), whose spheroid's surface brightness is well-fit by a S'ersic pro- Graham & Guzm'an 2003; Balcells et al. 2003), and define a sequence of increasing M/L and bulge fraction (Cappellari 2012) - until dwarf spheroidal systems appear around M B ∼ -14 mag. Above this mass galaxies are typically old with high S'ersic indices (Graham et al. 2001; Kauffmann et al. 2003a), have core-S'ersic surface brightness profiles (Graham & Guzm'an 2003), are bulge-dominated and show only a narrow range in M/L (Cappellari 2012). Tremonti et al. (2004) also report that the mass-metallicity relation flattens above this mass scale. The results discussed above are all consistent with a scenario in which galaxies with spheroid masses below the rough 3 × 10 10 M /circledot divide grow predominantly through gas-rich processes involving significant in-situ star formation whereas the growth of those galaxies above this mass scale is dominated by 'dry' processes involving little gas and the direct accretion of stars via mergers. This is similar to a scenario adopted by Shankar et al. (2012) based on their semi-analytic model of black hole and galaxy formation. In the high-mass regime where little gas is involved, black hole growth will be dominated by the binary merger of supermassive black holes during galaxy mergers. In a dry merger, to first order, the masses of the two progenitor galaxies will simply be added, as will the masses of their supermassive black holes. Thus, in this regime we can roughly expect a linear relationship between supermassive black hole mass and host stellar mass, as observed. In the low mass regime gas plays a more significant role, complicating the picture. Galaxies can grow from the direct cooling of gas from their immediate surroundings, and even when mergers do play a significant role the resulting growth (of both stellar mass and black hole mass) will not be simply additive as the 'wet' mergers will likely be accompanied by significant star formation. The slope of ∼ 2 . 2 in the M BH -M sph , ∗ relation that we find for S'ersic galaxies implies that black holes have grown more rapidly than their host spheroids. This is consistent with the growth of these galaxies being dominated by gaseous processes (e.g., gas-rich mergers, cooling of hot gas, secular evolution: see Hopkins & Hernquist 2009, for a detailed discussion of the full range of processes) that efficiently channel material onto the central black hole (see e.g., Marconi et al. 2004) but are less effective at growing the host galaxy spheroid. One such example of a supermassive black hole growing more rapidly than its host galaxy has recently been observed by Seymour et al. (2012). This steepening of the supermassive black hole mass - host spheroid stellar mass relation has been reproduced in a number of simulations (Cirasuolo et al. 2005; Dubois et al. 2012; Neistein & Netzer 2013, figures 5, 3 and 8 respectively), though these authors have typically focused on the agreement with the classical single loglinear relation in massive galaxies, rather than the downturn at lower mass in their simulated data.", "pages": [ 8, 9 ] }, { "title": "5. CONCLUSIONS", "content": "The results in Section 3 represent a major revision to the accepted picture of the relationship between supermassive black holes and their host galaxies. Our results have significant implications for: studies of how supermassive black holes and their hosts co-evolve over cosmic time; the detection of intermediate-mass supermassive black holes; and models of feedback from active galactic nuclei due to supermassive black hole growth. We have examined the bent nature of the supermassive black hole mass - spheroid stellar mass relation for a large sample of 75 galaxies. We find that, when separating galaxies into core-S'ersic and S'ersic galaxies based on their central light profiles, they follow significantly different supermassive black hole scaling relations whose slopes are not consistent with one another. For the coreS'ersic galaxies M BH ∝ M 0 . 97 ± 0 . 14 sph , ∗ , whereas for the S'ersic galaxies M BH ∝ M 2 . 22 ± 0 . 58 sph , ∗ . These results are consistent with the expectation from a single log-linear supermassive black hole mass - host velocity dispersion relation combined with a bent relationship between host spheroid luminosity and host velocity dispersion. We find that the supermassive black hole mass is an approximately constant 0.55% of a core-S'ersic galaxy's spheroid stellar mass. The non-log-linear M BH -M sph , ∗ relation implies that, for S'ersic galaxies, M BH is not a constant fraction of the host spheroid mass, but represents an increasing fraction with increasing M sph , ∗ . This research was supported by Australian Research Council funding through grants DP110103509 and FT110100263.", "pages": [ 9 ] }, { "title": "REFERENCES", "content": "Leigh, N., Boker, T., & Knigge, C. 2012, MNRAS, 424, 2130 Liu, F. S., Xia, X. Y., Mao, S., Wu, H., & Deng, Z. G. 2008, MNRAS, 385, 23", "pages": [ 10 ] } ]

[ { "title": "ABSTRACT", "content": "I present a generalized power-law diagnostic that allows to identify the presence of active galactic nuclei (AGN) in infrared (IR) galaxies at z > 1, down to flux densities at which the extragalactic IR background is mostly resolved. I derive this diagnostic from the analysis of 174 galaxies with S ν (24 µ m) > 80 µ Jy and spectroscopic redshifts z spec > 1 in the Chandra Deep Field South, for which I study the rest-frame UV/optical/near-IR spectral energy distributions (SEDs), after subtracting a hot-dust, power-law component with three possible spectral indices α = 1 . 3, 2.0 and 3.0. I obtain that 35% of these 24 µ m sources are power-law composite galaxies (PLCGs), which I define as those galaxies for which the SED fitting with stellar templates, without any previous power-law subtraction, can be rejected with > 2 σ confidence. Subtracting the power-law component from the PLCG SEDs produces stellar-mass correction factors < 1 . 5 in > 80% of cases. The PLCG incidence is especially high (47%) at 1 . 0 < z < 1 . 5. To unveil which PLCGs host AGN, I conduct a combined analysis of 4Ms X-ray data, galaxy morphologies, and a greybody modelling of the hot dust. I find that: 1) 77% of all the X-ray AGN in my 24 µ m sample at 1 . 0 < z < 1 . 5 are recognised by the PLCG criterion; 2) PLCGs with α = 1 . 3 or 2.0 have regular morphologies and T dust > ∼ 1000 K, indicating nuclear activity. Instead, PLCGs with α = 3 . 0 are characterised by disturbed galaxy dynamics, and a hot interstellar medium can explain their dust temperatures T dust ∼ 700 -800 K. Overall, my results indicate that the fraction of AGN among 24 µ m sources is between ∼ 30% and 52% at 1 . 0 < z < 1 . 5. Subject headings: infrared: galaxies - galaxies: evolution - galaxies: high-redshift", "pages": [ 1 ] }, { "title": "A GENERALIZED POWER-LAW DIAGNOSTIC FOR INFRARED GALAXIES AT z > 1: ACTIVE GALACTIC NUCLEI AND HOT INTERSTELLAR DUST", "content": "K. I. Caputi 1 Draft version August 28, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The study of the individual dust-obscured sources that make the extragalactic infrared (IR) background (Puget et al. 1996; Dole et al. 2006) has made enormous progress over the last decade, since the launch of the Spitzer Space Telescope (Werner et al. 2004), and successively the Akari Telescope (Murakami et al. 2007) and the Herschel Space Observatory (Pilbratt 2003). In particular, the scientific output of these missions has revealed the importance of powerful, dust-obscured starformation and nuclear activity in shaping galaxy evolution at high redshifts ( z > 1). This activity was dominated by luminous and ultra-luminous infrared galaxies (LIRGs and ULIRGs; Le Floc'h et al. 2005; Caputi et al. 2007), which had a main role in the global star formation history of the Universe (e.g. Hopkins & Beacom 2006), and the process of massive galaxy buildup at z > 1 (Caputi et al. 2006a). The availability of multi-wavelength ancillary data from deep galaxy surveys has been crucial to investigate the presence and properties of IR galaxies at high z . Most redshift estimates and the derivation of other parameters, such as the galaxy stellar mass, rely on the fitting of spectral energy distribution (SED) templates to broad-band photometry that traces the galaxy rest UV/optical and near-IR light ( λ rest < ∼ 3 µ m). The derivation of reliable values for these galaxy parameters requires a proper wavelength coverage of the photometric data, and also considering the possible variations that 1 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands. Email: [email protected] the galaxy SEDs may have, through the choice of sufficiently representative galaxy templates, and the analysis of departures of these SEDs from models of pure stellar evolution. A long-standing problem regarding the composition of the extragalactic IR background is understanding to which extent dust-obscured active galactic nuclei (AGN) are part of the IR galaxy population at high redshifts. A direct way to reveal nuclear activity is searching for X-ray detections (e.g. Rigby et al. 2004; Polletta et al. 2006; Fiore et al. 2008; Symeonidis et al. 2011; Matsuta et al. 2012), but dustobscured AGN can remain undetected even in typically deep X-ray maps. Spectral line diagnostics at optical and IR wavelengths are also useful to recognise AGN, but they are usually limited to relatively small samples of bright IR sources (e.g. Armus et al. 2007; Sajina et al. 2007; Caputi et al. 2008; Desai et al. 2008; Nardini et al. 2008; Hern'an-Caballero et al. 2009; Petric et al. 2011; Hwang et al. 2012). An alternative method that has proven to be quite efficient to reveal AGN in large samples of dust-obscured galaxies is the use of mid-IR colour-colour diagrams (Lacy et al. 2004; Stern et al. 2005), based on photometry at 3 . 6 -8 . 0 µ m, taken with the Spitzer Infrared Array Camera (IRAC; Fazio et al. 2004). The segregated locus that some AGN occupy in these diagrams is related to the fact that pure AGN SEDs are characterised by a power-law shape at rest-frame optical/near-IR wavelengths. In fact, the presence of a pure IRAC power-law SED has been proposed by some authors as a criterion to select AGN-dominated galaxies among Spitzer 24 µ mand IRAC-selected sources (Alonso-Herrero et al. 2006; Donley et al. 2007). But AGN selection through IRAC colour-colour diagrams has well-known limitations: it is complete only for bright IR sources (e.g. Lacy et al. 2007), and the resulting samples are typically contaminated by star-forming galaxies (Donley et al. 2012). This is especially the case at high redshifts, as the galaxy stellar emission is shifted into the IRAC bands, producing similar IRAC colours to those of AGN at lower redshifts. In addition to the identification, there is a second problem, which consists in understanding what fraction of the galaxy light is due to the AGN component at different wavelengths. At mid- and far-IR wavelengths (3 < ∼ λ rest < ∼ 1000 µ m), disentangling the star-formation/AGN contributions is required to properly derive the on-going, obscured star formation rates in IR galaxies. This issue has been tackled in different studies of LIRGs and ULIRGs at high z , by analysing mid-IR spectroscopic data, or mid-/far-IR broad-band data (e.g. Fadda et al. 2010; Gruppioni et al. 2010; Barthel et al. 2012; Hanami et al. 2012; Mullaney et al. 2012; Pozzi et al. 2012). In the rest-frame optical/near-IR (0 . 3 < ∼ λ rest < ∼ 3 µ m), a power-law component associated with an AGN distorts the light of the underlying stellar populations in the host galaxy, and may affect the derived galaxy stellar mass. For galaxies that are identified with X-ray luminous AGN, the SED fitting with stellar templates, and derivation of stellar masses are usually avoided (e.g. Caputi et al. 2006b), or attempted only after subtracting empirical AGN templates (e.g. Merloni et al. 2010; Mainieri et al. 2011). In all other cases, however, the derivation of stellar masses is of common practice, even when there is the suspicion that the rest-frame near-IR light could be partly contaminated by an AGN component (for example, in the case of an IRAC-band excess in the galaxy SED). Quantifying the importance of this effect in IR-selected galaxies is then necessary to understand the reliability of the derived host galaxy properties. In this work I present the analysis of the rest-frame optical/near-IR SEDs of 174 24 µ m-selected galaxies with S ν (24 µ m) ≥ 80 µ Jy and secure spectroscopic redshifts z spec > 1. My aim is to investigate whether, and when, a power-law component makes a relevant contribution to the galaxy SED, i.e. the subtraction of a power-law component produces a significant improvement of the galaxy SED fitting with stellar templates. Hereafter, I will refer to these galaxies as power-lawcomponent galaxies (PLCGs). Note that this criterion is more relaxed than the IRAC power-law shape imposed by other authors to select potential IR AGN candidates (Alonso-Herrero et al. 2006), and therefore, the sample analysed here should also include galaxies for which the AGN emission is less dominant over the underlying host galaxy light. This paper is organised as follows. In Section § 2, I give details on the sample selection and compilation of multi-wavelength photometry. In Section § 3, I present the results of the SED analysis, and quantify the effect on the derived stellar masses. Later, in Section § 4, I investigate the nature of the PLCGs at z > 1. I make use of ultra-deep X-ray data to obtain an independent diagnostic of nuclear activity among the 24 µ m sources, and understand in which cases the significant SED power-law component is related to the presence of an AGN. I also analyse the PLCG morphologies, and derive the greybody temperatures that are necessary to explain the SED power law. In Section § 5, I investigate how the latest proposed IRAC colour-colour criteria deal with the selection of PLCGs and other X-ray-detected AGN. Finally, in Section § 6, I summarise my findings and present some concluding remarks. All magnitudes and colours quoted in this paper are total and refer to the AB system (Oke & Gunn 1983). I adopt a cosmology with H 0 = 70kms -1 Mpc -1 , Ω M = 0 . 3 and Ω Λ = 0 . 7. All stellar masses refer to a Salpeter (1955) initial mass function (IMF) over star masses of (0 . 1 -100) M /circledot .", "pages": [ 1, 2 ] }, { "title": "2. SAMPLE SELECTION AND MULTI-WAVELENGTH DATASETS", "content": "The Great Observatories Origins Deep Survey (GOODS) programme (Giavalisco et al. 2004) comprises a wide range of deep galaxy surveys conducted with main astronomical facilities, including the Spitzer and Hubble Space Telescopes , the Chandra X-ray Observatory , and the largest optical/near-IR ground-based telescopes. In particular, the Spitzer images for the GOODS fields have been collected as part of the GOODS Spitzer Legacy Programme (PI: M. Dickinson), and include deep data from both IRAC and the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004). The 174 24 µ m-selected galaxies analysed here have been extracted from the publicly released 24 µ m catalogue of the GOODS-South (GOODS-S) field, which has a flux density limit of S ν (24 µ m) = 80 µ Jy, down to which this catalogue is highly reliable and complete 2 . Sources down to this flux density limit make ∼ 70% of the extragalactic 24 µ m background (Papovich et al. 2004; Dole et al. 2006). To decide which 24 µ m sources would be part of the analysis sample, I applied the following two criteria: 1) the source should have a secure (good-quality flag) spectroscopic redshift ( z spec ) from any of the multiple spectroscopic galaxy surveys of the GOODS-S field (Le F'evre et al. 2004; Szokoly et al. 2004; Teplitz et al. 2007; Vanzella et al. 2008; Popesso et al. 2009; Fadda et al. 2010; Kurk et al. 2013); 2) the redshift should be z spec > 1. I searched for spectroscopic counterparts of the 24 µ m sources within a 1.5 arcsec matching radius, and considered only one-to-one identifications. The final sample contains 174 24 µ m sources with z spec > 1, with similar numbers of galaxies at 1 . 0 < z < 1 . 5 and z ≥ 1 . 5 (89 and 85 sources, respectively). Restricting the SED study only to galaxies with secure spectroscopic redshifts ensures that all the results and conclusions in this paper are free from the uncertainties that are usually introduced by photometric redshift determinations at high redshifts. On the other hand, the multiple GOODS-S spectroscopic datasets correspond to a wide variety of selection criteria, and even include some IR spectra (Fadda et al. 2010), so the sub-sample of 24 µ m galaxies with spectroscopic redshifts is reasonably representative of the entire sample of S ν (24 µ m) > 80 µ Jy galaxies at z > 1. This has been verified by comparing the S ν (24 µ m) / S ν (i band) colour distributions of the current spectroscopic sample and the overall GOODS-S 24 µ m sample with z > 1, with spectroscopic and photometric redshifts (Caputi et al. 2006a,b; 2007). The S ν (24 µ m) flux densities versus spectroscopic redshifts z spec of the 174 galaxies are shown in Fig. 1. The dashed curve in this plot separates the LIRG and ULIRG regimes at bolometric IR luminosities L IR = 10 12 L /circledot . To compute total IR luminosities, I have considered the Bavouzet et al. (2008) monochromatic-to-total IR luminosity conversion, re-calibrated for galaxies with νL 8 µ m ν > 10 10 L /circledot (see Caputi et al. 2007). As in this work, the k-correction factors are based on a mixture of IR star-forming galaxy templates (Lagache et al. 2004) and empirical IR spectra. Although, in a strict sense, this LIRG/ULIRG separation curve corresponds to starforming galaxies, it is suitable for the purpose of illustrating the appoximate regions of the LIRG and ULIRG regimes in the S ν (24 µ m)z diagram. Independently, I made use of the publicly released GOODS-S IRAC maps and the SEXTRACTOR software (Bertin & Arnouts 1996) to extract a catalogue of IRAC sources at 3.6 and 4 . 5 µ m. I measured the photometry at 5.8 and 8 . 0 µ musing SEXTRACTOR in dual-map mode. In order to construct a multi-wavelength optical/nearIR catalogue for the IRAC-selected sources, I searched for counterparts of these sources in the GOODS-S Very Large Telescope (VLT) ISAAC near-IR images, and the Hubble Space Telescope Advanced Camera for Surveys (ACS) maps, within a matching radius of 0.5 arcsec. The finally compiled catalogue includes photometry in 11 bands: B , V , i , z , J , H , K s , [3.6], [4.5], [5.8] and [8.0]. All the magnitudes in this catalogue are total and have been corrected for galactic extinction. To obtain total magnitudes, I considered aperture magnitudes in circles of 4-arcsec (IRAC) and 2-arcsec (ISAAC and ACS) diam- er, and applied the corresponding aperture corrections. Within the catalogue of IRAC-selected sources with multi-wavelength photometry, I identified all the 174 24 µ m galaxies with spectroscopic redshifts z spec > 1. The SED fitting analysis explained in next section is based on this photometry. A total of 22 galaxies (12.5% of the sample) are out of the field of view in all the ACS bands, or at least one of the ISAAC bands. For this minority of sources, the SED fitting is based on 7,8,9 or 10 bands, depending on the case. For all the remaining galaxies, the SED fitting is based on the 11-band photometry. In case of a non-detection, a 2 σ upper limit has been considered for the flux density in the corresponding band.", "pages": [ 2, 3 ] }, { "title": "3.1. The Incidence of a Power-Law Component in the SEDs of Infrared Galaxies", "content": "I considered that the flux density S ν ( λ ) of each galaxy at observed wavelengths λ ≤ 8 µ m can be decomposed as: where the two terms on the right-hand side correspond to a stellar component and a power-law component with spectral index α , respectively. The power-law term is normalised to the 8 µ m flux density, so the constant b can vary between 0 and 1. This decomposition is similar to that proposed by Hainline et al. (2011) to analyse the rest optical/near-IR SEDs of sub-millimetre galaxies at z ∼ 2. For each of the 174 galaxies in my 24 µ m sample, I produced a suite of 31 photometric sets S stell . ν ( λ ) in the BVizJHK s [3.6][4.5][5.8][8.0] bands, corresponding to different power-law subtractions. I considered that α could take three, and b ten possible values: α = 1 . 3 , 2 . 0 and 3 . 0, and b = 0 . 0 , 0 . 1 , 0 . 2 , . . . , 1 . 0. The case with the identity line indicate the galaxies for which the SED fitting with stellar templates improves after the subtraction of a power-law component. This is the case for the majority of the 24 µ m galaxies. In particular, in 60 out of 174 cases (35%), the best SED-fitting solution on the original photometry, without a power-law subtraction, can be rejected with > 2 σ confidence. These 60 galaxies are, according to my definition, the PLCGs in the 24 µ m sample with z spec > 1. Interestingly, around two-thirds of the PLCGs (42 out of 60) lie at z spec < 1 . 5 (Fig. 4). This implies that ∼ 47% of all the LIRGs and ULIRGs at 1 . 0 < z < 1 . 5 are PLCGs, i.e. they have a significant power-law component in their rest-frame near-IR SEDs. The percentage of PLCGs at 1 . 0 < z spec < 1 . 5 is much higher than the known percentage of AGN among LIRGs and ULIRGs at these redshifts, and raises the question of whether the power-law component is actually related to an AGN presence in all these cases. I investigate this issue further in Section § 4. At z > 1 . 5, instead, the fraction of PLCGs among 24 µ m galaxies (the vast majority of which are ULIRGs) is of only ∼ 21%. I repeated the same SED modelling using the 2003 version of the Bruzual & Charlot stellar template library, and found very similar results: ∼ 38% of the 24 µ m sources with z spec > 1 are classified as PLCG. The resulting minimum χ 2 values are very similar to those obtained with the 2007 templates (the median of the minimum χ 2 differences is 0.009). All the following analysis will be based on the 2007 template library run, but no conclusion in this paper would change if I adopted the results obtained with the 2003 templates. The host galaxy properties, as determined from the best-fitting stellar templates, are similar for PLCGs and non-PLCGs. For both sub-samples, the best-fitting star formation histories correspond to single stellar populations or τ models with τ ≤ 0 . 1 Gyr in > 80% of cases. After the power-law subtraction, these models provide the best fitting for 93% of the PLCGs. The best-fitting extinction values for PLCGs (before power-law subtraction) and non-PLCGs are also very similar: the median and r.m.s. of the distributions are A V = 1 . 4 ± 0 . 6 and 2 b = 0 . 0 and any α value corresponds to the original photometry S stell . ν ( λ ) = S ν ( λ ). The three adopted α values are representative of the IR power-law indices that characterise different types of AGN, with different hot-dust components (e.g. Alonso-Herrero et al. 2003, 2006; Polletta et al. 2006; Honig et al. 2010). I performed the SED modelling on the 31 photometric sets S stell . ν ( λ ) of each galaxy, using a customised χ 2 -minimisation SED-fitting code that incorporates the 2007 version of the Bruzual & Charlot synthetic stellar template library, with solar metallicity (Bruzual & Charlot 2003; Bruzual 2007). These templates correspond to different star formation histories: a single stellar population, and different exponentiallydeclining star-formation histories with τ = 0 . 1 through 5 Gyr. In all cases, the redshift of the galaxy has been fixed to the known z spec value. To account for internal extinction, I convolved the stellar templates with the Calzetti et al. (2000) reddening law, allowing for 0 . 0 ≤ A V ≤ 3 . 0 with a step of 0.1. Stellar masses are obtained in the output of the same SED-fitting code. The technique is illustrated in Figure 2: once a powerlaw component is subtracted from the original photometry (open circles), a new photometric dataset is obtained (filled circles). For each galaxy, I have tested 31 possible power-law subtractions (including the original photometry), as explained above. I have then fitted stellar templates to all these photometric sets, and compared the resulting minimum χ 2 values. Figure 3 shows the minimum reduced χ 2 value after power-law subtraction (i.e. the absolute minimum obtained considering all the different ( α , b ) combinations), versus the minimum reduced χ 2 value obtained with the original photometry, for each galaxy. The points lying on the identity line correspond to galaxies for which the original photometry, without a power-law subtraction, produces the absolutely best fitting. The points below", "pages": [ 3, 4 ] }, { "title": "A V = 1 . 4 ± 0 . 7, respectively. After power-law subtraction, the PLCGs have A V = 1 . 0 ± 0 . 6.", "content": "Figure 5 shows the extinction A V versus spectroscopic redshifts for the 60 PLCGs, classified according to their power-law indices α . The non-PLCGs are also shown as a reference. A bit more than a half of the PLCGs (55%) are characterised by a power-law component with α = 3, while the remaining PLCGs have lower best-fitting power-law indices, i.e. α = 1 . 3 or 2. These indices are related to the temperature of the hot-dust component in the galaxy, which is higher for lower α values (cf. Section § 4.4). The effect of a decrease in the best-fitting A V value is especially evident for the α = 1 . 3 PLCGs: a shallow power-law component can mimic the effect of additional reddening. After power-law subtraction, the majority of the α = 1 . 3 and 2 PLCGs have A V ≤ 1, while most of the α = 3 PLCGs have A V > 1. The normalisation factor of the power-law component is b ≥ 0 . 50 for virtually all the PLCGs, which means that at least 50% of the 8 µ m flux density is in the powerlaw component. For around a half of the PLCGs, this contribution is ≥ 80%. This confirms that the PLCG definition adopted here, based on the > 2 σ improvement of the SED fitting, truly selects sources for which the power-law component makes an important contribution 2 to the rest-frame near-IR light of the galaxy. For galaxies at z ∼ 1 . 4, the emission from polycyclic aromatic hydrocarbons (PAHs) at rest λ rest ∼ 3 . 3 µ m will enter the observed 8 µ m band. Although this is a relatively faint PAH emission feature, it can potentially mimic the effect of a steep power-law component in the galaxy SED. To assess the importance of this effect on the PLCG selection, I have performed the SED fitting on additional photometric sets for each galaxy, with powerlaw subtractions corresponding to larger spectral indices, namely, α = 3 . 5 and 4. Within the total sample, only 13 galaxies appear to be PLCGs with such large bestfitting spectral index. However, their redshifts are not concentrated around z ∼ 1 . 4 or any other specific redshift, so the steeper α values are unlikely the effect of emission features. In fact, 11 out of these 13 galaxies have been recognised as PLCGs with α = 3 in my original analysis. Only two galaxies appear as new PLCGs: one at z spec = 2 . 810 (corresponding to an X-ray luminous AGN), and another one at z spec = 1 . 550 (for which the 3 . 3 µ m PAH emission could be partly within the 8 µ mfilter wavelength coverage). These results indicate that: a) the 3 . 3 µ m PAH emission plays a minor role in the overall IR SED for most IR galaxies; b) considering α > 3 values for the power-law subtraction produces a virtually negligible effect on the identified PLCG sample. To investigate whether the high fraction of PLCGs is truly a characteristic of the 24 µ m galaxy sample, I performed a similar SED fitting analysis on a control sample of 247 galaxies selected from the IRAC catalogue described in Section § 2, also with secure z spec > 1, but which are not 24 µ m-detected (i.e. S ν (24 µ m) < 80 µ Jy). These galaxies have been selected to have a similar stellar mass distribution as the 24 µ m galaxies ( M stell . > ∼ 10 10 M /circledot , based on the SED modelling of the original photometry). For each of these galaxies, I have produced 31 sets of photometric variations in the same manner explained above, and fitted the resulting SEDs with stellar templates in all cases. The comparison of minimum χ 2 values with and without power-law subtraction is shown in Fig. 6. For the control sample, the percentage of galaxies that significantly improve their SED fitting with stellar templates after a power-law subtraction is a factor of two smaller than for the 24 µ m galaxies: a pure stellar SED fitting can be discarded with > 2 σ confidence in only 18% of cases (44 out of 247 galaxies). These 44 PLCGs in the control sample have best-fitting α values shared in similar proportions as for the 24 µ m-detected PLCGs: 50% have best-fitting α = 3, while the other 50% have α = 1 . 3 or 2. These results obtained on the control sample indicate that the incidence of PLCGs among mid-IR-selected galaxies is twice more important than among other similarly massive galaxies at z > 1.", "pages": [ 5, 6 ] }, { "title": "3.2. The Effect of a Power-Law Subtraction on the Derived Stellar Masses", "content": "Figure 7 shows the distribution of corrected-to-original stellar mass ratios for the 60 24 µ m-selected PLCGs, classified according to their best-fitting power-law index. The corrected stellar masses are those derived from the SED modelling with stellar templates after the bestfitting power-law subtraction. The inspection of these distributions shows that the correction on the stellar masses is non-trivial: the exact correction needs to be derived on a case-by-case basis. However, for > 80% (90%) of the PLCGs the correction is within a factor of ∼ 1 . 5 (2.0). The widest correctionfactor distributions are those corresponding to power-law spectral indices α = 1 . 3 and 2, for which the stellar mass ratios range between < ∼ 0 . 1 and ∼ 1 . 8. The median values of these stellar mass ratios are 0.67 and 0.85, respectively. The corrections on the stellar masses are the smallest for the case α = 3 (the median of the ratio distribution is 0.92). This behaviour is expected, as the subtraction of such a steep power-law has a rapidly declining effect on the photometry at wavelengths shorter than 8 µ m, i.e the overall SED shape is not substantially modified except for a single photometric point. It may a priori seem surprising that the SED fitting for some galaxies yields larger stellar masses after the PL subtraction. This effect is produced because the best-fitting stellar template properties (star formation history, age and extinction) are usually different for the PL-subtracted photometry to those for the original pho- tometry, and, in some cases, the resulting overall normalisation factors (i.e. mass-to-light ratios) are larger. For 5 out of the 60 PLCGs, all with best-fitting α = 1 . 3, the corrected stellar mass is less than 1 / 10 of the stellar mass derived from the original photometry. These cases correspond to 'pure power-law sources', and all except one of them are identified as X-ray luminous AGN with L X > 10 43 erg s -1 in the Chandra 4 Ms catalogue for the GOODS-S field (Xue et al. 2011). The only source that is not X-ray detected is the highest-redshift of these pure power-law sources, with S ν (24 µ m) ≈ 286 µ Jy and z spec = 3 . 356, so it is likely an intrinsically luminous AGN as well. For these sources, a stellar mass calculation based on the original photometry should clearly be avoided. But, as said above, in all other cases the correction to the stellar mass is within a factor of two. This result validates the approach of obtaining approximate stellar mass estimates for e.g. low-luminosity AGN, using the original uncorrected photometry, that is followed by some authors (e.g. Silverman et al. 2009; Rosario et al. 2013).", "pages": [ 6 ] }, { "title": "4.1. Independent AGN Diagnostics: X-ray Detections", "content": "In order to investigate whether the PLCGs actually host AGN, I searched for counterparts of these sources in the publicly available, ultra-deep Chandra 4Ms X-ray catalogue for the GOODS-S field (Xue et al. 2011). This diagnostic is very useful to assess to which extent the PLCG criterion can be used to identify AGN in the general case that such deep X-ray catalogues are not available (to date, X-ray maps of such a depth have only been obtained for the GOODS-S field). Fig. 8 shows the S ν (24 µ m)z diagram for the 24 µ m galaxies in the z spec > 1 sample that are identified with X-ray sources in the Chandra 4Ms catalogue. The PLCGs are highlighted with couloured circles. There are a total of 59 24 µ m galaxies identified with an X-ray source, 48 out of which correspond to X-ray-classified AGN, and 11 are X-ray 'normal galaxies'. The Xray classification is based on the intrinsic X-ray luminosities, X-ray colours (hardness ratios), and derived neutral gas column densities (e.g. Bauer et al. 2004; Xue et al. 2011). Although the X-ray-based AGN classification is among the most secure, the criteria typically adopted to classify X-ray sources could still result in some low-luminosity or highly-obscured AGN being classified as normal galaxies (see Xue et al. 2011 and references therein). The analysis of Fig. 8 shows the following. In starburst galaxies, supernova explosions can produce hot-gas bubbles with significant X-ray emission. The resulting shock waves can heat the surrounding dust grains up to sublimation temperatures T > ∼ 1000 K. However, if the X-ray luminosities of X-ray galaxies were only a product of supernova explosions, they would correlate with the IR luminosities, as both trace the on-going star formation rates (Mas-Hesse et al. 2008). This is not the case for the PLCGs at z < 1 . 5 , for which two sources can differ in more than a factor of three in IR luminosity, but both be classified as X-ray normal galaxies with similar X-ray luminosities. This becomes clear by inspection of Fig. 4, where some bright 24 µ m sources at 1 . 0 < z < 1 . 5 are neither PLCGs nor X-ray galaxies at all. Thus, the SED power-law excess of these PLCGs is not simply tracing high star formation rates. At z < 1 . 5, it is interesting to note the different distributions of best-fitting spectral indices α among the X-ray AGN and normal galaxies that are recognised as PLCGs: 9 out of 13 (69%) X-ray AGN / PLCGs have α = 1 . 3 or 2, while only 2 out of 7 (29%) of the X-ray galaxy / PLCGs have α = 1 . 3 or 2. Or, equivalently, only 31% of the Xray AGN classified as PLCGs have α = 3, while 71% of the X-ray galaxies classified as PLCGs have α = 3. I will carry on this discussion in Section § 4.3, where I will argue that the spectral index α is a good discriminator of PLCGs of different nature. To investigate these two possible explanations, I consider those 24 µ m galaxies at z ∼ 2 that form part of the Fadda et al. (2010) IR galaxy sample. Pozzi et al. (2012) have studied the mid-/far-IR SED decomposition of these galaxies, and found that nine out of 24 of them have a significant AGN component. Only three out of these nine sources are recognised as PLCGs within my sample. This result suggests that data beyond observed 8 µ m are necessary to reveal more effectively an AGN presence in galaxies at z ∼ 2. In Section § 4.4, I will show that very hot dust temperatures ( T dust > ∼ 1300 K) are necessary to generate a power-law SED component detectable in the IRAC bands at z > 1 . 5. Such hot dust is typically found in AGN inner dusty tori (Barvainis 1987; Mor et al. 2012), but some AGN are characterised by lower dust temperatures, or the inner torus may be less exposed (e.g. Schartmann et al. 2005). This will restrict the fraction of AGN that can be identified with my PLCG criterion, which is based on SED fitting up to observed 8 µ m, at z ∼ 2.", "pages": [ 6, 7 ] }, { "title": "4.2. PLCGs Non-Detected in X-Rays: Stacking Analysis", "content": "To probe whether the X-ray non-detection of half of the PLCGs at 1 . 0 < z < 1 . 5 is simply due to the sensitivity limit of the Chandra 4Ms X-ray survey, I performed a stacking analysis of these sources in X rays. Fig. 9 shows the stacked X-ray images of the PLCGs that are not individually detected in X rays at 1 . 0 < z < 1 . 5. In the hard X-ray band (2.0-8.0 keV), the stacking does not produce any detection, while in the soft band (0.5-2.0 keV) a detection with a > 6 σ significance is obtained. This stacked signal confirms that these sources are X-ray emitters, but their fluxes are below the sensitivity limits of the 4Ms Chandra survey. Nuclear activity cannot be discarded as an agent partly responsible for the soft X-ray emission in these sources, but it is probably not the main cause: on average they have log 10 ( f X /f R ) ≈ -2 . 7, too low a value for an AGN classification (Xue et al. 2011). Around two thirds of the stacked PLCGs have spectral index α = 3 which, as I discuss below, should mainly correspond to star-forming galaxies with hot interstellar media (ISM) at 1 . 0 < z < 1 . 5. The stacking of the α = 3 PLCGs alone yields a 4 . 8 σ detection in the (0.5-2.0 keV) band, which corresponds to an average flux density f X ≈ 1 . 9 × 10 -17 erg cm -2 s -1 (after correction for galactic extinction; see Stark et al. 1992). At the median redshift of these sources z med = 1 . 13, this corresponds to a luminosity L X ≈ 1 . 3 × 10 41 erg s -1 , assuming a photon index Γ = 2, which is appropriate for star-forming galaxies (e.g. Lehmer et al. 2008). Using the relation calibrated by Mineo et al. (2013), I find that this corresponds to an instantaneous star formation rate SFR ≈ (37 ± 4) M /circledot yr -1 . This is significantly lower than the SFR derived from the average total IR luminosities of these sources, which is SFR ≈ 90 M /circledot yr -1 , as obtained using the SFRL IR relation given by Kennicutt (1998). This difference indicates the presence of internal absorption. To reconcile both SFR values, I find that these galaxies should have an average column density N H ≈ 4 . 5 × 10 21 cm -2 . In any case, as commented above, one has to be careful by considering the PLCG X-ray luminosity as a tracer of the instantaneous star formation rate, as many of these galaxies are likely composite star-forming/AGN systems. Also, the X-ray emission may be the result of energetic processes in the galaxy ISM that are not simply related to the on-going star formation rate. Some of these effects probably account for the dispersion observed in the L X -SFR relation.", "pages": [ 7, 8 ] }, { "title": "4.3. Morphologies of PLCGs and X-Ray Galaxies at 1 . 0 < z < 1 . 5", "content": "Further clues about the nature of PLCGs, particularly at 1 . 0 < z < 1 . 5, can be obtained by comparing their morphologies. Figure 10 shows HST /ACS z -band postage stamps of 15 PLCGs at 1 . 0 < z < 1 . 5, grouped in three different categories: X-ray detected sources, classified as AGN and Galaxies, and PLCGs that are not individually X-ray detected. The three classes display objects with different morphologies, including galaxies with compact and regular matter concentrations, and galaxies with very disturbed morphologies. The latter are seemingly the product of galaxy mergers, or are characterised by other violent processes, such as the radial ejection of material (see the X-ray galaxy at the right end of the middle row). The striking conclusion that can be extracted from Fig.10 is the trend between the PLCG morphology and the value of the best-fitting spectral index α : those PLCGs with α = 1 . 3 or 2.0 show quite a regular morphology, while the PLCGs with α = 3 . 0 are irregular and show signs of different kinds of galaxy interactions. This trend is independent of the X-ray detection and classification. This fact suggests that the spectral index α contains information about the nature of the source: in the PLCGs with α = 1 . 3 or 2.0, the SED power-law component is very likely the signature of nuclear activity, while in the α = 3 PLCGs the power law appears to be the consequence of a hot ISM produced by the violent galaxy dynamics. Shocks of hot gas can compress and heat up the ISM dust to several hundred Kelvin, and even up to sublimation temperatures ( T > ∼ 1000 K; Osterbrock & Ferland 2006). This hot dust, not related to an AGN origin, will also manifest itself as a power-law component in the rest near-IR galaxy SED. Of course, in some cases, an AGN and a host galaxy with a disturbed ISM may co-exist, producing X-ray confirmed AGN that are PLCGs with α = 3 (upper right stamps in Fig. 10; see also Kocevski et al. 2012 for a discussion on AGN morphology).", "pages": [ 8 ] }, { "title": "4.4. Estimated Temperatures of the PLCG Hot-Dust SED Components", "content": "I estimated the dust temperatures necessary to produce a power-law component in the galaxy SEDs, assuming a greybody function for the dust emission: where the second factor on the right-hand side of equation (2) corresponds to the black-body emission at temperature T , and h and k are the Planck's and Boltzmann's constants, respectively. I assumed β = 1 . 5, as it is common in the literature. Fig. 11 shows the observed flux density ratios S ν (8 µ m) / S ν (4 . 5 µ m) produced by greybody emitters at different temperatures, as a function of redshift. The horizontal solid lines indicate the fixed, observed flux density ratios produced by power-laws with the three spectral indices considered in this paper, i.e. α = 1 . 3, 2.0 and 3.0. Two main results can be derived from this simple diagram. Firstly, to generate a power-law component with α = 1 . 3 or 2 at 1 . 0 < z < 1 . 5, temperatures as high as 900 -1000 K are needed. Instead, for a power-law with spectral index α = 3 . 0, temperatures T ∼ 700 -800 K are sufficient. This may explain why most of the X-ray AGN, for which dust temperatures close to the sublimation point are expected, are associated with PLCGs with spectral indices α = 1 . 3 or 2, at 1 . 0 < z < 1 . 5. The PLCGs with spectral index α = 3 . 0 can be explained with a shock-heated ISM, with no need of dust getting close to the sublimation point (the exact dust sublimation temperature depends on the dust composition, but it is typically T > ∼ 1000 K). The existence of hot interstellar dust in IR galaxies has been previously discussed in the literature. Particularly, Lu et al. (2003) found that a colour dust temperature of ∼ 750 K can account for the near-IR excess observed in some local IR normal galaxies, similarly to what I find here for LIRGs at 1 . 0 < z < 1 . 5. Secondly, at redshifts z > 1 . 5 -2 . 0, producing a powerlaw component in the observed IRAC SED requires hot dust with temperatures T > ∼ 1300 K, for any of the spectral indices. This mostly explains why the incidence of PLCGs is lower at z > 1 . 5, as only sources with such an extremely hot dust component will be classified as PLCG with the criterion adopted in this paper.", "pages": [ 8, 9 ] }, { "title": "5. IRAC COLOUR-COLOUR DIAGNOSTIC", "content": "The use of Spitzer /IRAC colour-colour diagrams has proven to be quite successful for identifying AGN among IR-selected galaxies (Lacy et al. 2004; Stern et al. 2005). Over the years, these colour selections have been refined to be more reliable for AGN identification among fainter IR sources (e.g. Donley et al. 2012). Here I analyse how the PLCGs, and other 24 µ m galaxies at z spec > 1, are classified in an IRAC colour-colour diagram. Fig. 12 shows the IRAC colour-colour diagram for all the 24 µ m galaxies at z spec > 1, with the PLCGs and X-ray sources highlighted. The colours shown here are overall colours, i.e. there is no separation of stellar and power-law SED components. One can see that the range of overall IRAC colours displayed is quite wide, with some 24 µ m galaxies having much redder IRAC colours than others. PLCGs, in particular, display a wider range of colours than the non-PLCGs, as there are very few non-PLCGs with very red IRAC colours log 10 ( S 8 . 0 /S 4 . 5 ) > ∼ 0 . 2 and log 10 ( S 5 . 8 /S 3 . 6 ) > ∼ 0 . 1. For PLCGs, I find no correlation between the IRAC colours and the spectral index of the SED power-law component. The solid lines in Fig. 12 delimit the revised colour region for AGN selection proposed by Donley et al. (2012). More than 80% of the 24 µ mgalaxies that lie within that region are PLCGs or X-ray AGN, indicating that indeed the proposed colour selection is highly reliable. However, these colour criteria are too strict and miss a substantial fraction of PLCGs, many of which are X-ray confirmed AGN. The incompleteness of this colour selection was discussed by Donley et al. (2012), who argued that their criteria quickly lose completeness for AGN with Xray luminosities L X < 10 44 erg s -1 . Within the current sample, all the X-ray-detected sources in the Donley et al. (2012) region have L X > ∼ 5 × 10 43 erg s -1 . There are three galaxies that lie within the Donley et al. AGN colour-colour region, but which are neither identified as PLCGs, nor X-ray detected. One of them is a z spec = 1 . 045 galaxy for which the SED fitting with a power-law subtraction just lies below the 2 σ criterion for PLCG identification. So, this galaxy could still have a buried AGN. The other two galaxies are at redshifts z spec = 2 . 442 and 2.577. For these, the stellar bump centred at rest-frame ∼ 1 . 6 µ mis within the observed 5 . 8 µ m filter, producing a red S 5 . 8 µ m /S 3 . 6 µ m colour. These two galaxies are close to the S 8 . 0 µ m /S 4 . 5 µ m lower boundary of the Donley et al. colour wedge, so their presence within the wedge may simply be due to photometric scattering (note the size of the error bars). Galaxy colour selections are attractive for their simplicity, but no criterion has been found to safely identify the bulk of the AGN population at any redshift. The PLCG criterion proposed here offers a suitable alternative, which is particularly successful at identifying AGN of different luminosities at 1 . 0 < z < 1 . 5. Note that it would not be enough to extend the Donley et al. (2012) to bluer colours, because most of the PLCGs display similar IRAC colours as other IR sources which do not have a significant power-law component in their SEDs, so an extended colour selection would produce a highly contaminated AGN sample.", "pages": [ 9, 10 ] }, { "title": "6. SUMMARY AND CONCLUSIONS", "content": "In this paper I have studied the rest-frame UV/optical/near-IR ( λ rest < ∼ 3 µ m) SEDs of 174 24 µ mselected galaxies with secure spectroscopic redshifts z spec > 1, analysing explicitly the presence and significance of a hot-dust, power-law component. I modelled the residual light produced after subtracting different possible power-law components, using stellar templates, on 11 broad-bands from the B -band through 8 µ m. The subtracted power-law components are characterised by three different spectral indices α = 1 . 3, 2.0 and 3.0, and different normalisation weights. I found that in 60 out of 174 (35%) cases, the SED fitting with stellar templates without any previous powerlaw subtraction can be rejected with > 2 σ confidence, i.e. 35% of the 24 µ m galaxies at z > 1 are characterised by a significant power-law component in their rest near-IR SED. I referred to these galaxies as PLCGs. This high percentage of PLCGs at z > 1 appears to be inherent to the IR bright galaxy population. A similar analysis performed on a control sample of 247 IRAC-selected galaxies with z spec > 1 and similar stellar masses as the 24 µ mgalaxies, but which are not 24 µ m-detected, shows that the incidence of PLCGs is of only 18%. The incidence of PLCGs among 24 µ mgalaxies is much higher at 1 . 0 < z < 1 . 5 (47%) than at z > 1 . 5 (21%). The lower incidence at z > 1 . 5 is explained by the fact that the presence of very hot dust ( T dust > ∼ 1300 K) is necessary to have a significant power-law component manifested at observed wavelengths λ ≤ 8 µ m. Therefore, only sources with the most extreme dust conditions would be recognised with the PLCG criterion at these higher redshifts. The impact of a significant power-law component in the rest near-IR SED on the derived galaxy stellar masses needs to be analysed on a case-by-case basis, but it is small in general. The correction for stellar masses is within a factor of 1.5 in > 80% of cases. Only in a small percentage ( < 10%) of the PLCGs, basically all corresponding to powerful AGN, the correction to the stellar mass is so large that a stellar mass calculation based on the original photometry should clearly be avoided. A key issue is understanding whether the PLCGs are truly related to the presence of an AGN, particularly at 1 . 0 < z < 1 . 5, where the PLCG incidence is very high. To investigate this, I studied different properties of these sources. Firstly, I looked for identifications of the PLCGs in the ultra-deep 4Ms X-ray catalogue for the CDFS. I found that 77% of the X-ray AGN within my 24 µ m sample with 1 . 0 < z spec < 1 . 5 are classified as PLCGs. This indicates that the PLCG criterion provides a high completeness in selecting AGN at these redshifts. Interestingly, 78% of the X-ray normal galaxies within my 24 µ msample with 1 . 0 < z spec < 1 . 5 are also PLCGs. But their PLCG classification or X-ray luminosities do not correlate with the 24 µ m luminosities, so the X ray emission or the power-law SED excess is not simply a tracer of high star formation rates. A stacking analysis of the PLCGs that are not individually X-ray detected at 1 . 0 < z < 1 . 5 reveals that they have on average similar properties as X-ray-detected normal galaxies. The stacked X-ray images yield a significant detection in the soft X-ray band (0 . 5 -2 . 0 keV), and no signal in the hard X-ray band (2 . 0 -8 . 0 keV). Around two thirds of the stacked PLCGs have a bestfitting power law with α = 3. The visual inspection of optical HST /ACS images has proven very useful to shed light on the nature of the PLCGs at 1 . 0 < z < 1 . 5. I found a clear trend relating the PLCG morphology and the spectral-index α of the best-fitting SED power-law component: PLCGs with α = 1 . 3 or 2.0 look regular and typically have a nuclear matter concentration. Instead, the PLCGs with α = 3 . 0 typically show irregular morphologies, indicating a disturbed galaxy dynamics, which in some cases suggest the presence of galaxy mergers. This morphology trend is independent of the PLCG X-ray detection and classification. The α = 3 . 0 power-law component appears then to be a signature of quite extreme ISM conditions, where the interstellar dust is heated to temperatures T > ∼ 700 K by the gas shocks produced in the disturbed ISM. Of course, this does not preclude the α = 3 . 0 PLCGs to also host AGN. Actually, the presence of an AGN is confirmed in some of them by the X-ray data. But an AGN presence does not appear to be necessary to produce a significant α = 3 . 0 power-law component. For the PLCGs with α = 1 . 3 or 2.0, instead, higher dust temperatures are necessary T > ∼ 900 K. This property, along with the regular morphologies, take to the direct conclusion that these sources must host an AGN. One could speculate that the α = 3 . 0 PLCGs may all be on the way to form an AGN as well, but this hypothesis cannot be tested with the current data. Considering that all the PLCGs with α = 1 . 3 or 2.0 in my sample contain AGN, and adding all the other X-ray classified AGN within my 24 µ m sample, I obtain that a total of 30% of the 24 µ m sources at 1 . 0 < z < 1 . 5 host AGN. This should be considered as a lower limit, as some of the α = 3 . 0 PLCGs that are not individually X-ray detected could also contain AGN. If all of them had AGN, that would give an upper limit of ∼ 52% on the AGN fraction among LIRGs and ULIRGs at 1 . 0 < z < 1 . 5. Very recently, different works have concluded on a higher fraction of AGN among IR galaxies than previously known at z < 1, obtaining percentages of 37-50% (e.g. Alonso-Herrero et al. 2012; Juneau et al. 2013). The results of my study are in line with such conclusions. At z > 1 . 5, a similar criterion as that used at z < 1 . 5 would indicate that ∼ 40% of the 24 µ m galaxies contain an AGN, although this figure is largely based on the X-ray detections, rather than the PLCG incidence, which is relatively low (for a comparison, Pozzi et al. 2012 derived a fraction of 35% based on the mid-/far-IR SED analysis of 24 ULIRGs at z ∼ 2). In this work, I have calibrated the PLCG analysis technique making use of galaxies with secure spectroscopic redshifts. A complete analysis of the applicability of this technique with the simultaneous determination of photometric redshifts ( z phot ) will be presented in a future paper. A preliminary run of my SED fitting code leaving the redshift as a free parameter indicates that, in fact, the consideration of a power-law subtraction produces some improvement in the overall z phot -z spec comparison, reducing the number of catastrophic outliers. This suggests that the identification of PLCGs can be done, and the power-law subtraction is actually recommendable, when attempting a z phot determination. Although less straightforward than a simple IRAC colour-colour selection, the PLCG identification criterion introduced here is much more complete to select AGN of different luminosities, especially in the redshift range 1 . 0 < z < 1 . 5. Keeping only those PLCGs with spectral index α = 1 . 3 and 2 should result in the best compromise between completeness and reliability, as most X-ray normal galaxies and X-ray non-detections are among the PLCGs with α = 3. In the absence of very deep X-ray data, the PLCG selection with spectral index segregation offers the most efficient method to identify the AGN presence in any large galaxy sample. Based on observations made with the Spitzer Space Telescope , which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Also based on observations undertaken at the European Southern Observatory (ESO) Very Large Telescope (VLT) under different programmes; the NASA/ESA Hubble Space Telescope , obtained at the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Inc. (AURA), under NASA contract NAS 5-26555; and the Chandra X-ray Observatory , which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. I thank Almudena Alonso-Herrero, Maurilio Pannella, Brigitte Rocca-Volmerange, and John Silverman for very useful discussions, and an anonymous referee for a very constructive report. I am also grateful to the GOODS Team for the public release of their superb data products. Facilities: Spitzer (IRAC,MIPS), HST (ACS), VLT (ISAAC, FORS2, VIMOS).", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "Donley, J. L., Rieke, G. H., P'erez-Gonz'alez, P. G., Rigby, J. R., Alonso-Herrero, A., 2007, ApJ, 660, 167 Fadda, D., Yan, L., Lagache, G. et al., 2010, ApJ, 719, 425 Fazio, G. G., Hora, J. L., Allen, L. E. et al., 2004, ApJS, 154, 10 Fiore, F., Grazian, A. Santini, P. et al., 2008, ApJ, 672, 94 Kurk, J., Cimatti, A., Daddi, E. et al., 2013, A&A, 549A, 63 Lacy, M., Storrie-Lombardi, L. J., Sajina, A. et al., ApJS, 154, 166 Lacy, M., Petric, A., Sajina, A., et al., 2007, AJ, 133, 186 Lagache, G., Dole, H., Puget, J.-L. et al., 2004, ApJS, 154, 112 Le F'evre, O., Vettolani, G., Paltani, S. et al., 2004, A&A, 428, 1043 Le Floc'h, E., Papovich, C., Dole, H. et al., 2005, ApJ, 632, 169 Lehmer, B. D., Brandt, W. N., Alexander, D. M. et al., 2008, ApJ, 681, 1163 Lu, N., Helou, G., Werner, M. W. et al., 2003, ApJ, 588, 199 Mainieri, V., Bongiorno, A., Merloni, A. et al., 2011, A&A, 535A, 80 Mas-Hesse, J. M., Ot'ı-Floranes, H., Cervi˜no, M., 2008, A&A, 483, 71 Matsuta, K., Gandhi, P., Dotani, T. et al., 2012, ApJ, 753, 104 Merloni, A., Bongiorno, A., Bolzonella, M. et al., 2010, ApJ, 708, 137 Mineo, S., Gilfanov, M., Sunyaev, R., 2013, MNRAS, submitted, arXiv: 1207.2157 Mor, R., Netzer, H., 2012, MNRAS, 420, 526 Mullaney, J. R., Pannella, M., Daddi, E. et al., 2012, ApJ, 419, 95 Murakami H., Baba, H., Barthel, P. et al., 2007, PASJ, 59S, 369 Nardini, E., Risaliti, G., Salvati, M. et al., 2008, MNRAS, 385, L130 Papovich, C., Dole, H., Egami, E.et al., 2004, ApJS, 154, 70 Petric, A. O., Armus, L., Howell, J. et al., 2011, ApJ, 730, 28 Pilbratt, G. L., 2003, SPIE, 4850, 586 Polletta, M., Wilkes, B. C., Siana, B. et al., 2006, ApJ, 642, 673 Popesso, P., Dickinson, M., Nonino, M. et al., A&A, 494, 443 Pozzi, F., Vignali, C., Gruppioni, C. et al., 2012, MNRAS, 423, 1909 Puget, J.-L., Abergel, A., Bernard, J.-P., Boulanger, F., Burton, W. B., D'esert, F.-X., Hartmann, D., 1996, A&A, 308, L5 Rieke, G. H., Young, E. T., Engelbracht, C. W. et al., 2004, ApJS, 154, 25 Rigby, J. R. et al., 2004, ApJS, 154, 160 Rosario, D. J., Mozena, M., Wuyts, S. et al., 2013, ApJ, 763, 59 Sajina, A. et al., 2007, ApJ, 664, 713 Salpeter, E. E., 1955, ApJ, 121, 161 Schartmann, M., Meisenheimer, K., Camenzind, M., Wolf, S., Henning, T., 2005, A&A, 437, 861 Silverman, J. D., Lamareille, F., Maier, C. et al., 2009, ApJ, 696, 396 Stark, A. A., Gammie, C., Wilson R. W. et al., 1992, ApJS, 79, 77 Stern, D., Eisenhardt, P., Gorjian, V. et al., 2005, ApJ, 631, 163 Symeonidis, M., Georgakakis, A., Seymour, N. et al., 2011, MNRAS, 417, 2239 Szokoly, G. P., Bergeron, J., Hasinger, G. et al., 2004, ApJS, 155, 271 Teplitz, H. I., Desai, V., Armus, L. et al., 2007, ApJ, 659, 941 Vanzella, E., Cristiani, S., Dickinson, M. et al., 2008, A&A, 478, 83 Werner, M. W. et al., 2004, ApJS, 154, 1 Xue, Y. Q., Luo, B., Brandt, W. N. et al., 2011, ApJS, 195, 10", "pages": [ 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "The Seyfert 1 galaxy Ark 564 was observed with Chandra high energy transmission gratings for 250 ks. We present the high resolution X-ray spectrum that shows several associated absorption lines. The photoionization model requires two warm absorbers with two different ionization states ( logU = 0 . 39 ± 0 . 03 and logU = -0 . 99 ± 0 . 13), both with moderate outflow velocities ( ∼ 100 km s -1 ) and relatively low line of sight column densities ( logN H = 20 . 94 and 20 . 11 cm -2 ). The high ionization phase produces absorption lines of O vii , O viii , Ne ix , Ne x , Mg xi , Fe xvii and Fe xviii while the low ionization phase produces lines at lower energies (O vi & O vii ). The pressure-temperature equilibrium curve for the Ark 564 absorber does not have the typical 'S' shape, even if the metallicity is super-solar; as a result the two warm-absorber phases do not appear to be in pressure balance. This suggests that the continuum incident on the absorbing gas is perhaps different from the observed continuum. We also estimated the mass outflow rate and the associated kinetic energy and find it to be at most 0 . 009% of the bolometric luminosity of Ark 564. Thus it is highly unlikely that these outflows provide significant feedback required by the galaxy formation models.", "pages": [ 1 ] }, { "title": "A Two-Phase Low-velocity Outflow in the Seyfert 1 Galaxy Ark 564", "content": "A. Gupta and S. Mathur 1 Astronomy Department, Ohio State University, Columbus, OH 43210, USA [email protected]", "pages": [ 1 ] }, { "title": "Y. Krongold", "content": "Instituto de Astronomia, Universidad Nacional Autonoma de Mexico, Mexico City, (Mexico)", "pages": [ 1 ] }, { "title": "F. Nicastro", "content": "Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02138, USA Osservatorio Astronomico di Roma-INAF, Via di Frascati 33, 00040, Monte Porzio Catone, RM, (Italy)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Outflows are ubiquitous in AGNs, manifested by high-ionization absorption lines in Xrays and UV (Reynolds 1997, Crenshaw et al. 2003, and references therein) and are perhaps related to the accretion process (e.g., Proga 2007). Understanding outflows is therefore as important as understanding accretion itself. The X-ray absorbers, commonly known as warm absorbers (WAs), have typical ionization parameter log ξ 1 of 0 -2 erg s -1 , a column density of N H = 10 20 -10 22 cm -2 and an outflow velocity of 100 -1000 km s -1 , produced by warm ionized gas ( T ∼ 10 4 -10 6 K; Krongold et al. 2003). The WAs are usually detected in the 0 . 3 -2 keV soft X-ray band by absorption lines of Oxygen (O viii , O vii , O vi ), Iron (Fe viixii and Fe xvii-xxii ), and other highly ionized elements. Transitions by C iv , N v , and O vi are observed in both the X-ray and UV spectra of these sources with similar outflow velocities, suggesting a connection between the narrow absorption line systems in the UV and the X-rays WA (Mathur et al. 1994; 1995, Kaspi et al. 2002, Krongold et al. 2003). Several phenomenological models have tried to explain AGN warm absorber spectra showing multiple velocity components of multiple lines, and after years of effort a consensus is growing. In majority of the cases, if not all, physical properties and kinematics of the absorber are well determined and it can be described by at least two discrete ionization components (Detmers et al. 2011, Holczer & Behar 2012). These components are consistent with the same outflow velocity and appear to be in pressure equilibrium, and so likely emerge from a multiphase wind (e.g., Krongold et al. 2003; 2005; 2007, Netzer et al. 2003, Cardaci et al. 2009, Andrade-Velazquez et al. 2010). The low-ionization phase (LIP) of the wind produces UV and X-ray absorption lines, but the high-ionization phase (HIP) is seen only in X-rays. Despite ubiquitous detection and successful modeling of WA spectra by multiple absorbing components, very little is still known about their geometry and dynamical strength. Where do these outflows originate? Proposed locations span a wide range, of a factor of 10 6 in radial distance from the central ionizing source: the accretion disk (as suggested by the accretion disk wind models; Proga & Kallman 2004), the broad line region (Kraemer et al. 2005), the obscuring torus (Dorodnitsyn et al. 2008, Krolik & Kriss 2001, Blustin et al. 2005) and to the narrow line region (Behar et al. 2003, Crenshaw et al. 2009). In principle, these outflows could potentially provide a common form of AGN feedback required by theoretical models of AGN-galaxy formation (Silk & Rees 1998, Wyithe & Loeb 2003, Fabian 2012, and references therein), although estimating their mass outflow rate and the kinetic energy outflow rate depends critically on the location of WAs. Some recent studies of WAs found that the typical outflow velocity is a small fraction of the escape velocity and that WAs do not carry sufficient mass/energy/momentum to be efficient agents of feedback (Mathur et al. 2009, Krongold et al. 2007; 2010). With the goal of self consistent analysis and modeling of grating spectra of WAs, we present here the results of our analysis of Chandra archival data of Ark 564. The analysis of one Chandra observation of this source done in 2000 (50 ks) has been published by Matsumoto et al. (2004); here we present new data of the three 2008 Chandra observations (250 ks total).", "pages": [ 2, 3 ] }, { "title": "2. Ark 564", "content": "Ark 564 is a bright, nearby, narrow-line Seyfert 1 (NLS1) galaxy, with z = 0 . 024684, V = 14 . 6 mag (de Vaucouleurs et al. 1991), and L 2 -10 keV = (2 . 4 -2 . 8) × 10 43 ergs s -1 (Turner et al. 2001, Matsumoto et al. 2004, and present work). It has been studied across all wavebands (e.g., Turner et al. 2001, Collier et al. 2001, Shemmer et al. 2001, Romano et al. 2004) and shows large amplitude flux variations on short time scales and a peculiar emission line-like feature near 1 keV (Leighly et al. 1999, Turner et al. 2001, Comastri et al. 2001). Matsumoto et al. analyzed the Chandra HETGS observation of Ark 564 (that carried out in 2000) with an exposure time of 50 . 2 ks. They modeled the hard X-ray spectrum with a power law of photo-index of 2 . 56 ± 0 . 06 and fit the soft excess below 1 . 5 keV with a blackbody, of temperature 0 . 124 ± 0 . 003 keV. They find some evidence for a two phase WA with ionization parameters log ξ ∼ 1 and log ξ ∼ 2 and column density of logN H = 21 cm -2 . They find that the 1 keV emission feature is not due to blends of several narrow emission lines and suggest that it could be an artifact of the warm absorber. Brinkmann et al. (2007) studied the spectral variability of the X-ray emission of the Ark 564 using the ∼ 100 ks XMMNewton observation and find that the 'power law plus bremsstrahlung' model describes the spectrum well at all times, with flux variations of both components. Papadakis et al. (2007) analyzed the XMM-Newton EPIC data from Ark 564 2005 observations. They found evidence for two phases of photoionized X-ray absorbing gas with ionization parameter logξ ∼ 1 and log ξ ∼ 2 and column densities of N H ∼ 2 and 5 × 10 20 cm -2 , similar to the results of Matsumoto et al. (2004). They also detect an absorption line at ∼ 8 . 1 keV in the low resolution CCD spectra and assuming that this line corresponds to Fe xxvi Kα , they suggest the presence of highly ionized, absorbing material of N H > 10 23 cm -2 outflowing with relativistic velocity of ∼ 0 . 17 c . Analyzing both the EPIC and RGS data from the same XMM-Newton 2005 observations, Dewangan et al. (2007) find two warm absorber phases with ionization parameters log ξ ∼ 2 and logξ < 0 . 3 and column densities of N H ∼ 4 and 2 × 10 20 cm -2 and outflow velocities of 300 and 1000 km s -1 respectively. Smith et al. (2008) analyzed the combined XMM-Newton RGS spectrum of Ark 564 obtained from 2000 to 2005. They found three separate phases of photoionized X-ray absorbing gas, with ionization parameters log ξ = -0 . 86 , 0 . 87 , 2 . 56 and column densities of N H = 0 . 89 , 2 . 41 , 6 . 03 × 10 20 cm -2 respectively, all with very low velocity ( -10 ± 100 km s -1 ). From the emission line analysis they found a flow velocity of -600 km s -1 and claimed that the X-ray absorption and emission originate in different regions. Ark 564 is also known to have a strong UV absorber, characterized by O vi , Si xii , Si iv and C iv absorption lines (Crenshaw et al. 2002, Romano et al. 2002).", "pages": [ 3, 4 ] }, { "title": "3. Observations and Data Reduction", "content": "Ark 564 was observed with the Chandra High Energy Transmission Grating Spectrometer (HETGS) on 2000 June for 50 ks and between 2008 August 26 to 2008 September 6 for total duration of 250 ks. It was also observed with Low Energy Transmission Grating Spectrometer (LETGS) on 2008 April 21 for 100 ks. Table 1 lists the observation log. Matsumoto et al. analyzed and presented the results of the 50 ks Chandra HETG observation done in 2000 June. Here we report on the Chandra archival 2008 observations of Ark 564 made with both HETGS and LETGS. The HETGS consists of two grating assemblies, a high-energy grating (HEG) and a medium-energy grating (MEG). The HEG bandpass is 0 . 8 -10 keV and the MEG bandpass is 0 . 5 -10 keV but the effective area of both instruments falls off rapidly at either end of the bandpass. We performed the spectral analysis over 5 -25 ˚ Arange. The LETG is combined with the Advanced CCD Imaging Spectrometer-Spectroscopic (ACIS-S) array or with the High Resolution Camera-Spectroscopic (HRC-S) array. The Ark 564 LETG observation was done with HRC-S array. The LETG/HRC-S has a band pass of 0 . 07 -7 . 3 keV or 1 . 7 -170 ˚ A, but due to the low S/N of data on either ends, we restricted our spectral fitting to 5 -40 ˚ A . We reduced the data using the standard Chandra Interactive Analysis of Observations (CIAO) software (v4.3) and Chandra Calibration Database (CALDB, v4.4.2) and followed the standard Chandra data reduction threads 2 . For the Chandra ACIS/HETG observations, we co-added the negative and positive first-order spectra and built the effective area files (ARFs) for each observation using the fullgarf CIAO script. Unlike ACIS, the HRC does not have the energy resolution to sort individual orders, and each spectrum contains contributions from all the diffraction orders. For the HRC/LETG observation we used the standard ARF files of orders 1 through 6 and convolved them with the relevant standard redistribution matrix files (RMF). For HETG observations, we generated the light curves in 2 ks bins for the energy band (0 . 3 -10 keV), as shown in figure 1. The time-average count rate varies from 0 . 24 cts s -1 to 0 . 32 cts s -1 among 2008 observations. However, except for the initial 20 ks of obsID 10575, the count rate of all observations are consistent with each other, with average value of 0 . 31 cts s -1 . We analyzed the spectra using the CIAO fitting package Sherpa . As noticed above, the HETG observations do not show large variations, so to increase the signal to noise (S/N) of the spectrum we co-added the spectra obtained with each observation and averaged the associated ARFs using the ciao script add grating spectra . This gave a total net exposure time of 250 ks for the MEG and HEG. We fit the MEG and HEG data simultaneously, and discuss the LETG spectral analysis separately in section 5. Throughout the paper we applied the χ 2 minimization technique in the spectral analysis and the reported errors are of 1 σ significance for one interesting parameter.", "pages": [ 4, 5 ] }, { "title": "4.1. Continuum Modeling", "content": "To model the intrinsic continuum of the source, we first fitted a simple absorbed (Galactic N H = 6 . 4 × 10 20 cm -2 ; Dickey et al. (1990)) power law with varying photon index and amplitude. A single absorbed power law could not fit the data over the entire range. We found an excess of flux in the spectrum at energies below ≈ 1 . 5 keV. Ark 564 is known to have a strong soft continuum (Leighly et al. 1999, Turner et al. 1999), so to fit this soft excess we added a black-body component to the above mentioned simple power-law. The fit improved significantly ( χ 2 /d.o.f. = 5882 / 3997, ∆ χ 2 = 515) and figure 2 shows the continuum model fit to the MEG spectrum. We also plot the data:fit residuals, which show strong absorption features consistent with the known WA of this source.", "pages": [ 5 ] }, { "title": "4.2. Local z = 0 Absorption", "content": "The spectra of Ark 564 show narrow absorption lines at zero redshift. The O i and O ii absorptions are attributed to the ISM (Wilms et al. 2000) and O vii , O viii and Fe xvii absorption lines arise in the circumgalactic medium (CGM) of our Galaxy or in the Local Group (Gupta et al. 2012, and references therein). We modeled all the statistically significant local absorption features with Gaussian components of fixed width of 1 m ˚ A; the line response function of the grating is folded in the RMF file in each case. The model of continuum plus local absorption features ('Model A') improves the fit by ∆ χ 2 = 108 for 10 fewer degrees of freedom. The measured equivalent width (EW) and statistical significance of each line are reported in Table 2 and labeled in figure 3.", "pages": [ 6 ] }, { "title": "4.3. Intrinsic Absorption", "content": "As reported above, the Ark 564 spectrum shows many strong intrinsic absorption features. We measured the position, EW and statistical significance of all intrinsic absorption lines by fitting negative Gaussians of fixed width of 1 m ˚ A(Fig. 3, Table 2). All the Ark 564 absorption features are blueshifted with respect to the source, implying moderate outflow velocities of 82 -239 km s -1 . We find a few lines with no identification at 19 . 811 ± 0 . 006 ˚ A , 19 . 850 ± 0 . 005 ˚ A and 20 . 255 ± 0 . 005 ˚ A in the observer frame. We marked these features with green underlines in figure 3.", "pages": [ 6 ] }, { "title": "4.4. Photoionization model fitting: PHASE", "content": "We used the Photoionization model fitting code PHotoionized Absorption Spectral Engine (PHASE; Krongold et al. 2003), to model the warm absorber features. The PHASE code self consistently models all the absorption features observed in the X-ray spectra of AGNs. At its simplest, an absorption-line spectrum can be fit with PHASE using only four input parameters: 1) the ionization parameter of the absorber U ; 2) the equivalent hydrogen column density N H ; 3) the outflow velocity of the absorbing material V out ; and 4) the micro-turbulent velocity V turb of the material. The abundances have been set at the Solar values (Grevesse et al. 1993). We used the Ark 564 spectral energy distribution (SED) from Romano et al. (2004) to calculate the ionization balance of the absorbing gas in PHASE. The SED constructed by Romano et al. is based on a quasi-simultaneous multiwavelength campaign, and thus is the most accurate overall SED for this source obtained to date. In the X-rays, however, we use our own fits, as this radiation is the one responsible for the ion- ization of the charge states producing absorption in the X-ray band during our observations. We further connect the UV and the X-ray data with a simple power law (a straight line in log-log space connecting the last UV point and the first X-ray point). The most prominent absorption lines in the HETGS Ark 564 spectrum are those of Mg xi , Ne x , Ne ix , Fe xvii , Fe xviii , O viii , O vii and O vi at an outflow velocity of ≈ -100 km s -1 . We add a single ionized absorbing component to Model A described above (we call it Model B) to characterize this WA component. This absorber has best fit parameters of logU = 0 . 39 ± 0 . 03, logN H = 20 . 94 ± 0 . 02 cm -2 , and an outflow velocity relative to systemic of -94 ± 13 km s -1 . The fit gives a significant improvement over model A (Fig. 4; χ 2 /d.o.f. = 4722 / 3982, ∆ χ 2 = 1073). This absorber fits the high ionization lines so we will refer to this absorber as the 'high-ionization phase (HIP)' component. The HIP component fits the absorption features produced by ions such as Mg xi , Ne x , Ne ix , Fe xviii , Fe xvii , O viii , and O vii . Matsumoto et al. (2004) also report an absorber of similar characteristics: ionization parameter log U ∼ 1 and absorption column N H = 10 21 cm -2 , derived using the column densities of O viii , Ne ix , Ne x and Mg xi . The single-absorber model does not fit the O vi absorption line and it also under-predicts the O vii absorption (Fig. 5, red curve). This suggests the presence of another absorber with lower ionization state. To fit the residual features we added another absorber to our previous model defining Model C. An absorber with ionization parameter logU = -0 . 99 ± 0 . 13, logN H = 20 . 11 ± 0 . 06 cm -2 , and an outflow velocity of ∼ -137 ± 37 km s -1 successfully fits the low ionization lines, including O vi and O vii (Fig. 5 & Fig. 6). We call this component 'lower-ionization phase (LIP)' absorber. This model gives a χ 2 /d.o.f. = 4674 / 3979; ∆ χ 2 = 48, significantly improving over the single-absorber model B. An F-test gives a higher than 99 . 999% confidence for the presence of this absorber. The ionic column densities predicted by our best fit two-ionized absorber model (Model C) are listed in Table 4. As can be inferred from Table 4, except for O vii , the LIP and HIP components predict absorption from different ions but at similar velocities. This suggest that the Ark 564 absorbers may be present in the same outflowing multi-phase medium. We come back to this in section 6.3. To extrapolate the two component ionized absorber model of the MEG spectrum of Ark 564 to lower wavelengths, we simultaneously fit the HEG and MEG spectra (Table 3, Fig. 7). The HEG intrinsic absorption features are well fitted with HIP absorber, including Si xiii and Mg ix lines which were not detected in the MEG spectrum due to low signal to noise. The best-fit model parameters of the MEG+HEG fit are consistent with the MEGonly fit. Though most of the Ark 564 intrinsic absorption features are well fitted with two warm absorbers, the model does not fit the unidentified absorption lines mentioned in section 4.3. The identification, evaluation and interpretation of these features is be discussed in detail in a companion paper (Gupta et al. 2013b).", "pages": [ 6, 7, 8 ] }, { "title": "5. LETGS Spectral Analysis", "content": "Ark 564 was also observed with LETGS in April 2008. Ramirez (2013) analyzed that observation and found a very weak absorption line of O vii K α but a strong feature at 18 . 62 ˚ A (at the wavelength of O vii K β ). For this reason they identified the absorption feature at 18 . 62 ˚ A as blueshifted O viii K α with velocity ∼ 5500 km s -1 . They further modeled the 17 -25 ˚ A spectral region with two high ionization absorbing components with log ( ξ ) ∼ 3, one at v ∼ 0 km s -1 and one at v ∼ 5500 km s -1 . Since we found no evidence of an outflow with velocity ∼ 5500 km s -1 in the 2008 HETGS spectra, we reanalyzed the LETGS observation to check for the consistency with our model derived from the HETG data. To fit the Ark 564 LETGS spectrum continuum, we used the same model as for HETG data (a power law plus black-body). The best fit power law photon-index and temperature of black-body are reported in Table 3. We observed that the flux (2 -10 keV) of the source varied from 2 . 48 ± 0 . 11 × 10 -11 erg s -1 cm -2 to 2 . 79 ± 0 . 15 × 10 -11 erg s -1 cm -2 between the HETG and LETG observations. To fit the intrinsic absorption features, we start with the two component ionized absorber model obtained for HETG spectra. This model fits the absorption features of Fe xviii , Fe xvii , Ne x , Ne ix reasonably well, but overestimates the O vii k α line strength ( χ 2 /do.f. = 1426 / 1190; Fig. 8 & Fig. 9). Allowing the PHASE parameters to vary freely, the fit improved considerably ( χ 2 /do.f. = 1370 / 1184; Fig. 10). The best fit parameters of LETGS WA model are reported in Table 3. This model fits the narrow absorption due to O vii Kα , but leaves the residuals at 19 . 1 ˚ A , corresponding to O vii Kβ (Fig. 9). We also tried to fit the LETG spectrum with models suggested by ? , but the fit was not good ( χ 2 /do.f. = 1936 / 1190). Though this model well fits the line at 19.1 ˚ A(observed frame), it also predicts other absorption lines which are inconsistent with the data. As noted by ? , the absorption line at 19 . 1 ˚ A is too strong to be by O vii Kβ and could be, in part, due to a transient high velocity outflow component (Gupta et al. 2013b, companion paper). Between the LETG and HETG warm absorber models, the HIP component parameters are consistent within errors. However, the LETG LIP has lower ionization parameter and higher column in comparison to HETG LIP. We also note that the column densities of highly ionized ions (O viii , Fe xvii , Fe xviii , Ne ix , Ne x , Mg xi , Mg xii and Si xii ) are higher while for those of less ionized ions (O vii and O vi ) are smaller in the HETG observation than in LETG (Table 4). Both the instruments have good response in the spectral region where these lines are detected, so the observed differences cannot be due to instrumental artifacts; what is observed is the real variability in the WA properties between the two observations from 2008 April (LETG) to August/September (HETG).", "pages": [ 8, 9 ] }, { "title": "6.1. The Connection between UV and X-ray Absorbers", "content": "Ark 564 was observed with HST (STIS) and FUSE during May-July 2000 and June 2001 respectively. Crenshaw et al. (2002) and Romano et al. (2002) detected intrinsic absorption lines blueshifted by ≈ -190 km s -1 and ≈ -120 km s -1 respectively, similar to that of the X-ray WAs. Crenshaw et al. modeled the UV data with a single absorber of ionization parameters logU of ∼ 0 . 033 and column density logN H of ∼ 21 . 2 cm -2 . The UV absorption model also predicted the column densities of O vii and O viii of < 2 . 2 × 10 17 cm -2 and < 1 . 1 × 10 16 cm -2 respectively (Romano et al. 2002). The O vii column density measured from the X-ray WA models (0 . 3 -1 . 3) × 10 17 cm -2 is consistent with UV upper limits. The total HIP+LIP O viii column density of (2 . 0 -2 . 9) × 10 17 cm -2 is an order of magnitude higher than the UV estimates. However, the LIP O viii column density = (3 . 0 -4 . 6) × 10 15 cm -2 is in agreement with UV models. The consistency between the X-ray LIP absorber outflow velocity, hydrogen column density, and ionic column densities with the UV absorber model suggests that both are the same absorber. The HIP absorber, on the other hand, is different from the UV absorber, as expected. Romano et al. from FUSE observations of Ark 564 measured the intrinsic O vi column density of (5 . 70 -6 . 01) × 10 15 cm -2 . The upper limit on O vi column density measured from UV data is much smaller than our lower limit on O vi column of 1 . 0 × 10 16 cm -2 in X-rays. This could be due to the variability of the WA between the two observations. We note, however, that X-ray and UV O vi column densities have been found to be discrepant in other AGN absorption systems (e.g., Krongold et al. 2003) and in redshift zero absorption (Williams et al. 2006).", "pages": [ 9 ] }, { "title": "6.2. Estimates on mass and energy outflows rates", "content": "Using the values of warm-absorber parameters such as column density, ionization parameter and outflowing velocities, we can give a rough estimate of the mass outflow rate ( ˙ M out ) and the kinetic energy carried away by the warm absorbing winds (i.e. kinetic luminosity, L K ). But before we can measure the mass and energy outflow rates, we must know the location of the absorber. However, in the equation for the photoionization parameter ( U ∝ L/n e R 2 ), the radius of the absorbing region (R) is degenerate with the density ( n e ). In principle we can put constraints on the distance R by measuring the density of the WA with variability analysis (e.g., Krongold et al. 2007). Using this technique, Krongold et al. managed to determine the absorber density in NGC4051 and so the distance. For Ark 564, no such study is available in literature and in the present work the source does not show any significant variability either. Therefore, we will only set limits on the mass and energy outflow rates using the expression derived in Krongold et al., ˙ M out ≈ 1 . 2 πm p N H v out r . The estimate of maximum distance from the central source can be derived assuming that the depth ∆ r of the absorber is much smaller than the radial distance of the absorber (∆ r << r ) and using the definition of ionization parameter ( U = Q ( H ) 4 πr 2 n H c ), i.e. r ≤ r max = Q ( H ) 4 πUN H c . In several papers lower limit on the absorber distance was determined assuming that the observed outflow velocity is larger than the escape velocity at r : i.e. r ≥ r min = 2 GM BH v 2 out . However, as shown in Mathur et al. (2009) WA outflow velocities are usually lower than the escape velocities, so cannot really be used to derive a lower limit on r . Using the best fit values of ionization parameter and column density, we estimated the upper limits on HIP and LIP absorber locations of r HIP < 40 pc and r LIP < 6 kpc respectively, which are not very interesting limits. Using the above equation and outflow velocities of 94 km s -1 and 137 km s -1 , we obtain the mass outflow rates of ˙ M out < 6 . 4 × 10 24 g s -1 and ˙ M out < 2 . 2 × 10 26 g s -1 for HIP and LIP absorbers respectively. Similarly we obtained the constraints on kinetic luminosity of the outflows of ˙ E K < 2 . 8 × 10 38 erg s -1 and ˙ E K < 2 . 1 × 10 40 erg s -1 for the HIP and LIP absorbers respectively. In comparison to the Ark 564 bolometric luminosity of 2 . 4 × 10 44 erg s -1 (Romano et al. 2002), the total kinetic luminosity of these outflows is ˙ E K /L bol < 0 . 0001% for the HIP and < 0 . 009% for the LIP. Thus it is very unlikely that these outflows significantly affect the local environment of the host galaxy. The AGN feedback models typically required 0 . 5 -5% of the bolometric luminosity of an AGN to be converted into kinetic luminosity to have a significant impact on the surrounding environment (Hopkins & Elvis 2010, Silk & Rees 1998, Scannapieco & Oh 2004).", "pages": [ 9, 10 ] }, { "title": "6.3. Pressure Balance between LIP and HIP", "content": "The presence of two different absorbing components with different temperatures but similar outflow velocity suggests that the absorber could arise from two phases of the same medium (e.g., Elvis et al. 2000, Krongold et al. 2003). This is further supported by the fact that multiple components of the ionized absorber are found in pressure balance (e.g., Krongold et al. 2003; 2005; 2007, Cardaci et al. 2009, Andrade-Velazquez et al. 2010, Zhang et al. 2010). This result has proven valid against different methodologies and codes used in the analysis (Krongold et al. 2013). To investigate whether or not the absorbing components of Ark 564 are in pressure balance, we generated the pressure-temperature equilibrium curve (also known as the 'S-curve' (Krolik et al. 1981), for the SED used in our analysis: Fig. 11). Interestingly, we find that the equilibrium curve of Ark 564 does not have the typical 'S' shape where multiple phases can exist in pressure equilibrium, because there are no regions of instability (for all points in the plane the derivative of the curve is positive). Accepted at face value, this result implies that the absorber in Ark 564 is not in pressure balance and thus is not forming a multiphase medium, a result at odds with previous evidence on warm absorbers. While we cannot rule out this possibility, there are several arguments pointing towards a multiphase medium. We note that the two different absorbing components in Ark 564 have the same kinematics, which suggests that they are related. If they share the location (the most reasonable assumption), there must be a gradient of pressure between them, given that the LIP pressure is over 5 times larger than that of the HIP. Therefore, these two components should move on the S-curve to form a single component in a time comparable to the free expansion time, given by t exp = ∆ R/V s (where ∆ R is the thickness of the absorber and V s the speed of sound in the medium). The flow time of the components (i.e. the time in which the components cross our line of sight to the source) is given to first order as t flow = R/V out (where R is the distance from the illuminated face of the absorbing material to the ionizing source and V out its outflow velocity). For the warm absorber in Ark 564 V s ∼ V out (specially for the HIP that is hotter). It follows that t exp /t flow = ∆ R/R < 1. Then, the free expansion time is smaller than the flow time, and the two phase should dissolve into a single component before moving out from our line of sight, which is clearly not consistent with the data. Thus, if the two phases are not in pressure balance they should not be connected, and the similar kinematics in this, and in many other sources, would have to be considered a coincidence. Alternatively, the sources might be in pressure balance forming a multiphase medium, but there might be additional heating and/or cooling processes acting on the gas, changing the shape of the S-curve, but not the ionization balance. The most obvious parameter for this is the gas metallicity. Komossa & Mathur (2001) showed that the shape of the equilibrium curve not only depends upon the SED of the source, but also on the metallicity of the absorber, which affects the cooling of the gas. They further show that super-solar abundances restore the equilibrium zone in steep spectrum sources and increase the pressure range where a multiphase equilibrium is possible. Fields et al. (2007) showed that this is indeed the case for the ionized absorber in Mkn 279. Since Ark 564 also has a steep spectrum and a monotonically rising 'S' curve, we generated a new pressure-temperature curve with super-solar metallicity of 10 solar, shown as a dashed curve in figure 11. This is a good assumption as supersolar metallicity has been suggested for this source (Romano et al. 2004). Even with super-solar metallicity, the 'S-curve' is very steep, without regions where multiple components can coexist in pressure equilibrium. Thus, even having high metallicity does not solve the problem of finding the warm absorber components out of pressure balance and below we speculate on possible reasons. The continuum X-ray spectrum of Ark 564 is not only steep (Γ > 2 . 4), it also has an additional prominent soft excess, similar to the behavior seen in sources with steep soft Xray spectra (e.g., NGC 4051, Komossa & Mathur 2001; Mrk279, Fields et al. 2007). In fact, additional modeling shows that the main reason for a steep 'S-curve' is the extra heating produced by the soft excess (particularly in the LIP). If the soft excess continuum is not impinging directly on the absorbing gas, perhaps because it is the result of reflection toward our line of sight, then the two components would be in pressure balance. Other possibilities to produce a multi-valued S-curve, and LIP/HIP components in pressure balance include a weaker IR radiation field illuminating the material than the one observed (Krolik and Kriss 2001) or additional (more exotic) sources of heating at high temperatures, such as those discussed in Krolik et al. (1981). Therefore, if warm absorbers are indeed a multiphase medium in pressure equilibrium, it is likely that the overall radiation field impinging on the gas is different than the one observed. This effect might be stronger in sources with steep soft X-ray spectra. We note, however, that this suggestion is speculative; we cannot prove it or rule it out. We also note that photoionization models of the Broad Line Region demand that the ionizing continuum is different than the one observed (Binette & Krongold 2008, and references therein). If warm absorbers are indeed in pressure balance, their S-curves can be used for a better understanding of the physical properties and the processes acting on the material (Komossa & Mathur 2001, Chakravorty et al. 2012, Krongold et al. 2013).", "pages": [ 10, 11, 12 ] }, { "title": "7. Summary", "content": "Our best fit model of intrinsic absorption of NLS1 galaxy Ark 564 requires a twophase warm absorber with two different ionization states (HIP and LIP). Both the absorbers are outflowing at low velocities of order of ∼ 100 km s -1 . The HIP absorber reproduces most of the spectral features observed in the HETG spectra (O viii , Ne ix , Ne x , Mg xi , Fe xvii and Fe xviii ) except for a few at lower energies (O vii and O vi ), which are modeled by the LIP component. The pressure-temperature equilibrium curve for the Ark 564 warm absorber does not have the typical 'S' shape, even if the metallicity is super-solar; as a result the two WA phases do not appear to be in pressure balance. We speculate that the continuum incident on the absorbing gas is perhaps different from the observed continuum. We observe clear variability in the WA properties between the 2008 HETG observations and previous observations which could be in response to the change in continuum or the absorbing clouds passing our sight-line; the large time gap between observations does not allow us to distinguish between the two possibilities. We also estimated the mass outflow rate and associated kinetic energy assuming a biconical wind model (Krongold et al. 2007) and find that it represents a tiny fraction of the bolometric luminosity of Ark 564. Thus it is highly unlikely that these outflows provide significant feedback required by the galaxy formation models. Acknowledgement: Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number TM9-0010X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. YK acknowledges support from CONACyT 168519 grant and UNAM-DGAPA PAPIIT IN103712 grant.", "pages": [ 12, 13 ] }, { "title": "REFERENCES", "content": "Andrade-Vel´azquez, M. et al. 2010, ApJ, 711, 888 Behar, E. et al. 2003, ApJ, 598, 232 Blustin, A. J., Page, M. J., Fuerst, S. V., Branduardi-Raymont, G., & Ashton, C. E. 2005, A&A, 431, 111 Binette, L. & Krongold, Y. 2008, A&A, 477, 413 Brinkmann, W., Papadakis, I. E., & Raeth, C. 2007, A&A, 465, 107B Cardaci, M. V. et al. 2009, A&A, 505, 541 Chakravorty, S., Misra, R., Elvis, M., Kembhavi, A.K. & Ferland, G. et al. 2010, MNRAS, 422, 637 Collier, S. et al. 2001, ApJ, 561, 146 Comastri, A. et al. 2001, A&A, 365, 400 Costantini, E., Gallo, L. C., Brandt, W. N., Fabian, A. C., & Boller, T. 2007, MNRAS, 378, Krolik, J. H., & Kriss, G. A. 2001, ApJ, 561, 684 Krongold, Y., Nicastro, F., Brickhouse, N.S., Elvis, M., Liedahl D.A. & Mathur, S. 2003, ApJ, 597, 832 Krongold, Y., Nicastro, F., Elvis, M., Brickhouse, N. S., Mathur, S., & Zezas, A. 2005, ApJ, 620, 165 Krongold, Y. et al. 2007, ApJ, 659, 1022 Krongold, Y., Binette, L., & HernNandez-Ibarra, F. 2010, ApJ, 724L, 203K Krongold, y. et al. 2012, in preparation Leighly, K.M. 1999, ApJS, 125, 317 Mathur, S., Wilkes, B., Elvis, M., & Fiore, F. 1994, ApJ, 434, 493 Mathur, S., Elvis, M., & Wilkes, B. 1995, ApJ, 452, 230 Mathur, S., Weinberg, D. H., & Chen, X. 2003, ApJ, 582, 82 Mathur, S., Stoll, R., Krongold, Y., Nicastro, F., Brickhouse, N., & Elvis, M. 2009, AIPC, 1201, 33 Matsumoto, C., Leighly, K. M., & Marshall, H. L. 2004, ApJ, 603, 456 McKernan, B., Yaqoob, T., & Reynolds, C. S. 2007, MNRAS, 379, 1359 Netzer, H. et al. 2003, ApJ, 599, 933 Papadakis, I. E., Brinkmann, W., Page, M. J., McHardy, I., & Uttley, P. 2007, A&A, 461, 931 Proga, D., & Kallman, T. R. 2004, ApJ, 616, 688 Proga, D. 2007, ASPC, 373, 267 Ramirez, J. 2013, A&A, 551, A95 Reynolds, C. S. 1997, MNRAS, 286, 513 Romano, P., Mathur, S., Pogge, R. W., Peterson, B. M., & Kuraszkiewicz, J. 2002, ApJ, 578, 64 Romano, P., Mathur, S., & Turner, T. J. 2004, ApJ, 602, 635 Yao, Y. et al. 2009, ApJ, 696, 1418", "pages": [ 13, 14, 15, 16 ] } ]

[ { "title": "ABSTRACT", "content": "Binary stars are thought to be a controlling factor in globular cluster evolution, since they can heat the environmental stars by converting their binding energy to kinetic energy during dynamical interactions. Through such interaction, the binaries determine the time until core collapse. To test predictions of this model, we have determined binary fractions for 35 clusters. Here we present our methodology with a representative globular cluster NGC 4590. We use HST archival ACS data in the F606W and F814W bands and apply PSF-fitting photometry to obtain high quality color-magnitude diagrams. We formulate the star superposition effect as a Poisson probability distribution function, with parameters optimized through Monte-Carlo simulations. A model-independent binary fraction of (6.2 ± 0 . 3)% is obtained by counting stars that extend to the red side of the residual color distribution after accounting for the photometric errors and the star superposition effect. A model-dependent binary fraction is obtained by constructing models with a known binary fraction and an assumed binary mass-ratio distribution function. This leads to a binary fraction range of 6.8% to 10.8%, depending on the assumed shape to the binary mass ratio distribution, with the best fit occurring for a binary distribution that favors low mass ratios (and higher binary fractions). We also represent the method for radial analysis of the binary fraction in the representative case of NGC 6981, which shows a decreasing trend for the binary fraction towards the outside, consistent with theoretical predictions for the dynamical effect on the binary fraction. Subject headings: Binary frequency, globular clusters, HST ACS, evolution", "pages": [ 2 ] }, { "title": "Binary Frequencies in a Sample of Globular Clusters. I. Methodology and Initial Results", "content": "Jun Ji and Joel N. Bregman Department of Astronomy, University of Michigan, Ann Arbor, MI 48109 [email protected], [email protected] Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The standard picture of globular clusters shows that a cluster composed of single stars will undergo core collapse after several relaxation times (Lynden-Bell & Wood 1968; Cohn 1980; Lynden-Bell & Eggleton 1980; Spitzer 1987). Since only about one-fifth of globular clusters show collapsed cores (Djorgovski & King 1986; Harris 1996; 2010 edition), certain heating mechanisms are needed to counteract the gravitational contraction and avoid core collapse. This energy is expected to come primarily from the 'burning' of binaries, i.e. the dynamical interactions of binaries with single stars or other binaries will convert the binding energy in the binaries to the single stars or other binaries, so as to heat the environment stars (Heggie 1975; Hut & Bahcall 1983; Goodman & Hut 1989; Hut et al. 1992). Even a small primordial binary fraction is sufficient to prevent core collapse for many relaxation times (Gao et al 1991; Fregeau et al. 2003), so the binary fraction is an essential parameter that can dramatically affect the evolution of globular clusters. The binaries remaining in globular clusters are mainly hard binaries, whose binding energy is greater than the average kinetic energy of a single star in that cluster (Hut et al. 1992). Most of the soft ones are destroyed during their first interactions with other stars (Sollima 2008), and would not provide the heating energy. Mass-transfer binaries are among the hardest binaries, and those with degenerate primaries can be bright X-ray sources (Hut et al. 1992; Heinke 2011), such as Low mass X-ray binaries (LMXBs). LMXBs are thought to be formed in the dense cores of globular clusters through dynamical exchange processes (Clark 1975; Bailyn 1995; Cohn et al. 2010), as they show a strong correlation between the collisional parameter and their frequency (Pooley et al. 2003). Some CVs show this correlation too, which suggests their dynamical origin (Pooley & Hut 2006). Although theoretical models and simulations are well developed for the evolution of globular clusters with binary burning process, sufficient observations to test these models are lacking. One reason is that it is very difficult to isolate individual stars in the high density region such as the cores of globular clusters from the ground based telescope. The other reason is that photometry errors are very large due to superposition of those unresolved stars. These two difficulties make the observations of binaries fraction challenging. A direct method to detect binaries in globular clusters is by spectroscopic observation to measure radial velocity variations, which can only be applied to red giant and sub-giant stars. This is because those stars are bright in magnitude and cool in temperature, so there are many strong absorption lines for cross-correlation. This will improve the accuracy of radial velocity measurement to below 1 km/s. The drawback for this method is that it requires large amounts of observing time over a several years. For some long period binaries (greater than 10 years), even the current observational accuracy is not enough to discover the small radial velocity change over a reasonable time (5 years). For binaries composed of two main sequence stars, radial velocity observation present other challenges. Not only are these stars fainter than giants, but they have fewer lines for spectral cross-correlations, thereby demanding high S/N spectra that can place unreasonable demands on even the largest ground-based telescopes. Another issue with ground-based spectroscopic observations is that, even for good seeing conditions, multiple unrelated stars can fall in a single spectral PSF. This restricts such studies to the outer regions of globular clusters, which may not be representative of the dynamically more active inner parts (Gunn & Griffin 1979; Pryor et al. 1988; Yan & Cohen 1996; Cote & Ficher 1996; Cote et al. 1996). A different direct method for detecting binaries is through observation of eclipsing binaries. This method still needs a substantial observing time investment to monitor many stars in globular clusters and search for the photometric variables through their light curves. While this method is biased by small orbital inclination short period binaries, it is another valuable method for investigating binary systems (Mateo et al. 1990; Yan & Mateo 1994; Yan & Reid 1996; Albrow et al. 2001). Another method, and the one used here, makes use of the accurate measurement of stars that form the main sequence in a color-magnitude diagram (CMD). For main sequence stars of a single age and metallicity, the width of the main sequence of single stars should be limited only by measurement error. Relative to the single star main sequence, a binary star will be either redder, for a lower mass secondary, or brighter (by 0.75 mag) but at the same color for two stars of equal mass. An ensemble of binaries forms a thickening to the red of the single star main sequence, which we can measure. The method is difficult to execute from the ground because of the precision required at faint magnitudes (S/N = 50-100 at V = 21) and in crowded fields. However, the Hubble Space Telescope (HST) has sufficient sensitivity and spatial resolution to resolve even the core of globular clusters. This greatly improves the photometric accuracy, and makes the measurement of the binary fraction possible as a function of position. This method can detect widely-separated binaries, but it is biased against binaries with low-mass secondaries. The benefit of this method is that, it requires relatively less observing time, and one can measure the global binary fraction without other assumptions. To date, only a few globular clusters have been studied for the binary fraction with HST by analysing their color-magnitude diagrams. Rubenstein & Bailyn (1997) measured that the binary fraction of NGC 6752 to be 15% - 38% in the inner core radius, and probably less than 16% beyond that. Bellazzini et al. (2002) also determined the binary fraction of NGC 288 within its half light radius to be (15 ± 5)% with HST WFPC2 data, depending on the adopted binary mass ratio function. Its binary fraction outside the half light radius is less than 10%, and most likely closer to 0%. Richer et al. (2004) measured the evolved cluster M4 with proper-motion selected WFPC2 data. They found the binary fraction within 1.5 core radius is about 2%, decreasing to about 1% between 1.5 and 8.0 core radii. Zhao & Bailyn (2005) studied two clusters, M3 and M13, with WFPC2 data, and they found that the binary fraction of M3 within one core radius lies between 6% and 22%, and falls to 1% - 3% between 1-2 core radii. The binary fraction of M13, however, was not constrained with their method. Davis et al. (2008) measured the binary fraction of the core collapsed cluster NGC 6397 with both ACS and WFPC2 data. They well constrained the binary fraction outside the half light radius to be (1 . 2 ± 0 . 4)% with 126 orbits of observation with ACS and proper-motion selected clean data, and constrained the binary fraction within the half light radius to be (5 ± 1)% with WFPC2 data. Sollima et al. (2007) performed the first sample study for the binary fractions in globular clusters. They used aperture photometry to construct the CMDs for 13 low-density, high galactic latitude globular clusters with ACS data. They found a minimum of 6% binary fraction within one core radius for all clusters in their sample, and global fractions ranging from 10 to 50 per cent depending on the clusters and the assumed binary mass-ratio model. In our survey, we compile a sample of 35 Galactic globular clusters, a larger sample than previous efforts, and one that takes advantage of a wealth of archived HST data. We use the PSF photometry instead of aperture photometry, so that we can analyze high-density clusters in addition to low-density clusters. We try to constrain the binary mass-ratio function depending on the data quality and the number of binaries found, which is the largest uncertainty in determining binary fraction in previous studies. We also analyze the binary fraction radial distribution and variation along the main-sequence. In this paper, the first of two, we present the techniques used in obtaining the binary fraction. We present the sample selection in § 2, data reduction and photometry method in § 3, the artificial star tests in § 4, the high mass-ratio binary fraction estimate in § 5, the global binary fraction estimate in § 6, the binary fraction radial analysis in § 7, and discussions in § 8.", "pages": [ 3, 4, 5, 6 ] }, { "title": "2. Sample Selection", "content": "To achieve the primary goal of determining the binary fraction, we need accurate color-magnitude diagrams (CMD). CMDs can be made from several different filters, such as a subset of B, V, R, and I, although for the main sequence in the late G to early M spectral region, the most accurate CMDs are composed from a subset of V, R, and I (V and I to be used when possible). For good photometric errors (0.02 mag rms), about 6,000 stars on the CMD will yield an uncertainty in the binary fraction of 1.5-2% (after Hut et al. (1992), Table 4). This type of photometric accuracy is achieved with WFPC2 V band images in 1500 seconds for V = 24.0 (m-M = 14.6 mag for a M1 star), or for V = 24.9 when using the ASC/WFC (m-M = 15.5 mag for a M1 star). We examined the HST observations taken for every Galactic globular cluster in the list of Harris (1996; 2010 edition) and most have some WFPC2 or ACS observations, but many of these were taken in snapshot mode and are not sufficiently long to meet our criteria. We find about 35 globular clusters that are sufficiently luminous, and with long enough exposures that we can extract useful information on binary fractions. Most of those data sets are from the HST Treasury program for globular clusters with ACS observations in F606W and F814W filters (Sarajedini et al. 2007). Table 1 shows the basic information for each cluster in this sample, including Galactic longitude (l), Galactic latitude (b), metallicity ([Fe/H]), foreground reddening E(B-V), absolute visual magnitude M V , collapsed core (y/n), core radius ( r c ), half light radius ( r h ), log relaxation time at half mass radius (log t rh ), age ( t Gyr ), and dynamical age ( t Gyr /t rh ). The dynamical ages in this sample range from dynamical young clusters ( t Gyr /t rh = 1 . 5) to dynamical old one ( t Gyr /t rh = 46 . 4)(see upper right panel in Figure 1). The metallicities in this sample range from metal poor cluster ([ Fe/H ] = -2 . 37) to metal rich one ([ Fe/H ] = -0 . 32)(see Figure 1, lower left panel). Figure 1 lower right panel shows the distribution of those clusters relative to the Galactic disk plane, among which 9 clusters are within the ± 15 · Galactic latitudes, and are potentially affected by non-member stars. Table 2 shows the HST ACS observation log for each cluster, including filter type, number of exposures used, exposure time per frame, and data set ID. We only used the long exposure frames here and did not include those short exposure frames, because we are only interested in the CMDs below the turn-off point.", "pages": [ 6, 7 ] }, { "title": "3. Data Reduction and Photometry", "content": "We retrieved the HST ACS archival data on F606W and F814W bands for each globular cluster, employing the most recent calibration frames on the fly. A drizzled imaged was produced ( Multidrizzle was applied to all FLT images), with care taken to achieve accurate alignment of the images. Standard practices were applied for bad pixel masking and exposure calibrations. The extraction of stellar magnitudes from the many point sources in these crowded fields depends on the algorithm used. In these fields, especially the crowded central parts of clusters, a star often lies on the wings of another star and these wings are not azimuthally symmetric, which complicates the subtraction. We used two versions of Dolphot (Dolphin 2000), an automated CCD photometry package for general use and for HST data (ACS, WFPC2, and WPC3). The later version uses a more refined library for the point spread function of point sources and we found significant improvement in the quality of PSF photometry for Dolphot V1.2 relative to V1.0. This can be found in Figure 2, where the CMD for NGC 4590 near the turn-off point is well-defined and the main sequence is narrower by using Dolphot V1.2 (right panel) compared to Dolphot V1.0 (left panel). The more accurate photometry leads to reduced uncertainties in the binary fraction determinations and allows us to probe to smaller binary mass ratios. From the output photometry files, the error weighted average magnitudes for both the F814W and the F606W filters were used, with the zero point set to the VEGA system. More details about the data reduction process can be found in Ji (2011). We note that in the work of Milone et al. (2012), a different extraction routine was developed for their program which appears to lead to slightly smaller errors. Not all parts of the resulting color-magnitude diagram are equally useful and only a limited range is used. We chose stars one magnitude below the turn-off point, since close to the turn-off point, binary sequences turn to merge with the main sequence and lose all of the binary information. At increasingly fainter magnitudes, the photometric uncertainty becomes large enough that those stars are not useful in constraining binary fraction models. Typically, this occurs about four magnitudes below the turn-off point. An example of these effects is shown in the CMD of NGC 4590 (Figure 3). The red dashed lines indicate the portion of CMD we used during the analysis (other lines are discussed below).", "pages": [ 8, 9 ] }, { "title": "4. Main Sequence Definition and Artificial Star Tests", "content": "Without knowing the photometric errors, completeness, and the rate of superposition of stars, we cannot have accurate measurement of binary fractions from the CMD. The artificial star tests, however, can help us understand the photometric errors and accuracy, as well as completeness (or star recovery percentage) at different region of clusters and at different magnitudes. It can also simulate the rate of superposition of stars if performed properly.", "pages": [ 9 ] }, { "title": "4.1. Defining the Main-sequence Ridge Line", "content": "To perform artificial star tests and later CMD analysis, we need to define the main-sequence ridge line (MSRL) of the observed CMD first. We constructed the MSRL from the data instead of fitting theoretical isochrones to the CMD, as the latter method is not sufficiently accurate to define the ridge line. To define the MSRL from the CMD, we used a moving box method. In each step, we defined a box with a height of 0.1 magnitude in the F606W axis, and a width of the entire range in the color (F606W-F814W) axis. Then we searched the peak value of the color histograms in that box with a color bin-size of 0.0015 magnitude. The color value at the peak and the middle point of the box in the F606W axis direction are the defined MSRL for stars in that box. The process was repeated in increments along the Main Sequence and the resulting line was then smoothed with the moving average method with a period of 50. In Figure 3, the green line is the defined MSRL.", "pages": [ 9 ] }, { "title": "4.2. Performing the Artificial Star Tests", "content": "With the defined MSRL, we performed artificial star tests in Dolphot. The F606W magnitudes of those fake stars were randomly generated while the F814W magnitudes were derived from the MSRL, so that all those fake stars are on the MSRL. The (X, Y) coordinates of those fake stars were randomly chosen from the area of the drizzled reference frame, but were not at the empty non-data CCD area. Each fake star in this list was added to each FLT frame one at a time, and was recovered using the same photometry process by Dolphot. The final photometric output file was screened using the same criteria as used to obtain the CMD, as described above; we added 100,000 fake stars for each image. Figure 4 shows two examples of the artificial star tests, NGC 5053 (low stellar density cluster) and NGC 1851 (high stellar density cluster). The green line is the input fake stars, which are all on the MSRL. The scattered black dots are the recovered fake stars (69,995 stars for NGC 5053 and 52,205 stars for NGC 1851).", "pages": [ 10 ] }, { "title": "4.3. Photometric Accuracy, Uncertainty, and Completeness", "content": "From the input and recovered fake star lists, we studed the photometric accuracy and uncertainty. We compared the photometry for two clusters, NGC 5053 (low stellar density cluster) and NGC 1851 (high stellar density cluster), as their HST ACS observations are very similar in exposure times: NGC 5053 (F606W: 340s × 5, F814W: 350s × 5) and NGC 1851 (F606W: 350s × 5, F814W: 350s × 5). In Figure 5, we plot the magnitude differences for input and recovered stars for F606W filter (1st row) and F814W filter (2nd row). These plots indicate photometric discrepancies, and all of them showing differences within ± 0 . 2 magnitudes. At fainter magnitudes, the magnitudes of the recovered stars tend to be fainter than the input ones. Row 3 and 4 in Figure 5 show the photometric uncertainties for F606W and F814W filters, respectively, while row 5 in Figure 5 shows the uncertainties in color (F606W-F814W). From those plots we can see that most stars in NGC 1851 have similar photometric accuracy and uncertainties as in NGC 5053, except that in NGC 1851 there are more stars with large errors, even for bright stars. Row 6 in Figure 5 shows the completeness curve (recovery percentage), which is the ratio of the total number of recovered stars (N(out)) to the total number of input stars (N(In)) in a particular magnitude interval. The completeness curve for NGC 5053 is quite flat down to the 26.5 magnitude, indicating an even star recovery percentage over a broad magnitude range. However, for NGC 1851, the star recovery percentage is not evenly distributed, with a relatively higher recovery percentage for brighter stars than for fainter ones. This can be seen from the fake CMDs in Figure 4, where there are fewer faint stars in NGC 1851 than in NGC 5053, even though they have similar exposure times. This indicates that faint stars are difficult to recover in dense stellar region due to high and non-uniform background caused by the multitude of stars.", "pages": [ 10, 11 ] }, { "title": "5. The High Mass-ratio Binary Fraction", "content": "A binary with a mass-ratio close to unity is easiest to identify because they binaries have the largest distances from the main-sequence ridge line. This is evident in an examination of the straightened CMD (i.e. the CMD minus the color of the MSRL), where the binaries are distributed to the right side of the main sequence below the turn-off point (Figure 6). The maximum distance is from equal-mass binaries, and this distance varies with the shape of the MSRL (the green line in Figure 6). The minimum separation from the main sequence is for binaries where the secondary is of low mass (mass-ratios approaching zero; see the blue solid line in Figure 6). When the photometric errors increase, the main-sequence stars eventually spread to the equal-mass binary region, diluting the binary signature (see Figure 6, right panel for example). This emphasizes the photometric accuracy needed to distinguish binaries from single main-sequence stars (such as stars in the region B of Figure 6).", "pages": [ 11 ] }, { "title": "5.1. The Model for the Superposition of Stars", "content": "The most important source of contamination in identifying binaries arises from the superposition of two unrelated stars that occurs by chance when the star cluster is projected from 3D to 2D. Two or more stars along the line of sight and within the minimum angular resolution will be measured as one star, and are indistinguishable on the CMD from real binaries. It is very difficult to screen for blended stars, but statistically, we can determine their distribution through Monte-Carlo simulations. The blending fraction is proportional to the radial stellar number density of a cluster, decreasing from the core to outside region. The general way to measure the blending fraction is through artificial star tests, in which fake stars from MSRL are added to real images and are recovered with the same photometric processes as real stars. Any fake stars overlapping with real stars will represent a blended source, and the recovered magnitudes will be their sum. As long as the added fake stars follow the radial light distribution of a globular cluster, it represents the similar blending fraction as that cluster. The drawback for this method is that it is computationally time-consuming. Consequently, we performed the Monte-Carlo simulations for an appropriate set of conditions and then modeled the results analytically, using Poisson statistics (see the Appendix). In each simulation, we added fake stars to the real star map, where the fake stars have the same luminosity function and light distribution as the real ones, so it should have a similar blending fraction as the observed cluster. To generate the fake star list, we chose fake stars with their V magnitudes sampled by the observed luminosity function, and their I magnitudes calculated from the MSRL (i.e. all the fake stars are on the MSRL and have the same luminosity function as the real stars). The number of total fake stars added is equal to that of the total observed stars to produce a similar condition. Each fake star was assigned a radius r from the center of the cluster, which was sampled by the radial light distribution of real stars, so that the radial light distribution of fake stars is similar to the real ones. Then the azimuth angle is randomly assigned to the fake star to calculate the coordinates (x,y). To that position (x,y), any real star with distance less than 1 resolution element (the minimum resolution size Dolphot used) will be considered as a blended star, and the total magnitude is added. To find those blended stars, we used our k-dimensional tree algorithm by comparing lists of the observed stars and fake stars. To those blended sources we added photometric errors from the observed stars (see Figure 5). We repeated this simulation 30 times to obtain an average number or an average residual color distribution of blended stars.", "pages": [ 12, 13 ] }, { "title": "5.2. The Model for the Field Star Population", "content": "Field stars, both faint foreground stars and bright background stars, can also contaminate the binary population on CMDs and thus affect the accuracy of the measurement of the binary fraction, especially for low Galactic latitude clusters. They affect the number of binaries in the binary region on the CMD, as well as the number of single stars on the main-sequence. High galactic latitude clusters do not have many contaminating field stars. For example, NGC 4590 ( b = 36 . 05 · ) only has 24 field stars (simulated from the model of Robin et al. (2003)) in the ACS field of view, or 0.06% of the total observed stars in that field. Low Galactic latitude clusters, however, have many contaminating field stars, such as NGC 6624 ( b = -7 . 91 · ), which has 51,325 field stars in the ACS field of view, about half of the total observed stars in that field. The best way to select cluster members is by proper motion, which, however, requires at least two epochs of HST observations separated by years. An alternative way is to construct the field star model from the theoretical model of the Galaxy to statistically account for those field stars. We used the Stellar population Synthesis model of the Galaxy (Robin et al. 2003) to simulate field stars at the cluster position, with the size of ACS CCD chips, 202 '' by 202 '' . The V and I Johnson-Cousin magnitudes of the generated field stars are corrected for the reddening first and are converted into ACS F606W and F814W magnitudes by the transformations of Sirianni et al. (2005), then are randomly added to the original images along with Poisson noise. Then we adopt the same photometry method with Dolphot to recover those field stars. The recovered field stars will have similar photometry errors as the cluster stars, as well as the completeness and blending effect.", "pages": [ 13, 14 ] }, { "title": "5.3. The Estimate of the High Mass-ratio Binary Fraction", "content": "To estimate the high mass-ratio binary fraction, we divided the MSRL-subtracted CMD into different regions to count stars (see Figure 6, left panel). In Figure 6, the green line is the equal-mass binary population. The red line on the right side of the main sequence is the binary population with a binary mass-ratio of 0.5. The binary mass-ratio of 0.5 is chosen because in most cases, this binary population is beyond the 3 σ photometric errors of the main-sequence stars. The dashed blue lines are the upper and lower limits of usable stars for both the F606W magnitude and the residual color. The main-sequence star (or single star) region S is defined between the red lines. The binary region B is where the residual color is greater than the 0.5 binary mass-ratio curve and less than 0.2 (i.e. on the right side of the red lines but on the left side of dash blue line). The residual region R is where the residual colors are beyond the blue side of the symmetric line for the 0.5 binary mass-ratio curve but greater than -0.2. We count stars separately in those regions for three types of star populations, observed stars, simulated blended stars, and simulated field stars. The simulated blended stars and field stars are generated with methods mentioned in Section 5.1 and 5.2. Region S contains mainly single stars (main-sequence stars), with some contaminating field stars. Region R contains main-sequence stars with large photometric errors, and with some contaminating field stars. Region B contains mainly binaries with mass-ratios greater than 0.5, with some blended stars and field stars. It also has some main-sequence stars with large photometric errors, and can be estimated with the number in region R. So the high mass-ratio binary fraction fb ( highq ) can be calculated by the expression: where n obs B , n blend B , and n field B are the star numbers in region B for the observed stars, blended stars, and field stars, respectively. n obs R and n field R are the star numbers in region R for observed stars and field stars, respectively. ( n obs R -n field R ) represents any residual main-sequence stars with large photometric errors. This number should be also subtracted from region B assuming a symmetric distribution. The quantities n obs t and n field t are the total star numbers in the dashed blue line box for observed stars and field stars, which is the whole restricted region we use during the analysis. The 1 σ error estimate for the binary fractions is estimated using Poisson errors and error propagation.", "pages": [ 14, 15 ] }, { "title": "6. The Global Binary Fractions", "content": "The high mass-ratio ( > 0 . 5) binary fraction discussion in Section 5 only accounts for those binaries with large deviations from the main seqeuence. Binaries with small mass ratios, however, are ignored, as they are too close to the main-sequence and are hidden by the photometric errors. In this section, we show that, to within certain limits, one can statistically recover even small mass-ratio binaries hidden in the main sequence as long as the photometric errors are and the rate of superposition of stars are understood.", "pages": [ 15 ] }, { "title": "6.1. The Star Counting Method", "content": "To avoid contamination from photometric errors, we followed three procedures. First, we only select stars 3 to 4 magnitudes below the turn-off point to exclude faint stars with larger photometric errors. Second, we add two gaussian models to represent the photometric errors during fitting, one for the main component with similar small errors, and the other for stars with larger errors. Tests show this to be a suitable strategy. Third, we introduce the parameter q cut in the binary model, which represents the minimum binary mass-ratio that can be extracted from the data. To estimate this, we first fit the residual color distribution with two Gaussian models, which represent the photometric errors of the main-sequence stars. The fit is only applied from 0.02 on the red side to all the blue side, since the blue side is not affected by both blending stars and real binaries, the broadening is only due to photometric errors. The fit is quite good (with χ 2 close to 1). The positive residual of the fitting on the red side is due to binaries, with contamination from blending stars and field stars. After subtracting the photometric errors, field stars, and blended stars, the residual color distribution on the red site is only from the contribution of physical binaries. We summed all the residuals on the red side and divided it by the total number of stars without the field stars, which gives the binary fraction including low mass-ratio binaries.", "pages": [ 15, 16 ] }, { "title": "6.2. The χ 2 Fitting Method", "content": "There is another way to account for low mass-ratio binaries, and one can even constrain the binary mass-ratio distribution function. Here, we constructed the binary population by fitting to an additional binary mass-ratio distribution function. We assume the binary mass-ratio distribution function has the following power-law form: f ( q ) ∝ q x , where q ≡ M s /M p , and with a minimum value of q min , and a maximum value of 1 (equal mass binaries). The minimum mass for the secondary star is set to 0.2 M /circledot (the observed lowest mass from luminosity function of F606W band), thus this will set a minimum value of q min not to be 0. First we assume here three different cases for the power of x , x = 0, a flat mass-ratio distribution; x = -1, which leads to a peak at low mass ratios; x = 1, which leads to a peak at high mass ratios. Whenever there are enough stars, we can fit for x rather than assign a value. To construct the physical binary population, we first assume a binary fraction of f b , or a total of N ∗ f b stars to be in the binary systems, where N is the total number of the observed stars. We then extract V magnitudes with the number N ∗ f b from the observed V magnitude luminosity function to be the V magnitudes of the primary stars. The V magnitudes of the secondary stars are calculated using a mass-ratio q extracted from an assumed mass-ratio distribution and an assumed mass-luminosity relationship: L ∝ M 3 . 5 . Their I magnitudes are derived from the MSRL, and the combined binary magnitudes are calculated. Each binary system has added to the photometric errors at their V and I magnitudes to simulate the photometric spreads. Finally, we can obtain the residual color distribution for the constructed binary population by subtracting the color of the MSRL. We repeat this process 30 times for each binary fraction f b , and construct the average profile for the physical binary population as the way we applied it to blended stars. Now, we have models for single stars (two gaussian models), field stars (see Section 5.2) , the superposition of stars (see Section 5.1), and binary model with the known fraction f b (see Section 6.2). The total sum of all these models will produce the final model that is compared to the observed residual color distribution profile with a χ 2 test. For a given cluster, the models for single stars, field stars, and the blending stars are fixed as they depend only on the observed luminosity function, the Galactic positions, and the observed light distribution. The only model that changes during the fitting process is the binary model, as it varies with the binary fraction f b , and the power law index x from the binary mass-ratio distribution function. We developed the bisection method to search the best-fit f b (i.e. fits with the minimum chi-square value) at each assumed x. In this method, we first chose a wide initial range of binary fraction values (initial range is from 0 to 1). The chi-square values (model comparing to observed counts) were calculated for three binary fraction values, left most, right most, and the middle. Then the chi-square values at the left and right were compared to the middle one, and the left most or right most binary fraction value will be replaced by the middle one if its chi-square value is greater than the middle one. So the search range for the binary fraction now is reduced by half, and we calculated the chi-square value at the middle for the new range, and repeated the process again until the difference of chi-square values or the binary fraction range approached limiting value. The final middle value of the binary fraction range is the best-fit binary fraction with the minimum chi-square value. For clusters with enough bins, we also fit the power of x. A fitting example is shown in Table 3, for NGC 4590. The first three rows in the table show the fitting results with the power x , and the minimum binary mass ratio q min fixed, only with the binary fraction as a free parameter in the fit. The error range in the table is estimated by changing the parameter so that the χ 2 value changes by 1.0 (or 1 σ confidence level). The best-fit value favors the model with the power x = -1, which also gives a higher value of binary fraction (10 . 8 ± 0 . 4)%. With the power x , and q min free to fit, the fit improves significantly ( χ 2 /dof = 88 . 4 / 82), with a binary fraction of (10 . 8 ± 0 . 3)%. Figure 7 shows the residual color distribution fitted by using only the Gaussian model (upper) and with the best-fit model (lower) for NGC 4590. The symmetric spread of the main-sequence is due to photometric errors, which can be fitted by the Gaussian model fairly well, as there are no large systematic residuals on the blue side. The asymmetric spread of the main-sequence is due to binary populations and blending of stars, which is shown as positive residuals on the red side on the upper panel. In the lower panel, models for binaries and blending of stars are included, and this best-fit model fits well to the observed data. Figure 8 shows the model components for the fitting (upper) and the enlarged view (lower) for NGC 4590. For this cluster, we estimate about 50 field stars in the field of view of HST, so they are negligible contaminant. In Table 4, we show the binary fractions estimated with the three methods discussed above, the high mass-ratio method (the second column), the star counting method (the third column), and the χ 2 fitting method (the fourth column), and we expect the binary fraction obtained with the global methods (the star counting method and the χ 2 fitting method) to be higher than the high mass-ratio method, as we approach lower mass-ratio values.", "pages": [ 16, 17, 18 ] }, { "title": "7. The Binary Fraction Radial Analysis", "content": "One prominent predicted effect of globular cluster dynamical evolution is mass segregation, which implies that massive stars tend to sink to the center of the core while light stars are redistributed to the outside of the cluster. Since a binary system contains two stars, it is more massive than a single star, so they tend to sink towards the cluster core by mass segregation. By performing radial analysis of binary fractions, we can test for this dynamical effect.", "pages": [ 19 ] }, { "title": "7.1. The Analysis Method", "content": "In this analysis, we divide the whole ACS field of view into three annular bins, with their centers at the cluster center obtained from Harris (1996; 2010 edition). The bin sizes were chosen (iteratively) so that each bin contains roughly one-third of the total stars recovered from the whole field, which leads to similar error bars for the binary fraction in each region.", "pages": [ 19 ] }, { "title": "7.2. The Radial CMD Qualities and the Example of the Results", "content": "For high density clusters, the CMD quality for the central bin is the worst among the three, which shows larger photometric spread and a lower faint star recovery rate than the other two. This is understandable, as the higher star not only increases the background level, making fainter stars more difficult to detect, but also increases the blending probability, making the PSF determination poorer. For low density clusters, we do not observe large variations in the CMD qualities. In Table 5 and Fig. 10, we show the results of the radial analysis on the binary fraction for NGC 6981 as an example. In Table 5, we list the sizes of the annular bins, binary fractions obtained by the high mass-ratio method, the counting method, and the χ 2 fitting method, and the dof/ χ 2 for the fitting method. In Fig. 10, we plot the binary fractions obtained from different methods and from different annular bins against their positions relative to the cluster center, in units of the half mass radius. We clearly see a decreasing trend of the binary fraction towards the outside of this cluster, an effect discussed for the sample of 35 globular clusters in Paper II.", "pages": [ 19, 20 ] }, { "title": "8. Additional Factors Affecting the Binary Fraction and Final Comments", "content": "In this section, we will discuss the additional factors that can affect the measurement accuracy of binary fractions in globular clusters using their CMDs.", "pages": [ 20 ] }, { "title": "8.1. The Photometric Errors", "content": "The key aspect to estimating the binary fraction using the CMD method is the highly accurate photometry for the CMDs. As the photometric errors become larger, the spread of the main-sequence becomes larger, which will smear out the signals from binaries with small mass-ratio. In order to decrease the photometric errors, we need to increase the exposure time. This is true for low density clusters, in which stars are quite isolated, and the PSF can be determined quite well. For most low density clusters in our sample, the photometric errors are approaching their theoretical limit. High density clusters, however, have very crowded central regions, where the high and varied background make the PSF determination uncertain. Most of the errors are not from low S/N ratios but from uncertainties in the PSFs and the backgrounds. Increasing the observing time for those crowded clusters would not help lower the photometric errors. Instead, developing more sophisticated PSF photometry algorithm for crowded region is needed, which is still not fully mature.", "pages": [ 20 ] }, { "title": "8.2. The Metallicity Dispersion", "content": "Theoretical modeling shows that the dispersion of the metallicity of a globular cluster can also cause a spread in color on the main sequence. Figure 9 shows that a metallicity ( Z ) difference of 0.002 (from red to blue line at certain V magnitude), equivalent to δ [ Fe/H ] = 0 . 30, can cause a spread of about 0.054 magnitude in color, and 0.284 in the V magnitude. So given the uncertainty in [Fe/H] of 0.03, the spread in color will be 0.005, and shift in V magnitude will be 0.028. The observed intrinsic color spread is smaller compared to the typical width in color of the main-sequence with the HST observations (about 0.012 for NGC 5053 in our sample), and the shift in V magnitude will not affect the color distribution. Thus the intrinsic metallicity dispersion can be negligible. For multi-population systems (such as NGC 2808), however, this will not be the case (Piotto et al. 2007; Pasquini et al. 2011).", "pages": [ 21 ] }, { "title": "8.3. The Differential Reddening", "content": "Observations on globular clusters located near the Galactic center can be affected by the existence of large and differential extinction of the foreground dust. Alonso-Garcia et al. (2011) discuss a technique to correct this effect. From our sample, most clusters are well above the Galactic plane and with the reddening E(B-V) less than 0.1, which are not important to spread the color of the main sequence comparing to their photometric errors. For clusters near the Galactic plane, however, the reddening can be very large. Along with the heavily contaminated field stars, the determination of binary fractions in those clusters are quite uncertain.", "pages": [ 21 ] }, { "title": "8.4. Comparison with Another Survey", "content": "At the same time as this work was being carried out, another group was working toward a similar goal and recently published their comprehensive work (Milone et al. 2012). Although both efforts follow established approaches that use the CMD, there are some differences, which we identify and compare the values derived from the two independent approaches. One important difference is the software used to obtain photometry in crowded fields, which makes extensive use of the psf libraries. We used DOLPHOT (V1.2) while Milone et al. (2012) used a the proprietary algorithms described by Anderson et al. (2008), developed specifically for crowded field photometry. We used the stellar field model of Robin et al. (2003) while Milone et al. (2012) used the model of Girardi et al. (2005). Both we and Milone et al. (2012) performed extensive artificial star tests although with slight differences in how completeness was defined and how finely the globular cluster stellar density was subdivided. Most other procedures were essentially identical, including spline ridge-line fitting to define the Main Sequence or the magnitude range of the Main Sequenced used for analysis. The two clusters described here have small reddening, so we did not need the sophisticated corrections applied by (Milone et al. 2012), nor were there multiple epochs of data to be considered. The two clusters that we discuss here were also analyzed by Milone et al. (2012) and we find general agreement, although the results are expressed slightly differently. For NGC 4590, our binary fractions for q > 0 . 5 and within the half mass radius was 6.2 ± 0.3 % while (Milone et al. 2012) obtain a somewhat lower value of 5.3 ± 0.7 %. For the total binary fraction, (Milone et al. 2012) doubles the f ( q > 0 . 5) value, which assumes a flat distribution in q , obtaining 10.6 ± 1.4 %. We fit a flat functional form to the CMD distribution and obtain a binary fraction of 9.4 ± 0.7 %. Since these are the same data sets, the differences, which are comparable to the uncertainties, most likely reflects systematic differences between the approaches. The other comparison that can be made is the radial distribution of NGC 6981, which we find drops by about a factor of 4-5 from a bin within r h to one that extends to 1.9 r h (from 9.6 ± 0.6% to 2.1 ± 0.3%). This decrease is similar to Milones mean result for their sample (Milone et al. 2012) but for this particular object, they find a smaller decline, from about 5% to 3%, with error bars of about 1% for each value. The reason for this difference is not obvious to us. In a companion paper, we will compare our sample to theirs, which will provide the statistical power to identify significant differences and systematic effects.", "pages": [ 22, 23 ] }, { "title": "9. Final Comments and Summary", "content": "Binary stars are thought to be a controlling factor in globular cluster evolution. To systematically study them, we conducted this survey of 35 Galactic globular clusters, taking advantage of the wealth of the HST data. In this paper, the first of two, we present the techniques used in obtaining their binary fractions. We used the PSF-fitting photometry with DOLPHOT (V1.2) to obtain high quality color-magnitude diagrams. We applied three different methods to estimate the binary fractions. The high mass-ratio method, a model-independent method, counts the number of binaries extending above a binary mass-ratio of 0.5 on the color-magnitude diagram. The star counting method also takes into account the low mass-ratio binaries after modeling the main-sequence population, star superposition, and the field stars. The χ 2 fitting method not only estimates the binary fraction, but also models the binary mass-ratio distribution. We showed a representative globular cluster NGC 4590, with a constrained binary fraction in the range of 6.2% to 10.8% by the three methods. To test the effect of globular cluster dynamical evolution, we introduced the binary fraction radial analysis with NGC 6981 as an example, which shows a decreasing trend of binary fraction towards the outside of this cluster. We also discussed the factors that could affect the accuracy of measuring the binary fraction with our methods. In Paper II, we will show the results of this survey, including accurate color-magnitude diagrams, the binary fractions within the core and the half mass radius obtained with three methods, the radial binary fraction analysis, and the potential binary candidate list for further observation. We will compare our observational results to the theoretical predictions of the globular cluster dynamical evolution.", "pages": [ 23, 24 ] }, { "title": "10. Acknowledgements", "content": "The authors would like to thank A.E. Dolphin for answering our many questions that arose when using the photometry package Dolphot V 1.2. We appreciate the many thoughtful suggestions from the referee, as well as from Mario Mateo, Jon Miller, Eric Bell, Sally Oey, and Patrick Seitzer. We gratefully acknowledge financial support through a HST grant from NASA.", "pages": [ 24 ] }, { "title": "REFERENCES", "content": "Alonso-Garcia, J., Mateo, M., Sen, B., Banerjee, M., & von Braun, K. 2011, AJ, 141, 146 Alonso-Garcia, J. 2010, Uncloaking Globular Clusters in the Inner Galaxy, (University of Michigan, PhD thesis, p.220) Albrow, M. D., Gilliland, R. L., Brown, T. M., Edmonds, P. D., Guhathakurta, P., & Sarajedini, A. 2001, ApJ, 559, 1060 Anderson, J., King, I. R., Richer, H. B., et al. 2008, AJ, 135, 2114 Bailyn, C. D. 1995, ARA&A, 33, 133 Bellazzini, M., Fusi Pecci, F., Messineo, M., Monaco, L., & Rood, R. T. 2002, AJ, 123, 1509 Clark, G. W. 1975, ApJL, 199, 143 Cohn, H. 1980, ApJ, 242, 765 Cohn, H. N., Lugger, P. M., Couch, S. M., Anderson, J., Cool, A. M., van den Berg, M., Bogdanov, S., Heinke, C. O., & Grindlay, J. E. 2010, ApJ, 722, 20 Cote, P., & Fischer, P. 1996, AJ, 112, 565 Cote, P., Pryor, C., McClure, R., Fletcher, J. M., & Hesser, J. E. 1996, AJ, 112, 574 Davis, D. S., Richer, H. B., Anderson, J., Brewer, J., Hurley, J., Kalirai, J. S., Rich, R. M., & Stetson, P. B. 2008, AJ, 135, 2155 Djorgovski, S., & King, I. R. 1986, ApJ, 305L, 61 Dolphin, A. E. 2000, PASP, 112, 1383 Fullton, L. K., Carney, B. W., Olszewski, E. W., Zinn, R., Demarque, P., Da Costa, G. S., Janes, K. A., & Heasley, J. N. 1996, ASPC, 92, 269 Fregeau, J. M., Grkan, M. A., Joshi, K. J., & Rasio, F. A. 2003, ApJ, 593, 772 Gao, B., Goodman, J., Cohn, H., & Murphy, B. 1991, ApJ, 370, 567 Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & da Costa, L. 2005, A&A, 436, 895 Goodman, J., & Hut, P. 1989, Nature, 339, 40 Gunn, J. E., & Griffin, R. F. 1979, AJ, 84, 752 Harris, W. E. 1996, AJ, 112, 1487 Heggie, D. C. 1975, MNRAS, 173, 729 Heinke, C. O. 2011, arXiv, 1101.5356 Hut, P., & Bahcall, J. N. 1983, ApJ, 268, 319 Hut, P., McMillan, S., Goodman, J., Mateo, M., Phinney, E. S., Pryor, C., Richer, H. B., Verbunt, F., & Weinberg, M. 1992a, PASP, 104, 981 Ji, J. 2011, Ph.D. Thesis, University of Michigan (Ann Arbor, MI) Lynden-Bell, D., & Eggleton, P. P. 1980, MNRAS, 191, 483 Lynden-Bell, D., & Wood, R. 1968, MNRAS, 138, 495 Mateo, M., Harris, H. C., Nemec, J., & Olszewski, E. W. 1990, AJ, 100, 469 Milone, A. P., Piotto, G., Bedin, L. R., et al. 2012, A&A, 540, A16 Pasquini, L., Mauas, P., Kaufl, H. U., & Cacciari, C. 2011, A&A, 531, 35 Piotto, G., Bedin, L. R., Anderson, J., King, I. R., Cassisi, S., Milone, A. P., Villanova, S., Pietrinferni, A., & Renzini, A. 2007, ApJL, 661, 53 Pooley, D., Lewin, W. H. G., Anderson, S. F., Baumgardt, H., Filippenko, A. V., Gaensler, B. M., Homer, L., Hut, P., Kaspi, V. M., Makino, J., Margon, B., McMillan, S., Portegies Zwart, S., van der Klis, M., & Verbunt, F. 2003, ApJL, 591, 131 Pooley, D., & Hut, P. 2006, ApJL, 646, 143 Pryor, C. P., Latham, D. W., & Hazen, M. L. 1988, AJ, 96, 123 Richer, H. B., Fahlman, G. G., Brewer, J., Davis, S., Kalirai, J., Stetson, P. B., Hansen, B. M. S., Rich, R. M., Ibata, R. A., Gibson, B. K., & Shara, M. 2004, AJ, 127, 2771 Robin, A. C., Reyl, C., Derrire, S., & Picaud, S. 2003, A&A, 409, 523 Rubenstein, E. P., & Bailyn, C. D. 1997, ApJ, 474, 701 Salaris M., & Weiss A., 2002, A&A, 388, 492 Sarajedini, A., Bedin, L. R., Chaboyer, B., Dotter, A., Siegel, M., Anderson, J., Aparicio, A., King, I., Majewski, S., Marn-Franch, A., Piotto, G., Reid, I. N., & Rosenberg, A. 2007, AJ, 133, 1658 Sirianni, M., Jee, M. J., Bentez, N., Blakeslee, J. P., Martel, A. R., Meurer, G., Clampin, M., De Marchi, G., Ford, H. C., Gilliland, R., Hartig, G. F., Illingworth, G. D., Mack, J., & McCann, W. J. 2005, PASP, 117, 1049 Sollima, A., Beccari, G., Ferraro, F. R., Fusi Pecci, F., & Sarajedini, A. 2007, MNRAS, 380, 781 Sollima, A. 2008, MNRAS, 388, 307 Spitzer, L. 1987, Dynamical Evolution of Globular Clusters (Princeton, NJ, Princeton University Press, 1987, 191 p) Yan, L., & Cohen, J. G. 1996, AJ, 112, 1489 Yan, L., & Mateo, M. 1994, AJ, 108, 1810 Yan, L., & Reid, M. 1996, MNRAS, 279, 751 Zhao, B., & Bailyn, C. D. 2005, AJ, 129, 1934", "pages": [ 25, 26, 27 ] }, { "title": "Appendix", "content": "The Superposition of stars is a contamination that has the same effect as real binaries on CMD. It is very difficult to screen for them, but statistically, we can estimate the number of blended stars. In this appendix, we will discuss the probability for different types of blended stars (i.e. unresolved doubles, triples, etc.) in globular clusters, which can provide a good estimate on the blending frequency for globular clusters at different stellar density.", "pages": [ 28 ] }, { "title": "1) Poisson Distribution", "content": "The probability that one star is blended with others depends on the projected 2D star number density as well as the angular resolution. It is a Poisson process and the probability can be described by the Poisson probability distribution function: where x is the companion number for the blends, i.e. x = 1 is for unresolved double stars (star with one companion), and x = 2 is for unresolved triple stars (star with two companions), etc. µ is the area ratio of the minimum resolved area to the mean occupied area per star in the reference frame.", "pages": [ 28 ] }, { "title": "2) Monte Carlo Blending Test", "content": "To test the hypothesis that the blending probability can be described by a Poisson distribution function, we performed the following Monte Carlo simulations. We randomly distributed N total stars in a fixed square area with each size of 100 pixels to form the reference frame. Secondly, we randomly added one test star to this reference frame. Then we counted how many reference stars are within the minimum resolution radius r min of the added test star, where r min = 1 to simulate the ACS image. If the count is more than 0, then the added test star will be considered as a blend. After counting, the test star was removed, and a new test star was randomly added to follow the above process. We added 5000 test stars in all for each simulation, and the final blending fraction is the total number of the blending stars divided by the total number (5000) of added test stars. Blends with different companion number were counted separately. We repeated this simulation 30 times (i.e. 30 different random distributions of the reference stars) for each star number density (i.e. each N total ) to get the mean blending fraction and the standard deviation. The simulation setup is shown in Figure 11. Results were compared to the Poisson probability distribution function (see Figure 12), where µ is defined as So for the fixed area with size L and the minimum resolution radius r min , µ only depends on the input star number N total . Here we calculated the blending probability for three different blending stars, x = 1 (unresolved double stars), x = 2 (unresolved triple stars), and x = 3 (unresolved quadruple stars), and compared to the Monte Carlo simulations (Figure 12). In Figure 12, we can see that the Poisson distribution curves match those MC data points fairly well. This indicates that as long as we know the projected star number density and the minimum resolution, we can estimate the blending fraction using the Poisson distribution function. For example, in our globular cluster sample, the maximum star number density is from NGC 7078, where we obtain 156,080 stars down to the 26th magnitude in the HST ACS CCD with the size of 4096 by 4096 pixels, which is equivalent to around 90 stars in the 100 by 100 pixel area of our Monte Carlo simulation setup (see the blue vertical dot-dash line in Figure 12). Even for this densest cluster, the blending fraction with one companion is less than 3%, and the blending fraction with two companions is much smaller, less than 0.04%. The typical star number recovered in our cluster sample is around 30,000 stars, which is equivalent to around 17 stars in the 100 by 100 pixel area. The blending fraction is less than 0.7% for blends with one companion, which is only a small fraction in the total binary fraction budget. The blending fraction with a higher number of companions is two orders of magnitude smaller. Note that here we used the average star number density to calculate the blending fraction for the whole field, which is not appropriate, as in globular clusters the star number density varies quickly along the radius. But as long as we only consider small range of radius, the density gradient will be small, and one can use this method to estimate the blending fraction in that area. The blending model in the fitting process, however, takes into account the stellar number density gradient, as it follows the observed stellar radial distribution. continued on next page", "pages": [ 28, 29, 30, 32 ] } ]

[ { "title": "ABSTRACT", "content": "MACHO-97-BLG-41 is a gravitational microlensing event produced by a lens composed of multiple masses detected by the first-generation lensing experiment. For the event, there exist two different interpretations of the lens from independent analyses based on two different data sets: one interpreted the event as produced by a circumbinary planetary system while the other explained the light curve with only a binary system by introducing orbital motion of the lens. According to the former interpretation, the lens would be not only the first planet detected via microlensing but also the first circumbinary planet ever detected. To resolve the issue using state-of-the-art analysis methods, we reanalyze the event based on the combined data used separately by the previous analyses. By considering various higher-order effects, we find that the orbiting binary-lens model provides a better fit than the circumbinary planet model with ∆ χ 2 ∼ 166. The result signifies the importance of even and dense coverage of lensing light curves in the interpretation of events. Subject headings: gravitational lensing: micro - planetary systems - binaries: general", "pages": [ 1 ] }, { "title": "REANALYSIS OF THE GRAVITATIONAL MICROLENSING EVENT MACHO-97-BLG-41 BASED ON COMBINED DATA", "content": "YOUN KIL JUNG 1 , CHEONGHO HAN 1 , 4 , ANDREW GOULD 2 , AND DAN MAOZ 3 1 Department of Physics, Institute for Astrophysics, Chungbuk National University, Cheongju 371-763, Korea 2 Department of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA and 3 School of Physics and Astronomy, Tel-Aviv University, Tel Aviv 69978, Israel Draft version April 24, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The last two decades have witnessed tremendous progress in gravitational microlensing experiments. On the observational side, improvement in both hardware and software has contributed to the great increase of the detection rate of lensing events from tens of events per year at the early stage of the experiments to thousands per year at the current stage. In addition, photometry based on difference imaging substantially improved the quality of photometry. Along with the observational progress, there also has been advance on the analysis side. A good example is the analysis of light curves of lensing events produced by multiple masses. The light curve of a single-lens event is described by a simple analytic equation with a small number of parameters and the lensing magnification varies smoothly with respect to the lensing parameters. As a result, observed light curves can be easily modeled by a simple χ 2 minimization method. However, when events are produced by multiple masses, modeling light curves becomes very complex not only because of the increased number of lensing parameters but also because of the non-linear variation of lensing magnification with respect to the parameters. The non-linearity of lensing magnifications is caused by the formation of caustics which denote positions on the source plane where the lensing magnification of a point source diverges. Caustics cause difficulties in lens modeling in two ways. First, they make it difficult to use a simple linearized χ 2 minimization method in modeling light curves because of the complexity of the parameter space caused by the singularity. Second, magnification computations for source positions on a caustic are numerically intensive. As a result, modeling a multiple lens event was a daunting task at the early stage of lensing experiments. However, with the introduction of efficient non-linear modeling methods such as the Markov Chain Monte Carlo (MCMC) algorithm, combined with advances in computer technology such as computer clusters or graphic processing units, precise and fast modeling became possible and now it is routine to model light curves in real time as lensing events progress (Dong et al. 2006; Cassan 2008; Kains et al. 2009; Bennett 2010; Ryu et al. 2010; Bozza et al. 2012). Furthermore, current modeling take into account various subtle higher-order effects. In this paper, we reanalyze the lensing event MACHO-97BLG-41 that is a multiple-lens event detected by the firstgeneration lensing experiment Massive Compact Halo Objects (MACHO: Alcock et al. 1993). For the event, there exist two different interpretations. Based mainly on the data obtained from the MACHO experiment, Bennett et al. (1999) interpreted the event as produced by a circumbinary planetary system where a planet was orbiting a stellar binary. On the other hand, Albrow et al. (2000), from independent analysis based on a different data set obtained by the Probing Lensing Anomalies NETwork (PLANET: Albrow et al. 1998) group, arrived at a different interpretation that the light curve could be explained without the introduction of a planet but rather by considering the orbital motion of the binary lens. According to the interpretation of Bennett et al. (1999), the lowestmass component of the triple-mass lens would be not only the first planet detected via microlensing but also the first circumbinary planet ever detected. Despite the importance of the event, the issue of its correct interpretation remains unresolved. Therefore, we revisit MACHO-97-BLG-41, applying state-of-the-art analysis methods to the combined data used separately by Bennett et al. (1999) and Albrow et al. (2000).", "pages": [ 1 ] }, { "title": "2. DATA", "content": "The data used for our analysis come broadly from two streams. The first stream comes from MACHO survey observations plus the Global Microlensing Alert Network (GMAN: Alcock et al. 1997) and Microlensing Planet Search (MPS: Rhie et al. 1999) follow-up observations. We refer to this data set as the 'MACHO data'. The other stream comes from observations conducted by the PLANET group. We denote this data set as the 'PLANET data'. In Table 1, we present tele- NOTE. - MSO: Mount Stromlo Observatory; CTIO: Cerro Tololo InterAmerican Observatory, SAAO: South Africa Astronomy Astronomical Observatory, ESO: European Southern Observatory. scopes, passbands, and the number of data of the individual data sets. Note that Bennett et al. (1999) analyzed only the MACHO data set while Albrow et al. (2000) conducted their analysis based only on the PLANET data set. In order to use data sets obtained from different observatories, we rescale error bars. For this, we first readjust error bars so that the cumulative distribution of χ 2 /dof ordered by magnifications matches to a standard cumulative distribution of Gaussian errors by introducing a quadratic error term (Bachelet et al. 2012). Then, error bars are rescaled so that χ 2 /dof becomes unity, where χ 2 is derived from the best-fit solution. During this normalization process, 3 σ outliers from the solution are removed to minimize their effect on modeling. In Figure 1, we present the light curve based on the combined MACHO + PLANET data. The light curve is characterized by two separate peaks at HJD ' = HJD -2450000 ∼ 619 and 654. We note that the models of Bennett et al. (1999) and Albrow et al. (2000) are consistent in the interpretation of the overall light curve including the peak at HJD ' ∼ 654. However, the two models differ in the interpretation of the peak at HJD ' ∼ 619. While Bennett et al. (1999) explained the first peak by introducing an additional lens component of a circumbinary planet, Albrow et al. (2000) described the peak by considering the orbital motion of the binary lens.", "pages": [ 1, 2 ] }, { "title": "3.1. Standard Binary-lens Model", "content": "To describe the observed light curve of the event, we begin with a standard binary-lens model. Basic description of a binary-lens event requires seven lensing parameters. The first three of these parameters describe the geometry of the lenssource approach including the time of the closest lens-source approach, t 0, the lens-source separation (normalized by the angular Einstein radius of the lens, θ E) at that moment, u 0, and the time scale for the source to cross the Einstein radius, t E (Einstein time scale). Another three parameters describe the binary nature of the lens including the projected separa- ion s (in units of θ E) and the mass ratio q between the binarylens components, and the source trajectory angle with respect to the binary axis α . The two peaks of the light curve of MACHO-97-BLG-41 are likely to be features involved with caustic crossings or approaches during which finite-source effect becomes important (Dominik 1995; Gaudi & Gould 1999; Gaudi & Petters 2002). To account for this effect, an additional parameter, the normalized source radius ρ ∗ = θ ∗ /θ E is needed, where θ ∗ is the angular source radius. In our standard binary-lens modeling, we additionally consider the limbdarkening variation of the source star surface brightness by introducing linear-limb-darkening coefficients, Γ λ . The surface brightness profile is modeled as S λ ∝ 1 -Γ λ (1 -3cos φ/ 2), where λ denotes the observed passband and φ is the angle between the normal to the surface of the source and the line of sight toward the source (Albrow et al. 1999). Based on the source type (subgiant) determined by the spectroscopic observation conducted by Lennon et al. (1997), we adopt coefficients from Claret (2000). The adopted values are Γ B = 0 . 793, Γ V = 0 . 666, Γ R = 0 . 575, and Γ I = 0 . 479 for data sets acquired with a standard filter system. For the MSO 1.3m data, which used a non-standard filter system, we adopt ( Γ B + Γ V ) / 2 for the B -band data and ( Γ R + Γ I ) / 2 for the R -band data. Although the two sets of solutions presented by Bennett et al. (1999) and Albrow et al. (2000) already exist, we separately search for solutions in the vast parameter space encompassing wide ranges of binary separations and mass ratios in order to check the possible existence of other solutions. From this, we find a unique solution with s ∼ 0 . 49 and q ∼ 0 . 48. See Table 1 for the complete solution of the model. We note that the solution is basically consistent with the results of Bennett et al. (1999) and Albrow et al. (2000) in the interpretation of the main part of the light curve including the peak at HJD ' ∼ 654. At the bottom panel of Figure 1, we present the residual of the standard binary model. It is found that there exist some significant residuals for the standard model. This is also consistent with the previous analyses that a basic binary model is not adequate to precisely describe the light curve.", "pages": [ 2, 3 ] }, { "title": "3.2. Higher-order Effects", "content": "The existence of residuals in the fit of the standard binary lens model suggests the need for considering higher-order effects. We consider the following effects. First, we consider the effect of the motion of an observer caused by the orbital motion of the Earth around the Sun. This 'parallax' effect causes the source trajectory to deviate from rectilinear, resulting in long-term deviations in lensing light curves (Gould 1992). The event MACHO 97-BLG-41 lasted ∼ 100 days, which is an important portion of the Earth's orbital period, i.e. 1 year, and thus the parallax effect can be important. Considering the parallax effect requires two additional parameters π E , N and π E , E , that are the two components of the lens parallax vector π E projected onto the sky along the north and east equatorial coordinates, respectively. The magnitude of the parallax vector corresponds to the relative lens-source parallax, π rel = AU( D -1 L -D -1 S ), scaled to the Einstein radius of the lens, i.e. π E = π rel /θ E (Gould 2004). Second, the orbital motion of a binary lens can also cause the source trajectory to deviate from rectilinear (Dominik 1998). The orbital motion causes further deviations in lensing light curves by deforming the caustic over the course of the event. The 'lens orbital' effect can be important for long time-scale events produced by close binary-lens events for which the event duration comprises an important portion of the orbital period of the lens system (Shin et al. 2013). To first order approximation, the lens orbital motion is described by two parameters, ds / dt and d α/ dt , that represent the change rates of the normalized binary separation and the source trajectory angle, respectively (Albrow et al. 2000). Third, we also check the possible existence of a third body in the lens system. Introducing an additional lens component requires three additional lensing parameters including the normalized projected separation, s 2, and the mass ratio, q 2, between the primary and the third body, and the position angle of the third body with respect to the line connecting the primary and secondary of the lens, ψ .", "pages": [ 3 ] }, { "title": "3.3. Result", "content": "We test models considering various combinations of the higher-order effects. In Table 2, we list the goodness of the fits and the best-fit parameters for the individual tested models. From the comparison of the models, we find the following results. First, we confirm the result of Albrow et al. (2000) that the consideration of the orbital effect substantially improves the fit. We find that the improvement is ∆ χ 2 ∼ 441 compared to the static binary model. When we additionally consider the parallax effect, the improvement of the fit, ∆ χ 2 ∼ 2 . 7, is very meager. This implies that among the two effects, the orbital motion of the lens is the dominant higher-order effect in explaining the residual from the standard model. In Figure 1, we present the model light curve on the top of the observed light curve and the residual of the model. In Figure 2, we also present the geometry of the lens system. Second, we find that the existence of a third body does not provide a fully acceptable fit. Our best-fit solution of three-body lens modeling is consistent with the solution of Bennett et al. (1999) in the sense that the third body is a circumbinary planet with a small mass ratio. See Table 2 for the best-fit parameters and Figure 2 for the lens system geometry. With the introduction of a planetary third body, the fit does improve from the standard binary model with ∆ χ 2 ∼ 270. The additional consideration of the parallax effect further improves the fit with ∆ χ 2 ∼ 9, but it is still substantially poorer than the orbiting binary-lens model with ∆ χ 2 ∼ 166. From the comparison of the residual (see Figure 1), it is found that the triple lens model cannot precisely describe the light curve during and before the first peak, 595 /lessorsimilar HJD ' /lessorsimilar 625. The key PLANET data that exclude the triple lens are from SAAO. Their smooth 'parabolic' decline signals a caustic exit along the axis of cusp. This is compatible with the triangular caustic generated by the close binary, but not with the quadrilateral caustic induced by the putative planet. In partic- ular, the near-symmetric shape of the quadrilateral caustic implies that a cusp-axis trajectory would have a near-symmetric excess flux in the approach to the first peak as after its exit, which is not seen in the pre-peak MSO data.", "pages": [ 3, 4 ] }, { "title": "4. CONCLUSION", "content": "We have conducted a reanalysis of the event MACHO-97BLG-41 for which there exist two different interpretations. From the analysis considering various higher-order effects based on the combined data sets used separately by the previous analyses, we find that the dominant effect for the deviation from the standard binary-lens model is the orbital motion of the binary lens. The result signifies the importance of even and dense coverage of lensing light curves for correct interpretation of gravitational lenses. For MACHO-97-BLG-41, the difference between the two previous interpretations partially stems from the poor coverage of the first peak that is important in the interpretation. Although a strategy based on survey plus followup observations can densely cover anomalies occurring at an expected time (e.g., peak of a high-magnification event) or long-lasting anomalies, it would be difficult to densely cover short-lasting anomalies arising abruptly at an unexpected moment. Since the first-generation lensing experiments, there has been great progress in survey experiments. The cadence of survey observations has increased from ∼ 1 - 2 per day to several dozens a day for the current lensing experiments (OGLE: Udalski (2003), MOA: Bond et al. (2001), Sumi et al. (2003), Wise: Shvartzvald & Maoz (2012)). Furthermore, a new survey based on a network of multiple telescopes (KMTNet: Korea Microlensing Telescope Network) equipped with large format cameras is planned to achieve a cadence of more than 100 per day. With improved coverage, the characterization of microlenses by future surveys will be more accurate. Work by CH was supported by Creative Research Initiative Program (2009-0081561)of National Research Foundation of Korea. AG was supported by NSF grant AST 1103471. DM and AG acknowledge support by the US Israel Binational Science Foundation.", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Albrow, M. D., Beaulieu, J. -P., Birch, P., et al. 1998, ApJ, 509, 687 Albrow, M. D., Beaulieu, J. -P., Caldwell, J. A. R., et al. 1999, ApJ, 522, 1022 Albrow, M. D., Beaulieu, J. -P., Caldwell, J. A. R., et al. 2000, ApJ, 534, 894 Alcock, C., Akerlof, C. W., Allsman, R. A., et al. 1993, Nature, 365, 621 Alcock, C., Allen, W. H., Allsman, R. A., et al. 1997, ApJ, 491, 436 Bachelet, E., Shin, I. -G., Han, C., et al. 2012, ApJ, 754, 73 Bennett, D. P. 2010, ApJ, 716, 1408 Bennett, D. P., Rhie, S. H., Becker, A. C., et al. 1999, Nature, 402, 57 Bond, I. A., Abe, F., Dodd, R. J., et al. 2001, MNRAS, 327, 868 Bozza, V., Dominik, M., Rattenbury, N. J., et al. 2012, MNRAS, 424, 902 Cassan, A. 2008, A&A, 491, 587 Claret, A. 2000, A&A, 363, 1081 Dominik, M. 1995, A&AS, 109, 597 Dominik, M. 1998, A&A, 329, 361 Dong, S., Depoy, D. L., Gaudi, B. S., et al. 2006, ApJ, 642, 842 Gaudi, B. S., & Gould, A. 1999, ApJ, 513, 619 Gaudi, B. S., & Petters, A. O. 2002, ApJ, 580, 468 Gould, A. 1992, ApJ, 392, 442 Gould, A. 2004, ApJ, 606, 319 Kains, N., Cassan, A., Horne, K., et al. 2009, MNRAS, 395, 787 Lennon, D. J., Mao, S., Reetz, J., et al. 1997, ESO Messenger, 90, 30 Rhie, S. H., Becker, A. C., Bennett, D. P., et al. 1999, ApJ, 522 1037 Ryu, Y.-H., Han, C., Hwang, K.-H., et al. 2010, ApJ, 723, 81 Shin, I. -G., Sumi, T., Udalski, A., et al. 2013, ApJ, 764, 64 Shvartzvald, Y. & Maoz, D. 2012, MNRAS, 419, 3631 Sumi, T., Abe, F., Bond, I. A., et al. 2003, ApJ, 591, 204 Udalski, A. 2003, Acta Astron., 53, 291", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "We present a measurement of the X-ray spectrum of the luminous X-ray binary in I Zw 18, the blue compact dwarf galaxy with the lowest known metallicity. We find the highest flux yet observed, corresponding to an intrinsic luminosity near 1 × 10 40 erg s -1 establishing it as an ultraluminous X-ray source (ULX). The energy spectrum is dominated by disk emission with a weak or absent Compton component and there is no significant timing noise; both are indicative of the thermal state of stellarmass black hole X-ray binaries and inconsistent with the Compton-dominated state typical of most ULX spectra. A previous measurement of the X-ray spectrum shows a harder spectrum that is well described by a powerlaw. Thus, the binary appears to exhibit spectral states similar to those observed from stellar-mass black hole binaries. If the hard state occurs in the range of luminosities found for the hard state in stellar-mass black hole binaries, then the black hole mass must be at least 85 M /circledot . Spectral fitting of the thermal state shows that disk luminosities for which thin disk models are expected to be valid are produced only for relatively high disk inclinations, /greaterorsimilar 60 · , and rapid black hole spins. We find a ∗ > 0 . 98 and M > 154 M /circledot for a disk inclination of 60 · . Higher inclinations produce higher masses and somewhat lower spins. Subject headings: black hole physics - galaxies: individual (blue compact dwarf, I Zw 18) - X-rays: binaries - X-rays: galaxies", "pages": [ 1 ] }, { "title": "A STATE TRANSITION OF THE LUMINOUS X-RAY BINARY IN THE LOW-METALLICITY BLUE COMPACT DWARF GALAXY I ZW 18", "content": "1 2 Philip Kaaret and Hua Feng Draft version October 9, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The early universe was deficient in heavy elements. Thus, understanding the metallicity dependence of Xray emission and compact object formation is important for our understanding of the early universe, particularly the mechanisms of reionization (Madau et al. 2004; Mirabel et al. 2011) and the effect of feedback from compact objects produced by star formation on the dynamics of early galaxies (Power et al. 2009). Recent models of the formation and evolution of stars and binaries with low metallicity suggest that their X-ray emission is significantly enhanced (Dray 2006; Dray & Tout 2007; Linden et al. 2010). These trends are supported by observations. Mapelli et al. (2010) demonstrated an increase in the number of X-ray bright HMXBs (for a given SFR) for moderately low metallicity galaxies. Kaaret et al. (2011) found a suggestion of enhanced X-ray luminosity (for a given SFR) in a sample of very-metal-poor blue compact dwarf galaxies. It has also been suggested that low metallicity environments may lead to the formation of unusually massive compact objects. Solar metallicity stars are expected to lose much of their mass via winds at the ends of their lives and produce remnants with masses less than about 20 M /circledot . Reduced metallicity decreases the effect of winds and leads to more massive remnants (Belczynski et al. 2010). This has been suggested to explain the increased occurrence of X-ray binaries with very high luminosities, the ultraluminous X-ray sources (Kaaret et al. 2001; Feng & Soria 2011), in low metallicity environments (Pakull & Mirioni 2003; Zampieri et al. 2004; Mapelli et al. 2009). Blue compact dwarf galaxies (BCDs) are physically small galaxies with blue optical colors and low metallicities (Kunth & Ostlin 2000; Wu et al. 2006). They are the best local analogs to early galaxies yet identified. I Zw 18 is a nearby BCD with the lowest known metallicity and the highest X-ray luminosity (Kaaret et al. 2011). The X-ray emission from I Zw 18 is dominated by a single X-ray binary and observations reported to date have shown a maximum luminosity below the Eddington luminosity of a 20 M /circledot compact object. Here, we report on a measurement of the X-ray spectrum of I Zw 18 that reveals the highest flux yet observed. The spectrum shows distinct curvature indicative of a black hole X-ray binary in the thermal state. We present our results on the X-ray spectrum in § 2 and discuss their implications for the nature of compact object in § 3.", "pages": [ 1 ] }, { "title": "2. OBSERVATIONS AND ANALYSIS", "content": "I Zw 18 was observed by the X-ray Multi-Mirror Mission (XMM-Newton) on 2002 April 10 for 32.8 ks and on 2002 April 16 for 28.9 ks. We reduced the data according to the standard procedures for imaging spectroscopy with the European Photon Imaging Camera (EPIC). Background flaring reduced the useful exposure for the first observation for the pn camera to 24.1 ks and for the two MOS cameras to 30.9 ks. Flaring was much worse for the second exposure and reduced the useful exposure for the pn to 5.3 ks. Thus, we chose to further analyze only the first observation. We extracted spectra using circular source regions with 30 '' radius centered on the X-ray source apparent in each image. This extraction region is larger than the extent of I Zw 18 and therefore includes all X-ray emission from the galaxy. Background subtraction was performed using a circular background region with a radius of 60 '' located Note . - The table includes: the model name, goodness of fit ( χ 2 ) and degrees of freedom (DoF), the intrinsic luminosity in the 0.3-10 keV band assuming a distance of 18.2 Mpc (Aloisi et al. 2007) and isotropic emission, absorption column density for the component with abundances fixed to Z/Z /circledot = 0 . 019, the disk temperature at the inner edge ( kT ) or the powerlaw cutoff energy ( E c ), and the photon index (Γ). on the same CCD chip. Together the 3 spectra contain about 3940 net counts. Spectral response files suitable for point source analysis were calculated using the most recent calibrations. The spectra were grouped to have a minimum of 16 counts per bin. We fitted the X-ray spectra using the xspec software package (Arnaud 1996). We used the 0.3-10 keV energy range for fitting. The Chandra X-ray Observatory observed I Zw 18 for 41 ks on 2000 February 8 and imaging and spectroscopic results were reported by Bomans & Weis (2002) and Thuan et al. (2004). The imaging shows that the X-ray emission was dominated by a single point source with at least 96% of the flux. Both groups found good fits with an absorbed power-law model, but residuals near 0.65 keV that they ascribed to an O III hydrogenlike line. Thuan et al. (2004) report an observed flux of 7 . 2 × 10 -14 erg cm -2 s -1 in the 0.5-10 keV band, a photon index of 2 . 01 ± 0 . 16, and an absorption column density, N H = (1 . 44 ± 0 . 38) × 10 21 cm -2 . We fitted the XMM spectra using an absorbed power- law model and found an observed flux of (2 . 78 ± 0 . 12) × 10 -13 erg cm -2 s -1 in the 0.5-10 keV band, a photon index of 2 . 31 ± 0 . 09, and an absorption column density, N H = (2 . 7 ± 0 . 3) × 10 21 cm -2 . Allowing the normalization to vary between the 3 detectors did not improve the fit, so we used the same normalization for all three detectors during all subsequent fitting. The marked increase in flux from the Chandra to the XMM observation shows that the emission is variable and likely predominantly due to a single X-ray binary. Thus, we consider only spectral models appropriate for X-ray binaries. Since the XMM extraction region includes the whole galaxy, it is possible that a second source, other than the one detected with Chandra, contributes a significant fraction of the flux observed in the XMM observation. Due to the low numbers of X-ray sources detected in similar galaxies (Kaaret et al. 2011), this is unlikely, but could be tested with a new Chandra observation. The absorption column density required for these spectral fits is well above the total Galactic H i column density towards I Zw 18 of N H = 2 . 5 × 10 20 cm -2 . Thus, most of the absorbing material likely resides in I Zw 18. The metallicity of I Zw 18 is measured via optical spectroscopy of H ii regions to be Z/Z /circledot = 0 . 019 (Izotov & Thuan 1999), where we have adopted a solar oxygen abundance of 12+log(O/H) = 8.9, while the abundances in the neutral interstellar medium are several times lower (Aloisi et al. 2003). The X-ray binary imaged with Chandra lies in a star formation region and the H ii region abundance is appropriate. The X-ray absorption model used above and also by Bomans & Weis (2002) and Thuan et al. (2004) assumes solar abundance and is likely an incorrect description of the true energy dependence of the absorption. In the fits below, we use two absorption components: one with solar abundances with a column density fixed to the Galactic H i column towards I Zw 18 (TBabs in xspec) and a second with a variable column density (TBvarabs), abundances fixed to Z/Z /circledot = 0 . 019, and redshift fixed to 0.00254. The results of fitting to various models are shown in Table 1. The only model which is excluded is the simple powerlaw, while the diskbb model is marginally excluded. A model consisting of the sum of these two components provides a good fit. The three best fitting models are statistically indistinguishable. They have an observed flux of 2 . 7 × 10 -13 erg cm -2 s -1 in the 0.3-10 keV band and the absorption column density within I Zw 18 is in the range 1.2-1.4 × 10 21 cm -2 . Lelli et al. (2012) mapped H i in I Zw 18 via the 21 cm line at 2 '' resolution and their ∗ maps show a total column density near 6 × 10 21 cm -2 at the X-ray source position. The lower N H values from the X-ray spectral fits are reasonable if the X-ray binary lies closer than the midplane of the galaxy or if the metallicity of the H i gas is lower than that of the H ii regions, suppressing X-ray absorption. We note that none of these model fits requires an emission line at 0.65 keV and we suggest that the line was due to an inappropriate choice of absorption model. The cutoff powerlaw model is empirical, but provides a simple analytical form, f ( E ) = E -Γ exp( -E/E c ). The improvement in the fit going from the powerlaw to the cutoff powerlaw indicates spectral curvature at high energies. The sum of a powerlaw and multicolor disk blackbody (diskbb) is often used to model the spectral of Galactic black hole X-ray binaries (Remillard & McClintock 2006). In our fitting, the photon index (Γ) was not well constrained, so we fixed Γ = 2. The powerlaw normalization was 2 . 3 × 10 -5 and the diskbb normalization was 7 . 8 × 10 -3 . The disk component produces 76% of the flux in the 2-10 keV band. The kerrbb model represents a sophisticated model of a thin accretion disk around a Kerr black hole including all relativistic effects and also self-irradiation of the disk due to light deflection (Li et al. 2005). The simpl model adds a Comptonization component calculated by convolution of the disk spectrum (Steiner et al. 2009). We fixed the source distance to 18.2 Mpc (Aloisi et al. 2007), the spectral hardening factor to 1.7 (Shimura & Takahara 1995), the torque at the inner disk boundary to zero, and included self irradiation and limb darkening in the model. The fit was insensitive to the black hole spin parameter ( a ∗ ), inclination ( i ), and Comptonization photon index (Γ). The best fitted spectrum for a ∗ = 0 . 9986, i = 60 · , Γ = 2 . 0 is shown in Fig. 1. The fitted parameters not included in Table 1 were: black hole mass, M BH = 249 +71 -52 M /circledot , mass accretion rate, ˙ M = (35 ± 3) × 10 18 g s -1 , and the Compton scattered fraction, 0 . 27 ± 0 . 13. We also performed fits with Γ = 2 . 6, to cover the range typically found in the thermal state, and none of the fit parameters changed significantly. We performed fits for spins over a range from a ∗ = 0 . 9 to the maximum achievable spin a ∗ = 0 . 9986 (Li et al. 2005) for inclinations of 60 · and 75 · . Lower inclinations and lower spins produce Eddington ratios, L/L Edd > 0 . 3, inconsistent with application of the kerrbb model. The spin was kept fixed for each individual fit. The χ 2 varied by less than 0.5 over this range. The Comptonized fraction did not vary significantly, with best fitted values in the range 0.26-0.33. The absorption column also did not vary significantly, with best fitted values in the range 1.0-1.2 × 10 21 cm -2 . The best fitted black hole mass ( M BH ) and accretion rate vary with both a ∗ and i . Fig. 2 show the variation in M BH and the ratio of disk luminosity to the Eddington luminosity ( L/L Edd ). For comparison with Gladstone et al. (2009), we fitted the XMM data in the 2-10 keV band with a powerlaw and a broken powerlaw, each without absorption. We note that our spectrum contains fewer counts, ∼ 4000, than the /greaterorsimilar 10 , 000 required by Gladstone et al. (2009)for inclusion in their sample. The powerlaw provides an adequate fit with χ 2 / DoF = 51 . 7 / 43 and a photon index Γ = 2 . 40 ± 0 . 16. The broken powerlaw provides only a slight improvement, χ 2 / DoF = 48 . 4 / 41 for an F-test value of 0.25. The fit is insensitive to the low energy photon index, so we fixed it to 1.4 (one less than Γ for the simple powerlaw). The break energy is then between the lower end of the fitting range, 2.0 keV, and 2 . 9 keV and the high energy photon index is 2 . 59 ± 0 . 25. We note that fitting the full energy range to a broken powerlaw model with absorption leads to a break energy of 1 . 9 ± 0 . 3 keV. We also fitted the data in the full energy range with the sum of a disk model (diskpn) and a Comptonization model (comptt) with absorption. We tied the Comptonization photon input temperature to the disk temperature and used an inner disk radius of 6 R G . The model produced a reasonable fit with χ 2 / DoF = 152 . 6 / 148 and a disk temperature kT = 0 . 5 +0 . 5 -0 . 2 keV. However, the other parameters were very poorly constrained. Only lower bounds were obtained for the Compton optical depth, τ > 0 . 05, and the plasma temperature, kT e > 1 . 14 keV. Gladstone et al. (2009) found significant improvement (F-test significance level of > 99%) of the best fitted model relative to a hot corona with kT e = 50 keV for the spectra of sources identified as in the ultraluminous state. Making the same comparison, we find ∆ χ 2 = 0 . 8 (DoF = 148) and an F-test significance level of 62%. Thus, the low temperature Compton component does not produce a statistically significant improvement in the fit, as required for the ultraluminous state. We reduced the Chandra data previously analyzed by Bomans & Weis (2002) and Thuan et al. (2004) and found a net 490 counts. Fitting with an absorbed powerlaw, we found results consistent within errors with those of Thuan et al. (2004). For comparison with the XMM-Newton results reported above, we also fitted the Chandra data with models with two absorption components, a Galactic component with solar abundances and N H = 2 . 5 × 10 20 cm -2 and a component for I Zw 18 well below any of those reported for the ultraluminous state. Also, a spectral model with a low temperature Compton component ( kT e = 1-3 keV) produces no statistically significant improvement in the fit relative to a model with a hot Compton component ( kT e =50 keV). These results suggest that the XMM spectrum is inconsistent with the criteria established for the ultraluminous state. However, we caution that the spectrum contains fewer counts than those used to establish the properties of the ultraluminous state. A higher quality spectrum with at least 3 × the number of counts would be needed to draw definitive conclusions. with Z/Z /circledot = 0 . 019 with variable N H . Fitting with a powerlaw, we found a good fit with χ 2 / DoF = 16 . 2 / 25, Γ = 1 . 78 ± 0 . 21, N H = 9 +10 -8 × 10 20 cm -2 , and a flux of 8 . 3 × 10 -14 erg cm -2 s -1 in the 0.3-10 keV band. Fitting with a cutoff powerlaw produced an essentially identical fit with a best fitted cutoff energy of 500 keV and a 90% confidence lower bound on the cutoff energy of 4.7 keV. Fitting with a diskbb model produced a poor fit with χ 2 / DoF = 36 . 7 / 25. Thus, we conclude there is no evidence for spectral curvature at high energies in the Chandra spectrum. We examined the timing properties of the XMM pn data. The 0.3-10 keV light curve binned in 450.8 s intervals appears constant, with χ 2 / DoF = 55 . 8 / 54. The rms power integrated over 0.01-1 Hz is less than 5% at 99% confidence. Due to the low number of counts, the Chandra data do not allow us to place useful constraints on the timing properties.", "pages": [ 1, 2, 3, 4 ] }, { "title": "3. DISCUSSION", "content": "The XMM-Newton observations described here show the highest flux yet observed from I Zw 18, corresponding to an intrinsic luminosity near 1 × 10 40 erg s -1 and establishing the X-ray binary in I Zw 18 as an ultraluminous X-ray source (ULX). The spectral shapes of the XMM versus Chandra data are different with the high flux (XMM) spectrum showing distinct curvature at high energies while the low flux (Chandra) spectrum shows no evidence of curvature, see Fig. 3. This is consistent with the state transitions seen in stellar-mass black hole X-ray binaries, specifically the transition between the hard X-ray spectral state and the thermal state. The lack of timing noise and low fraction ( < 30%) of powerlaw flux in the diskbb plus powerlaw model reported in Table 1 for the high flux state are consistent with its identification as the thermal state (Remillard & McClintock 2006). Gladstone et al. (2009) has suggested the existence of an 'ultraluminous' state based on curvature at high energies in the spectra of several ULXs. Broken powerlaw fits to the XMM spectra of I Zw 18 produce break energies State transitions have been reported in ULXs that have been modeled and interpreted as being unlike those of stellar-mass black hole X-ray binaries. (Pintore & Zampieri 2012) analyzed multiple XMM observations of the two ULXs in NGC 1313 and classified their spectra into two states: the 'thick corona' state and the 'very thick corona' state. The spectra in both states are dominated by a coronae with kT e in the range 16 keV. Other authors have reported on spectral evolution in ULXs with the common characteristic that a Compton component with kT e > 1 keV, contributes a major and usually dominant fraction of the flux (Feng & Kaaret 2009; Kajava et al. 2012). In the XMM spectra of I Zw 18, a Compton component modeled as a powerlaw produces a low fraction of the X-ray flux and a Compton component with a low temperature ( kT e = 1-3 keV) produces no statistically significant improvement in the fit. Thus, the spectral state transitions in most ULXs appear different from that seen in I Zw 18. While rare, the thermal state and transitions between the hard state and the thermal state are sometimes found in ULXs. The strongest case for such a transition is for M82 X-1 which shows both a spectral transition and a simultaneous change in the X-ray timing properties of the source as seen in stellar-mass black hole X-ray binaries (Feng & Kaaret 2010). Spectral evidence for the hard/thermal state transition has also been presented for the extremely luminous X-ray binary in ESO 243-49 (Servillat et al. 2011) and for M82 X37.8+54 (Jin et al. 2010). Spectral evidence for the thermal state was found for NGC 247 X-1 (Jin et al. 2011) and at lower luminosities in CXOM31 J004253.1+411422 (Middleton et al. 2012). Detection of a state transition similar to those seen in stellar-mass black hole X-ray binaries strengthens interpretation of the hard spectrum seen in the Chandra observation as evidence that the source was in the hard state. The maximum luminosities observed from stellar-mass black holes in the hard state are less than 0 . 3 L Edd (Rodriguez, Corbel, & Tomsick 2003; Zdziarski et al. 2004; Yuan et al. 2007; Miyakawa et al. 2008). The luminosity of the binary in I Zw 18 while in the hard state was 3 . 3 × 10 39 erg s -1 . This corresponds to L < 0 . 3 L Edd only if M BH > 85 M /circledot . In the thermal state, the inner radius of the accretion disk is set by the mass and spin of the black hole. As described above, we found that the spectral fits with simpl*kerrbb model were insensitive to the black hole spin ( a ∗ ) and inclination ( i ), thus, we considered a range of values for a ∗ and i . For i < 60 · , the source is super-Eddington for all spins. The accretion disk is ex- to be geometrically thin, and thus the kerrbb model is expected to be valid, only for L/L Edd < 0 . 3 (McClintock et al. 2006), so we considered only fits with i > 60 · . Fig. 2 shows the dependence of black hole mass ( M BH ) and disk luminosity relative to Eddington ( L/L Edd ) on a ∗ for i = 60 · . We find that L/L Edd ≤ 0 . 3 (within errors) for a ∗ > 0 . 98 with M BH = 195 +57 -41 M /circledot . For higher spins, L/L Edd decreases and M BH increases. For i = 75 · , we find that L/L Edd ≤ 0 . 3 for a ∗ > 0 . 92 with M BH = 288 +78 -57 M /circledot . Again, L/L Edd decreases and M BH increases for higher spins. In conclusion, the X-ray spectrum from the binary in I Zw 18 while in the high flux state can be interpreted in terms of a non-rotating black hole with super-Eddington emission and a mass similar to that of the most massive known stellar-mass black hole (Silverman & Filippenko 2008; Prestwich et al. 2007). However, if the nature of the state transition seen from the source is similar to those seen from stellar-mass black hole X-ray binaries, including the luminosity threshold for the transition, then the compact object is likely a near maximally-rotating black hole with an unusually high mass M BH > 85 M /circledot . This is close to the maximum mass of 75 M /circledot predicted for black holes formed via stellar collapse in a low metallicity ( Z/Z /circledot = 0 . 019) environment (Belczynski et al. 2010). Modeling of the thermal state spectrum suggests a higher mass, M BH > = 154 M /circledot , in the IMBH range. We note that the X-ray source is coincident with a massive star cluster, sites suggested as potential places of original for IMBHs (Portegies Zwart et al. 2004), particularly in lowmetallicity environments (Mapelli et al. 2013). Due to their similarities to early galaxies, identification of an IMBH in a BCD would be of particular interest for our understanding of the formation of early galaxies and supermassive black holes (Reines et al. 2011). We thank the referee for comments that improved the paper. This research has made use of data obtained from the Chandra Data Archive and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa.", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Aloisi, A., Savaglio, S., Heckman, T.M. et al. 2003, ApJ, 595, 760 Aloisi, A. et al. 2007, ApJ, 667, L151 Arnaud, K.A. 1996, ASP Conf. Proc. 101, 17 Bomans, D.J., & Weis, K. 2002, ASP Conf. Proc. 262, 141 Belczynski, K. et al. 2010, ApJ, 714, 1217 Dray, L.M. 2006, MNRAS, 370, 2079", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "We investigate the saturation level of hydromagnetic turbulence driven by the magnetorotational instability in the case of vanishing net flux. Motivated by a recent paper of Bodo, Cattaneo, Mignone, & Rossi, we here focus on the case of a non-isothermal equation of state with constant thermal diffusivity. The central aim of the paper is to complement the previous result with closure parameters for mean-field dynamo models, and to test the hypothesis that the dynamo is affected by the mode of heat transport. We perform computer simulations of local shearing-box models of stratified accretion disks with approximate treatment of radiative heat transport, which is modeled via thermal conduction. We study the effect of varying the (constant) thermal diffusivity, and apply different vertical boundary conditions. In the case of impenetrable vertical boundaries, we confirm the transition from mainly conductive to mainly convective vertical heat transport below a critical thermal diffusivity. This transition is however much less dramatic when more natural outflow boundary conditions are applied. Similarly, the enhancement of magnetic activity in this case is less pronounced. Nevertheless, heating via turbulent dissipation determines the thermodynamic structure of accretion disks, and clearly affects the properties of the related dynamo. This effect may however have been overestimated in previous work, and a careful study of the role played by boundaries will be required. Subject headings: accretion, accretion disks - dynamo - magnetohydrodynamics (MHD) - turbulence", "pages": [ 1 ] }, { "title": "DYNAMO EFFECTS IN MAGNETOROTATIONAL TURBULENCE WITH FINITE THERMAL DIFFUSIVITY", "content": "OLIVER GRESSEL NORDITA, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 106 91 Stockholm, Sweden Draft version March 23, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "There are few concepts in classical physics that are equally fundamental as the conservation of angular momentum. One formidable consequence of this law is the formation of gaseous accretion disks around a wide range of astrophysical objects. Yet to explain observed luminosities based on the release of gravitational binding energy (King et al. 2007), a robust mechanism is required to circumvent the consequences of angular momentum conservation. Being sufficiently ionised, these discs harbor dynamically important magnetic fields, which render the disk unstable to a mechanism called magnetorotational instability (MRI, Balbus & Hawley 1998), releasing energy from the differential rotation and converting it into turbulent motions. Ultimately, these motions are what is powering the redistribution of angular momentum on large scales. In the absence of externally imposed large-scale fields, a separate dynamo mechanism may be required to replenish sufficiently coherent fields to drive sustained MRI turbulence (Vishniac 2009). This requirement becomes particularly apparent in models that neglect the vertical structure of the disk. In this case a convergence problem has been encountered (Pessah et al. 2007; Fromang & Papaloizou 2007; Bodo et al. 2011), which has been attributed to the lack of an outer scale of the turbulence (Davis et al. 2010). An alternative explanation has been suggested by Kitchatinov & Rudiger (2010), who point out the problem of resolving the radial fine-structure related to non-axisymmetric MRI modes (required to circumvent Cowling's 'no dynamo' theorem). Convergence can naturally be recovered when accounting for the vertical stratification of the disk (Shi et al. 2010; Oishi & Mac Low 2011). Together with rotational anisotropy, the introduced vertical inhomogeneity can produce a pseudo-scalar leading to a classical mean-field dynamo (Brandenburg et al. 1995a). In a previous [email protected] paper (Gressel 2010, hereafter G10), we have inferred meanfield closure parameters from isothermal stratified MRI simulations (also see Brandenburg 2008) and demonstrated that the 'butterfly' pattern (see e.g. Miller & Stone 2000; Shi et al. 2010; Simon et al. 2012) typical for stratified MRI can in fact be described in such a framework. In the current paper, we aim to extend this line of work towards a more realistic thermodynamic treatment, including heating from turbulent dissipation (Gardiner & Stone 2005; Piontek et al. 2009) and crude heat transport. This new effort is largely initiated by a recent paper of Bodo, Cattaneo, Mignone, & Rossi (2012), hereafter BCMR. The authors make the intriguing suggestion that the treatment of the disk thermodynamics will have a strong effect on the dynamo and accordingly on the saturation level of the turbulence. This is in contrast to the result of Brandenburg et al. (1995b), who find that turbulent transport is not affected by the presence of convection. As a first step to improve the realism of MRI simulations compared to the commonly applied isothermal equation of state, BCMR assume thermal conduction with constant thermal diffusivity, κ . They then identify a critical value κ c below which the primary mode of vertical heat transport changes from being predominantly conductive to being dominated by turbulent convective motions. One motivation of the present work is to scrutinize these models and at the same time obtain mean-field closure coefficients for the two suggested regimes, thereby confirming the assumption that the thermodynamic treatment of the disk affects underlying dynamo processes. A second focus of our paper will be the authors' choice of impenetrable vertical boundary conditions, which we find to have profound implications for the resulting vertical disk equilibrium.", "pages": [ 1 ] }, { "title": "2. MODEL AND EQUATIONS", "content": "The simulations presented in this paper extend the simulation of Gressel (2010) to include a more realistic treat- ment of thermodynamic processes as pioneered by Bodo et al. (2012). Simulations are carried out using the secondorder accurate NIRVANA-III code (Ziegler 2004), which has been supplemented with the HLLD Riemann solver (Miyoshi & Kusano 2005) for improved accuracy. We solve the standard MHD equations in the shearing-box approximation (Gressel & Ziegler 2007) employing the finite-volume implementation of the orbital advection scheme as described in Stone & Gardiner (2010); for interpolation we use the Fourier method by Johansen et al. (2009). We here neglect explicit viscous or resistive dissipation terms but include an artificial mass diffusion term (as described in Gressel et al. 2011) to circumvent time-step constraints due to low density regions in upper disk layers. We remark that, owing to the totalvariation-diminishing (TVD) nature of our numerical scheme and the total energy formulation, heating via dissipation of kinetic and magnetic energy at small scales is accounted for even in the absence of explicit (or artificial) dissipation terms. Written in a Cartesian coordinate system ( ˆ x , ˆ y , ˆ z ) and with respect to conserved variables ρ , ρ v , and the total energy e = /epsilon1 + 1 2 ρ v 2 + 1 2 B 2 , with /epsilon1 being the thermal energy density, the equations read: with the total pressure given by p /star ≡ p + 1 2 B 2 , and a fixed external potential Φ( z ) = 1 2 Ω 2 z 2 . The inertial acceleration a i ≡ 2Ω( q Ω x ˆ x -ˆ z × v ) arises due to tidal and Coriolis forces in the local Hill system, rotating with a fixed Ω ≡ Ω 0 ˆ z , and where the shear-rate q ≡ dlnΩ / dln R has a value of -3 / 2 for Keplerian rotation. We furthermore assume an adiabatic equation of state, such that the gas pressure relates to the thermal energy density as p = ( γ -1) /epsilon1 , with the ratio of specific heats γ = 5 3 , as appropriate for a mono-atomic dilute gas. Finally, the temperature, T , appearing in the conductive energy flux, is obtained via the ideal-gas law p = ρT , where we chose units such that the factor ¯ µ m H / k B relating to the gas constant disappears. Following the approach taken by BCMR, we adopt a thermal conductivity, k , in terms of a constant diffusivity coefficient κ , related via For the isothermal run, we do not evolve Equation (3) but instead obtain the gas pressure via p = ρT 0 , with fixed temperature T 0 = 1 . Note that, in our units, T 0 differs from BCMR by a factor of two, owing the alternative definition of the initial hydrostatic equilibrium, which is in our case. For all simulations, we adopt the same box size of H × πH × 6 H at a numerical resolution of 32 × 96 × 192 grid cells in the radial, azimuthal, and vertical direction, respectively. This corresponds to a linear resolution of ∼ 32 /H in all three space dimensions. As typical for shearing box simulations, we initialize the velocity field with the equilibrium solution v = q Ω x ˆ y , and adiabatically perturb the density and pressure by a white-noise of 1% rms amplitude. The magnetic configuration is of the zero-net-flux (ZNF) type with a basic radial variation To obtain a uniform transition into turbulence, we further scale the vertical field with a factor ( p ( z ) /p (0)) 1 / 2 resulting in β P = const ( = 1600 initially). 1 Owing to the divergence constrain, this of course can only be done by introducing a corresponding radial field at the same time. In practice, we specify a suitable vector potential. Horizontal boundary conditions (BCs) are of the standard sheared-periodic type, and we correct the hydrodynamic fluxes to retain the conservation properties of the finitevolume scheme (Gressel & Ziegler 2007). For the vertical boundaries, we implement stress-free BCs (i.e., ∂ z v x = ∂ z v y = 0 ) with two different cases for the treatment of the vertical velocity component v z , namely: (i) impenetrable, and (ii) allowing for outflow (but preventing in-fall of material from outside the domain). We will demonstrate that this distinction will have profound implications for the resulting density and temperature profiles within the box. To counter-act the severe mass loss occurring in the case of open BCs, we continuously rescale the mass density, keeping the velocity and thermal energy density intact. 2 Unlike in earlier work (Gressel et al. 2012), which was adopting an isothermal equation of state, we here do not restore towards the initial profile, but simply rescale the current profile. This is, of course, essential to allow for an evolution of the vertical disc structure, owing to heating via turbulent dissipation. We remark that such a replenishing of material can be thought of as a natural consequence of radial mass transport within a global disk. As in BCMR, we use ∂ z B = 0 , B x = B y = 0 as boundary condition for the magnetic field, and impose a constant temperature T = T 0 at the top and bottom surfaces of the disk. The latter choice is motivated by the assumption that the upper disk layers are likely optically thin, and there exists a thermal equilibrium with their surroundings. As in previous work, we compute the density and thermal energy of the adjacent grid cells in the z direction to be in hydrostatic equilibrium.", "pages": [ 1, 2 ] }, { "title": "3. RESULTS", "content": "The main motivation of this paper is to reproduce, as closely as possible, the results of BCMR, where we then aim to establish mean-field dynamo effects for the contrasting cases of efficient versus inefficient thermal conduction, as studied there. Moreover, we shall begin to explore the impact of vertical boundary conditions, by studying the somewhat more realistic case allowing for a vertical outflow of material. We have performed in total four simulations: the two main simulations ('M2' and 'M3') adopt thermal diffusivity of κ = 0 . 12 , and κ = 0 . 004 , respectively, representing the regimes of efficient and inefficient thermal transport (at molecular level), respectively. For both these simulations, and for a third isothermal reference run (model 'M1'), we adopt outflow boundary conditions. To assess the impact of the imposed vertical BCs (see column 'vBC' in Table 1), and to make direct contact with previous work, we adopt a fourth model, 'M4', with a value κ = 0 . 004 , and impenetrable boundaries at the top and bottom of the domain (labeled 'wall' in the following).", "pages": [ 2, 3 ] }, { "title": "3.1. Comparison with BCMR", "content": "We ran the different models for approximately 300 orbital times, 2 π Ω -1 ; note that BCMR use time units of Ω -1 instead. All time averages are taken in the interval t = [50 , 300] . For the isothermal reference run, we obtain a time-averaged Maxwell stress, of (0 . 54 ± 0 . 14) × 10 -2 , which is comparable to the κ = 0 . 12 case with (0 . 63 ± 0 . 17) × 10 -2 . In the case κ = 0 . 04 with outflow boundaries we find a somewhat higher value of (0 . 81 ± 0 . 21) × 10 -2 , which is however significantly lower than in the otherwise identical case with impenetrable boundaries with (1 . 53 ± 0 . 34) × 10 -2 . While stresses are indeed increased (by approximately 30%) at lower conductivity, clearly, the effect of the treatment of the boundaries is much more significant. Due to the strong fluctuations, the relative amplitudes are best seen in Figure 2, where we plot time-averaged vertical profiles of M xy normalized to the initial midplane gas pressure, p 0 . Compared to the isothermal case M1, the thermally conductive model M2 has a very similar vertical structure. The low conductivity model M3, where turbulent overturning motions dominate, is not too different in terms of its vertical profile either. Markedly, model M4, with impenetrable vertical BCs, shows maxima of M xy around | z | = 1 . 5 H . This is similar to the profiles shown in figure 8 of BCMR, but note that their model shows a strong peak within 0 . 5 H of the vertical boundaries, while we only observe a very thin boundary layer in our model. At the end of their result section, BCMR point-out that the spatio-temporal behavior of the dynamo is remarkably different between the κ = 0 . 12 , and κ = 0 . 004 cases (see their figure 9). If we compare to run M4 with impenetrable boundaries (uppermost panel of Figure 3), we indeed find a similarly irregular butterfly diagram. BCMR argue that they find 'no evidence for cyclic activity or pattern propagation' in the conductive regime. In contrast to this, looking at Figure 3, it appears that even model M4 at some level shows a propagating dynamo wave. More importantly, models M2, and M3 (which presumably are in the conductive, and convective regimes, respectively) show extremely similar dynamo patterns and cycle frequency (middle and lower panels in Figure 3). This again suggests that the vertical boundaries have a profound effect on the mechanism of vertical heat transport and, as a consequence, on the dynamo. To establish the fact that our model M4 is indeed comparable to the corresponding simulation of BCMR, we now look at vertical profiles of the gas density and temperature. For the latter, BCMR had found a peculiar 'tent' shape (see their figure 4), i.e. a linear dependence of T on z , joined together at z = 0 by a parabolic segment. In Figure 4, we compile the T ( z ) profiles, and such a tent-like profile can indeed be seen for model M4. However, in the case of low thermal diffusivity, κ = 0 . 004 , we observe a much weaker deviation from the isothermal profile with outflow boundary conditions. Along with the tent-shaped temperature profile, BCMR found that heat transport from convection would erase the vertical density stratification and lead to a constant density near the disk midplane (see their figure 5). We observe a very similar density profile for model M4, which moreover shows strong density peaks at the domain boundary. These peaks are a consequence of enforcing the hydrodynamic fluxes to be zero at the domain boundaries. Unlike for impenetrable vertical boundaries, model M3 shows a more regular Gaussian density profile. For models M1-M3, the width of the bellshaped density profiles is consistent with the trend in temperature and reflects hydrostatic equilibrium. Apparently, such an equilibrium cannot be obtained if solid-wall boundaries are applied. BCMR conjecture that when κ crosses a critical value of κ c /similarequal 0 . 02 , the vertical heat transport changes from being mainly conductive to being predominantly convective. This was illustrated by their figure 6, where they compared the conductive heat flux with the mean turbulent heat flux where horizontal lines as usual indicate averages over the x and y directions. In Figure 6, we show the corresponding quantities for our non-isothermal runs: as in previous plots, model M4 agrees very well with the result of BCMR, but again differs significantly from the otherwise identical model M3 with outflow boundary conditions. Compared to M4, the net transport of heat is much reduced in the case of models M2, and M3. For clarity we plot these curves separately in the inset of Figure 6 with magnified ordinate. Much as expected, model M2 is dominated by a positive (i.e. outward) conductive heat flux, whereas the turbulent flux is negligible. Unlike model M4 with solid-wall boundaries, the low-conductivity model, M3, only shows a very moderate level of convective heat flux, and notably one of the opposite sign. This negative flux appears to largely balance its positive conductive counterpart, implying low levels of net-conductive heat transport towards the disk surface and net-convective transport towards the midplane. It is instructive to note that the heating predominantly occurs at z /similarequal ± 2 H , where the velocity and magnetic field fluctuations are highest, and not, as one might think, near the midplane.", "pages": [ 3, 4 ] }, { "title": "3.2. Mean-field coefficients", "content": "One objective of this work was to test the dependence of the dynamo on the mechanism of vertical heat transport. One way of doing this is to establish mean-field closure parameters for the new class of models with constant thermal diffusivity. Like in previous work, we utilize the test-field (TF) method (Schrinner et al. 2005, 2007) to measure coefficients such as the α effect, turbulent pumping, and eddy diffusivity. The used method (Brandenburg 2005) is 'quasikinematic' (Rheinhardt & Brandenburg 2010) in the sense that it has been found to remain valid into the non-kinematic regime in the absence of magnetic background fluctuations (Brandenburg et al. 2008) - whether this covers MRI is a topic of discussion. We here only briefly recapitulate the general framework of mean-field MHD and refer the reader to G10 for a more detailed description. A recent review about meanfield dynamos can be found in Brandenburg et al. (2012). For the shearing-box approximation, due to its periodic character in the horizontal direction, there are no characteristic gradients expected in the radial or azimuthal directions. The natural mean-fields are accordingly those, which only vary in the vertical direction. With respect to the velocity u = v -q Ω x ˆ y , the mean-field induction equation reads where we have ignored a contribution due to microscopic magnetic diffusivity. Note that the explicit x dependence in the shear term, q Ω x ˆ y , drops out once the curl operation is applied. Furthermore, B z = B z ( t = 0) = 0 because of flux conservation in the periodic box. In this description, turbulence effects due to correlated velocity and magnetic field fluctuations are embodied in the mean electromotive force which is typically parametrized as (Brandenburg 2005): Given explicit knowledge of the rank-two α and ˜ η tensors, this closure allows to formulate the mean-field induction equation (11) in terms of mean quantities alone, leading to the classical α Ω dynamo description, first applied to MRI turbulence by Brandenburg et al. (1995a). Tensor coefficients representing the closure parameters are presented in Figure 7 for model M2, with a high value κ = 0 . 12 of the thermal diffusivity. The obtained profiles are largely similar to the corresponding curves of the isothermal model M1 (not shown), which moreover agree 3 with previous isothermal results reported in G10. Notably, for model M2, the contrast between the disc midplane and the upper disc layers is somewhat less pronounced compared to the purely isothermal case. This trend is continued when going to lower thermal diffusivity (see Figure 8 below). Unlike reported in G10, we now find α xx and α yy to be predominantly of the same sign, which would argue in favor of a kinematic (rather than magnetic) origin of the effect. Note however the significant fluctuations in α xx , which cast some doubt on whether this coefficient can be meaningfully determined in the presence of shear. On the other hand, a fluctuating α should not be disregarded as a possible source of a mean-field dynamo (Vishniac & Brandenburg 1997). Before we proceed, we briefly discuss the remaining coefficients. In panel (b) of Figure 7, we show the off-diagonal tensor elements of the α tensor, which are dominantly symmetric, i.e. α xy /similarequal α yx . We remark that for the classical diamagnetic pumping effect, one would require anti -symmetric parity. The observed symmetry may however be interpreted as differential pumping, i.e. transporting radial and azimuthal field in opposite directions. For reference, we plot the mean vertical velocity u z (see dashed line), which additionally transports the mean field and hence leads to the characteristic acceleration in the butterfly diagram. The diagonal parts of the ˜ η tensor are shown in panel (c), where we also plot the rms velocity fluctuation (dashed line). Apart from the boundary layers, the turbulent diffusivity agrees well with the theoretical expectation that is, assuming a coherence time τ = 0 . 03 Ω -1 of the turbulence. Given the dominance of the azimuthal field, the coefficients ˜ η xx , and ˜ η yy are surprisingly isotropic (unlike predicted by Vainshtein, Parker, & Rosner 1993). It is however interesting to note that while ˜ η xy , shown in panel (d) of Figure 7, is identical to the diagonal elements of the diffusivity tensor in panel (c), its counterpart ˜ η yx is much smaller. With negative shear and both coefficients positive, the dynamo based on the Radler (1969) effect is decaying (cf. the dispersion relation in Brandenburg 2005, appendix B). This does however not exclude the possibility that ˜ η yx has an effect on the overall pattern propagation. We now proceed to the corresponding coefficients for the case of inefficient thermal conduction. In Figure 8 we accordingly show the α and ˜ η tensor components for model M3. We recall that in the quasi-isothermal case (cf. Figure 7) the α xx and α yy coefficients showed a trend to flatten and even reverse their slope near the midplane (also cf. figure 9 in Brandenburg 2008). This was reasoned to be related to a negative α effect due to magnetic buoyancy (Brandenburg 1998; Rudiger & Pipin 2000). In contrast, here the α xx curve shows a more monotonic dependence on z , indicating that such magnetic effects may be less pronounced in this case. Such a trend appears consistent with reduced magnetic buoyancy in the case of a stiffer effective equation of state. Moving on to panel (b), we note that α xy is now suppressed and even shows a slight trend to change its sign - indicating a possible significance of diamagnetic pumping in the adiabatic case. For the turbulent diffusivity plotted in panel (c), we observe a significant deviation from the classical expectation (dashed line), resulting in a nearly constant η T ( z ) . We conclude that the equation of state and the means by which energy is transported to the upper disk layers indeed have subtle effects on the inferred dynamo tensors. Which of the differences seen between Figures 7 and 8 is in the end responsible for the enhanced dy-n ty seen in model M3, will require further careful study. For completeness, in Figure 9, we also show the results for model M4, where dynamo coefficients are larger by a factor of several. This can be considered consistent with the much higher Maxwell stresses observed in this case. Unlike for model M3, ˜ η ( z ) now shows a pronounced z dependence, and α yy , and α yx are in fact negative 4 (for | z | < 2 H ) as suggested by Brandenburg (1998).", "pages": [ 4, 5, 6 ] }, { "title": "4. CONCLUSIONS", "content": "The primary goal of our work was to make contact with recent results by BCMR, and to complement their work with a direct measurement of mean-field dynamo effects via the TF method. Given that we have used a very similar numerical method and applied identical parameters (i.e. for model M4), it should not surprise the reader that we can satisfactorily confirm all aspects of the corresponding simulation by BCMR. Minor discrepancies arise with respect to the boundary layers, which may be related to the detailed treatment of the hydrostatic equilibrium there. Accordingly, we can confirm their main result, namely that - in the presence of impenetrable vertical boundary conditions - one observes a transition from a conductively dominated vertical heat transport to a state that is regulated by convective overturning motions. This transition obviously depends on the value of the applied constant thermal diffusivity κ . Like reported in BCMR, in the κ = 0 . 004 case, we observe a flat density profile (even with a slight minimum at the disk midplane), and a 'tent'-shaped temperature profile - presumably established by convective heat transport as a result of Rayleigh-Taylor-type instability. This case is also associated with a much increased dynamo activity (by an order of magnitude in the TF coefficients, not shown here), resulting in an overall Maxwell stress that is increased by a factor of three compared to the isothermal reference model. Unlike speculated by BCMR, we however do not think that the enhanced dynamo activity is related to the magnetic boundary conditions. This is despite the fact that such a connection indeed exists for the unstratified case (Kapyla & Korpi 2011), where the different magnetic boundary conditions serve to create an inhomogeneity in an otherwise translationally symmetric system. We rather attribute the different dynamo regime to the overall different hydrodynamic state - which however appears largely influenced by the choice of impenetrable boundary conditions. Aseparate set of models (M1-M3) with a more natural condition allowing the gas to flow out of the domain shows much less dramatic effects when going to the low thermal diffusivity regime. Naturally, one arrives at moderately hotter disk interiors along with more spread-out, yet still Gaussian density profiles. Dynamo TF coefficients are somewhat altered in this case, along with a roughly 30% higher turbulent Maxwell stress. Establishing a link (Blackman 2010) between the turbulent transport coefficients in the momentum equation (i.e. the Maxwell and Reynolds stresses) and the induction equation (i.e. the α effect, turbulent diffusion, etc.) will be key to understand magnetized accretion in a quantitative manner, and derive powerful closure models in the spirit of Ogilvie (2003) or Pessah et al. (2006). A possible direct extension of the existing models with varying amounts of thermal conductivity may be to cross-correlate 〈 M xy 〉 with e.g. α yy for various values of κ . Such a connection has been suggested by Brandenburg (1998) and been derived in the quasi-linear regime by Rudiger & Pipin (2000). A complication in this endeavor however arises from the fact that the TF coefficients are likely measured in a magnetically affected, i.e. quenched state. Given the dramatic effect of open versus closed vertical boundaries demonstrated in this paper, it will be of prime interest to study the connection between the disk and the launching of a magnetically driven wind (Ogilvie 2012), including a possible influence of the wind on the disk dynamo. The amount of recent work (Fromang et al. 2012; Moll 2012; Bai & Stone 2012; Lesur et al. 2013) illustrates the importance of this issue. In terms of the vertical boundaries imposed on the temperature, including a transition into an opticallythin disk corona will be important. Then a radiative boundary condition consistent with black-body radiation can be applied. To conclude, we want to emphasize that, clearly, the presented simulations can only be regarded as a first step towards a realistic treatment of the disk thermodynamics. Ideally, full-blown radiative transfer should be employed, and simulations of radiation dominated accretion disks (Blaes et al. 2011) demonstrate that this has indeed become feasible. The author acknowledges the anonymous referee for providing a well-informed report and wishes to thank Axel Brandenburg and Gianluigi Bodo for useful comments on an earlier draft of this manuscript. This work used the NIRVANA-III code developed by Udo Ziegler at the Leibniz Institute for Astrophysics (AIP). Computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Centre for High Performance Computing (PDC-HPC).", "pages": [ 6, 7 ] }, { "title": "REFERENCES", "content": "Bai, X.-N., & Stone, J. M. 2012, submitted to ApJ, arXiv:1210.6661 Balbus, S. A., & Hawley, J. F. 1998, RvMP, 70, 1 Brandenburg, A., Nordlund, ˚ A., Stein, R. F., & Torkelsson, U. 1995b, in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 462, Small-Scale Structures in Three-Dimensional Hydrodynamic and Gardiner, T. A., & Stone, J. M. 2005, in AIP Conf. Series, Vol. 784,", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "We performed the first numerical simulations of magnetorotational instability from a sub-magnetarclass seed magnetic field in core collapse supernovae. As a result of axisymmetric ideal MHD simulations, we found that the magnetic field is greatly amplified to magnetar-class strength. In saturation phase, a substantial part of the core is dominated by turbulence, and the magnetic field possesses dominant large scale components, comparable to the size of the proto-neutron star. A pattern of coherent chanel flows, which generally appears during exponential growth phase in previous local simulations, is not observed in our global simulations. While the approximate convergence in the exponential growth rate is attained by increasing spatial resolution, that of the saturation magnetic field is not achieved due to still large numerical diffusion. Although the effect of magnetic field on the dynamics is found to be mild, a simulation with a high-enough resolution might result in a larger impact. Subject headings: supernovae: general - magnetohydrodynamics (MHD) - Instabilities - methods: numerical - stars: magnetars", "pages": [ 1 ] }, { "title": "GLOBAL SIMULATIONS OF MAGNETOROTATIONAL INSTABILITY IN THE COLLAPSED CORE OF A MASSIVE STAR", "content": "H. Sawai 1 , S. Yamada 2 , and H. Suzuki 1 Not to appear in Nonlearned J., 45.", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The explosion mechanism of core-collapse supernovae (CCSNe) is still unresolved despite persistent efforts by many researchers over several decades. Recent state-ofart simulations show that the neutrino heating mechanism assisted by hydrodynamical instabilities revives the accretion shock, i.e., the explosion results in. However, estimated explosion energies are smaller than canonical value of order 10 51 erg (e.g., Marek & Janka 2009; Suwa et al. 2010). Meanwhile, effects of magnetic field on the explosion dynamics has been studied well for the decade. Numerical simulations assuming a strong poloidal magnetic field (typically 10 12 -10 13 G at the pre-collapse phase), and rapid rotation in most cases, show that magnetic force assists in driving the energetic explosion (e.g., Yamada & Sawai 2004; Obergaulinger et al. 2006; Burrows et al. 2007; Takiwaki et al. 2009; Sawai et al. 2013). The magnetic fields assumed in these simulations are so strong that the conservation of magnetic flux during collapse results in the field strength of /greaterorsimilar 10 15 G for the proto-neutron star surface. This is comparable to inferred surface magnetic fields of magnetar candidates (see Woods & Thompson 2006, for review of magnetars). At present, however, the strength of the magnetic field at the pre-collapse stage and the origin of strong magnetic field of magnetar are very uncertain. Stellar evolution simulations by Heger et al. (2005), which implement Tayler-Spruit dynamo model, show that the pre-collapse strength of the poloidal magnetic field in 15 M /circledot star is only 10 6 G. Meanwhile, recent observations report that some OB stars in the main sequence stage possess an ∼ 1 kG surface magnetic field (e.g., Wade et al. 2012), which corresponds to magnetar-class magnetic flux. Ferrario & Wickramasinghe (2006) car- [email protected] ried out a population synthesis calculation from main sequence stars to neutron stars, assuming the magnetic flux is conserved in the post-main-sequence evolution (fossil field hypothesis). Their result implies that ∼ 10% of OB stars have magnetar-class magnetic flux, while the majority have ∼ 1-2 orders of magnitude weaker one. Even when the pre-collapse magnetic flux is weak, corresponding to the magnetic field of /lessorsimilar 10 13 G for the proto-neutron star surface, MHD instabilities may amplify it to magnetar-class strength. So far, there have been a small number of works focusing on this issue. Thompson & Duncan (1993) argued that convective dynamo in proto-neutron stars generates a magnetar-class magnetic field. Simulations performed by Endeve et al. (2012) shows that standing accretion shock instability amplifies the magnetic field around the proto-neutron star surface from ∼ 10 12 G to ∼ 10 14 G. For rapidly rotating progenitors, magnetorotational instability (MRI, Balbus & Hawley 1991) may greatly amplify the magnetic field (Akiyama et al. 2003). Local box simulations characterizing a post-bounce core show that an initially weak magnetic field, ∼ 10 12 -10 13 G inside the proto-neutron star, exponentially grows due to MRI (Obergaulinger et al. 2009; Masada et al. 2012). 3 In simulations of MRI from a sub-magnetar-class seed magnetic field, /lessorsimilar 10 13 G, a quite fine grid size compared to the scale of the iron core, e.g., ∼ 1000 km, is required: The wavelength of maximum growing mode around the surface of the proto-neutron star is 3 In the simulations performed by Obergaulinger et al. (2009), although the size of the simulation box is small, ∼ 1 km, compared to the iron core, the global gradients of physical quantities are taken into account, by which they refer these simulations semi-global. where v A is Alfv'en velocity. In order to attain such a high resolution, local simulations are adopted in previous works (Obergaulinger et al. 2009; Masada et al. 2012). However, in local simulations, it is problematic to prepare a suitable background state: Although an initially stationary background is usually used in local simulations, post-bounce cores are dynamical. Additionally, local simulations are incapable of studying the global dynamics. To relieve these problems, global simulations are desirable. In this letter, we report the first global simulations of MRI in CCSNe from a seed magnetic field of submagnetar-class flux. To ease high computational cost demanded in global simulations, we carried out computations in axisymmetry and in limited radial range, 50500 km. Particular attention is paid to, (i) whether MRI amplify the magnetic field to magnetar-class strength even in the dynamical background of CCSNe, and (ii) whether the amplified magnetic field is strong enough to affect the dynamics.", "pages": [ 1, 2 ] }, { "title": "2. NUMERICAL METHODS", "content": "In the simulations, we solve the following ideal MHD equations, employing a time-explicit Eulerian MHD code, Yamazakura (Sawai et al. 2013): in which notations of the physical variables follow custom. Here, Φ is Newtonian mono-pole gravitational potential. Computations are done with polar coordinates in two dimension, assuming axisymmetry and equatorial symmetry. We adopt a tabulated nuclear equation of state produced by Shen et al. (1998a,b). Neutrinos are not dealt with in our simulations. The electron fraction, which is required to obtain the pressure of a fluid element from the EOS table, is given by a prescription suggested by Liebendorfer (2005). We should note that the Liebendorfer's prescription is only valid until bounce. Sawai et al. (2013) discussed that adopting this prescription in the post-bounce phase may lead to underestimation of pressure by upto 20%. Before simulating MRI, we first follow the collapse of a 15 M /circledot progenitor star (S. E. Woosley 1995, private communication) for the central region of 4000 km radius, until ∼ 100 ms after bounce (basic run). After the central density reaches 10 12 g cm -3 , the core is covered with N r × N θ = 720 × 30 numerical grids, where the spa- resolution is 0.4-23 km. The core is assumed to be rapidly rotating with the initial angular velocity profile of where r 0 = 1000 km and Ω 0 = 3 . 9 rad s -1 , corresponding to a millisecond proto-neutron star after collapse. The same dipole-like magnetic field configuration as one employed by Sawai et al. (2013) is initially assumed with the typical field strength of ∼ 10 11 G around the pole inside a radius of 1000 km, with which the strength of ∼ 10 13 G is obtained for the proto-neutron star surface. The initial rotational energy and magnetic energy divided by the gravitational binding energy are 5 . 0 × 10 -3 and 5 . 3 × 10 -7 , respectively. To conduct high resolution global simulations to capture the growth of MRI in the core, we restrict the numerical domain to a radial range of 50 < ( r/ km) < 500 (MRI run). The initial condition of MRI run is set by mapping the data of basic run onto the above domain, at 5 ms after bounce. At that time, the density at the inner-most grids is (1 . 5-2 . 2) × 10 11 g cm -3 . The inner and outer radial boundary conditions for MRI run are given by the data of basic run, except for B r . The boundary values of B r are given so that the divergence free condition of the magnetic field is satisfied. The grid spacing is such that the radial and angular grid sizes are same, viz. ∆ r = r ∆ θ , at the inner and outer most cells. We perform MRI runs with four different grid resolutions, where the grid sizes of the inner most cells, ∆ r min , (and the numbers of grids, N r × N θ ,) are 12.5 m (8900 × 6400), 25 m (4700 × 3200), 50 m (2300 × 1600), and 100 m (1200 × 800). Note that with above parameters, typical MRI growth wavelength around the inner boundary is several 100 m at the beginning of MRI runs. We stop each MRI run, before the shock surface reaches the outer boundary.", "pages": [ 2 ] }, { "title": "3. RESULTS", "content": "In each MRI run, we found a clear exponential growth of the poloidal magnetic energy during t ≈ 4-18 ms, where t = 0 ms corresponds to the beginning of MRI run (see left panel of Figure 1). The growth timescale is larger for higher grid resolution, but almost converges to τ MRI,num ≈ 8 ms at MRI run with ∆ r min = 25 m. Afterward, the poloidal magnetic energy gradually increases, and roughly saturates around t ≈ 50 ms, in which the saturation energy is larger for higher grid resolution. The evolution of toroidal magnetic energy does not show clear exponential growth in each MRI run. Although the growth rate is initially larger compared with that of the poloidal magnetic energy, it becomes smaller after the exponential growth of the poloidal magnetic energy sets in. At saturation, in MRI run with ∆ r min = 12 . 5 m, the poloidal magnetic energy is twice as large as the toroidal magnetic energy. We also follow the variation of the poloidal magnetic field structure, for MRI run with ∆ r min = 12 . 5 m. Around t ≈ 4 ms, when the exponential growth begins, magnetic field lines around the pole in the vicinity of the inner boundary start to bend. During the exponential growth phase, the magnetic field lines in this region are further stretched, and filaments of strong magnetic field, ∼ 10 14 G, appear (see panel (a) of Figure 2 for t = 11 ms). After the exponential growth ceases, the topology of magnetic field lines becomes rather tangled, which implies that the flow becomes turbulent (see panel (b) for t = 20 ms). Subsequently, filaments of strong magnetic field and the turbulent flow pattern, which first appear around the pole, also emerge in other regions. This phase corresponds to the duration of the gradual increase of the poloidal magnetic energy, t ≈ 18-50 ms, observed in Figure 1. After the poloidal magnetic energy saturates ( t ≈ 50 ms), a considerable fraction inside a radius of 150 km is dominated by turbulent flow, where the strength of the magnetic field reaches ∼ 10 14 -10 15 G in filamentous flux tubes (see panel (c) for t = 70 ms). The bent magnetic field lines, the exponential growth of the poloidal magnetic field, the fact the growth rate is larger for higher gird resolution, and the turbulence after the exponential growth, all invoke the occurrence of MRI in our simulations. As seen in panel (e) of Figure 2, the number of grids covering the maximum growing wavelength in MRI run with ∆ r min = 12 . 5 m is more than 20 in most areas, although it is less than five in some limited locations in the vicinity of the pole and equator. Thus, our highest-resolution MRI run seems almost capable to fairly capture the linear growth of MRI. According to Akiyama et al. (2003), the growth timescale obtained by the linear theory is where In panel (f) of Figure 2, the distribution of τ MRI,th for MRI run with ∆ r min = 12 . 5 m at 4 ms is depicted. It is found that, except for the very vicinity of the pole, where the grid resolution is not enough, the growth timescale is shortest, τ MRI,th ∼ 10 ms, around the pole in the vicinity of the inner boundary. This is consistent with the fact that magnetic field amplification first occurs there. Additionally, the numerically estimated growth timescale, τ MRI,num ≈ 8 ms (Figure 1), approximately coincides with the above theoretical value. We also estimated τ MRI,th without the effect of the gradient of entropy and the lepton fraction, by setting N = 0 in Equation (7), but found insignificant difference from the original one, which indicates that convection does not much contribute to amplify the magnetic field. With all the above facts, we conclude that amplification of poloidal magnetic field found in our simulations is mainly due to MRI. Since axisymmetry is assumed in our simulations, the toroidal component of the magnetic field is not directly amplified by MRI. The growth of the toroidal magnetic energy, which is slower than that of the poloidal magnetic energy, seems due to winding of magnetic field lines by differential rotation. In local simulations performed by Obergaulinger et al. (2009), the pattern of coherent channel flows appears during the exponential growth phase. In our simulations, on the other hand, such a pattern is never observed. Although chanel-like configurations are locally observed in the exponential growth phase (see panel (d) of Figure 2), they are incoherent, and soon disrupted in timescale of milliseconds. Such immediate disruption of channel flows seems to be caused by dynamical flow motions in the background, which is difficult to be taken into account in local simulations. In left panel of Figure 3, the energy spectra of the poloidal magnetic energy are plotted for MRI run with ∆ r min = 12 . 5 m at different evolutionary phases. Each spectrum is derived for radial wave numbers, k r , and for a region 50 < ( r/ km) < 100 and 15 · < θ < 60 · . At t = 4 ms, the spectrum shows the dominance of a large scale, /greaterorsimilar 50 km, just reflecting the structure of the background magnetic field. During the linear growth phase, t ≈ 4-18 ms, smaller scale structures, /lessorsimilar 10 km, grow fast compared with the larger ones. Consistently, we found that the maximum growing wavelength of MRI around the pole is generally ∼ 0 . 1-10 km during this phase. At the end of the simulation (saturation phase), the spectrum shows almost flat distribution for r /greaterorsimilar 5 km, and steep decay for r /lessorsimilar 1 km. In the range between them, a slope close to ∝ k -5 / 3 is observed, which seems to correspond to the inertial region of Kologomorov's theory. Such spectrum affirms that the flow is turbulent. Also, it indicates that large scale components of the magnetic field, comparable to the size of the proto-neutron star, are produced in our simulations. Since the maximum growing wavelength of MRI is generally smaller than this scale, as mentioned above, the large scale components are likely to be a result of inverse cascade. We also compare the average spectra of the poloidal magnetic energy for t = 65-70 ms among MRI runs (right panel of Fig 3). It is found that in a lower resolution run, the spectrum tends to turn the steep decay at larger scale, which seems to be due to a larger numerical diffusion. It is likely that this causes smaller spectral energy for a low resolution run over whole scales, and smaller saturation level observed in Figure 1. Figure 4 shows that magnetic pressure reaches /greaterorsimilar 10% of matter pressure in some locations, which implies that the magnetic field mildly affects the dynamics. Note, however, that we did not obtain the convergence of the saturation magnetic field, and a larger saturation level is attained for a higher resolution run (Figure 1). This suggest that the simulation with a high-enough resolution might result in a larger impact of the magnetic field on the dynamics.", "pages": [ 2, 3, 6 ] }, { "title": "4. DISCUSSION AND CONCLUSION", "content": "We have carried out the first global simulations of MRI in CCSNe from a sub-magnetar-class seed magnetic field, ∼ 10 13 G around the surface of the proto-neutron star, and rapid rotation, in which two-dimensional ideal MHD equations are solved with assumptions of axisymmetry and equatorial symmetry. As a result of computations, we found that MRI greatly amplifies the magnetic field to magnetar-class strength, ∼ 10 14 -10 15 G, and that the magnetic field at saturation phase is dominated by large scale structure. Although, the impact of the magnetic field on the dynamics found to be mild, we do not obtain the convergence of the saturation magnetic field strength, which is larger for a higher resolution run. A simulation with a higher resolution is necessary to assess the actual impact. We also found that the evolution of flow pattern in our global simulations are quite different from those appear in local simulations by Obergaulinger et al. (2009). Since MRI and turbulence are intrinsically nonaxisymmetric phenomena, three-dimensional simulations are, in fact, necessary to lead to conclusive results. Note, however, that even the world's best computers may only capable to simulate global MRI from a magnetar-class seed magnetic field in three dimension. Obergaulinger et al. (2009) found that the main difference between two-dimensional and three dimensional simulations is that the former results in a larger saturation magnetic field compared with the latter. They argued that the smaller saturation level in three dimension is caused by the disruption of coherent chanel flows before they grow well due to non-axisymmetric parasitic instabilities. Such difference might not appear in global simulations, since our simulations show that coherent chanel flows do not develop even in two dimension. While the present simulations set the position of the inner boundary at r = 50 km, we found in basic run that most rotational energy is reserved inside a radius of 50 km. It is expected that a simulation with the inner boundary located at a smaller radius would result in a larger saturation magnetic field. What we obtained in this study may be a lower limit. Although, in our simulations, a sub-magnetar-class magnetic field is amplified to magnetar-class strength due to MRI, we should be cautious in concluding that MRI can be the origin of magnetar fields. Before that, further investigations may be required on the sustainability of a large scale, strong magnetic field until, e.g., the formation of the neutron star. Additionally, it is worth investigating whether magnetar-class strength is also attained by MRI from a weaker seed magnetic field, e.g., < 10 12 G around the proto-neutron star surface. Note that previous local simulations of MRI, including ones in the context of accretion disks, show that a weaker seed magnetic field results in a lower saturation level (e.g., Obergaulinger et al. 2009; Okuzumi & Hirose 2011). The dependence of the saturation magnetic field on the rotational velocity, which also has been discussed in many local simulations (e.g., Masada et al. 2012), is another issue to be studied in future works. H.S. is grateful to Kenta Kiuchi for useful advise on MPI parallelization. Numerical computations in this work were carried out at the Yukawa Institute Computer Facility. This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan (20105004, 21840050, 24244036.)", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Akiyama, S., Wheeler, J. C., Meier, D. L., & Lichtenstadt, I. 2003, ApJ, 584, 954 Balbus, S. A. 1995, ApJ, 453, 380 Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214 Burrows, A., Dessart, L., Livne, E., Ott, C. D., & Murphy, J. 2007, ApJ, 664, 416 Endeve, E., Cardall, C. Y., Budiardja, R. D., et al. 2012, ApJ, 751, 26 Ferrario, L. & Wickramasinghe, D. T. 2006, Mon. Not. Astron. Soc., 367, 1323 Heger, A., Woosley, S. E., & Spruit, H. C. 2005, ApJ, 626, 350 Kitiashvili, I. N., Abramenko, V. I., Goode, P. R., et al. 2012, arXiv:1206.5300 Liebendorfer, M. 2005, ApJ, 633, 1042 Marek, A., & Janka, H.-T. 2009, ApJ, 694, 664 Masada, Y., Takiwaki, T., Kotake, K., & Sano, T. 2012, ApJ, 759, 110 Obergaulinger, M., Aloy, M. A., & Muller, E. 2006, A&A, 450, 1107 Obergaulinger, M., Cerd'a-Dur'an, P., Muller, E., & Aloy, M. A. 2009, A&A, 498, 241 Obergaulinger, M., & Janka, H.-T. 2011, arXiv:1101.1198 Okuzumi, S., & Hirose, S. 2011, ApJ, 742, 65 Shen, H., Toki, H., Oyamatsu, K., & Sumiyoshi, K. 1998a, Nuclear Physics A, 637, 435 Shen, H., Toki, H., Oyamatsu, K., & Sumiyoshi, K. 1998b, Progress of Theoretical Physics, 100, 1013 Sawai, H., Yamada, S., Kotake, K., & Suzuki, H. 2008, ApJ in press Suwa, Y., Kotake, K., Takiwaki, T., et al. 2010, PASJ, 62, L49 Yamada, S., & Sawai, H. 2004, ApJ, 608, 907 Takiwaki, T., Kotake, K., & Sato, K. 2009, ApJ, 691, 1360 Thompson, C., & Duncan, R. C. 1993, ApJ, 408, 194 Wade, G. A., Grunhut, J., Grafener, G., et al. 2012, MNRAS, 419, 2459", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Upon formation, degenerate He core white dwarfs are surrounded by a radiative H-rich layer primarily supported by ideal gas pressure. In this Letter, we examine the effect of this H-rich layer on mass transfer in He+C/O double white dwarf binaries that will eventually merge and possibly yield a Type Ia supernova (SN Ia) in the double detonation scenario. Because its thermal profile and equation of state differ from the underlying He core, the H-rich layer is transferred stably onto the C/O white dwarf prior to the He core's tidal disruption. We find that this material is ejected from the binary system and sweeps up the surrounding interstellar medium hundreds to thousands of years before the SN Ia. The close match between the resulting circumstellar medium profiles and values inferred from recent observations of circumstellar absorption in SNe Ia gives further credence to the resurgent double detonation scenario. Subject headings: binaries: close- novae, cataclysmic variables- nuclear reactions, nucleosynthesis, abundances- supernovae: general- white dwarfs", "pages": [ 1 ] }, { "title": "CIRCUMSTELLAR ABSORPTION IN DOUBLE DETONATION TYPE Ia SUPERNOVAE", "content": "KEN J. SHEN 1,2,5 , JAMES GUILLOCHON 3 , AND RYAN J. FOLEY 4,6 Accepted for publication in The Astrophysical Journal Letters", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Type Ia supernovae (SNe Ia) are responsible for nearly half of the heavy element production in the Universe (Timmes et al. 1995) and serve as probes of cosmic acceleration (Riess et al. 1998; Perlmutter et al. 1999). However, the nature of their progenitor systems is still a mystery. While researchers agree that SNe Ia involve the explosions of C/O white dwarfs (WDs), three main possibilities for the companion to the exploding WD remain in contention: a H-rich donor (typically referred to as a 'single degenerate' system; Whelan & Iben 1973; Nomoto 1982), another C/O WD ('double degenerate'; Iben & Tutukov 1984; Webbink 1984), and a He WD or Heburning star ('double detonation'; Livne 1990). Despite concerted effort, no consensus has yet been reached. Recent theoretical work (Schwab et al. 2012; Shen et al. 2012), in an update of initial studies (e.g., Nomoto & Iben 1985), suggests that the long term evolution of a double degenerate system does not yield a SN Ia but instead leads to the formation of a C-rich giant and then possibly to a collapse to a neutron star (Saio & Nomoto 1985). Furthermore, a wide variety of work, both theoretical (e.g., Shen & Bildsten 2007; Ruiter et al. 2009; Kasen 2010) and observational (e.g., Di Stefano 2010; Gilfanov & Bogdán 2010), suggests that single degenerate systems cannot be the dominant progenitor channel. While the double detonation scenario has been comparatively less well-studied, recent work (Fink et al. 2007, 2010; Sim et al. 2010) has resurrected interest in them and spurred ongoing research. One point seemingly in favor of single degenerate progenitors has come from recent observations of circumstellar material (CSM) surrounding 10 -30% of SNe Ia at distances of 0 . 1 -1 pc from the explosion and velocities of 50 -150 km s -1 (Patat et al. 2007; Blondin et al. 2009; Simon et al. 2009; Sternberg et al. 2011; Foley et al. 2012). The authors have concluded that the presence of CSM this close to the explosion implies some form of a H-rich single degenerate progenitor, since several variants of this scenario predict mass loss histories that might mimic the observed CSM distributions. However, we show in this Letter that H-rich material is also ejected hundreds to thousands of years prior to the merger of a He WD and a C/O WD, which may lead to a SN Ia in the double detonation scenario. For ambient interstellar medium (ISM) densities appropriate for spiral galaxies, this ejecta and the swept-up ISM yield a CSM profile that matches observed distances, velocities, and column densities. We also find that the lower ISM density in elliptical galaxies inhibits the formation of neutral Na in the CSM, which could explain the non-detection of neutral Na absorption in SNe Ia in ellipticals without dust lanes. We begin in Section 2 by describing the binary evolution that produces a He WD with a thin surface H layer. We derive the properties of the initially stable H-rich mass transfer from the He WD to the C/O WD and perform a time-dependent calculation of this accretion onto the C/O WD in Section 3. Prior to the merger of the He WD with the C/O WD, the H-rich accretion yields multiple ejection events akin to classical novae, which expel material into the surrounding ISM. We model the resulting CSM in Section 4, and conclude in Section 5.", "pages": [ 1 ] }, { "title": "2. EVOLUTIONARY SCENARIO AND PROGENITOR CHARACTERISTICS", "content": "Double detonations may arise in systems with a C/O WD primary via stable He mass transfer from a He-burning star (Livne 1990) or low mass He WD donor (Fink et al. 2007), or by unstable mass transfer from a higher mass He WD (Guillochon et al. 2010) or C/O WD donor (Pakmor et al. 2013). The subclass of double detonation progenitors we consider in this work are binaries with a relatively high mass 0 . 3 -0 . 45 M glyph[circledot] He WD and a 0 . 9 -1 . 2 M glyph[circledot] C/O WD. This range of He WD masses is predicted to yield a He detonation either via an accretion stream instability in the lead-up to the merger or during the disruption of the He WD (Guillochon et al. 2010; Dan et al. 2012), or possibly during the viscous evolution of the merger remnant (Schwab et al. 2012). Detonations of either - /circledot lower or higher mass C/O WDs underproduce or overproduce 56 Ni, respectively, as compared to typical SNe Ia (Sim et al. 2010; Ruiter et al. 2013). Throughout this Letter, we focus on a fiducial binary with a 0 . 40 M glyph[circledot] He WD and a 1 . 0 M glyph[circledot] C/O WD. Binaries with different components will yield quantitatively, but likely not qualitatively, different results, which we defer to future work. We construct our fiducial 0 . 40 M glyph[circledot] He WD with the stellar evolution code MESA 7 (Paxton et al. 2011) by truncating the evolution of a 1 . 0 M glyph[circledot] star with an initial metallicity of Z = 0 . 01 as it ascends the red giant branch (RGB). Chemical diffusion is active throughout the calculation. When the H-deficient core mass reaches 0 . 399 M glyph[circledot] , mass loss is turned on to rapidly remove the convective H envelope, yielding a 0 . 40 M glyph[circledot] He core WD. Upon formation, He WDs possess a thin, 2 × 10 -4 -2 × 10 -3 M glyph[circledot] H-rich remnant surface layer. This layer evolves under the action of cooling, chemical diffusion, and nuclear burning, yielding a radiative layer containing essentially pure H on top of a degenerate He core polluted with 14 N. We refer readers to previous studies (e.g., Althaus et al. 2001; Kaplan et al. 2012) for a comprehensive description of He WD evolution and instead describe our two fiducial He WD models, which evolve for 10 8 and 3 × 10 9 yr after the truncation of the RGBprior to the onset of mass transfer to the C/O WD. These two timescales are chosen to parameterize our uncertainties in the double WD binary's separation following common envelope evolution. The elapsed time between the formation of the He WD and the onset of mass transfer depends strongly on this separation and can range from a Hubble time for an initial separation of a = 1 . 3 × 10 11 cm to 10 8 yr for a separation of 3 . 8 × 10 10 cm. The initial separation and resulting merger timescale are critical for determining the mass transfer history once accretion begins. The H layer on an older He WD will be colder /circledot and thinner, and such a system will have a shorter delay between the onset of H-rich mass transfer and the disruption of the He WD. The cooling and contraction of the WD, as well as a small amount of residual H-burning, also affect the abundance structure within the surface layers. Figure 1 shows abundance profiles for our two He WDs. Since 3 He and 14 N act as catalysts for H-burning in classical and recurrent novae (Townsley & Bildsten 2004; Shen & Bildsten 2009), a proper calculation of the mass transfer history onto the C/O WD, which we describe in the next section, should take into account both the changing thermal properties and abundance profiles of the aging He WD.", "pages": [ 1, 2 ] }, { "title": "3. TIME DEPENDENT MASS TRANSFER RATE AND MASS EJECTION HISTORY", "content": "When mass transfer begins, the He WD's radiative H layer is transferred first. We follow the He WD donor's response to this mass transfer by removing mass at a constant rate of 10 -7 M glyph[circledot] yr -1 in MESA (see Kaplan et al. 2012 for a similar analysis of lower mass He WDs). The choice of mass removal rate in this calculation is essentially arbitrary because the thermal timescale in the H layer is > 10 6 yr; thus, for any accretion rate > 10 -10 M glyph[circledot] yr -1 , the thermal conditions in the bulk of the envelope are fixed at the start of mass transfer. Calculations with constant mass removal rates of 10 -5 and 10 -9 M glyph[circledot] yr -1 were also run as a check, with negligible differences in the results. The donor radius vs. removed mass is shown in the top panel of Figure 2 for our two models as labeled. The younger model remains hotter and thus has a larger radius. Both models have initial radii that are significantly larger than the zerotemperature analytic relation ( dotted line ; Nauenberg 1972). Once the 10 -3 -10 -2 M glyph[circledot] ideal gas, radiative layer has been stripped away, the degenerate He core expands upon further mass loss. The bottom panel of Figure 2 shows the absolute value of the differential radial response, ξ ≡ d ln R / d ln M . It is large and positive during the removal of the radiative H layer and approaches 0 as this layer is exhausted. As the radius of the mass-losing donor expands, ξ becomes negative and eventually approaches the value for a zero-temperature low mass WD( ξ glyph[similarequal] -1 / 2), shown as the dotted line. In addition to depending on the donor radius and differential response, the actual accretion rate must also take into account the efficiency of angular momentum exchange between the accretor and the orbit (Nelemans et al. 2001; Marsh et al. 2004). Assuming conservative mass transfer and angular momentum loss only due to gravitational wave radiation, the mass transfer rate is The extra term f ( q ), which accounts for the efficiency of angular momentum exchange (Verbunt & Rappaport 1988), has little effect on the pre-merger evolution. For our binary, the derived mass transfer rates for perfectly efficient and inefficient angular momentum feedback differ by less than 10% while ξ > 10. For ξ =1, their ratio is 1 . 82. Since mass transfer in our fiducial binary proceeds at all times via direct impact accretion instead of disk accretion (Lubow & Shu 1975), we assume that angular momentum feedback is inefficient for the remainder of this Letter.", "pages": [ 2, 3 ] }, { "title": "3.1. Accretion onto the C/O WD", "content": "Using equation (1), the donor's radial response shown in Figure 2, and a relation between the donor's Roche lobe and the orbital separation (Eggleton 1983), we can derive the actual mass transfer rate as a function of time. We show this accretion rate history in the top panel of Figure 3 for our two models. The accretion rate is plotted vs. the time prior to the WD merger, which we assume occurs when the accretion rate reaches the 1 . 0 M glyph[circledot] C/O WD's Eddington rate of 2 × 10 -5 M glyph[circledot] yr -1 . 8 8 Weassume that super-Eddington accretion will be driven from the system via dynamical friction with the donor. The resulting evolution of the Roche overfill factor (e.g., Marsh et al. 2004) implies a time of 1 -10 yr between the onset of super-Eddington accretion and the donor's disruption. However, We model this time-dependent mass transfer onto a 1 . 0 M glyph[circledot] C/O WD in MESA, taking into account the changing composition of accreted material shown in Figure 1, but ignoring the C/O WD's own relatively small H- and He-rich layers for simplicity. For faster convergence, the optical depth of the outer zone is moved inwards to 10 3 . As H-rich material piles up on the surface of the C/O WD, the density and temperature at the base of the accreted layer increase until convective H-burning is ignited, as in a classical nova. This causes the WD's radius to expand until it overflows its Roche radius. We assume this material is driven out of the system by its interaction with the companion He WD via a common envelope (e.g., Livio et al. 1990), and that the ejection velocity is roughly equal to the He WD's circular velocity of glyph[similarequal] 1500 km s -1 . The bottom panel of Figure 3 shows the mass ejection history prior to the disruption of the He WD. A total of 3 -6 × 10 -5 M glyph[circledot] of material is ejected from the binary in multiple ejections over the course of 200 -1400 yr prior to the merger. The evolution of this ejecta and the swept-up ISM is the subject of the next section.", "pages": [ 3 ] }, { "title": "4. EJECTA - ISM INTERACTION", "content": "The interaction of an expanding shell of material with the surrounding ISM has been studied in detail with respect to SN remnants (e.g. Chevalier 1977), classical and recurrent novae (e.g., Moore & Bildsten 2012), and tidal tails from double WD mergers (Raskin & Kasen 2013). Analytic solutions for such evolution are well-known; however, our situation is complicated by the existence of multiple ejection episodes, which will interact with previously shocked ISM. We thus defer discussion of analytic results to future work and instead present numerical hydrodynamic simulations. We employ a 1D, spherically symmetric, Eulerian hydrodynamics code that follows the zone-centered evolution of mass density, ρ , momentum density, ρ v , and energy density ρ v 2 / 2 + ρ u , where u is the specific internal energy. The equation of state only allows for ideal gas pressure because the medium is optically thin. The mean molecular weight is assumed to be the solar value, µ = 0 . 6, for the entire domain, which has a 10 15 cm spatial resolution. Fluxes are calculated with piecewise upwinded finite differencing. Artificial viscosity is included via the prescription of Tscharnuter & Winkler (1979). An optically-thin cooling function, Λ , is included for T > 10 4 Kbyfitting to the results of Gnat & Sternberg (2007). The qualitative results in this section have been verified by a similar Lagrangian hydrodynamics code. We calculate the interaction of our two fiducial binaries' ejecta with 10 2 K ISM at three mass densities: 0 . 1, 1, and 10 mp cm -3 , where mp is the proton mass. Ejection episodes are initiated by increasing the density in the inner r pert = 3 × 10 16 cm to a constant value of 3 M ej / 4 π r 3 pert , where M ej is the ejecta mass, and setting the velocity to v ej = 1500 km s -1 , which is roughly the He WD's circular velocity. The perturbed material's initial radius is chosen such that the ejecta is still essentially freely expanding. Each ejection event is evolved until the next ejection, as prescribed by the ejection history in Figure 3, after which the inner zones are again perturbed in a similar manner. In Figure 4, we show the mass density vs. distance from the SN at the time of merger for 6 simulations. We find that uncertainties such as the efficiency of angular momentum feedback and the actual Eddington rate may change this merger timescale. when the SN Ia occurs, the outgoing shock is at a distance of 0 . 1 -0 . 6 pc and the just-shocked ISM has velocities of 50 -300 km s -1 . These velocities correspond to post-shock temperatures of 6 × 10 4 -2 × 10 6 K. Thus, without substantial cooling, there will be a negligible amount of blueshifted, neutral Na. Significant cooling occurs in only one of our six simulations, which has the highest ISM density and the longest delay between the first ejection and merger. However, there will likely be significant clumping in the ejecta (e.g., due to Rayleigh-Taylor instabilities), which is unresolvable in our 1D simulation. Properly capturing the clumping will require multi-dimensional simulations, which we defer to future work. Since the cooling timescale, t cool ∼ kT / n Λ , is inversely proportional to density, clumping will enhance cooling in the shocked material and may allow for increased formation of neutral Na. For now, we approximate regions where neutral Na may exist in the CSM by the conditions T < 10 4 K or t cool < the simulation age at merger; these regions are shown in Figure 4 as thick lines. Note that while clumping will alter the CSM's density, it will not significantly affect its position or velocity. In Figure 5, we show the normalized differential column density per velocity bin for regions that might contain neutral Na, using the same approximation as in Figure 4. The three models that contain any blueshifted, possibly neutral, Na are shown in the three panels as labeled. This metric approximates the neutral Na absorption line profile and shows systemic velocity offsets of 50 -120 km s -1 . For comparison, the red dashed line in the middle panel shows the normalized average of the Na D1 and D2 absorption profiles for SN 2007le 84 d after B -band maximum (Simon et al. 2009). The velocity profiles of the possibly neutral Na resemble those seen in observations, and the Na column densities (4 × 10 11 -4 × 10 12 cm -2 ) overlap with those derived from SNe Ia showing variable absorption lines. However, our column densities are a factor of a few higher than the observed mean of Sternberg et al. (2011). Given our ad hoc inclusion of the effects of clumping and the unconsidered complications of time-dependent photoionization and recombination following the SN, our derived neutral Na column densities are merely suggestive. Future multi-dimensional work will enable more accurate predictions and possibly correct this mismatch. No narrow Na absorption has yet been detected in a SN Ia in an elliptical galaxy without obvious dust lanes (Foley et al. 2012). Our results agree with this finding for several reasons. Elliptical galaxies have lower ISM densities, which yield a smaller column of shocked material at the time of the explosion. Furthermore, the lower ISM density implies both lower post-shock densities as well as less shock deceleration, which means higher CSM temperatures when the SN Ia occurs. These factors increase the cooling timescale and decrease the amount of neutral Na.", "pages": [ 3, 4 ] }, { "title": "5. CONCLUSIONS", "content": "In this Letter, we have considered the effect of the H-rich layer that surrounds a He WD on the He WD's interaction with a C/O WD companion prior to a SN Ia. We have calculated its structure (Section 2), its impact on mass transfer and its ejection (Section 3), and the ejecta's interaction with the surrounding ISM (Section 4). We have found that if a SN Ia occurs when the He and C/O WDs merge, the characteristics of the CSM at the time of the explosion match recent observations of neutral Na surrounding 10 -30% of SNe Ia in spiral galaxies. We have also found that the lower ISM density in elliptical galaxies inhibits the formation of significant neutral Na in the CSM, which may be why these features have not been detected in such SNe Ia. SN ejecta have been observed to collide with surrounding CSM in several SNe Ia (e.g., Hamuy et al. 2003; Dilday et al. 2012; Silverman et al. 2013). This interaction requires significant material within glyph[lessorsimilar] 10 16 cm, which is difficult to produce in our model unless the SN Ia occurs < 2 yr after an ejection event. More detailed modeling and inclusion of the superEddington accretion phase just before the merger may help to shed light on these observations. While our results are promising, future studies are necessary to strengthen the findings. Further work will include an exploration of a range of WD masses and ages, simulations of the ejecta - ISM interaction in multiple dimensions, which will allow for clumping and non-spherical ejection, and calculations of the time-dependent photoionization and recombination after the SN Ia's UV flash. The effects of tidal heating, which will likely be significant for these extremely close binaries (e.g., Piro 2011), should also be considered. We thank Jason Dexter, Dan Kasen, Rodolfo Pérez, Eliot Quataert, Cody Raskin, and Jeff Silverman for discussions. KJS is supported by NASA through Einstein Postdoctoral Fellowship grant number PF1-120088 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Althaus, L. G., Serenelli, A. M., & Benvenuto, O. G. 2001, MNRAS, 323, 471 Chevalier, R. A. 1977, ARA&A, 15, 175 Dan, M., Rosswog, S., Guillochon, J., & Ramirez-Ruiz, E. 2012, MNRAS, 422, 2417 Di Stefano, R. 2010, ApJ, 712, 728 Dilday, B. et al. 2012, Science, 337, 942 Eggleton, P. P. 1983, ApJ, 268, 368 Fink, M., Hillebrandt, W., & Röpke, F. K. 2007, A&A, 476, 1133 Fink, M., Röpke, F. K., Hillebrandt, W., Seitenzahl, I. R., Sim, S. A., & Kromer, M. 2010, A&A, 514, A53 Foley, R. J. et al. 2012, ApJ, 752, 101 Gilfanov, M., & Bogdán, Á. 2010, Nature, 463, 924 Gnat, O., & Sternberg, A. 2007, ApJS, 168, 213 Hamuy, M. et al. 2003, Nature, 424, 651 Iben, Jr., I., & Tutukov, A. V. 1984, ApJS, 54, 335 Kaplan, D. L., Bildsten, L., & Steinfadt, J. D. R. 2012, ApJ, 758, 64 Kasen, D. 2010, ApJ, 708, 1025 Livio, M., Shankar, A., Burkert, A., & Truran, J. W. 1990, ApJ, 356, 250 Livne, E. 1990, ApJ, 354, L53 Marsh, T. R., Nelemans, G., & Steeghs, D. 2004, MNRAS, 350, 113 Nauenberg, M. 1972, ApJ, 175, 417 Nomoto, K. 1982, ApJ, 253, 798 Nomoto, K., & Iben, Jr., I. 1985, ApJ, 297, 531 Pakmor, R., Kromer, M., & Taubenberger, S. 2013, submitted (arXiv:1302.2913) Patat, F. et al. 2007, Science, 317, 924 Paxton, B., Bildsten, L., Dotter, A., Herwig, F., Lesaffre, P., & Timmes, F. 2011, ApJS, 192, 3 Perlmutter, S. et al. 1999, ApJ, 517, 565 Piro, A. L. 2011, ApJ, 740, L53 Raskin, C., & Kasen, D. 2013, ApJ, submitted (arXiv:1304.4957) Riess, A. G. et al. 1998, AJ, 116, 1009 Ruiter, A. J., Belczynski, K., & Fryer, C. 2009, ApJ, 699, 2026 Ruiter, A. J. et al. 2013, MNRAS, 429, 1425 Schwab, J., Shen, K. J., Quataert, E., Dan, M., & Rosswog, S. 2012, MNRAS, 427, 190 -. 2009, ApJ, 692, 324 Shen, K. J., Bildsten, L., Kasen, D., & Quataert, E. 2012, ApJ, 748, 35 Silverman, J. M. et al. 2013, ApJ, submitted (arXiv:1304.0763) Timmes, F. X., Woosley, S. E., & Weaver, T. A. 1995, ApJS, 98, 617 Communications, 18, 171 Verbunt, F., & Rappaport, S. 1988, ApJ, 332, 193 Webbink, R. F. 1984, ApJ, 277, 355 Whelan, J., & Iben, I. J. 1973, ApJ, 186, 1007", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "This paper examines the radiative transfer of synchrotron radiation in the presence of a magnetic field configuration resulting from the compression of a highly disordered magnetic field. It is shown that, provided Faraday rotation and circular polarization can be neglected, the radiative transfer equations for synchrotron radiation separate for this configuration, and the intensities and polarization values for sources that are uniform on large scales can be found straightforwardly in the case where opacity is significant. Although the emission and absorption coefficients must, in general, be obtained numerically, the process is much simpler than a full numerical solution to the transfer equations. Some illustrative results are given and an interesting effect, whereby the polarization increases while the magnetic field distribution becomes less strongly confined to the plane of compression, is discussed. The results are of importance for the interpretation of polarization near the edges of lobes in radio galaxies and of bright features in the parsec-scale jets of AGN, where such magnetic field configurations are believed to exist. (Note: The original ApJ version of this paper contained two errata, which are corrected in this version of the paper. First, Fig. 2 was plotted as a mirror image of the correct version, reflected about the line log 10 ( ν/ν 0 ) = 0. Second, as stated, Equation A16 is valid for γ = 2, not γ = 3.) Subject headings: radiation mechanisms: non-thermal - polarization - galaxies: jets - galaxies: magnetic fields", "pages": [ 2 ] }, { "title": "T. V. Cawthorne", "content": "Jeremiah Horrocks Institute, University of Central Lancashire, Preston, Lancashire, PR1 2HE, U.K. and", "pages": [ 1 ] }, { "title": "P. A. Hughes", "content": "Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1042, U.S.A. Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction.", "content": "Models for relativistic jet production in active galactic nuclei strongly favor ordered magnetic fields within thousands of gravitational radii of the central supermassive black hole (Pudritz et al. 2012) even when complex flows and instability are admitted (McKinney et al. 2012; Porth 2013). Such fields are often assumed to persist to the parsec and even kiloparsec scale, and indeed might be required to explain observations as diverse as transverse gradients in Faraday rotation measure (Pudritz et al. 2012) and the extraordinary stability of flows such as that revealed by radio and X-ray observations of Pictor A (Wilson et al. 2001). Nevertheless, compelling evidence exists that a substantial fraction of the magnetic field energy is in a random component, from the sub-parsec to kiloparsec scales. Following an analysis of cm-band single-dish data by Jones et al. (1985), which revealed a magnetic field structure capable of explaining the 'rotator events' seen in time series data of Stokes parameters Q and U , activity in a number of AGN has been successfully modeled by shocks that compress an initially tangled magnetic field, increasing the percentage polarization during outburst (Hughes et al. 1989b, 1991). Such a model has recently been extended to incorporate oblique shocks (Hughes et al. 2011). Tangled magnetic fields carried through conical shock structures have been explored by Cawthorne (2006), and this picture has been successfully used to explain the characteristics of a stationary jet feature in 3C 120 (Agudo et al. 2012) and has been suggested as an explanation of multiwavelength variations of 3C 454.3 (Wehrle et al. 2012). On the larger (kiloparsec) scale, Laing & Bridle (2002) have pioneered analysis of the magnetic field structure of jets, most recently concluding (Laing et al. 2006) that the jet in 3C 296 has a random but anisotropic magnetic field structure. The spectral, spatial, and temporal behavior of the Stokes parameters Q and U provides a powerful diagnostic of the magnetic field structure, and thus indirectly, of the flow character in such jets, and the degree of linear polarization for a compressed, tangled magnetic field (due, for example, to a shock) was explored in the optically thin limit by Hughes et al. (1985). (An earlier paper, Laing (1980), also considered this kind of structure in the limit of an infinitely strong compression.) However, at least on the parsec and sub-parsec scale these flows exhibit opacity. Indeed, the 'core' seen in low frequency ( ν ≤ 10 GHz) VLBI maps is widely interpreted as being the ' τ = 1-surface': the location of the transition from optically thin to optically thick emission at the observing frequency of the map (Marscher 2006). This location within the jet has a special significance for jet studies, as it is by definition the surface from which propagating components first appear as distinct features on the map; there is compelling evidence that γ -ray flares arise close to the mm-wave core (Marscher et al. 2010), understanding the origin of which requires knowledge of the flow conditions there. At these higher frequencies, the interpretation of the core as the τ = 1-surface is certainly complicated by the presence of stationary features (possibly recollimation shocks) which, even if responsible for the core in some sources, must lie close to regions of significant opacity (Marscher 2006). It would therefore be of great value to have a description of the polarized emission from compressed, tangled magnetic fields in the presence of opacity. In earlier work, Crusius & Schlickeiser (1986, 1988) calculated the emitted synchrotron intensity (Stokes I) for a purely random magnetic field. Crusius-Waetzel et al. (1990) extended the discussion to polarization (Stokes I, Q & U) but the considered field geometry remained purely random, with zero mean polarization, the focus of the study being observable fluctuations - in particular the rms deviations from the means - for various models of the magnetic field turbulent structure, with finite coherence scale and a finite number of magnetic cells. In a recent major study Lazarian & Pogosyan (2012) admitted a mean magnetic field, but were concerned with axisymmetric turbulence that leads to anisotropic intensity (Stokes I) fluctuations. The primary goal was to facilitate probing Galactic MHD turbulence. Our work, on the other hand, considers a compressed random field, such as would result from a shock, or subsonic disturbance, leading to non-zero mean polarization, and computes that in the limit N cells → ∞ , so that only the smooth, mean behavior is described. For a parsec-scale jet magnetic field of 10 -5 T (O'Sullivan & Gabuzda 2009) the gyroradius of an electron with γ = 10 2 is 1 . 7 × 10 4 m, more than twelve orders of magnitude less than the system scale, thus permitting a very small scale turbulent field - effectively an infinite number of cells within a telescope beam - without the cell scale approaching the gyroradius. Typical radio source hotspot fields are 10 -9 T (Donahue et al. 2003); thus the ratio of system scale (kpc) to gyroradius is only one order of magnitude less for particles of the same energy, and no more than three orders of magnitude less for particles of an energy radiating in the same radio waveband. This still comfortably permits a small scale turbulent field with effectively an infinite number of cells within the volume without the cell scale approaching the gyroradius. In no case would our approximation make it necessary to consider turbulent cell sizes so small that jitter radiation (Medvedev 2000; Fleishman 2006) is important.", "pages": [ 3, 4, 5 ] }, { "title": "2. Propagation of synchrotron radiation through a compressed random field.", "content": "This section demonstrates that the radiative transfer equations for the propagation of synchrotron radiation through a compressed random field separate, provided circular polarization and Faraday rotation can be neglected. The resulting absorption and emission coefficients are obtained in Appendix A. The approach follows those of Appendix A in Hughes, Aller & Aller (1985) and Chapter 3 from Pacholczyk (1970). In order to obtain consistency between these two works, the coordinate system used in Hughes, Aller & Aller have been relabelled as follows. Before compression, the direction of the magnetic field is defined by reference to the ( a ' , b ' , c ' ) coordinate system as shown in Fig. 1 (left panel). The polar angle θ separates the a ' axis and the direction of the local magnetic field, and the azimuthal angle φ separates the c ' axis and the projection of the field onto the b ' , c ' plane. In this system, B a ' = B 0 cos θ , B b ' = B 0 sin θ sin φ and B c ' = B 0 sin θ cos φ . After compression, such that unit length parallel to the a ' axis is reduced to length K , the requirement that magnetic flux is conserved yields a magnetic field with components B a ' = B 0 cos θ , B b ' = B 0 sin θ sin φ/K and B c ' = B 0 sin θ cos φ/K . A rotation of the coordinate system through angle δ about the b ' axis gives the a, b, c coordinate system (chosen so that the observer lies on the -c axes) in terms of which the local magnetic field is These results can be obtained from Hughes, Aller & Aller (1985) by making the substitutions ( x →-c ' , y → b ' , z → a ' , x ' →-c , y ' → b , z ' → a , and /epsilon1 →-δ ). Following Pacholczyk (1970) Equation 3.66 and assuming that the circular polarization and Faraday rotation are negligible, the radiative transfer equations are written in terms of I ( a ) and I ( b ) , the intensities measured by dipoles aligned with the a and b axes, respectively, and the Stokes parameter U ( ab ) : Here, χ H is the angle between the a axis and the projection of the magnetic field onto the plane of the sky, as shown in Fig. 1 (right diagram). κ (1) and κ (2) are, respectively, the absorption coefficients for planes of (electric field) polarization perpendicular and parallel to the projected magnetic field. Likewise, /epsilon1 (1) and /epsilon1 (2) are, respectively, the emission coefficients for planes of (electric field) polarization perpendicular and parallel to the projected magnetic field. The polarization-averaged absorption coefficient is defined by For a power-law distribution of radiating electrons such that the density of electrons in the energy interval dE is N ( E ) dE = N 0 E -γ dE , the emission and absorption coefficients for a region with uniform field are given by Pacholczyk (1970) as where the constants C and D are given in Appendix B. Inside the square brackets, the plus sign refers to polarization (1) and the minus sign to polarization (2).", "pages": [ 5, 7, 8 ] }, { "title": "2.1. Separation of the transfer equations.", "content": "Equations 4 to 6 contain a term describing the contribution to I ( a ) , ( b ) and U ( ab ) due to polarized absorption, which depends on ( κ (1) -κ (2) ) sin 2 χ H . For the power-law distribution of particles considered here it is always true that κ (1) > κ (2) , Thus in the (12) frame, these contributions are always in the same sense. For a uniform magnetic field this term is zero because polarized absorption orthogonal to the field does not contribute to the and so so that ( κ (1) -κ (2) ) sin 2 χ H ∝ B a B b ( B 2 a + B 2 b ) ( γ -2) / 4 . Averaging this expression over all θ and φ gives mode parallel to the field, and vice versa. If we consider a partially compressed random magnetic field as equivalent to the sum of a uniform component orthogonal to the sense of compression, plus a superposed random distribution of field elements, the latter will not reintroduce contributions from these difference terms, as their random distribution guarantees that they do not modify the polarized component of the radiation. Equivalently, while a compressed magnetic field exhibits a preferred sense - the plane of compression and the projected magnetic field elements will be distributed with a narrow dispersion in χ H about the projection of this direction on the plane of the sky, on average there will be as many elements with χ H > 0 as there are with χ H < 0, with no net effect upon the radiation field. A more formal demonstration of this result follows. From Equations 4 and 5, From Fig. 1 (right diagram), From Equations 1 to 3, it is clear that B a is an even function of φ while B b is an odd function of φ , so that the integrand in Equation 14 is an odd function of φ . Integrating with respect to φ from -π to π therefore yields the result zero. Therefore Hence, dI ( a ) /ds depends only on I ( a ) , dI ( b ) /ds depends only on I ( b ) and dU ( ab ) /ds depends only on U ( ab ) . In this case, the equations separate and have straightforward solutions. Very similar similar arguments apply to the term ( /epsilon1 (1) -/epsilon1 (2) ) sin 2 χ in Equation 6, which describes the contribution of polarized emission to U ( ab ) , so that < ( /epsilon1 (1) -/epsilon1 (2) ) sin 2 χ H > = 0. Assuming that the magnetic field is disordered on scales small compared those over which the radiation field changes significantly, the emission and absorption coefficients can be averaged over initial magnetic field direction and the radiative transfer Equations 4 to 6 thus simplify to for which, in a uniform source, the following solutions can be obtained straightforwardly: where I ( a ) ( s = 0), I b ( s = 0) and U ( ab ) ( s = 0), are the values incident upon the source, s is the path length through the source, and", "pages": [ 8, 9, 10 ] }, { "title": "3. Results.", "content": "Appendix A shows how the emission coefficients < /epsilon1 ( a ) , ( b ) > and absorption coefficients < κ ( a ) , ( b ) > can be expressed in terms of the function F ( a ) , ( b ) γ (Equations A5 to A7 and Equation A11). The integrals in these expressions are not, in general, analytically tractable, but since F ( a ) , ( b ) γ =3 has a simple solution (Equations A14, A15 and A16), simple formulae exist for the emission coefficients when γ = 3 and for the absorption coefficients when γ = 2. A rough analytical approximation to F ( a ) , ( b ) γ =2 is given by Equations A17 and A 18 and correction factors are plotted in Fig. 5. These allow computation of intensities and polarization in the case γ = 2 without resort to a computer. Expressions for the constants C , D and µ are given in Appendix B, and their values are given in Table 1 for some values of γ in the range of greatest interest. Results illustrating how the emergent polarization varies with frequency ν and line of sight angle δ are presented below. The integrals were performed numerically using Simpson's rule with 50 evaluations per integral. Comparison between results that can be obtained analytically and the corresponding values obtained numerically suggests that the latter are accurate four significant figures at least.", "pages": [ 11 ] }, { "title": "3.1. Polarization as a function of frequency.", "content": "If the source is uniform on scales over which the intensity changes significantly, then the solutions given by Equations 19, 20 and 21 apply. It is convenient to define a characteristic frequency, ν 0 , at which the polarization averaged opacity is unity, i.e., (Note that ν 0 will be a function of K , δ and γ .) Then, from Equations A11 and A12, the opacities in polarizations a and b are The intensities are then given by Equations 19, 20 and B9 and the degree of polarization is then The spectral variation of the degree of polarization is illustrated in Fig. 2 for three values of δ , two values of γ , and K = 0 . 2. The figure illustrates the transition from optically thin emission, where the polarization fraction is generally high and the ( E field) polarization direction is parallel to the a axis (Π > 0), to optically thick emission, where the polarization is generally lower, and the polarization direction is parallel to the b axis (Π < 0). The polarization decreases as δ , the angle of inclination between the line of sight and the plane of compression, increases, and the disordered component of the magnetic field becomes more apparent. In the optically thin (high frequency) limit, the degree of polarization is while in the optically thick (low frequency) limit, the degree of polarization is These values are plotted as a function of compression factor K , for various values of the inclination angle δ , in Fig 3. As expected, in the optically thin limit, the degree of polarization decreases monotonically with increasing K , and with increasing δ . The value of Π in the optically thick limit generally decreases in magnitude as K increases, though for δ less than about 10 · , the value of Π thick has a turning point at about K = 0 . 2. It is, at first sight, surprising that, as K increases from zero and the field becomes more isotropic (or less strongly confined to the plane of compression), the degree of polarization actually increases. This occurs because, although both emission and absorption coefficients for the two polarizations become closer, as clearly they should, the values of /epsilon1/κ for the two polarizations initially diverge. The reason is that while K is very small and increasing, both /epsilon1 ( a ) //epsilon1 ( b ) and κ ( a ) /κ ( b ) decrease, but the ratio of the /epsilon1 values decreases more strongly than that of the κ values. This occurs because the contribution to the coefficients from the component of field perpendicular to the plane of compression (which, in relative terms, is increasing) is greater for the emission coefficients than the absorption coefficients, because the latter depend more sensitively on magnetic field. This subtle effect can be more easily understood with reference to a similar but simpler magnetic field geometry, as shown in Appendix C.", "pages": [ 11, 12, 13 ] }, { "title": "3.2. Polarization as a function of inclination.", "content": "The dependence of the degree of polarization upon δ , the angle of inclination between the line of sight and plane of compression, is described below. If the emitting plasma is confined between two planes, each parallel to the plane of compression and separated by a distance w , then the path length through the plasma is L = w/ sin δ . The opacity is characterised by the value τ 0 = < κ > ( δ = 90 · , K, γ ) w , the polarization averaged opacity when the line of sight is perpendicular to the plane of compression. The value of τ 0 is given by Then, if δ is varied while K , γ and ν remain fixed, The intensities I ( a ) and I ( b ) and the degree of polarization are then given by Equations 28 and 29. The results are shown in Fig. 4, in which Π, the degree of polarization, is plotted against δ , for a compression factor K = 0 . 2, and values of τ 0 = 0 . 25 , 1 . 0 , 4 . 0. The results show that, as δ decreases from 90 · , the polarization first rises as the partial order of the magnetic field becomes more apparent, but then starts to fall, as opacity begins to take effect. As δ decreases further, Π changes from positive to negative in value (i.e. the polarization angle changes by 90 · ), and the degree of polarization approaches the optically thick limit shown in Fig 3. As τ 0 increases in value, the maximum (positive) value of Π decreases and the frequency at which Π changes from positive to negative increases.", "pages": [ 13, 15 ] }, { "title": "4. Summary of results.", "content": "The radiative transfer equations for synchrotron radiation have been shown to separate for the case of propagation through a compressed, random magnetic field, provided Faraday rotation and circular polarization can be neglected. Although, in general, the emission and absorption coefficients must be computed numerically, this is still much simpler than a full numerical solution of the coupled equations. Expressions for the emission and absorption coefficients are given in Appendix A. Exact analytical expressions result only for the emission coefficients when (energy index) γ = 3, and for the absorption coefficients when γ = 2. A rough approximation, together with a plot of correction factors, is given to allow calculation of the emission coefficient for γ = 2. This allows the solution to be found for a source that is uniform on large-scales, for γ = 2, without resort to a computer. Some illustrative results are presented, showing the variation of polarization with frequency, and with inclination of the plane of compression to the line of sight. The optically thin and thick limits to fractional polarization are plotted against compression factor, K , for various inclination angles. When the inclination angle δ < 10 · , the optically thick limit reveals an unusual trend in which, for very small K , the polarization increases as K increases, i.e., as the magnetic field becomes less strongly confined to the plane of compression. This effect is discussed in the context of a simpler magnetic field model in Appendix C.", "pages": [ 16 ] }, { "title": "5. Acknowledgments", "content": "TVC thanks the director of the Jeremiah Horrocks Institute at the University of Central Lancashire for a sabbatical semester, during which most of the present work was undertaken. He also thanks Dr. J.-L. Gomez and the Instituto di Astrofisica de Aldalucia in Granada for their generous hospitality during a part of the sabbatical. This work arose from discussions with Dr. Gomez during that visit. PAH was partially supported by NASA Fermi GI grant NNX11AO13G during this work. The authors thank the anonymous referee for a number of very useful comments on the manuscript. (The authors also thank Mr. Christopher Kaye for identifying the two errors in the original, published version of the paper (as described in the abstract).)", "pages": [ 16, 17 ] }, { "title": "A. Computation of the emission and absorption coefficients.", "content": "Following the approach of Hughes, Aller & Aller (1985), it is convenient to define the functions M and N such that Furthermore, for a 1-D adiabatic compression, the particle density per unit energy, N 0 = N 0 ( K ) ∝ K -( γ +2) / 3 (e.g., Hughes, Aller, & Aller (1989a)). Then, from Equation 22, the emission coefficient /epsilon1 ( a ) becomes", "pages": [ 17 ] }, { "title": "Similarly, from Equation 23", "content": "and The values of κ ( a ) and κ ( b ) averaged over the initial magnetic field direction are thus where Averaging over the initial magnetic field direction, the emission coefficients become where and The absorption coefficients are given by Equations 9, 24 and 25. It is convenient to express κ ( a ) and κ ( b ) in terms of the polarization averaged absorption coefficient, κ (Equation 7), so that Unfortunately, the integrals appearing above are not, in general, analytically tractable. However, it is straightforward to evaluate F ( a ) , ( b ) γ if γ = 3 and H ( a ) , ( b ) γ if γ = 2. The results are These results allow analytical calculation of the emission coefficients if γ = 3 or the absorption coefficients if γ = 2. In an attempt to provide a means of calculating intensities without the aid of a computer various approximate solutions to the integrals for the F and H functions were attempted. The more sophisticated approaches, such as rational function approximations, were not successful. The best results overall were obtained by setting γ = 2 and making the rather crude approximation that ( K 2 ( M + N )) 1 / 4 /similarequal 1 in Equations A6 and A7. In that case, The approximation for F ( a ) are accurate to within 20%, while that for F ( b ) is accurate to within 30%. While this is not very helpful by itself, suitable correction factors, T ( a ) , ( b ) , where are plotted in Fig. 5. In combination with Equations A14 and A15, these results allow intensities and degrees of polarization to be determined for energy index γ = 2, without resort to a computer.", "pages": [ 18, 19 ] }, { "title": "B. Constants.", "content": "Pacholczyk's treatment of synchrotron radiation in a uniform magnetic field involves a large number of physical constants which are helpful in the derivations he performs. However, here, it is the constants of proportionality, C and D , (which are actually functions of γ ) that are of greatest interest and expressions for them are given here. Additionally, the formulae are converted from the obsolete CGS system to SI. Substituting expressions for the constants c 1 , c 3 and c 5 into Equation 3.49 from Pacholczyk (1970) yields an expression for the emission coefficient in CGS units: where, -e is the electron charge, m is the electron mass, c is the speed of light in free space, H ⊥ is the component of magnetic field intensity perpendicular to the line of sight, and the numerical term β is given by The plus sign in Equation B1 refers to polarization 1 ( E perpendicular to the magnetic field) and the minus sign to polarization 2 ( E parallel to the field). It is now straightforward to convert this expression to SI, resulting in the formula where /epsilon1 0 is the permittivity of free space and B ⊥ = ( B 2 a + B 2 b ) 1 / 2 . It follows that the constant C in all expressions for the emission coefficients is given by Similarly, substituting for c 6 and c 1 in Equation 3.51 from Pacholczyk (1970) yields an expression for the absorption coefficients where Again, this is easily converted to SI, giving It follows (by comparison with Equation 9) that the constant D in the expressions for the absorption coefficients is given by One further constant of importance appears in the term /epsilon1/κ , which appears when the expressions for the emission and absorption coefficients are substituted into the uniform source solutions, Equations 19 and 20. where ν L = eB 0 / (2 πm ) is the cyclotron frequency in magnetic field B 0 and µ is the numerical value given by Numerical values of α , β and µ are given for common values of γ in Table A1. This section presents a magnetic field model that is similar to, but simpler than, that discussed in the main text of this paper. The aim is to illustrate more clearly the origin of the unusual behaviour of Π thick shown in Fig. 3 (lower panel) in which, as K increases from zero, (reducing the anisotropy of the magnetic field) then for δ = 0, Π thick actually increases. This behaviour is more easily understood in the case of a source in which the magnetic field is in the plane of the sky. It consists of a large number of cells, a fraction (1 -x ) of which have magnetic field B parallel to the b axis with value B 0 /K , and a fraction x have B parallel to the a axis with value B 0 ( K < 1). The fact that this field configuration doesn't satisfy ∇ . B = 0 does not detract from its usefulness for the present purpose. Since the magnetic field is in the sky plane in one of two orthogonal directions, Equations 8 and 9 give the emission and absorption coefficients for the fraction (1 -x ) of cells with B parallel to the b axis as where, for γ = 2, s = ( γ +1) / ( γ +7 / 3) = 9 / 13, r = ( γ +2) / ( γ +10 / 3) = 3 / 4 and the upper and lower symbols in the plus or minus sign refer to polarizations ( a ) and ( b ) respectively. For the fraction x of cells with magnetic field parallel to the a axis, the (1 + r ) factors become (1 -r ) and vice versa, and similarly for (1 ± s ). The magnetic field becomes B 0 . For these cells, the emission and absorption coefficients are therefore The total contribution to /epsilon1 ( a ) is therefore Similarly, the remaining coefficients are In the optically thick limit, the degree of polarization is where Q = ( /epsilon1 ( a ) κ ( b ) ) / ( /epsilon1 ( b ) κ ( a ) ). If K → 0, Π thick is as for a uniform field, i.e. negative, and 0 < Q < 1. Then, if Q increases with increasing K , | Π thick | decreases. If Q decreases with increasing K , then | Π thick | increases. Substituting from Equations, C5 to C9, where S = (1 -s ) / (1 + s ), R = (1 -r ) / (1 + r ), and X = x/ (1 -x ). If K /lessmuch 1, then, neglecting terms of order K 3 and higher which will decrease with increasing K if dQ/dK < 0, i.e. if or to two significant figures, if γ = 2. So provided K is very small, as K increases, Q decreases and | Π thick | increases, while Π thick is negative. This happens because as K increases, the emission process tends to favour I ( b ) over I ( a ) (i.e. /epsilon1 ( b ) increases more than /epsilon1 ( a ) ). However, the absorption process (or the mean free path) favours I ( a ) over I ( b ) (because κ ( b ) increases more than κ ( a ) ). If K is small, the effect on Q due to the change in emission coefficients ( ∝ K 3 / 2 ) dominates that due to the change in absorption coefficients because ( ∝ K 2 ) when K /lessmuch 1. Comparison with of Inequality C13 with the position of the turning point on the δ = 0 curve from the lower panel of Fig. 3, shows that, in the compressed random field model, | Π thick | increases with K over a more limited range of K , from K = 0 to about 0 . 2, rather than 0 . 34. This discrepancy arises because, in the model compressed random field model, when K is small, the field parallel to the b axis is like a plate of spaghetti, much of it points toward us, reducing the emission coefficients of this component by sin 3 / 2 θ and the absorption coefficients by sin 2 θ , where θ is the inclination of the field to the line of sight. The result is to replace K in the above expressions by K/ sin θ . This will tend to make the condition on K more stringent than given by Inequality C13.", "pages": [ 21, 22, 23, 24, 25 ] }, { "title": "REFERENCES", "content": "Lazarian, A., & Pogosyan, D. 2012, ApJ, 747, 5 Marscher, A. P. 2006, Relativistic Jets: The Common Physics of AGN, Microquasars, and Gamma-Ray Bursts, 856, 1 Marscher, A. P., Jorstad, S. G., Larionov, V. M., et al. 2010, ApJ, 710, L126 McKinney, J. C., Tchekhovskoy, A., & Blandford, R. D. 2012, MNRAS, 423, 308 Medvedev, M. V. 2000, ApJ, 540, 704 O'Sullivan, S. P., & Gabuzda, D. C. 2009, MNRAS, 400, 26 Pacholczyk, A. G. 1970, Radio astrophysics, San Francisco: Freeman, 1970 Porth, O. 2013, MNRAS, 429, 2482 Pudritz, R. E., Hardcastle, M. J., & Gabuzda, D. C. 2012, Space Sci. Rev., 169, 27 Wehrle, A. E., Marscher, A. P., Jorstad, S. G., et al. 2012, ApJ, 758, 72 Wilson, A. S., Young, A. J., & Shopbell, P. L. 2001, ApJ, 547, 740", "pages": [ 26, 27 ] } ]

[ { "title": "ABSTRACT", "content": "We present simulations modeling closed regions of the solar corona threaded by a strong magnetic field where localized photospheric vortical motions twist the coronal field lines. The linear and nonlinear dynamics are investigated in the reduced magnetohydrodynamic regime in Cartesian geometry. Initially the magnetic field lines get twisted and the system becomes unstable to the internal kink mode, confirming and extending previous results. As typical in this kind of investigations, where initial conditions implement smooth fields and flux-tubes, we have neglected fluctuations and the fields are laminar until the instability sets in. But previous investigations indicate that fluctuations, excited by photospheric motions and coronal dynamics, are naturally present at all scales in the coronal fields. Thus, in order to understand the effect of a photospheric vortex on a more realistic corona, we continue the simulations after kink instability sets in, when turbulent fluctuations have already developed in the corona. In the nonlinear stage the system never returns to the simple initial state with ordered twisted field lines, and kink instability does not occur again. Nevertheless field lines get twisted, but in a disordered way, and energy accumulates at large scales through an inverse cascade. This energy can subsequently be released in micro-flares or larger flares, when interaction with neighboring structures occurs or via other mechanisms. The impact on coronal dynamics and CMEs initiation is discussed. Keywords: magnetohydrodynamics (MHD) - Sun: corona - Sun: coronal mass ejections (CMEs) -Sun: magnetic topology - turbulence", "pages": [ 1 ] }, { "title": "FIELD LINES TWISTING IN A NOISY CORONA: IMPLICATIONS FOR ENERGY STORAGE AND RELEASE, AND INITIATION OF SOLAR ERUPTIONS", "content": "A. F. Rappazzo 1 , † , M. Velli 2 , and G. Einaudi 3 1 Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Delaware 19716, USA 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA 3 Berkeley Research Associates, Inc., 6537 Mid Cities Avenue, Beltsville, MD 20705, USA Draft version April 2, 2021", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Photospheric convection and the coronal magnetic field play a key role in heating the solar corona. For the magnetically confined (closed) regions of the corona it has long been suggested that small heating events, dubbed nanoflares , continuously deposit energy at the small scales and can contribute a substantial fraction of the total heating (Parker 1972, 1988, 1994). The slow photospheric motions (with a timescale τ c ∼ 8 minutes, magnitude v c ∼ 1 kms -1 , and correlation scale /lscript c ∼ 10 3 km ) transfer energy from the photosphere into the corona shuffling the footpoints of the magnetic field lines. The work done by the denser photospheric plasma on the magnetic field lines footpoints injects energy into the corona, mostly as magnetic energy. The perturbations generated at the photospheric level propagate along the loop at the Alfv'en speed. In a coronal loop the Alfv'en velocity associated to the strong axial magnetic field B 0 is v A = B 0 / √ 4 πρ 0 ∼ 2 × 10 3 kms -1 ( ρ 0 is the average mass density), and considering a typical loop length L ∼ 4 × 10 4 km we obtain for the Alfv'en crossing time τ A = L/v A ∼ 20 s . The crossing time is therefore significantly smaller than the photospheric timescale: τ A << τ c . Furthermore as magnetic pressure largely exceeds plasma pressure the plasma β is small ( << 1). Because of the fast Alfv'en timescale and low beta the dynamics of the magnetically confined solar corona are generally approximated as a quasi-static evolution through a sequence of equilib- ies leading the system from an equilibrium to the next one through relaxation (Taylor 1974, 1986; Heyvaerts & Priest 1984). Within this framework many works have studied the relaxation dynamics in detail (e.g., Yeates et al. 2010; Pontin et al. 2011; Wilmot-Smith et al. 2011). On the other hand it has been shown that in some instances the validity of that picture is not valid, e.g., in reduced magnetohydrodynamic (MHD) simulations of the Parker model for coronal heating (Rappazzo et al. 2007, 2008; Rappazzo, Velli & Einaudi 2010). In fact that approximation is attained neglecting the velocity and plasma pressure in the MHD equations whose solution is then bound to be a static force-free equilibrium. But the self-consistent evolution of the plasma pressure and velocity, although small compared with the dominant axial magnetic field B 0 (and therefore very close in first approximation to the equilibrium solution of a uniform axial magnetic field), does not bind the system to force-free equilibria and allows the development of turbulent nonlinear dynamics with formation of field-aligned current sheets where a significant heating occurs. Most numerical simulations of a simple model coronal loop in Cartesian geometry threaded by a strong magnetic field shuffled at the top and bottom plates by photospheric motions have used as boundary velocity an incompressible field with all wavenumbers of order ∼ 4 excited (Einaudi et al. 1996; Dmitruk & G'omez 1997; Georgoulis et al. 1998; Dmitruk & G'omez 1999; Einaudi & Velli 1999; Dmitruk et al. 2003; Rappazzo et al. 2007, 2008; Rappazzo & Velli 2011): in real space this corresponds to distorted vortices with length-scale ∼ 10 3 km one next to the other filling the whole photospheric plane (Rappazzo et al. 2008, Figures 1 and 2). This configuration does not give rise to instabilities. The system transitions smoothly from the linear to the nonlinear stage where integrated physical quantities like energies and dissipation fluctuate around a mean value in what is best described as a statistically steady state and the energy deposited at the small scales is approximately in the nanoflare range for the numerous heating events. This disordered vortical forcing mimics a uniform and homogeneous convection and the resulting coronal dynamics give rise to a basal background coronal heating within the lower limit of the observational constraint. With this kind of forcing the system is not able to accumulate a significant amount of magnetic energy to be subsequently released in more substantial heating events like microflares and flares . It is therefore pivotal to implement different kinds of photospheric forcings to understand the role of convective motions in coronal heating and the physical mechanisms and conditions for a significant storage of magnetic energy and its release . Shearing or twisting the field lines might appear as the most straightforward way to make the system accumulate energy. To this end, in recent work (Rappazzo, Velli & Einaudi 2010) we have implemented a 1D velocity forcing with a sinusoidal shear flow at the boundary, with v y ( x ) ∝ sin (2 πkx//lscript ), wavenumber k = 4, spanning the whole photospheric plane ( /lscript is the cross-section length). Initially the magnetic field that develops in the coronal loop is a simple map of the photospheric velocity field, i.e., b y ( x ) ∝ ( t/τ A ) sin (2 πkx//lscript ), with its intensity growing linearly in time. A sheared magnetic field is known to be subject to tearing instability (Furth et al. 1963), in fact magnetic energy accumulates until a tearing instability sets in, magnetic energy is released and the system transitions to the nonlinear stage. On the other hand continuing the simulation we found that, once the system has become fully nonlinear the dynamics are fundamentally different: the magnetic shear is not recreated . Once fluctuations are present, the orthogonal magnetic field ( b x and b y ) is organized in magnetic islands with X and O points, the nonlinear terms do not vanish any more and energy can be transported efficiently from large to small scales where it is dissipated. In the nonlinear stage the dynamics are very similar whether the forcing velocity is a shear flow or made of disordered vortices, and magnetic energy is not stored efficiently, therefore larger releases of energy are not possible. A shear photospheric flow can give rise to a sheared coronal magnetic field only in the unlikely condition that no relevant perturbations are present in the corona, or for very strong shear flows with velocity higher than the typical photospheric velocity of 1 kms -1 , when the perturbations naturally present in the corona can be neglected. All photospheric forcings described so far fill the entire photospheric plane. Spatially localized velocity fields, like a single vortex that does not fill the entire plane, might be able to induce a higher storage of magnetic energy in the corona. A vortex twists the magnetic field lines, and the resulting helical magnetic structure is kink unstable and widely used to model coronal loops (Baty & Heyvaerts 1996; Velli et al. 1997; Lionello et al. 1998; Browning et al. 2008; Hood et al. 2009). In these studies a twisted magnetic field lines structure is used as initial condition, but this is assumed to have been induced by photospheric motions shuffling their footpoints, while the field lines are actually line-tied to a motionless photosphere . To understand the dynamics of this photospherically driven system we have performed numerical simulations applying localized vortices at the photospheric boundary. Mik'ıc et al. (1990); Gerrard et al. (2002) performed boundary forced simulations, but they stop just after kink instability sets in. In this paper we continue the simulations for longer times. This allows us to understand the dynamics of the system both when initially only an axial uniform magnetic field is present and a smooth ordered flux-tube with twisted field lines gets formed and kink instability sets in, and the later dynamics when the localized boundary vortex twists the footpoints of a disordered magnetic field where magnetic fluctuations and small scales are already present and field lines are no longer smooth. Furthermore, highly twisted magnetic structures, such as flux ropes , are broadly used to model the initiation of coronal mass ejections (CMEs, e.g., see Low (2001); Amari & Aly (2009); Chen (2011); Torok et al. (2011), and references therein). To advance our understanding of eruption initiation it is therefore important to understand if and under which conditions photospheric motions can self-consistently generate flux ropes (Amari et al. 1999), or if these structures can only be advected into the corona from sub-photospheric regions via emerging flux. The paper is organized as follows. In § 2 we describe the basic governing equations and boundary conditions, as well as the numerical code used to integrate them. In § 3 we discuss the initial conditions for our simulations and briefly summarize the linear stage dynamics more extensively detailed in Rappazzo et al. (2008). The results of our numerical simulations are presented in § 4, while the final section is devoted to our conclusions and discussion of the impact of this work on coronal physics.", "pages": [ 1, 2 ] }, { "title": "2. GOVERNING EQUATIONS", "content": "We model a coronal loop as an axially elongated Cartesian box with an orthogonal cross section of size /lscript and an axial length L embedded in an homogeneous and uniform axial magnetic field B 0 = B 0 ˆe z aligned along the z -direction. Any curvature effect is neglected. The dynamics are integrated with the equations of RMHD (Kadomtsev & Pogutse 1974; Strauss 1976; Montgomery 1982), well suited for a plasma embedded in a strong axial magnetic field. In dimensionless form they are given by: where u ⊥ and b ⊥ are the velocity and magnetic fields components orthogonal to the axial field and p is the plasma pressure. The gradient operator has components only in the perpendicular x -y planes while the linear term ∝ ∂ z couples the planes along the axial direction through a wave-like propagation at the Alfv'en speed c A . Incompressibility in RMHD equations follows from the large value of the axial magnetic fields (Strauss 1976) and they remain valid also for low β systems (Zank & Matthaeus 1992; Bhattacharjee et al. 1998) such as the corona. Furthermore Dahlburg et al. (2012) have recently performed fully compressible simulations of a similar Cartesian coronal loop model, showing that the inclusion of thermal conductivity and radiative losses, that transport away the heat produced by the small scale energy dissipation, keep the dynamics in the RMHD regime. To render the equations nondimensional, we have first expressed the magnetic field as an Alfv'en velocity [ b → b/ √ 4 πρ 0 ], where ρ 0 is the density supposed homogeneous and constant, and then all velocities have been normalized to the velocity u ∗ = 1 kms -1 , the order of magnitude of photospheric convective motions. Lengths and times are expressed in units of the perpendicular length of the computational box /lscript ∗ = /lscript and its related crossing time t ∗ = /lscript ∗ /u ∗ . As a result, the linear terms ∝ ∂ z are multiplied by the dimensionless Alfv'en velocity c A = v A /u ∗ , where v A = B 0 / √ 4 πρ 0 is the Alfv'en velocity associated with the axial magnetic field. We use a computational box with an aspect ratio of 10, which spans Our forcing velocities have a linear scale of ∼ 1 / 4 that corresponds to the convective scale of ∼ 1 , 000 km in conventional units, thus the box extends (4 , 000 km ) 2 × 40 , 000 km . The index n in the diffusive terms (1)-(2) is called dissipativity and for n > 1 these correspond to so-called hyperdiffusion (e.g., Biskamp 2003). For n = 1 standard diffusion ( Re 1 = Re ) is recovered and in this case the kinetic and magnetic Reynolds numbers are given by: where c is the speed of light, and numerically they are given the same value Re = Re m . Note . - The numerical grid resolution is n x × n y × n z . The next columns indicate respectively the value of the hyperdiffusion coefficient Re 4 and the simulation time span. In the simulations presented in this paper we use hyperdiffusion with n = 4. Hyperdiffusion is used because the implemented boundary velocity forcings and the magnetic flux tubes induced initially are localized to a small area of the computational box, and the dynamics would be dramatically diffusive with standard diffusion at a reasonable resolution (see next section § 3 for a more detailed discussion). Our parallel code RMH3D solves numerically Equations (1)-(3) written in terms of the potentials of the orthogonal velocity and magnetic fields in Fourier space, i.e., we advance the Fourier components in the x - and y -directions of the scalar potentials. Along the z -direction, no Fourier transform is performed so that we can impose non-periodic boundary conditions ( § 3), and a central second-order finite-difference scheme is used. In the x -y plane, a Fourier pseudospectral method is implemented. Time is discretized with a third-order RungeKutta method. For a more detailed description of the numerical code see Rappazzo et al. (2007, 2008).", "pages": [ 2, 3 ] }, { "title": "3. BOUNDARY CONDITIONS AND LINEAR STAGE DYNAMICS", "content": "Magnetic field lines are line-tied to the top and bottom plates ( z = 0 and L ) that represent the photospheric surfaces. Here we impose, as boundary condition, a velocity field that convects the footpoints of the magnetic field lines. Along the x and y directions periodic boundary conditions are implemented. All simulations (see Table 1) employ a circular vortex applied at the top plate z = L . The velocity potential for this vortex is centered in the interval I = [1 / 2 -1 / 8 , 1 / 2 + 1 / 8] of linear extent /lscript c = 1 / 4 and vanishes outside: The velocity is linked to the potential by u ⊥ = ∇ ϕ × ˆe z and its components are: in the interval I and vanish outside. As shown in Figure 1 Equations (7)-(9) describe a counter-clockwise vortex centered in the middle of the plane z = L and has circular streamlines, with a slight departure from a perfectly circular shape toward the edge of the interval I . Averaging over the surface I the velocity rms is 〈 ( u L ) 2 〉 I = 1 / 2, the same value of the boundary velocity fields used in our previous works (Rappazzo et al. 2007, 2008; Rappazzo, Velli & Einaudi 2010; Rappazzo & Velli 2011). In all simulations a vanishing velocity is imposed at the bottom plate z = 0: At time t = 0 we start our simulations with a uniform and homogeneous magnetic field along the axial direction B 0 = B 0 ˆe z . The orthogonal component of the velocity and magnetic fields are zero inside our computational box u ⊥ = b ⊥ = 0, while at the top and bottom planes the vortical velocity forcing is applied and kept constant in time. We briefly summarize and specialize to the case considered in this paper the linear stage analysis described in more detail in Rappazzo et al. (2008). In general for an initial interval of time smaller than the nonlinear timescale t < τ nl , nonlinear terms in Equations (1)-(3) can be neglected and the equations linearized. For simplicity we will at first neglect also the diffusive terms and consider their effect later in this section. The solution during the linear stage with a generic boundary velocity forcing u L , and u 0 = 0, (respectively at the top and bottom planes z = L and 0) is given by: where τ A = L/v A is the Alfv'en crossing time along the axial direction z . The magnetic field grows linearly in time, while the velocity field is stationary and the order of magnitude of its rms is determined by the boundary velocity profile. Both are a mapping of the boundary velocity field u L . For a generic forcing the solution (11)-(12) is valid only during the linear stage, while for t > τ nl when the fields are big enough the nonlinear terms cannot be neglected. Nevertheless there is a singular subset of velocity forcing patterns for which the generated coronal fields (11)-(12) have a vanishing Lorentz force and the nonlinear terms vanish exactly. This subset of patterns is characterize by having the vorticity constant along the streamlines (Rappazzo et al. 2008). In this case magnetic energy grows quadratically in time until some instability eventually sets in. Two kind of velocity patterns can be identified: a) 1D patterns with their streamlines all parallel to each other, like a shear flow , or b) a radial pattern with circular streamlines, like a circular vortex . Since in the linear stage the coronal fields are a mapping of the boundary velocity (11)-(12), a shear flow induces a sheared magnetic field subject to tearing instabilities (Rappazzo, Velli & Einaudi 2010), while the vortical flows considered in this paper twist the field lines into helices subject to kink instabilities. The vortex (8)-(9) is not perfectly circular as the streamlines depart from an exact round shape toward the edge (Figure 1), but as we show in § 4.1 field line tension adjusts the induced coronal orthogonal field lines in a round shape. So far we have neglected the diffusive terms in the RMHD Equations (1)-(3). In § 4 we show that for this kind of problem the use of hyperdiffusion is crucial, otherwise the dynamics are dominated by diffusion. Overlooking this numerical fact can result in misleading conclusions (Klimchuk et al. 2009, 2010), upon which we will comment in § 4. Here we want to understand the diffusive effects on the linear dynamics, i.e., when nonlinear terms are negligible or artificially suppressed by a low numerical resolution. We now consider the effect of standard diffusion (case n = 1 in eqs. (1)-(2)) on the solutions (11)-(12): these are the solutions of the linearized equations obtained from (1)-(2) retaining also the diffusive terms. In the linear regime , as the magnetic field grows in time (11), the diffusive term ( ∇ 2 ⊥ b ⊥ ∝ b ⊥ //lscript 2 ) becomes increasingly bigger until diffusion balances the magnetic field growth, and the system reaches a saturated equilibrium state. Including diffusion the magnetic field evolves as i.e., for times smaller than the diffusive timescale τ R Equation (11) is recovered with the field growing linearly in time, while for times bigger than τ R the field asymptotes to its saturation value. The diffusive timescale associated with the Reynolds number Re is τ R = /lscript 2 c Re/ (2 π ) 2 where /lscript c is the length-scale of the forcing pattern, that for the pattern (7)-(9) is given by /lscript c ∼ /lscript/ 4 where /lscript is the orthogonal computational box length. The total magnetic energy E M and ohmic dissipation rate J will then be given by For times smaller than the diffusive timescale τ R both quantities grow quadratically in time, while for t /greaterorsimilar 2 τ R they asymptote to their saturation value E sat M and J sat : Magnetic energy saturates to a value proportional to the square of both the Reynolds number and the Alfv'en velocity, while the heating rate saturates to a value that is proportional to the Reynolds number and the square of the axial Alfv'en velocity. Even though we use grids with ∼ 512 2 points in the x-y plane, the timescales associated with ordinary diffusion are small enough to affect the large-scale dynamics, inhibiting the development of instabilities and nonlinearity. The diffusive time τ n at the scale λ associated with the dissipative terms used in Equations (1)-(2) is given by For n = 1 the diffusive time decreases relatively slowly toward smaller scales, while for n = 4 it decreases far more rapidly. As a result for n = 4 we have longer diffusive timescales at large spatial scales and diffusive timescales similar to the case with n = 1 at the resolution scale. Numerically we require the diffusion time at the resolution scale λ min = 1 /N , where N is the number of grid points, to be of the same order of magnitude for both normal and hyper-diffusion, i.e., Then for a numerical grid with N = 512 points that requires a Reynolds number Re 1 = 800 with ordinary diffusion we can implement Re 4 ∼ 10 19 (table 1), removing diffusive effects at the large scales and allowing, if present, the development of kink instabilities and nonlinear dynamics.", "pages": [ 3, 4, 5 ] }, { "title": "4. NUMERICAL SIMULATIONS", "content": "In this section we present the results of the numerical simulations summarized in Table 1. Simulations A and B have the same parameters, but simulation B employs a lower resolution to achieve a very long duration. In all simulations the vortical velocity pattern (7)-(9) is applied at the top plate z = L , and a vanishing velocity at the bottom plate z = 0. Initially no perpendicular magnetic or velocity field is present inside the computational box b ⊥ = u ⊥ = 0, and the system is threaded only by the constant and uniform field B 0 = B 0 ˆe z . The computational box has an aspect ratio of 10, with /lscript = 1 and L = 10.", "pages": [ 5 ] }, { "title": "4.1. Run A", "content": "We present here the results of run A, a simulation performed with a numerical grid of 512 × 512 × 208 points, and hyperdiffusion coefficient Re 4 = 10 19 with diffusivity n = 4. The Alfv'en velocity is v A = 200kms -1 , corresponding to a nondimensional ratio c A = v A /u ∗ = 200. The total duration is ∼ 1 , 900 axial Alfv'en crossing times τ A = L/v A . the total ohmic dissipation rate and S , the power injected from the boundary by the work done by convective motions on the field lines' footpoints (see Equation (22)), along with some saturation curves for magnetic energy (14). Additionally Figure 4 shows snapshots of the magnetic field lines of the orthogonal component b ⊥ and electric current j = j z , the leading order component in RMHD ordering (Strauss 1976), at selected times in the mid-plane z = 5. The circular vortical velocity field (8)-(9) applied at the top boundary ( z = 10) initially induces velocity and magnetic fields in the computational box that follow the linear behavior given by Equations (11)-(12), i.e., they are a mapping of the velocity at the boundary with the magnetic field increasing linearly in time (Figure 4, times t = 0 . 61 τ A and 80 . 64 τ A ). In the linear stage ( t /lessorsimilar 83 τ A ) magnetic energy is well-fitted (Figure 2) by the linear curve (14) in the limit Re → ∞ , i.e., in the absence of diffusion (indeed in this limit the curve can be obtained directly from the linear Equation (11)). This is because we are using hyperdiffusion that effectively gets rid of diffusion at the large scales. Two other magnetic energy linear diffusive saturation curves are drawn for Re = 800 and 400, typical Reynolds numbers used in our previous simulations with standard diffusion n = 1 and orthogonal grids with respectively 512 2 and 256 2 grid points (see, e.g., Rappazzo et al. 2008). Their saturation level is very low compared to the magnetic energy values when kink instability develops ( t ∼ 83 τ A ) and in the following nonlinear stage. This is because the vortex and induced magnetic field occupy only a limited volume elongated along z at the center of the x-y plane: at these scales diffusion dominates with these resolutions using standard diffusion. For this reason the use of hyperdiffusion is crucial to study this problem, otherwise diffusion dominates and a balance between the injection of energy from the boundary and its numerical removal by diffusion is reached very soon, inhibiting the development of kink instability and nonlinear dynamics. This diffusive linear regime was reached in previous simulations by Klimchuk et al. (2009, 2010), where four similar vortices were applied at the boundary. Therefore their conclusion that nonlinear dynamics or instabilities (not to mention turbulence) cannot develop in such physical systems is simply a numerical issue: this can be overcome adopting hyperdiffusion as we have done here or, alternatively, implementing grids with much higher resolutions that require impractically large numerical resources. The localized boundary vortex (shown with a colored contour in Figure 5) generates a mostly poloidal magnetic field confined to the axial volume in correspondence of the vortex, resulting in helical field lines for the total magnetic field (Figure 5, time t = 60 . 57 τ A ). Outside this volume the poloidal field vanishes and only the axial field B 0 is present. Amp'ere's law then guarantees that the total net current is zero. As shown in Figure 4 in the linear stage ( t = 0 . 61 τ A , 80 . 64 τ A ) there is a stronger upflowing current concentrated in the middle, and a weaker ring-shaped down-flowing current distributed at the edge of the flux-tube. This magnetic configuration is well known to be kink unstable, and is similar to the NC (Null Current) forcefree model studied by Lionello et al. (1998). The main differences are that their axial field B 0 is not uniform, dropping by ∼ 50% outside the flux tube, and that the field lines are line-tied to a motionless photosphere. They performed a linear stability analysis of this configuration finding that there is a critical axial loop length L crit beyond which the system is unstable and has a constant growth rate γτ A ∼ 0 . 02. They also examined other equilibria with net current finding a similar qualitative behavior, with variations for the critical length and growth rates. Lionello et al. (1998) found that for the NC case the ratio of the axial critical length over the cross-length of the flux-tube is L crit //lscript c ∼ 9. In the case considered here the ratio of the axial length ( L = 10) over the cross- ngth of the flux tube (the extent of the boundary vortex /lscript c = 1 / 4, Equation (7)) is L//lscript c = 40, therefore it is fully in the unstable region. Of course at a given length (beyond the critical length) there is also a critical twist beyond which the configuration is unstable. In our simulations the system is continuously forced at the boundary, and in the linear stage the twist grows linearly in time (from Equation (11), as the twist is proportional to b ⊥ /B 0 ), thus such a critical twist is certainly attained. In our case the 'equilibrium' solution is not static but is given by the linear solution (11), indicated here with b lin , with the magnetic field growing linearly in time while mapping the boundary vortex. Thus we compute the perturbed magnetic energy as We find that in the linear stage this quantity grows exponentially in time, obtaining for the perturbed magnetic field a growth rate γτ A ∼ 0 . 02, as Lionello et al. (1998) for their NC equilibrium model. This growth rate is also confirmed by the fact that kink instability sets in at t ∼ 83 τ A (Figures 2 and 3) and 1 /γ ∼ 50 τ A . As mentioned in § 3 the forcing boundary vortex departs from an exact circular shape at its edges where its vorticity is not exactly constant along the streamlines, thus there is a small Lorenz force for the resulting magnetic field (11). This small difference in the linear field acts as a perturbation. Additionally Lionello et al. (1998) found out that configurations with zero net current are unstable to the internal kink mode (opposed to the global kink mode for configurations with a net current), for which magnetic perturbations and the radial displacement of the plasma column are confined within the original flux tube. This is found also in our simulation as shown in Figure 4 at the onset of the nonlinear stage at t = 83 . 85 τ A , when the plasma displaces inside the flux tube toward its edge where a strong current sheet forms. The internal kink mode releases almost 90% of the accumulated energy around time t ∼ 83 . 5 τ A (Figure 2) in correspondence of the big ohmic dissipative peak shown in Figure 3. The released energy is ∆ E ∼ 10 3 × 10 22 erg = 10 25 erg , in the micro-flare range (the factor to convert energy into dimensional units, given our normalization choice discussed in § 3, is 10 22 , i.e., 1 → 10 22 erg ). As a result of the kink instability magnetic reconnection occurs (Figure 4, t = 85 . 05 τ A ) and the magnetic field lines get substantially unwind as shown in Figure 5 (times t = 60 . 57 τ A and 100 . 78 τ A ) with field lines twisting only ∼ 180 · after the instability. In summary, during the linear stage, the transition to and the first phase of the nonlinear regime, the analysis of Lionello et al. (1998) is fully confirmed also for the photospherically driven case considered here: the system forced by a circular vortex is unstable to an internal kink mode , releases most of the stored magnetic energy and magnetic reconnection untwists the field lines. Linear calculations (Baty 2001) show that similar dynamics are expected also for different configurations with different aspect ratios and magnetic guide field values, except for those that fall below the instability threshold. The phenomenology described so far is also in agree- ment with that of three-dimensional simulations with a realistic geometry (Amari & Luciani 2000). In particular strong nonlinearities persist right after the instability occurs ( t = 85 . 05 τ A and 100 . 78 τ A ), when the system cannot be described as a constantα force-free state. An inverse cascade of magnetic energy is observed, as the orthogonal magnetic field acquires longer scales and the overall volume occupied by twisted field lines increases, as shown in Figure 4 just before ( t = 85 . 05 τ A ) and after ( t = 100 . 78 τ A ) the instability. In Amari & Luciani (2000) this corresponds also to an inverse cascade of magnetic helicity, corresponding in the RMHD case to an inverse cascade of the square potential ψ (see the end of this section and our discussion in § 5 for more about this quasi-invariant analogous to helicity in RMHD). On the other hand, at later times the dynamics are certainly surprising when, in the fully nonlinear stage, fluctuations created by the kink instability are present in the corona. For t > 100 τ A magnetic energy increases steadily, while kinetic energy remains small (Figure 2). This is in contrast to all our previous simulations with space-filling boundary motions, either distorted vortices (Rappazzo et al. 2007, 2008; Rappazzo & Velli 2011) or shear flows (Rappazzo, Velli & Einaudi 2010), when in the nonlinear regime a magnetically dominated statistically steady state was reached where integrated quantities would fluctuate around an average value (with velocity fluctuations smaller than magnetic fluctuations). In our case ohmic dissipation J and the integrated Poynting flux S do reach a statistically steady state (Figure 3). The integrated Pointing flux is the power entering the system at the boundaries as a result of the work done by photospheric motions on the footpoints of magnetic field lines ( u L is the photospheric forcing velocity). But in contrast to our previous results, here the power does not balance on the average the dissipation rate, its average is slightly higher resulting in the magnetic energy growth shown in Figure 2. In physical space the dynamics are surprising in two ways. First , after the kink instability, even though we continue to stir the field lines' footpoints with the same vortex, no further kink instability develops. Analogously to the shear flow case (Rappazzo, Velli & Einaudi 2010), once the system transitions to the nonlinear stage the magnetic fluctuations generated during the instability do not have a vanishing Lorentz force. In fact around t ∼ 100 τ A , at the end of the big dissipative event, the topology of the orthogonal component of the magnetic field is characterized by circular, but distorted , field lines (Figure 4). Naturally the Lorentz force does not vanish now and the vorticity is not constant along the streamlines. Nonlinear terms do not vanish as they do during the linear stage for t < 83 τ A . When they vanish magnetic energy can be stored, without getting dissipated, into an ordered flux-tube with helical field lines (Figure 5, t = 60 . 57 τ A ), and matching perfectly round orthogonal magnetic field lines (Figure 4, t = 80 . 64 τ A ). But now nonlinearity continuously transfers energy from large to small scales where it is dissipated. In physical space small scales are not uniformly distributed, but they are organized in field-aligned current sheets. These, once formed during the onset of the nonlinear stage, persist throughout the subsequent dynamics (as shown in Figures 4 and 6), with the energy cascade continuously feeding them. Second , the photospherical vortical motions do not give rise to an orderly helical flux-tube as in the linear stage (Figure 5, t = 60 . 57 τ A ). However, magnetic field lines get twisted, but in a disordered way (Figure 5 and 4, t ≥ 100 . 78 τ A ). A new phenomenon occurs: on longer timescales the magnetic field acquires longer spatial scales (Figure 4), the volume where field lines are twisted increases (Figure 5), while the current exhibits always a small filling factor occupying a small fraction of the volume (Figure 6). To better understand these phenomena we need to investigate the energy dynamics in Fourier space. We consider the spectra in the orthogonal x -y plane integrated along the z direction. As they are isotropic in the Fourier k x -k y plane we compute the integrated 1D spectra, so that for the total magnetic energy E M we obtain: where n indicates the shell in k -space with wavenumber k = ( k, l ) ∈ Z 2 included in the range n -1 < ( k 2 + l 2 ) 1 / 2 ≤ n , and N is the maximum wavenumber admitted by the numerical grid (corresponding to the smallest resolved orthogonal scale). The time averaged magnetic and kinetic energy spectra as a function of wavenumber are shown in Figure 7, the inset shows the spectrum of the boundary vortex' kinetic energy (see Equation (7)). Photospheric motions therefore inject energy at wavenumbers between 2 and 7 (see Equation (22)), the system is magnetically dominated and the power-laws exhibited at higher wavenumbers, in the inertial range, are similar to those obtained with previous space-filling boundary forcings (Rappazzo et al. 2008; Rappazzo, Velli & Einaudi 2010; Rappazzo & Velli 2011), with the spectrum of magnetic energy much steeper than that of kinetic energy. However the time-average of the low-wavenumber modes hides an interesting dynamics. Figure 8 shows the first five magnetic energy modes as a function of time. While modes with wavenumbers n ≥ 3 after the kink instability fluctuate around a mean value, the first two modes n = 1 , 2 grow steadily with mode n = 1 becoming prevalent. This shows that an inverse cascade takes place. While the direct cascade transfers energy from the injection scale toward small scales (current sheets) where energy is dissipated, analogously the inverse cascade transfers energy toward the large scales (modes 1 and 2) where no dissipative process is at work and consequently energy accumulates . In physical space this process gives rise to the large scales that the magnetic field acquires in the orthogonal direction, shown in Figures 4 and 5, discussed previously. In the RMHD system with boundary conditions as we apply here there is no strict invariant known to follow an inverse cascade, such as magnetic helicity in 3D MHD or the square of the vector potential in 2D MHD (Biskamp 2003; Berger 1997; Brandenburg & Matthaeus 2004). RMHD resembles the 2D MHD case in the sense that though the square of the vector potential is not conserved, the terms violating conservation arise only from the boundaries in the axial direction. A dynamical magnetic inverse cascade mechanism is therefore still active, impeded only by the inputs coming from photospheric motions at the boundary, and this explains the accumulation of magnetic energy at the largest transverse scales.", "pages": [ 5, 6, 10, 11 ] }, { "title": "4.2. Run B", "content": "The simulation described in the previous section (run A) has a duration of ∼ 1200 τ A , but this time span leaves undetermined the behavior of the low wavenumber modes over longer time scales. Indeed these modes keep growing, as shown in Figure 8, resulting in a steady growth of total magnetic energy, shown in Figure 2. To understand the long-time dynamics of the system, we have performed another simulation, run B, with the same physical parameters of run A, but half the orthogonal resolution (Table 1), extending the duration up to ∼ 11000 τ A . Figure 9 shows that over longer times the energy of the system is prevalently in mode 1, i.e., the largest possible scale. But this mode does not grow indefinitely and over these much longer time-scales it reaches a statistically steady state, fluctuating around its mean value. The largest energy fluctuations shown in Figure 9 result in energy drops of ∼ 2000, that in dimensional units correspond to a micro-flare with ∆ E ∼ 2 × 10 25 erg , releasing about twice the amount of energy released by the kink instability around t ∼ 85 τ A in run A (compare with Figure 2). Notice that the kink instability does not appear in Figure 9 because the sampling time interval for the modes is too long in run B, but it is clearly shown in the r.m.s. of the energies (not shown) and in the dissipation rate (see inset in Figure 10). Comparing Figure 9 with Figure 10, where the ohmic dissipation rate J is shown as a function of time, displays another interesting result. The large and sharp energy drops shown in Figure 9 correspond to large dissipative peaks in Figure 10, e.g., at times t ∼ 4750 τ A and t ∼ 10300 τ A , but it is also possible to have equally large but more gradual energy drops, e.g., between times t ∼ 6750 τ A and t ∼ 7750 τ A , without a corresponding single large dissipative peak but rather a cluster of smaller peaks. In physical space we have already seen in run A that initially the inverse cascade corresponds to a perturbed magnetic field that occupies an increasingly larger volume (Figure 4) until all the field lines in the box get twisted (Figure 5). In run B we observe that successively, once the computational box has been filled with perpendicular magnetic field, the rising amplitude of modes 1 corresponds to an increase of the magnetic field intensity, while the fluctuations in the energy mode are due to magnetic reconnection events. In fact due to the periodic boundary conditions in x and y the same system repeats indefinitely along these directions. When the orthogonal magnetic field reaches the boundary it starts to interact with the neighboring structures (i.e. with itself coming from the other side). The magnetic energy drops in mode 1 correspond to magnetic reconnection events that make the system oscillate between the different possible configurations with energy contained at the (large) scales of mode 1 shown in Figure 11 (there is no preferred orthogonal direction for the system at this scale). While the periodic boundary conditions limit the interactions of large-scale twisted magnetic structures it is clearly shown that interaction with such other magnetic copies of itself is one of the ways in which the accumulated energy can be released. Further possibilities and the dynamics of these interactions will be the subject of future works.", "pages": [ 11, 12 ] }, { "title": "5. CONCLUSIONS AND DISCUSSION", "content": "In this paper we have investigated the dynamics of a closed coronal region driven at its boundary by a localized photospheric vortex. Such small vortical motions with scales typical of photospheric convection ( ∼ 1000 km) have been recently observed in the photosphere (Brandt et al. 1988; Bonet et al. 2008, 2010), and can induce relevant dynamics in the solar corona (Velli & Liewer 1999; Wedemeyer-Bohm et al. 2012; Panasenco, Martin, & Velli 2013). A 'straightened out' closed region of the solar corona is modeled as an elongated Cartesian box where the top and bottom plates mimic the photosphere, and the dynamics are integrated with the Reduced MHD equations (Kadomtsev & Pogutse 1974; Strauss 1976), well suited for a plasma threaded by a strong axial magnetic field. The initial condition consists simply of a uniform axial magnetic field. Its field lines are originally straight and its footpoints are line-tied at both ends in the top and bottom photospheric plates. The photospheric vortex drags the field lines' footpoints twisting the magnetic field lines (Figure 5, t = 60 . 57 τ A ). Even though the vortex that we employ is not perfectly circular (Figure 1, Equations (7)-(9)) in the linear stage the field lines' tension straightens out in a round shape the orthogonal magnetic field lines (Figure 4, t = 0 . 61 τ A and t = 80 . 64 τ A ), in this way the Lorentz force vanishes in the planes and the system is able to accumulate energy. The small departure from a round shape (at its edge) of the boundary vortex introduces a small perturbation in the coronal field. The system is then unstable to the internal kink mode (Figure 4, t = 80 . 64 τ A ), and releases about 90% of the accumulated energy in a dissipative event (Figures 2 and 3). The energy released in this event is of the order of a micro-flare with ∆ E ∼ 10 25 erg. These results are in agreement with those of Lionello et al. (1998), that consider similar initial conditions, performs a refined linear analysis, but does not employ a boundary forcing, i.e., the field lines are linetied to a motionless photosphere. Therefore the initial linear stage and the development of the kink instability are in agreement with previous works that have always employed large-scale smooth fields with no broad-band fluctuations as initial conditions, both in the case of field lines line-tied to a motionless photosphere (Baty & Heyvaerts 1996; Velli et al. 1997; Lionello et al. 1998; Browning et al. 2008; Hood et al. 2009) and with a boundary driver (Mik'ıc et al. 1990; Gerrard et al. 2002). On the other hand in the solar corona perturbations are continuously injected from the lower atmospheric layers. Numerical simulations (e.g., Rappazzo et al. 2008) confirm that especially in closed regions, where waves cannot escape toward the interplanetary medium, broadband magnetic fluctuations of the order of a few percent of the strong axial magnetic field (not infinitesimal perturbations as classically used in instabilities studies) are naturally present. Therefore in order to gain a first insight of the coronal dynamics when the magnetic field is already structured, i.e., there are finite magnetic fluctuations (small but not infinitesimal) with small scales and current sheets, we continue the simulation after kink instability develops. In fact right after kink instability the magnetic energy is small (Figure 2), with b ⊥ /B 0 ∼ 5%, but it is already structured with current sheets (Figure 4, t = 100 . 78 τ A ) and a broad band spectrum (Figure 7). The boundary vortex continues to twist the magnetic field lines, but in a disordered way (Figure 5, t = 202 . 15 1211 . 78 τ A ). The presence of an already structured magnetic field allows nonlinear dynamics to develop: once current sheets and small scales are present, an energy cascade continues to feed them, as shown by the energy spectra in Figure 7. Therefore current sheets do not disappear, and the continuous transfer of energy from the large to the small scales prevents the field lines to increase their twist beyond ∼ 180 · . The twist remains approximately constant in the nonlinear stage as shown in Figure 5 ( t = 202 . 15 - 1211 . 78 τ A ). Furthermore because the current is now concentrated in thin current sheets (Figures 4 and 6) kink instabilities do not develop. We had already observed a similar behavior in our previous simulations that employed space-filling boundary drivers. In particular when the field lines were sheared by a 1D boundary forcing (Rappazzo, Velli & Einaudi 2010) the coronal field was sheared only in the linear stage, but after that a multiple tearing instability developed and in the coronal field magnetic fluctuations and current sheets were formed, the continuous shearing motions at the boundary were not able to recreate a sheared coronal field and further instabilities were not observed. But in the simulations presented in this paper a new phenomenon occurs. Although the field lines' twist is approximately constant in the nonlinear stage, the volume where field lines are twisted increases, and the magnetic field acquires larger scales (Figures 4 and 5). Besides a direct cascade that transfers energy from the large to the small scales where it is dissipated in current sheets, an inverse cascade takes place, transferring energy from the injection scale toward larger scales, where no dissipation takes place and energy can accumulate. The analysis of the magnetic energy modes (Figures 8 and 9) shows indeed that on long time-scales most of the energy is stored at the largest possible scale (mode 1). The inverse cascade is able to store a significant amount of magnetic energy. Although magnetic helicity is not defined in RMHD, the integral of the magnetic square potential ψ is approximately conserved (see discussion in last paragraph of § 4.1). The inverse cascade of magnetic energy corresponds also to an inverse cascade of the square potential, as clearly shown in Figure 4 where the field lines are the contour of ψ (and ψ ≥ 0). In future compressible simulations we expect to observe for magnetic helicity (well defined in 3D MHD) dynamics similar to those shown here for magnetic energy, i.e., an increase of magnetic helicity (injected from the boundary) and its inverse cascade, in analogy to the the inverse helicity cascade observed by Amari & Luciani (2000). Because of the periodic boundary conditions along x and y the system is virtually repeated along these directions. When the field lines get twisted in the entire computational box, this twisted structure interacts with these neighboring twisted structures. This interaction is the only condition that limits the growth of magnetic energy, giving rise to impulsive magnetic reconnection events, that now is not inhibited by the circular topology of the orthogonal magnetic field lines of a single structure. These events make the system oscillate between the many possible configurations with energy in mode 1 (two of these are shown in Figure 11). The associated energy drops shown in Figure 9 are also in the micro-flare range with ∆ E ∼ 2 × 10 25 erg, twice the value of the energy released initially by the kink instability. Although in the presented simulations the generated magnetic structures interact only with similar structures repeated by the periodic boundary conditions along x and y , we can infer that the interaction of a single twisted magnetic structure with other magnetic structures can give rise to similar release of energy. A more general investigations of the interaction between twisted magnetic structures is under way to understand under which conditions the interaction leads to energy storage and/or release, and to determine quantitatively these properties. Previous simulations that employed a spacefilling photospheric forcing (Rappazzo et al. 2008; Rappazzo, Velli & Einaudi 2010) were not able to accumulate a significant amount of energy to be successively released in micro or larger flares. Those photospheric motions, that mimic a uniform and homogeneous convection, give instead rise to a basal background coronal heating rate in the lower range of the observational constraint (10 6 erg cm -2 s -1 ) and a million degree corona (Dahlburg et al. 2012). In the case of a space-filling boundary driver we had also observed that the inverse cascade is inhibited (see Rappazzo et al. 2008, § 5.4) for typically strong DC magnetic fields. An inverse cascade is possible only for weak guide fields (see Rappazzo et al. 2008, § 5.4), a condition applicable only to limited regions of the corona. We conclude that in presence of line-tying and a strong guide field, inverse cascade can be a good mechanism to store energy, but only if the boundary motion is localized in space as the vortex used here, and not space-filling. Subsequently the interaction of this magnetic structures with others can release the accumulated energy. In general photospheric motions will be a superposition of approximately homogenous space-filling convective motions and localized vortical and also shearing motions (e.g., see Dahlburg et al. 2009, for a localized shear case). While the space-filling motions give rise to a basal background coronal heating (e.g., Rappazzo et al. 2008), localized motions can give rise to higher impulsive releases of energy in the micro-flare range and above, contributing to coronal heating while increasing the temporal intermittency of the energy deposition and of its associated radiative emissions. In future works we will consider cases with localized motions superimposed to a homogenous space-filling convection-mimicking velocity field to determine, among other things, how stronger the localized velocity has to be respect to the background motions in order to develop dynamics similar to those presented in this paper. As mentioned in the introduction, highly (and orderly) twisted magnetic structures, such as flux ropes , are used to initiate solar eruptions (e.g., see Torok et al. (2011) for a recent application, and the reviews by Low (2001); Chen (2011) for further examples of this model). Kinklike instabilities developing in these flux-ropes give rise to an explosive dynamics leading to the formation of a CME. We have shown that kink-unstable flux ropes are not formed in the corona by boundary vortical motions, unless a very strong vortex is applied and the coronal magnetic fluctuations can then be neglected. Therefore, although flux ropes can be formed in the complex dynamics in and around a prominence region (Amari et al. 1999), given the ubiquitous presence of magnetic fluctuations in the solar corona, the development of kink-like instabilities may be strongly limited. While the dynamics of the induced CME can be a good approximation, we conclude that such models offer a poor model of the initiation process for which more realistic models are called for (Amari et al. 2011). Generally speaking, in a realistic 3D geometry one might expect that the growth of energy in the transverse field leads to an inflation and rise of a magnetic loop due to the curvature, which we have neglected here. This effect was included by Amari et al. (1996), who showed that twisting the footpoints of a curved flux rope leads to its gradual expansion and the system rises to larger solar radii. In our simulations the twist does not increase (the overall field lines twist is limited to 180 · ), remaining roughly constant in the nonlinear stage. It is left to future work to understand under which conditions such a system, including curvature, has dynamics similar to those of Amari et al. (1996), or whether different dynamics are possible (see also Gerrard et al. 2004), and how the dynamics develop in a 3D geometry considering small or large-scales photospheric vortices. This work was carried out in part at the Jet Propulsion Laboratory under a contract with NASA. This research supported in part by the NASA Heliophysics Theory program NNX11AJ44G, and by the NSF Solar Terrestrial and SHINE programs (AGS-1063439 & AGS-1156094), by the NASA MMS and Solar probe Plus Projects. Simulations have been performed through the NASA Advanced Supercomputing SMD awards 11-2331 and 123188.", "pages": [ 12, 13, 14 ] }, { "title": "REFERENCES", "content": "(Cambridge: Cambridge Univ. Press) Rappazzo, A. F., Velli, M., & Einaudi, G. 2010, ApJ, 722, 65 Velli, M., Lionello, R., & Einaudi, G. 1997, Sol. Phys., 172, 257", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "We have performed a detailed pulse phase resolved spectral analysis of the cyclotron resonant scattering features (CRSF) of the two Be/X-ray pulsars A0535+26 and XTE J1946+274 and the wind accreting HMXB pulsar 4U 1907+09 using Suzaku observations. The CRSF parameters vary strongly over the pulse phase and can be used to map the magnetic field and a possible deviation form the dipole geometry in these sources. It also reflects the conditions at the accretion column and the local environment over the changing viewing angles. The pattern of variation with pulse phase are obtained with more than one continuum spectral models for each source, all of which give consistent results. Care is also taken to perform the analysis over a stretch of data having constant spectral characteristics and luminosity to ensure that the results reflect the variations due to the changing viewing angle alone. For A0535+26 and XTE J1946+274 which show energy dependent dips in their pulse profiles, a partial covering absorber is added in the continuum spectral models to take into account an additional absorption at those phases by the accretion stream/column blocking our line of sight. Subject headings: X-rays: binaries- X-rays: individual: A0535+26- individual: XTE J1946+274- individual: 4U 1907+09- stars: pulsars: general", "pages": [ 2 ] }, { "title": "Pulse phase dependent variations of the cyclotron absorption features of the accreting pulsars A 0535+26, XTE J1946+274 and 4U 1907+09 with Suzaku", "content": "Chandreyee Maitra 1 and Biswajit Paul Raman Research Institute, Sadashivnagar, Bangalore-560080, India [email protected]; [email protected] Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The X-ray binary sources in which the compact object is a highly magnetized neutron star, often with a massive companion are called accretion powered pulsars. Due to the strong magnetic field of the neutron star, the matter here flows along the magnetic field lines to the poles of the system, forming an X-ray emitting accretion column above it (Pringle & Rees 1972; Davidson & Ostriker 1973; Lamb et al. 1973). Another important consequence of the strong magnetic fields ( ∼ 10 12 G) are the cyclotron resonant scattering features (CRSFs) formed by the resonant scattering of photons by the electrons which are quantized into Landau levels forming absorption like features at multiples of E c = 11 . 6 keV × 1 1+ z × B 10 12 G , E c being the centroid energy, z the gravitational redshift and B the magnetic field strength of the neutron star. The CRSFs thus provide a direct tool to measure the magnetic field strength of the neutron star. It was first discovered in the spectrum of Her X-1 (Trumper et al. 1977; Truemper et al. 1978) and about 20 sources with CRSFs have been discovered so far (Pottschmidt et al. 2012). The CRSFs which are found mostly in high mass X-ray binaries, about a half of which are transient sources, lie between the energy range of 10-60 keV. In addition to the magnetic field strength, the CRSFs also provide crucial information on the emission geometry and its physical parameters like the electron temperature, optical depth etc. Pulse phase resolved spectroscopy of the cyclotron parameters is an especially useful tool to probe the emission geometry at different viewing angle as the neutron star rotates. It can further be used to map the magnetic field geometry of the neutron star. Since the CRSFs also show variations with luminosity and spectral changes, to perform pulse phase resolved analysis, care should be taken to obtain the results solely due to the changing viewing angle by averaging over the data stretch with similar counts and spectral ratios. Proper continuum modeling of the energy spectrum also plays an important role in phase resolved analysis. Suzaku , with its broadband energy coverage is most ideally suited in this regard. A0535+26 is a Be/X-ray binary pulsar which was discovered during a giant outburst in 1975 by Ariel V (Rosenberg et al. 1975). It consists of a 103 s pulsating neutron star with a O9.7IIIe optical companion HDE245770 (Bartolini et al. 1978) in an eccentric orbit of e=0.47, with orbital period of 111 days (Finger et al. 2006). The distance to the source is ∼ 2 kpc (Giangrande et al. 1980; Steele et al. 1998). Upto six giant outbursts have been detected in this source so far, the latest ones during 2009/2010 (Caballero et al. 2011a,b). The last giant outburst was followed by two smaller outbursts with a periodicity of 115 days which is longer than its orbital period. Precursors to the giant outburst was also observed with the same periodicity (Mihara et al. 2010). CRSFs at ∼ 45 keV and ∼ 100 keV were discovered in this source during the 1989 giant outburst with HEXE (Kendziorra et al. 1994). The second harmonic at ∼ 110 keV was confirmed with OSSE during the 1994 outburst (Grove et al. 1995), although the presence of the fundamental at ∼ 45 keV was dubious. It was later confirmed during the 2005 outburst with Integral , RXTE (Caballero et al. 2007) and Suzaku observations (Terada et al. 2006). XTE J1946+274 is a transient Be/X-ray binary pulsar discovered by ASM onboard RXTE (Smith et al. 1998), and CGRO onboard BATSE (Wilson et al. 1998) during a giant outburst in 1998, revealing 15.8 s pulsations. The optical counterpart was identified as an optically faint B ∼ 18.6 mag, bright infrared (H ∼ 12.1) Be star (Verrecchia et al. 2002). The source has a moderately eccentric orbit of 0.33 with an orbital period of 169.2 days (Paul et al. 2001; Wilson et al. 2003). After the initial giant outburst and several short outbursts at periodic intervals, the source went into quiescence for a long time until the recent outburst in 2010 (Caballero et al. 2010b). A CRSF was discovered at ∼ 35 keV from the RXTE data of the 1998 outburst observations (Heindl et al. 2001). 4U 1907+09 is a persistent wind accreting high mass X-ray binary discovered in the Uhuru surveys (Giacconi et al. 1971; Schwartz et al. 1972), having a highly redenned companion star (O8-O9 Ia) of magnitude 16.37 mag and a mass loss rate of ˙ M = 7 ∗ 10 -6 M /circledot yr -1 (Cox et al. 2005). It has a moderately eccentric (e=0.28) orbit of 8.3753 days (in 't Zand et al. 1998). It is a slowly rotating neutron star with period of ∼ 440 s, and has showed several episodes of torque reversals with a steady spin-down from 1983 to 1998 (in 't Zand et al. 1998; Mukerjee et al. 2001), a much slower spin-down from 1998 to 2003 (Baykal et al. 2006), a torque reversal between 2004 and 2005 (Fritz et al. 2006) and a second torque reversal between 2007 and 2008 (Inam et al. 2009), which restored the source to the same spin-down rate before 1998. A CRSF at ∼ 19 keV was reported using data from the Ginga /observations (Makishima & Mihara 1992; Makishima et al. 1999) and was later confirmed from the BeppoSaX observations with the discovery of a harmonic at ∼ 36 keV. (Cusumano et al. 1998). Rivers et al. (2010) performed a time and phase resolved analysis of the Suzaku observations of the source made during 2006 and 2007. Here we present the results obtained from a pulse phase resolved spectroscopic analysis of these three sources with a motivation to investigate the variation pattern of the cyclotron parameters with pulse phase. The pulse phase dependence of the CRSF parameters are presented for the first time for A0535+26 and XTE J1946+274, whereas a more detailed result is presented for 4U 1907+09 which is in agreement with the earlier results of Rivers et al. (2010). The analysis is done taking into account various factors which might smear the pulse phase dependence results.The results presented in this paper are one of the most detailed results on pulse phase resolved measurements of CRSF available so far.", "pages": [ 3, 4, 5 ] }, { "title": "2. Observations & Data Reduction", "content": "There are two sets of scientific instruments onboard Suzaku. The X-ray Imaging Spectrometer XIS (Koyama et al. 2007) consisting of three front illuminated CCD detectors (FI :XIS0, XIS2, XIS3) and one back illuminated CCD detector (BI: XIS1) work in the 0.2-12 keV range and the Hard X-ray Detector (HXD) made of PIN diodes (Takahashi et al. 2007) and GSO crystal scintillator detectors cover the energy bands of 10-70 keV and 70-600 keV respectively. Suzaku (Mitsuda et al. 2007) observed A0535+26 twice, once on September 14-15 2005 during the decline of the second normal outburst of 2005, and again on August 24 2009 during the decline of the 2009 normal outburst. We have chosen the 2009 observation (Obs. Id-404054010) for our analysis because of the longer duration (exposure ∼ 52 ks) and its 'HXD nominal' pointing position which is more suitable for CRSF studies, although the count rates were comparable for both the observations. The XIS's were operated in the ' 1 4 window' 'burst' clock data mode which has a total time resolution of 2 s. XTE J1946+274 was observed on October 11 2010 (Obs. Id-405041010) just after the peak of the September/October 2010 normal outburst. The source was observed for ∼ 51 ks in the 'HXD nominal' pointing position, and the XIS's were operated in the ' 1 4 window' 'normal' clock data mode which has a time resolution of 2 s. 4U 1907+09 was also observed twice with Suzaku , once on May 2006, and again on April 2007. We have chosen the 2007 observation (Obs. Id-402067010) for our analysis because of similar reasons as in the case of A0535+26, i.e. longer exposure of ∼ 158 ks and the 'HXD nominal' pointing position. The XIS's were operated in 'normal' clock data mode with no window option which has a time resolution of 8 s. The XIS data was reduced and extracted from the unfiltered XIS events, which were reprocessed with the CALDB version 20120428. We checked for any significant photon pile-up effect in the reprocessed XIS event files. To perform pile-up estimation, we examined the Point Spread Function (PSF) of the XISs and obtained the count rate at the image peak per CCD exposure as given by Yamada & Takahashi (Yamada et al. 2012) 1 . Crab data is assumed to be free from pile-up and has a value of 36 ct/sq arcmin/s/CCD exposure at the image peak. Following their procedure, the XIS data of A0535 +26 and XTE J1946+274 had values of 2-3 ct/sq arcmin/s/CCD exposure at the image peaks, and showed no evidence of significant pile-up. 4U 1907+09 on the other hand showed a case of moderate photon pile-up. The value obtained at the image center was higher than the Crab Nebula count rate of 36 ct/sq arcmin/s/CCD exposure . The radius at which this value equals 36 in the PSF is about 15-16 arcsec, and hence 15 pixels were removed from the image center to account for this effect. For the extraction of XIS light curves and spectra from the reprocessed XIS data, a 4 ' diameter circular region was selected around the source centroid for A0535+26 and XTE J1946 +274, and an additional central 15 ( ∼ 16 '' ) pixels were removed in the case of 4U 1907+09 to discard the maximum pile up affected regions. Background light curves and spectra were also extracted by selecting regions of the same size away from the source. The XIS count rate was 3.6 c/s, 3.1 c/s and 8.1 c/s for A0535+26, XTE J1946 +274 and 4U 1907+09 respectively. 4U 1907+09 had a ∼ 12 % loss in count rate after the removal of the photons from the central region due to the pile-up correction. Response files and effective area files were generated by using the FTOOLS task 'xisresp'. The HXD/PIN light curves and spectra were extracted after reprocessing the unfiltered event files 2 . The HXD/PIN background was created by adding the simulated 'tuned' non X-ray background event files (NXB) corresponding to the month and year of the respective observations Fukazawa et al. (2009) 3 to the the cosmic X-ray background, which was simulated as suggested by the instrument team 4 after applying appropriate normalizations for both cases. The corresponding response files were obtained from the Suzaku guest observatory facility. 5", "pages": [ 5, 6, 7, 8 ] }, { "title": "3.1. Timing analysis: Light curves & Hardness ratio & pulse period determination", "content": "We performed timing analysis after applying barycentric corrections to the event data files using the FTOOLS task 'aebarycen'. Light curves were extracted with a time resolution of 2 s for the XISs (0.2-12 keV), and 1 s for the HXD/PIN (10-70 keV) respectively. For XTE J1946+274 which has a short pulse period, light curve with the resolution of 10 ms was extracted from the HXD/PIN data to search for the pulse period. We applied pulse folding and χ 2 maximization technique to search for pulsations in the XIS data for A 0535+26 and PIN/HXD data for XTE J1946+274. The best estimate of the period was found to be 103 . 47 ± 0 . 09 s for A0535+26. This value is consistent with the pulse period determined from the INTEGRAL IBIS data during the same outburst at MJD 55054.995 (Caballero et al. 2010a) assuming the spin down value determined from the same. For XTE J1946+274, the best-fit period was estimated to be 15 . 75 ± 0 . 11 s. Orbital correction of the pulse arrival times was not required for both the sources having a very long orbital period. The XIS and PIN light curves of the sources binned with its pulse period for A 0535+26 and 10 pulsar periods for XTE J1946+274, are shown in Figure 1. The light curves show more or less constant count rate, and do not have any particular trend of variation. For each figure, the third panel shows the hardness ratios (ratio of PIN counts to XIS counts) which is also more or less constant throughout the observation duration and does not have any signatures of spectral variability which might affect the results of pulse phase resolved spectroscopy.", "pages": [ 8, 9 ] }, { "title": "3.2. Energy Dependence of the pulse profiles", "content": "We created the energy resolved pulse profiles for the entire stretch of observations by folding the light curves in different energy bands with the obtained pulse period. The pulse profiles in the energy range of 0.3-12 keV were created using all the three XISs (0, 1 & 3), and in the 10-70 keV range were created from the PIN data. The energy dependence of the pulse profiles in A 0535+26 are shown in Figure 2. The pulse profiles are complex in structure with narrow dips in the low energy ranges ≤ 12 keV which morphed to become a simpler, more sinusoidal profile at higher energies. The following characteristics are observed with a careful examination of the profiles. The energy dependence is very similar to that found during the 2005 Suzaku observation (Naik et al. 2008). The profile is however, very different from the simple sinusoidal profile at all energies found during the quiescence phase of the source (Mukherjee & Paul 2005; Negueruela et al. 2000), or the double peaked profile extending upto higher energies during its giant outbursts (Mihara 1995; Kretschmar et al. 1996). The energy dependence of the pulse profiles of XTE J1946+274 is shown in Figure 3. The pulse profiles show a clear double peaked structure which extends upto the high energies. The following characteristics can be observed in more detail. The energy dependence of the pulse profiles of XTE J1946+274 is very similar to that investigated by Wilson et al. (2003) during the 1998 outburst of the source.", "pages": [ 9, 10 ] }, { "title": "3.3.1. Pulse phase averaged spectroscopy", "content": "We performed pulse phase averaged spectral analysis of A 0535+26 and XTE J1946+274 using spectra from the three front illuminated CCDs (XISs-0 and 3), the back illuminated CCD (XIS-1) and the PIN. We performed spectral fitting using XSPEC v12.7.0. The XIS spectra were fitted from 0.8-10 keV and the PIN spectrum from 10-70 keV. The energy range of 1.75-2.23 keV was neglected due to an artificial structure in the XIS spectra around the Si edge and Au edge. After appropriate background subtraction, the spectra were fitted simultaneously with all parameters tied, except the relative instrument normalizations which were kept free. The XIS spectra were rebinned by a factor of 6 from 0.8-6 keV and 7-10 keV, and by a factor of 2 between 6-7 keV. The PIN spectrum of A 0535+26 was rebinned by a factor of 2 upto 22 keV, by 4 upto 45 keV, and 6 upto 70 keV. Due to comparatively inferior statistics in the PIN spectrum of XTE J1946+274, higher rebinning factors of 2, 6, and 10 were applied in the above mentioned energy ranges. In HMXB accretion powered pulsars, the continuum emission can be interpreted to arise by Comptonization of soft X-rays in the plasma above the neutron star surface. It is usually modeled phenomenologically with a powerlaw and cutoff at high energies (White et al. 1983; Mihara 1995; Coburn 2001). The most widely used empirical models are the high energy cutoff (highecut) or the Fermi Dirac cutoff (fdcut) (Tanaka 1986) with the powerlaw component, or cutoff powerlaw (cutoffpl) model. Other models include the negative-positive exponential powerlaw component (NPEX) (Mihara 1995), and a more physical comptonization model 'CompTT' (Titarchuk 1994). We tried to fit the energy spectra with all the continuum models mentioned above, available as a standard or local package in XSPEC and carried out further analysis with only the models which gave best fits for the respective sources.", "pages": [ 11 ] }, { "title": "A 0535+26 :", "content": "For A0535+26 the best fits were obtained with the NPEX, powerlaw and the 'CompTT' model (assuming spherical geometry for the comptonizing region). The powerlaw model however did not require a 'highecut' to fit the energy spectra. Including the GSO spectra in the fitting, the relative normalization of the GSO with respect to XIS showed that the flux in the GSO band (50-200 keV) was overestimated ∼ 4 times without the inclusion of a 'highecut' in the spectrum. As inclusion of the GSO spectrum is not possible for phase resolved studies due to its limited statistics, and a spectrum of an accretion powered pulsar without a cutoff at higher energies is not viable, we have carried out further analysis with the 'NPEX' and 'CompTT' models. We applied a partial covering absorption model 'pcfabs' in both the cases along with the Galactic line of sight absorption, to take into account the intrinsic absorption evident at certain pulse phases. This is evident in the pulse profiles and is a feature local to the neutron star. The narrow Fe k α feature found at 6.4 keV was modeled by a gaussian line. In addition, a deep and wide feature found at ∼ 45 keV was modeled with a Lorentzian profile, which is the CRSF found previously in this source (Caballero et al. 2007). For A 0535+26, the CRSF has been reported before at the same energy, even in a Suzaku observation (Terada et al. 2006). So we do not comment on its detection significance here. We also tried a gaussian profile to model the CRSF feature. Since the centroid energy of the Lorentzian description is not coincident with the minimum of the line profile (Nakajima et al. 2010), apart from a slight offset between the centroid energies of the Lorentzian and gaussian profile,the other parameters like the depth and width are consistent between the two models. The fits are also similar. We however considered a Lorentzian profile for the CRSFs for the rest of the paper after verifying the consistency between the Lorentzian and Gaussian profiles. The CRSF parameters were also consistant within error bars for both the continuum models, the centroid energy being only slightly higher for the 'powerlaw' model. The reduced χ 2 obtained for the models were 1.25 and 1.26 for 839 and 840 d.o.f respectively with no systematic residual pattern.", "pages": [ 12, 13 ] }, { "title": "XTE J1946+274:", "content": "For XTE J1946+274, best fits with similar values of reduced χ 2 were obtained with the 'highecut', 'NPEX' and 'CompTT' model. Similar to A 0535+26, the local absorption of the neutron star was taken into account by the model 'pcfabs', and a gaussian line was also used to account for the narrow Fe k α feature found at 6.4 keV. A deep and wide residual was found at ∼ 38 keV, at the same energy as the CRSF discovered by Heindl et al. (2001). As discussed previously, the CRSF was modeled with a Lorentzian profile. The 'highecut' and 'NPEX' models gave consistant values of the CRSF parameters, but the 'CompTT' model required a much shallower and narrow profile. Moreover, we were unable to constrain all the parameters of the 'CompTT' well for this source, probably due to the poorer quality of the PIN data. We have thus carried out the further analysis of this source with the two former models. For the best fitting models, the reduced χ 2 was 1.09 and 1.11 respectively for 826 d.o.f. Without the inclusion of the CRSF, the difference in χ 2 was 150 and 119 respectively for the same models. The best-fitting values for the spectral models for both the sources are given in Table 1. Figure 5 shows the best-fit spectra for both the sources along with the residuals before and after including the CRSF, thus showing the presence of the feature clearly. Muller et al. (2012) however have reported the analysis of the RXTE,INTEGRAL and Swift observations during the same outburst of this source. Instead of a line at 36 keV, they found a weak evidence of a CRSF at ∼ 25 keV. It may be worthwhile mentioning in this context that the Suzaku PIN data has better sensitivity than INTEGRAL ISGRI at this energy range, and hence may be better suited for CRSF detection. However we have carefully checked the statistical significance and possible systematic errors associated the CRSF. Statistical significance : To estimate the detection significance of the CRSF we tried to fit the PIN spectrum alone with the 'highecut' model with its powerlaw index frozen to the value obtained from the best fitting broadband spectrum. The addition of the CRSF improved the χ 2 from 51.56 to 28.13 for 20 d.o.f corresponding to an F value of 16.7, and a F-test false alarm probability of 6 × 10 -4 . Possible systematic errors : At first, we used the the earth occultation data to check the reproducibility of the NXB (Fukazawa et al. 2009). We extracted the spectra using the earth occultation data in three energy bands centering the CRSF and compared ratio of the count rates with the NXB. The ratio obtained were 1.3, 1.2 and 1.2 at 10 -28, 28 -48 and 48 -70 keV respectively indicating the lack of any energy dependent feature that can be introduced by the simulated X-ray background. We also included a systematic uncertainty of 3% on the PIN spectrum to check the detection of the CRSF. The line was still detected, but the uncertainty in the depth of the feature increased by 23%. The detection of pulse phase dependence of this feature as discussed in section 3.3.2 is also in favor of its presence since the background data is not expected to vary over the pulse phase. Finally, to verify the existence of the CRSF in a model independent manner, we divided the PIN spectrum of a pulse phase with the deepest CRSF, by the same of a pulse phase with the shallowest CRSF detected (see section 3.3.2, pulse phase resolved spectroscopy for the corresponding spectra). Figure 4 shows the ratio plot of the two spectra. Although the quality of the data is not good after 40 keV, the dip at ∼ 30-35 keV is clearly seen indicating the presence of the CRSF.", "pages": [ 13, 14 ] }, { "title": "3.3.2. Pulse phase resolved spectroscopy", "content": "For the phase-resolved analysis we extracted the source spectra for both the XIS's and the PIN data after applying phase filtering in the FTOOLS task XSELECT. The same background spectra and response matrices as used for the phase- averaged spectra were however used in both the cases. The spectra were also fitted in the same energy range and rebinned by the same factor as in phase-averaged case. The Galactic absorption ( N H1 ) column density and the Fe line width were frozen to the phase-averaged values for the two respective models. Phase resolved spectroscopy of the cyclotron parameters : For investigating the pulse phase-resolved spectroscopy of the two CRSFs, phase resolved spectra were generated with their phases centered around 25 independent bins but at thrice their widths. This resulted in 25 overlapping bins out of which only 8 were independent. We however froze the width of the CRSF to the phase-averaged value of the respective models, and varied the rest of the continuum as well as the line parameters with pulse phase. This was due to our inability to constrain all the parameters because of limited statistics. Figure 6 shows the variation of the cyclotron parameters of the sources using the best fit models as a function of pulse phase. For both the sources, the different continuum models used result in a very similar pattern of variation of the parameters. This gives us a reasonable amount of confidence on the obtained results. The following features are evident from the Figure 6. The results are compared with respect to the high energy PIN profile (10-70 keV). A 0535+26 : where the pulse profile picks up. XTE J1946+274 Phase resolved spectroscopy of the continuum parameters: A dependence of the continuum energy spectrum on the pulse phase is implied from the strong energy dependence of the pulse profiles, as seen in Figure 2 and Figure 3. A partial covering absorption model in which the absorber is phase locked with the neutron star is required to explain the narrow energy dependent dips in the pulse profiles. This was also our main motivation in applying the partial covering absorption 'pcfabs' to model the continuum energy spectra. We generated the phase resolved spectra with 25 independent phase bins to investigate the pulse phase-resolved spectroscopy of the continuum parameters for A 0535+26. Due to the short spin period of XTE J1946+274, 25 independent phase bin extraction was not possible, specially for the XIS data. We proceeded with the extracting of 25 overlapping but 8 independent phase bins for extraction of both XIS and PIN data as was done for the phase resolved spectroscopy of the CRSF parameters. The cyclotron parameters of the corresponding phase bins were frozen to the best-fit values obtained from the results of investigation of the cyclotron line parameters using 25 overlapping phase bins. Figures 8 & 9 shows phase resolved continuum parameters using the best-fit spectral models as a function of the pulse phase for A 0535+26 and XTE J1946+274 respectively. The results obtained as seen from the Figure from both the models are as follows:", "pages": [ 15, 16, 17 ] }, { "title": "4.1. Timing analysis: Light curves & Hardness ratio & pulse period determination", "content": "4U 1907+09 is a variable X-ray source showing flaring and dipping activity in the timescales of minutes to hours (in 't Zand et al. 1997). We performed timing analysis after applying barycentric corrections to the event data files using the FTOOLS task 'aebarycen'. Light curves were extracted with a time resolution of 8 s (full window mode of the XIS data) for the XISs (0.2-12 keV), and 1 s for the HXD/PIN (10-70 keV) respectively. We applied pulse folding and χ 2 maximization technique to search for pulsations in the XIS data. The source having an eccentric orbit with a short orbital period, proper correction of the pulse arrival times are required to accurately determine the pulse period. However, the orbital ephemeris of this source is not known with high accuracy (in 't Zand et al. 1998). Thus to account for the orbital motion of the binary, we included a dp dt term in the fitting, starting with an initial guess consistant with the parameters of the binary, and iterating for different values of dp dt to get the maximum χ 2 . The best fit period corresponding to this was 441 . 113 ± 0 . 035 s MJD 54209.43189 with dp dt = 3 . 1 × 10 -6 . This value obtained is marginally higher than that found by Rivers et al. (2010) (441 . 03 ± 0 . 03). However they have not mentioned, taking into account the orbital correction of the pulse arrival times in their work which might be a reason for this discrepancy. Figures 1 shows the XIS and PIN light curves along with the hardness ratio. As can be seen from the figure, the light curves show two flaring features in between and a dip in the last ∼ 10 ks of the observation. These features were also mentioned in Rivers et al. (2010), while performing time resolved spectroscopy of the same Suzaku observation, and were probed further by them to investigate the spectral variability with time. The flares may, however also affect our results of pulse phase resolved spectroscopy. We have thus compared the pulse profiles and the energy spectra in these stretches individually with that from the rest of the observation. Though the pulse profiles look very similar in all the stretches, the energy spectra is harder with an increased absorption in the last stretch of the observation containing the dip. The main aim of this work being pulse phase resolved spectroscopy to probe the CRSF parameters, we excluded the stretch of the observation coincident with the dip in the light curve for further analysis. The arrows in Figures 1 indicate the length of the observation chosen for this work. Pulse profile for this duration of observation was also created in the XIS and PIN energy bands as before for A 0535+26 and XTE J1946+274. Due to the absence of low energy dips in this source however, the energy dependence of the pulse profiles was not investigated further.", "pages": [ 18, 19 ] }, { "title": "4.2. Pulse phase averaged spectroscopy", "content": "Phase averaged spectroscopy was carried out in the same procedure as in A 0535+26 and XTE J1946+274. Best fits were obtained with the 'highecut', 'NPEX' and 'compTT' model with comparable values of reduced χ 2 and similar residual patterns. Rivers et al. (2010) also obtained similar results with the 'highecut', 'fdcut' and 'NPEX' model. A comparison between the NPEX model parameters obtained in our analysis and those reported in Rivers et al. (2010) reveal a softer less absorbed spectra obtained by us. This is expected, since we have excluded the the last stretch of data from our analysis which had a more harder and absorbed spectra. Two gaussian lines were also used to model the narrow Fe k α and Fe k β feature found at 6.4 and 7.1 keV respectively. In addition, a relatively shallow and narrow feature found at ∼ 18 keV was modeled with a Lorentzian profile which is the CRSF previously detected in this source (Makishima & Mihara 1992; Makishima et al. 1999). As also discussed in Rivers et al. (2010), the first harmonic of the CRSF at ∼ 36 keV could not be detected in the PIN spectra probably due to the statistical limitation of the data in this energy range. The CRSF parameters obtained with the 'NPEX' and 'CompTT' models were consistant within error bars with that found by Rivers et al. (2010) who performed phase resolved spectroscopy in 6 independent bins using the gaussian absorption model (keeping in mind that the centroid energy of the Lorentzian description is not coincident with the minimum of the line profile (Nakajima et al. 2010)). The 'highecut' model however required a deeper CRSF to fit the spectra. The reduced χ 2 obtained for the models were 1.62, 1.51 and 1.69 for 832, 837 and 838 d.o.f for highecut, NPEX and CompTT respectively with no systematic residual patterns. Due to the compatibility of the CRSF parameters obtained with the 'NPEX' and CompTT models, we have carried out further phase resolved analysis using these two models. Figure 5 shows the best-fit spectra for 4U 1907+09 along with the residuals before and after including the CRSF, thus showing the presence of the feature clearly. The CRSF in this source is very strong and has also been reported in the same Suzaku observation before (Rivers et al. 2010). We therefore do not comment on its detection significance.", "pages": [ 19, 20 ] }, { "title": "4.3. Pulse phase resolved spectroscopy", "content": "For investigating the pulse phase-resolved spectroscopy of the CRSF, we generated phase resolved spectra with 25 overlapping but 8 independent phase bins and used the same analysis procedure as discussed previously for A 0535+26 and XTE J1946+274. We were however able to constrain the phase dependent variation of all the CRSF parameters for this source, probably due to the longest observation duration available for it and a low cyclotron energy compared to the other sources. Figure 6 shows the variation of the cyclotron parameters of the source using the best fit models as a function of pulse phase. As in the case of the previous sources, the similar pattern of variation obtained for the different continuum models used give us considerable amount of confidence on the obtained results. The following characteristics can be observed in more detail from Figure 6. As before, the variations are compared with respect to the high energy PIN profile.", "pages": [ 20, 21 ] }, { "title": "5. Discussions & Conclusions", "content": "In the present work we have presented the results of detailed pulse phase resolved spectroscopy of the CRSF parameters of A 0535+26, XTE J1946+274 and 4U 1907+09 using long Suzaku observations. Pulse phase dependence of the CRSF parameters are obtained for the first time in A0535+26 and XTE J1946+274 and a more detailed and careful analysis has been done in 4U 1907+09 which is consistant with the earlier result obtained using the same observation (Rivers et al. 2010). The analysis is done taking into account various factors which might smear the pulse phase dependence results as mentioned in earlier sections. The strength of our results lie in the fact that we have obtained similar pattern of variation of the CRSF parameters for all the three sources with more than one continuum model.", "pages": [ 21 ] }, { "title": "5.1. pulse phase dependence of the cyclotron parameters", "content": "Results of pulse phase resolved spectroscopy of the cyclotron parameters have been presented previously in some sources, for example in Her X-1 (Soong et al. 1990; Enoto et al. 2008; Klochkov et al. 2008), 4U 0115+63 (Heindl et al. 2000), Vela X-1 (Kreykenbohm et al. 1999, 2002; La Barbera et al. 2003; Maitra & Paul 2013), 4U 1538-52 (Robba et al. 2001), Cen X-3 (Suchy et al. 2008), and more recently in GX 301-2 (Suchy et al. 2012), 1A 1118-61 (Suchy et al. 2011; Maitra et al. 2012) and 4U 1626-67 (Iwakiri et al. 2012). By modeling the pulse phase dependence with different continuum models, we have been able to establish the robustness of the results. In the process of trying to fit the energy spectrum with different continuum models we have also noticed certain trends in continuum model fitting. The 'highecut' model being a very simple model with less number of parameters, is a good choice to model the continuum in case of moderate or poor statistics. This is evident in the case of XTE J1946+274. The 'CompTT' on the hand which is a more physical description of the spectra and has a reasonable number of free parameters is better for continuum fitting specially for phase resolved spectroscopy if the statistical quality of the data is reasonably good. This is probably the reason why it failed to constrain the continuum well in the case of XTE J1946+274. The 'NPEX' model approximates the photon number spectrum for an unsaturated Comptonization (Sunyaev & Titarchuk 1980; M'esz'aros 1992), and has a clear physical meaning inspite of being a phenomenological model. It is useful for all the three sources with significantly different statistical quality. By assuming certain physics and geometry of the line forming region, the CRSF feature has been modeled analytically and with simulations by Araya & Harding (1999); Araya-G'ochez & Harding (2000); Schonherr et al. (2007) and more recently by Nishimura (2008, 2011); Mukherjee & Bhattacharya (2012). Although these models predict variations in the depth, width and the centroid energy of the CRSF features with the changing viewing angle at different pulse phases, a variation in the CRSF parameters as large as 30% as found in our results needs to take into account either a possible deviation or distortion from the simple dipole geometry of the magnetic field (Schonherr et al. 2007; Mukherjee & Bhattacharya 2012) , a gradient in the field itself (Nishimura 2008), or a different geometry of the accretion column (Kraus 2001). A detailed modeling taking into account these factors would provide us a detailed information about the geometry and emission patterns of the sources. However simpler interpretations can be done, since the correlation of the deepest and shallowest CRSFs with the pulse profile of the source can provide some idea about the beaming pattern of the source at that luminosity. Following this, the trend of shallowest lines near the pulse peak and deepest near the off-pulse as found in A 0535+26 and XTE J1946+274, favors a pencil beam geometry. On the other hand, deepest and widest lines found near the peak and shallowest and narrowest near the off-pulse as found in 4U 1907+09 favors a fan beam geometry for the emission. These results may be further complicated by assuming the contribution from both the magnetic poles of the neutron star in contrary to one of them, either due to gravitational light bending or particular geometry of the system allowing the view of both the poles. Modeling of the variations of the CRSF parameters with pulse phase is ongoing. Detailed discussions on the same will be made in a future work. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center On line Service, provided by NASA/Goddard Space Flight Center. Chandreyee Maitra would like to thank Carlo Ferrigno for providing the 'fdcut' and the 'newhighecut' local models.", "pages": [ 22, 23 ] }, { "title": "REFERENCES", "content": "Araya, R. A., & Harding, A. K. 1999, ApJ, 517, 334 Araya-G´ochez, R. A., & Harding, A. K. 2000, ApJ, 544, 1067 Bartolini, C., Guarnieri, A., Piccioni, A., Giangrande, A., & Giovannelli, F. 1978, IAU Circ., 3167, 1 Baykal, A., ˙ Inam, S. C¸ ., & Beklen, E. 2006, MNRAS, 369, 1760 Caballero, I., Kretschmar, P., Santangelo, A., et al. 2007, A&A, 465, L21 Caballero, I., Pottschmidt, K., Barragan, L., et al. 2010a, arXiv:1003.2969 Caballero, I., Pottschmidt, K., Bozzo, E., et al. 2010b, The Astronomer's Telegram, 2692, Caballero, I., Pottschmidt, K., Santangelo, A., et al. 2011a, arXiv:1107.3417 Caballero, I., Ferrigno, C., Klochkov, D., et al. 2011b, The Astronomer's Telegram, 3204, 1 Coburn, W. 2001, Ph.D. Thesis, W. A., Rothschild, R. E., et al. 2002, ApJ, 580, 394 Cox, N. L. J., Kaper, L., & Mokiem, M. R. 2005, A&A, 436, 661 Cusumano, G., di Salvo, T., Burderi, L., et al. 1998, A&A, 338, L79 Davidson, K., & Ostriker, J. P. 1973, ApJ, 179, 585 Enoto, T., Makishima, K., Terada, Y., et al. 2008, PASJ, 60, 57 Finger, M. H., Camero-Arranz, A., Kretschmar, P., Wilson, C., & Patel, S. 2006, Bulletin of the American Astronomical Society, 38, 359 Fritz, S., Kreykenbohm, I., Wilms, J., et al. 2006, A&A, 458, 885 Fukazawa, Y., Mizuno, T., Watanabe, S., et al. 2009, PASJ, 61, 17 Giacconi, R., Kellogg, E., Gorenstein, P., Gursky, H., & Tananbaum, H. 1971, ApJ, 165, L27 Giangrande, A., Giovannelli, F., Bartolini, C., Guarnieri, A., & Piccioni, A. 1980, A&AS, 40, 289 Grove, J. E., Strickman, M. S., Johnson, W. N., et al. 1995, ApJ, 438, L25 Heindl, W. A., Coburn, W., Gruber, D. E., et al. 2000, American Institute of Physics Conference Series, 510, 173 Heindl, W. A., Coburn, W., Gruber, D. E., et al. 2001, ApJ, 563, L35 Inam, S. C¸ ., S¸ahiner, S¸., & Baykal, A. 2009, MNRAS, 395, 1015 in 't Zand, J. J. M., Strohmayer, T. E., & Baykal, A. 1997, ApJ, 479, L47 in 't Zand, J. J. M., Baykal, A., & Strohmayer, T. E. 1998, ApJ, 496, 386 Iwakiri, W. B., Terada, Y., Mihara, T., et al. 2012, ApJ, 751, 35 Kendziorra, E., Kretschmar, P., Pan, H. C., et al. 1994, A&A, 291, L31 Klochkov, D., Staubert, R., Postnov, K., et al. 2008, A&A, 482, 907 Koyama, K., Tsunemi, H., Dotani, T., et al. 2007, PASJ, 59, 23 Kraus, U. 2001, ApJ, 563, 289 Kretschmar, P., Pan, H. C., Kendziorra, E., et al. 1996, A&AS, 120, 175 Kreykenbohm, I., Kretschmar, P., Wilms, J., et al. 1999, A&A, 341, 141 Kreykenbohm, I., Coburn, W., Wilms, J., et al. 2002, A&A, 395, 129 Kreykenbohm, I., Wilms, J., Kretschmar, P., et al. 2008, A&A, 492, 511 La Barbera, A., Santangelo, A., Orlandini, M., & Segreto, A. 2003, A&A, 400, 993 Lamb, F. K., Pethick, C. J., & Pines, D. 1973, ApJ, 184, 271 Makishima, K., & Mihara, T. 1992, Frontiers Science Series, 23 Makishima, K., Mihara, T., Nagase, F., & Tanaka, Y. 1999, ApJ, 525, 978 Maitra, C., Paul, B., & Naik, S. 2012, MNRAS, 2231 Maitra, C., & Paul, B. 2013, ApJ, 763, 79 M'esz'aros, P. 1992, High-energy radiation from magnetized neutron stars., by M'esz'aros, Mihara, T. 1995, Ph.D. Thesis, Mihara, T., Yamamoto, T., Sugizaki, M., Nakajima, M., & Maxi Team 2010, The First Year of MAXI: Monitoring Variable X-ray Sources, Mitsuda, K., Bautz, M., Inoue, H., et al. 2007, PASJ, 59, 1 Mukerjee, K., Agrawal, P. C., Paul, B., et al. 2001, ApJ, 548, 368 Mukherjee, U., & Paul, B. 2005, A&A, 431, 667 Mukherjee, D., & Bhattacharya, D. 2012, MNRAS, 420, 720 Muller, S., Kuhnel, M., Caballero, I., et al. 2012, arXiv:1209.1918 Nishimura, O. 2008, ApJ, 672, 1127 Nishimura, O. 2011, ApJ, 730, 106 Naik, S., Dotani, T., Terada, Y., et al. 2008, ApJ, 672, 516 Nakajima, M., Mihara, T., & Makishima, K. 2010, ApJ, 710, 1755 Negueruela, I., Reig, P., Finger, M. H., & Roche, P. 2000, A&A, 356, 1003 Paul, B., Agrawal, P. C., Mukerjee, K., et al. 2001, A&A, 370, 529 Pottschmidt, K., Suchy, S., Rivers, E., et al. 2012, American Institute of Physics Conference Series, 1427, 60 Pringle, J. E., & Rees, M. J. 1972, A&A, 21, 1 Robba, N. R., Burderi, L., Di Salvo, T., Iaria, R., & Cusumano, G. 2001, ApJ, 562, 950 Rivers, E., Markowitz, A., Pottschmidt, K., et al. 2010, ApJ, 709, 179 Rosenberg, F. D., Eyles, C. J., Skinner, G. K., & Willmore, A. P. 1975, Nature, 256, 628 Schonherr, G., Wilms, J., Kretschmar, P., et al. 2007, A&A, 472, 353 Schwartz, D. A., Bleach, R. D., Boldt, E. A., Holt, S. S., & Serlemitsos, P. J. 1972, ApJ, 173, L51 Smith, D. A., Takeshima, T., Wilson, C. A., et al. 1998, IAU Circ., 7014, 1 Soong, Y., Gruber, D. E., Peterson, L. E., & Rothschild, R. E. 1990, ApJ, 348, 641 Suchy, S., Pottschmidt, K., Wilms, J., et al. 2008, ApJ, 675, 1487 Suchy, S., Pottschmidt, K., Rothschild, R. E., et al. 2011, ApJ, 733, 15 Suchy, S., Furst, F., Pottschmidt, K., et al. 2012, ApJ, 745, 124 Sunyaev, R. A., & Titarchuk, L. G. 1980, A&A, 86, 121 Steele, I. A., Negueruela, I., Coe, M. J., & Roche, P. 1998, MNRAS, 297, L5 Tanaka, Y. 1986, IAU Colloq. 89: Radiation Hydrodynamics in Stars and Compact Objects, 255, 198 Takahashi, T., Abe, K., Endo, M., et al. 2007, PASJ, 59, 35 Terada, Y., Mihara, T., Nakajima, M., et al. 2006, ApJ, 648, L139 Titarchuk, L. 1994, ApJ, 434, 570 Truemper, J., Pietsch, W., Reppin, C., et al. 1978, ApJ, 219, L105 Trumper, J., Pietsch, W., Reppin, C., & Sacco, B. 1977, Eighth Texas Symposium on Relativistic Astrophysics, 302, 538 Verrecchia, F., Israel, G. L., Negueruela, I., et al. 2002, A&A, 393, 983 White, N. E., Swank, J. H., & Holt, S. S. 1983, ApJ, 270, 711 Wilson, C. A., Finger, M. H., Wilson, R. B., & Scott, D. M. 1998, IAU Circ., 7014, 2 Wilson, C. A., Finger, M. H., Coe, M. J., & Negueruela, I. 2003, ApJ, 584, 996 Yamada, S., Uchiyama, H., Dotani, T., et al. 2012, PASJ, 64, 53 . Time (s)", "pages": [ 24, 25, 26, 27, 28, 29 ] } ]

[ { "title": "ABSTRACT", "content": "IGR J17480 -2446 is an accreting X-ray pulsar in a low-mass X-ray binary harbored in the Galactic globular cluster Terzan 5. Compared with other accreting millisecond pulsars, IGR J17480 -2446 is peculiar in its low spin frequency (11 Hz), which suggests that it might be a mildly recycled neutron star at the very early phase of mass transfer. However, this model seems to be in contrast with the low field strength deduced from the kiloHertz quasi-periodic oscillations observed in IGR J17480 -2446. Here we suggest an alternative interpretation, assuming that the current binary system was formed during an exchange encounter either between a binary (which contains a recycled neutron star) and the current donor, or between a binary and an isolated, recycled neutron star. In the resulting binary, the spin axis of the neutron star could be parallel or anti-parallel with the orbital axis. In the later case, the abnormally low frequency of IGR J17480 -2446 may result from the spin-down to spin-up evolution of the neutron star. We also briefly discuss the possible observational implications of the pulsar in this scenario. Subject headings: neutron stars: X-ray binary -stars: evolution -binary: IGR J17480 -2446", "pages": [ 1 ] }, { "title": "On the formation of the peculiar low-mass X-ray binary IGR J17480 -2446 in Terzan 5", "content": "Long Jiang and Xiang-Dong Li 1 Department of Astronomy, Nanjing University, Nanjing 210093, China 2 Key laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Millisecond pulsars are thought to be old neutron stars having been recycled in lowmass X-ray binaries (LMXBs) (Aplar et al. 1982). According to this scenario, material is transferred to the neutron star from its companion when it fills its Roche lobe, spinning up the neutron star to the period of milliseconds, and reducing the surface magnetic field by several orders of magnitude, from ∼ 10 11 -10 12 Gto ∼ 10 8 -10 9 G(Bhattacharya & van den Heuvel 1991, for a review). Currently there are 14 accreting millisecond pulsars (Patruno & Watts 2012). The first accreting millisecond pulsar is SAX J1808.4 -3658, which rotates at a frequency of 401 Hz (Wijnands & van der Klis 1998). Spectral modeling of the iron line constrained its magnetic field to be ∼ 3 × 10 8 G at the poles (Cackett et al. 2009; Papitto et al. 2009), similar to the value ∼ 2 × 10 8 G derived from the spin-down rate measured between outbursts (Hartman et al. 2008, 2009). With a spin frequency of 599 Hz, IGR J00291+5934 is another source showing both spin-up and spin-down during and between outbursts respectively, suggesting the magnetic field to be ∼ 2 × 10 8 G (Hartman et al. 2011). Both SAX J1808.4 -3658 and IGR J00291+5934 are composed by an accreting pulsar and a brown dwarf companion. The 435 Hz accreting pulsar XTE J1751 -305 has a helium white dwarf companion. Its magnetic field was derived from its spin-down to be ∼ 4 × 10 8 G, while its spin-up during the 2005 outburst was strongly affected by timing noise (Papitto et al. 2008). From the aforementioned examples, one sees that the spin frequencies and magnetic fields reported are consistent with the prediction of the recycling scenario. A new accreting pulsar IGR J17480 -2446 was detected with the International GammaRay Astrophysics Laboratory in 2010 October (Bordas et al. 2010). This system is located in the core of Terzan 5, one of the densest globular clusters in the Galaxy. It looks like a typical LMXB, with an orbital period of 21.3 h. The binary mass function was estimated to be ∼ 0 . 021275(5) M /circledot , indicating a companion star of mass larger than 0 . 4 M /circledot (Papitto et al. 2011). The most remarkable feature of IGR J17480 -2446 is that its rotating frequency is only 11 Hz (Strohmayer & Markwardt 2010), too slow compared with the known spin frequencies ( ∼ 185 -600 Hz ) of accreting millisecond pulsars. Estimating the magnetic field of IGR J17480 -2446 is not straightforward. Assuming that the inner radius of the accretion disk lies between the neutron star's radius and the corotation radius when the source shows pulsations, Papitto et al. (2011) and Cavecchi et al. (2011) evaluated the magnetic field in the range from ∼ 2 × 10 8 G to ∼ 2 × 10 10 G. Miller et al. (2011) used the results of a relativistic iron line fit to estimate the magnetic field at the poles to be B ∼ 10 9 G. Papitto et al. (2012) estimated the magnetic field in the range between ∼ 5 × 10 9 G and ∼ 1 . 5 × 10 10 G from the spin-up rate during outbursts. Finally, assuming the kiloHertz quasi-periodic oscillation (kHz QPO) frequency as an orbital frequency at the inner disk radius, one can get a lower limit of the radius. If the disk is truncated at the magnetospheric radius, the upper limit of the magnetic field of the neutron star can be derived. Barret (2012) detected highly significant QPOs soon after the source had moved from the atoll state to the Z state at frequencies between 800 and 870 Hz, and suggested the surface magnetic field be less than 5 × 10 8 G (see also Altamirano et al. 2012). The above investigations indicate that there is a possibility that the magnetic field of IGR J17480 -2446 is similar to other accreting millisecond pulsars. If this is the case, there is interesting implication for its magnetic field evolution. It is controversial whether there is long-term evolution of the magnetic fields of rotation-powered neutron stars. However, magnetic field decay in accreting neutron stars has been widely accepted, and the mechanisms include accelerated Ohmic decay, vortex-fluxoid interactions, and magnetic burial or screening (Payne et al. 2008, for a review). Shibazaki et al. (1989) proposed a phenomenological form relating magnetic field evolution with accreted mass ∆ m (see also Romani 1990), where B 0 is the initial magnetic field, and m ∗ is a constant. By fitting to observations of LMXBs, Shibazaki et al. (1989) found m ∗ ∼ 10 -4 M /circledot . van den Heuvel & Bitzaraki (1995) showed that there is a remarkable correlation between the magnetic fields and the orbital periods of binary radio pulsars with nearly circular orbits and low-mass helium white dwarf companions. This relation is consistent with increasing decay of neutron star magnetic field with increasing amount of matter accreted: neutron stars with magnetic fields below a few 10 9 G have accreted material of (0 . 5 f ) M /circledot , where f ∼ 0 . 5 -1 is the accretion efficiency. From the measured masses of neutron stars in binary systems, Zhang et al. (2011) also found that the average mass of millisecond radio pulsars is indeed ∼ 0 . 2 M /circledot heavier that that of other long-period pulsars. The abnormally low rotation frequency suggests that IGR J17480 -2446 could be exactly in the process of becoming an accreting millisecond pulsar. Indeed, observations show that it is spinning-up at a rate ˙ ν ≈ 1 . 4 × 10 -12 Hzs -1 (Cavecchi et al. 2011; Patruno et al. 2012). Patruno et al. (2012) proposed that IGR J17480 -2446 is a mildly recycled pulsar which has started a spin-up phase lasting less than a few 10 5 yr. This means that IGR J17480 -2446 is in an exceptionally early RLOF phase. A potential problem of this scenario is that the incipient RLOF mass transfer may cause little field reduction according to Eq. (1) (see discussion in Section 3). To account for this, Patruno et al. (2012) assumed that the neutron star underwent two phase of evolution, i.e., the magneto-dipole spin-down and the wind accretion spin-down before the current RLOF spin-up phase. During the wind accretion phase the neutron star magnetic field B decayed to be ∼ 10 10 G, in proportion to the rotation rate, due to the flux-line vortex line coupling (Srinivasan et al. 1990). However, this model of magnetic field decay seems not to be compatible with observations. For example, the symbiotic X-ray pulsar GX 1+4 is believed to possess very strong magnetic field B ∼ 3 × 10 13 G with very low spin frequency 6 . 3 × 10 -3 Hz (Cui 1997). Other long-period X-ray pulsars such as 4U 2206+54 (Finger et al. 2010; Reig et al. 2012), GX301 -2 (Doroshenko et al. 2010) and SXP 1062 (Fu & Li 2012) are even though to be accreting magnetars with B > 10 14 G. Alternatively, if the magnetic field of IGR J17480 -2446 is ∼ 10 8 -10 9 G, Eq. (1) implies that it should have accreted a sufficient amount of mass (at least 0 . 1 M /circledot ), and it will be difficult to explain its slow spin in the traditional recycling scenario. Considering the fact that Terzan 5 is one of the densest and metal-richest clusters in our Galaxy (Cohn et al. 2002; Ortolani et al. 2007), with 35 rotation-powered millisecond pulsars discovered so far (Ransom et al. 2005; Hessels et al 2006; Pooley et al. 2010), we suggest that the companion star of IGR J17480 -2446 is not the original one in its primordial binary, and that the neutron star has undergone close encounter during which the primordial binary system broke up and formed a triple system. In the end of the short-interval triple phase, the neutron star captured the current companion and lost its first donor, which had spun it up to the spin of milliseconds (along with the magnetic field decayed to ∼ 10 8 -10 9 G). When the second mass transfer occurred, if the spin angular momentum of the neutron star was reversed to the orbital angular momentum of the current companion, the neutron star was spin-down first. This phase lasted ∼ 10 8 yr until the spin angular momentum reduced to zero and succeeded by the current spin-up. Since the second spin-up epoch started just recently, it is not abnormal to detect the system with slow spin. The structure of this work is as follows. In the following section we briefly review the exchange encounter processes in the globular cluster, and estimate the formation rate of neutron stars that have evolved from the reversed-to-parallel accretion channel. In Section 3 we describe the possible evolution of IGR J17480 -2446 in some detail. The observational implications are discussed in Section 4.", "pages": [ 1, 2, 3, 4 ] }, { "title": "2. The chance of exchange encounters in Terzan 5", "content": "Terzan 5 is reported as the densest globular cluster with a central mass density ∼ (1 -4) × 10 6 M /circledot pc -3 (Lanzoni et al. 2010). It is composed by two different populations of stars with sub-solar metallicity ( Y = 0 . 26 and Z = 0 . 01) and an age of 12( ± 1) Gyr, and with supra-solar metallicity ( Y = 0 . 29 and Z = 0 . 03) and an age of 6( ± 2) Gyr. Given the high density and old ages of Terzan 5, its X-ray binaries are likely to be formed during close encounter processes: a neutron star captured tidally during a close encounter with a single star or took the place of one member of a binary star in an exchange encounter. In our case we assume that the current donor of IGR J17480 -2446 has exchanged its original companion in the progenitor binary. We set m 1 as the mass of the first donor, which has lost most of its matter and was ejected after the encounter, m 2 as the mass of the neutron star, and m 3 as the mass of the incoming object, assumed to be a main sequence star (all the masses are in the units of M /circledot ). Following Heggie et al. (1996) we write the semi-analytical exchanging cross section as follows, where M 12 = m 1 + m 2 , M 13 = m 1 + m 3 , M 23 = m 2 + m 3 , M 123 = m 1 + m 2 + m 3 , k = 3 . 70 + 7 . 49 µ 1 -1 . 89 µ 2 -15 . 49 µ 2 1 -2 . 93 µ 1 µ 2 -2 . 92 µ 2 2 +3 . 07 µ 3 1 +13 . 15 µ 2 1 µ 2 -5 . 23 µ 1 µ 2 2 + 3 . 12 µ 3 2 , µ 1 = m 1 /M 12 , µ 2 = m 3 /M 123 . Here ¯ a is the averaged orbital separation of the binary in units of AU, and v ∞ is the velocity dispersion of the cluster in the units of km s -1 . This exchange process results in a binary system consisting of a main-sequence companion and a recycled neutron star 1 . The encounter rate is roughly where n t and n bin are the number densities (in units of pc -3 ) of the target stars and the original binaries, respectively. We first discuss the number density of the target objects. We assume that all the stars in the globular cluster were formed more or less simultaneously, and the initial mass distribution is given by a power law function: d N = C 0 m -1 -x d m (Salpeter 1955; Verbunt 1988), where both the normalization constant C 0 and the power index x need to be derived from the observational data. For Terzan 5 we set the value of x to be in the range of 1 -2 (cf. Verbunt & Hut 1987; Verbunt 1988). The normalization constant C 0 is dependent on the total stellar mass in the globular cluster, or the mean mass density ρ . The number density of the target stars ( n t ) can be derived to be where m up and m low are the upper and lower mass limits of the target stars respectively, while m max and m min are for all stars in the globular cluster. We take the turnoff mass as the upper limit of the stellar mass in the cluster, which can be calculate from the main sequence lifetime of a star with mass m (Eggleton 2006), for 0 . 25 ≤ m ≤ 50. Setting m min = 0 . 1 and m max = 1 . 2 (i.e., the turnoff mass of stars with age of 6 Gyr), with the reported central mass density of Terzan 5 ∼ (1 -4) × 10 6 M /circledot pc -3 (Lanzoni et al. 2010), we calculate the number density of stars of mass 0 . 5 -1 . 2 M /circledot 2 to be n t ∼ (1 . 8 -7 . 2) × 10 5 pc -3 for x = 2, and ∼ (4 . 7 -18 . 8) × 10 5 pc -3 for x = 1. So in the following we take n t ∼ 5 × 10 5 pc -3 as a rough estimate. The number density of the binaries n bin can be estimated by using the total number of binaries with millisecond pulsars ( N b ) divided by the volume ( V ) of the cluster core. The number of binary systems which have undergone exchange encounters with a reversed-spinning neutron star is where f ( m ) = m 7 / 2 3 M 1 / 6 123 M 1 / 6 23 M -5 / 2 13 M -1 / 3 12 e k , and T p is the time interval between the formation of the original binary and the encounter, which can be roughly set as ∼ 3 × 10 9 yr, the half age of the metal-rich population in Terzan 5. A factor of 1 / 2 is added to account for the fact that the orbit angular momentum of the later binary can be either parallel or anti-parallel with the spin of the neutron star. Taking typical values for the parameters in Eq. (5), i.e., m 1 /similarequal 0 . 3, m 2 /similarequal 1 . 4, m 3 /similarequal 0 . 8, v ∞ ∼ 10, and ¯ a ∼ 0 . 02, we obtain N ∼ 0 . 5 N b , suggesting that a considerable fraction of the millisecond binary pulsars in Terzan 5 may have experienced the specified dynamical interaction. There are 35 known millisecond pulsars (Ransom et al. 2005; Hessels et al 2006; Pooley et al. 2010) in this globular cluster, and the total number of millisecond pulsars may be ∼ 150 (Bagchi et al. 2011). It is not surprising that all the binary pulsars may have been formed by dynamical interactions, and probably half of them might have experienced exchange encounters that leave a reversed-spinning neutron star in the new binary. We may expect that other globular cluster also harbour systems like IGR J17480 -2446. For example, in the globular cluster NGC 6440, the central density is ∼ 5 × 10 5 M /circledot pc -3 (Webbink 1985), so around 5% of the millisecond pulsars might have undergone the exchange evolution.", "pages": [ 4, 5, 6, 7 ] }, { "title": "3. Spin evolution of the neutron star", "content": "In the last section we argue that the abnormality of IGR J17480 -2446 may be explained by assuming that the current donor is not the member of the original binary but a captured object during a close encounter. In the following we will discuss the spin evolution of the neutron star in detail. Due to accretion in the original binary, the neutron star's spin may reach the equilibrium period (Bhattacharya & van den Heuvel 1991), where B 8 is the neutron star magnetic field in units of 10 8 G, ˙ m 17 is the accretion rate in units of 10 17 gs -1 , and R 6 is the neutron star radius in units of 10 6 cm. It is known that, after the neuron star has accreted ∼ 0 . 1 M /circledot and its magnetic field has decreased to be /lessorsimilar 10 9 G, its spin period will be insensitive to its initial value (e.g., Wang et al. 2011). The required accretion time to reach the equilibrium period is (Aplar et al. 1982), where P 3 is the spin period in units of 3 ms, and I 45 is the momentum of inertia in units of 10 45 g cm 2 . This time is considerably shorter than the total duration of mass transfer in LMXBs (usually /greaterorsimilar a few 10 8 yr), so it is likely that when the exchange encounter occurred, the neutron star had already been spun up to be a millisecond pulsar. After the capture of the new companion star and the formation of the current binary, the neutron star first spun down due to magnetic dipole radiation, but this would not change its spin period significantly, since the evolutionary time for a rotation-powered millisecond pulsar is usually /greaterorsimilar a few 10 9 yr. When the captured star started to fill its RL due to stellar expansion or due to shrinking of the RL caused by the loss of orbital angular momentum, the neutron star would experience the second mass transfer that further altered its spin evolution. If the orbital angular momentum and the neutron star's spin angular momentum were parallel, accretion onto the neutron star would change its spin period to a new equilibrium period determined by the current mass accretion rate. However, since P eq is weakly dependent on the accretion rate as seen in Eq. (6), the values of the equilibrium periods should be close to each other 3 . If the orbital and spin angular momenta were anti-parallel, the subsequent evolution was composed of a spin-down phase followed by a spin-up phase. The magnitude of the period derivation in both phases can be described as (Bhattacharya & van den Heuvel 1991) This gives the spin-down time, and the spin-up time to the current period, where P 100 = P/ 100 ms. The real value of t up could be even larger, since IGR J17480 -2446 currently appears as a transient source (Papitto et al. 2011; Patruno et al. 2012). The typical evolutionary lifetime of a LMXB is roughly t ev ∼ ∆ m/ ˙ m ∼ 3 × 10 8 ˙ m -1 17 yr for an average accreted mass of ∼ 0 . 3 M /circledot . The observed number of the IGR J17480 -2446-like systems can then be roughly estimated as or N obs ∼ 0 . 6 for N ∼ 0 . 5 N b ∼ 75, which suggests there could be at most one such system in Terzan 5. Obviously the rarity of IGR J17480 -2446 originates from its very short duration of the current spin-up phase, and it will become a millisecond pulsar again a few 10 7 yr later. It is also noted that, according to Eq. (10), the accreted mass to accomplish the spin-up to 11 Hz is ∼ 0 . 002 M /circledot . With this amount of mass, Eq. (1) suggests that the magnetic field would have decayed only from ∼ 10 12 G to ∼ 5 × 10 10 G if this were the first phase of mass accretion. Specifically, there was enough accreted matter to spin up the neutron star, but it would be insufficient to substantially reduce the magnetic field to ∼ 10 8 -10 9 G. In our proposed scenario, this problem does not appear since the neutron star had already been recycled before the exchange encounter.", "pages": [ 7, 8 ] }, { "title": "4. Discussion and conclusions", "content": "The newly discovered accreting millisecond pulsar IGR J17480 -2446 in the globular cluster Terzan 5 has surprisingly low spin frequency, and has been suggested to be a mildly recycled pulsar that started a spin-up phase in an exceptionally recent time. Here we propose an alternative explanation if the magnetic field of IGR J17480 -2446 is as low as other accreting millisecond pulsars, taking into account the dense environment of IGR J17480 -2446. In dense globular clusters, when a single star interacts with a binary (the neutron star could either be a member of the binary or the single object), the most probable result is that one of the binary components is replaced by the single star if it is the lightest one (Krolik et al. 1984). In our case the resulting binary will be composed by a recycled neutron star and a relatively massive companion star. The high density of the globular cluster Terzan 5 supports the possibility of such triple-object close encounter. The low spin frequency of IGR J17480 -2446 may be explained as the result of reversed-to-parallel evolution of the neutron star's spin. In Patruno et al. (2012), the system is assumed to be in its incipient mass transfer process, while in this work, since the donor has transferred some more material through RLOF, the mass of the companion may be ∼ 0 . 1 -0 . 2 M /circledot less than its initial mass. Accordingly, the neutron star has experienced twice accretion phases, so its mass may be considerably higher than its initial value. However, it seems difficult to distinguish our model and Patruno et al. (2012) in these respects, since detail calculations (e.g. Lin et al. 2011) show that the evolutions of LMXBs are rather complicated, depending on the initial masses of the component stars, the initial orbital periods, and the processes of mass and orbital angular momentum transfer and loss. The neutron star magnetic field may serve as a distinct feature. We assume that the neutron star has experienced long time accretion, so its magnetic filed has reached the bottom field, ∼ 10 8 -10 9 G, considerably lower than the expected value of Patruno et al. (2012). Both the spin evolution and the kHz QPO frequencies can present constraints on the magnitude of the magnetic field, if the mass accretion rate of the neutron star can be accurately determined. Finally it is pointed out that this work is based on the specified relation between the magnetic field and accreted mass described by Eq. (1). As we know that the mechanism for accretion-induced field decay is still uncertain and there may be other forms. For example, in Kiel et al. (2008), it is assumed that the magnetic field decays exponentially with the amount of mass accreted: where k is a scaling parameter that determines the rate of decay. For choices of k = 3000 and 10000, as suggested by Kiel et al. (2008), an accretion of only ∼ 0 . 002 M /circledot can decrease the magnetic field to ∼ 2 × 10 9 G or < 10 8 G. Thus, with Eq. (12) the small amount of accretion required to spin up the neutron star to 11 Hz would also be sufficient to highly suppress the magnetic field. If it can someday be established, via other means, that the capture by the neutron star of a second companion is necessary, or possibly not needed for IGR J17480 -2446, then this might point toward either Eq. (1) or (12) as the more valid expression for the field decay in accreting neutron stars. We are grateful to an anonymous referee for helpful comments. This work was supported by the Natural Science Foundation of China under grant number 11133001 and the Ministry of Science, the National Basic Research Program of China (973 Program 2009CB824800), and the Qinglan project of Jiangsu Province.", "pages": [ 8, 9, 10 ] }, { "title": "REFERENCES", "content": "Alpar, M. A., Cheng, A. F., Ruderman, M. A. & Shaham, J. 1982, Nature, 300, 728 Altamirano, D., Ingram, A., van der Klis, M., et al. 2012, arXiv:1210.1494 Bagchi, M., Lorimer, D. R. & Chennamangalam, J. 2011, MNRAS, 418, 477 Barret, D. 2012, ApJ, 753, 84 Bhattacharya, D. & van den Heuvel, E. P. J. 1991, Phys. Rep., 203, 1 Bordas, P., Kuulkers, E., Alfonso-Garz´on, J., et al. 2010, ATel, 2919, 1 Boyles, J., Lorimer, D. R., Turk, P. J., et al. 2011, ApJ, 742, 51 Cackett, E. M., Altamirano, D., Patruno, A., et al. 2009, ApJ, 694, L21 Cavecchi, Y., Patruno, A., Haskell, B., et al. 2011, ApJ, 740, L8 Cohn, H., Lugger, P. M., Grindlay, J. E. & Edmonds, P. D. 2002, ApJ, 571, 818 Cui, W. 1997, ApJ, 482, L163 Degenaar, N. & Wijnands, R. 2011, MNRAS, 414, L50 Doroshenko, V., Santangelo, A., Suleimanov, V., et al. 2010, A&A, 515, 10 Eggleton, P. 2006, Evolutionary Processes in binary and multiple stars, Cambridge University Press, p.36 Finger, M. H., Ikhsanov, N. R., Wilson-Hodge, C. A. & Patel, S. K. 2010, ApJ, 709, 1249 Fu, L. & Li, X.-D. 2012, ApJ, 757, 171 Hartman, J. M., Patruno, A., Chakrabarty, D., et al. 2008, ApJ, 675, 1468 Hartman, J. M., Patruno, A., Chakrabarty, D., et al. 2009, ApJ, 702, 1673 Hartman, J. M., Galloway, D. K. & Chakrabarty, D. 2011, ApJ, 726, 26 Heggie, D. C., Hut, P. & McMillan, S. L. W. 1996, ApJ, 467, 359 Hessels, J. W. T., Ransom, S. M., Stairs, I. H., et al. Science, 2006, 311, 1901 Kiel, P. D., Hurley, J., Bailes, M., & Murray, J. R. 2008 MNRAS, 388, 393 Krolik, J. H., Meiksin, A. & Joss, P. C. 1984, ApJ, 282, 466 Lanzoni, B., Ferraro, F. R., Dalessandro, E., et al. 2010, ApJ, 717, 653 Lin, J., Rappaport, S., Podsiadlowski, Ph., Nelson, L., Paxton, B., & Todorov, P. 2011, ApJ, 732, 70 Miller, J. M., Maitra, D., Cackett, E. M., et al. 2011, ApJ, 731, L7 Ortolani1, S., Barbuy, B., Bica, E., et al. 2007, A&A, 470, 1043 Papitto, A., D'A'ı, A., Motta, S., et al. 2011, A&A, 526, L3 Papitto, A., Menna, M. T., Burderi, L., et al. 2008, MNRAS, 383, 411 Papitto, A., Di Salvo, T., D'A'ı, A., et al. 2009, A&A, 493, L39 Papitto, A., Di Salvo, T., Burderi, L., et al. 2012, MNRAS, 423, 1178 Patruno, A., Alpar, M. A., van der Klis, M. & van den Heuvel, E. P. J. 2012, ApJ, 752, 33 Patruno, A. & Watts, A. L. 2012, to appear in 'Timing neutron stars: pulsations, oscillations and explosions', T. Belloni, M. Mendez, C.M. Zhang Eds., ASSL, Springer (arXiv:1206.2727) Payne, D. J. B., Vigelius, M., & Melatos, A. 2008, in A decade of accreting millisecond X-ray pulsars. AIP Conference Proceedings, Vol. 1068, p. 144 Pooley, D., Homan, J., Heinke, C., et al. 2010, ATel, 2974, 1 Ransom, S. M., Hessels, J. W. T., Stairs, I. H., et al. 2005, Science, 307, 892 Reig, P., Torrej'on, J. M. & Blay, P. 2012, MNRAS, 425, 595 Romani R. W. 1990, Nature, 347, 741 Salpeter, E. E. 1955, ApJ, 121, 161s Shibazaki N., Murakami T., Shaham J., & Nomoto K., 1989, Nature, 342, 656 Srinivasan, G., Bhattacharya, D., Muslimov, A. G. & Tsygan, A. J. 1990, Curr. Sci., 59, 31 Strohmayer, T. E. & Markwardt, C. B. 2010, ATel, 2929, 1 van den Heuvel, E. P. J. & Bitzaraki, O. 1995, A&A, 297, L41 Verbunt, F. 1988, Adv. Space Res., 8, 529 Verbunt, F. & Hut, P. 1987, in The origin and evolution of nutron stars, IAU Symp. No.125, eds. Helfand, D. J. & Huang, J.-H., Reidel, Dordrecht, p.187 Wang, J., Zhang, C.-M., Zhao, Y.-H., et al. 2011, A&A, 526, A88 Webbink, R. F. 1985, in Dynamics of star clusters, IAU Symp. No.113, eds. Goodman, J. & Hut, P. Reidel, Dordrecht, p.541 Wijnands, R. & van der Klis, M. 1998, Nature, 394, 344 Zhang, C.-M. & Kojima, Y. 2006, MNRAS, 366, 137 Zhang, C.-M. Wang, J., Zhao, Y.-H., Yin, H.-X., Song, L.-M. et al. 2011, A&A, 527, A83", "pages": [ 10, 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "The process by which the mass density profile of certain galaxy clusters becomes centrally concentrated enough to produce high strong lensing (SL) cross-sections is not well understood. It has been suggested that the baryonic condensation of the intra-cluster medium (ICM) due to cooling may drag dark matter to the cores and thus steepen the profile. In this work, we search for evidence of ongoing ICM cooling in the first large, well-defined sample of strong lensing selected galaxy clusters in the range 0 . 1 < z < 0 . 6. Based on known correlations between the ICM cooling rate and both optical emission line luminosity and star formation, we measure, for a sample of 89 strong lensing clusters, the fraction of clusters that have [OII] λλ 3727 emission in their brightest cluster galaxy (BCG). We find that the fraction of line-emitting BCGs is constant as a function of redshift for z > 0 . 2 and shows no statistically significant deviation from the total cluster population. Specific star formation rates, as traced by the strength of the 4000 ˚ A break, D 4000 , are also consistent with the general cluster population. Finally, we use optical imaging of the SL clusters to measure the angular separation, R arc , between the arc and the center of mass of each lensing cluster in our sample and test for evidence of changing [OII] emission and D 4000 as a function of R arc , a proxy observable for SL cross-sections. D 4000 is constant with all values of R arc , and the [OII] emission fractions show no dependence on R arc for R arc > 10 '' and only very marginal evidence of increased weak [OII] emission for systems with R arc < 10 '' . These results argue against the ability of baryonic cooling associated with cool core activity in the cores of galaxy clusters to strongly modify the underlying dark matter potential, leading to an increase in strong lensing cross-sections. Subject headings: cooling flows - galaxies: clusters: strong lensing - techniques: spectroscopic", "pages": [ 1 ] }, { "title": "SEARCHING FOR COOLING SIGNATURES IN STRONG LENSING GALAXY CLUSTERS: EVIDENCE AGAINST BARYONS SHAPING THE MATTER DISTRIBUTION IN CLUSTER CORES", "content": "Peter K. Blanchard 1 , 2 , Matthew B. Bayliss 2 , 3 , Michael McDonald 4 , 5 , H˚akon Dahle 6 , Michael D. Gladders 7 , 8 , Keren Sharon 9 , Richard Mushotzky 10 Draft version September 10, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Galaxy clusters that exhibit strong lensing in their cores are some of the rarest objects in the Universe and the global strong lensing cross-section for galaxy cluster-scale structures is dominated by a small fraction of the total galaxy cluster population. In strong lensing (SL) galaxy clusters, theory and simulations predict that certain astrophysical factors play a role in increasing SL cross-sections. N -body simulations predict that dark matter concentrations in strong lensing clusters should be significantly larger than most other clusters (Hennawi et al. 2007; Meneghetti et al. 2010) [email protected] and that triaxiality and clumpiness in the cores could be significant in producing SL clusters (Hennawi et al. 2007). While many strong lensing clusters have high mass, Dalal et al. (2004) showed that the central mass concentration rather than the mass itself is a more important determinant of how giant arcs are produced by cluster-scale mass distributions. However, many studies have found that simple dissipationless (i.e. dark matter only) cosmological simulations tend to underpredict the abundance of SL galaxy clusters by an order of magnitude or more indicating that all factors such as triaxiality and substructure contributing to large strong lensing cross-sections have not been taken into account (e.g. Bartelmann et al. 1998; Luppino et al. 1999; Zaritsky & Gonzalez 2003; Gladders et al. 2003; Li et al. 2006). Additional factors that may contribute to large cross-sections include dark matter condensation due to cooling baryons (Rozo et al. 2008; Mead et al. 2010), central galaxies and substructure (Flores et al. 2000; Meneghetti et al. 2000, 2003; Hennawi et al. 2007; Meneghetti et al. 2010), triaxiality of cluster mass profiles (Oguri et al. 2003; Dalal et al. 2004; Hennawi et al. 2007; Meneghetti et al. 2010), major mergers that increase the cross-section on short timescales (Torri et al. 2004; Fedeli et al. 2006; Hennawi et al. 2007), structure along the line of sight not related to the lens or source (Wambsganss et al. 2005; Hilbert et al. 2007; Puchwein & Hilbert 2009), and the properties of the background galaxies (Hamana & Futamase 1997; Wambsganss et al. 2004; Bayliss et al. 2011a; Bayliss 2012). Wambsganss et al. (2004) and Dalal et al. (2004) showed that increasing the source redshifts in simulations increases SL cross-sections. Failure to account for realistic source redshift distributions has been demonstrated to have a factor of ∼ 10 × effect on giant arc abundances (Bayliss 2012). Using SL clusters to test predictions from theories and cosmological models has historically been limited by the lack of large, well-defined lens samples. The first homogeneously selected cluster lens samples had sizes N /revsimilar 5 (Le Fevre et al. 1994; Zaritsky & Gonzalez 2003; Gladders et al. 2003) and thus too small to have statistical power, but this is now changing as we move solidly into a new era of wide-field imaging surveys - such as the SDSS (e.g., Hennawi et al. 2008; Kubo et al. 2009; Diehl et al. 2009; Kubo et al. 2010; Bayliss et al. 2011b; Oguri et al. 2012), the Canada-France-Hawaii-Telescope Legacy Survey (CFHTLS; Cabanac et al. 2007), and the Second Red Sequence Cluster Survey (RCS2; Bayliss 2012). One reasonable physical scenario that could contribute to the accumulation of mass in the cores of strong lensing galaxy clusters involves baryonic cooling. The hot intracluster medium (ICM) in clusters cools by losing energy in the form of X-ray radiation. In this picture, in order to maintain hydrostatic equilibrium, the cool gas flows inward establishing a cooling flow (e.g. Fabian 1994). In some galaxy clusters, the cooling rate in the center is anomalously high to the point that the cooling time is shorter than the Hubble time. Classical estimates suggest cooling rates of about 1000 M /circledot /yr which should lead to equally high star formation rates. However, such dramatic amounts of star formation are not observed so there must be some mechanism, such as feedback from active galactic nuclei, which can offset the energy loss from cooling (McNamara & Nulsen 2007; Fabian 2012; McNamara & Nulsen 2012). Even so, there is often a small amount of cooling gas fueling star formation in these 'cool core' clusters, representing the residual in the feedback/cooling balance, at typical levels of 1-10 M /circledot /yr (O'Dea et al. 2008; McDonald et al. 2011b), but can be as high as > 100 M /circledot /yr (McNamara et al. 2006; O'Dea et al. 2008; McDonald et al. 2012b). Several studies have found that cool core clusters also contain optical emission-line nebulae in the central regions (Hu et al. 1985; Johnstone et al. 1987; Heckman et al. 1989; Edwards et al. 2007; Hatch et al. 2007; McDonald et al. 2010, 2011a). In addition, star formation rates (SFR) in the brightest cluster galaxies (BCGs) of cool core clusters are known to be higher than SFRs in non-cool cores (Johnstone et al. 1987; McNamara & O'Connell 1989; Allen 1995) and excess IR emission has been found to be proportional to H α emission suggesting both may be due to star formation as a result of the cooling intracluster medium (O'Dea et al. 2008). By studying UV and H α emission of extended filaments in cool cores, McDonald et al. (2011b, 2012a) found that in the majority of clusters (with Perseus as a notable exception), the warm gas is primarily photoionized by massive, young stars, with small contributions most likely from slow shocks. Both the optical emission and the star formation seem to be related to the X-ray properties of the ICM, such as the X-ray cooling rate, suggesting that cooling gas from the intracluster medium is the source of the warm ionized gas and the fuel for star formation (e.g., Edge 2001; O'Dea et al. 2008; McDonald et al. 2010, 2011a,b; Tremblay et al. 2012). This link suggests that the presence of either warm ionized gas or ongoing star formation in the BCG may indicate that the ICM is cooling rapidly in the cluster core. Recent work demonstrates that the evolution of cool core clusters matches the evolution of optically emitting nebulae, suggesting that optical emission-line nebulae may serve as an effective tracer for cool cores (Donahue et al. 1992; McDonald 2011; Samuele et al. 2011). This is significant because at high redshifts it is difficult to determine the cooling rate since the X-ray flux is very low for most of the sources. This paper makes use of the correlation between optical emission-line luminosity and cool core strength, the former having the advantage of being measureable from the ground via even modest aperture telescopes. In this work we use observations of a sample of 89 strong lensing galaxy clusters with BCG spectra available from the SDSS to test for evidence that baryonic cooling is contributing strongly to the high surface mass density of strong lensing galaxy clusters. This paper is organized as follows. In Section 2 we describe the strong lensing cluster sample and the data analyzed. In Section 3, we describe our analysis methods and present the evolution of [OII] line emission and 4000 ˚ A break ratio for our sample compared to the total cluster population. Section 4 provides a discussion of the results and the paper concludes with a summary in Section 5. In this paper we assume Ω M = 0 . 27, Ω Λ = 0 . 73, and H 0 = 71 km s -1 Mpc -1 (Hinshaw et al. 2009).", "pages": [ 1, 2 ] }, { "title": "2.1. Strong Lensing Selected Cluster Sample", "content": "To minimize systematic effects and to allow statistically robust analysis it is important that we have a large, uniformly selected sample of SL clusters. In an attempt to obtain a well-understood sample, a systematic search for strong lensing galaxy clusters in the SDSS DR7 was carried out (Hennawi et al. 2008). Followup observations and analyses of subsets of the Sloan Giant Arcs Survey (SGAS) sample have been previously published (Bayliss et al. 2010; Koester et al. 2010; Bayliss et al. 2011b,a; Oguri et al. 2012). In brief, candidate galaxy clusters in the SDSS data were selected at optical wavelengths using the red sequence algorithm (Gladders & Yee 2000). Each candidate optically selected galaxy cluster was visually inspected by four experts who each assigned a numerical score based on the presence or absence of any evidence of strong lensing in the images. The score scale ranges from 0 to 3 where 3 means there is obvious strong lensing and 0 means no evidence for lensing. The final score is the average of each individual score from each person. Follow-up observations were obtained so that the purity of the sample, the number of candidate strong lenses that actually are SL clusters, could be understood. These efforts have produced the first sample of hundreds of candidate strong lensing galaxy clusters, which will be described in full detail in a forthcoming publication (M. D. Gladders et al., in preparation). We are using this new large sample of SL clusters to conduct the first systematic search for observational evidence of enhanced gas cooling in strong lensing galaxy clusters. The completeness and purity as a function of score for this SL sample is well understood and the majority of the sample clusters have deep, optical followup observations (98% follow-up for score > 1.5 and 75% follow-up for score > 1.0). We remove from the sample those clusters for which the score is below 1.3 to prevent clusters that do not clearly exhibit strong lensing from contaminating our conclusions because the purity of the sample as a function of score drops off strongly between mean scores of 1.5 and 1.0. After a visual inspection of deep follow-up images of each cluster, we also removed 6 SL cluster candidates that cannot be visually confirmed at high confidence as lenses. As described in detail in the next section, we then match the remaining clusters to the SDSS spectroscopic catalog using updated coordinates from follow-up data.", "pages": [ 2, 3 ] }, { "title": "2.2. Matching SL Cluster Coordinates to BCG Spectra", "content": "In order to obtain the spectra for the BCGs of interest from the SDSS data set, the SL sample cluster coordinates were matched with the MPA-JHU release of spectrum measurements from SDSS DR7. The SL sample coordinates come from a visual inspection of the field where the centers of mass of the clusters are approximated by eye. When an arc forms around an obvious BCG, the centroid of the BCG is assigned as the center of mass. However, in cases where there is no obvious BCG, the centroid of the arc itself is used. As a result, we first look for exact matches to BCG spectra and then manually inspect the near-match cases. Coordinates of the SL clusters were compared to the positions in the spectral catalog to find the separation between each sample object and all the objects in the spectral database. The match for each sample cluster is then the object in the spectral database that is the smallest distance away from that SL cluster. 84 SL clusters had lensing centers that matched those of spectra in the database to within 1.5 arcseconds, where the 1.5 arcsecond cut is motivated by the size of the SDSS spectroscopic fiber aperture (3 arcsecond diameter). Figure 1 shows the histogram of distances for the matching process between the SL cluster sample and the spectroscopic database. It is clear that some SL clusters do not have matches with the spectra file. Images of those non-matches for which the match distance is greater than 1.5 arcseconds but less than 2 arcminutes were manually inspected to determine if there are any appropriate bright cluster member galaxies with spectra. For example, some clusters may have multiple bright galaxies in the core, all located close to the center of mass of the cluster. The galaxy corresponding to the SL sample coordinates might not have a spectrum but another galaxy nearby, that is also part of the central mass distribution, might have one. As mentioned above, some of the SL coordinates are actually centroids of giant arcs so the corresponding BCG with a spectrum must be manually determined. 5 of the moderately matching systems were included in the final sample. The final sample thus results in 89 clusters. The spectroscopic redshift distribution of these clusters is shown in Figure 2.", "pages": [ 3 ] }, { "title": "2.3. Optically Selected Galaxy Cluster Catalog", "content": "To compare our results to the total cluster population, we are using the GMBCG catalog (Hao et al. 2010) which was also used by McDonald (2011) who studied the evolution of optical line emission in the total population. The GMBCG catalog was created by searching for BCGs and the red sequence to find galaxy clusters from SDSS DR7 producing a catalog of over 55,000 galaxy clusters in the redshift range 0.1 < z < 0.55. The spectroscopic redshift distribution of all the GMBCG clusters with SDSS spectra is shown in Figure 2. We can also compare the range in cluster masses spanned by the SL and GMBCG samples using a galaxy cluster richness estimator. Most of the SL clusters in our sample have measured richnesses from the GMBCG catalog. For the remaining clusters we used a similar procedure to that used by Hao et al. (2010) to measure richness so that we could compare the richness distribution between the two samples. We note that the richness distribution of our SL sample represents a subset of the richness distribution of the GMBCG catalog weighted towards higher richness. The mean richness and 1 -σ uncertainties of the SL sample is 22 +48 -11 and the mean and uncertainty of the GMBCG is 12 +6 -11 . The richnesses of the SL sample span the range 2 -87 and the GMBCG richnesses range from 8 to 143 with only 0 . 1% greater than 87. The SL sample is drawn from the full range of GMBCG richnesses with a preference for higher richness as expected from simulations (Hennawi et al. 2007; Meneghetti et al. 2010).", "pages": [ 3, 4 ] }, { "title": "2.4. SDSS Data", "content": "The relevent data include spectroscopic redshifts, emission line flux measurements and 4000 ˚ Abreak ratios. The lines of interest in this work are H β , [OII] λλ 3727, and [OIII] λλ 5007. We do not use H α because at z /revsimilar 0 . 4 it is redshifted out of the wavelength coverage of SDSS. The [OII] line is a good tracer of star formation rates over the redshift range of our sample (Kennicutt 1998; Kewley et al. 2003) and stays within the wavelength coverage of SDSS which is 3800 ˚ A - 9200 ˚ A 11 . Also, in our redshift range the [OII] line stays blueward of bright sky lines that exist redward of 7300 ˚ A, which can cause residuals from sky subtraction. An important property of the SDSS spectroscopic database is that the 3 '' spectroscopic fiber aperture encompasses different physical regions on the sky at different redshifts, as the angular diameter distance changes with redshift. For nearby BCGs, for example, the fiber aperture will only encompass a fraction of the total area of the BCG, and would therefore fail to detect any line emission from extended, often filamentary, regions beyond the physical radius probed by the SDSS fiber. McDonald (2011) showed that above a redshift of about 0.3, the fiber aperture encompasses nearly the total H α emission from a sample of galaxy clusters. But for z < 0 . 3 it is essential that an aperture correction be performed. Two aperture corrections were used in this work. McDonald (2011) derived a universal L Hα ( r ) profile based on low-z, well-resolved systems to determine the fraction of emission outside the aperture. The second correction assumes that the mean H β luminosity should be constant with distance. Thus, the only change in H β luminosity should be due to the aperture encompassing different physical diameters. As in McDonald (2011), we find that the two different aperture corrections produced consistent results.", "pages": [ 4 ] }, { "title": "3.1. Evolution of Emission in the Range 0 . 1 < z < 0 . 6", "content": "To understand the evolution of [OII] line emission in strong lensing galaxy clusters we must determine the fraction of SL galaxy clusters that exhibit [OII] line emission as a function of some redshift bin. To do this, for each SL galaxy cluster we calculate the probability, assuming Gaussian statistics, that the line luminosity is above a certain threshold. Following McDonald (2011), the condition for strong [OII] emission is L [ OII ] > 3 . 1 × 10 40 erg s -1 and the condition for weak [OII] emission is 7 . 8 × 10 39 erg s -1 < L [ OII ] < 3 . 1 × 10 40 erg s -1 . In a given redshift bin, the fraction of SL galaxy clusters with weak or strong [OII] emission is given by the average of the individual probabilities for each cluster in that bin. To avoid confusing optical line emission from warm gas in BCGs with AGN activity, if [OIII]/H β > 3 (i.e. Seyfert galaxy where [OII] emission is not necessarily from star formation) for a particular cluster, the cluster is classified as non-emitting and the probability of being an [OII] emitter is set to zero. 14 of the SL clusters in the sample fall into this category of non-emitting. Figure 3 shows the fraction of SL galaxy clusters with all, weak, and strong [OII] emission in the central galaxy. This is the evolution of [OII] emission for 89 SL galaxy clusters in the range 0 . 1 < z < 0 . 6. The over-plotted gray areas represent the evolution of emission for the GMBCG catalog from McDonald (2011). The statistical agreement between the SL sample and the GMBCG catalog indicates that the fraction of central galaxies in SL clusters with bright [OII] emission as a function of redshift differs little from the general cluster population. The trend of a constant fraction of optical line emission for z > 0 . 2 in the general cluster population appears to be mirrored in the strong lensing cluster sample. If a large fraction of SL galaxy clusters showed strong [OII] emission, then this would suggest that baryonic cooling plays an important role in increasing SL cross-sections. Instead, we find no evidence for an enhancement in [OII] emission and thus, baryonic cooling, in strong lensing selected clusters. The mean [OII] fractions (for each of the all, weak, and strong cases) that we compute for both the SL sample and the GMBCG catalog at z > 0 . 2 are given in Table 1.", "pages": [ 4 ] }, { "title": "3.2. Probing Star Formation Using the 4000 ˚ A Break Ratio", "content": "As a check on our results we can investigate the specific star formation rate (sSFR), another tracer of ongoing cooling, in each strong lensing BCG and compare it to the rate in the total population. This sSFR must be independent from our flux measurements to contain new information so we use the 4000 ˚ A break index provided by the MPA-JHU data release as a tracer for specific star formation rate. The 4000 ˚ Abreak index is the ratio of the mean flux in the range 4000 ˚ A - 4100 ˚ A to the mean flux in the range 3850 ˚ A - 3950 ˚ A (Brinchmann et al. 2004). Objects with low star formation, and thus few young, blue stars, will have strong 4000 ˚ A break ratios. In Figure 4 we plot the mean 4000 ˚ A break of our SL clusters in five redshift bins as well as the mean 4000 ˚ A break of the GMBCG catalog. There is no deviation from the GMBCG catalog, indicating that SL clusters exhibit the same specific rate of star formation as the general population of BCGs. This is consistent with our results above that found that [OII] line emission in SL clusters deviates little from the total population. To understand the break strength distribution of the SL sample we also plot a histogram of the distribution in Figure 4. The lack of strong bimodality suggests that the SL sample clusters are not forming many stars in their cores. Typical star forming galaxies tend to have break strengths of /revsimilar 1.3 (Kauffmann et al. 2003). The division between star forming and non-star forming galaxies occurs around a break strenth of 1.6 (Kauffmann et al. 2003). The SL cluster BCGs have break strength values indicating they are predominantly non-star forming. The vertical lines in the histogram of Figure 4 indicate the 4000 ˚ A break strength values for various classical cool cores and non-cool cores. PKS0745, A1795, and A1835 are strong cool cores whereas A2029 is a non-cool core. The strong cool cores tend to have values well below the SL sample while A2029 has a value /revsimilar 0.1 away from the SL sample mean indicating that SL sample clusters are not exhibiting the typical break strength values of cool core clusters.", "pages": [ 5 ] }, { "title": "3.3. [OII] Emission Fraction As a Function of Strong Lensing Cross-Section", "content": "To better characterize the above results, we investigate whether or not clusters with a larger strong lensing cross-section show stronger emission. We use an observationally defined quantity, R arc , for each cluster lens as a proxy for strong lensing cross-section. We define R arc as the radial separation between the arcs and the center of mass of the SL clusters. R arc is an observable that is simple to measure for our entire sample, and which provides an approximate estimate of the Einstein radius. The Einstein radius describes the critical curve for a given strong lens, and is defined analytically as the location in the lens plane where the formal magnification of a source distorted by a lens goes to infinity (Schneider et al. 1992). In the simplest case of a spherically symmetric lensing potential and perfect alignment between the source, lens, and observer, the source is reimaged into a ring described by the Einstein radius. The radius of this ring is the Einstein radius, θ E , and is given by: where G is Newton's gravitational constant, M is the mass of the lensing cluster, c is the speed of light, D LS is the distance between the lens and the source, D L is the distance between the observer and the lens, and D S is the distance between the observer and the source. Physically realistic lensing systems have critical curves with much more complex morphologies, but the Einstein radius for such systems can still be defined and measured as the radius of a circle which has the same area on the sky as the area contained within the critical curve. The size of the critical curve provides a measurement of the 'strong-ness' of a strong lens, where the SL cluster population consists of a broad range of structures ranging from the rarest super-lenses with extreme strong lensing cross-sections, to the more numerous marginal strong lenses. Detailed strong lensing reconstructions of the critical curves for our cluster lens sample is observationally unfeasible as it would require extensive follow-up observations. However, rather than model the critical curve for each SL cluster, it is also possible to define a simple observable quantity by fitting an ellipse to a multiply imaged source - or giant arc - and measure the radius corresponding to a circle with an area equal to the area of the fitted ellipse (R arc ). Tests in simulations show that this quantity has a large intrinsic uncertainty when used to estimate the Einstein radius for an individual lens system, but that on average it correlates with Einstein radius (Puchwein & Hilbert 2009). We can therefore use R arc for our SL cluster sample to sort lenses approximately by the size of their strong lensing cross-section. This sorting allows us to probe whether baryonic cooling processes may be helping to drive up strong lensing cross-sections within a subset of the total cluster lens population. We estimate the radial separation, R arc , of the arcs from the center of mass of the cluster in each SL cluster from optical follow-up images taken with the Mosaic Camera (MOSCA) on the 2.5m Nordic Optical Telescope. In each image the center of mass of the cluster (usually the BCG) as well as the arcs are located. The fitting program mpfitellipse (More 1978; Markwardt 2009) is then used to fit an ellipse to the curvature of the arcs to recover a rough estimate of the critical curve for each cluster lens. For the measured radial separations to be useful as a way to sort and compare members of our sample, they must be scaled to remove the distance dependence of each measurement. This is accomplished by scaling each measurement by: where D LS , D L , D S are the relevant distance values for each particular cluster, and D L 0 S 0 , D L 0 , D S 0 are values for a fiducial lens configuration. Because the source redshifts for many of our individual SL systems are unknown, we use the typical source redshift as measured in the literature, z s = 2 ± 1 (Bayliss et al. 2011a; Bayliss 2012). The source redshift uncertainty for each individual lens system produces a systematic uncertainty in the final scaled R arc values for our SL cluster sample, but this uncertainty is quite small ( ∼ +3% -8% for a lens redshift of 0.3, the median of our sample) and does not impact our results. With the scaled R arc measurements, one can then determine the [OII] emission fraction as a function of R arc ( /revsimilar θ E ). Figure 5 shows the fraction of SL clusters with all, weak, and strong [OII] emission in four bins of R arc . In this plot only those SL clusters with z > 0 . 2 were included because this is where there is no evidence of changing [OII] emission fractions. From Figure 5 it seems that there is no statistically significant dependence of [OII] emission on Einstein radii above about 10 arcseconds. Below this value there is a slight increase in the fraction of weak [OII] emitters whereas the strong [OII] fraction is consistent with no dependence. For weak emission the data points in the bins below 10 arcseconds deviate from the GMBCG mean by about 1σ and for total emission the data points deviate from the GMBCG mean by less than 1σ and are thus not statistically robust deviations. Figure 5 also shows the 4000 ˚ A break strengths as a function of R arc , which show no deviation from the GMBCG mean break strength and no evidence for variation in the break strength as a function of R arc . Clusters with large Einstein radii exhibit optical tracers of baryonic cooling in their cores with the same frequency as clusters with small Einstein radii, and also as the total cluster population.", "pages": [ 5, 6, 7 ] }, { "title": "4. DISCUSSION", "content": "Figure 3 demonstrates that the fraction of strong lensing galaxy clusters over the range 0 . 2 < z < 0 . 6 with [OII] line-emitting BCGs is constant and shows no statistically significant deviation from the total cluster population, suggesting that baryonic cooling is not enhanced in SL clusters over the general cluster population. Figure 4 supports this conclusion by showing that there is no evolution in 4000 ˚ A break ratios and that they match the mean ratio in an optically selected sample of galaxy clusters - the GMBCG catalog. Furthermore, the typical D 4000 value for the SL sample is consistent with non-cool cores that are not forming many stars in the BCG. If ongoing cooling were playing a continuing role in generating efficient SL clusters then we would expect to see some evidence of enhanced cooling in the form of intermediate temperature (10 4 K) gas or ongoing star formation (e.g., Edge 2001; O'Dea et al. 2008; McDonald et al. 2010, 2011a,b; Tremblay et al. 2012), as traced by optical emission or the 4000 ˚ A break in the cluster cores. We find no evidence for such an enhancement; instead, our analysis suggests that cool cores are no more prevalent in strong lensing clusters than in the general cluster population. Our results argue that baryonic cooling associated with cool core activity is not an efficient mechanism for dramatically increasing strong lensing cross-sections in galaxy clusters. Rozo et al. (2008) and Mead et al. (2010) found that simulations which include baryonic cooling can increase strong lensing cross-sections of simulated galaxy clusters by factors of /revsimilar 2-3. These scenarios require a 'runaway' cooling flow which causes dark matter to condense in the core by sufficient amounts to alter the total matter density profile and the strong lensing properties of the cluster. Since runaway cooling flows are not observed, it is evident that other factors, like AGN feedback, act on sufficiently short timescales to prevent runaway cooling and unrealistically cuspy gas density profiles (Best et al. 2005; McNamara & Nulsen 2007; Fabian 2012; McNamara & Nulsen 2012). Otherwise, we would observe the effects of this runaway cooling in the form of massive starbursts. This feedback scenario is consistent with recent studies (e.g. Mead et al. 2010; Killedar et al. 2012) that found that simulations which include models of AGN feedback, together with cold dark matter and gas dynamics, show less significant increases in strong lensing cross-sections. This agreement between observational and simulation-based results is encouraging, and suggests that the current generation of cosmological simulations include feedback models that are sufficiently sophisticated to recover the impact of baryonic processes on the total matter distribution in cluster cores. Our results are also interesting in the context of recent work in which the slopes of the central density profiles in a small sample of relaxed clusters were estimated from multi-wavelength observations (Newman et al. 2013). The selection of the clusters studied by Newman et al. (2013) complicates a direct comparison between their conclusions and the results of our work, which uses a large generic strong lensing selection. Newman et al. (2013) found that the observed density profiles of their seven clusters are in good agreement with the predictions from dark matter (DM) only simulations, measuring total density profiles in the cores of seven clusters with slopes that match cold dark matter (CDM) simulations. They argue that dynamical heating is a possible mechanism for offsetting any effects that baryonic contraction might have on the matter distribution in massive cluster cores. It makes sense that the results of such a mechanism would be observable in a sample of clusters that was chosen specifically to be dynamically relaxed and undisturbed, where the total matter distribution in the cores (baryonic+DM) has had the opportunity to virialize. However, the strong lensing selection of the SGAS clus- ter lens sample does not preferentially select for relaxed systems, and in fact there is evidence suggesting that dynamically disturbed and merging systems should be well-represented in a strong lensing selected cluster sample (Torri et al. 2004; Oguri et al. 2013). The matter distribution in the cores of such a sample should not necessarily be expected to have the same average profile properties as a sample that is selected to be relaxed. Having noted the different selection criteria for our sample and that of Newman et al. (2013), we do note that there is broad agreement between our results and those of Newman et al. (2013) in that neither result favors a scenario in which baryonic cooling is acting to steepen the matter distributions in the cores of clusters. It therefore follows that it is not reasonable to invoke baryonic cooling as a dominant explanation for the apparent discrepancies between observed and predicated arc abundances (Bartelmann et al. 1998; Luppino et al. 1999; Zaritsky & Gonzalez 2003; Gladders et al. 2003; Li et al. 2006). We note that in Figure 5 there is a marginal increase (at the ∼ 1 -σ level) in the fraction of strong lensing clusters with R arc < 10 '' exhibiting weak [OII] emission. The observable R arc correlates strongly with Einstein Radius, which itself correlates with the total mass of the cluster lens, so that the R arc < 10 '' bin will include, on average, the lower-mass cluster lenses in our sample. This marginal increase is in qualitative agreement with the suggestion that baryonic cooling could be responsible for small excesses in the concentration parameters measured for lower-mass and smaller Einstein radius strong lensing selected clusters by Oguri et al. (2012). However, neither the increase in optical line emission that we measure, nor the excess concentrations in Oguri et al. (2012) are statistically robust (i.e. > 2σ ), and we refrain from claiming that the combination of these two results can be interpreted as strong evidence for cooling baryons driving up concentrations in low-mass or small Einstein radius strong lensing clusters. These marginal excesses in optical line emission and concentration could, however, reflect consistency with the expectation from simulations that gas cooling may more strongly affect clusters with lower masses where the cooling mass in the core can comprise a larger fraction of the total mass (Rozo et al. 2008; Killedar et al. 2012).", "pages": [ 7, 8 ] }, { "title": "5. SUMMARY", "content": "In this work, we searched for optical line emission and recent star formation in a sample of 89 strong lensing galaxy clusters to probe whether or not baryonic cooling processes significantly affect the mass density profiles of clusters. Using published SDSS spectral data for the BCGs of the SL clusters we have calculated the fraction of SL clusters with [OII] line emission as a function of redshift. We find that the evolution of [OII] line emission in the SL sample is constant for z > 0 . 2 and that there is no statistically significant difference between the SL sample and the general cluster population. The 4000 ˚ A break ratio in the SL sample also matches the general population, indicating that the average specific star formation rate is similar between the two populations. We also sorted the SL cluster sample by R arc - an observable that correlates strongly with Einstein radius - to look for trends in the optical tracers of gas cooling as a function of the individual lens cross-sections. We find that [OII] line emission fractions and 4000 ˚ A break ratios showed no significant dependence on Einstein radius, suggesting that baryonic cooling does not play a large role increasing strong lensing cross-sections among either the small or large strong lensing cross-section end of the total cluster lens population. The results of this work combined with the well-studied correlations between ICM cooling and BCG star formation and line emission argue strongly that baryonic cooling associated with cool core activity does not significantly influence the dark matter distribution to steepen the mass density profile in the cores of strong lensing galaxy clusters. This work is supported in part by the National Science Foundation Research Experiences for Undergraduates (REU) and Department of Defense Awards to Stimulate and Support Undergraduate Research Experiences (ASSURE) programs under grant number 0754568 and by the Smithsonian Institution. Part of this work was based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. M. B. B. acknowledges support from the NSF Astronomy Division under grant number AST-1009012. M. M. acknowledges support provided by NASA through a Hubble Fellowship grant from STScI. M.D.G. thanks the Research Corporation for support of this work through a Cottrell Scholars award. We would like to thank Jonathan McDowell and Marie Machacek for helpful feedback on early drafts of this paper.", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "McCarthy, P. 1989, ApJ, 338, 48 Hinshaw, G., et al. 2009, ApJS, 180, 225 Hu, E. M., Cowie, L. L., & Wang, Z. 1985, ApJS, 59, 447 Johnstone, R. M., Fabian, A. C., & Nulsen, P. E. J. 1987, MNRAS, 224, 75 Kennicutt, Jr., R. C. 1998, ARA&A, 36, 189", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "We report the first detection of a redshift-depth enhancement of background galaxies magnified by foreground clusters. Using 300 000 BOSS-Survey galaxies with accurate spectroscopic redshifts, we measure their mean redshift depth behind four large samples of optically selected clusters from the SDSS surveys, totalling 5 000 -15 000 clusters. A clear trend of increasing mean redshift towards the cluster centers is found, averaged over each of the four cluster samples. In addition we find similar but noisier behaviour for an independent X-ray sample of 158 clusters lying in the foreground of the current BOSS sky area. By adopting the mass-richness relationships appropriate for each survey we compare our results with theoretical predictions for each of the four SDSS cluster catalogs. The radial form of this redshift enhancement is well fitted by a richness-to-mass weighted composite Navarro-Frenk-White profile with an effective mass ranging between M 200 ∼ 1 . 4-1 . 8 × 10 14 M /circledot for the optically detected cluster samples, and M 200 ∼ 5 . 0 × 10 14 M /circledot for the X-ray sample. This lensing detection helps to establish the credibility of these SDSS cluster surveys, and provides a normalization for their respective mass-richness relations. In the context of the upcoming bigBOSS, Subaru-PFS, and EUCLID-NISP spectroscopic surveys, this method represents an independent means of deriving the masses of cluster samples for examining the cosmological evolution, and provides a relatively clean consistency check of weak-lensing measurements, free from the systematic limitations of shear calibration. Keywords: galaxies: clusters: general - gravitational lensing: weak - cosmology: observations dark matter", "pages": [ 1 ] }, { "title": "CLUSTER LENSING PROFILES DERIVED FROM A REDSHIFT ENHANCEMENT OF MAGNIFIED BOSS-SURVEY GALAXIES", "content": "Jean Coupon Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan", "pages": [ 1 ] }, { "title": "Tom Broadhurst 1", "content": "Department of Theoretical Physics, University of Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain and IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36-5, E-48008 Bilbao, Spain", "pages": [ 1 ] }, { "title": "Keiichi Umetsu", "content": "Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan Draft version September 29, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Our current paradigm predicts that the properties of dark energy will be imprinted in the matter power spectrum and its time evolution. Although probing the large scale structure in the universe is a promising strategy to investigate the different dark-energy model candidates, it is becoming increasingly challenging to detect the relatively subtle signatures of the various lesser ingredients and higher order effects, forcing us to observe many independent physical phenomena over gigantic volumes, to reach sufficient joint statistical precision. In the era of large scale cosmological surveys, gravitational lensing is one of the most promising methods for establishing the fluctuations of matter in the universe, as lensing is purely gravitational and can be largely assumed to be independent of the unknown nature of the dark matter. One may employ two-dimensional (Kilbinger et al. 2013) or tomographic (Benjamin et al. [email protected] 1 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan 2012) cosmic shear to measure the integrated lensing effect of matter fluctuations along the line-of-sight up to very large scales (Taylor et al. 2004; Massey et al. 2007). Or, one may use cluster abundances (Rozo et al. 2010) to put constraints on the evolution of the dark matter halo mass function, which is known to be a very sensitive function of the evolution of the mean mass density and of the dark energy properties. Masses can be derived from the radial shear profile around clusters in the weak lensing regime (Sheldon et al. 2009; Leauthaud et al. 2010; Oguri & Hamana 2011; Umetsu et al. 2012), from X-ray (Rykoff et al. 2008), SZ effect (Marriage et al. 2011), or satellite kinematics (More et al. 2011). Even though lensing from stacked clusters undoubtedly leads to a less noisy mass estimate among all these methods, accurate statistics will be limited, however, by our ability to measure galaxy shapes down to the required precision. Recently, Heymans et al. (2012) demonstrated that for a survey like the CFHTLS-wide, spreading over 154 deg 2 , the level of systematics in shape measurement, as given by the state-of-the art shape measurement technique (Lensfit, Miller et al. 2013), was lower than sta- tistical errors. For upcoming experiments such as the Hyper Suprime Cam (HSC, Miyazaki et al. 2012) survey, the Dark Energy Survey (DES 2 ), the Large Synoptic Survey Telescope project (LSST 3 ) or EUCLID 4 survey (Laureijs et al. 2011) which will probe volumes orders of magnitude larger than the CFHTLS-wide, the limiting requirements will be much more stringent. Lens magnification provides independent observational alternatives and complementary means to weak shape measurement (Van Waerbeke et al. 2010; Hildebrandt et al. 2011). In practice magnification bias has been used to improve the mass profiles of individual massive clusters, and stacked samples of clusters from recent dedicated cluster surveys. Here the number counts of flux limited samples of background sources is reduced in surface density by the magnified sky area, and compensated to some extent by fainter objects magnified above the flux limit (Broadhurst et al. 1995, hereafter BTP95) with a net magnification-bias in the surface density depending on the slope of the background counts. For very red background galaxies behind clusters the net effect is a clear 'depletion' of red background galaxies towards the cluster center (Broadhurst et al. 2005, 2008) and is used in combination with weak shear to enhance the precision of individual cluster mass profiles (Umetsu & Broadhurst 2008; Umetsu et al. 2011b). A strength of this method resides in the fact that no shape information is used and is therefore less prone to systematic errors, as opposed to shear measurement for which complex corrections are required (Kaiser et al. 1995; Umetsu & Broadhurst 2008) In the statistical regime, for deep wide area surveys, progress has been made recently in a similar way using distant Lyman-break galaxies behind stacked samples of foreground clusters, resulting in a claimed positive magnification-bias enhancement of the observable surface density of background galaxies at small radius. In practice, some cross contamination of cluster members and foreground galaxies with the background will complicate such measurements. Until recently, magnification bias studies (Ford et al. 2012; Hildebrandt et al. 2013) have faced such a high statistical error that their conclusions were not affected by these systematics. However in the future a careful treatment of potential sources of systematics will be mandatory. With redshift information for large samples of background galaxies, cluster magnification may be tackled more fully, as the luminosity function of background galaxies is magnified both in the density and luminosity directions, in a characteristic redshift dependent way. For flux limited redshift samples, the redshift distribution becomes skewed to higher mean redshift as a consequence of magnification, and the mean luminosity is also modified owing to the curvature (non-power law) of the luminosity function (BTP95). The latter effect has been claimed in the case of magnified QSO's behind galaxies in the SDSS survey (M'enard et al. 2010). In this paper, we are interested in the enhanced redshift depth of background galaxies magnified by foreground clusters and averaged over large cluster samples recently identified within the SDSS survey. We measure the mean redshift of BOSS galaxies behind SDSS clusters (York et al. 2000; Schlegel et al. 2009; Dawson et al. 2013), using four different catalogues of optically detected clusters found from very different independent methods, including the maxBCG (Koester et al. 2007), the GMBCG (Hao et al. 2010), the Szabo et al. (2011) and the Wen et al. (2012) samples. We compare the depth magnification to the expected lensing signal using mass-richness relationships calibrated from X-ray and weak shear measurements. In addition we use the MCXC X-ray cluster sample assembled by Piffaretti et al. (2011) from a number of independent ROSAT observations. In Section § 2 we present the data sets used in this study; the BOSS sample, the four galaxy cluster catalogs and the X-ray sample. In a third section, we describe our model, and in a fourth section we present our measurement technique, followed by our results presented and discussed in Section § 5. In Section § 6 we present a brief error analysis and we conclude in Section § 7. Everywhere we assume a flat ΛCDM cosmology with Ω m = 0 . 258, Ω λ = 0 . 742, and h 100 = 0 . 72 (Hinshaw et al. 2009). All masses are expressed in unit of M /circledot .", "pages": [ 1, 2 ] }, { "title": "2.1. The lens samples", "content": "The Sloan Digital Sky Survey (SDSS, York et al. 2000) is an optical ( ugriz filters) photometric and spectroscopic survey which represents the largest probe of the local universe to date. In addition to its unprecedented volume observed, the success of the SDSS project is based upon an accurate photometric calibration, an ingenious synergy between photometric and spectroscopic observing strategies, and a handy and well documented database. In the published literature, several methods have been used to construct cluster samples from the SDSS database. In this study, we thus focus on the following four publicly-available cluster catalogs based on optical identifications. First, the maxBCG cluster catalog (Koester et al. 2007) is a cluster sample constructed from 7 500 deg 2 of photometric data in the SDSS. To detect candidate clusters, the method identifies a local overdensity of cluster members along the red-sequence . The richness N 200 is defined as the number of galaxy members which are brighter than 0 . 4 L /star and lying on the red-sequence inside the radius of r 200 , within which the mean interior density is 200 times the critical density of the universe. The public maxBCG catalog consists of cluster candidates with 10 members or more. Since the maxBCG sample was constructed from an earlier SDSS data release, the probed area does not fully overlap with the final SDSS coverage in the Galactic south cap, where BOSS galaxies have been targeted. Therefore, not all of the 3 000 deg 2 of BOSS data available to date overlap with the maxBCG cluster sample. Keeping all galaxycluster pairs within one deg, the subsample used in this study includes 6 308 clusters. Second, more recently, Hao et al. (2010) proposed a slightly different optical-based detection method, the Gaussian Mixture Brightest Cluster Galaxy (GMBCG), which relies on both the presence of the brightest cluster galaxy (BCG) and the red-sequence to identify clusters. Among the differences with maxBCG cluster algorithm finder is the way to estimate the cluster redshift, where for GMBCG it is estimated solely from the BCG photometric redshift (or spectroscopic redshift when available). It is claimed by Hao et al. (2010) that the GMBCG catalog is volume limited out to z = 0 . 4; however, to ensure no spurious correlation between false cluster detection and background sources, secure foreground-background sample separation is necessary. For this reason we limit the GMBCG sample to the range 0 . 1 < z < 0 . 3, taking into account the typical photometric redshift error. This redshift selection, in conjunction with rejection of clusters lying outside the BOSS area, leads to a total of 4 631 clusters with richness N 200 greater than 10. Although an improved weighted richness estimator is available, we employ the scaled richness estimator, which is defined in the same way as for the maxBCG catalog. Third, an alternative method of Dong et al. (2008) does not rely on the red-sequence, but on the peak locations in the likelihood map generated from the convolution of the galaxy distribution in redshift space with aperture matched filters. These filters are constructed from the assumed cluster density profile and galaxy luminosity functions. This method has been applied to the SDSS (Szabo et al. 2011), and the overlap with BOSS allows us to use 5 646 galaxy clusters below redshift 0.3. A noticeable feature of this catalog is that the richness is computed as the sum of the luminosity of all galaxies above 0 . 4 L /star divided by L /star , and is therefore not directly comparable to the definition used by the two previous data samples. The publicly available version includes all clusters with an estimated richness Λ 200 of 20 or higher. In the rest of the paper, we will refer to this sample as 'AMF'. Fourth, we use the catalog produced by Wen et al. (2012). In this method, a friend-of-friend algorithm is applied to luminous galaxies using a linking length of 0.5 Mpc in the transverse separation and a photometric redshift difference within 3 σ along the line-of-sight direction. The center is assumed to be the position of the BCG, identified from a global BCG sample, as the brightest galaxy physically linked to the candidate cluster. In a similar fashion to the AMF catalog, the richness is computed from the total luminosity of all galaxies brighter than 0 . 4 L /star , in units of L /star . The sample is limited to a richness threshold of 12. The algorithm was applied to the latest SDSS-III data release, resulting in an increase of factor two in area. With a larger overlap with BOSS, especially in the south Galactic region, we obtain 15 112 clusters within one degree off the edge of the background BOSS sample. In the following, this sample will be referred to as 'WHL12', and the richness denoted Λ 200 for consistency with the notation adopted for the AMF catalog. Finally, we complete our set of foreground cluster catalogs with the meta-catalogue of X-ray detected clusters (MCXC) assembled by Piffaretti et al. (2011). This data set is a compilation of X-ray detected clusters from a number of publicly available data collected by the ROSAT satellite. The sample extends to redshift ∼ 0 . 6 with a wide range of masses. Here we relax the maximum redshift range to z = 0 . 35 as the majority of clusters have secure spectroscopic redshift measurements and do not suffer from false detection caused by projection effects. The number of clusters in common over the DR9 (see below) released BOSS area is currently 158.", "pages": [ 2, 3 ] }, { "title": "2.2. The Source Sample", "content": "Our background sample is extracted from the first data release 'DR9' of the BOSS spectroscopic survey. BOSS aims at measuring the scale of baryon acoustic oscillations at redshift z = 0 . 5 as a sensitive cosmological test. The first data release, covering 3 000 deg 2 , was made publicly available in summer 2012 through the SDSS III/BOSS DR9 data release 5 . We use the 'CMASS' spectroscopic sample for which the targets were selected in the range 17 . 5 < i AB < 19 . 9, using color selection techniques to ensure homogeneous sample in mass between redshifts 0 . 43 and 0 . 7. We refer to Maraston et al. (2012) for a complete study of the galaxy stellar mass function and the verification of the homogeneity of the data across the full redshift range. The BOSS DR9 data release comprises about 3 000 deg 2 in the south and north galactic areas. We further select only primary spectroscopic galaxies, and remove all galaxies with uncertain redshift measurements, by imposing the flag zwarning=0 . The total number of galaxies used as background sources is 316 220. Fig. 1 illustrates the data coverage used in this study. The foreground cluster sample is shown in blue and the background BOSS galaxy sample in red. We show the maxBCG clusters as an example, and note that the WHL12 sample has a better overlap with BOSS in the south Galactic part. The two other catalogues have a similar coverage compared to maxBCG, and the MCXC has a full coverage over the BOSS area but is not spatially homogeneous because it is drawn from several independent X-ray surveys. We summarize in Table 1 cluster sample properties.", "pages": [ 3 ] }, { "title": "3. MODEL", "content": "In this section we outline the formalism that describes the systematic effect of lens magnification modifying the source selection function, by foreground galaxy clusters. Lensing magnification is caused by both isotropic and anisotropic focusing of light rays due to the presence of massive foreground objects acting as gravitational lenses. The former effect is described by the convergence, κ ( r ) = Σ( r ) / Σ crit , the projected mass density Σ( r ) in units of the critical surface mass density, where D l , D s , and D ls are the proper angular diameter distances from the observer to the lens, the observer to the source, and the lens to the source. The latter effect is due to the gravitational shear γ ( r ) = γ 1 + iγ 2 with spin2 rotational symmetry (see Bartelmann & Schneider 2001). Since gravitational lensing conserves surface brightness, the apparent flux of background sources increases in proportion to the magnification factor. This shift in magnitude implies that the limiting luminosity at any background redshift lies effectively at a fainter limit given by L lim ( z ) /µ ( z ), hence increasing the surface density of magnified sources behind foreground lenses. On the other hand, the number of background sources per unit area decreases due to the expansion of sky area. These two effects compete with each other, and the effective variation in the source number density n eff , known as the magnification bias (BTP95), depends on the steepness of the source number counts as a function of the flux limit F . For background sources at redshift z , the magnification bias is expressed as where n 0 is the unlensed number density of background sources, L the limiting luminosity of the background sample, and µ the magnification, For the case of weak gravitational lensing, the magnitude shift induced by magnification is sufficiently small, so that the source number count can be locally approximated by a power law at the limiting luminosity. This simplifies Eq. 2 to with β the logarithmic slope of the luminosity function Φ evaluated at the limiting luminosity: ∣ We retrieve the well-known result that if the count slope β ( z, L ) is greater than unity, the net effect of magnification bias is to increase the source number density, or decrease otherwise (BTP95). If the count slope is unity, the net magnification effect on the source counts vanishes. In the strict weak-lensing limit, the magnification bias is directly related to the projected mass distribution as n eff ( z ) /n 0 ( z ) ≈ 2[ β ( z, L ) -1] κ ( z ). 6 The limiting magnitude and the count slope vary with redshift, so that the integrated magnification-bias effect will translate into an enhancement in mean source redshift as The limiting luminosity can be computed from the apparent limiting magnitude for a given survey as where i AB = 19 . 9 is the limiting magnitude of BOSS, d L the luminosity distance, and K , the K -correction: In our analysis, we restrict the source redshift range to 0 . 43 < z < 0 . 7, where the BOSS target selection is established to be uniform (Dawson et al. 2013). We assume a Schechter (1976) luminosity function with the parameters measured in the VIMOS VLT Deep Survey (VVDS, Le F'evre et al. 2005) by Ilbert et al. (2005). The authors provide a detailed measurement of the luminosity function in all UBVRI optical bands of the survey over a wide redshift range, 0 . 2 < z < 2 . 0. This survey has the advantage of being considerably deeper than the BOSS survey, with redshift completeness to a fainter limiting magnitude and therefore describes better the underlying luminosity function, in terms of the slope as a function of redshift and magnitude required here for our lensing predictions. To minimize the uncertainties on the K -correction, we take the rest-frame V -band luminosity function measured at redshift ∼ 0 . 5, which best matches the redshifted galaxies observed in the i -band at redshift 0.5, so that this choice allows us to neglect the second term in Eq. 8. Finally, to account for luminosity evolution as function of redshift, we employed the parametrization of Faber et al. (2007), such that our assumed Schechter parameters are: Note that no precise normalization of the luminosity function is required for our analysis but only the gradient of the logarithmic slope β at the limiting luminosity as a function of background redshift, which has been well determined from the VVDS, and confirmed by other deep probes of similar volume such as the COMBO-17 photometric survey (Wolf et al. 2003) or DEEP2 spectroscopic survey (Willmer et al. 2006). In particular, the M /star parameter measured in the B -band is found to be in excellent agreement among those three deep surveys at z = 0 . 5 (Faber et al. 2007, Fig. 7). The VVDS field-of-view is one deg 2 , comprising 11 034 redshift measurements, allowing us to determine the respective levels of Poisson uncertainty and cosmic variance into the Schechter parameters we require for our predictions. We quantify in Sec. 6.6 the impact of these assumptions on the size of the mean redshift depth enhancement predicted. In particular we show that these model uncertainties arising from the parameterisation of the luminosity function are substantial compared to the detected signal, however smaller than our statistical errors on the measurements from the current DR9 release. To date, a number of studies (Umetsu et al. 2011b; Hildebrandt et al. 2011; Ford et al. 2012) have measured a magnification-bias signal by assuming an effective single-plane source redshift for a given background sample and comparing their number density to a random sample. Here we do not need to approximate the depth as we have precise spectroscopic redshift measurements for all individual sources, providing a direct estimate for the enhancement of mean source redshift behind clusters with respect to that of the total sample. Our method developed here has two main advantages over the standard magnification-bias because we are simply measuring an averaged depth enhancement of the background galaxies, rather than the lensing induced change in the surface density of galaxies. The latter effect requires careful correction for screening by foreground/cluster galaxies. Secondly, since we probe a relatively narrow redshift range of galaxies having a wellunderstood selection function than for typical faint background source counts used in the magnification bias work located at z ∼ 1 -3. Therefore, our method enables in practice a relatively better understood determination of the statistical lensing effect for the foreground clusters we are examining. To quantify and characterize the cluster mass distribution, we compare the observed cluster lensing profiles with analytic spherical halo density profiles. In the present study, we consider (1) the Navarro et al. (1997, hereafter NFW) and (2) the singular isothermal sphere (SIS) models. The former is a theoreticallyand observationally-motivated model of the internal structure of cluster-sized halos (e.g. Okabe et al. 2010; Umetsu et al. 2011a), ρ ( r ) ∝ ( r/r s ) -1 (1 + r/r s ) -2 with r s the characteristic radius at which the logarithmic density slope is isothermal. The latter model provides a simple, one-parameter description of isothermal density profiles, ρ ( r ) ∝ r -2 . The two-parameter NFW model can be specified by the degree of concentration, c 200 = r 200 /r s , and the halo mass, the total mass enclosed within a sphere of radius r 200 , within which the mean interior density is 200 times the critical mass density at the cluster redshift, ρ crit ( z ). Here we employ the mean concentration-mass relation of Bhattacharya et al. (2013), derived from ΛCDM cosmological N -body simulations covering a wide halo mass range of 2 × 10 12 -2 × 10 15 M /circledot h -1 and a wide redshift range of z = 0-2. We use their fitting formula for the full-sample relation c 200 ( M 200 , z ). The NFW profile is thus specified by M 200 alone. The use of the Bhattacharya et al. (2013) relation is motivated by recent detailed cluster lensing work (Coe et al. 2012; Umetsu et al. 2012; Okabe et al. 2013) finding a good agreement with their predictions for high-mass clusters ( M 200 ∼ 10 15 M /circledot ) at z = 0 . 2-0 . 4. We employ the radial dependence of the projected NFW lensing profiles given by Wright & Brainerd (2000). For the SIS model, the magnification is given by with θ E = 4 π ( σ v /c ) 2 D ls /D s the Einstein radius. Here, σ v is the one-dimensional velocity dispersion related to the halo mass, Finally, the magnification factor is plugged into Eqs. 4 and 5, so that the depth magnification is computed as a function of the physical distance at the cluster redshift as The total effect of magnification on the redshift distribution n ( z ) of BOSS source galaxies is shown in Fig. 2.", "pages": [ 3, 4, 5 ] }, { "title": "4. MEASUREMENTS", "content": "Given the sparse source density around foreground clusters, we must measure the signal around stacked clusters to increase the signal-to-noise ratio. To ease the computation of the mean redshift of BOSS galaxies around clusters, we employed a modified version of the Davis & Peebles (1983) two-point cross-correlation estimator: where L is the lens cluster sample, S the source BOSS galaxy sample weighted by their redshift and R the unweighted BOSS galaxy sample. Pairs are computed at the redshift of the cluster as function of physical scale. We can write the previous equation explicitly: where the subscripts l and s denote the lens and source samples, respectively. Equation 15 then reduces to as defined by Eq. 14. In practice we perform the measurements using the software Swot (Coupon et al. 2012), a fast two-point correlation code optimized for large data sets. The algorithm makes use of large scale approximations, tree-code structured data, and parallel computing to considerably accelerate pair counting. Correlating ∼ 300 000 background objects with ∼ 6 000 foreground objects takes about five minutes on a desktop computer. Error bars are estimated by generating foreground cluster positions over the BOSS area: using 20 000 points with random projected positions on the sky and a volume-weighted random redshift in the range 0 . 1 < z < 0 . 3 , we compute w ( r ) in a similar fashion as for the data samples, using the same binning. We repeat the process 100 times and compute the dispersion of the ensemble. Error bars for the signal of each cluster sample are further multiplied by √ 20 000 /N clusters to account for Poisson error. Additionally, the significance of the detection for each sample is computed using the rescaled covariance matrix. For optically detected cluster samples, we compare our measurements to the expected theoretical signal by first converting the richness to the mass, using the maxBCG mass-richness relationship calibrated from X-ray and weak lensing by Rozo et al. (2009): and we further convert M 500 into M 200 using the method described in Hu & Kravtsov (2003), and the concentration-mass relationship of Bhattacharya et al. (2013). We employ the same relation for GMBCG, as the richness definition is identical to maxBCG. As for the WHL12 sample, the authors chose a different definition for the richness (see Sec. 2.1), but also provide an independent X-ray/lensing calibrated mass-richness relationship (Wen et al. 2010): Since the richness definition assumed for AMF is similar to that of WHL12 we use the same parametrisation for these two samples. For the X-ray cluster sample, an estimation of the M 500 mass is directly provided, however due to the low angular resolution of ROSAT, X-ray mass estimates may be underestimated. To correct for this, we matched the MCXCclusters with the local cluster sample presented in Vikhlinin et al. (2009), and compared the value of M 500 : Then we simply convert M ' 500 into M 200 as described above. It is found that the MCXC-based value is about 25% smaller than those derived by Vikhlinin et al. (2009) at the effective mass of our subsample M 200 ∼ 5 × 10 14 M /circledot . Finally to perform a strict comparison between the stacked measured signal and the prediction, we compute the composite halo-mass theoretical signal:", "pages": [ 5, 6 ] }, { "title": "5. RESULTS", "content": "We show in Fig. 3 our measurements for the four maxBCG, GMBCG, AMF, and WHL12 SDSS cluster samples. Similarly, Fig. 4 shows our results for the X-ray MCXC cluster sample. As described in Sec. 4, the errors are obtained from the standard deviation of 100 subsamples with 20 000 randomized cluster positions over the BOSS area, and rescaled for each individual sample depending on the number of clusters to account for Poisson uncertainty. The detection significance, shown in Table 1 for each individual sample, is computed for all data points with 10 cluster-galaxy pairs or more, using the covariance matrix of the random samples up to r = 10 . 0Mpc. The resulting covariance matrix is displayed in Fig. 5: the higher so that and correlation at large scales translates into the fact that the same background galaxy may be used for several clustergalaxy pairs. On the other hand, there is little correlation at smaller scales ( < 1 Mpc) due to the relatively sparse distribution of foreground clusters (approximately, two per sq. deg). For each case, the red-dashed line represents the composite-NFW lensing signal expected for the respective cluster sample, obtained assuming a mass-richness relationship calibrated from X-ray and weak-lensing measurements taken from the published literature: given by Rozo et al. (2009) for maxBCG and GMBG (Eq. 21) and by Wen et al. (2012) for AMF and WHL12 (Eq. 24), whereas the M 200 masses for MCXC have been translated from their M 500 values assuming the NFW form. For comparison, we also plot the expected lensing signal obtained assuming a single effective mass 〈 M 200 〉 , for the NFW (black solid line) and SIS (black-dotted dashed line) profiles. At small scales, there is a small difference between the composite versus single NFW profiles, which however is not significant due to the large statistical errors. As for the SIS case, overall, it is likely that the lensing signal at small scales is overestimated. Here, it is reassuring to see that the NFW model provides a better description of the observed lensing signals because the lensing-based mass-richness relationships were calibrated References . - (1) Koester et al. (2007); (2) Hao et al. (2010); (3) Szabo et al. (2011); (4) Wen et al. (2012) ; (5)Piffaretti et al. (2011). assuming the NFW form in all cases for the maxBCG relation (Johnston et al. 2007; Mandelbaum et al. 2008; Sheldon et al. 2009), and most cases for the WHL12 relation (Wen et al. 2010, and references therein). These results demonstrate the robustness of our measurements and appear to be in excellent agreement with previous lensing studies. Although the large statistical errors prevent us from accurately testing both the amplitude and slope of the mass-richness relationship for the current BOSS sample, it is interesting to compare the results between the different cluster samples. With a detection significance at the nearly 5 σ level, the maxBCG and AMF samples seem to give the highest-confidence detections among our cluster samples. The AMF catalog catalogue has the highest minimum richness cut (Λ 200 ≥ 20) but the averaged signal appears to be relatively higher so that the lower number of clusters above this richness cut is almost compensated by a greater lensing signal due to their higher cluster masses. On the other hand, the measured lensing signal in the WHL12 sample is shown to be relatively low compared to the model prediction based on the same mass-richness relationship, despite its larger sample size and greater overlap with current BOSS sky area. This could be a consequence of a relatively higher impurity of the cluster sample, due perhaps to a higher rate of false detections, which will lead to an underestimation of the geometric lensing signal as found here. Similarly, we obtained for GMBCG a relatively low level of detection significance, 2 . 4 σ . It is interesting to note that Wen et al. (2012) find a lower matching rate of their WHL12 clusters with GMBCG, compared to that with maxBCG. This could indicate that these two samples have a higher level of contamination by false cluster detections. Another interesting point is that the cluster redshift scatter between WHL12 and GMBCG can be as high as several times the photometric redshift errors, as shown by their comparison (Wen et al. 2012, Table 8). A more detailed and careful analysis would be certainly required to confirm such a conclusion. We shall come back to this issue and its consequence for magnification bias contamination in Sec. 6.4. The MCXC X-ray sample has the largest statistical errors due to the small number, 158, of clusters (Fig. 4). However, it is worth noting that there is a likely excess of the lensing signal at r /similarequal 2-10Mpc ( > r 200 ) with respect to the NFW predictions, which cannot be explained by simply increasing the halo masses. In the context of ΛCDM, this large-scale excess signal can be naturally explained by the two-halo term contribution due to largescale structure associated with the central clusters (see Oguri & Hamana 2011). We note that, in contrast to the gravitational shear, magnification is sensitive to the sheet-like mass distribution, and therefore can be used as a powerful tool to probe the two-halo term in projection space.", "pages": [ 6, 7, 8 ] }, { "title": "6.1. Sources Sample Variance", "content": "As described in Sec. 4, our error estimate simply reflects the statistical variation of the mean redshift of sources across the BOSS area. This estimator primarily provides a measure of the source sample variance given a sample of foreground clusters, because it measures the fluctuations of the number counts across the field. We display the magnification signal as measured for AMF (for which we find the highest level of detection significance) in the top panel of Fig. 6, compared to the mean signal from the random samples shown in the bottom panel. The random-sample signal is consistent with null detection, indicating that the 100 random samples are sufficiently large for realistic estimates of the errors.", "pages": [ 8, 9 ] }, { "title": "6.2. Spectroscopic Redshift Errors", "content": "The errors on individual source redshifts might also affect our statistical lensing measurements if only a small number of cluster-galaxy pairs are used at small scales ( < ∼ 0 . 05Mpc). However, the mean spectroscopic error is ∆ z = 1 . 3 × 10 -4 , so that our measured signal is tow orders of magnitude larger than that. Hence, we conclude that this should not have a significant impact on our measurements.", "pages": [ 9 ] }, { "title": "6.3. Photometric Calibration", "content": "The primary requirement of our analysis method is an homogeneous target selection across the survey field. In particular, a spatial variation of the photometric calibration would affect both the cluster detections and the BOSS target selection. In our analysis, we can rely on the accurate and homogeneous calibration of the SDSS data (Fukugita et al. 1996; Gunn et al. 1998). We have checked that a variation of a few 0.01 magnitude would cause a mean redshift change less than δz = 5 × 10 -4 , which is again significantly smaller than the measured signal.", "pages": [ 9 ] }, { "title": "6.4. False Cluster Detections", "content": "Perhaps the most important source of systematics in magnification bias methods originates from the potential contamination by physically associated lens-source pairs that create a spurious correlation signal indistinguishable from the lensing signal. Since we use spectroscopic redshift information for the background, lowredshift contamination by background sources can be neglected here. Hence, the only possible source for such systematics would arise from erroneously-detected distant clusters. Let us consider a galaxy overdensity (such as rich clusters or filamentary structure) located in the source range 0 . 43 < z < 0 . 7 whose member galaxies were misattributed to foreground galaxies with lower photometric redshifts, which can be incorrectly detected as a low redshift cluster. Since cluster-sized overdensities can have a significant impact on the redshift distribution, the resulting mean redshift could be noticeably different from that of the whole sample. The maxBCG and GMBCG samples would have the lowest chance of being affected by this effect, because their detection algorithms rely on bright red galaxies, for which we expect a very low rate of photometric redshift outliers (defined as being off by a few sigmas) due to a clear measurable Balmer break in the redshift range considered here, and the high rate of spectroscopic redshifts for these objects. On the other hand, for the WHL12 and AMF samples which make use of the full galaxy population with photometric redshifts, such misidentification is more likely to happen. Interestingly, Fig. 4 in Oyaizu et al. (2008) shows a photometric-to-spectroscopic redshift comparison of the CC2 sample that was used for AMF. A closer look at the published figure suggests that very few galaxies (one can count only six galaxies out of ∼ 10 000 - less than 0.1%) scatter from the redshift range [0 . 43 : 0 . 7] to [0 . 1 : 0 . 3]. Taking into account the increased scatter in overdense regions, a threshold richness of ∼ 10 in cluster catalogs implies that only those clusters which are richer than a few hundreds would be large enough to cause such a false detection at low redshift. Also rare, but more likely to occur than in the previous case, is massive filaments aligned with the line-of-sight, which can lead to enhanced low-z contamination. This might cause a false cluster detection and also impact the mean redshift estimation of the background sample. Again from Fig. 4 in Oyaizu et al. (2008), it is clearly seen that the scattered galaxies lie below the mean redshift of BOSS galaxies. This means that the expected direction of this effect would be to decrease the mean redshift and bias low our measurements. This could constitute a plausible explanation for the low signal level measured from WHL12, since they employ photometric redshifts in their detection algorithm. It is worth noting that such large overdensities would be easy to spot and mask out in a spectroscopic background sample, when the statistical significance of the BOSS survey improves.", "pages": [ 9 ] }, { "title": "6.5. Cluster Miscentering", "content": "Misidentification of cluster centers is another potential source of systematic errors for cluster lensing measurements at small scales. Recently, George et al. (2012) studied the impact of the choice for the cluster center on the measured lensing signal based on X-ray galaxy groups detected in the COSMOS field. Their findings show that at small scales ( < 0 . 1 Mpc), choosing the BCG over the most massive galaxy close to the X-ray peak position (their center definition yielding the best result) reduces the signal by about 20% within 75 Kpc. Johnston et al. (2007) demonstrated that the lensing convergence κ , which is locally related to the magnification to first order, is less affected by cluster miscentering than the weak shear, and that smoothing due to miscentering effects nearly vanishes at twice the typical positional offset from the cluster mass centroid. This indicates that our results would not be affected by the miscentering effects beyond a radius of r = 0 . 15Mpc. In our present analysis, the statistical uncertainty is too large to be able to estimate the degree of miscentering or the likely level of correction from our data. Still, it is worth noting that our measurements show the lensing signal decreases at small scales for three out of the four samples, although these cluster samples are not entirely independent with sizeable proportions of clusters in common (Wen et al. 2012).", "pages": [ 9, 10 ] }, { "title": "6.6. Model Uncertainties", "content": "In this work, while we attempt to demonstrate the feasibility and great potential of this new method, we are limited by statistical uncertainties and thus unable to constrain both the amplitude and slope of the massrichness relationship. When BOSS is completed, it will increase the respective size of background sample and the large area of sky covered - which means we can also increase the size of the foreground cluster samples by a factor of three - leading to an anticipated factor-three increase in the signal-to-noise ratio. This will then allow us to subdivide cluster samples into bins of richness, thus complementing the standard shear-based mass measurements. A further substantial improvement is expected in near future with bigBOSS and the NIR spectroscopic survey of EUCLID, which will increase the sample size of background galaxies by one order of magnitude. More importantly, EUCLID will obtain a sufficiently large number of spectra of background sources at z > 1, pushing the foreground cluster sample to higher redshift than shear measurement will allow. Similarly, the ambitious Subaru Prime Focus Spectrograph (PFS, Sugai et al. 2012) survey with its powerful NIR capability promises to provide an unprecedented large highredshift sample of spectroscopic redshifts. To fully exploit the greater statistical power in future surveys, it is critical to control systematics. In particular, the luminosity function slopes of source samples must be precisely known, in order to accurately extract the lensing signal from observations. To do this, a homogeneous selection of background sources is crucial. To compute theoretical predictions, we have implicitly assumed that mass is proportional to luminosity when constructing the above composite cluster lensing mass profiles. In details we must expect that the mass-to-light ratio varies as a function of mass, redshift, and galaxy type. In Coupon et al. (2012), we fit the mass-to-light ratio as function of mass and redshift for blue and red galaxies, using the 30-band COSMOS data (Ilbert et al. 2009): in the mass bin 10 10 . 5 -10 11 . 5 M /circledot , we find the slope of the luminosity function changes from 0.44 to 0.69 depending of the galaxy type. This implies that, at most, the change in luminosity keeping a constant mass would be about 0.1 magnitude from redshift 0.4 to 0.7. Given the parametrization of the luminosity function employed here we can test how the slope of the number counts would be affected by simply shifting the limiting magnitude by 0.1. In practice, since we are concerned only with the relatively bright end of the galaxy distribution, it is equivalent to a shift in M /star parameter. Furthermore, two additional sources of systematic errors could have affected the determination of the source luminosity function by Ilbert et al. (2005): the Poisson error and the cosmic variance. The former was estimated by Ilbert et al. (2005), and given as ∆ M /star ∼ 0 . 25, which is correlated with the faint-end slope α , so that this estimate may be rather pessimistic. For the later, the effect of cosmic variance on the count slope is more difficult to assess. To obtain a crude estimate of this effect we consider the value of ∆ M /star = 0 . 02, which was estimated by Loveday et al. (2012) for the GAMA survey from jackknife resampling of nine subregions each with 16 deg 2 . Although GAMA is a lower-redshift survey compared to VVDS, the volume of each subregion is larger than the volume probed over the 1 deg 2 field-of-view of VVDS at z = 0 . 5, so that we have adopted a conservative estimate of 0.1. Finally we add in quadrature all three sources of systematic uncertainty, and reexamine our model predictions in the two extreme cases, M /star ± 0 . 29. In Fig. 6, the resulting error region is indicated by the blue-shaded area which is systematically less than the statistical error for the current dataset.", "pages": [ 10 ] }, { "title": "7. CONCLUSIONS", "content": "Using over 300 000 BOSS galaxies we have measured the mean redshift of background galaxies behind large samples of SDSS galaxy clusters. Our results show a net increase of the mean redshift behind the clusters compared to that of the total sample, in line with reasonable expectations for the effect of lens magnification. We have tested four different cluster catalogs, the maxBCG, GMBCG, AMF and WHL12 samples, with detection significance ranging from 2.8 σ s (GMBCG) to 4.9 σ s (AMF), where the level of systematic errors is expected to be negligible compared to our statistical errors. In order to speed-up and ease the measurements of the mean redshift as function of physical scale around clusters, we employed a sophisticated code ( Swot ) that performs rapid parallel calculations and accurate error estimations, making practical the handling of future samples containing several millions of objects. Based on precise measurements of luminosity functions from a number of deep spectroscopic surveys, and on an accurate modelling of dark matter halo profiles, we have compared our results to theoretical predictions. We constructed the expected signal by summing the individual contribution for each cluster given its richness from two independently established mass-richness relationships corresponding to two kinds of richness definition, and each calibrated with X-ray and weak lensing data. For three samples out of four, the agreement with the NFW profile is excellent. The mean masses of the clusters derived from our basic stacking analysis vary between the four large SDSS cluster samples in the range 1 . 18-1 . 81 14 M /circledot , after allowing for the mass-richness relations appropriate for each cluster sample and with a mean radial profile consistent with the observed radial trend towards higher mean redshift at smaller cluster ra- dius. Only WHL12 showed a marginal deviation from the expected level, suggesting an underestimated signal. Given the high density of candidate clusters in this catalog compared to the others, this effect may indicate a somewhat larger contamination of spurious clusters that dilute the lensing signal in proportion to the fraction of false detections. Further investigation of possible sources of systematic error will be feasible with a more thorough understanding of the source population of the BOSS survey, in terms of colour and magnitude selection and Eddington bias affecting the very bright end of the BOSS luminosity function, arising from photometric error. With the current sample, given our relatively large statistical errors we conclude that the systematic errors in the measurements should be negligible at present but that with the completion of the BOSS survey we will reach an increased level of precision that may warrant closer scrutiny of the details of the source selection function and possible redshift uncertainties for the foreground cluster populations. We believe that the method is very robust against systematic errors, even when we increase the statistical power in future experiments such as bigBOSS, PFS and EUCLID (which will observe background sources 25 times as dense and over 5 times the area of this study) with millions of spectroscopic/grism redshifts, at which point this method will be powerful in its own right for defining cluster mass-concentration relations and the evolution of the cluster mass function. These results may be compared with independent estimates of magnification bias from faint number counts and with cosmic shear measurements for which the sources of systematic error are very different. We acknowledge Yen-Ting Lin for fruitful discussions and comments, and for providing us with the calibration formula for MCXC masses. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/ . SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. The work is partially supported by the National Science Council of Taiwan under the grant NSC97-2112-M001-020-MY3 and by the Academia Sinica Career Devel- opment Award.", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "Bartelmann, M. & Schneider, P. 2001, Phys. Rep., 340, 291 Benjamin, J., Van Waerbeke, L., Heymans, C., et al. 2012, ArXiv e-prints Okabe, N., Smith, G. P., Umetsu, K., Takada, M., & Futamase, T. 2013, ArXiv e-prints 2010, ApJ, 723, L13 1033 647, 853 120, 1579", "pages": [ 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "We present Chandra high energy transmission grating spectra of the narrowline Seyfert-1 galaxy Ark 564. The spectrum shows numerous absorption lines which are well modeled with low velocity outflow components usually observed in Seyfert galaxies (Gupta et al. 2013). There are, however, some residual absorption lines which are not accounted for by low-velocity outflows. Here we present identifications of the strongest lines as Kα transitions of O vii (two lines) and O vi at outflow velocities of ∼ 0 . 1 c . These lines are detected at 6 . 9 σ , 6 . 2 σ , and 4 . 7 σ respectively and cannot be due to chance statistical fluctuations. Photoionization models with ultra-high velocity components improves the spectral fit significantly, providing further support for the presence of relativistic outflow in this source. Without knowing the location of the absorber, its mass and energy outflow rates cannot be well constrained; we find ˙ E ( outflow ) /L bol lower limit of ≥ 0 . 006% assuming a bi-conical wind geometry. This is the first time that absorption lines with ultra-high velocities are unambiguously detected in the soft X-ray band. The presence of outflows with relativistic velocities in AGNs with Seyfert-type luminosities is hard to understand and provides valuable constraints to models of AGN outflows. Radiation pressure is unlikely to be the driving mechanism for such outflows and magneto-hydrodynamic may be involved.", "pages": [ 1 ] }, { "title": "Discovery of Relativistic Outflow in the Seyfert Galaxy Ark 564", "content": "A. Gupta and S. Mathur 1 Astronomy Department, Ohio State University, Columbus, OH 43210, USA [email protected]", "pages": [ 1 ] }, { "title": "Y. Krongold", "content": "Instituto de Astronomia, Universidad Nacional Autonoma de Mexico, Mexico City, (Mexico)", "pages": [ 1 ] }, { "title": "F. Nicastro", "content": "Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02138, USA Osservatorio Astronomico di Roma-INAF, Via di Frascati 33, 00040, Monte Porzio Catone, RM, (Italy)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "There have been reports of ultra-high velocity outflows in the X-ray spectra of radioquiet AGNs and quasars. These outflows are identified through blue-shifted Fe xxv and/or Fe xxvi absorption lines (at rest-frame energies of 7-10 keV) from highly ionized gas (log ξ 1 = 3 -6 erg s -1 cm), with column densities as large as N H = 10 22 -10 24 cm -2 , and with relativistic velocities of 0.1c-0.3c (Pounds et al. 2003a;b, Tombesi et al. 2010, and references therein). The mass outflow rate of these high velocity outflows were comparable to the accretion rate and their kinetic energy was a significant fraction of the bolometric luminosity (Tombesi et al. 2012). These outflows can provide effective feedback that is required by theoretical models of galaxy formation to solve a number of astrophysical problems ranging from cluster cooling flows to structures of galaxies. Although high velocity outflows are detected in a large number of sources, in all the cases identification are based on Fe K-shell transitions which fall in a region of the spectrum where instrumental resolution is much lower than in the soft X-ray band. The significance of the absorption line detections is often questioned and with only a few lines observed, accurate parametrization of the photoionized plasma becomes difficult. While the observed transitions are from highly ionized gas (log ξ = 3 -6 ers s -1 cm), photoionization models predict that in such a plasma in addition to highly ionized iron, lighter elements should also be present, strongest of which are S, Si and O (e.g., Sim et al. 2010, Krongold et al. 2003). Absorption lines from these highly ionized elements lie in the soft X-ray band, so warm absorber signatures of relativistic outflows at soft X-rays should be present, but were never unambiguously observed until now (see § 5.3 for existing evidence). For complete understanding of the properties of high velocity outflows it is necessary to detect transitions of other ions such as of O, Ne, Si and S as predicted by models. In this paper, we report the serendipitous discovery of high velocity outflows in the soft X-ray band in Ark 564, identified during our detailed analysis of the Chandra archival data of this source (Gupta et al. 2013, hereafter Paper I). Primary goal of Paper I was to self consistently analyze and model the grating spectra of typical ( V out = 100 -1000 km s -1 ) warm absorbers in Ark 564. Here we present the discovery of warm absorbers (WA) with ultra-high velocities.", "pages": [ 2 ] }, { "title": "2. Data and Spectral Analysis", "content": "Ark 564 is a bright, nearby ( z = 0 . 024684), narrow-line Seyfert 1 (NLS1) galaxy, with luminosity of L 2 -10 keV = (2 . 4 -2 . 8) × 10 43 ergs s -1 (Turner et al. 2001, Matsumoto et al. 2004, Paper I). In Paper-I we have discussed the Chandra observation and data reduction of this source which we briefly summarize here. Ark 564 was observed with the Chandra High Energy Transmission Grating Spectrometer (HETGS) in 2008 in three separate exposures totaling 250 ks. We followed the standard procedure to extract spectra. We used the software package CIAO (Version 4.3) and calibration database CALDB (Version 4.4.2) developed by the Chandra X-ray center. We co-added the negative and positive first-order spectra and built the effective area files (ARFs) for each observation using the fullgarf CIAO script. The HETG-MEG spectra were analyzed using the CIAO fitting package Sherpa . We co-added the spectra obtained with HETG-MEG and averaged the associated ARFs. We fitted the continuum with absorbed power law plus black body component and modeled all the statistically significant local absorption ( z = 0) features with Gaussian components. Further we used the photoionization model fitting code PHotoionized Absorption Spectral Engine (PHASE; Krongold et al. 2003), to fit the typical warm absorber features. The best fit model of intrinsic absorption requires two warm absorbers with two different ionization states (log U 2 = 0 . 38 ± 0 . 02 and log U = -1 . 3 ± 0 . 13), both with moderate outflow velocities ( ∼ 100 km s -1 ) and relatively low line of sight column densities ( ∼ N H = 10 20 cm -2 ). For detailed spectral analysis and best fit parameters, we refer readers to Paper-I. Though most of the intrinsic absorption features in the source spectrum are well fitted with two warm absorbers, there are residual absorption line-like features in the spectral regions of 19 . 0 -21 . 0 ˚ A , 17 . 0 -17 . 5 ˚ A and 13 . 5 -13 . 8 ˚ A each with individual significance of 1 . 7 -6 . 9 sigma (Fig. 1). To check for the consistency of these residuals and to confirm that these are not the artifacts of co-adding spectra, we inspect individual Ark 564 HETGSMEG spectra from the 2008 observations. After fitting the data with continuum, Galactic and two-phase WAs model (Model-A) as noted above, the same residuals are found in the same spectral region in all the three observations (Fig. 2). In the following sections, we discuss the identifications, statistical significance and possible origin of these absorption lines.", "pages": [ 3 ] }, { "title": "3.1. Identification of the Absorption Lines", "content": "The strongest residual features in the co-added spectrum are present at 19 . 805 ± 0 . 005 ˚ A , 19 . 845 ± 0 . 001 ˚ A and 20 . 250 ± 0 . 011 ˚ A (observed frame, Fig. 1). Errors refer to 1 σ confidence level throughout the paper, unless noted otherwise. First we try to identify the features at 19 . 805 ± 0 . 005 ˚ A and 19 . 845 ± 0 . 001 ˚ A . There is no known instrumental feature near these energies (Chandra Proposers' Observatory Guide, or POG). The z=0 lines are already included in the model; there are no permitted lines with oscillator strength > 0 . 0001 at wavelength of either 19 . 805 , 19 . 845 or 20 . 250 ˚ A . At the observed energies, there would be no intervening system with z WHIM < z Ark 564 from the warm-hot intergalactic medium (WHIM; Mathur et al. 2003, Nicastro et al. 2005). Therefore, we assume that these absorption lines are intrinsic to the source. We identify these lines based on a combination of chemical abundance and line strength (the oscillator strength f > 0 . 1). Given the detected wavelength and assuming a very broad range of inflow/outflow velocities -60 , 000 to 60 , 000 km s -1 the likely candidates are O vii Kβ at λ rest = 18 . 62 ˚ A , O viii Kα at λ rest = 18 . 96 ˚ A , Ca xvii kβ at λ rest = 19 . 56 ˚ A , Ca xvi kβ at λ rest = 20 . 617 ˚ A , Ar xiv kβ at λ rest = 21 . 15 ˚ A , Ca xvi Kα at λ rest = 21 . 45 ˚ A and O vii kα at λ rest = 21 . 602 ˚ A . Considering that argon and calcium are orders of magnitude less abundant than oxygen, and because O vii Kα and O viii Kα are by far the strongest possible lines, the most likely candidates are O viii Kα with inflow velocities of 0 . 019 c and 0 . 021 c , or O vii Kα with outflow velocities of 0 . 105 c and 0 . 103 c . To distinguish between the two possibilities of inflow and outflow, we search for possible associations of other lines such as O viii kβ , O vii kβ , O vi and/or O v . We do not find any possible association for inflows. The absorption feature at 20 . 25 ± 0 . 01 ˚ A corresponds to O vi kα ( λ rest = 22 . 037 ˚ A ) at the outflow velocity of 0 . 103 c . We also found an absorption feature at 17 . 085 ± 0 . 011 ˚ A with 1 . 7 σ significance, corresponding to O vii kβ line at the outflow velocity of 0 . 105 c . The detection of O vi Kα and O vii Kβ lines at the same velocity favors the outflow scenario.", "pages": [ 4 ] }, { "title": "3.2. Statistical Significance of Absorption Lines", "content": "To investigate whether the apparent absorption lines could be due to statistical fluctuations, we calculate the probability of detection of individual lines due to random statistical fluctuation. First we fit the lines with negative gaussians of fixed width 0.001 ˚ A, folding through the detector response and leaving other parameters (centroid and amplitude) free to vary. The addition of three gaussian lines at 19 . 805 ± 0 . 006 ˚ A , 19 . 845 ± 0 . 005 ˚ A and 20 . 250 ± 0 . 005 ˚ A to our previous model (Model A) improves the fit statistic by ∆ χ 2 = 120 , ∆ d.o.f. = 6, an improvement at more than 99.99% confidence by the F-test (Bevington and Robinson 1992). We measured the equivalent width (EW) of lines at 19 . 805 ˚ A , 19 . 845 ˚ A and 20 . 250 ˚ A of 15 . 6 ± 2 . 5 m ˚ A, 16 . 5 ± 2 . 4 m ˚ A and 16 . 1 ± 3 . 4 m ˚ A respectively. Errors are at 1 sigma confidence level and are calculated using the ' projection ' command in Sherpa . Thus the three absorption lines are detected with 6 . 2 σ , 6 . 9 σ and 4 . 7 σ significance respectively. Further, using the Gaussian probability distribution, we looked for the probability of detection of these lines by chance. For the lines detected with 6 . 2 σ , 6 . 9 σ and 4 . 7 σ significance, the probability of false detection is 2 . 8 × 10 -10 , 2 . 6 × 10 -12 and 1 . 3 × 10 -6 respectively. There are, however, 4801 wavelength bins in our spectrum (these are all the wavelength bins in the wavelength range of 1 -25 ˚ A , beyond which data is of poor quality. The rest of the HETG spectrum was never used in any of the analysis, even in Paper I). Therefore the probability of finding absorption lines at the observed significance anywhere in the spectrum due to random statistical fluctuations is 0 . 0001%, 0 . 000001% and 0 . 6% respectively. For the outflow system with velocity of 0 . 105 c , we also detected the O vii K β line with chance probability of 0 . 04. Thus the combined chance detection probability of detecting both O vii K α and O vii K β lines is 4 × 10 -6 %. Similarly for the other system (at 0 . 103 c ) the combined chance detection probability of detecting both O vii K α and O vi K α is 4 × 10 -8 %. Thus we conclude that the detected absorption lines are not due to random statistical fluctuations, but are signatures of outflows.", "pages": [ 4, 5 ] }, { "title": "3.3. Other Absorption lines at blueshift of ≈ 0 . 1 c", "content": "After confirming the high detection significance of O vii and O vi absorption lines identified with high velocity outflows, we searched the entire spectrum for other ionic transitions at similar blueshifts. We detected two other absorption features at 17 . 351 ± 0 . 009 and 13 . 625 ± 0 . 011 ˚ A with EWs of 11 . 8 ± 2 . 7 and 5 . 0 ± 1 . 7 m ˚ A (Fig. 3, Table 1). The feature at 13 . 63 ˚ A is also detected in the HEG spectrum. The tentative identification of these features are O viii Kα and Fe xvii Kα at outflow velocities ∼ 0 . 110 c and ∼ 0 . 114 c respectively. The outflow velocities of these systems are close to each other and to the above reported systems of high velocity outflow.", "pages": [ 5 ] }, { "title": "4. Photoionization Modeling of High Velocity Outflows", "content": "To determine the physical properties of the absorber responsible for producing highly blueshifted absorption features and to check for the physical consistency of line identifications, we modeled the negative Gaussian lines in the above fits with a photoionization model based on the code PHASE (Krongold et al. 2003). The PHASE code models selfconsistently more than 3000 X-ray bound-bound and bound-free transitions imprinted by photoionized absorbers, given the ionization state of the absorber, its column density, and its internal turbulent motion. The parameters of the code are: 1) ionization parameter U ; 2) equivalent hydrogen column density N H ; 3) outflow velocity of the absorbing material V out , and 4) micro-turbulent velocity V turb of the material. The abundances have been set at the Solar values (Grevesse et al. 1993). Usually, the micro-turbulent velocity is not left free to vary, because different transitions due to ionized gas are heavily blended, or because different velocity components are also blended and cannot be resolved (e.g. Krongold et al. 2003, 2005 for NGC 3783). In the case of Ark 564, with the HETGS resolution, it is possible to distinguish two velocity components (see below). So, despite the fact that the individual absorption lines cannot be resolved, we have left the micro-turbulent velocity free to vary. We used the Ark 564 spectral energy distribution (SED) from Romano et al. (2004) to calculate the ionization balance of the absorbing gas in PHASE, same as in Paper-I. A PHASE component (which we call system 1) with ionization parameter of log U = -0 . 60 ± 0 . 38 and column of log N H = 19 . 8 ± 0 . 2 cm -2 successfully reproduced the O vii K α and K β lines at ∼ 19 . 805 and ∼ 17 . 08 ˚ A , at the blueshift of 32365 ± 38 km s -1 with respect to source (Fig. 4). The addition of PHASE component 'System 1' to Model-A significantly improves the fit (∆ χ 2 = 15 , ∆ d.o.f. = 4). According to F-test, the absorber is present at confidence level of 99 . 87%. A second low ionization parameter component (we call this system 2) with log U = -1 . 2 ± 0 . 21 and column of log N H = 20 . 0 ± 0 . 3 cm -2 is also required to fit other high velocity outflow absorption features such as of O vii and O vi , with outflow velocity of 31735 ± 59 km s -1 (Fig. 4). Inclusion of this component improves the fit (∆ χ 2 = 31, ∆ d.o.f. = 4) over the previous model at a confidence level of more than 99.999%, according to F-test. The best fit PHASE parameters are reported in Table 2 and the best fit model are shown in Fig. 5. Successfully modeling the residuals with two PHASE components robustly confirms the presence of high velocity outflows in Ark 564.", "pages": [ 6 ] }, { "title": "5.1. Comparison with Theoretical Models", "content": "Several models suggest radiation pressure as the driving mechanism to produce the typical low velocity outflows observed in Seyfert galaxies (Proga & Kallman 2002; 2004, Krolik & Kriss 1995, Dorodnitsyn et al. 2008). But in radiation driven disk-wind models outflow velocities depend on AGN luminosity and these models cannot account for relativistic velocities in AGNs with Seyfert-type luminosities (Barai et al. 2011). Relativistic outflows in the UV have been detected only in most luminous broad absorption line quasars. Indeed, Laor & Brandt (2002) and Ganguly et al. (2008) have shown that the maximum outflow velocity is proportional to L 0 . 6 , close to what is expected from radiation pressure driven winds. Observations of relativistic outflows in X-rays in moderate luminosity AGNs therefore pose intriguing puzzles; as shown in figure 6, the relativistic outflows (from Tombesi et al. 2011) lie above the Ganguly et al. line of maximum velocity. The relativistic outflow we find in Ark 564, shown as a /star in figure 6, also lies above the line. Thus these high velocity outflows appear not to be driven by radiation line pressure mechanism. King (2010) shock wind models produce winds with velocities v ∼ 0 . 1 c , but in quasars accreting at Eddington limits. In this model a high velocity ionized outflow collides with the ISM of the host galaxy, losing much of its energy by efficient cooling resulting in a strongly shocked gas. In the multi-dimensional Monte Carlo simulations of AGN disk-wind models of Sim et al., outflow velocities are assumed to be at escape velocities, and not from ab-initio calculations (Sim et al. 2008; 2010). It is possible that magnetic fields are important for launching such winds, as in jets (Sim et al. 2010); observations of relativistic outflows in moderate luminosity AGNs provide valuable constraints to the theory of magneto-hydrodynamic winds. The magneto-hydrodynamic accretion-disk wind models of Fukumura et al. (2010a;b) predict high-velocity ( v out ≤ 0 . 6 c ) outflows from ab-initio calculations. These models, however, explain only the high-ionization high-velocity outflows, similar to those observed by Tombesi et al. (2012). In these models, ultra-high velocities are produced when UV to X-ray spectral slope is steep ( α OX ≤ -2), i.e. the AGNs are relatively UV bright (or X-ray faint). Another feature of these models is that density scales as n ( r ) ∝ r -1 , leading to a clear prediction that the outflow velocity depends on ionization parameter. Perhaps some of the model assumptions can be modified to explain the ultra-high velocity warm absorbers we present here.", "pages": [ 7 ] }, { "title": "5.2. Mass and Energy Outflow Rates", "content": "In Paper-I we have showed that the typical warm-absorbers in Ark 564 ( v out ≈ 100 km s -1 ) are not energetic enough to provide the effective feedback as required by theoretical models. Here we estimate the total mass and kinetic energy outflow rate of relativistic outflows in Ark 564 to determine if these can possibly be important for feedback. To measure the mass and energy outflow rates, we must know the location of the absorber which is measured only in a few sources (e.g. NGC 4051, Krongold et al. 2007). Since we do not know the location of the high velocity outflow in Ark 564, we estimate the mass and energy outflow rates in several different ways. It is often assumed that the observed outflow velocity is the escape velocity at the launch radius r : i.e. r = 2 GM BH v 2 out . There is no justification for this assumption, as shown by Mathur & Stoll (2009). Nonetheless, making this assumption provides us with a lower limit on the absorber location. Romano et al. (2004) determine the central black hole mass of Ark 564 to be M BH ≤ 8 × 10 6 M /circledot . This leads to the minimum distance of system 1 and system 2 absorbers of r min = 84 r s and r min = 88 r s respectively (in units of the Schwarzschild radius; for Ark 564 r s = 7 . 8 × 10 -7 pc). The estimate of maximum distance from the central source can be derived assuming that the depth ∆ r of the absorber is much smaller than the radial distance of the absorber (∆ r << r ) and using the definition of ionization parameter ( U = Q ( H ) 4 πr 2 n H c ), i.e. r ≤ r max = Q ( H ) 4 πUN H c . Using the best fit values of ionization parameter and column density, we estimated the upper limits on system 1 and system 2 absorber locations of r max = 5 . 4 kpc and r max = 13 . 6 kpc respectively, which are not very interesting limits. For a bi-conical wind, the mass outflow rate is ˙ M out ≈ 1 . 2 πm p N H v out r (Krongold et al. 2007). Substituting r with r min and using outflow velocities of 32365 km s -1 and 31735 km s -1 , we obtain lower limit on mass outflow rates of ˙ M out ≥ 4 . 1 × 10 20 g s -1 and ˙ M out ≥ 4 . 2 × 10 20 g s -1 for system 1 and system 2 absorbers respectively. Similarly we obtained the constraints on kinetic luminosity of the outflows of ˙ E K ≥ 7 . 2 × 10 39 erg s -1 and ˙ E K ≥ 7 . 1 × 10 39 erg s -1 for the system 1 and system 2 absorbers respectively. In comparison to the Ark 564 bolometric luminosity of 2 . 4 × 10 44 erg s -1 , the total kinetic luminosity of these high velocity outflows is ˙ E K /L bol ≥ 0 . 006%. This lower limit is much smaller than the ratio of ˙ E K /L bol ∼ 0 . 5 -5% expected if the outflow is to be important for feedback (Hopkins & Elvis 2010, Silk & Rees 1998, Scannapieco & Oh 2004, di Matteo et al. 2005).However, with very large uncertainties between r min and r max , the ratio between the mechanical power of these outflows and the bolometric luminosity cannot be well constrained.", "pages": [ 8 ] }, { "title": "5.3. Comparison with other high velocity outflows", "content": "Before this work, the high velocity outflows were detected mostly with Fe xxv and/or Fe xxvi absorption lines in hard X-ray band ( § 1). In a handful of quasar such as PG11211+143 (Pounds et al. 2003a), PG0844+349 (Pounds et al. 2003b) and MR 2251-178 (Gibson et al. 2005) high velocity outflows were detected at soft X-ray energies, but these detections were either based on absorption lines of low statistical significance or have strong contamination from absorption lines from the halo of our Galaxy. Here we present the detection of high velocity outflows in Ark 564; this is the first time such outflows are found in the soft X-ray band in a Seyfert galaxy. We firmly establish the presence of high velocity WAs first through identifying number of ionic transitions at similar velocity and further successfully modeling these features with photoionization models. Papadakis et al. (2007) also detected an absorption line at 8.1 keV in the low resolution CCD spectra of Ark 564 and assuming that this line corresponds to Fe xxvi , they suggested the presence of highly ionized, absorbing material of logN H > 23 cm -2 outflowing with relativistic velocity of 0.17c. If the presence of such a feature and its identification are correct, then this is suggestive of a velocity gradient with higher charge states such as Fe xxvi at higher velocity. Interestingly, the two 'other' lines we identified in Ark 564 ( § 3.0.3) also have somewhat higher velocity than System 1 and System 2 and have higher charge states. This is exactly as expected from the models of King (2010) and Fukumura et al. (2010a;b). Recently Tombesi et al. (2013) presented the connection between ultra-fast outflows (UFOs) and soft X-ray WAs. They strongly suggest that these absorbers represent parts of a single large-scale stratified outflow and they continuously populate the whole parameter space (ionization, column, velocity), with the WAs and the UFOs lying always at the two ends of the distribution (Fig. 7). The Ark 564 low-velocity WAs (Paper-I) and UFOs (Papadakis et al.) are in well agreement with linear correlation fits from Tombesi et al. 2013. However, our low-ionization low-column high-velocity outflows in the Ark 564 probe a completly different parameter space. Figure 7 clearly shows that Ark 564 high-velocity outflows have ionization parameter and column density as of typical WAs, but much higher velocity, probing a distinct region in velocity versus ionization/column parameter space.", "pages": [ 9 ] }, { "title": "6. Conclusion", "content": "We report on the discovery of high velocity outflows in the NLS1 Galaxy Ark 564. These absorbers are identified through multiple absorption lines of O vi , O vii Kα , O vii Kβ , O viii and Fe xvii at blueshifts of ∼ 0 . 1 c (with respect to the source) detected in the Chandra HETG-MEG spectra. The two observed velocity components are well fitted with two photoionization model components. Both absorbers have low ionization parameter of log U = -0 . 60 ± 0 . 38 and -1 . 2 ± 0 . 21 and low column densities of log N H = 19 . 8 ± 0 . 2 and 20 . 0 ± 0 . 3 cm -2 and are required at high significance of 99.87% and > 99.99% respectively. Without knowing the location of the absorber, its mass and energy outflow rates cannot be well constrained; we find ˙ E ( outflow ) /L bol lower limit of ≥ 0 . 006% assuming a bi-conical wind geometry. Determining the absorber location is therefore very important for providing meaningful constraints. This can be achieved through studying the response of absorption lines to continuum variations. This is the first time that absorption lines with ultra-high velocities are unambiguously detected in the soft X-ray band. The presence of outflows with relativistic velocities in AGNs with Seyfert-type luminosities is hard to understand and provides valuable constraints to models of AGN outflows. Radiation pressure is unlikely to be the driving mechanism for such outflows and magneto-hydrodynamic may be involved. Finding such relativistic outflows in several other AGNs and measuring their mass/energy outflow rates is therefore important. Acknowledgement: Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number TM9-0010X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. YK acknowledges support from CONACyT 168519 grant and UNAM-DGAPA PAPIIT IN103712 grant.", "pages": [ 9, 10 ] }, { "title": "REFERENCES", "content": "Turner, T. J. et al. 2001, ApJ, 561, 131 . .", "pages": [ 12, 14, 18 ] } ]

[ { "title": "ABSTRACT", "content": "Three of the most important and most puzzling features of the Sun's atmosphere are the smoothness of the closed field corona, the accumulation of magnetic shear at photospheric polarity inversion lines (PIL), and the complexity of the slow wind. We propose that a single process, helicity condensation, is the physical mechanism giving rise to all three features. A simplified model is presented for how helicity is injected and transported in the closed corona by magnetic reconnection. With this model we demonstrate that helicity must condense onto PILs and coronal hole boundaries, and estimate the rate of helicity accumulation at PILs and the loss to the wind. Our results can account for many of the observed properties of the closed corona and wind. Subject headings: Sun: magnetic field - Sun: corona", "pages": [ 1 ] }, { "title": "Helicity Condensation as the Origin of Coronal and Solar Wind Structure", "content": "S. K. Antiochos NASA Goddard Space Flight Center, Greenbelt, MD, 20771 [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "A classic, but puzzling feature of the Sun's high-temperature ( > 1MK) atmosphere is its apparent lack of complexity. High-resolution XUV and X-ray images of the closed-field corona, such as those from the Transition Region and Coronal Explorer (TRACE) mission, invariably show a smooth collection of loops (e.g. Schrijver et al. 1999). If the underlying photospheric flux distribution is highly structured with several polarity regions, then the topology of the loops in the corona will appear complex in XUV images, but this is only due to seeing through multiple flux systems. The surprising result is that the loops in any one flux system, such as a bipolar active region, are generally not observed to be twisted or tangled. Since the coronal field is line-tied to the photosphere, which is undergoing continuous chaotic motions due to the convective flows, it would seem that geometrical complexity should eventually appear in the coronal field. Instead, the magnetic field appears to remain laminar and not far from a potential state over most of the corona. Extrapolations of the field do indicate the presence of coronal currents (Leka et al. 1996; Tian et al. 2005), but these are large-scale volumetric currents that produce only a global shear or twist rather than field line tangling (Schrijver 2007). The observation that loops are untangled, at least on present observable scales, is especially surprising given that the corona is being heated continuously. The standard theory for the heating is that the energy is due to stressing of the coronal field by the random motions of the photospheric footpoints. This is the basic idea of Parker's nanoflare model and similar theories (Parker 1972, 1983, 1988; van Ballegooijen 1986; Mikic et al. 1989; Berger 1991; Rappazzo et al. 2008). The photospheric motions are postulated to tangle and braid the field lines, producing small scale current sheets, which then release their energy via reconnection. A great deal of work has been done applying this nanoflare scenario to coronal observations with considerable success (Klimchuk 2006). The problem with such reconnection-heating models is that any helicity injected into the corona as a result of the motions is expected to survive, because reconnection in a high Lundquist-number system like the corona conserves magnetic helicity (Taylor 1974; Berger 1984). Consequently, even if it is injected on scales below present-day resolution, < 1 arcsecond, the helicity should build up and appear as twisting or tangling of the large-scale coronal field. Note also that even if the degree of tangling required for the heating is small for example, Parker estimates a misalignment angle between the reconnecting stressed field and the initial potential state of only 20 · or so (Parker 1983) - any net helicity injected by the stressing should continue to accumulate and eventually produce large-scale observable effects. We conclude, therefore, that both the basic observations of photospheric motions and the reconnection theories for coronal heating imply that the coronal field should have a complex geometry, in direct disagreement with observations. There are, at least, two seemingly likely explanations for this disagreement. The first is that photospheric motions produce equal and opposite helicity everywhere, so that no net helicity is injected into the corona. Nanoflare reconnection then simply cancels out the positive and negative helicities. This explanation, however, has both observational and theoretical difficulties. Numerous observations imply that photospheric motions, including flux emergence, do inject a net helicity into each hemisphere. For example, observations of prominence structure (Martin et al. 1992; Rust 1994; Zirker et al. 1997; Pevtsov et al. 2003) and of active region vector fields (Seehafer 1990; Pevtsov et al. 1995) indicate a strongly preferred sign for the helicity injected into each solar hemisphere, possibly related to the differential rotation (DeVore 2000). Furthermore, theory and numerical simulations (Linton et al. 2001) have shown that, for magnetic flux tubes with parallel axial fields, reconnection occurs only if the tubes have the same sign of helicity, the so-called co-helicity case as discussed by Yamada et al. (1990). Flux tubes with opposite helicity only bounce when they collide (Linton et al. 2001); therefore, reconnection cannot cancel out positive and negative injected helicity in interacting coronal loops. This point will be clarified in Figure 2 below. The other possible explanation for the lack of helicity buildup is that the heating is due not to reconnection, but to true diffusion, in which case helicity is not conserved. This hypothesis was proposed by Schrijver (2007) to explain the TRACE images. He argued that continual reconnection induced by the rapidly varying field of the magnetic carpet (Harvey 1985; Schrijver et al. 1997) causes the chromosphere and transition region to act like a high-resistivity layer. Coronal loop field lines can slip along this layer and, thereby, lose their tangles. This explanation, however, also has theoretical and observational difficulties. Reconnection is not physically equivalent to diffusion, no matter how frequent the reconnection. Diffusion does destroy helicity and will relax a magnetic field back down to its minimum energy potential state, but reconnection can relax the system down only to some state compatible with total helicity conservation, such as a linear force-free state (Taylor 1974, 1986). In fact, for the line-tied corona, we have argued that helicity imposes very stringent constraints on the possible end state of a system undergoing reconnection relaxation (Antiochos et al. 2002). Furthermore, observations imply that whenever helicity is, indeed, present in the corona, it does not show evidence for diffusive decay. The largest concentration of coronal helicity is in filaments and prominences, or more precisely, in the strongly sheared field that defines a filament channel (e.g. Tandberg-Hanssen 1995; Mackay et al. 2010). Although there is still debate over the exact topology of the filament channel magnetic field; in particular, whether it is a sheared arcade (Antiochos et al, 1994) or a twisted flux rope, the models agree that for a physically realistic 3D topology all the filament flux must be connected to the photosphere. Consequently, if the chromosphere or transition region really did contain a high-resistivity layer, the filament channel shear would simply disappear by field-line slippage. Such slippage is never observed; if anything, filament shear seems to increase continuously until it is ejected from the corona with a filament eruption/CME. From the discussion above, we conclude that a net helicity is injected into each coronal hemisphere by the photosphere, and that reconnection preserves this helicity. But, in that case, where does the helicity go? In a sense, the answer is obvious: The helicity injected into the closed-field corona must end up as the magnetic shear in filament channels. These are the only locations in the corona where the magnetic field is strongly non-potential and, hence, has a strong helicity concentration. Although this answer is intuitively appealing, it seems extremely unlikely. It naturally raises the long-standing questions: What exactly are filament channels and how do they form? Along with laminar coronal loops, filament channels are also classic but puzzling features of the Sun's atmosphere. These structures consist of low-lying magnetic flux centered about photospheric polarity inversion lines (PIL), in which the chromospheric and coronal magnetic field lines run almost parallel to the inversion line rather than perpendicular, as expected for a potential field (Rust 1967; Leroy et al. 1983; Martin 1998). Direct measurements of the filament vector field both in the photosphere (Kuckein et al. 2012) and corona (Casini et al. 2003) show that the component parallel to the PIL is dominant. The channels have narrow widths, of order 10 Mm, but their lengths can be greater than a solar diameter for PILs that encircle the Sun. Filament channels are very common, invariably appearing about any long-lived PIL, both in active regions and quiet Sun. It should be emphasized that the channels are much more common than observable filaments and prominences, which require the presence of substantial amounts of cold plasma as well as the magnetic shear. Two general mechanisms have been proposed for filament channel formation. One mechanism is flux emergence, specifically the emergence of a sub-photospheric twisted flux rope. Most simulations of flux rope emergence find that the resulting structure in the corona is a sheared arcade localized near the PIL (e.g. Mancester 2001; Fan 2001; Magara & Longcope 2003; Archontis 2004; Leake et al. 2010; Fang et al. 2012). The basic process is straightforward; the twist component of the sub-photospheric flux rope emerges to become the overlying quasi-potential arcade in the corona, while the axial sub-surface component emerges to become the shear field of the filament channel. It is interesting to note that, in general, the resulting filament channel in the corona is not a twisted flux rope, but a sheared arcade, because the concave-up portion of the flux-rope field lines stays trapped below the surface even in 3D (Fang et al. 2012). Although flux emergence can yield a sheared arcade, there are major theoretical and observational difficulties with this process as the general mechanism for filament channel formation. First, the simulations have yet to show that flux emergence agrees quantitatively with the amount of flux and degree of shear measured in observed filament channels. In fact, Leake et al. (2010) argue that only a small amount of axial flux emerges, at least in 2.5D simulations, far too small to account for the magnetic free energy in observed filament channels. Similar conclusions have been reached from recent 3D simulations, as well (Fang et al. 2012). A much greater problem for the model is that filament channels are frequently observed to form in regions where there is no apparent flux emergence, such as at PILs between decaying regions and high latitude PILs (e.g. Mackay et al. 2010). Moreover, the emergence of a simple bipolar active region rarely produces a filament channel at its PIL. The channel usually forms only well after the end of the flux emergence, when the active region has decayed and dispersed to interact with surrounding flux regions. Consequently, flux emergence cannot be the only mechanism for filament channel formation. The second, and perhaps, the most popular mechanism that has been proposed for filament channel formation is flux cancellation (Martin 1998). The basic picture is that large-scale shear due to differential rotation or flux emergence concentrates at PILs as opposite-polarity photospheric flux converges and cancels there. Note that this mechanism inherently requires reconnection at the photospheric PIL in order to form low-lying loops that can sink and disappear and concave-up loops that can rise into the corona (van Ballegooijen & Martens 1989). A fundamental difficulty with such reconnection, however, is that it produces a twisted flux rope in the corona just like flare reconnection produces the highly twisted flux rope of a CME. On the other hand, high resolution observations of filaments both from the ground (Lin et al. 2005) and space (Vourlidas et al. 2010) show a field geometry consisting of long, parallel strands, with no evidence of twist or tangling. In fact, empirical models for filaments derived solely from observations, have a laminar field geometry exactly like that of the TRACE loops , except that the field lines are stretched out and primarily horizontal rather than arched (Martin ref). It has been suggested that the large twist component resulting from reconnection may diffuse away (van Ballegooijen 2004), but as argued above, any diffusion would also decrease the shear component, contrary to observations. Note also that the twist produced by flare reconnection, which is physically identical to flux cancellation reconnection, is never observed to diffuse away, but is measured to persist out to 1 AU (Kumar & Rust 1996; Qiu et al. 2007). In addition to the lack of observed twist, the prevalence of filament channels poses severe difficulties for the flux cancellation model and, indeed, for any model. As stated above, filament channels are ubiquitous, appearing over all types of PILs ranging from the most complex and strongest active regions to very quiet high-latitude regions. In fact, it is not uncommon to observe a filament channel that continues unbroken over a PIL that passes from an active region into neighboring quiet region with the cold material, itself, transitioning smoothly from a typical low-lying active region filament to a high-lying quiet sun filament (e.g. Su & van Ballegooijen 2012). Given these observations, it seems improbable that filament channels are due to some phenomenon in the plasma-dominated photosphere, because the dynamics there are insensitive to the magnetic field structure and, in particular, to whether a PIL is present or not. This is especially true in the weak-field regions where quiescent prominences typically form (Klimchuk 1987). It seems much more likely that filament channel formation is due to some generic process occurring in the magnetically-dominated corona and upper chromosphere. We propose that the origin of filament channels is the reconnection-driven evolution of helicity injected into the closed-field corona. This hypothesis seems counterintuitive, because coronal-loop helicity is injected on small scales more-or-less uniformly throughout the corona, whereas filament channels are coherent structures, localized only around PILs and extending to very large scale. In this paper, we describe a process, helicity condensation, that performs exactly the required transformation of small-scale coronal loop helicity into large-scale filament-channel shear. Helicity condensation keeps coronal loops laminar while shearing filament channels. Furthermore, helicity condensation may be responsible for much of the complex structure and dynamics observed in the slow solar wind. We argued above that the helicity injected into the corona must end up in filament channels, but in regions containing coronal holes, another possibility is that some helicity is ejected out into the wind by the opening of closed flux at the coronal hole boundary. Such helicity transfer is implicitly present in the S-Web model for the slow wind (Antiochos et al. 2011, 2012), which postulates that this wind is due to continual dynamics of the open-closed flux boundary. If so, then helicity condensation also will play a major role in the origin and properties of the slow solar wind. We describe below the basic process of helicity condensation and derive estimates of its effectiveness in the Sun's corona.", "pages": [ 1, 2, 3, 4, 5, 6 ] }, { "title": "2. A Model for Helicity Injection and Transport", "content": "In order to understand how magnetic helicity is likely to evolve in the corona, we must first consider the injection process. Assume, for the moment, that the photospheric flux distribution consists of only two polarity regions as shown in Figure 1: a negative northern hemisphere and a positive south, so that all the flux closes across the equatorial PIL (dashed line). The yellow arches in the figure denote two arbitrary small flux tubes corresponding to coronal loops or to the strands inside an observable coronal loop. The quasi-random photospheric motions will introduce small-scale structure and inject helicity to this coronal field. Helicity will also be injected by large-scale motions, such as differential rotation, and by flux emergence/cancellation, such as the magnetic carpet, but for simplicity let us model the injection as due to the continual small-scale photospheric motions, in particular, the granular or supergranular flows. Note that if the magnetic carpet dynamics do not change the net coronal flux, their effect on the coronal helicity can be captured by effective photospheric motions. Furthermore, recent analysis of high-resolution Solar Dynamics Observatory (SDO) data indicates that the bulk of the helicity injected into active regions is due to photospheric motions rather than flux emergence (Liu & Schuck 2012). Following the arguments of Sturrock & Uchida (1981), the energy and, certainly, the helicity injected into coronal loops by stochastic horizontal flows at the photosphere will be primarily in the form of twist. Therefore, we model the motions as a set of randomly located and randomly occurring rotations that have fixed spatial and temporal scales. The true photospheric motions are more complex than a set of fixed-scale rotations, but we are interested only in that part of the flow that injects helicity to the corona. Note also that there is some evidence for exactly the pattern of photospheric rotations of Fig. 1 in measurements of the vorticity of the supergranulation (Duvall & Gizon 2000; Gizon & Duvall 2003; Komm et al. 2007). It has been known since the time of Hale (1927) that sunspot whirls have a clear hemispheric preference, counterclockwise in the north and clockwise in the south (Pevtsov et al. 1995), indicating a preferred sense for the helicity of the subsurface solar motions. The same helicity preference, negative in the north and positive in the south, has been well documented to occur in all types of coronal magnetic structures ranging from quiet Sun field to active region complexes (Pevtsov & Balasubramaniam 2003) and has been observed out in the heliospheric magnetic field (Bieber et al. 1987). As shown in Figure 1, this hemispheric 'rule' is in the same sense as would be expected from the surface differential rotation, but the actual mechanism is still not clear. In any case, we expect there to be a preferred sense to the helicity injecting rotations as illustrated in Figure 1. Note that this is only a preference; a fraction of the rotation in each hemisphere could well have the 'unpreferred' sense. The motions shown in Figure 1 have a number of interesting implications for the coronal field. Assuming, for simplicity, that the rotations are solid body, have size d , and have magnitude, Θ, then each rotation of a photospheric flux tube with axial flux, produces in the corona a twist flux, where B p is the average normal field at the photosphere. In open field regions (not shown in the Figure), this twist flux simply propagates outward, resulting in a net helicity to the turbulence in the fast wind (e.g. Leamon et al 1998). We will discuss the implications for the slow wind below. In the closed field regions, however, the coronal loops acquire a twist component to their magnetic field, as shown in the Figure. If the loop is perfectly symmetric about the equator, then on average, the twists imposed by the two footpoints cancel out so that no net helicity is injected. Basically, the loop is twisted at one end, but untwisted at the other. On the other hand, if the loop has both footpoints in one hemisphere, as is usually the case when the PIL is not exactly at the equator, then the twist from each footpoint will add. Even if the loop is transequatorial, we do not expect any symmetry for a real coronal loop, so a net twist will still be produced by the footpoint motions. Note also, that the effect of any unpreferred-sense rotations (clockwise in the north and counterclockwise in the south) is only to decrease the rate of twisting. To first order, the unpreferred rotations simply untwist the loops, but it should be emphasized that since the rotations are time varying, they can create higher order topological structure in the field even if the net injected helicity vanishes. All higher order topological features, however, such as the braiding of three flux tubes, are not conserved by reconnection (e.g. Pontin et al. 2011) and are not expected to build up in the corona. Consider now the interaction of the two twisted flux tubes of Fig. 1 due to some random motion that causes them to collide. In fact, the twist itself will cause the flux tubes to expand and interact. Since their main axial fields are parallel, only the twist components of the flux tubes can reconnect. If the tubes have the same sense of twist (helicity), then at the contact point between the tubes, their twist components will be oppositely directed and, hence, will reconnect. This is illustrated in the three sequences of Fig. 2, where the red and blue circles correspond to field lines of the twist magnetic component. In the top sequence the twist components are in the same sense, so they are oppositely directed at their contact point; consequently, they will reconnect there. Another way of understanding this result is to note that for interacting tubes with the same helicity, the photospheric rotations have a stagnation point between them. It is well known that such stagnation point flows lead to exponentially growing separation of magnetic footpoints and, hence, to exponentially growing currents in the corona (e.g. Antiochos & Dahlburg 1997), which can drive efficient reconnection. The effect of this reconnection is to spread the twist component over the flux of the two tubes, in other words, the two tubes merge into one globally twisted tube as illustrated in the Figure and as found in simulations of flux-tube collisions (Linton et al. 2001). On the other hand, if the tubes have opposite twist (helicity), then at their point of interaction the twist components are parallel. As illustrated in the second sequence of Figure 2, there is no reconnection in this case. The tubes simply 'bounce' (Linton et al. 2001). This result emphasizes the point made in Antiochos & Dahlburg (1997) that reconnection in the solar corona is highly constrained by line-tying. As a result, the coronal magnetic field cannot simply relax to a minimum energy Taylor (1974) state, which for the opposite twist case corresponds to the initial potential field. The third sequence in Figure 2 illustrates the effect of continued interaction of samehelicity flux tubes. If a larger-scale merged flux tube reconnects with another tube, the result is further merging of the flux and the spread of the twist to even larger scale. This reflects the well-known result from turbulence studies that magnetic helicity tends to cascade upward in scale (e.g. Biskamp 1993). A key point is that the scale referred to in our cascade process is the amount of axial flux, which is closely related to, but not identical to the spatial scale. As argued directly below, the helicity cascades up to the largest possible flux scale, which corresponds to all the axial flux inside a single polarity region. Let us consider the end result of this reconnection-driven helicity cascade. Assume a flux system, as in Figure 3, with simple topology given by a PIL (heavy dashed curve in the Figure) and a separatrix curve somewhere on the photosphere (light dashed curve) that defines all the flux that closes across the PIL. There must be additional PILs on the photosphere, but these are not shown. If the separatrix curve lies in the north, then the flux system is entirely in the north and the twist injected by the photosphere will be predominately counter-clockwise. Since only the relative footpoint motions are significant, we can assume without loss of generality that all the twist is imparted inside the PIL as shown in the Figure. The expected evolution of this twist is seen in Figure 4, which shows a top view of the flux system. As a result of reconnection, the helicity 'condenses' onto the largest scale in the flux system, the PIL, since this encompasses all the flux in the system. The PIL defines the boundary of the polarity region. We conclude, therefore, that the net effect of the many small-scale photospheric twists and the coronal reconnection is to impart a coherent, global twist of the whole flux system that concentrates at the PIL. This global twist is not a true physical motion; the large-scale flux system does not actually rotate as a coherent body, but the photospheric helicity injection and subsequent transport by reconnection does result in an effective global rotation of the magnetic field. The key point is that such a rotation of the whole flux system corresponds to a coherent localized shear all along the PIL, exactly what is needed to explain the formation of filament channels. Such an effective motion produces a channel consisting of field lines that are sheared but smooth and laminar, with no twist or tangles in agreement with high-resolution observations of prominence threads (Lin et al. 2005; Vourlidas et al. 2010). The coronal reconnection in our helicity condensation model results in a structure that is the direct opposite to that of flux cancellation reconnection, which invariably produces a highly twisted flux rope at the PIL. Another important point is that the helicity condensation mechanism is unaffected by the shape of the PIL, in particular whether the PIL contains so-called switchbacks where it forms a sharp zigzag. As long as the flux system defined by the PIL is primarily in one hemisphere, helicity condensation will form a filament channel with the same chirality all along that PIL and with roughly the same amount of shear. This result is in contrast to the predictions of some of the flux cancellation models (van Ballegooijen et al. 1998), but is in good agreement with observations (Pevtsov et al. 2003). Furthermore, since the photospheric con- vection in either quiet or active regions is not observed to change significantly with phase of the solar cycle, the model predicts that the hemispheric helicity rule should hold independent of solar cycle. Again, this conclusion appears to be in good agreement with observations (Pevtsov et al. 2003), unlike some flux cancellation models (Mackay & van Ballegooijen 2001).", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "2.1. Rate of Helicity Cascade", "content": "Prominences typically form on time scales of several days, which sets a constraint that any model must satisfy; therefore, we calculate below the rate of filament channel formation predicted by the helicity condensation model. Let us consider a polarity region as in Figure 4 with scale L that is large compared to the helicity injection scale d . The rate of helicity injection η into a flux tube Φ d by a photospheric twist of average angular velocity V d /d is given by the product of the rate of change of the twist flux of Eq. (2) and the axial flux Φ d , (which stays constant): (e.g. Berger 2000). The total helicity injection rate at the scale d over the whole flux system region with scale L is therefore given by: where Φ L is the flux of the whole system. Following Kolmogorov's classic theory for hydrodynamic turbulence (Kolmogorov 1941), we assume a constant helicity transfer rate at any scale λ . Therefore: which implies that: It should be noted that unlike V d , which is the actual velocity of the photospheric flows, the quantity V L does not represent a true plasma velocity. It is only an effective velocity for the transfer of twist to the largest scale by coronal reconnection. The physical plasma velocities will likely be dominated by reconnection jets and will have both larger magnitude and smaller scale than V L . The physical velocities and kinetic energy are expected to cascade downward, not upward, in scale. Consequently, the velocity spectrum V λ derived in Eq. 6 may not be directly observable, but the effective velocity V L is indeed physically significant. It quantifies the rate at which magnetic helicity 'condenses' out of the corona at the largest scale of the flux system and, hence, V L corresponds to the effective shear velocity along the PIL. Note also, that the helicity cascade process derived above is somewhat different than the usual hydrodynamic turbulence in which velocity is injected statistically uniformly at some scale and then cascades down to where it is dissipated, usually at a kinetic scale. In such an energy cascade, the injection of kinetic energy at a global scale results in the slow increase of thermal energy (temperature) approximately uniformly throughout the system. In our cascade, however, helicity is injected statistically uniformly at some intermediate scale and then simply piles up at the largest global scale. Even though the helicity injection (i.e., photospheric motions) are uniform, the cascade produces a localized spatial structure in the corona. Helicity condensation in the Sun's atmosphere is a striking example of selforganization in a complex system. The time scale for filament channel formation can now be calculated directly from Eq. (6). We note that for the photospheric helicity injection the important parameter is the product of the velocity and the coherence scale of that velocity. Granules typically have V d ∼ 1 km/s and d ∼ 700 km, while supergranules have: V d ∼ . 25 km/sec and d ∼ 30 , 000 km. Eq. (6) implies that the product V d d is the important quantity; therefore, supergranules are expected to dominate the helicity injection. Taking the whole flux system to have scale d/L ∼ 10 -100 implies that a shear of order 1,000 - 10,000 km will build up in ∼ 10 5 s, where we assume that equal helicity is injected at both ends of a flux tube. These results indicate that a high-latitude filament channel with typical shear scales of 100,000 km will form in several days or so, which is consistent with observations (Tandberg-Hanssen 1995; Mackay et al. 2010). Note also that we expect that the width of the helicity condensation region to be of order the width of the elemental photospheric rotation, ∼ 15 , 000 km for supergranules, which again is consistent with the observed widths of filaments (Tandberg-Hanssen 1995; Mackay et al. 2010). The supergranular rate of helicity injection estimated above is also consistent with estimates of solar helicity loss to the wind. Taking the system size L to be of order the solar radius ∼ 10 11 cm, and the average field strength at the photosphere to be ∼ 10 G, we derive from Eq. (4) and the numbers above, a helicity injection rate over the solar surface of ∼ 10 38 Mx/s. Over the course of a full solar cycle, this yields a total helicity loss of ∼ 3 × 10 46 Mx, which agrees well with the inferred losses from observations of CMEs and the wind (DeVore 2000).", "pages": [ 10, 11 ] }, { "title": "2.2. Implications of the Model", "content": "We conclude from the derivation above that helicity condensation can account for filament channel formation, at least, in regions that do not exhibit strong flux emergence. The mechanism can also account for the observed smoothness of coronal loops. Let τ be the time scale required for the system to establish a steady state (except, of course, at the largest scale L where no steady-state is possible). We expect that τ is determined by the twist required to produce intense current sheets, of order a full rotation or so (Antiochos 1998), and not by the rate of reconnection. The driving velocity is only ∼ 0.1% of the coronal Alfven speed, so that the reconnection need not be fast in order to keep pace with the driving. For a given τ , the twist angle produced by the effective velocities of Eq. (6) scales as: This result implies that the corona will exhibit the most structure at the scale at which the twist is injected (presumably the supergranular scale), and at the largest scale where the helicity piles up, the whole length of the PIL. The so-called coronal cells recently discovered by Sheeley & Warren (2012) appear to be evidence for just this type of structure separation. These authors observe that the large-scale corona breaks up into three distinct structures: flux tubes twisted on a scale of 30,000 km or so, long filment channels along PILs that typically span the whole Sun, and coronal holes. Our helicity condensation model is in excellent agreement with these observations. An important issue that is raised by the observations and that we have yet to discuss is the effect on the model of a coronal hole or, more generally, of an open field region. Note, also, that even if no coronal hole is present, the simple picture of Figs. 3 and 4 is topologically incomplete. The PIL cannot be the only boundary that defines the closed negative polarity region. At the very least, there must be a point somewhere in the region where a magnetic spine line connects up to a null point Lau & Finn (1990); Antiochos (1990); Priest & Titov (1996); thereby, making this field line effectively open. Of course, real solar polarity regions tend to have much more complexity often containing intricate open field areas and corridors (Antiochos et al. 2011). Assume that the polarity region of Figs. 3 contains a coronal hole, as illustrated in Figure 5. The presence of the coronal hole introduces subtleties to the calculation of helicity evolution, because the helicity of a truly open field that extends to infinity is not physically meaningful. An open field can have arbitrary helicity due to linkages at infinity where the field vanishes, but the topology does not. Therefore, let us consider instead a system where all the field lines remain closed so the helicity is well-defined throughout the evolution, and let us model the coronal hole as a region where no photospheric twists are imposed; hence, no helicity is injected into this region. In an actual coronal hole twist is injected by photospheric motions, exactly as in closed field regions, but the twist propagates away at the Alfven speed and presumably has no effect on the subsequent evolution in the low corona. Consequently, we can simply model this region as being untwisted. The analysis of the helicity cascade in the large annular flux region bounded by the PIL and the coronal hole boundary proceeds exactly as above. The only difference is that the total twisted area L 2 is replaced by L 2 -H 2 where H is the scale of the coronal hole. Therefore, the largest scale for the helicity is given by: and the effective velocity for helicity condensation is given by Eq (6) above except that L is replaced by L ' . Also, the twist spectrum, Eq (7) is unchanged. As long as H << L , the presence of the coronal hole has minimal effect on the filament channel formation process, and on the smoothness of coronal loops. It is evident from Fig. 5, however, that helicity condensation does have an effect on the magnetic field near the coronal hole boundary. We note that twist, or more accurately magnetic shear, also condenses at this boundary, but curiously enough the shear has the sense opposite to that at the PIL. This result may seem physically unlikely, but in fact it is mandated by helicity conservation. The key point is that the shear flux Φ 1 that condenses onto the PIL encircles all the photospheric flux in the polarity region, including that in the coronal hole region ('CH'). Therefore, the helicity H 1 due to this shear flux is given by: where Φ L ' is the amount of closed photospheric flux inside the PIL and Φ CH is the amount of photospheric flux in the coronal hole, which can be arbitrary compared to Φ L ' . But the CH flux is not twisted and never contributes to the helicity injection; hence, it should not affect the helicity condensation at the PIL. The only way to ensure that the CH flux has no effect is to have a shear flux Φ 2 that condense at the CH boundary and is exactly equal and oppositely directed to that at the PIL. Such a shear flux encircles only the CH flux and, thereby, adds a helicity contribution: which exactly cancels out that between the PIL shear flux and the CH. Fig. 5 shows that reconnection would produce just this required shear flux at the CH boundary. In other words, helicity condensation predicts that a filament channel should form at coronal hole boundaries, at the same rate as at the PIL but with the opposite handedness. These results clearly have major implications for observations. At PILs, the magnetic shear builds up until eventually, it is ejected as a prominence eruption/CME. We expect that a similar process of buildup and ejection occurs at the CH boundaries, but with much less explosive dynamics. The closed field lines near the CH boundary consist of very long, high-lying loops that form the outer shell of the streamer belt. Consequently, it requires far less shear and free energy to open up these loops than to eject the filament channel. We expect that helicity condensation at CH boundaries results in continual small bursts of flux opening and closing there, as is required by models for the slow wind (Antiochos et al. 2011, 2012). An interesting prediction is that the helicity of the closed flux opening at the CH boundary should be opposite to that of the photospheric injection into the coronal hole open field lines, i.e., into the fast wind. It may be possible to test this prediction with in situ measurements. In summary, we argue that a single deeply-profound process, helicity condensation, can explain three long-standing observational challenges in solar/heliospheric physics: the formation of filament channels, the smoothness of coronal loops, and the origin of the slow wind. Furthermore, the model implies major new predictions for solar structure and dynamics. We look forward to many more theoretical and observational studies of helicity condensation in the Sun's corona and wind. This work has been supported, in part, by the NASA TR&T Program. The work has benefited greatly from the authors' participation in the NASA TR&T focused science teams on multiscale coupling and the slow solar wind. The author thanks C. R. DeVore, J. T. Karpen, and J. A. Klimchuk for invaluable scientific discussions and J. T. Karpen for help with the graphics.", "pages": [ 11, 12, 13, 14 ] }, { "title": "REFERENCES", "content": "Antiochos, S. K. 1987, ApJ, 312, 886 Antiochos, S. K. 1990, J. Italian Astron. Soc., 61, 369 Antiochos, S. K., Dahlburg, R. B., & Klimchuk, J. A. 1994, ApJ, 420, L41 Antiochos, S. K. & Dahlburg, R. B. 1997, Sol. Phys., 174, 5 Antiochos, S. K. 1998, ApJ, 502, L181 Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485 Antiochos, S. K., Karpen, J. T., & DeVore, C. R. 2002, ApJ, 575, 578 Antiochos, S. K., Miki´c, Z., Titov, V. S., Lionello, R., & Linker, J. A. 2011, ApJ, 731, 112 Antiochos, S. K., Linker, J. A., Lionello, R., Miki´c, Z., Titov, V. S., & Zurbuchen, T. H. 2011, Space Sci. Rev., 172, 169 Archontis, V. 2004, ApJ, 615, 685 Berger, M. A. 1984, Geophys. & Astrophys. Fluid Dynamics, 30, 79 Berger, M. A. 1991, A&A, 252, 369 Berger, M. A. 2000, Encyclopedia of Astronomy and Astrophysics, ed. P. Murdin, (Bristol: IOP) Bieber, J. W., Evenson, P. A., & Matthaeus, W. H. 1987, ApJ, 315, 700 Biskamp, D. 1993, Nonlinear magnetohydrodynamics, Ch. 7, (New York: Cambridge Uni- versity Press) Casini, R. Lopez Ariste, A., Tomczyk, S., & Lites, B. W. 2003, ApJ, 598, L67 DeVore, C, R. 2000, ApJ, 539, 944 Duvall, T. L., Jr. & Gizon, L. 2000, Sol. Phys., 192, 177 Fan, Y. 2001, ApJ, 554, L111 Fang, F., Manchester, W. B., van der Holst, B. & Abbett, W. 2012, ApJ, 745, 37 Gizon, L. & Duvall, T. L., Jr. 2003, in ESA Special Publication, Vol. 517, GONG 2002, Local and Global Helioseismology: The Present and Future, ed. H. Sawaya-Lacoste (Noordwijk: ESA), 43 Hale, G. E. 1927, Nature, 119, 708 Klimchuk, J. A. 1987, ApJ, 323, 368 Klimchuk, J. A. 2006, Sol. Phys., 234, 41, DOI: 10.1007/s11207-006-0055-z Kolmogorov, A. N. 1941, Dokl. Akad. Nauk. SSSR, 30, 299 Komm, R., Howe, R., Hill, F., Miesch, M., Haber, D., & Hindman, B. 2007, ApJ, 667, 571 Kuckein, C. Martnez Pillet, V., & Centeno, R 2012, A&A, 539, A131 Kumar, A. & Rust, D. M. 1996, J. Geophys. Res., 101, 15667 Leake, J. A., Linton, M. G., & Antiochos, S. K., 2010, ApJ, 722, 550 Liu Y. & Schuck, P., 2012, ApJ, (in press) Harvey, K. L. 1985, Aust. J. Phys., 38, 875 Lau, Y.-T. & Finn, J. M. 1990, ApJ, 350, 672 Leamon, R. J., Matthaeus, W. H., Smith, C. W., & Wong, H. K. 1998, ApJ, 507, L181 Leka, K., Canfield, R., McClymont, A., & van Driel-Gesztelyi, L. 1996, ApJ, 462, 547 Leroy, J.-L., Bommier, V., & Sahal-Brechot, S. 1983, Sol. Phys., 83, 135 Lin, Y., Engvold, O., Rouppe van der Voort, L., Wiik, J. E. & Berger, T. E. 2005, Sol. Phys., 226, 239 Linton, M. G., Dahlburg, R. B., & Antiochos, S. K. 2001, ApJ, 553, 905 Magara, T. & Longcope, D. W. 2003, ApJ, 586, 630 Manchester, IV, W. 2001, ApJ, 547, 503 Martin, S. F., Marquette, W. H., & Bilimoria, R. 1992, ASP Conf. Ser. 27, The Solar Cycle, ed. K. L. Harvey, (San Francisco: ASP), 53 Martin, S. F. 1998, Sol. Phys., 182, 107 Mackay, D. H., & van Ballegooijen, A. A. 2001, ApJ, 560, 445 Mackay, D. H., Karpen, J. T., Ballester, J. L., Schmieder, B., & Aulanier, G. 2010, Space Sci. Rev., 151, 333 Mikic, Z., Schnack, D. D., & van Hoven, G. 1989,ApJ, 338, 1148 Parker, E. N. 1972, ApJ, 174, 499 Parker, E. N. 1983, ApJ, 264, 642 Parker, E. N. 1988, ApJ, 330, 474 Pevtsov, A. A., Canfield, R. C., & Metcalf, T. R. 1995, ApJ, 440, L109 Pevtsov, A. A., Balasubramaniam, K. S., & Rogers, J. W. 2003, ApJ, 595, 500 Pevtsov, A. A. & Balasubramaniam, K. S. 2003, Advances Space Res., 32, 1867 Pontin, D. I., Wilmot-Smith, A. L., Hornig, G., & Galsgaard, K. 2011, A&A, 525, A57 Priest, E. R., & Titov, V. S. 1996, Phil. Trans. R. Soc., 354, 2951 Qiu, J., Hu, Q., Howard, T. A. & Yurchyshyn, V. B. 2007, ApJ, 659, 758 Rappazzo, A. F., Velli, M., Einaudi, G., & Dahlburg, R. B. 2008, ApJ, 677, 1348 Rust, D. M. 1967, ApJ, 150, 313 Rudt, D. M. 1994, Geophys. Res. Lett., 21, 241 Schrijver, C. J., Title, A. M., van Ballegooijen, A. A., Hagenaar, H. J., and Shine, R. A. 1997, ApJ, 487, 424 Schrijver, C. J. et al. 1999, Sol. Phys., 198, 325 Schrijver, C. J. 2007, ApJ, 662, L119 Sheeley, N. R. & Warren, H. P. 2012, ApJ, 749, 40 Seehafer, N. 1990, Sol. Phys., 125, 219 Su, Y. & van Ballegooijen, A. A. 2012, ApJ, 757, 168 Sturrock, P. A. & Uchida, Y. 1981, ApJ, 246, 331 Tandberg-Hanssen, E 1995, The Nature of Solar Prominences, (Dordrecht: Kluwer) Zirker, J. B., Martin, S. F., Harvey, K., & Gaizauskas, V. 1997, Sol. Phys., 175, 27", "pages": [ 15, 16, 17, 18 ] } ]

[ { "title": "ABSTRACT", "content": "A new symplectic N-body integrator is introduced, one designed to calculate the global 360 · evolution of a self-gravitating planetary ring that is in orbit about an oblate planet. This freely-available code is called epi int , and it is distinct from other such codes in its use of streamlines to calculate the effects of ring self-gravity. The great advantage of this approach is that the perturbing forces arise from smooth wires of ring matter rather than discreet particles, so there is very little gravitational scattering and so only a modest number of particles are needed to simulate, say, the scalloped edge of a resonantly confined ring or the propagation of spiral density waves. The code is applied to the outer edge of Saturn's B ring, and a comparison of Cassini measurements of the ring's forced response to simulations of Mimas' resonant perturbations reveals that the B ring's surface density at its outer edge is σ 0 = 195 ± 60 gm/cm 2 which, if the same everywhere across the ring would mean that the B ring's mass is about 90% of Mimas' mass. Cassini observations show that the B ring-edge has several free normal modes, which are long-lived disturbances of the ring-edge that are not driven by any known satellite resonances. Although the mechanism that excites or sustains these normal modes is unknown, we can plant such a disturbance at a simulated ring's edge, and find that these modes persist without any damping for more than ∼ 10 5 orbits or ∼ 100 yrs despite the simulated ring's viscosity ν s = 100 cm 2 /sec. These simulations also indicate that impulsive disturbances at a ring can excite long-lived normal modes, which suggests that an impact in the recent past by perhaps a cloud of cometary debris might have excited these disturbances which are quite common to many of Saturn's sharp-edged rings. Subject headings: planets: rings", "pages": [ 2 ] }, { "title": "An N-body Integrator for Gravitating Planetary Rings, and the Outer Edge of Saturn's B Ring", "content": "Joseph M. Hahn Space Science Institute c/o Center for Space Research University of Texas at Austin 3925 West Braker Lane, Suite 200 Austin, TX 78759-5378 [email protected] 512-992-9962 Joseph N. Spitale Planetary Science Institute 1700 East Fort Lowell, Suite 106 Tucson, AZ 85719-2395 [email protected] 520-622-6300 Submitted for publication in the Astrophysical Journal on December 28, 2012 Revised April 26, 2013 Accepted June 1, 2013", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "A planetary ring is often coupled dynamically to a satellite via orbital resonances. The ring's response to resonant perturbations varies with the forcing, and if the ring is for instance composed of low optical depth dust, then the ring's response will vary with the satellite's mass and its proximity. But in an optically thick planetary ring, such as Saturn's main A and B rings or its many dense narrow ringlets, the ring is also interacting with itself via self gravity, so its response is also sensitive to the ring's mass surface density σ 0 (Shu 1984; Melita & Papaloizou 2005; Hahn et al. 2009). So by measuring a dense ring's response to satellite perturbations, and comparing that measurement to a model for the ring-satellite system, one can then infer the ring's physical properties, such as its surface density σ 0 , and perhaps other quantities too (Melita & Papaloizou 2005; Tiscareno et al. 2007; Hahn et al. 2009). Recently Hahn et al. (2009) developed a semi-analytic model of the outer edge of Saturn's B ring, which is confined by an m = 2 inner Lindblad resonance with the satellite Mimas. The resonance index m also describes the ring's anticipated equilibrium shape, with the ring-edge's deviations from circular motion expected to have an azimuthal wavenumber of m = 2. So the B ring's expected shape is a planet-centered ellipse, which has m = 2 alternating inward and outward excursions. The model of Hahn et al. (2009) also calculates the ring's equilibrium m = 2 response excited by Mimas, but that comparison between theory and observation was done during the early days of the Cassini mission when that spacecraft's measurement of the ring-edge's semimajor axis a edge was still rather uncertain. It turns out that the ring's inferred surface density is very sensitive to how far the B ring's outer edge extends beyond the resonance, which was quite uncertain then due to the uncertainty in a edge , so the uncertainty in the ring's inferred σ 0 was also relatively large. Now however a edge is known with much greater precision, so a re-examination of this system is warranted. Cassini's monitoring of the B ring also reveals that the ring's outer edge exhibits several normal modes, which are unforced disturbances that are not associated with any known satellite resonances. Figure 1 illustrates this phenomenon with a mosaic of images that Cassini acquired of the B ring's edge on 28 January 2008. Spitale & Porco (2010) have also fit a kinematic model to four years worth of Cassini images of the B ring; that model is composed of four normal modes having azimuthal wavenumbers m = 1 , 2 , 2 , 3 that steadily rotate over time at distinct rates. In the best-fitting kinematic model there are two m = 2 modes, one that is forced by and corotating with Mimas, as well as a free m = 2 mode that rotates slightly faster. The amplitudes and orientations of all the modes as they appear in the 28 January 2008 data is also shown in Fig. 2. Note that although the B ring's outer edge, as seen in Fig. 1, might actually resemble a simple m = 2 shape on 28 January 2008, at other times the ring-edge's shape is much more complicated than a simple m = 2 configuration, yet at other times the ring-edge is relatively smooth and nearly circular; see for example Fig. 1 of Spitale & Porco (2010). This behavior is due to the superposition of the normal modes that are rotating relative to each other, which causes the B ring's edge to evolve over time. Since this system is not in simple equilibrium, a time-dependent model of the ring that does not assume equilibrium is appropriate here. So the following develops a new N-body method that is designed specifically to track the time evolution of a self-gravitating planetary ring, and that model is then applied to the latest Cassini results. Section 2 describes in detail the N-body model that can simulate all 360 · of a narrow annulus in a self-gravitating planetary ring using a very modest number of particles. Section 3 then shows results from several simulations of the outer edge of Saturn's B ring, and demonstrates how a ring's observed epicyclic amplitudes and pattern speeds can be compared to N-body simulations to determine the ring's physical properties. Results are then summarized in Section 5.", "pages": [ 2, 3, 4 ] }, { "title": "2. Numerical method", "content": "The following briefly summarizes the theory of the symplectic integrator that Duncan et al. (1998) use in their SYMBA code and Chambers (1999) use in the MERCURY integrator to calculate the motion of objects in nearly Keplerian orbits about a point-mass star. That numerical method is adapted here so that one can study the evolution of a self-gravitating planetary ring that is in orbit about an oblate planet.", "pages": [ 4 ] }, { "title": "2.1. symplectic integrators", "content": "The Hamiltonian for a system of N bodies in orbit about a central planet is where body i has mass m i and momentum p i = m i v i where v i = ˙ r i is its velocity and V ij is the potential such that f ij = -∇ r i V ij is the force on i due to body j where ∇ r i is the gradient with respect to coordinate r i , and the index i = 0 is reserved for the central planet whose mass is m 0 . Next choose a coordinate system such that all velocities are measured since V ij = V ji . This Hamiltonian has three parts, and the following will employ spatial coordinates such that all r i are measured relative to the central planet. This combination of planetocentric coordinates and barycentric velocities is referred to as 'democratic-heliocentric' coordinates in Duncan et al. (1998) and 'mixed-center' coordinates in Chambers (1999). In the above, H A is the sum of two-body Hamiltonians, H B represents the particles' mutual interactions, and H C accounts for the additional forces that arise in this particular coordinate system that are due to the central planet's motion about the barycenter. Hamilton's equations for the evolution of the coordinates r i and momenta p i for particle i ≥ 1 are ˙ r i = ∇ p i H and ˙ p i = -∇ r i H . So a particle that is subject only to Hamiltonian H B during short time interval δt would experience the velocity kick which of course is i 's response to the forces exerted by all the other small particles in the system. And since H C is a function of momenta only, a particle subject to H C during time δt will see its spatial coordinate kicked by due to the planet's motion about the barycenter. Now let ξ i ( t ) represent any of particle i 's coordinates x i or momenta p i ; that quantity evolves at the rate (Goldstein 1980) where the brackets are a Poisson bracket, and the operator A is defined such that Aξ i = [ ξ i , H A ], with operators B and C defined similarly. The solution to Eqn. (6) for ξ i evaluated at the later time t +∆ t is formally (Goldstein 1980), but this exact expression is in general not analytic and not in a useful form. However Duncan et al. (1998) and Chambers (1999) show that the above is approximately which indicates that five actions that are to occur as the system of orbiting bodies are advanced one timestep ∆ t by the integrator. First ( i. ) the operator e B ∆ t/ 2 acts on ξ i ( t ), which increments ( i.e. kicks) particle i 's velocity v i by Eqn. (4) due to the system's interparticle forces with δt = ∆ t/ 2. Then ( ii. ) the e C ∆ t/ 2 operator acts on the result of substep ( i. ) and kicks the particle's spatial coordinates r i according to Eqn. (5) due to the central planet's motion about the barycenter. Then in substep ( iii. ) the e A ∆ t operation advances the particle along its unperturbed epicyclic orbit about the central planet during a full timestep ∆ t , with this substep is referred to below as the orbital 'drift' step. Step ( iv. ) is another coordinate kick δ r i and the last step ( v. ) is the final velocity kick. In a traditional symplectic N-body integrator the planet's oblateness is treated as a perturbation whose effect would be accounted for during steps ( i. ) and ( v. ) which provide an extra kick to a particle's velocity every timestep. Those kicks cause a particle in a circular orbit to have a tangential speed that is faster than the Keplerian speed by the fractional amount that is of order ∼ J 2 ( R/r ) 2 ∼ 3 × 10 -3 where J 2 /similarequal 0 . 016 is Saturn's second zonal harmonic and r/R ∼ 2 is a B ring particle's orbit radius r in units of Saturn's radius R . The particle's circular speed is super-Keplerian, and if its coordinates and velocities were to be converted to Keplerian orbit elements, its Keplerian eccentricity would also be of order e ∼ 3 × 10 -3 . This putative eccentricity should be compared to the observed eccentricity of Saturn's B ring, which is the focus of this study and is of order e ∼ 10 -4 , about 30 times smaller than the particle's Keplerian eccentricity. The main point is, that one does not want to use Keplerian orbit elements when describing a particle's nearly circular motions about an oblate planet because the Keplerian eccentricity is dominated by planetary oblateness whose effects obscures the ring's much smaller forced motions. To sidestep this problem, the following algorithm uses the epicyclic orbit elements of Borderies-Rappaport & Longaretti (1994) which provide a more accurate representation of an unperturbed particle's orbit about an oblate planet. Note that this use of epicyclic orbit elements effectively takes the effects of oblateness out of the integrator's velocity kick steps ( i. ) and ( v. ) and places oblateness effects in the integrator's drift step ( iii. ), which is preferable because the forces in the B ring that are due to oblateness are about ∼ 10 4 times larger than any satellite perturbation. The following details how these epicyclic orbit elements are calculated and are used to evolve the particle along its unperturbed orbit during the drift substep.", "pages": [ 5, 6, 7, 8 ] }, { "title": "2.2. epicyclic drift", "content": "This 2D model will track a particle's motions in the ring plane, so the particle's position and velocity relative to the central planet can be described by four epicyclic orbit elements: semimajor axis a , eccentricity e , longitude of periapse ˜ ω , and mean anomaly M . For a particle in a low eccentricity orbit about an oblate planet, the relationship between the particle's epicyclic orbit elements and its cylindrical coordinates r, θ and velocities v r , v θ are which are adapted from Eqns. (47-55) of Borderies-Rappaport & Longaretti (1994). These equations are accurate to order O ( e 2 ) and require e /lessmuch 1. Here Ω 0 ( a ) is the angular velocity of a particle in a circular orbit while κ 0 ( a ) is its epicyclic frequency and the frequency η 0 ( a ) is defined below, all of which are functions of the particle's semimajor axis a . Also keep in mind that when the following refers to the particle's orbit elements, it is the epicyclic orbit elements that are intended 1 , which are distinct from the osculating orbit elements that describe pure Keplerian motion around a spherical planet. But these distinctions disappear in the limit that the planet becomes spherical and the orbit frequencies Ω 0 , κ 0 , and η 0 all converge on the mean motion √ Gm 0 /a 3 , where G is the gravitational constant and m 0 is the central planet's mass; in that case, Eqns. (9) recover a Keplerian orbit to order O ( e 2 ). The three orbit frequencies Ω 0 , κ 0 , and η 0 appearing in Eqns. (9) are obtained from spatial derivatives of the oblate planet's gravitational potential Φ, which is where R p is the planet's effective radius, J 2 k is one of the oblate planet's zonal harmonics, and P 2 k (0) is a Legendre polynomial with zero argument. For reasons that will be evident shortly, these calculations will only preserve the J 2 term in the above sum, so and the orbital frequencies are ∣ where the additional frequency β 0 ( a ) is needed below. During the particle's unperturbed epicyclic drift phase its angular orbit elements M and ˜ ω advance during timestep ∆ t by amount where the frequencies Ω and κ in Eqns. (13) differ slightly from Eqns. (12) due to additional corrections that are of order O ( e 2 ): (Borderies-Rappaport & Longaretti 1994). Borderies-Rappaport & Longaretti (1994) also show that the above equations have three integrals of the motion: the particle's specific energy E , its specific angular momentum h , and its epicyclic energy I 3 . Those integrals are Advancing the particle along its epicyclic orbit require converting its cylindrical coordinates and velocities into epicyclic orbit elements. To obtain the particle's semimajor axis, solve the angular momentum integral h ( a ) = a 2 Ω 0 , which is quadratic in a so where g = ( rv θ ) 2 / 2 Gm 0 R p . Note though that if the J 4 and higher oblateness terms had been preserved in the planet's potential, then the angular momentum polynomial would be of degree 4 and higher in a , for which there is no known analytic solution. That equation could still be solved numerically, but that step would have to be performed for all particles at every timestep, which would slow the N-body algorithm so much as to make it useless. So only the J 2 term is preserved here, which nonetheless accounts for the effects of planetary oblateness in a way that is sufficiently realistic. To calculate the particle's remaining orbit elements, use Eqn. (15c) to obtain the I 3 integral which then provides its eccentricity via Then set x = e cos M and y = e cos M and solve Eqns. (9a) and (9d) for x and y : which then provides the mean anomaly via tan M = y/x . To summarize, the epicyclic drift step uses Eqns. (15-18) to convert each particle's cylindrical coordinates into epicyclic orbit elements. The particles' orbit frequencies Ω( a, e ) and κ ( a, e ) are obtained via Eqns. (12) and (14), and Eqns. (13) are then used to advance each particle's orbit elements M and ˜ ω during timestep ∆ t , with Eqns. (9) used to convert the particles' orbit elements back into cylindrical coordinates.", "pages": [ 8, 9, 10, 11 ] }, { "title": "2.3. velocity kicks due to the ring's internal forces", "content": "The N-body code developed here is designed to follow the dynamical evolution of all 360 · of a narrow annulus within a planetary ring, and it is intended to deliver accurate results quickly using a desktop PC. The most time consuming part of this algorithm is the calculation of the accelerations that the gravitating ring exerts on all of its particles, so the principal goal here is to design an algorithm that will calculate these accelerations with sufficient accuracy while using the fewest possible number of simulated particles.", "pages": [ 11 ] }, { "title": "2.3.1. streamlines", "content": "The dominant internal force in a dense planetary ring is its self gravity, and the representation of the ring's full 360 · extent via a modest number of streamlines provides a practical way to calculate rapidly the acceleration that the entire ring exerts on any one particle. A streamline is the closed path through the ring that is traced by those particles that share a common initial semimajor axis a . The simulated portion of the planetary ring will be comprised of N r discreet streamlines that are spaced evenly in semimajor axis a , with each streamline comprised of N θ particles on each streamline, so a model ring consists of N r N θ particles. Simulations typically employ N r ∼ 100 streamlines with N θ ∼ 50 particles along each streamline, so a typical ring simulation uses about five thousand particles. Note though that the assignment of particles to a given streamline is merely labeling; particles are still free to wander over time in response to the ring's internal forces: gravity, pressure, and viscosity. But as the following will show, the simulated ring stays coherent and highly organized throughout the run, in the sense that particles on the same streamline do not pass each other longitudinally, nor do adjacent streamlines cross. Because the simulated ring stays so highly organized, there is no radial or transverse mixing of the ring particles, and the particles will preserve over time membership in their streamline 2 .", "pages": [ 11 ] }, { "title": "2.3.2. ring self gravity", "content": "The concept of gravitating streamlines is widely used in analytic studies of ring dynamics (Goldreich & Tremaine 1979; Borderies et al. 1983a, 1986; Longaretti & Rappaport 1995; Hahn et al. 2009), and the concept is easily implemented in an N-body code. Because the simulated portion of the ring is narrow, its streamlines are all close in the radial sense. Consequently the gravitational pull that one streamline exerts on a particle is dominated by the nearest part of the streamline, with that acceleration being quite insensitive to the fact that the more distant and unimportant parts of the perturbing streamline are curved. So the perturbing streamline can be regarded as a straight and infinitely long wire of matter whose linear density is λ /similarequal m p N θ / 2 πa to lowest order in the streamline's small eccentricity e , where m p is the mass of a single particle. The gravitational acceleration that a wire of matter exerts on the particle is where ∆ is the separation between the particle and the streamline. However the particles in that streamline only provide N θ discreet samplings of a streamline that is after all slightly curved over larger spatial scales. So to find the distance to nearest part of the perturbing streamline, the code identifies at every timestep the three perturbing particles that are nearest in longitude to the perturbed particle. A second-degree Lagrange polynomial is then used to fit a smooth continuous curve through those three particles (Kudryavtsev & Samarin 2013), and this polynomial provides a convenient method for extrapolating the perturbing streamline's distance ∆ from the perturbed particle. This procedure is also illustrated in Fig. 3, which shows that the radial and tangential components of that acceleration are and to lowest order in the perturbing streamline's eccentricity e ' , where v ' r and v ' θ are the radial and tangential velocity components of that streamline. Equation (20) is then summed to obtain the gravitational acceleration that all other streamlines exerts on the particle. To obtain the gravity that is exerted by the streamline that the particle inhabits, treat the particle as if it resides in a gap in that streamline that extends midway to the adjacent r particles. The nearby portions of that streamline can be regarded as two straight and semiinfinite lines of matter pointed at the particle whose net gravitational acceleration is where ∆ + and ∆ -are the particle's distance from its neighbors in the leading (+) and trailing (-) directions. The radial and tangential components of that streamline's gravity are where v r , v θ are the perturbed particle's velocity components. A major benefit of using Eqn. (19) to calculate the ring's gravitational acceleration is that there is no artificial gravitational stirring. This is in contrast to a traditional Nbody model that would use discreet point masses to represent what is really a continuous ribbon of densely-packed ring matter. Those gravitating point masses then tug on each other in amounts that very rapidly in magnitude and direction as they drift past each other in longitude, and those rapidly varying tugs will quickly excite the simulated particles' dispersion velocity. As a result, the particles' unphysical random motions tend to wash out the ring's large-scale coherent forced motions, which is usually the quantity that is of interest. So, although Eqn. (19) is only approximate because it does not account for the streamline's curvature that occurs far away from a perturbed ring particle, Eqn. (19) is still much more realistic and accurate than the force law that would be employed in a traditional global N-body simulation of a planetary ring, which out of computational necessity would treat a continuous stream of ring matter as discreet clumps of overly massive gravitating particles.", "pages": [ 12, 13, 14 ] }, { "title": "2.3.3. ring pressure", "content": "A planetary ring is very flat and its vertical structure will be unresolved in this model, so a 1D pressure p is employed here. That pressure p is the rate-per-length that a streamline segment communicates linear momentum to the adjacent streamline orbiting just exterior to it, with that momentum exchange being due to collisions occurring among particles on adjacent streamlines. So for a small streamline segment of length δ/lscript that resides somewhere in the ring's interior, the net force on that segment due to ring pressure is δf = [ p ( r -∆) -p ( r )] δ/lscript since p ( r -∆) is the pressure or force-per-length exerted by the streamline that lies just interior and a distance ∆ away from segment δ/lscript , and p ( r ) is the force-per-length that segment δ/lscript exerts on the exterior streamline. And since force δf = A p δm where δm = λδ/lscript is the segment's mass, the acceleration on a particle due to ring pressure is since the ring's surface density σ = λ/ ∆. Formulating the acceleration in terms of pressure differences across adjacent streamlines is handy because the model can then easily account for the large pressure drop that occurs at a planetary ring's edge, which can be quite abrupt when the ring's edge is sharp. For a particle orbiting at the ring's innermost streamline, the acceleration there is simply A p = -p ( r ) /λ since there is no ring matter orbiting interior to it so p ( r -∆) = 0 there; likewise the acceleration of a particle in the ring's outermost streamline is A p = p ( r -∆) /λ . Pressure is exerted perpendicular to the streamline and hence it is predominantly a radial force, so by the geometry of Fig. 3 the radial component of the acceleration due to pressure is A p,r /similarequal A p while the tangential component A p,θ /similarequal -A p v r /v θ is smaller by a factor of e , where v r and v θ are the perturbed particle's radial and tangential velocities. This accounts for the pressure on the particle due to adjacent streamlines. The acceleration on the particle due to pressure gradients in the particle's streamline is simply A p = -( ∂p/∂θ ) / ( rσ ). This acceleration points in the direction of the particle's motion, so the radial and tangential components of that acceleration are A p,r /similarequal A p v r /v θ and A p,θ /similarequal A p . Acceleration due to pressure requires selecting an equation of state (EOS) that relates the pressure p to the ring's other properties, and this study will treat the ring as a dilute gas of colliding particles for which the 1D pressure is p = c 2 σ where c is the particles dispersion velocity. However alternate EOS exist for planetary rings, and that possibility is discussed in Section 4.2. A simple finite difference scheme is used to calculate the pressure gradient in Eqn. (23) in the vicinity of particle i in streamline j that lies at at longitude θ i,j . Lagrange polynomials are again used to evaluate the adjacent streamlines' planetocentric distances r i,j -1 and r i,j +1 along the particle's longitude θ i,j , so the pressure gradient at particle i in streamline j is ∣ where the pressures in the adjacent streamlines p i,j +1 and p i,j -1 are also determined by interpolating those quantities to the perturbed particle's longitude θ i,j . The surface density σ i,j in the vicinity of particle i in streamline j is determined by centering a box about that particle whose radial extent spans half the distance to the neighboring streamlines, so If however streamline j lies at the ring's inner edge where j = 0 then the surface density there is σ i, 0 = λ 0 / ( r i, 1 -r i, 0 ) while the surface density at the outermost j = N r -1 streamline is σ i,N r -1 = λ N r -1 / ( r i,N r -1 -r i,N r -2 ).", "pages": [ 14, 15 ] }, { "title": "2.3.4. ring viscosity", "content": "Viscosity has two types, shear viscosity and bulk viscosity. Shear viscosity is the friction that results as particles on adjacent streamlines collide as they flow past each other. The friction due to this shearing motion causes adjacent streamlines to torque each other, so shear viscosity communicates a radial flux of angular momentum through the ring. A particle on a streamline experiences a net torque and hence a tangential acceleration when there is a radial gradient in that angular momentum flux. And if there are additional spatial gradients in the ring's velocities that cause ring particles to converge towards or diverge away from each other, then these relative motions will cause ring particles to bump each other as they flow past, which transmits momentum through the ring via the pressure forces discussed above. However the ring particles' viscous bulk friction tends to retard those relative motions, and that friction results in an additional flux of linear momentum through the ring. Any radial gradients in that linear momentum flux then results in a radial acceleration on a ring particle. The 1D radial flux of the z component of angular momentum due to the ring's shear viscosity is derived in Appendix A: (see Eqn. A16) where ν s is the ring's kinematic shear viscosity and ˙ θ = v θ /r is the angular velocity. The quantity F is the rate-per-length that one streamline segment communicates angular momentum to the adjacent streamline orbiting just exterior, so the net torque on a streamline segment of length δ/lscript is δτ = [ F ( r -∆) -F ( r )] δ/lscript but δτ = rA ν,θ δm where δm = λδ/lscript so the tangential acceleration due to the ring's shear viscosity is Again this differencing approach is useful because it easily accounts for the large viscous torque that occurs at a ring's sharp edge since A ν,θ = -F ( r ) /λr at the ring's inner edge and A ν,θ = F ( r -∆) /λr at the ring's outer edge. Appendix B shows that the radial flux of linear momentum due to the ring's shear and bulk viscosity is (Eqn. B7) where ν b is the ring's bulk viscosity. This quantity is analogous to a 1D pressure so the corresponding acceleration is in the ring's interior and A ν,r = -G ( r ) /λ or A ν,r = G ( r -∆) /λ along the ring's inner or outer edges. To evaluate the partial derivatives that appear in the flux equations (26) and (28), Lagrange polynomials are again used to determine the angular and radial velocities ˙ θ and v r in the adjacent streamlines, interpolated at the perturbed particle's longitude, with finite differences used to calculate the radial gradients in those quantities.", "pages": [ 16, 17 ] }, { "title": "2.3.5. satellite gravity", "content": "All ring particles are also subject to each satellite's gravitational acceleration, A s = Gm s / ∆ 2 , where m s is the satellite's mass and ∆ is the particle-satellite separation. Satellites also feel the gravity exerted by all the ring particles, as well as the satellites' mutual gravitational attractions. And once all of the accelerations of each ring particle and satellite are tallied, each body is then subject to the corresponding velocity kicks of steps ( i. ) and ( v. ) that are described just below Eqn. (8).", "pages": [ 17 ] }, { "title": "2.4. tests of the code", "content": "The N-body integrator developed here is called epi int , which is shorthand for epicyclic integrator , and the following briefly describes the suite of simulations whose known outcomes are used to test all of the code's key parts. Forced motion at a Lindblad resonance: numerous massless particles are placed in circular orbits at Mimas' m = 2 inner Lindblad resonance. In this test, Mimas' initially zero mass is slowly grown to its current mass over an exponential timescale τ s = 1 . 6 × 10 4 ring orbits, which excites adiabatically the ring particle's forced eccentricities to levels that are in excellent agreement with the solution to the linearized equations of motion, Eqn. (42) of Goldreich & Tremaine (1982). Similar results are also obtained for the particle's response to Janus' m = 7 inner Lindblad resonance, which is responsible for confining the outer edge of Saturn's A ring. These simulations test the implementation of the integrator's kick-step-drift scheme as well as the satellite's forcing of the ring. Precession due to oblateness: this simple test confirms that the orbits of massless particles in low eccentricity orbits precess at the expected rate, ˙ ˜ ω ( a ) = Ω -κ = 3 2 J 2 ( R p /a ) 2 Ω( a ), due to planetary oblateness J 2 . Ringlet eccentricity gradient and libration: when a narrow eccentric ringlet is in orbit about an oblate planet, dynamical equilibrium requires the ringlet to have a certain eccentricity gradient so that differential precession due to self-gravity cancels that due to oblateness. And when the ringlet is composed of only two streamlines then this scenario is analytic, with the ringlet's equilibrium eccentricity gradient given by Eqn. (28b) of Borderies et al. (1983b). So to test epi int 's treatment of ring self-gravity, we perform a suite of simulations of narrow eccentric ringlets that have surface densities 40 < σ < 1000 gm/cm 2 with initial eccentricity gradients given by Eqn. (28b), and integrate over time to show that these pairs of streamlines do indeed precess in sync with no relative precession, as expected, over runtimes that exceed of the timescale for massless streamlines to precess differentially. And when we repeat these experiments with the ringlets displaced slightly from their equilibrium eccentricity gradients, we find that the simulated streamlines librate at the frequency given by Eqn. (30) of Borderies et al. (1983b), as expected. Density waves in a pressure-supported disk: this test examines the model's treatment of disk pressure, and uses a satellite to launch a two-armed spiral density wave at its m = 2 ILR in a non-gravitating pressure supported disk. The resulting pressure wave has a wavelength and amplitude that agrees with Eqn. (46) of Ward (1986), as expected. Viscous spreading of a narrow ring: in this test epi int follows the radial evolution of an initially narrow ring as it spreads radially due to its viscosity, and the simulated ring's surface density σ ( r, t ) is in excellent agreement with the expected solution, Eqn. (2.13) of Pringle (1981).", "pages": [ 17, 18 ] }, { "title": "3. Simulations of the Outer Edge of Saturn's B Ring", "content": "The semimajor axis of the outer edge of Saturn's B ring is a edge = 117568 ± 4 km, and that edge lies ∆ a 2 = 12 ± 4 km exterior to Mimas' m = 2 inner Lindblad resonance (ILR) (Spitale & Porco 2010, hereafter SP10). Evidently Mimas' m = 2 ILR is responsible for confining the B ring and preventing it from viscously diffusing outwards and into the Cassini Division. Mimas' m = 2 ILR excites a forced disturbance at the ring-edge whose radiuslongitude relationship r ( θ ) is expected to have the form r ( θ, t ) = a edge -R m cos m ( θ -˜ ω m ) where R m is the epicyclic amplitude of the mode whose azimuthal wavenumber is m and whose orientation at time t is given by the angle ˜ ω m ( t ). This forced disturbance is expected to corotate with Mimas' longitude, and such a pattern would have a pattern speed ˙ ˜ ω m = d ˜ ω m /dt that satisfies ˙ ˜ ω m = Ω s where Ω s is satellite Mimas' angular velocity. SP10 have analyzed the many images of the B ring's edge that have been collected by the Cassini spacecraft, and they show that this ring-edge does indeed have a forced m = 2 shape that corotates with Mimas as expected. But they also show that the B ring's edge has an additional free m = 2 pattern that rotates slightly faster than the forced pattern. SP10 also detect two additional modes, a slowly rotating m = 1 pattern as well as a rapidly rotating m = 3 pattern. These findings are confirmed by stellar occulation observations of the B ring's outer edge that also detect additional lower-amplitude m = 4 and m = 5 modes (Nicholson et al. 2012). The following will use the N-body model to investigate the higher amplitude m = 1 , 2, and 3 modes seen at the B ring's edge. But keep in mind that only the m = 2 forced pattern has a known driver, namely, Mimas' m = 2 ILR, while the nature of the perturbation that launched the other three free modes in the B ring is quite unknown. So to study the B ring's behavior when those free modes are present, an admittedly ad hoc method is used. Specifically, the simulated ring particles' initial conditions are constructed in a way that plants a free m = 1 , 2, or 3 pattern at the simulated ring's edge at time t = 0. The N-body integrator then advances the system over time, which then reveals how those free patterns evolve over time. And to elucidate those findings most simply, the following subsections first consider the B ring's m = 1, 2, and 3 patterns in isolation. All simulations use a timestep ∆ t = 0 . 2 / 2 π = 0 . 0318 orbit periods, so there are 31.4 timesteps per orbit of the simulated B ring, and nearly all simulations use oblateness J 2 = 0 . 01629071, which is the same value we used in previous work (Hahn et al. 2009). And lastly, these simulations also zero the viscous acceleration that is exerted at the simulated B ring's innermost and outermost streamlines, to prevent them from drifting radially due to the ring's viscous torque. This is in fact appropriate for the simulation's innermost streamline, since in reality the viscous torque from the unmodeled part of the B ring should deliver to the inner streamline a constant angular momentum flux F that it then communicates to the adjacent streamline, so the viscous acceleration A ν,θ ∝ ∂F/∂r at the simulation's inner edge really should be zero. But zeroing the viscous acceleration of outer streamline might seem like a slight-of-hand since it should be A ν,θ = F/λr according Section 2.3.4. But setting A ν,θ = 0 is done because, if not, then the outermost streamline will slowly but steadily drifts radially outwards past Mimas' m = 2 ILR, which also causes that streamline's forced eccentricity to slowly and steadily grow as the streamline migrates. This happens because the model does not settle into a balance where the ring's positive viscous torque on its outermost streamline is opposed by a negative torque exerted by the satellite's gravity. We also note that the semi-analytic model of this resonant ring-edge, which is described in Hahn et al. (2009), also had the same difficulty in finding a torque balance. So to sidestep this difficulty, this model zeros the viscous acceleration at the outermost streamline, which keeps its semimajor axis static as if it were in the expected torque balance. This then allows us to compare simulations to the B ring's forced m = 2 pattern to that measured by the Cassini spacecraft. The validity of this approximation is also assessed below in Section 4.1.", "pages": [ 18, 19, 20 ] }, { "title": "3.1. the forced and free m = 2 patterns", "content": "SP10 detect a forced m = 2 pattern at the B ring's outer edge that has an epicyclic amplitude R 2 = 34 . 6 ± 0 . 4 km, and that forced pattern corotates with the satellite Mimas. They also detect a free pattern whose epicyclic amplitude is 2 . 7 km larger, so the forced and free patterns are nearly equal in amplitude, and the free pattern rotates slightly faster than the forced pattern by ∆ ˙ ˜ ω 2 = 0 . 0896 ± 0 . 0007 degrees/day (SP10). The radius-longitude relationship for a ring-edge that experiences these two modes can be written where R 2 is the epicyclic amplitude of the forced pattern that corotates with Mimas whose longitude is θ s ( t ) at time t , and ˜ R 2 is the epicyclic amplitude of the free pattern with ˜ ω 2 ( t ) being the free pattern's longitude. The N-body integrator epi int is used to simulate the forced and free m = 2 patterns that are seen at the outer edge of the B ring, for simulated rings having a variety of initial surface densities σ 0 . These simulations use N r = 130 streamlines that are distributed uniformly in the radial direction with spacings ∆ a = 5 . 13 km, so the radial width of the simulated portion of the B ring is w = ( N r -1)∆ a = 662 km. Each streamline is populated with N θ = 50 particles that are initially distributed uniformly in longitude θ and in circular coplanar orbits. These simulations use a total of N r N θ = 6500 particles, which is more than sufficient to resolve the m = 2 patterns seen here. These systems are evolved for t = 41 . 5 years, which corresponds to 3 . 2 × 10 4 orbits, and is sufficient time to see the simulation's slightly faster free m = 2 pattern lap the forced m = 2 pattern several times. The execution time for these high resolution, publication-quality simulations is 1.5 days on a desktop PC, but sufficiently useful preliminary results from lower-resolution simulations can be obtained in just a few hours. The B ring's viscosity is unknown, so these simulations will employ a value for the kinematic shear viscosity ν s and bulk viscosity ν b that are typical of Saturn's A ring, ν s = ν b = 100 cm 2 /sec (Tiscareno et al. 2007). The simulated particles' dispersion velocity c is also chosen so that the ring's gravitational stability parameter Q = cκ/πGσ 0 = 2, since Saturn's main rings likely have 1 /lessorsimilar Q /lessorsimilar 2 (Salo 1995). Mimas' mass is m s = 6 . 5994 × 10 -8 Saturn masses, and its semimajor axis a s is chosen so that its m = 2 inner Lindblad resonance lies ∆ a res = 12 . 2 km interior to the simulated B ring's outer edge. This model only accounts for the J 2 = 0 . 01629071 part of Saturn's oblateness, so the constraint on the resonance location puts the simulated Mimas at a s = 185577 . 0 km, which is 38 km exterior to its real position. Starting the ring particles in circular orbits provides an easy way to plant equalamplitude free and forced m = 2 patterns in the ring. This creates a free m = 2 pattern that at time t = 0 nulls perfectly the forced m = 2 pattern due to Mimas. However the free pattern rotates slightly faster than the forced pattern, so the ring's epicyclic amplitude varies between near zero and ∼ 2 R 2 as the rotating patterns interfere constructively or destructively over time. This behavior is illustrated in Fig. 4 which shows results from a simulation of a B ring whose undisturbed surface density is σ 0 = 280 gm/cm 2 . The wire diagrams show the ring's streamlines via radius versus longitude plots, with dots indicating individual particles, and the adjacent grayscale map shows the ring's surface density at that instant. Figure 4 shows snapshots of the system at five distinct times that span one cycle of the ring's circulation: at time t = 26 . 4 yr when the ring's outermost streamline is nearly circular due to the forced and free patterns being out of phase by nearly 180 · /m = 90 · and interfering destructively, to time t = 28 . 2 yr when the forced and free patterns are in phase and interfere constructively, to nearly circular again at time t = 30 . 0 yr. The circulation cycle seen in Fig. 4 repeats for the duration of the integration, which spans about 10 cycles. The gray lines in Fig. 4 show the semimajor axes a of all particles on each streamline; note that all particles on a given streamline preserve a common semimajor axes, and this is also true of their eccentricities e . In the simulations shown here, the two orbit elements a and e do not vary with the particle's longitude θ . This however is distinct from the particles' angular orbit elements M and ˜ ω , which do vary linearly with longitude θ along each streamline. Recall that the epi int code does not in any way force or require particles to inhabit a given streamline. The streamline concept is only used when calculating the forces that all of the ring's streamlines exert on each particle, which the symplectic integrator then uses to advance these particles forwards in time. Although a particle's e and a are in principle free to drift away from that of the other streamline-members, that does not happen in the simulations shown here; evidently the particles' a and e evolve slowly in the orbit-averaged sense, with that time-averaged evolution being independent of longitude θ . This accounts for why all particles on the same streamline have the same evolution in a and e . This time-averaged evolution is also a standard assumption that is routinely invoked in analytic models of planetary rings (see cf . Goldreich & Tremaine 1979; Borderies et al. 1986; Hahn et al. 2009), and the simulations shown here confirm the validity of that assumption. A suite of seven B ring simulations is performed for rings whose undisturbed surface densities range over 120 ≤ σ 0 ≤ 360 gm/cm 2 . Results are summarized in Fig. 5 which shows the forced epicyclic amplitude R 2 (solid curve) and the free epicyclic amplitude ˜ R 2 (dashed curve) from each simulation. These amplitudes are obtained by fitting Eqn. (30) to the simulated B ring's outermost streamline assuming that the free pattern there rotates at a constant rate, ˜ ω 2 ( t ) = ˜ ω 0 + ˙ ˜ ω 2 t where ˜ ω 0 is the free pattern's angular offset at time t = 0 and ˙ ˜ ω 2 is the free mode's pattern speed. Equation (30) provides an excellent representation of the ring-edge's behavior over time, and that equation has four parameters R 2 , ˜ R 2 , ˜ ω 0 , and ˙ ˜ ω 2 that are determined by least squares fitting. The observed epicyclic amplitude of the B ring's forced m = 2 component is R 2 = 34 . 6 ± 0 . 4 km (SP10), and the gray bar in Fig. 5 indicates that the outer edge of the B ring has a surface density of about σ 0 = 195 gm/cm 2 . And if we naively assume that the ring's surface density is everywhere the same, then its total mass of Saturn's B ring is about 90% of Mimas' mass. Figure 5 also shows that the amplitude of the forced pattern R 2 gets larger for rings that have a smaller surface density σ 0 , due to the ring's lower inertia, with the forced response varying roughly as R 2 ∝ σ -0 . 67 0 . This also makes lighter rings more difficult to simulate, because their larger epicyclic amplitudes also causes the ring's streamlines to get more bunched up at periapse. For instance in the σ 0 = 280 gm/cm 2 simulation of Fig. 4, the ring's edge at longitudes θ = θ s and θ = θ s ± π are overdense by a factor of 3 at time t = 28 . 2 yr, which is when the force and free patterns add constructively. Streamline bunching in lighter rings is even more extreme, which is also more problematic, because streamlines that are too compressed can at times cross in these overdense sites, and the simulated ring's subsequent evolution becomes unreliable. To avoid the streamline crossing that occurs in simulations of lower surface density, the model also grows the mass of Mimas exponentially over the timescale τ s that takes values of 0 . 41 ≤ τ s ≤ 6 . 2 years, with faster satellite growth ( τ s = 0 . 41 yrs or 320 B ring orbits) occurring in simulations of a heavy B ring having σ 0 ≥ 280 gm/cm 2 and slower growth ( τ s = 6 . 2 yrs or 4800 B ring orbits) for the lighter σ 0 ≤ 240 gm/cm 2 ring simulations. The satellite growth timescale τ s controls the amplitude of the free pattern ˜ R 2 , with the ring having a smaller free epicyclic amplitude ˜ R 2 when τ s is larger; see the dashed curve in Fig. 5. Indeed, when the satellite grows over a timescale τ s /greatermuch 6 . 2 yrs ( i.e. τ s /greatermuch 4800 orbits), the ring responds adiabatically to forcing by the slowly growing Mimas, and shows only a forced m = 2 pattern that corotates with Mimas, with the free m = 2 pattern having a negligible amplitude. Consequently, only the σ 0 = 280 , 320, and 360 gm/cm 2 simulations in Fig. 5 are faithful in their attempt to reproduce a B ring whose free epicyclic amplitude ˜ R 2 is slightly larger than the forced amplitude R 2 . However the lower-surface density simulations have free patterns whose amplitudes are smaller than the forced patterns, and these simulated rings have outer edges whose longitude of periapse librate about Mimas' longitude, rather than circulate. Also of interest here is the so-called radial depth of the m = 2 disturbance, ∆ a e/ 10 , which is defined as the semimajor axis separation between the ring's outer edge and the streamline whose mean eccentricity is one-tenth that of the edge's eccentricity. For these m = 2 simulations the radial depth is ∆ a e/ 10 = 154km, so the radial width of the simulated part of the ring is w = 4 . 3∆ a e/ 10 .", "pages": [ 20, 21, 25, 26, 27 ] }, { "title": "3.1.1. sensitivity to resonance location and other factors", "content": "The surface density σ 0 that is inferred from the amplitude of the ring's forced motion R 2 is very sensitive to the uncertainty in the ring's semimajor axis, which is δa edge . For example, when the B ring is simulated again but with its outer edge instead extending further out by δa edge = 4 km, those simulations show that the ring's forced amplitude R 2 is larger by about 6 km, which requires increasing σ 0 by δσ 0 = 60 gm/cm 2 so that the simulated R 2 is in agreement with the observed value. Similarly, when the simulated ring's edge is moved inwards by δa edge = 4 km, the forced amplitude R 2 is smaller and the ring's surface density σ 0 must be decreased by δσ 0 to compensate. So the surface density of the B ring-edge is σ 0 = 195 ± 60 gm/cm 2 , and this value represents the mean surface density of outer ∆ a e/ 10 /similarequal 150km that is most strongly disturbed by Mimas' m = 2 resonance. These results are also in excellent agreement with the semi-analytic model of Hahn et al. (2009), which calculated only the ring's forced motion. However these results are very insensitive to the model's other main unknown, the ring's viscosity ν . For instance, when we re-run the σ 0 = 200 gm/cm 2 simulation with the ring's shear and bulk viscosities increases as well as decreased by a factor of 10, we obtain the same forced response R 2 . So these findings are insensitive to range of ring viscosities considered here, 10 < ν < 1000 cm 2 /sec.", "pages": [ 27, 28 ] }, { "title": "3.1.2. free m = 2 pattern", "content": "The dotted curve in Fig. 6 shows the simulations' free m = 2 pattern speeds ˙ ˜ ω 2 , which is also sensitive to the ring's undisturbed surface density σ 0 . The purpose of this subsection is to illustrate how a free normal mode can also be used to determine the ring's surface density. Although these result will not be as definitive as the value of σ 0 that was inferred from the ring's forced pattern, due to a greater sensitivity to the observational uncertainties, the following illustrates an alternate technique that in principle can be used to infer the surface density of other rings, such as the many narrow ringlets orbiting Saturn that also exhibit free normal modes. But first note the models' large discrepancy with the observed free m = 2 pattern speed reported in SP10, which is the upper horizontal bar in Fig. 6. This discrepancy is not due to the δa edge = ± 4km uncertainty in the ring-edge's semimajor axis, which makes the simulated ring particles' mean angular velocity uncertain by the fraction δ Ω / Ω = 1 . 5 δa edge /a edge /similarequal 0 . 005%. We find empirically that the simulations' pattern speeds are also uncertain by this fraction, so δ ˙ ˜ ω 2 /similarequal 0 . 02 deg/day, which is the vertical extent of the gray band around the simulated data in Fig. 6. Rather, this discrepancy is indirectly due to the absence of the J 4 and higher terms from the N-body simulations. To demonstrate this, repeat the σ 0 = 200 gm/cm 2 simulation with J 2 boosted slightly by factor f /star = 1 . 0395013 so that the second zonal harmonic is J /star 2 = f /star J 2 = 0 . 016934294. This increases the simulated B ring-edge's angular velocity slightly to Ω edge = 758 . 8824 deg/day, which is in fact the ring's true angular velocity at a = a edge when the higher order J 4 and J 6 terms are also accounted for 3 . And since Saturn's gravitational force there is a edge Ω 2 edge , this means that Saturn's gravity on the simulated particles at r = a edge is in fact the true value. Note that boosting J 2 to the slightly larger value J /star 2 also requires bringing the simulated Mimas inwards and just interior to its true semimajor axis by 2km. Which speeds up both the forced and free pattern speeds, and is why this simulation's free m = 2 pattern speed ˙ ˜ ω 2 , which is the cross in Fig. 6, is in better agreement with the observed pattern speed. So the discrepancy between all the other simulated and observed pattern speeds ˙ ˜ ω 2 is due to those models' not accounting for the additional gravity that is due to the J 4 and higher terms in Saturn's oblate figure. Compensating for the absence of those oblateness effects requires altering the simulated satellite's orbits slightly, which in turn alters the forced and free pattern speeds slightly. But the following will show that these two patterns' relative speeds are quite insensitive to the particular value of J 2 and the absence of the J 4 and higher terms. The best way to compare simulated to observed free m = 2 patterns is to consider the free m = 2 pattern speed relative to the forced pattern speed, which is the satellite's mean angular velocity Ω sat . That frequency difference is ∆ ˙ ˜ ω 2 = ˙ ˜ ω 2 -Ω sat , and is plotted versus ring surface density σ 0 in Fig. 7. Black dots are from the simulation and the light gray band indicates the δ ˙ ˜ ω 2 /similarequal 0 . 02 deg/day spread that results from the δa edge = ± 4 km uncertainty in the ring-edge's semimajor axis. The relatively large uncertainty in a edge means that one can only conclude from Fig. 7 that σ 0 /lessorsimilar 210 gm/cm 2 . If however the uncertainty in a edge were instead δa edge = ± 1 km, then the uncertainty in ∆ ˙ ˜ ω 2 would be 4 times smaller (darker gray band), which would have allowed us to determine the ring surface density with a much smaller uncertainty of only ± 20 gm/cm 2 . The lesson here is that if one wishes to use models of free patterns to infer σ 0 in, say, narrow ringlets, then one will likely need to know the ring-edge's semimajor axis with a precision of δa edge /similarequal ± 1 km. The cross in Fig. 7 indicates that the the free m = 2 pattern speed relative to the forced is unchanged when Saturn's oblateness is boosted to J /star 2 . And to demonstrate that this kind of plot is rather insensitive to oblateness effects, the white dot in Fig. 7 shows that these relative pattern speeds change only very slightly even when J 2 is set to zero. Note though that there will be instances where there is no forced mode with which to compare pattern speeds. In that case it will be convenient to convert the free pattern speed ˙ ˜ ω m = Ω ps into a radius by solving the Lindblad resonance criterion for the resonance radius r = a m , where κ ( r ) is the ring particles' epicyclic frequency (Eqn. 12b), and /epsilon1 = +1( -1) at an inner (outer) Lindblad resonance. So for the simulated B ring's free m = 2 mode, Eqn. (31) is solved for the radius r = ˜ a 2 of the /epsilon1 = +1 inner Lindblad resonance that is associated with this mode. That quantity is to be compared to a nearby reference distance, which in this case would be the semimajor axis of the B ring's outer edge a edge . Results are shown in Fig. 8, which shows the simulations' distance from the B ring's outer edge to the free m = 2 pattern's ILR , ∆ a 2 = a edge -˜ a 2 , plotted versus ring surface density σ 0 . Heavier rings have a faster pattern speeds (Fig. 6-7), and so the pattern's resonance resides at a higher orbital frequency Ω( r ) and thus must lie further inwards of the ring's outer edge in order to satisfy the resonance condition, Eqn. (31). Figure 8 has the same information content as Fig. 7, which is why it also tells us that the B ring's outer edge has σ 0 /lessorsimilar 210 gm/cm 2 . However a plot like Fig. 8 will also provide the best way to interpret the B ring's free m = 3 mode, which is examined below in subsection 3.2. Lastly, note that the free m = 2 patterns seen in these simulations persist for 3 × 10 4 orbits or 40 years without any sign of damping, despite the ring's viscosity ν = 100 cm 2 /sec. This is illustrated in Fig. 9, which plots the ring-edge's epicyclic amplitude over time for the nominal σ 0 = 200 gm/cm 2 simulation. Indeed we have also rerun this simulation using a viscosity that is ten times larger and still saw no damping. These experiments reveal a possibly surprising result, that a free pattern can persist at a ring-edge for a considerable length of time, likely hundreds of years or longer, and Section 4.1 will show that this longevity is due to the viscous forces being several orders or magnitude weaker than the ring's other interval forces. So one possible interpretation of the free modes seen at the B ring and at other ring edges is that they are relics from past disturbances in Saturn's ring that may have happened hundreds or more years ago. This possibility is discussed further in Section 4.3.", "pages": [ 28, 30, 31, 32 ] }, { "title": "3.2. the free m = 3 pattern", "content": "The B ring's free m = 3 mode has an epicyclic amplitude of ˜ R 3 = 11 . 8 ± 0 . 2 km, a pattern speed ˙ ˜ ω 3 = 507 . 700 ± 0 . 001 deg/day, and the inner Lindblad resonance associated with this pattern speed lies ∆ a 3 = 24 ± 4 km interior to the B ring's outer edge (SP10). To excite a free m = 3 pattern at the ring-edge, place a fictitious satellite in an orbit that has an m = 3 inner Lindblad resonance ∆ a 3 = 24 km interior to the ring's outer edge. Noting that the satellite Janus happens to have an m = 3 resonance in the vicinity, about 2000 km inwards of the B ring's edge, these simulations use a Janus-mass satellite to perturb the simulated ring for about 1650 orbits (about 2 years), which excites an m = 3 pattern at the ring's outer edge. The satellite is then removed from the system, which converts the pattern into a free normal mode, and epi int is then used to evolve the now unperturbed ring for another 1 . 8 × 10 4 orbits (about 23 years). Figure 10 plots the ring-edge's epicyclic amplitude, where it is shown that the free mode persists at the B ring's outer edge, undamped over time, despite the simulated ring's viscosity of ν = 100 cm 2 /sec. A suite of such B ring simulations is performed, with ring surface densities 120 ≤ σ 0 ≤ 360 gm/cm 2 and all other parameters identical to the nominal model of Section 3.1 except where noted in Fig. 11 caption. The pattern speed Ω ps = ˙ ˜ ω 3 of the m = 3 normal mode is then extracted from each simulation, with those speeds again being slightly faster in the heavier rings. Those pattern speeds are then inserted into Eqn. (31) which is solved for the radius of the inner Lindblad resonance ˜ a 3 , each of which lies a distance ∆ a 3 = a edge -˜ a 3 inwards of the ring's outer edge, and those distances are plotted in Fig. 11 versus ring surface density σ 0 . The simulated distances ∆ a 3 are compared to the observed edge-resonance distance reported in SP10, which indicates a ring surface density 160 ≤ σ 0 ≤ 310 gm/cm 2 . This finding is consistent with the the results from the m = 2 patterns, but this constraint on σ 0 is again rather loose due to the δa edge = ± 4 km uncertainty in the ring-edge's semimajor axis. But our purpose here is to show how one might use models of free normal modes to infer the surface density of other rings and narrow ringlets, which again will likely require knowing the ring-edge's semimajor axis to ± 1 km or better. Also note that the radial depth of this m = 3 disturbance is ∆ a e/ 10 = 50 km, about three times smaller than the radial depth of the m = 2 disturbance.", "pages": [ 35 ] }, { "title": "3.3. the free m = 1 pattern", "content": "The B ring's free m = 1 mode has an epicyclic amplitude of ˜ R 1 = 20 . 9 ± 0 . 4 km and a pattern speed ˙ ˜ ω 1 = 5 . 098 ± 0 . 003 deg/day that is slightly faster than the local precession rate, and the inner Lindblad resonance that is associated with this pattern speed lies ∆ a 1 = 253 ± 4 km interior to the B ring's outer edge (SP10). Several simulations of the B ring's m = 1 pattern are evolved for model rings having surface densities of 120 ≤ σ 0 ≤ 360 gm/cm 2 . To excite the m = 1 pattern at the simulated ring's edge, again arrange a fictitious satellite's orbit so that its m = 1 ILR lies ∆ a 1 = 253 km interior to the B ring's edge, which is the site where the resonance condition (Eqn. 31) is satisfied when the satellite's mean angular velocity matches the ring particles' precession rate, Ω s = ˙ ˜ ω = Ω ps . The simulated ring is perturbed by a satellite whose mass is about 20% that of Mimas, for 1 . 6 × 10 4 orbits or 21 years, which excites a forced m = 1 pattern at the ring's edge that corotates with the satellite. The satellite is then removed, which converts the forced m = 1 pattern into a free pattern, and the ring is evolved for another 6 . 4 × 10 4 orbits or 83 years. For each simulation the free pattern speed is measured, and Eqn. (31) is then used to convert the free pattern speed into a resonance radius ˜ a 1 , which is displayed in Fig. 12 that shows that resonance's distance from the ring's outer edge, ∆ a 1 = a edge -˜ a 1 . As the figure shows, the free m = 1 pattern rotates slightly faster in the heavier ring and thus the associated m = 1 ILR must lie further inwards in order to satisfy the resonance condition Ω ps = ˙ ˜ ω = 3 2 J 2 ( R p /a ) 2 Ω. Again there is no damping of the free m = 1 pattern, which stays localized at the ring's outer edge over the simulation's 83 yr timespan, despite the simulated ring's viscosity ν = 100 cm 2 /sec. The radial depth of this m = 1 disturbance is much greater than the others, ∆ a e/ 10 = 614 km, which is about four times larger than the m = 2 disturbance. Comparing Fig. 12 to Figs. 8 and 11 also shows that the LR associated with the m = 1 disturbance lies about 10 times further from the ring-edge than the m = 1 and m = 2 resonances. Which is why the m = 1 simulation uses streamlines whose width ∆ a is ∼ 10 × larger, since a wider portion of the B ring-edge must be simulated in order to capture this disturbances' deeper reach into the B ring. Note also that the ± 4 km uncertainty in this resonance's position relative to the B ring edge, which is entirely due to the uncertainty in the B ring-edge's semimajor axis, is in this case relatively small. Which is why the ring's free m = 1 mode can also be used to probe its surface density with some precision (unlike the free m = 2 and m = 3 modes), and is consistent with a B ring surface density of σ 0 /similarequal 200 gm/cm 2 ,", "pages": [ 38 ] }, { "title": "3.4. convergence tests", "content": "A number of simulations have also been performed, which repeat the ring simulations using various particle numbers N r and N θ and various widths w of the simulated ring. We find that the results reported here do not change significantly when the simulated ring is populated densely with enough particles, and when the radial width of the simulated B ring is sufficiently wide. Those convergence tests reveal that the number of particles along each streamline must satisfy N θ ≥ 20 m , that the radial width of each streamline should satisfy ∆ a ≤ 0 . 04∆ a e/ 10 , and that the total width of the simulated ring should satisfy w > 4∆ a e/ 10 . All of the simulations reported here satisfy these requirements.", "pages": [ 40 ] }, { "title": "4. Discussion", "content": "This section re-examines the model's treatment of viscous effects at the ring's edge, and also describes related topics that will be considered in followup work.", "pages": [ 40 ] }, { "title": "4.1. the ring's internal forces", "content": "Figure 13 plots the accelerations that the ring's internal forces-gravity, pressure, and viscosity-exert on each ring particle. These accelerations are from the nominal σ 0 = 200 gm/cm 2 simulation that is described in Figs. 5-9, and these accelerations are plotted versus each particle's distance from the ring's edge, so those forces obviously get larger closer to the ring's disturbed outer edge. But the main point of Fig. 13 is that the ring's self gravity is the dominant internal force in the ring, exceeding the pressure force by a factor of ∼ 100 at the ring's outer edge and by a larger factor elsewhere. Those pressure forces are also about ∼ 10 × larger than the ring's viscous forces. But recall that those simulations had zeroed the viscous acceleration that the ring exerts on its outermost streamline (Section 3), when that acceleration should instead be A ν,θ = F/λr as indicated by the large blue dot at the right edge of Fig. 13. Note though that the neglected viscous acceleration of the ring's edge is still about ∼ 1000 × smaller than that due to ring gravity and ∼ 10 × smaller than that due to ring pressure. So this justifies neglecting, at least for the short-term t ∼ 100 yr simulations considered here, the much smaller viscous forces at the ring's outer edge. Nonetheless this study's neglect of the small viscous force at the ring's outer edge implies that this model does not yet account for the B ring's radial confinement by Mimas' m = 2 ILR. So there appears to be some missing physics that will be necessary if one is interested in the ring's resonant confinement or the ring's long-term evolution over t /greatermuch 100 yr timescales. The suspected missing physics is described below.", "pages": [ 40, 41 ] }, { "title": "4.1.1. unmodeled effects: the viscous heating of a resonantly confined ring-edge", "content": "The model's inability to confine the B ring's outer edge at Mimas' m = 2 ILR may be a consequence of the ring's kinematic viscosity ν being treated here as a constant parameter everywhere in the simulated ring. Although treating ν as a constant is a simple and plausible way to model the effects of the ring's viscous friction, it might not be adequate or accurate if one wishes to simulate the resonant confinement of a planetary ring. This is because the ring's viscosity transports both energy and angular momentum radially outwards through the ring. So if the ring's outer edge is to be confined by a satellite's m th Lindblad resonance, the satellite must absorb the ring's outward angular momentum flux, which it can do by exerting a negative gravitational torque at the ring's edge. But Borderies et al. (1982) show via a simple Jacobi-integral argument that resonant interactions only allow the satellite to absorb but a fraction of the energy that viscosity delivers to the ring-edge. Consequently the ring's viscous friction still delivers some orbital energy to the ring-edge where it accumulates and heats up the ring particles' random velocities c . And if collisions among particles are the main source of the ring's viscosity, then ν s /similarequal c 2 τ/ 2Ω(1 + τ 2 ) where τ ∝ σ is the ring's optical depth (Goldreich & Tremaine 1982). In this case viscous heating would increase c as well as ν s at the ring's edge. The enhanced dissipation there should also increase the angular lag φ between the ring-edge's forced pattern and the satellite's longitude (see Eqn. 83b of Hahn et al. 2009). Which will also be important because the gravitational torque that the satellite exerts on the ring-edge varies as sin φ (Hahn et al. 2009), and that torque needs to be boosted if the satellite is to confine the spreading ring. To model this phenomenon properly, the epi int code also needs to employ an energy equation, one that accounts for how viscous heating tends to increase the ring particles' dispersion velocity c and viscosity ν s nearer the ring's edge. The increased dissipation and the resulting orbital lag will allow the satellite to exert a greater torque on the ring which, we suspect, will enable the satellite to resonantly confine the simulated ring's outer edge. The derivation of this energy equation and its implementation in epi int are ongoing, and those results will be reported on in a followup study.", "pages": [ 43 ] }, { "title": "4.2. an alternate equation of state", "content": "The EOS adopted here is appropriate for a dilute gas of colliding ring particles whose mutual separations greatly exceed their sizes. This should be regarded as a limiting case since ring particles can of course be packed close to each other in the ring. But Borderies et al. (1985) consider the other extreme limiting case, with close-packed particles that reside shoulder to shoulder in the ring. In that case the ring is expected to behave as an incompressible fluid whose volume density ρ = σ/ 2 h stays constant. So when some perturbation causes ring streamlines to bunch up and increases the ring's surface density σ , the ring's vertical scale height h also increases as ring particles are forced to accumulate along the vertical direction. This in turn increases the ring's pressure due to the larger gravitational force along the vertical direction. Borderies et al. (1985) show that infinitesimal density waves in an incompressible disk are unstable and grow in amplitude over time. This phenomenon is related to the viscous overstability, and Longaretti & Rappaport (1995) show that it can distort a narrow eccentric ringlet's streamlines in a way that accounts for its m = 1 shapes. Borderies et al. (1985) also suggest that unstable density waves can be trapped between a Lindblad resonance and the B ring's outer edge, which might explain the normal modes seen there, and Spitale & Porco (2010) use this concept to estimate the ring's surface density there. But keep in mind that this instability only occurs when the ring particles are densely packed to the point of being incompressible, which requires the ring to be very thin and dynamically cold. We have shown here that the amplitude of the B ring's forced motions indicates that the ring-edge has a surface density σ /similarequal 200 gm/cm 2 . So if this ring is incompressible and composed of icy spheres having a mean volume density of ρ = σ/ 2 h /similarequal 0 . 5 gm/cm 3 , this then requires a B ring thickness of only h ∼ 2 meters, which is rather thin compared to other estimates (Cuzzi et al. 2010). Similarly the ring particles' dispersion velocity c must be small compared to that expected for a dilute particle gas, so c /lessmuch ( h Ω ∼ 0 . 3 mm/sec), which again is cold compared to all other estimates for Saturn's rings (Cuzzi et al. 2010). The upshot is that an incompressible EOS requires the ring to be very thin and dynamically cold, likely much colder and thinner than is generally thought. Consequently we are optimistic that the compressible EOS used here, p = σc 2 , is the appropriate choice for simulations of the outer edge of Saturn's B ring. Nonetheless in a followup investigation we do intend to encode the incompressible EOS into epi int , to see if the BGT instability can account for the higher m ≥ 2 free modes that are seen at the outer edge of the B ring and in many other narrow ringlets.", "pages": [ 43, 44 ] }, { "title": "4.3. impulse origin for normal modes", "content": "The simulations of Section 3 used a fictitious temporary satellite to excite the free modes that occur at many Saturnian ring edges. These simulations used an admittedly ad hoc method-the sudden appearance and disappearance of a satellite-to excite these modes. Nonetheless these models demonstrate that transient and impulsive events can excite normal modes at ring edges, and those simulations show that normal modes can persist at the ring's edge for hundreds of years after the disturbance has occurred. Which suggests that an impulsive event in the recent past, perhaps an impact into Saturn's rings, might be responsible for exciting the normal modes that are seen at the outer edge of the B ring, as well as the normal modes that are also seen along the edges of several narrower ringlets (French et al. 2010; Hedman et al. 2010; French et al. 2011; Nicholson et al. 2012) The possibility that normal modes are due to an impact is motivated by the discovery of vertical corrugations in Saturn's C and D rings (Hedman et al. 2007, 2011) and in Jupiter's main dust ring (Showalter et al. 2011). These vertical structures are spirals that span a large swath of each ring, and they are observed to wind up over time due to the central planet's oblateness. Evolving the vertical corrugations backwards in time also unwinds their spiral pattern until some moment when the affected region is a single tilted plane. Unwinding the Jovian corrugation shows that that disturbance occurred very close to the date when the tidally disrupted comet Shoemaker-Levy 9 impacted Jupiter in 1994, which suggests an impact from a tidally disrupted comet as the origin of these ring-tilts (Showalter et al. 2011). However a single sub-km comet fragment cannot tilt a large ∼ 2 × 10 5 km-wide planetary ring. But a disrupted comet can produce an extended cloud of dust, and if that disrupted dust cloud returns to the planet with enough mass and momentum, then it might tilt a ring that at a later date would be observed as a spiral corrugation. However the tidal disruption of comet about a low-density planet like Saturn is more problematic, because tidal disruption only occurs when the comet's orbit is truly close to parabolic and not too hyperbolic, and with periapse just above the planet's atmosphere (Sridhar & Tremaine 1992; Richardson et al. 1998). But it is easy to envision an alternate scenario that might be more likely, with a small km-sized comet originally in a heliocentric orbit coming close enough to Saturn to instead strike the main A or B rings. This scenario is more probable because the cross-section available to orbits impacting the main rings is significantly larger than those resulting in tidal disruption. The impacting comet's considerably greater momentum will nonetheless carry the impactor through the dense A or B rings, but the collision itself is likely energetic enough to shatter the comet. And if that collision is sufficiently dissipative, then the resulting cometary debris will then stay bound to Saturn, and in an orbit that will return that debris back into the ring system on its next orbit. Small differences among the orbits of individual debris particles' means that, when the debris encounters the rings again, that impacting debris will be spread across a much larger footprint on the ring, which presumably will allow any dense rings or ringlets to absorb the debris' mass and momentum in a way that effectively gives the ring particles there a sudden velocity kick ∆v in proportion to the comet debris density ρ and velocity v r relative to the ring matter. But if comet Shoemake-Levy 9's (SL9) impact with Jupiter is any guide, then impact by a cloud of comet debris could last as long week of time, which might tend to smear this effect out due to the ring's orbital motion. But that effect would be offset if the debris train's dust cloud is also rather clumpy, like the SL9 debris train was. Indeed, it is possible that this scenario might also account for the spiral corrugations of Saturn's C and D ring. It is also conceivable that an inclined cloud of impacting comet debris might also excite the vertical analog of normal modes-long-lived vertical oscillations of a ring's edge. This admittedly speculative scenario will be pursued in a followup study, to determine whether debris from an impact-disrupted comet can excite the normal modes seen at ring edges, and to determine the mass of the progenitor comet that would be needed to account for these modes' observed amplitudes.", "pages": [ 44, 45, 46 ] }, { "title": "5. Summary of results", "content": "We have developed a new N-body integrator that calculates the global evolution of a self-gravitating planetary ring as it orbits an oblate planet. The code is called epi int , and it uses the same kick-drift-step algorithm as is used in other symplectic integrators such as SYMBA and MERCURY . However the velocity kicks that are due to ring gravity are computed via an alternate method that assumes that all particles inhabit a discreet number of streamlines in the ring. The use of streamlines to calculate ring self gravity has been used in analytic studies of rings (Goldreich & Tremaine 1979; Borderies et al. 1983a, 1986), and the streamline concept is easily implemented in an N-body code. A streamline is the closed path through the ring that is traced by particles having a common semimajor axis. All streamlines are radially close to each other, so the gravitation acceleration due to a streamline is simply that due to a long wire, A = 2 Gλ/ ∆ where λ is the streamline's linear density and ∆ is the particle's distance from the streamline. Which is very useful since particles are responding to the pull of smooth wires rather than discreet clumps of ring matter so there is no gravitational scattering. Which means that only a modest number of particles are needed, typically a few thousand, to simulate all 360 · of a scalloped ring like the outer edges of Saturn's A and B ring. Only a few thousand particles are also needed to simulate linear as well as nonlinear spiral density waves, and execution times are just a few hours on a desktop PC. Another distinction occurs during the particles' unperturbed drift step when particles follow the epicyclic orbit of Longaretti & Rappaport (1995) about an oblate planet, rather than the usual Keplerian orbit about a spherical planet. This effectively moves the perturbation due to the planet's oblate figure out of the integrator's kick step and into the drift step. The code also employs hydrodynamic pressure and viscosity to account for the transport of linear and angular momentum through the ring that arises from collisions among ring particles. Another convenience of the streamline formulation is that it easily accounts for the large pressure drop that occurs at a ring's sharp edge, as well as the large viscous torque that the ring exerts there. The model also accounts for the mutual gravitational perturbations that the ring and the satellites exert on each other. The epi int code is written in IDL, and the source code is available for download at http://gemelli.spacescience.org/~hahnjm/software.html . This integrator is used to simulate the forced response that the satellite Mimas excites at its m = 2 inner Lindblad resonance (ILR) that lies near the outer edge of Saturn's B ring. That resonance lies ∆ a 2 = 12 ± 4 km inwards of the ring's edge, and simulations show that the ring's forced epicyclic amplitude varies with the ring's surface density σ 0 as R 2 ∝ σ 0 . 67 0 . Good agreement with Cassini measurements of R 2 occurs when the simulated ring has a surface density of σ 0 = 195 ± 60 gm/cm 2 (see Fig. 5), where the uncertainty in σ 0 is dominated by the δa edge = 4 km uncertainty that Spitale & Porco (2010) find in the ring-edge's semimajor axis. This σ 0 is the mean surface density over that part of the B ring that is disturbed by this resonance, whose influence in the ring extends to a radial distance of ∆ a e/ 10 ∼ 150 km from the B ring's outer edge. And if we naively assume that this surface density is the same everywhere across Saturn's B ring, then its total mass is about 90% of Mimas' mass. Cassini observations reveal that the outer edge of Saturn's B ring also has several free normal modes that are not excited by any known satellite resonances. Although the mechanism that excites these free modes is uncertain, we are nonetheless able to excite free modes in a simulated ring via various ad-hoc methods. For instance, a fictitious satellite's m th Lindblad resonance is used to excite a forced pattern at the ring edge. Removing that satellite then converts the forced patten into a free normal mode that persists in these simulations for up to ∼ 100 years or ∼ 10 5 orbits without any damping, despite the simulated ring having a kinematic viscosity of ν = 100 cm 2 /sec; see Fig. 10 for one example. Alternatively, starting the ring particles in circular orbits while subject to Mimas' m = 2 gravitational perturbation excites both a forced and a free m = 2 pattern that initially null each other precisely at the start of the simulation. But the forced patten corotates with Mimas' longitude while the free pattern rotates slightly faster in a heavier ring, which suggests that a free mode's pattern speed can also be used to infer a ring's surface density σ 0 . However the free pattern speed is also influenced by the J 4 and higher terms in the oblate planet's gravity field, which are absent from this model which only accounts for the J 2 component. So the simulated pattern speed cannot be compared directly to the observed pattern speed; see Fig. 6. To avoid this difficulty, the resonance condition (Eqn. 31) is used to calculate the radius of the Lindblad resonance that is associated with the free normal mode. Plotting the distances of the simulated and observed resonances from the B ring's edge (Figs. 8, 11, and 12) then provides a convenient way to compare simulations to observations of free modes in a way that is insensitive to the planet's oblateness. Simulations of the B ring's free m = 2 and m = 3 patterns are consistent with Cassini measurements of the B ring's normal modes when the simulated ring-edge again has a surface density of σ 0 ∼ 200 gm/cm 2 , which is a nice consistency check. But these particular measurements do not provide tight constraint on the ring's σ 0 , due to the fact that the m = 2 and m = 3 Lindblad resonances only lie ∆ a m ∼ 25 km from the outer edge of a ring whose semimajor axis a is uncertain by δa edge = 4 km. However the B ring's free m = 1 normal mode does lie much deeper in the ring's interior, ∆ a 1 = 253 ± 4, so the uncertainly in its location is fractionally much smaller, and this normal mode does confirm the σ 0 /similarequal 200 gm/cm 2 value that was inferred from simulations of the B ring's forced response R 2 . One of the goals of this study is to determine whether simulations of free modes can be used to determine the surface density and mass of a narrow ringlet. Such ringlets show a broad spectrum of free normal models over 0 ≤ m ≤ 5 (French et al. 2010; Hedman et al. 2010; French et al. 2011; Nicholson et al. 2012), and the answer appears to be yes since free pattern speeds do increase with σ 0 . However Section 3.1.2 shows that the semimajor axes of the ringlet's edges likely need to be known to a precision of δa edge ∼ 1 km in order for a free mode to provide a useful measurement of the ringlet's σ 0 . The origin of these free modes, which are quite common along the edges of Saturn's broad rings and its many narrow ringlets, is uncertain. Borderies et al. (1985) show that, if a planetary ring's particles are packed shoulder to shoulder such that the ring behaves like an incompressible fluid, then that ring is unstable to the growth of density waves, a phenomenon also termed viscous overstability, and they suggest that the B ring's normal modes might be due to unstable waves that are trapped between a Lindblad resonance and the ring's edge. To study this further, we will in a followup study adapt epi int to employ an incompressible equation of state, to see if the viscous overstability can in fact account for the free normal modes seen along the Saturnian ring edges. Although the current version of epi int does not account for the origin of these free modes, one can still plant a free mode along the edge of a simulated ring by temporarily perturbing a ring at a fictitious satellite's Lindblad resonance, and then removing that satellite, which creates an unforced mode that persists undamped at the ring-edge for more than ∼ 10 5 orbits or ∼ 100 yrs despite the simulated ring having a kinematic viscosity of ν = 100 cm 2 /sec. Because this forcing is suddenly turned on and off, this suggests that any sudden or impulsive disturbance of the ring can excite normal modes, with those disturbances possibly persisting for hundreds or maybe thousands of years. And in Section 4.3 we suggest that the Saturnian normal modes might be excited by an impact with a collisionally disrupted cloud of comet dust. This is a slight variation of the scenario that Hedman et al. (2007) and Showalter et al. (2011) propose for the origin of corrugated planetary rings, and in a followup investigation we intend to determine whether such impacts can also account for the normal modes seen in Saturn's rings. And lastly, we find that epi int 's treatment of ring viscosity has difficulty accounting for the radial confinement of the B ring's outer edge by Mimas' m = 2 inner Lindblad resonance. This model employs a kinematic shear viscosity ν s that is everywhere a constant, which causes the simulation's outermost streamline to slowly but steadily drift radially outwards. Which in turn causes the ring's forced epicyclic amplitude R 2 to slowly grow over time, and makes difficult any comparison to Cassini's measurement of R 2 . To sidestep this difficulty, the model zeros the torque that the simulated ring exerts on its outermost streamline, which does allow the ring to settle into a static configuration that can be compared to Cassini observations and yields a measurement of the ring's surface density σ 0 . This approximate treatment is also examined in in Section 4.1, which shows that the viscous acceleration of the ring-edge, had it been included in the simulation, is still orders of magnitude smaller than that due to ring self gravity. So this study of the dynamics of the B ring's forced and free modes is not adversely impacted by this approximate treatment. But this does mean that the B ring's radial confinement is still an unsolved problem, and Section 4.1.1 suggests that this might be a consequence of treating ν s as a constant. Borderies et al. (1982) show that viscosity's outward transport of energy should also heat the ring's outer edge and increase the ring particles' dispersion velocity c there. And if collisions among ring particles are the dominant source of ring viscosity, then ν s ∝ c 2 and viscous dissipation would be enhanced at the ring edge, which in turn would increase the angular lag between the ring's forced response and the Mimas' longitude. That then would increase the gravitational torque that that satellite exerts on the ring-edge. So in a followup study we will modify epi int to address this problem in a fully self-consistent way, to see if enhanced dissipation at the ring-edge also increases Mimas' gravitational torque there sufficiently to prevent the B ring's outer edge from flowing viscously beyond that satellite's m = 2 inner Lindblad resonance.", "pages": [ 46, 47, 48, 49 ] }, { "title": "Acknowledgments", "content": "J. Hahn's contribution to this work was supported by grant NNX09AU24G issued by NASA's Science Mission Directorate via its Outer Planets Research Program. The authors thank Denise Edgington of the University of Texas' Center for Space Research (CSR) for composing Fig. 3, and J. Hahn thanks Byron Tapley for graciously providing office space and the use of the facilities at CSR. The authors are also grateful for the helpful suggestions provided by an anonymous reviewer.", "pages": [ 49, 50 ] }, { "title": "A. Appendix A", "content": "The following calculates the flux of angular momentum that is communicated via a disk's viscosity. The disk is flat and thin and has a vertical halfwidth h and constant volume density ρ that is related to its surface density σ via ρ = σ/ 2 h . The disk is assumed viscous, and its gravity is ignored here since this Appendix is only interested in the angular momentum flux that is transported solely by viscosity. The density of angular momentum in the disk is /lscript = r × ρ v , and the vertical component along the z = x 3 axis is /lscript 3 = x 1 ρv 2 -x 2 ρv 1 in Cartesian coordinates x = x 1 and y = x 2 where ρ and v i are functions of position and time, so the time rate of change of /lscript 3 is The time derivatives in the above are Euler's equation, where the Π ik are the elements of the momentum flux density tensor where p is the pressure and σ ' ik are the elements of the viscous stress tensor (Landau & Lifshitz 1987). Inserting Eqn. (A3) into (A1) yields where the vector is the flux density of the i component of linear momentum and ˆ x k is the unit vector along the x k axis. Equation (A4) can be rewritten but note that Π 1 · ∇ x 2 -Π 2 · ∇ x 1 = Π 21 -Π 12 = σ ' 12 -σ ' 21 = 0 since the viscous stress tensor is symmetric (Eqn. A11), so where Integrating Eqn. (A7) over some volume V that is bounded by area A yields by the divergence theorem, so Eqn. (A9) indicates that F 3 is the flux of the x 3 component of angular momentum out of volume V that is being transported by advection, pressure, and viscous effects. This Appendix is interested in the part of F 3 that is due to viscous effects, which will be identified as F ' 3 and is obtained by replacing Π ik in Eqn. (A3) with -σ ' ik so This is the 2D flux of the x 3 component of angular momentum that is transported by the disk's viscosity whose horizontal components in Cartesian coordinates are F ' 3 = F ' 1 ˆ x 1 + F ' 2 ˆ x 2 where F ' 1 = x 2 σ ' 11 -x 1 σ ' 21 and F ' 2 = x 2 σ ' 12 -x 1 σ ' 22 . However this Appendix desires the radial component of F ' 3 are some site r, θ in the disk, which is F ' r = F ' 1 cos θ + F ' 2 sin θ . The elements of the viscous stress tensor are (Landau & Lifshitz 1987) where η is the shear viscosity, ζ is the bulk viscosity, and δ ik is the Kronecker delta. Inserting this into F ' r and replacing x 1 = r cos θ and x 2 = r sin θ then yields The horizontal velocities are v 1 = v r cos θ -v θ sin θ and v 2 = v r sin θ + v θ cos θ when written in terms of their radial component v r and tangential component v θ = r ˙ θ . The derivatives in Eqn. (A12) are when written in terms of cylindrical coordinates, and the combinations of derivatives in Eqn. (A12) are Inserting these into Eqn. (A12) then yields a result that is thankfully much more compact, noting that the second term in Eqn. (A15) may be neglected since the azimuthal gradient is much smaller than the radial gradient for the disks considered here. This is the radial component of the disk's 2D viscous angular momentum flux density, so the 1D viscous angular momentum flux density is Eqn. (A15) integrated through the disk's vertical cross section: where ν s = η/ρ is the disk's kinematic shear viscosity.", "pages": [ 50, 51, 52 ] }, { "title": "B. Appendix B", "content": "The flux density of x 1 -type momentum is Π 1 (see Eqn. A5) while the flux density of x 2 -type momentum is Π 2 , so the flux density of radial momentum is G = cos θ Π 1 +sin θ Π 2 and the radial component of this momentum flux density is where ˆ r is the unit vector in the radial direction. The part of that momentum flux that is transported solely by viscous effects will be called G ' r and is again obtained by replacing the Π ik in the above with -σ ' ik : Equations (A13) provide the combination and inserting this plus Eqn. (A14a) into Eqn. (B3) then yields but the ∂v θ /∂θ term is again neglected in the streamline approximation. This is the 2D radial momentum flux due to viscous transport, so the vertically integrated linear momentum flux due to viscosity is", "pages": [ 52, 53 ] }, { "title": "REFERENCES", "content": "Borderies, N., Goldreich, P., & Tremaine, S. 1982, Nature, 299, 209 -. 1986, Icarus, 68, 522 Borderies-Rappaport, N. & Longaretti, P.-Y. 1994, Icarus, 107, 129 Chambers, J. E. 1999, MNRAS, 304, 793 Cuzzi, J. N., Burns, J. A., Charnoz, S., Clark, R. N., Colwell, J. E., Dones, L., Esposito, L. W., Filacchione, G., French, R. G., Hedman, M. M., Kempf, S., Marouf, E. A., Murray, C. D., Nicholson, P. D., Porco, C. C., Schmidt, J., Showalter, M. R., Spilker, L. J., Spitale, J. N., Srama, R., Sremˇcevi'c, M., Tiscareno, M. S., & Weiss, J. 2010, Science, 327, 1470 Duncan, M. J., Levison, H. F., & Lee, M. H. 1998, AJ, 116, 2067 French, R. G., Marouf, E. A., Rappaport, N. J., & McGhee, C. A. 2010, AJ, 139, 1649 French, R. G., Nicholson, P. D., Colwell, J., Marouf, E. A., Rappaport, N. J., Hedman, M., Lonergan, K., McGhee-French, C., & Sepersky, T. 2011, in EPSC-DPS Joint Meeting 2011, 624 Goldreich, P. & Tremaine, S. 1979, AJ, 84, 1638 -. 1982, ARA&A, 20, 249 Goldstein, H. 1980, Classical mechanics (2nd ed.) (Reading, Massachusetts:Addison-Wesley) Hahn, J. M., Spitale, J. N., & Porco, C. C. 2009, ApJ, 699, 686 Hedman, M. M., Burns, J. A., Evans, M. W., Tiscareno, M. S., & Porco, C. C. 2011, Science, 332, 708 Hedman, M. M., Burns, J. A., Showalter, M. R., Porco, C. C., Nicholson, P. D., Bosh, A. S., Tiscareno, M. S., Brown, R. H., Buratti, B. J., Baines, K. H., & Clark, R. 2007, Icarus, 188, 89 Hedman, M. M., Nicholson, P. D., Baines, K. H., Buratti, B. J., Sotin, C., Clark, R. N., Brown, R. H., French, R. G., & Marouf, E. A. 2010, AJ, 139, 228 Kudryavtsev, L. D. & Samarin, M. K. 2013, Lagrange interpolation formula, Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php? title=Lagrange interpolation formula&oldid=17497 Landau, L. D. & Lifshitz, E. M. 1987, Fluid mechanics, 2nd Ed. (Elsevier) Longaretti, P.-Y. & Rappaport, N. 1995, Icarus, 116, 376 Melita, M. D. & Papaloizou, J. C. B. 2005, Celestial Mechanics and Dynamical Astronomy, 91, 151 Nicholson, P. D., French, R. G., & M., H. M. 2012, contributed talk at AAS/DDA conference, 1, 1 Pringle, J. E. 1981, ARA&A, 19, 137 Richardson, D. C., Bottke, W. F., & Love, S. G. 1998, Icarus, 134, 47 Salo, H. 1995, Icarus, 117, 287 Showalter, M. R., Hedman, M. M., & Burns, J. A. 2011, Science, 332, 711 Shu, F. H. 1984, in IAU Colloq. 75: Planetary Rings, ed. R. Greenberg & A. Brahic, 513-561 Spitale, J. N. & Porco, C. C. 2010, AJ, 140, 1747 Sridhar, S. & Tremaine, S. 1992, Icarus, 95, 86 Tiscareno, M. S., Burns, J. A., Nicholson, P. D., Hedman, M. M., & Porco, C. C. 2007, Icarus, 189, 14 Ward, W. R. 1986, Icarus, 67, 164", "pages": [ 53, 54, 55 ] } ]

[ { "title": "ABSTRACT", "content": "The attenuation of high-energy gamma-ray spectrum due to the electron-positron pair production against the extragalactic background light (EBL) provides an indirect method to measure the EBL of the universe. We use the measurements of the absorption features of the gamma-rays from blazars as seen by Fermi Gamma-ray Space Telescope to explore the EBL flux density and constrain the EBL spectrum, star formation rate density (SFRD) and photon escape fraction from galaxies out to z = 6. Our results are basically consistent with the existing determinations of the quantities. We find a larger photon escape fraction at high redshifts, especially at z = 3, compared to the result from the recent Ly α measurements. Our SFRD result is consistent with the data from both gamma-ray burst and UV observations in 1 σ level. However, the average SFRD we obtain at z /greaterorsimilar 3 matches the gamma-ray data better than the UV data. Thus our SFRD result at z /greaterorsimilar 6 favors that it is sufficiently high enough to reionize the universe. Subject headings: galaxies: evolution - gamma rays: galaxies - stars: formation", "pages": [ 1 ] }, { "title": "THE EXTRAGALACTIC BACKGROUND LIGHT FROM THE MEASUREMENTS OF THE ATTENUATION OF HIGH-ENERGY GAMMA-RAY SPECTRUM", "content": "Yan Gong 1 and Asantha Cooray 1 1 Department of Physics & Astronomy, University of California, Irvine, CA 92697 Draft version June 25, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The extragalactic background light (EBL) is the cumulative radiation from the stars and active galactic nuclei (ANGs) through the universe history. It is tightly related to the starlight from direct emission and dust re-radiation which are dominant in the ultraviolet (UV) to near-infrared (near-IR) and mid-IR to submillimeter bands, respectively (Baldry & Glazebrook 2003; Fukugita & Peebles 2004; Franceschini et al. 2008; Finke et al. 2010; Stecker et al. 2012; Inoue et al. 2013). Hence the measurements of EBL at different redshifts are important and fundamental for the understanding of the star formation history and galaxy formation and evolution. However, direct EBL observations is limited by large uncertainties due to the foreground contamination that needs to be accounted for (Hauser & Dwek 2001). This makes the measurements of the EBL very difficult especially at high redshifts. The high-energy gamma-rays interact with the EBL photons and generate the electron-positron pairs (Nikishov 1962; Gould & Schreder 1966; Fazio & Stecker 1970; Stecker et al. 1992). This effect can cause an absorption feature in the observed gamma-ray spectrum above a critical energy relative to its intrinsic spectrum. The evolution of the EBL therefore can be derived by the observations of the attenuated spectrum of the highenergy gamma-rays at different redshifts, which provides an indirect but feasible measurement of the EBL evolution. The difficulty of this method is how to determine the intrinsic spectrum of the gamma-ray sources and distinguish the absorption feature caused by the EBL from the intrinsic variation in the gamma-ray spectrum. Ackermann et al. (2012) reported the gamma-ray absorption feature in a sample of gamma-ray blazars in the redshift range 0 . 03 < z < 1 . 6 using the Large Area Telescope (LAT) of the Fermi Gamma-ray Space Tele- scope 1 . This sample contains 150 blazars of the BL Lacertae (BL Lac) type in the 3-500 GeV band, providing the constraints on the EBL spectrum from UV to the nearIR band. To determine the absorbed spectrum, they analyzed the sample in three independent redshift bins z < 0 . 2, 0 . 2 ≤ z < 0 . 5 and 0 . 5 ≤ z < 1 . 6, and excluded the possibility that the attenuation is caused by the intrinsic properties of the gamma-ray sources. We make use of this data set to constrain the EBL spectrum from UV to near-IR band, and extract the information of the photon escape fraction and the star formation history. Our EBL model is based on the work of Finke et al. (2010), which is dependent on a model for the stellar evolution, initial mass function (IMF), photon escape fraction, and the star formation rate density (SFRD). The Markov Chain Monte Carlo (MCMC) method is adopted in our constraint process, and we derive the average values and standard deviations for the EBL, photon escape fraction and SFRD from the obtained MCMC chains. We assume the flat ΛCDM with Ω M = 0 . 27, Ω b = 0 . 046 and h = 0 . 71 for the calculation throughout the paper (Hinshaw et al. 2012).", "pages": [ 1 ] }, { "title": "2. EBL MODEL", "content": "We calculate the EBL spectrum based on the model proposed by Finke et al. (2010), where the EBL spectrum is evaluated by the initial mass function ξ ( M ∗ ), the comoving star formation rate density ˙ ρ ∗ ( z ), and the photon escape fraction f esc ( λ, z ) which denotes the fraction of photons that can escape a galaxy without absorption by interstellar dust. It gives an analytic relation between the EBL spectrum and the IMF, SFRD and f esc , and hence provides a good way to constrain these quantities using the EBL observational data. We adopt the IMF model from Chabrier (2003) where it is expressed in two parts, where M ∗ is the stellar mass, A = 0 . 158, M c ∗ = 0 . 079 M /circledot , σ = 0 . 69, B = 4 . 43 × 10 -2 and x = 1 . 3. As this IMF model is consistent with the other models (e.g. Baldry & Glazebrook 2003; Razzaque et al. 2009), we fix the values of these parameters when performing our MCMC fits. We make use of the SFRD model proposed by Cole et al. (2001), which takes the form where ˙ ρ ∗ ( z ) is in hM /circledot yr -1 Mpc -3 , and a , b , c and d are free parameters. At low redshifts with z /lessorsimilar 2, the SFRD can be constrained by the current observational data, and we take a z ≤ 2 = 0 . 0118, b z ≤ 2 = 0 . 08, c z ≤ 2 = 3 . 3 and d z ≤ 2 = 5 . 2 (Hopkins & Beacom 2006, hereafter HB06) 2 . At z > 2, the uncertainty of SFRD becomes very large because of uncertain dust extinction at the high redshifts. Hence we treat these four parameters as free and constrain them in our MCMC fits at z > 2. Note that we don't consider the contribution from quiescent galaxies and AGNs in our model, since they are not the main components of the EBL which can contribute about 10% ∼ 13% emission to the total intensity (Dominguez et al. 2011). The photon escape fraction f esc should be a function of both the photon energy and redshift, and it is still not well determined by current observations. For simplicity, we assume f esc is linearly increasing with the wavelength λ , which agree well with results from the observations of nearby galaxies (Driver et al. 2008). If the slope is independent of the redshift, then the photon escape fraction can be written as Here m , n and p are free parameters and are needed as free parameter to fit in the MCMC process. When the photon energy is greater than 13 . 6 eV, we set f esc = 0, since most of the photons in this energy range would be absorbed by the neutral hydrogen in the galaxies. Next we can estimate the comoving luminosity density /epsilon1 j /epsilon1 (in units of W/Mpc 3 ) where /epsilon1 = hν/m e c 2 is the dimensionless photon energy. In our model, we consider both of the emission from stars and the re-radiation from interstellar dust, i.e. we have /epsilon1 j /epsilon1 = /epsilon1 j star /epsilon1 + /epsilon1 j dust /epsilon1 . The comoving luminosity density from starlight at redshift z is given by ∣ ∣ Here we take M min ∗ = 0 . 1 M /circledot , M max ∗ = 100 M /circledot , 2 These values are obtained using the IMF from Baldry & Glazebrook (2003) which is well consistent with the IMF we use in Chabrier (2003). z max = 6, and ˙ N ∗ ( /epsilon1, M ∗ , t ∗ ) = πR 2 ∗ c n ∗ ( /epsilon1, T ∗ ) is the number of photons emitted per unit time per unit energy from a star with radius R ∗ at lifetime t ∗ . The n ∗ ( /epsilon1, T ∗ ) is the stellar photon emission spectrum where T ∗ is the stellar temperature. To estimate these quantities, i.e. R ∗ , T ∗ and n ∗ , we assume the Planck spectrum for the starlight. This approximation is in a good agreement with the results given by Bruzual & Charlot (2003) for stars with solar metallicity between 1 and 10 Gyr of age, which dominate the emission between 0.1 and 10 µ m (Finke et al. 2010). We use a model of the Hertzsprung-Russell diagram to take into account different stellar evolution phases from the main-sequence to post main-sequence (e.g. the Hertzsprung gap, the giant branch, the horizontal branch, the asymptotic giant branch and the white dwarf) which take a stellar mass 0 . 1 ≤ M ∗ ≤ 100 M /circledot (see Eggleton et al. 1989 for details). We also assume all stars have the solar metallicity, and it is constant over the cosmic history and stellar mass (Finke et al. 2010). The radiation of dust which dominates the mid- and far-infrared bands is not important in our analysis here, since the gamma-ray sample we use is in the 3-500 GeV band. The process of photon-photon interaction between these gamma-rays and EBL photons would mainly occur in the near-infrared or higher energy EBL bands where the EBL photons are emitted directly from stars. Here we take the same dust emission model with three dust components used by Eq. (11) of Finke et al. (2010). The proper EBL spectrum (or EBL intensity, in units of nWm -2 sr -1 ) at energy /epsilon1 p and redshift z can be derived by ∣ ∣ where /epsilon1 p = (1+ z ) / (1 + z ' ) /epsilon1 ' is the proper dimensionless photon energy at z . Also, it is easy to get the proper EBL energy density if we notice ρ b = (4 π/c ) /epsilon1 p I which is in units of erg/cm 3 . After obtaining the EBL spectrum, we can further estimate the optical depth for gamma-ray absorption with observed energy E γ emitted at redshift z 0 where d l/ d z is the cosmological line element, E b = /epsilon1 p × m e c 2 is the proper photon energy of the EBL background at z, u = cos( θ ) where θ is the angle of incidence, and σ γγ is the cross-section of the pair production. The n b ( E b , z ) = ρ b /E b is the proper number density of EBL photons at z. The E min is the minimum threshold energy of EBL photons that can interact with a gamma-ray of observed energy E γ where m e is the mass of electron. Then the intrinsic gamma-ray spectrum is modified to be Therefore, we can use the observations of attenuation of gamma-ray spectrum to compare with our theoretical model and constrain the free parameters that describe the SFRD and f esc in the model.", "pages": [ 1, 2, 3 ] }, { "title": "3. MCMC FITTING", "content": "We employ the MCMC method to perform the constraints. There are several advantages for this method, and the most important one is the time cost of the computations linearly increase with the number of the free parameters. Thus this method is suitable to fit a large set of parameters in an acceptable computation time. The Metropolis-Hastings algorithm is used in the MCMC process to decide possibility of accepting a new chain point (Metropolis et al. 1953; Hastings 1970). We use a Gaussian sampler with adaptive step size to estimate the proposal density matrix (Doran & Muller 2004), and assume the uniform prior probability distribution for the parameters. The χ 2 distribution is employed to calculate the likelyhood function which is given by where N is the number of the data, τ th γγ is the theoretical optical depth given by Eq. (5), and τ obs γγ and σ i are the optical depth and errors from the observations. We use the observational data in Ackermann et al. (2012), where they provide the measurements of the absorption feature derived from 150 blazars in the 3-500 GeV band of the Fermi-LAT survey (see the data points in the left panel of Fig. 1). This sample covers the redshift range from 0.03 to 1.6, and gives the absorption feature in three redshift bins with the central redshifts z c /similarequal 0.1, 0.35 and 1, respectively. We finally have 18 data points (6 in each redshift range), and we fit them all in three redshift bins with our model. We have seven free parameters in our model: a , b , c and d in the SFRD given in Eq. (1), and m , n and p in the f esc shown in Eq. (2). As we assume uniform prior for the parameters in the MCMC process, their ranges are set as a ∈ (0 , 0 . 1), b ∈ (0 , 1), c ∈ (0 , 6), d ∈ (0 , 10), m ∈ ( -4 , 4), n ∈ ( -2 , 2) and p ∈ (0 , 3). These ranges are chosen according to the relevant models (Hoplins & Beacom 2006; Driver et al. 2008) and empirical experience. We perform some pre-runs to check these ranges and make sure that they have the correct physical meaning and there is no the other maximum out of these ranges. Note that we fix a , b , c and d to be the values in the HB06 model when z ≤ 2, since the SFRD is relatively well-determined in this redshift range by the current observations as we discuss in the last section. We run 8 parallel chains and obtain about 10 5 points sampled in each chain after the convergence determined by the criterion of Gelman & Rubin (1992). After the burn-in process and thinning the chains, we merge them into one chain and finally collect about 10000 points that are used to investigate the probability distributions of the parameters and statistical quantities of the components in the model. More details of our MCMC method can be found in Gong & Chen (2007).", "pages": [ 3 ] }, { "title": "4. RESULTS", "content": "In the left panel of Fig. 1, we show the data of the attenuation of the gamma-ray spectrum by the EBL background in the three redshift bins, and the best fits of our model are denoted in red dashed lines. We fit 18 data points in the three redshift bins simultaneously and per- form a global fitting for all seven free parameters in our model. The best-fits and 1 σ errors (68.3% confidence level) of the parameters derived from the 1-D probability distribution function (PDF) are a = 0 . 055 +0 . 041 -0 . 050 , b = 0 . 57 +0 . 43 -0 . 54 , c = 3 . 9 +2 . 0 -3 . 3 , d = 4 . 0 +5 . 5 -3 . 8 , m = 0 . 32 +0 . 18 -0 . 11 , n = 1 . 4 +0 . 4 -0 . 5 and p = 2 . 2 +0 . 8 -1 . 4 . The data are measured in three redshift bins z < 0 . 2, 0 . 2 < z < 0 . 5 and 0 . 5 < z < 1 . 6. We take central redshifts of z = 0 . 1, z = 0 . 35 and z = 1 to perform our theoretical calculation. Note that our fitting results are only from the UV to near-IR bands of the EBL out to 1 µ m at z = 0. The EBL spectrum at z = 0 derived from our MCMC chains are shown in the right panel of Fig. 1. We calculate the EBL flux density for each chain point (which is a 7-D point and contains the values of seven free parameters in our model) at different wavelengths, and estimate the average values (blue solid line) and standard deviations (1 σ , blue region). The EBL data evaluated from the galaxy counts (red triangles) and direct measurements (green circles) are also shown for comparison (Gilmore et al. 2012; Abramowski et al. 2013). The purple region gives the 1 σ statistical contour derived from different energy bands of High Energy Stereoscopic System (H.E.S.S.) (Abramowski et al. 2013). We find the EBL spectrum from our MCMC chains are consistent with these observational results at λ /greaterorsimilar 0 . 4 µ m (vertical dashed line). For λ /lessorsimilar 0 . 4 µ m, the energy of gamma-ray which can interact with the EBL photons is less than 200 Gev (see Eq. (6)), and the data points of optical depth τ γγ are close to 0 as shown in the left panel, which can not give stringent constraints on the EBL at z = 0 in this regime. In Fig. 2, we show the f esc ( λ, z ) at three redshifts z = 0, z = 3 and z = 6. The f esc are calculated by each MCMC chain point at different wavelengths and redshifts, and the average values and standard deviations (1 σ ) are shown in blue solid line and shaded region respectively. The vertical black dotted line denotes the hydrogen ionization energy at 13.6 eV ( ∼ 912 ˚ A), and note that the f esc is set to be 0 when the photon energy is greater than 13.6 eV in our calculation since most of these photons would be absorbed by the neutral hydrogen gas in galaxies. The red circles with error bars are the Ly α escape fraction derived from the Ly α luminosity function around these three redshifts(Hayes et al. 2011, Blanc et al. 2011 and references therein). We find our f esc ( λ, z ) results well agree with these results at z = 0 and 6, but are higher at z = 3. The green points and lines in the upper panel are the average photon escape fraction and errors derived from 10,000 nearby galaxies (Driver et al. 2008). Our result is lower than theirs, especially around λ = 0 . 5 µ m, but consistent at longer wavelengths with λ /greaterorsimilar 1 µ m. In Fig. 3, we show the SFRD at different redshifts. The blue dashed line and the shaded region are the average SFRD values and standard deviations (1 σ ) evaluated by the MCMC chains, respectively. The black solid line denotes the model from Hopkins & Beacom (2006) with the IMF of Baldry & Glazebrook (2003). As addressed in Section 2, we fix SFRD to be the HB06 model at z ≤ 2 in our MCMC process, since it can be well constrained by the current observations. We find the average SFRD (blue dashed line) from our MCMC chains is higher than servations (Bouwens et al. 2012) with flatter slope, but it agrees with the data from the gamma-ray burst measurements shown in red triangles (Kistler et al. 2009) and green circles (Robertson & Ellis 2012, 'low-Z SFR' model). Also, we should note that our result is consistent with both of the gamma-ray and UV data in 1 σ level given the large uncertainty. This implies our SFRD result favors that it is alone sufficient to reionize the universe (Madau et al. 1999).", "pages": [ 3, 4, 5 ] }, { "title": "5. DISCUSSION AND CONCLUSION", "content": "We explore the EBL spectrum at near-IR to UV bands by fitting the gamma-ray attenuation data detected by Fermi-LAT measurements shown in Ackermann et al. (2012). Our EBL model is based on earlier work of Finke et al. (2010). This model can fit the gamma-ray attenuation data well in all three redshift bins within 0 . 03 ≤ z ≤ 1 . 6. After obtaining the MCMC chains, we derive the average values and standard deviations of the EBL spectrum νI ν ( λ, z = 0), f esc ( λ, z ) and ˙ ρ ∗ ( z ) respectively from chain points. Also, we compare our results with the corresponding observational data, and find they are basically consistent with these observations in the regime of the gamma-ray attenuation data used in the constraints. The f esc we get agrees with the data of Ly α escape fraction measurements at z = 0 and 6, but a bit larger at z = 3. Also, it is smaller than the result of Driver et al. (2008) around λ = 0 . 5 µ m at z = 0. For the star formation history, we obtain a higher average SFRD (blue dashed line in Fig. 3) at z /greaterorsimilar 3 with flatter slope than the result from the HB06 model and UV data. But note that our results in fact still consistent with theirs in 1 σ level given the large constraint uncertainty. On the other hand, our average SFRD matches the results given by GRB measurements very well, and this indicates that our SFRD has a trend to favor that the star formation alone at high redshifts could reionize the universe. Recently, Orr et al. (2011) and Yuan et al. (2012) also perform constraints on the EBL spectrum using the gamma-ray observations from Fermi and ground-based air Cherenkov telescopes with 12 and 7 blazars respectively. Their results around near-IR band are consistent with ours in 1 σ level but are higher than ours in the optical band. However, our EBL spectrum in the optical band agrees well with that from Dominguez et al. (2011) in which they use the observed galaxy luminosity function and galaxy SED-type fractions to derive the EBL spectrum. This work was supported by NSF CAREER AST0645427.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Ackermann, M., et al. 2012, Science, 338, 1190 Baldry, I. K. & Glazebrook, K. 2003, ApJ, 593, 258 Blanc, G. A., Adams, J. J., Gebhardt, K., et al. 2011, 736, 31 Chabrier, G. 2003, Publ. Astron. Soc. Pac., 115, 763-796", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "M66 and M96 in the Leo I Group are nearby spiral galaxies hosting Type Ia Supernovae (SNe Ia). We estimate the distances to these galaxies from the luminosity of the tip of the red giant branch (TRGB). We obtain V I photometry of resolved stars in these galaxies from F 555 W and F 814 W images in the Hubble Space Telescope archive. From the luminosity function of these red giants we find the TRGB I -band magnitude to be I TRGB = 26 . 20 ± 0 . 03 for M66 and 26 . 21 ± 0 . 03 for M96. These values yield distance modulus ( m -M ) 0 = 30 . 12 ± 0 . 03(random) ± 0 . 12(systematic) for M66 and ( m -M ) 0 = 30 . 15 ± 0 . 03(random) ± 0 . 12(systematic) for M96. These results show that they are indeed the members of the same group. With these results we derive absolute maximum magnitudes of two SNe (SN 1989B in M66 and SN 1998bu in M96). V -band magnitudes of these SNe Ia are ∼ 0.2 mag fainter than SN 2011fe in M101, the nearest recent SN Ia. We also derive near-infrared magnitudes for SN 1998bu. Optical magnitudes of three SNe Ia (SN 1989B, SN 1998bu, and SN 2011fe) based on TRGB analysis yield a Hubble constant, H 0 = 67 . 6 ± 1 . 5(random) ± 3 . 7(systematic) km s -1 Mpc -1 . This value is similar to the values derived from recent WMAP9 results, H 0 = 69 . 32 ± 0 . 80 km s -1 Mpc -1 . and from Planck results, H 0 = 67 . 3 ± 1 . 2 km s -1 Mpc -1 , but smaller than other recent determinations based on Cepheid calibration for SNe Ia luminosity, H 0 = 74 ± 3 km s -1 Mpc -1 . Subject headings: galaxies: distances and redshifts - galaxies: individual (M66, M96) - galaxies: stellar content - supernovae: general - supernovae: individual (SN 1989B, SN 1998bu)", "pages": [ 1 ] }, { "title": "THE TIP OF THE RED GIANT BRANCH DISTANCES TO TYPE IA SUPERNOVA HOST GALAXIES. II. M66 AND M96 IN THE LEO I GROUP", "content": "Myung Gyoon Lee and In Sung Jang Astronomy Program, Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 151-742, Korea Draft version September 18, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Type Ia Supernovae (SNe Ia) are a powerful tool to investigate the expansion history of the universe, because their peak luminosity is as bright as a galaxy and is known as an excellent standard candle. Since the discovery of the acceleration of the universe based on the observations of SNe Ia, higher than ever accuracy of their peak luminosity is needed to investigate various problems in cosmology (Freedman & Madore 2010; Riess et al. 2011; Lee & Jang 2012; Tammann & Reindl 2013). We started a project to improve the accuracy of the calibration of the peak luminosity of SNe Ia by measuring accurate distances to nearby resolved galaxies that host SNe Ia. We derive accurate distances to the SN Ia host galaxies using the method to measure the luminosity of the tip of the red giant branch (TRGB) (Lee et al. 1993). We presented the result of the first target, M101, a wellknown spiral galaxy hosting SN 2011fe that is the nearest SN Ia since 1972 (Lee & Jang 2012 (Paper I)). This paper is the second of the series, presenting the results for M66 and M96 in the Leo I Group. M66 (NGC 3627, SAB(s)b) and M96 (NGC 3368, SAB(rs)ab) are nearby bright spiral galaxies hosting SNe Ia: SN 1989B in M66 (Evans & McNaught 1989; Wells et al. 1994) and SN 1998bu in M96 (Villi et al. 1998; Suntzeff et al. 1999; Jha et al. 1999; Hernandez et al. 2000; Spyromilio et al. 2004). M66 has been host to other three SNe as well: SN II 1973R (Ciatti & Rosino 1977), SN imposter SN 1997bs (Van Dyk et al. 2000), and SN II-L 2009hd (Elias-Rosa et al. 2011). They are considered to be the members of the compact Leo I Group that includes three subgroups: the Leo Triplet (M66, M65, and NGC 3628), the M96 Group (including M96 (NGC 3368), M95 (NGC 3351), and M105 (NGC 3379)), and the NGC 3607 Group (de Vaucouleurs 1975; Saha et al. 1999). The Leo I Group has played an important role as a stepping stone for calibration of the secondary distance indicators, because it includes both early and late type galaxies at the distance closer than the Virgo cluster and because it hosts SNe Ia. In particular M66 and M96 have been used as important calibrators for the absolute magnitudes of SNe Ia and the Tully-Fisher relation (Saha et al. 1999; Suntzeff et al. 1999; Saha et al. 2006; Jha et al. 2007; Hislop et al. 2011; Tammann & Reindl 2013). Harris et al. (2007a) derived a value for the distance to the Leo I Group, ( m -M ) 0 ≈ 30 . 10 ± 0 . 05 ( ≈ 10 . 5 Mpc), from the mean of the known distances to five brightest galaxies in the group (M66, M95, M96, M105, NGC 3351 and NGC 3377). Often the member galaxy candidates without known distances are assumed to be at the same distance, but it is still important to derive a precise distance to each member galaxy candidate for investigating various aspects of the member galaxies. Unfortunately recent estimates of the distances to M66 and M96 based on resolved stars show a large range (Hislop et al. 2011; Tammann & Reindl 2013). Saha et al. (1999) found 68 Cepheids in M66 from F 555 W and F 814 W images obtained with the Hubble Space Telescope (HST) /Wide Field Planetary Camera 2 (WFPC2) and derived a distance modulus of ( m -M ) 0 = 30 . 22 ± 0 . 12 from the photometry of 25 good Cepheids. Later Cepheid estimates range from ( m -M ) 0 = 29 . 70 ± 0 . 07 (Willick & Batra 2001) to 30 . 50 ± 0 . 09 (Saha et al. 2006), showing as much as 0.8 mag differences. On the other hand, Mould & Sakai (2009a) presented a distance modulus ( m -M ) 0 = 29 . 82 ± 0 . 10 using the TRGB method from F 555 W and F 814 W images obtained with the HST /Advanced Camera for Surveys (ACS) . Furthermore Tully et al. (2009) presented an even smaller TRGB distance estimate, ( m -M ) 0 = 29 . 60 ± 0 . 09. Thus there is a significant difference between the Cepheid distances and TRGB distances as well as among the estimates of each method. In the case of M96, Tanvir et al. (1995) found 7 Cepheids from HST /WFPC2 F 555 W and F 814 W images and derived a distance modulus of ( m -M ) 0 = 30 . 32 ± 0 . 16. Later Cepheid estimates showed a significant spread, ranging from ( m -M ) 0 = 29 . 94 ± 0 . 13 (Willick & Batra 2001) to 30 . 42 ± 0 . 15 (Kochanek 1997). Surprisingly Mould & Sakai (2009b) presented a much smaller TRGB distance estimate ( m -M ) 0 = 29 . 65 ± 0 . 28 derived from the HST images. Thus the difference between the Cepheid distances and TRGB distance is as much as 0.3 to 0.7 mag and the range of the Cepheid distances is about 0.4. In this study we use the well-known TRGB method to estimate the distances to M66 and M96 from the images available in the HST archive. The TRGB method is an efficient and precise primary distance indicator for resolved galaxies so that it is an excellent tool for calibration of more powerful distance indicators such as SN Ia and Tully-Fisher relations (Lee et al. 1993; Sakai et al. 1996; Jang et al. 2012; Tammann & Reindl 2013). Section 2 describes how we derive photometry of the point sources in the images and § 3 presents color-magnitude diagrams of the resolved stars in each galaxy, and derive distances to each galaxy using the TRGB method. We discuss implications of our results in § 4, and summarizes primary results in the final section.", "pages": [ 1, 2 ] }, { "title": "2. DATA REDUCTION", "content": "Table 1 lists the information of the HST /ACS images we used for the TRGB analysis in this study: F 555 W and F 814 W images of M66 and M96 (Proposal ID: 10433). We made drizzled images for each filter combining the flat fielded images in the HST archive using Tweakreg and AstroDrizzle task in DrizzlePac provided by the Space Telescope Science Institute (http://www.stsci.edu/hst/HST overview/drizzlepac/). Total exposure times for F 555 W and F 814 W are, respectively, 2224 s and 8872 s for M66, and 2280 s and 9112 s for M96. In Figure 1 we illustrate the locations of the HST fields in the gray scale maps of i -band Sloan Digital Sky Survey images of M66 and M96. The HST fields cover the west region of each galaxy off from the galaxy center. Two known SNe Ia (SN 1989B and SN 1998bu) are located close to the center of each galaxy and are not covered by these images, as marked in Figure 1. Instrumental magnitudes of point sources in the images were obtained using the DAOPHOT package in IRAF (Stetson 1994), as done for M101 in Lee & Jang (2012). Details are described in Lee & Jang (2012). Mean values for the aperture correction errors are 0.02 mag for both filters. The instrumental magnitudes were converted into the standard Johnson-Cousins V I magnitudes, using the information in Sirianni et al. (2005). The average errors for this transformation are 0.02 mag. We adopted the standard Johnson-Cousins V I magnitudes for transformation to compare our results with others in the literature and combine our results with those for other galaxies sometimes based on F606W images.", "pages": [ 2 ] }, { "title": "3.1. Photometry of Resolved Stars", "content": "The HST /ACS fields cover disk regions with spiral arms in each galaxy. We need to select resolved old red giants for the analysis of the TRGB method. Therefore we selected an outer region avoiding arms in each field, as marked by the hatched region in Figure 1. Thus chosen regions have the lowest sky background level in the images. Color-magnitude diagrams (CMDs) of the resolved stars in the selected regions in M66 and M96 are plotted in Figure 2. It shows that most of the resolved stars in each galaxy are red giants belonging to the thick slanted feature, which is a red giant branch (RGB). The brightest part of the RGB is seen at I ≈ 26 . 2 mag in each galaxy, which corresponds to the TRGB. We adopted the foreground reddening values, E ( B -V ) = 0 . 028 for M66 and 0.022 for M96 in Schlegel et al. (1998); Schlafly & Finkbeiner (2011). These values yield A I = 0 . 049 and E ( V -I ) = 0 . 040 for M66 and A I = 0 . 038 and E ( V -I ) = 0 . 031 for M96. We assumed that internal reddening for the old red giants is zero.", "pages": [ 2 ] }, { "title": "3.2. TRGB Distance Measurement", "content": "We estimated the distances to M66 and M96 from the photometry of the resolved stars using the TRGB method, as described in Lee & Jang (2012). Figure 3(a) and (c) plot the I -band luminosity functions of the red giants obtained counting the stars inside the box as marked in Figure 2. In Figure 3 an abrupt discontinuity is seen at I ≈ 26 . 2 mag for each galaxy, which is also noticed in the CMDs. This matches the TRGB in each galaxy. We performed a quantitative analysis of the TRGB measurement using the edge-detecting algorithm (Sakai et al. 1996; M'endez et al. 2002; Mouhcine et al. 2010). When the I -band luminosity function of the stars is given by N ( I ) and σ I is the mean photometric error, the edge-detection response function is given by E ( I ) (= N ( I + σ I ) -N ( I -σ I )). The values of the TRGB magnitudes were determined from the peak values of the edge-detection response function. Figure 3(b) and (d) illustrate the edge-detection response functions for M66 and M96, respectively. The edge-detection response function for each galaxy shows a strong peak at the position corresponding to the TRGB. We estimated the measurement errors for the TRGB magnitudes using bootstrap resampling method as described in Lee & Jang (2012). Thus estimated TRGB magnitudes are I TRGB = 26 . 20 ± 0 . 03 for M66 and 26 . 21 ± 0 . 03 for M96, both of which are almost the same. We obtained a median color value of the TRGB from the color of the brightest part of the RGB: ( V -I ) TRGB = 1 . 97 ± 0 . 05 for M66 and 1 . 93 ± 0 . 04 for M96. For calculating distance moduli from apparent TRGB magnitudes we adopted a relation Rizzi et al. (2007) derived: M I , TRGB = -4 . 05( ± 0 . 02) + 0 . 217( ± 0 . 01)(( V -I ) 0 ,TRGB -1 . 6). After correction for foreground reddening, we derived distance modulus : ( m -M ) 0 = 30 . 12 ± 0 . 03 for M66 and ( m -M ) 0 = 30 . 15 ± 0 . 03 for M96 (where 0.03 is a measurement error). We derived a value of the systematic error to be 0.12, from the combination of the TRGB magnitude error, aperture correction error, and standard transformation error, as described in Lee & Jang (2012). Thus derived distance to these galaxies are 10 . 57 ± 0 . 15 ± 0 . 58 Mpc for M66 and 10 . 72 ± 0 . 15 ± 0 . 59 Mpc for M96. Our distance estimates for M66 and M96 are summarized in Table 2.", "pages": [ 2, 4 ] }, { "title": "4.1. Comparison with Previous Distance Estimates", "content": "There are numerous previous estimates for the distances to M66 and M96 based on various methods (TRGB, Cepheids, Tully-Fisher relations, surface brightness fluctuation (SBF), planetary nebula luminosity functions (PNLFs), and SNe Ia), as listed in Tables 3 and 4. We compare our estimates for the distances to M66 and M96 with these previous estimates. Figure 4 shows a comparison of distance measurements for each galaxy in this study and previous studies. We derived a probability density curve for each measurement with a normalized Gaussian function centered at the distance modulus value with a width equal to the measurement error. Comparison of the TRGB distances derived in this study and previous studies (Mould & Sakai 2009a; Tully et al. 2009) shows significant differences. Our distance estimate for M66 is 0.3 mag larger than that of Mould & Sakai (2009a) (( m -M ) 0 = 29 . 82 ± 0 . 10) and 0.5 mag larger than that of Tully et al. (2009) (( m -M ) 0 = 29 . 60 ± 0 . 09). In the case of M96, our distance estimate is 0.5 mag larger than that of Mould & Sakai (2009b) (( m -M ) 0 = 29 . 65 ± 0 . 18). These differences are explained in terms of the TRGB magnitude differences: the two previous studies derived much brighter magnitudes for the TRGB than this study. Mould & Sakai (2009a) and Tully et al. (2009) presented I TRGB = 25 . 83 and 25.56, respectively, for M66, which are, respectively, 0.37 mag and 0.64 mag brighter than the our value. Similarly Mould & Sakai (2009b) presented I TRGB = 25 . 66 for M96, which is 0.55 mag brighter than our value. What caused these differences is not clear, but the previous measurements might have been affected by younger stars in the disk of each galaxy. Note that we used only the stars in the arm-free regions in each galaxy to reduce the contamination due to younger stars for our analysis. Our distance estimate is consistent with some of the previous estimates based on other distance indicators (Cepheids, Tully-Fisher relations, SBF, and SN Ia). However, the spread in the previous measurements for each method is significant and the errors for each measurement are mostly larger than ours. It is expected that our results will be useful for improving the calibration of these other distance indicators in the future.", "pages": [ 4 ] }, { "title": "4.2. The Membership of the Leo I Group", "content": "The distance estimates derived in this study show that M66 and M96 are at the same distance and that they are located at the same distance as the mean distance to the Leo I Group (Harris et al. 2007a). These confirm that M66 and M96 are indeed the members of the Leo I Group. M96 is the brightest member of the Leo I Group. However it is not located at the center of the M96 Group. An E1 galaxy M105 resides at the center of the M96 Group, and M96 is 48 ' at the south-west of the group center. M96 has a large pseudo bulge (Nowak et al. 2010), and appears to be connected to a tidal feature extended out from the well-known giant HI ring surrounding a pair of M105 and NGC 3384 (SB0) (Schneider et al. 1983; Schneider 1989). Whether this giant gas ring around M105/NGC 3384 is primordial or formed via collision of disk galaxies (M105/NGC 3384 and M96) has been controversial (Thilker et al. 2009; Michel-Dansac et al. 2010). Precise distance estimates of M96 and M105/NGC 3384 will be useful to investigate the origin of this giant ring, because the relative distances (as well as velocities) are critical constraints for simulation models (Michel-Dansac et al. 2010). Here we compare the distance to M96 with that of M105. Harris et al. (2007b) estimated the I -band magnitude of the TRGB for M105 to be I TRGB = 26 . 10 ± 0 . 10 from the HST /ACS F 606 W and F 814 W images of a field 630 '' west and 173 '' north of the galaxy center, and derived a distance modulus ( m -M ) 0 = 30 . 10 ± 0 . 16 adopting the foreground reddening of A I = 0 . 05 ± 0 . 02 and the absolute TRGB magnitude given in Bellazzini et al. (2004), M I,TRGB = -4 . 05 ± 0 . 12. This value is nearly the same as the TRGB distance to M96 derived in this study, showing that M105 and M96 are at the same distance. The radial velocities of M96 and M105 are also very similar ( 897 ± 4 km s -1 and 911 ± 2 km s -1 , respectively), while they are ∼ 200 km s -1 larger than that of NGC 3384, 704 ± 2 km s -1 . These results indicate that the three galaxies (M96, M105, and NGC 3384) are close enough to interact with each other. This supports the collisional scenario that the giant gas ring was formed when M96 collided with NGC 3384/M105 (Michel-Dansac et al. 2010).", "pages": [ 4 ] }, { "title": "4.3. The Calibration of the Absolute Magnitudes of SNe Ia and the Hubble Constant", "content": "The distances to M66 and M96 derived in this study can be used to improve the calibration of the absolute magnitudes of SNe Ia. Tables 5 and 6 list, respectively, the V -band maximum magnitudes of SN 1989B and SN 1998bu derived in this study and previous studies (Gibson et al. 2000; Sandage et al. 2006; Tammann & Reindl 2013). Recently Tammann & Reindl (2013) derived M V, max = -19 . 45 ± 0 . 15 for SN 1989B and M V, max = -19 . 38 ± 0 . 16 for SN 1998bu from the photometry in the literature (Suntzeff et al. 1999; Jha et al. 1999; Hernandez et al. 2000; Wells et al. 1994), adopting a mean TRGB distance of the Leo I Group, ( m -M ) 0 = 30 . 39 ± 0 . 10. These values were obtained after correcting for the Galactic extinction, host galaxy extinction, and decline rates (∆ m 15 ). These values will become fainter by 0.27 and 0.24 mag if the A List of Distance Measurements for M66 a The Extragalactic Distance Database (EDD) (Tully et al. 2009). TRGB distances to M66 and M96 derived in this study are used: M V, max = -19 . 18 ± 0 . 11 for SN 1989B and -19 . 14 ± 0 . 12 for SN 1998bu. Other previous estimates (Gibson et al. 2000; Sandage et al. 2006) are affected in the similar way, yielding M V, max = -19 . 46 ± 0 . 17 for SN 1989B and -19 . 38 ± 0 . 11 for SN 1998bu in Gibson et al. (2000), and M V, max = -19 . 17 ± 0 . 06 for SN 1989B and -19 . 11 ± 0 . 06 for SN 1998bu in Sandage et al. (2006). SN 2011fe in M101 is the nearest recent SN Ia with modern photometry so that it is an excellent object for calibration of SNe Ia. Lee & Jang (2012) derived maximum magnitudes of SN 2011fe from the photometry in the literature, adopting a new TRGB distance derived from the weighted mean of nine fields in M101, M V, max = -19 . 38 ± 0 . 05(random) ± 0 . 12(systematic). Thus V -band magnitudes of SN 1989B and SN 1998bu are ∼ 0 . 2 mag fainter than that of SN 2011fe. This difference is similar to the dispersion of the absolute mag- des of SNe Ia, 0.14 (Tammann & Reindl 2013). It is noted that the internal extinction for SN 2011fe is known to be negligible, A V = 0 . 04 (Patat et al. 2011), while those for SN 1989B and 1998bu are not, as listed in Tables 7 and 8, respectively. The values for A V derived in the previous studies range from 0 . 82 ± 0 . 08 to 1 . 33 ± 0 . 14 for SN 1989B and from 0 . 74 ± 0 . 11 to 1 . 06 ± 0 . 11 for SN 1998bu (Reindl et al. 2005; Wang et al. 2006; Jha et al. 2007; Tammann & Reindl 2013). Therefore the errors due to internal extinction for SN 1989B and SN 1998bu are expected to be larger than that for SN 2011fe. Further studies to derive better estimates for internal extinction for both SNe are needed in the future. Near-infrared (NIR) photometry of SN 1998bu in M96 is available in the literature so that SN 1998bu plays as one of the important calibrators for NIR magnitudes of SNe Ia. Tammann & Reindl (2013) derived JHK s maximum magnitudes at each band of SN 1998bu from the previous photometry (Jha et al. 1999; Suntzeff et al. 1999; Hernandez et al. 2000; Wood-Vasey et al. 2008) : J = 11 . 55 ± 0 . 03, H = 11 . 59 ± 0 . 03, and K s = 11 . 42 ± 0 . 03. They adopted a value for internal extinction of A V = 0 . 74 ± 0 . 11. Corresponding extinctions in NIR bands are A J = 0 . 19 ± 0 . 03, A H = 0 . 12 ± 0 . 02, and A K S = 0 . 08 ± 0 . 01. If we apply internal extinctions presented above and adopt our new TRGB distance, we obtain NIR absolute magnitudes of SN 1998bu : M J, max = -18 . 79 ± 0 . 05, M H, max = -18 . 68 ± 0 . 05, and M K s , max = -18 . 81 ± 0 . 04. Lee & Jang (2012) derived JHK s magnitudes of SN 2011fe in M101 from the photometry in Matheson et al. (2012), adopting a new TRGB distance they derived: M J, max = -18 . 79 ± 0 . 04(random) ± 0 . 12(systematic), M H, max = -18 . 55 ± 0 . 04(random) ± 0 . 12(systematic), and M K s , max = -18 . 66 ± 0 . 05(random) ± 0 . 12(systematic). Thus absolute J magnitude of SN 1998bu is the same as that of SN 2011fe, while H,K s magnitudes of SN 1998bu are ∼ 0 . 14 mag brighter than those of SN 2011fe. We derive weighted mean values of SN 1989bu and SN 2011fe from these: M J, max = -18 . 79 ± 0 . 03, M H, max = -18 . 60 ± 0 . 03, and M K s , max = -18 . 75 ± 0 . 03. It is noted that these values are 0 . 2 ∼ 0 . 4 mag brighter than recent calibrations of the NIR magnitudes for SNe Ia available in the literature (Krisciunas et al. 2004; Folatelli et al. 2010; Barone-Nugent et al. 2012; Kattner et al. 2012). Recently several calibrations of the NIR absolute magnitudes of SNe Ia were published, but they show a large spread with ∼ 0 . 2 mag differences (Wood-Vasey et al. 2008; Folatelli et al. 2010; Burns et al. 2011; Mandel et al. 2011; Kattner et al. 2012; Barone-Nugent et al. 2012; Matheson et al. 2012). Further studies are needed to understand these large differences in the NIR magnitudes of SNe Ia. The relations between the Hubble constant and the absolute magnitude of SNe Ia are given by log H 0 = 0 . 2 M V,max + 5 + (0 . 688 ± 0 . 004) in Reindl et al. (2005) or by the equations (2) and (4) in Gibson et al. (2000). Using these relations we derive the Hubble constant : H 0 = 69 . 1 ± 3 . 2(random) km s -1 Mpc -1 for SN 1989B, H 0 = 71 . 0 ± 2 . 6(random) km s -1 Mpc -1 for SN 1998bu, and H 0 = 65 . 0 ± 2 . 1(random) km s -1 Mpc -1 for SN 2011fe. A weighted mean of these three measurement is H 0 = 67 . 6 ± 1 . 5(random) ± 3 . 7(systematic) km s -1 Mpc -1 . Note that this value for the Hubble constant is similar to the recent estimates based on the cosmic microwave background radiation maps in WMAP9 data, H 0 = 69 . 32 ± 0 . 80 km s -1 Mpc -1 (Bennett et al. 2012) and Planck data H 0 = 67 . 3 ± 1 . 2 km s -1 Mpc -1 (Planck Collaboration et al. 2013), but smaller than other recent determinations based on Cepheid calibration for SNe Ia luminosity, H 0 = 74 ± 3 km s -1 Mpc -1 (Riess et al. 2011; Freedman et al. 2012) .", "pages": [ 4, 5, 7, 8 ] }, { "title": "5. SUMMARY", "content": "We present V I photometry of the resolved stars in two spiral galaxies M66 and M96 that host SNe Ia in the Leo I Group, derived from HST /ACS F 555 W and F 814 W images. Then we estimate the distances to these two galaxies applying the TRGB method to this photometry. We summarize main results in the following. This paper is based on image data obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). The authors would like to thank WonKee Park for technical support in image processing. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) (No. 2012R1A4A1028713).", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "Ajhar, E. A., Tonry, J. L., Blakeslee, J. P., Riess, A. G., & Schmidt, B. P. 2001, ApJ, 559, 584 Barone-Nugent, R. L., Lidman, C., Wyithe, J. S. B., et al. 2012, MNRAS, 425, 1007 9 Bellazzini, M., Ferraro, F. R., Sollima, A., Pancino, E., & Origlia, L. 2004, A&A, 424, 199 Bennett, C. L., Larson, D., Weiland, J. L., et al. 2012, arXiv:1212.5225 Burns, C. R., Stritzinger, M., Phillips, M. M., et al. 2011, AJ, 141, 19 Ciardullo, R., Feldmeier, J. J., Jacoby, G. H., et al. 2002, ApJ, 577, 31 Ciatti, F., & Rosino, L. 1977, A&A, 56, 59 de Vaucouleurs, G. 1975, in Stars and Stellar Systems 9 : Galaxies and the Universe, ed. A. Sandage, M. Sandage, & J. Kristian (Chicago : Univ. Chicago Press), 557 Dolphin, A. E., & Kennicutt, R. C., Jr. 2002, AJ, 123, 207 a Same as Table 5. Hernandez, M., Meikle, W. P. S., Aparicio, A., et al. 2000, MNRAS, 319, 223 Hislop, L., Mould, J., Schmidt, B., et al. 2011, ApJ, 733, 75 Jang, I. S., Lim, S., Park, H. S., & Lee, M. G. 2012, ApJ, 751, L19 Jensen, J. B., Tonry, J. L., Barris, B. J., et al. 2003, ApJ, 583, 712 Jha, S., Garnavich, P. M., Kirshner, R. P., et al. 1999, ApJS, 125, 73 Lee, M. G., & Jang, I. S. 2012, ApJ, 760, L14 (Paper I) Phillips, M. M., Lira, P., Suntzeff, N. B., et al. 1999, AJ, 118, 1766 Pierce, M. J. 1994, ApJ, 430, 53 Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2013, arXiv:1303.5076 Riess, A. G., Macri, L., Casertano, S., et al. 2011, ApJ, 730, 119 Rizzi, L., Tully, R. B., Makarov, D., et al. 2007, ApJ, 661, 815 Russell, D. G. 2002, ApJ, 565, 681", "pages": [ 8, 9, 10 ] } ]

[ { "title": "ABSTRACT", "content": "The high-mass X-ray binary SMC X-1 is an eclipsing binary with an orbital period of 3.89 d. This system exhibits a superorbital modulation with a period varying between ∼ 40 d and ∼ 65 d. The instantaneous frequency and the corresponding phase of the superorbital modulation can be obtained by a recently developed time-frequency analysis technique, the Hilbert-Huang transform (HHT). We present a phase-resolved analysis of both the spectra and the orbital profiles with the superorbital phase derived from the HHT. The X-ray spectra observed by the Proportional Counter Array onboard the Rossi X-ray Timing Explorer are fitted well by a blackbody plus a Comptonized component. The plasma optical depth, which is a good indicator of the distribution of material along the line of sight, is significantly anti-correlated with the flux detected at 2 . 5 -25 keV. However, the relationship between the plasma optical depth and the equivalent width of the iron line is not monotonic: there is no significant correlation for fluxes higher than ∼ 35 mCrab but clear positive correlation when the intensity is lower than ∼ 20 mCrab. This indicates that the iron line production is dominated by different regions of this binary system in different superorbital phases. To study the dependence of the orbital profile on the superorbital phase, we obtained the eclipse profiles by folding the All Sky Monitor light curve with the orbital period for different superorbital states. A dip feature, similar to the pre-eclipse dip in Her X-1, lying at orbital phase ∼ 0 . 6 -0 . 85, was discovered during the superorbital transition state. This indicates that the accretion disk has a bulge that absorbs considerable X-ray emission in the stream-disk interaction region. The dip width is anti-correlated with the flux, and this relation can be interpreted by the precessing tilted accretion disk scenario. Subject headings: accretion disks - stars: individual (SMC X-1) - X-rays: binaries -X-rays: individual (SMC X-1)", "pages": [ 1 ] }, { "title": "Superorbital Phase-Resolved Analysis of SMC X-1", "content": "Chin-Ping Hu, Yi Chou, Ting-Chang Yang, Yi-Hao Su Graduate Institute of Astronomy, National Central University, Jhongli 32001, Taiwan Hu: [email protected], Chou: [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "SMC X-1, first discovered by Leong et al. (1971), is an eclipsing high-mass X-ray binary (HMXB) consisting of a neutron star with a mass of 1.06 M /circledot (van der Meer et al. 2007) and a B0 I supergiant companion, Sk 160, with a mass of 17.2 M /circledot (Reynolds et al. 1993). This system exhibits X-ray pulsation with a 0.71 s period (Lucke et al. 1976), an orbital eclipse every 3.89 d (Schreier et al. 1972), and superorbital modulation (Gruber & Rothschild 1984) with a period varying between ∼ 40 d and ∼ 65 d (Trowbridge et al. 2007; Hu et al. 2011). The superorbital modulation is interpreted as an obscuring effect caused by a precessing warped and tilted accretion disk (Wojdowski et al. 1998). After the launch of the All Sky Monitor (ASM) onboard the Rossi X-ray Timing Explorer (RXTE) , the variation of the superorbital modulation period of SMC X-1 was studied by various time-frequency techniques, e.g., the Morlet wavelet transform (Rib'o et al. 2001), the dynamic power spectrum (Clarkson et al. 2003), the slide Lomb-Scargle periodogram (Trowbridge et al. 2007), and the Hilbert-Huang transform (Hu et al. 2011). Among these techniques, the HHT, proposed by Huang et al. (1998), can provide a well-defined instantaneous frequency, as well as the corresponding phase, of the superorbital modulation. The X-ray spectra are good indicators of the properties of the central X-ray source and the environment of the binary system. Woo & Clark (1995) studied the variation in the spectral properties for a complete orbital cycle and identified the wind dynamics in this binary system using the data collected by Ginga . Wojdowski & Clark (2000) analyzed the X-ray spectrum observed by the Advanced Satellite for Cosmology and Astrophysics during an eclipse and found that it is inconsistent with the line-driven wind model proposed by Blondin & Woo (1995). Vrtilek et al. (2005) reported the results of spectral analysis of eight Chandra observations, which covered different orbital phases and superorbital states. They found that the spectra in the superorbital high state are independent of the orbital phase, whereas the low-state spectra depend strongly on the orbital phase. Combining timing and spectral analysis could facilitate to further study of the nature of this binary system. For example, Naik & Paul (2004) applied energy-resolved timing analysis to the BeppoSAX observations and found that the pulse profiles of the soft thermal component and the hard power-law component are different. Hickox & Vrtilek (2005) made a similar analysis and presented pulse phase-resolved spectral analysis of SMC X-1 using Chandra and XMM-Newton observations. The variations in the pulse profiles of different spectral models were attributed to a twisted, pulsar-illuminated inner accretion disk. Because the Chandra and XMM-Newton observations covered parts of the superorbital phase, the long-term variation in the pulse profile can roughly support the disk and beam geometry. If the number of observations is large enough to cover most of the superorbital modulation, it is possible to further study the variation in the spectral properties versus the superorbital modulation in detail with phase-resolved spectral analysis. For example, Leahy (2001) studied the spectral variation of the 35 d superorbital modulation of Her X-1 using Ginga observations. Naik & Paul (2003) analyzed the variations in the spectral parameters, especially the iron line intensity and equivalent width (EW), in different superorbital states of LMC X-4 and Her X-1. Because the X-ray observations of SMC X-1 made by the RXTE cover most of the superorbital phase, the only difficulty in performing phase-resolved spectral analysis is that the period of the superorbital modulation is not stable, so the corresponding phase is hard to define. However, this difficulty can be solved by using an advanced time-frequency analysis method, the HHT, which can provide a well-defined phase function for modulation with a variable period. The orbital profile, in addition to the spectral behavior, was found to vary with the superorbital modulation. Trowbridge et al. (2007) studied the orbital profile of different superorbital phases, but they had insufficient statistics to demonstrate a relationship between the orbital profile and superorbital phase. The increased number of RXTE/ASM observations since then, combined with the superorbital phase defined by the HHT, should provide a statistically sound basis for studying orbital profile variations. We present our studies of the superorbital phase-resolved analysis of the X-ray spectra and the variation in the orbital profile in this paper. In section 2, we briefly introduce the observations made by the Proportional Counter Array (PCA) onboard the RXTE and the light curve collected by the ASM. The spectral models, the variation in the spectral parameters versus the superorbital phase, and the variation in the orbital profile with the superorbital phase, are described in Section 3. Finally, we discuss our results in section 4, in particular the variation in the iron line EW and the mechanism that causes the variation in the X-ray dip.", "pages": [ 1, 2, 3 ] }, { "title": "2.1. RXTE PCA", "content": "The PCA onboard the RXTE consists of five proportional counter units (PCUs) with an energy range of 2-60 keV. The field of view is limited by the collimator to ∼ 1 · . SMC X-1 was observed frequently by the PCA from 1996 August 28 to 2004 January 30. red Between 1997 and 1998, the PCA made 1 -3 observations on SMC X-1 per month. These observations provided ∼ 40 data ponits randomly distributed over the superorbital cycle. Other 5 series of consecutive observations made in 1996, 2000, 2003, and 2004 covered /lessorsimilar 0 . 5 superorbital cycles. All the data are archived on the website of the High Energy Astrophysics Science Archive Research Center (HEASARC) of the National Aeronautics and Space Administration (NASA). The spectra were analyzed using all of the Standard-2 mode data except for those between 1998 October 16 and 1998 December 4, which were contaminated by an outburst from the nearby X-ray pulsar XTE J0111.27317 (Chakrabarty et al. 1998). Furthermore, two PCA observations (OBSID 30125-05-01-00 and 30125-05-03-02) were rejected because they contained insufficient photons to yield meaningful spectral fittings. The Standard-2 mode data provide 129 energy channels with an energy range of 2 to 60 keV and 16 s timing resolution; however, only the energy channels between ∼ 2.5 and 25 keV were used for spectral fitting. For the low-energy boundary, we ignored the first three channels because the PCA calibration is not accurate for channels 1 -3. Among the five PCUs, we extracted only the spectra from PCU2 because it was always in operation. The spectra, response matrices, and background models were created using the FTOOLS analysis software. Furthermore, a 1% systematic error was added to the error of each spectrum as in the manipulation by ˙ Inam et al. (2010), which was suggested by Wilms et al. (1999). The time span covers August 1996 to January 2004, which corresponds to RXTE gain epochs 3 to 5. To correct the problem with gain-drift of RXTE , we normalized the flux of individual observations to the nearest epoch Crab observation.", "pages": [ 3, 4 ] }, { "title": "2.2. RXTE ASM", "content": "Although the PCA observations cover most of the superorbital states, they are still insufficient for studying the relationship between the orbital and superorbital profiles. Thus, we used ASM data to investigate the variation in the orbital profiles. The ASM onboard the RXTE continuously swept the entire sky once every 90 min for the entire RXTE lifetime. Its energy range is 1.3 to 12.1 keV. The summed band dwell light curve collected since MJD 50,134 was analyzed for orbital profile variations. Because the ASM gain of ASM has changed moderately in the last two years (Levine et al. 2011), the light curve is slightly noisy after MJD 55,200 and even the superorbital modulation cannot be recognized after MJD 55,600. Thus, the data collected after MJD 52,000 were excluded from this study.", "pages": [ 4 ] }, { "title": "3.1. High and Low State X-ray Spectra", "content": "To obtain the variation in the spectral parameters with respect to the superorbital phase, we first removed the data within orbital phase of 0.87 to 1.13 according to the orbital ephemeris proposed by Wojdowski et al. (1998) to avoid variations caused by eclipses. We then fitted the combined spectra for both the high and low state with different models in order to select the model that best describes the PCA spectra. The superorbital phases from Hu et al. (2011) were adopted to define four states according to the superorbital profile: the high state (0.19 - 0.54), the low state (0.73 - 1.05), and two transitions: the ascending state (0.05 - 0.19) and descending state (0.54 0.73). In previous studies, we noted that the flux in the same state may differ with the superorbital cycle; e.g., the high state count rate of the 40-d superorbital cycle is generally lower than that of the 65-d cycle because the modulation amplitude and period are anti-correlated (Hu et al. 2011). Because the number of X-ray photons is sufficiently high in the high state, it is improper to combine all the high state observations from different superorbital cycles, which may have different spectral properties. Thus, we combined only the spectra in a series of consecutive observations made in December 2003 to obtain the spectral model of the high state. For the low state, combining all the uneclipsed observations is acceptable because the count rate is relatively low. We did not combine the spectra in the transition phases because the spectral properties change dramatically during these states. The spectra, together with the corresponding background and RMF files, were combined using the addspec command. The XSPEC v12.8.0 package of HEAsoft was applied for spectral fitting. We tried two spectral models to describe the spectra of both the high and low states. Model 1 contains a blackbody component plus a simple power law with a high-energy cutoff (Woo & Clark 1995; Naik & Paul 2004; Vrtilek et al. 2005; Hickox & Vrtilek 2005). Model 2 contains a blackbody component plus a Comptonized component (Naik & Paul 2004; Vrtilek et al. 2005), which describes the Comptonization of soft blackbody photons in a hot plasma (Titarchuk 1994). A Gaussian line with a central energy of 6.4 keV, which corresponds to the central energy of the K α emission line of iron atoms, was added during spectral fitting. As a result, Model 1 can be described as follows: where σ ( E ) is the photoelectric cross section, n H is the equivalent hydrogen column density, f bb ( E ) is the blackbody emission, f pl ( E ) is the power law model, f cut ( E ) is a multiplicative model that represents a high-energy exponential cutoff, and f Fe ( E ) is a Gaussian iron emission line. Model 2 can be described as follows: where f comp ( E ) is the Comptonized component, which was implemented as compTT in the XSPEC package, and the input soft photon energy is set to the same value as the blackbody temperature. The best-fit parameters are shown in Table 1. The reduced χ 2 revealed that using the inverse Comptonized model provided significantly better fitting than using the power law with a high-energy cutoff. Thus, we selected Model 2 to describe the spectra from all the individual PCA observations in further analysis. The folded count rate spectra, corresponding models, and residuals are shown in Figure 1. From the observed parameters, we found that both n H and the plasma optical depth ( τ ) in the low state are significantly higher than those in the high state. In addition, the EW of iron line was also calculated by using the eqwidth command in XSPEC . The plasma temperature ( kT e ), line width ( σ line ), and EW are lower in the low state than in the high state.", "pages": [ 4, 5 ] }, { "title": "3.2. Superorbital Phase-Resolved Variation in Spectral Parameters", "content": "Before the phase-resolved spectral analysis, we examined the correlation between the PCA observations and the RXTE /ASM data on the basis of the phase obtained from the HHT. We divided the superorbital phase into 20 bins and only one of them contained no PCA observation. The mean flux of individual bins obtained by spectral fitting of PCA observations, together with the folded ASM light curve with the same 20 bins, are shown in Figure 2. The linear correlation coefficient between the PCA flux and the corresponding ASM count rate is 0.98 with a null hypothesis probability of 3 . 7 × 10 -7 , which indicates a very strong correlation. This correlation means that the PCA observations can almost reproduce the ASM ones according to the definition of the superorbital phase by the HHT. To obtain the variation in the spectral parameters that could reveal the emission and absorption properties of the different phases of disk precession, we folded all the spectral parameters according to the superorbital phase. Figure 3 shows the variations in the unabsorbed flux, n H , τ , kT e , and the EW of the iron line in different superorbital phases. The unabsorbed flux, which is normalized by the nearest epoch Crab observation, shows similar properties in the binned PCA observations and folded ASM light curve, e.g., an asymmetric superorbital profile. In the high state, the n H values are consistent with those in ˙ Inam et al. (2010) and the τ values also fall in a narrow range. We further examined the correlations and found that both n H and τ are anti-correlated with the flux. The linear correlation coefficient between n H and the flux is -0 . 69 with a null hypothesis probability of 5 . 4 × 10 -16 , whereas the linear correlation coefficient between τ and the flux is -0 . 81, with a null hypothesis probability of 5 . 8 × 10 -22 . Although both n H and τ show strong anti-correlations, n H is strongly influenced by the soft X-ray band, to which the PCA is insensitive. Furthermore, the n H values show great diversity during the low and ascending states. In contrast, the τ values show less diversity and have a stronger anti-correlation with the flux. Thus, we chose τ as the indicator of the material in the line of sight. The variation in τ and n H in the descending state seems to differ from that in the ascending state. This may represent either sampling bias caused by insufficient statistics or an indication of different absorption properties on the ascending/descending sides of the warp region. On the other hand, the linear correlation coefficient between kT e and the flux is 0 . 66 with a null hypothesis probability of 3 . 9 × 10 -15 , which indicates a strong positive correlation. This is not surprising because the reprocessing region is farther from the central X-ray source in the low state than in the high state. Another interesting parameter that is related to the flux is the EW of the iron line. The correlation between the EW and the flux is relatively complex, as shown in the upper panel of Figure 4. We found that the correlations in the high-intensity and low-intensity regions are probably different. The linear correlation coefficient between the EW and the flux for those data points with fluxes higher than 35 mCrab is -0 . 19 with a null hypothesis probability of 0 . 06, which indicates a marginal anti-correlation. However, when the flux is lower than 19 mCrab, the linear correlation coefficient between the EW and the flux is -0 . 59 with a null hypothesis probability of 1 . 9 × 10 -4 , which indicates a much stronger anti-correlation. The relation between the EW and τ is plotted in the lower panel of Figure 4. The distributions of those two data groups are obviously distinct, and the correlations are also different significantly. The linear correlation coefficient between the EW and τ at fluxes higher than 35 mCrab is 0 . 13 with a null hypothesis probability of 0.22, which indicates no significant correlation. On the other hand, the linear correlation coefficient between the EW and τ at fluxes lower than 19 mCrab is 0.64 with a null hypothesis probability of 5 . 3 × 10 -5 , which shows a strong positive correlation. The different correlations between the EW and τ indicate different origins of the iron line production, which will be discussed in section 4.", "pages": [ 5, 6, 7 ] }, { "title": "3.3. Superorbital Phase-Resolved Variation in Orbital Profile", "content": "In addition to the spectral behavior, the variation in the orbital profile with the superorbital phase is also an interesting issue for further study. Because the sampling of PCA observations is insufficient to investigate variations in the orbital profile, we used the data collected by the ASM to conduct this portion of our study. Each superorbital cycle was first equally divided into 20 subsets according to the superorbital phase. All the subsets of the same superorbital phase were then folded with the orbital ephemeris provided by Wojdowski et al. (1998). We therefore obtained 20 superorbital phase-resolved eclipsing profiles, as shown in Figure 5 (a). The eclipse profiles for the ascending, high, descending, and low states are shown in Figure 5 (b) - (e), respectively. The profile of the high state resembles that of a typical total eclipsing X-ray binary with a sharp eclipse, whereas those of the ascending, descending, and low states show greater variation in the uneclipsed region. For the ascending and descending states, we eliminated the data points near the high state within ∼ 0 . 05 superorbital cycles so that the most interesting characteristic of the orbital profile, the dip-like feature at orbital phase ∼ 0 . 6 -0 . 85, would be more visible. A broad dip feature between orbital phases 0.5 and 0.85 can also be seen in the ascending state, whereas a narrower dip feature during orbital phases 0.65 and 0.85 can be observed in the descending state. In the low state, the dip feature is unclear owing to low photon statistics. The dip feature is believed to represent absorption by the bulge in the accretion stream-disk interaction region. Because the variation in the orbital profile is strongly related to the superorbital phase, a twodimensional folded light curve is a good way to investigate the relationship between the orbital and superorbital profiles, as suggested by Trowbridge et al. (2007). We first defined a data window in the superorbital phase domain with a size of 0.05 cycles and folded the data points in the window with the orbital ephemeris. The window was then moved forward by a step of 0.01 cycles to obtain the next orbital profile. This process was repeated until the end of the data set. Finally, all the profiles were combined into a three-dimensional map, as shown in Figure 6. The uneclipsed count rates of all the orbital profiles were normalized to 1, and the resulting map was smoothed by a Gaussian filter to enhance the eclipse and dip features. From the dynamic folded light curve, we found that a sharp eclipse feature can be seen throughout both transition states and the high state ( ∼ 0 . 05 -0 . 75). In addition, a major dip appears in the descending and early low state. It is centered at orbital phase ∼ 0 . 7 and increases in width as the flux decreases. The dip feature in the ascending state is less visible and wider than that in the descending state, and the relationship between the dip width and flux is harder to obtain. In the deep low state, the eclipse and dip features cannot be recognized because of limited ASM sensitivity.", "pages": [ 7 ] }, { "title": "4. Discussion and Conclusions", "content": "SMC X-1 exhibits an obvious superorbital modulation, the period of which changes dramatically with time. The RXTE /PCA observations covered most of the superorbital phases, so the data provided fruitful information on the spectral properties of different superorbital states. ˙ Inam et al. (2010) analyzed all the spectra observed by the PCA. All the n H values, including those in different orbital and superorbital states, were found to increase as the X-ray flux decreased. This may be due to absorption by the companion or a warped region in the accretion disk. However, further details of the relationship between the spectral indices and superorbital phases were still unknown. The most difficult challenge, the definition of superorbital phases of variable periodicity, is solved by using the HHT. This research demonstrated phase-resolved spectral analysis of the uneclipsed observations. First, we found that the combined spectra are better described by the inverse Comptonized component than by a power-law with a high-energy cutoff. In addition, both n H and τ show an anti-correlation with the flux, but the correlation between τ and the flux is more significant than that between n H and the flux. Thus, we chose τ as an indicator of absorption by the material along the line of sight in the PCA energy range. Comparing n H and τ , we found that the EW of the iron line exhibits a more complex relationship with the flux. The correlation between the EW and the flux when the flux is lower than 19 mCrab shows similar behavior to that of LMC X-4 obtained by Naik & Paul (2003), who explained the variation in the EW by the presence of two producing regions. In the high-intensity state, the iron line is dominated by emissions near the central region of the compact object, and the EW is almost constant. When the disk precesses to the low-intensity state, the central region is almost obscured by the inner warp, and the iron line is dominated by another region far from the compact object. Unfortunately, the variation in n H with the superorbital phase of LMC X-4 is not available. In SMC X-1, we use another indicator, τ , to represent the materials that reprocessed the X-rays. From the relations between the EW and τ we found a positive correlation when the flux was lower than 19 mCrab, which contains the low state, the early ascending state, and possibly the late descending state, although no samples were available. This indicates that SMC X-1 also contains a weaker iron-line emitting region far from the central neutron star. In the high-intensity state, the EW of SMC X-1, like that of LMC X-4, remains in a relatively stable region and does not show a significant correlation with the flux and τ . TSMC X-1 differs from LMC X-4 in that the EWs of SMC X-1 in the high-intensity region are larger. The EW values of SMC X-1 when the flux is larger than 35 mCrab are similar to that of Her X-1 obtained by Leahy (2001), who studied the variations in the iron line EW during the 35-d superorbital cycle of Her X-1 using Ginga observations. The EW of Her X-1 in the main high state has a mean value of 0.48 keV and a standard deviation of 0.12 keV. In our case, those EWs at fluxes higher than 35 mCrab have a mean value of 0.49 keV and a standard deviation of 0.11 keV, which is consistent with that of Her X-1. Although the flux varies dramatically from 70 mCrab to 35 mCrab, the obscuring effect rather than the absorption effect dominates the variation in the flux: thus, neither τ nor the EW is strongly correlated with the flux. However, when the flux drops to less than ∼ 20 mCrab, the iron line production is dominated by another region more distant than the inner warp of the accretion disk. At the same time, absorption by materials far from the central region dominates the flux variation. Thus, both τ and the EW show strong anti-correlations with the flux. Using Chandra observations, Vrtilek et al. (2005) shows that the iron line of SMC X-1 consists of at least two components: a 6.4 keV K α line superposing on a broad Fe line. The Fe line in this study is a combination of those components due to limited spectral resolution of RXTE , and the variation of EW may indicates varying contributions of them. Thus, we could not identify how the individual component varies. The variation of all the line components on both the superorbital and orbital phases can be achieved after the X-ray observatories with high spectral resolutions, like Chandra and XMM-Newton , make enough amount of observations. From the variation in the orbital profile (Figures 5 and 6), we found an absorption dip at orbital phase ∼ 0 . 6 -0 . 85, the width of which increases as the count rate decreases in the transition and low states. Dips in X-ray binary systems are believed to be caused by absorption of the central X-ray emission in the impact region of the accretion stream and disk. Although SMC X-1 is an HMXB system, the steady high X-ray intensity could indicate the existence of a stream-fed accretion disk (Woo & Clark 1995). In the superorbital high state, the inclination angle of the tilted disk is low, and we can observe the central X-ray source directly during the uneclipsed phase. As the disk precesses to the transition state, the inclination angle becomes higher; the central X-ray source begins to be gradually obscured by the inner warped region of the accretion disk, and the X-ray intensity start to decrease. At the same time, the bulge in the outer rim of the accretion disk is also lifted, becoming closer to our line of sight. Because the bulge is co-rotating with the binary system, we see the periodic absorption dip at orbital phase ∼ 0 . 6 -0 . 85. Woo & Clark (1995) studied the light curve and spectral variations in 1.3 orbital cycles of Ginga observations. The extended and asymmetric eclipse transitions agree with the line-driven stellar wind model proposed by Blondin & Woo (1995), although the distribution of circumstellar material was further modified by Wojdowski & Clark (2000) and Wojdowski et al. (2008). However, we found that the orbital profile varies greatly with the superorbital phase. Thus, the variation in the orbital profile in our analysis is more likely related to the precession of the accretion disk than to the distribution of circumstellar material. Woo & Clark (1995) suggested that the dip feature is caused by the absorption of the accretion stream on the basis of the observed variation in n H in orbital phase 0.9 but this feature could not be directly obtained in their light curve (see Figure 2 in Woo & Clark (1995)). Instead, a tiny dip can be marginally obtained in orbital phase ∼ 0 . 7. We could not identify the superorbital state of the Ginga observation because it occurred before the launch of the RXTE , but it is unlikely to be the low state owing to the high count rate. If the Ginga observation was made during the superorbital high state, it implies that the narrow, shallow dip can be observed even in the high state. High-state dips could not be observed in the ASM light curve, probably because of the limited sensitivity. We look forward to the data collected by the Monitor of All-sky X-ray Image ( MAXI ), which has better sensitivity, for verification. Moon et al. (2003) and Trowbridge et al. (2007) also mentioned the dip feature of SMC X-1 and associated it with the light curve dips of Her X-1, although no further studies of the SMC X-1 dip were made. The dips of Her X-1 were first discovered by Giacconi et al. (1973), and they can be further divided into two groups, pre-eclipse dips and anomalous dips (Moon & Eikenberry 2001). A series of extensive studies have been made since then, e.g., Shakura et al. (1998); Moon & Eikenberry (2001); Igna & Leahy (2011) and references therein. The dips obtained in orbital phase ∼ 0 . 6 -0 . 85 of SMC X-1 may be associated with the pre-eclipse dip distributed in orbital phase 0 . 7 -0 . 9 of Her X-1. Interestingly, the pre-eclipse dip of Her X-1 would migrate toward earlier orbital phases when the disk precesses. We did not detect this migration behavior but obtained the variation in the dip width of SMC X-1. Thus, the mechanism of the dip of SMC X-1 is probably not as complex as that of Her X-1. Future extensive studies with observations of higher sensitivity could unveil the detailed properties of the dips of SMC X-1. This research made use of the RXTE /PCA data provided by the High Energy Astrophysics Science Archive Research Center of NASA's Goddard Space Flight Center. The data collected by the ASM are provided by the ASM/ RXTE teams at MIT and at the RXTE SOF and GOF at NASA's GSFC. This research was supported by grant NSC 100-2119-M-008-025 and NSC 1012112-M-008-010 from the National Science Council of Taiwan.", "pages": [ 8, 9, 10 ] }, { "title": "REFERENCES", "content": "Blondin, J. M., & Woo, J. W. 1995, ApJ, 445, 889 Chakrabarty, D., et al. 1998, Int. Astron. Union Circular, 7048, 1 Clarkson, W. I., Charles, P. A., Coe, M. J., Laycock, S., & Tout, M. D. 2003, MNRAS, 339, 447 Giacconi, R., et al. 1973, ApJ, 184, 227 Gruber, D. E., & Rothschild, R. E. 1984, ApJ, 283, 546 Hickox, R. C., & Vrtilek, S. D. 2005, ApJ, 633, 1064 Hu, C.-P., et al. 2011, ApJ, 740, 67 Huang, N. E., et al. 1998. Proc. R. Soc. Lond. A, 454, 903 Igna, C. D., & Leahy, D. A. 2011, MNRAS, 418, 2283 ˙ Inam, S. C¸ ., Baykal, A., & Beklen, E. 2010, MNRAS, 403, 378 Leahy, D. A. 2001, ApJ, 547, 449 Leong, C., Kellogg, E., Gursky, H., Tananbaum, H., & Giacconi, R. 1971, ApJ, 170, L67 Levine, A. M., Chakrabarty, D., Corbet, R. H. D., & Harris, R. J. 2011, ApJS, 196, 6 Lucke, R., et al. 1976, ApJ, 206, L25 Moon, D. S., & Eikenberry, S. S. 2001, ApJ, 552, L135 Moon, D. S., Eikenberry, S. S., & Wasserman, I. M. 2003, ApJ, 582, L91 Naik, S., & Paul, B. 2003, A&A, 401, 265 Naik, S., & Paul, B. 2004, A&A, 418, 655 Reynolds, A. P., Hilditch, R. W., Bell, S. A., & Hill, G. 1993, MNRAS, 261, 337 Rib´o , M., Peracaula, M., Paredes, J. M., N´u˜nez, J., & Otazu, X. 2001, in Proc. of Fourth INTE- GRAL Workshop, Exploring the Gamma-Ray Universe, ed. A. Gim´enez,V. Reglero, & C. Winkler (Noordwijk: ESA Publications Division), 333 Schreier, E., et al. 1972, ApJ, 178, L71 Shakura, N. I., Ketsaris, N. A., Prokhorov, M. E., & Postnov, K. A. 1998, MNRAS, 300, 992 Titarchuk, L. 1994, ApJ, 434, 313 Trowbridge, S., Nowak, M. A., & Wilms, J., et al. 2007, ApJ, 670, 624 van der Meer, A., Kaper, L., van Kerkwijk, M. H., Heemskerk, M. H. M., & van den Heuvel, E. P. J. 2007, A&A, 473, 523 Vrtilek, S. D., Raymond, J. C., Boroson, B., & McCary, R. 2005, ApJ, 626, 307 Wilms, J., Nowak, M. A., Dove, J. B., Fender, R. P., & Di Matteo, T. 1999, ApJ, 552, 460 Wojdowski, P. S., Clark, G. W., & Levine, A. M. 1998, ApJ, 502, 253 Wojdowski, P. S., & Clark, G. W. 2000, ApJ, 541, 963 Wojdowski, P. S., Liedahl, D. A., & Kallman, T. R. 2008, ApJ, 673, 1023 Woo, J. W., & Clark, G. W. 1995, ApJ, 445, 896", "pages": [ 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "The neutrino-cooled accretion disk, which was proposed to work as the central engine of gamma-ray bursts, encounters difficulty in interpreting the X-ray flares after the prompt gamma-ray emission. In this paper, the magnetic coupling between the inner disk and the central black hole is taken into consideration. For mass accretion rates around 0 . 001 ∼ 0 . 1 M /circledot s -1 , our results show that the luminosity of neutrino annihilation can be significantly enhanced due to the coupling effects. As a consequence, after the gamma-ray emission, a remnant disk with mass M disk /lessorsimilar 0 . 5 M /circledot may power most of the observed X-ray flares with the rest frame duration less than 100 seconds. In addition, a comparison between the magnetic coupling process and the Blandford-Znajek mechanism is shown on the extraction of black hole rotational energy. Subject headings: accretion, accretion disks - black hole physics - gamma-ray burst: general - magnetic fields", "pages": [ 1 ] }, { "title": "Neutrino-cooled Accretion Model with Magnetic Coupling for X-ray Flares in GRBs", "content": "Yang Luo, Wei-Min Gu, Tong Liu, and Ju-Fu Lu Department of Astronomy and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen, Fujian 361005, China", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The launch of Swift satellite has led to tremendous discoveries of gamma-ray bursts (GRBs) (see M'esz'aros 2006; Gehrels et al. 2009, for a review). A surprising one is that large X-ray flares are common in GRBs and occur at times well after the initial prompt emission (Romano et al. 2006; Falcone et al. 2007; Chincarini et al. 2007; Bernardini et al. 2011). The X-ray flare is an episodic phenomena showing a sudden brightness at the late afterglow stage. From the spectral and temporal analysis of X-ray flares, it is strongly suggested that X-ray flares may have a common origin as the prompt gamma-ray pulses and are related to the late time activity of the central engine (Bernardini et al. 2011; Romano et al. 2006). For the energy reservoir of powering GRBs or X-ray flares, it is believed that they are produced through an ultra-relativistic jet, i.e., a neutrino annihilation-driven jet or a Poynting flux-dominated jet. For the former jet, neutrinos annihilate above the disk and form a hot fireball which subsequently expand to accelerate the jet by its thermal pressure (Popham et al. 1999; Di Matteo et al. 2002; Chen & Beloborodov 2007). On the other hand, in the case of a Poynting flux-dominated jet, the jet derives its energy from the rotational energy of the central black hole (BH) via the Blandford-Znajek (BZ) process (Blandford & Znajek 1977; McKinney & Gammie 2004; Tchekhovskoy et al. 2008). Several mechanisms were proposed to explain the episodic phenomenon of X-ray flares, including fragmentation of a rapidly rotating core (King et al. 2005), magnetic regulation of the accretion flow (Proga & Zhang 2006), fragmentation of the accretion disk (Perna et al. 2006), differential rotation in a post-merger millisecond pulsar (Dai et al. 2006), transition from thin to thick disk (Lazzati et al. 2008), He-synthesis-driven winds (Lee et al. 2009), the propagation instabilities in GRB jets (Lazzati et al. 2011), and the episodic, magnetically dominated jets (Yuan & Zhang 2012). Apart from the energy and the episodic activity of X-ray flares, their evolution has also been investigated, which shows that the average energy released in the form of X-ray flares overlaid on the power-law decay of the afterglow of GRBs (Lazzati et al. 2008; Margutti et al. 2011). In the present work, we will concentrate on another issue, i.e., whether or not the remnant disk after the prompt gamma-ray emission can power X-ray flares through neutrino annihilation. The luminosity of neutrino annihilation produced by the accretion disk is sensitive to the accretion rate. In the late stage of disk evolution, the accretion rate will probably be quite low (e.g., ∼ 0 . 01 M /circledot s -1 ) and the annihilation luminosity will drop sharply according to previous numerical calculations (e.g., Popham et al. 1999). In other words, the observed X-ray flares, if produced by the annihilation mechanism, will still require a relatively high accretion rate. We take the second flare of GRB 070318 as an example, which has an isotropic luminosity of 2 . 96 × 10 48 erg s -1 and a rest frame duration of 80.6 s (see Table 1). In order to produce such a luminosity by neutrino annihilation, an accretion rate around 0 . 06 M /circledot s -1 is required for a rapidly rotating BH ( a ∗ = 0 . 9) due to our numerical calculations in Section 3, which is consistent with the results in Popham et al. (1999). Subsequently, the required remnant disk mass is around 4 . 8 M /circledot , which is obviously beyond what the progenitor can provide (e.g., Shibata & Taniguchi 2008; Pannarale et al. 2011). In this work, we will investigate the neutrino-cooled disk with the magnetic coupling (MC) between the inner disk and the BH, where the magnetic field connects the accretion disk and the central BH (Li 2002; Wang et al. 2002; Uzdensky 2005). In such a model, the angular momentum and the energy are transported from the BH horizon to the accretion disk through a closed magnetic field. Accordingly, this transported energy will be released and therefore increase the radiation of the disk (Li 2002). The MC effects on the structure and radiation of neutrino-cooled disks have first been studied by Lei et al. (2009). They showed that both the neutrino luminosity and the annihilation luminosity will increase significantly owing to the MC process. In the present work, we will focus on relatively low accretion rates 0 . 001 ∼ 0 . 1 M /circledot s -1 to study the possibility of the remnant disk to power X-ray flares. We would point out that, although such type of magnetic fields is one of the possible field geometries discussed by McKinney (2005, Figure 2) and has been studied by a few previous works, the existence of such magnetic fields remains a problem according to MHD simulations. We will discuss this issue in the last section. The paper is organized as follows. Equations are presented in Section 2. The structure and neutrino radiation of the disk are calculated in Section 3. A comparison of our numerical results with the observed X-ray flares is shown in Section 4. Conclusions and discussion are made in Section 5.", "pages": [ 1, 2, 3 ] }, { "title": "2. Equations", "content": "For neutrino-dominated accretion, the disk is extremely hot and dense and the neutrino radiation can balance the viscous dissipation. The structure and radiation of such a disk have been widely investigated in previous works (Popham et al. 1999; Di Matteo et al. 2002; Gu et al. 2006; Chen & Beloborodov 2007; Liu et al. 2007; Lei et al. 2009; Pan & Yuan 2012; Liu et al. 2013). Some simulations showed that the accretion flow is very dynamic and the inner radius of the flow changes with time. In addition, the flow is subject to various HD and MHD instabilities so that it is non-asymmetric. In order to avoid the complexity of solving partial differential equations, the present work is still based on the assumption of a steady and axisymmetric accretion flow. In such case, the basic equations of the neutrino-cooled accretion disk including the MC effects may refer to Section 2 of Lei et al. (2009), and the relativistic effects of the spinning BH were shown in Riffert & Herold (1995). For simplicity, we define the gravitational radius as r g ≡ GM/c 2 , the dimensionless radius as x ≡ r/r g , and the dimensionless spin parameter as a ∗ ≡ cJ/GM 2 . The disk is assumed to be Keplerian rotating, thus the angular velocity of the flow is expressed as In the present work, we investigate the properties of the neutrino-cooled disk with magnetic field lines connecting the BH with the inner disk. Such a MC process may have substantial effects on the energy and angular momentum balance of the disk (Li & Paczy'nski 2000; Li 2002; Wang et al. 2002; Janiuk & Yuan 2010; Kov'acs et al. 2011). In addition, Uzdensky (2005) showed that the inner part of the disk is magnetically coupled to the BH, but the magnetic field cannot be stable in the outer region. Here, we follow the assumptions of Wang et al. (2003) that the MC process is constrained by a critical polar angle θ 0 , and the magnetic field varies as a power law with the disk radius. The magnetic torque exerted to the disk from the BH horizon can be expressed as where θ 0 is a critical polar angle. In the scenario (e.g., Wang et al. 2002, Figure 1), the MC process exists in the range θ 0 < θ < π/ 2. On the contrary, for the space with 0 < θ < θ 0 , the BZ process may occur and therefore some accreted materials may be pushed away, particularly in the inner region of the disk. In the present study, we will focus on the MC process and a constant accretion rate is assumed for simplicity. The critical angle is calculated by where ξ = r/r ms , χ ms = √ r ms /r g , ξ out = r out /r ms , and T 0 = 3 . 26 × 10 45 ( B H / 10 15 G) 2 ( M/M /circledot ) 3 g cm 2 s -2 . β = Ω / Ω H is the ratio of the angular velocity of the disk (Eq. (1)) to the angular velocity at the horizon, where Ω H = ( ca ∗ / 2 r g ) / (1 + √ 1 -a 2 ∗ ). In addition, under the equipartition assumption (e.g., McKinney 2005; Lei et al. 2009), the magnetic field strength at the horizon can be estimated as B 2 H = 8 πc ˙ M/r 2 g . The energy equation is written as where Q + vis is the viscous heating rate, including the contributions of the gravitational potential Q + G and the MC process Q + MC = -T MC / (4 πr ) · d Ω /dr . The quantities Q -adv and Q -ν are respectively the advective cooling rate and the neutrino cooling rate. Here, we neglect the radiation of photons since they are trapped in the disk. The neutrino cooling generally consists of the four processes: the electron-positron pair annihilation, the bremsstrahlung emission of nucleons, the plasmon decay, and the Urca process (e.g., Liu et al. 2007). In addition, for the equation of state, the total pressure consists of five terms, i.e., the gas pressure, the radiation pressure, the degeneracy pressure, the neutrino pressure, and the magnetic pressure. The detailed description of the neutrino cooling and the pressure can be found in some previous papers (e.g., Popham et al. 1999; Gu et al. 2006; Janiuk & Yuan 2010). For the term of magnetic pressure, following Lei et al. (2009), the ratio of magnetic pressure to total pressure is assumed to be 0.1.", "pages": [ 3, 4 ] }, { "title": "3.1. Disk structure", "content": "In our calculations we fix M = 3 M /circledot , α = 0 . 1, a ∗ = 0 . 9, n = 3, and r out = 100 r g . The numerical results of structure and radiation are shown in Figure 1, where the solid lines and the dashed lines represent the solutions with and without the MC process, respectively. We choose three accretion rates, i.e., ˙ M = 0 . 005, 0 . 05, and 0 . 5 M /circledot s -1 for the study. Figures 1a, 1b, and 1c show respectively the radial profiles of the mass density, the temperature, and the neutrino cooling efficiency defined as the ratio of the neutrino cooling rate to the viscous heating rate Q -ν /Q + vis . It is seen that, for the outer region with r /greaterorsimilar 20 r g , the solid and dashed lines are identical to each other, which implies that the MC effects in this region are negligible. On the contrary, for the inner region with r /lessorsimilar 20 r g , the solid lines and the dashed lines are apparently separate, which indicates that the MC effects are substantial. For the inner region, Figure 1a shows that the density with MC is obviously larger than that without MC. The physical understanding is as follows. The MC process can transfer angular momentum from the BH to the inner disk. As a consequence, the additional transferred angular momentum will work as a barrier to prevent the flow from radial acceleration, and therefore the accreted matter will accumulate in this region and the mass density will significantly increase. On the other hand, for ˙ M = 0 . 5 M /circledot s -1 , it is shown by Figure 1a that the solid line drops inwards even faster than the dashed line. The reason is that, for large ˙ M , the disk will become optically thick to neutrinos. Thus, most of the generated neutrinos are trapped in the disk instead of escaping away. The total pressure, which includes the neutrino pressure, will therefore increase significantly and the disk will probably become geometrically thick. Accordingly, the mass density will drop sharply due to the increased vertical height and radial velocity. The neutrino trapping can be also indicated by Figure 1c, where the solid line steeply drops inwards for ˙ M = 0 . 5 M /circledot s -1 , indicating that most neutrinos are trapped in the disk rather than being radiated.", "pages": [ 5 ] }, { "title": "3.2. Annihilation luminosity", "content": "In the scenario of hyper-accretion disks, the GRBs are powered by the neutrino and anti-neutrino annihilation above the surface of disks. Since the X-ray flares and the gammaray emission have the same power origin, the flares may still be powered by the neutrino annihilation. The disk luminosity is calculated from the marginal stable orbit r ms to the outer boundary r out . The neutrino luminosity from the disk is calculated as The total annihilation luminosity is calculated by the integration over the whole space outside the disk, following the method in previous works (e.g., Ruffert et al. 1997; Popham et al. 1999). Figure 2 shows the variations of the neutrino luminosity L ν ( L ' ν ) and the annihilation luminosity L ν ¯ ν ( L ' ν ¯ ν ) with the mass accretion rate ˙ M . Similar to Figure 1, the solid lines and the dashed lines represent the solutions with and without the MC process, respectively. The upper solid (dashed) line corresponds to L ν ( L ' ν ), and the lower solid (dashed) line corresponds to L ν ¯ ν ( L ' ν ¯ ν ). It is seen that, both the neutrino luminosity and the annihilation luminosity with the MC process are significantly larger than those without the MC process. In particular for relatively low accretion rates such as ˙ M ∼ 0 . 01 M /circledot s -1 , L ν ¯ ν is larger than L ' ν ¯ ν by up to four orders of magnitude. The physical reason is that, in addition to the gravitational energy, the MC process can efficiently extract the BH rotational energy into neutrino radiation. Moreover, the apparent difference between the lower solid line and the lower dashed line implies that it is quite possible for a remnant low-mass disk to power X-ray flares.", "pages": [ 5, 6 ] }, { "title": "3.3. Efficiency of BH rotational energy extraction", "content": "We compare the efficiency of BH rotational energy extraction between the MC process and the BZ mechanism. The former efficiency is defined as For the BZ mechanism, the magnetic field lines, which are dragged in by the accretion disk, accumulate around the BH horizon and then get twisted by the BH space-time, which enable the extraction of BH rotational energy. The efficiency of the BZ process was discussed in Tchekhovskoy et al. (2011), where Φ 0 = Φ BH / ( ˙ Mcr 2 g ) 1 / 2 is the dimensionless magnetic flux threading the BH, κ is a numerical constant related to the magnetic field geometry (we adopt κ = 0 . 044 here), and f (Ω H ) is set to be 1. The above equation shows that the BZ efficiency is relevant to the dimensionless magnetic flux Φ 0 . The value of Φ 0 , however, is quite uncertain and may be related to the accretion rate. It can be regarded as the ability for the accretion disk to drag magnetic fields into the BH horizon. Recent simulations show that Φ 0 can be as large as several tens and the disk can be depicted as a magnetic arrest disk. At a high Φ 0 , a large amount of magnetic flux is transported to the center and the efficiency can be larger than 1. Here we simply assume a constant magnetic flux, Φ 0 = 50, which is close to the simulation result Φ 0 ≈ 47 (Tchekhovskoy et al. 2011). Equations (6) implies that η MC is independent of the accretion rate ˙ M . The variation of η MC and η BZ with the spin parameter a ∗ is shown in Figure 3. It is seen that, for a ∗ /greaterorsimilar 0 . 5, η MC can be significantly larger than the efficiency of normal accretion process 0 . 06 /lessorsimilar η /lessorsimilar 0 . 42, and can be even larger than 1 for extremely spinning BH. Such a result indicates that the MC process is quite an efficient mechanism to extract the BH rotational energy. The efficiency η MC decreases sharply with decreasing a ∗ for a ∗ /lessorsimilar 0 . 4. The reason is that the angular velocity of BH will be less than that of the disk at ISCO for a ∗ /lessorsimilar 0 . 36, and therefore the MC process will become weak for BH spin below this critical value. It is also seen that, for a ∗ /greaterorsimilar 0 . 4, η MC is several times larger than η BZ , whereas for a ∗ /lessorsimilar 0 . 3, η BZ is significantly larger than η MC due to a sharp decrease of η MC with decreasing a ∗ . In other words, for fast spinning BH systems, MC may be more powerful than BZ on the extraction of rotational energy, whereas for slow spinning BH systems, BZ is likely to be more powerful. Another different effect between MC and BZ is that, the BZ process can directly transfer the rotational energy into the jet, whereas the MC process can only transfer the rotational energy into the disk, and the jet power is also related to another process, i.e., neutrino annihilation. Thus, for the efficiency of powering the jet, the BZ process will probably be much more efficient. For a comparison, the dotted line shows the efficiency of radiation due to neutrino annihilation η ν ¯ ν ( ≡ L ν ¯ ν / ˙ Mc 2 ) for a typical accretion rate ˙ M = 0 . 05 M /circledot s -1 . It is seen that η BZ is significantly larger than η ν ¯ ν for any a ∗ . Thus, the BZ process is more efficient to power a jet than the MC process. It is also possible for the BZ process to work as the central engine to power the X-ray flares.", "pages": [ 6, 7 ] }, { "title": "4. Comparison with observations", "content": "In order to compare our numerical results with the observations of X-ray flares, we compile a sample of 21 GRBs with 43 flares as shown in Table 1, which includes all the flares with available redshift in Table 1 of Chincarini et al. (2010). The width of the flares is calculated in the rest frame: w res = w/ (1 + z ), where z is the redshift, w is the observed width, and w res is the width in the rest frame. The isotropic energy of a flare E flare can be estimated from the fluence S : E flare = 4 πD 2 l S/ (1 + z ), where D l is the luminosity distance. Thus, the average, isotropic luminosity can be obtained as L iso = E flare /w res . A comparison of our numerical results with the observations is shown in Figure 4, which includes all the flares in Table 1. Similar to Figures 1 and 2, the solid lines represent the results with MC, whereas the dashed lines represent the results without MC. The upper and lower solid (dashed) lines correspond to the disk mass M disk = 0 . 5 M /circledot and 0.05 M /circledot , respectively. The four theoretical lines are calculated by w res = M disk / ˙ M together with the relationship between ˙ M and L ν ¯ ν obtained in Section 3. Figure 4 shows that most of the flares locate above the upper dashed line, which means that for the case without MC, it will generally require a remnant disk mass M disk significantly larger than 0.5 M /circledot , which may be unpractical. However, it is seen that most of the flares exist between the two solid lines, which indicates that if MC works, a reasonable remnant disk with 0 . 05 M /circledot /lessorsimilar M disk /lessorsimilar 0 . 5 M /circledot is able to power nearly all the plotted flares. As mentioned in Section 1, taking the second flare of GRB 070318 as an example, the model without MC requires M disk ≈ 4 . 8 M /circledot . On the contrary, if the MC effects are taken into account, the luminosity of 2 . 96 × 10 48 erg s -1 corresponds to ˙ M ≈ 0 . 0032 M /circledot s -1 (according to our numerical calculations shown by the lower solid line in Figure 2). Therefore, the rest frame duration 80.6 s only requires M disk ≈ 0 . 26 M /circledot for powering this flare.", "pages": [ 7, 8 ] }, { "title": "5. Conclusions and discussion", "content": "In this paper, we have studied the neutrino-cooled disks by taking into account the MC process between the central BH and the inner disk. We have shown that, for mass accretion rates around 0 . 001 ∼ 0 . 1 M /circledot s -1 , the luminosity of neutrino annihilation can be enhanced by up to four orders of magnitude due to the MC effects. As a consequence, the remnant disk with M disk /lessorsimilar 0 . 5 M /circledot may power most of the observed X-ray flares with the rest frame duration less than 100 seconds. We would point out that, for a few X-ray flares with extremely long duration, the neutrino-cooled disk cannot work as the central engine, even though MC is included. For example, GRB 050724 has three flares with a redshift of 0.258. The third flare of this source has a rest frame width of 3 . 1 × 10 5 seconds and the luminosity 6 . 7 × 10 43 erg s -1 . According to our numerical results in Figure 2, the accretion rate should be around 1 . 7 × 10 -4 M /circledot s -1 for the case including MC. Thus, it requires a remnant disk M disk > 50 M /circledot , which is obviously unphysical. On the other hand, X-ray flares with peak time less than and larger than 1000 s may have different origin (Margutti et al. 2011). The mechanism for powering the flares with peak time larger than 1000 s is worthy for further studies, but is beyond the scope of the present paper. Another issue we would like to stress is related to the configuration of the large-scale magnetic field in accretion disks. Although the MC process has been studied by quite a few previous works, such type of magnetic fields, however, has not been found in MHD simulations. Thus, it remains unclear whether the MC process can occur between the inner disk and the central BH. On the other hand, some simulations showed that the BZ mechanism can be a solution to the GRB's central engine (e.g., Tchekhovskoy et al. 2008). In the scenario of the Poynting flux-dominated jet, the efficiency of extracting the BH rotational energy mainly depends on the magnetic flux being dragged in (Tchekhovskoy et al. 2011; McKinney et al. 2012). The theoretical analysis showed that it requires a geometrically thick disk to transport a large mount flux into the center (Lubow et al. 1994; Rothstein & Lovelace 2008; Beckwith et al. 2009; Cao 2011). Simulations also confirmed that a thick disk can efficiently transport magnetic flux (McKinney et al. 2012). For a neutrino-cooled disk, neutrinos play a vital role to release the dissipation heat and the disk is likely to be geometrically thin (e.g., Shibata et al. 2007). Then, for the Poynting flux-dominated jet, it remains a problem whether the accretion flow can accumulate adequate magnetic fields to the inner region to power the jet. In this work, the flow is assumed to be steady and the mass accretion rate is a free parameter. In other words, for a given accretion rate, we will obtain a corresponding solution. On the other hand, the simulations of Tchekhovskoy et al. (2011) found a correlated variation between the accretion rate and the magnetic flux Φ BH . In this spirit, a varying strength of MC process may also have effects on the variation of accretion rate. Such a study requires further time-dependent calculations. We thank Raffaella Margutti, Shujin Hou, Da-Bin Lin, and Mou-Yuan Sun for beneficial discussions, and the referee for helpful suggestions. This work was supported by the National Natural Science Foundation of China under grants 11073015, 11103015, 11222328, and 11233006.", "pages": [ 8, 9 ] }, { "title": "REFERENCES", "content": "Beckwith, K., Hawley, J. F., & Krolik, J. H. 2009, ApJ, 707, 428 Bernardini, M. G., Margutti, R., Chincarini, G., Guidorzi, C., & Mao, J. 2011, A&A, 526, A27 Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433 Cao, X. 2011, ApJ, 737, 94 Chen, W.-X., & Beloborodov, A. M. 2007, ApJ, 657, 383 Chincarini, G., Moretti, A., Romano, P., et al. 2007, ApJ, 671, 1903 Chincarini, G., Mao, J., Margutti, R., et al. 2010, MNRAS, 406, 2113 Dai, Z. G., Wang, X. Y., Wu, X. F., & Zhang, B. 2006, Science, 311, 1127 Di Matteo, T., Perna, R., & Narayan, R. 2002, ApJ, 579, 706 Falcone, A. D., Morris, D., Racusin, J., et al. 2007, ApJ, 671, 1921 Gehrels, N., Ramirez-Ruiz, E., & Fox, D. B. 2009, ARA&A, 47, 567 Gu, W.-M., Liu, T., & Lu, J.-F. 2006, ApJ, 643, L87 Janiuk, A., & Yuan, Y.-F. 2010, A&A, 509, A55 King, A. R., et al. 2005, ApJ, 630, L113 Kov'acs, Z., Gergely, L., & Biermann, P. L. 2011, MNRAS, 416, 991 Lazzati, D., Blackwell, C. H., Morsony, B. J., & Begelman, M. C. 2011, MNRAS, 411, L16 Lazzati, D., Perna, R., & Begelman, M. C. 2008, MNRAS, 388, L15 Lee, W. H., Ramirez-Ruiz, E., & L'opez-C'amara D. 2009, ApJ, 699, L93 Lei, W. H., Wang, D. X., Zhang, L., et al. 2009, ApJ, 700, 1970 Li, L.-X. 2002, ApJ, 567, 463 Li, L.-X., & Paczy'nski, B. 2000, ApJ, 534, L197 Liu, T., Gu, W.-M., Xue, L., & Lu, J.-F. 2007, ApJ, 661, 1025 Liu, T., Xue, L., Gu, W.-M., & Lu, J.-F. 2013, ApJ, 762, 102 Lubow, S. H., Papaloizou, J. C. B., & Pringle, J. E. 1994, MNRAS, 267, 235 Yuan, F., & Zhang, B. 2012, ApJ, 757, 56 /s32 /s32 /s32 /s32 /s32 /s32", "pages": [ 9, 10, 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "We report observations of an X class flare on 2011 September 6 by the instruments onboard the Solar Dynamics Observatory ( SDO ). The flare occurs in a complex active region with multiple polarities. The Extreme-Ultraviolet (EUV) Variability Experiment (EVE) observations in the warm coronal emission reveal three enhancements, of which the third one corresponds to an EUV late phase. The three enhancements have a one-to-one correspondence to the three stages in flare evolution identified by the spatially-resolved Atmospheric Imaging Assembly (AIA) observations, which are characterized by a flux rope eruption, a moderate filament ejection, and the appearance of EUV late phase loops, respectively. The EUV late phase loops are spatially and morphologically distinct from the main flare loops. Multi-channel analysis suggests the presence of a continuous but fragmented energy injection during the EUV late phase resulting in the warm corona nature of the late phase loops. Based on these observational facts, We propose a three-stage magnetic reconnection scenario to explain the flare evolution. Reconnections in different stages involve different magnetic fields but show a casual relationship between them. The EUV late phase loops are mainly produced by the least energetic magnetic reconnection in the last stage. Subject headings: Sun: corona - Sun: flares- Sun: UV radiation - Sun: magnetic topology", "pages": [ 1 ] }, { "title": "PRODUCTION OF THE EXTREME-ULTRAVIOLET LATE PHASE OF AN X CLASS FLARE IN A THREE-STAGE MAGNETIC RECONNECTION PROCESS", "content": "Y. Dai 1,2 , M. D. Ding 1,2 , and Y. Guo 1,2 1 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China; [email protected] 2 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China Accepted for publication in the Astrophysical Journal Letters", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "It is widely accepted that flares are a result of the rapid release of magnetic energy stored in the solar corona. Among the radiative output from a flare, emission from the Extreme-Ultraviolet (EUV) spectral range covers a substantial fraction, thus serving as an important tool to diagnose the dynamics and evolution of the flare. Observations of the Sun in EUV have been carried out for more than half a century (e.g., Friedman 1963). However, previous observations often suffered from the limited spectral range. The situation has been greatly improved since the recently launched Solar Dynamics Observatory ( SDO ; Pesnell et al. 2012) mission provides both spectroscopic observations with full EUV spectral coverage and simultaneous imaging observations in multiple EUV bandpasses. One of the intriguing phenomena discovered by SDO is an 'EUV late phase' in some flares (Woods et al. 2011), which is seen as a second peak in the warm coronal ( ∼ 3 MK) emissions (such as Fe XVI ) several minutes to a few hours after the GOES soft X-ray (SXR) peak. There are, however, no significant enhancements of the SXR or hot coronal ( ∼ 10 MK) emissions in the EUV late phase, and spatially-resolved observations show that the secondary warm coronal emission comes from a set of longer loops rather than the original flaring loops. Up to now, there are only a few reports of the EUV late phase in literatures, and the origin of the EUV late phase is still not fully understood. Some authors thought that the EUV late phase is due to a second energy injection late in the flare (e.g., Woods et al. 2011; Hock et al. 2012b), while others proposed that it is mainly a cooling-effect extending from the initial main flare heating (e.g., Liu et al. 2013). To clarify this question, in this Letter we present SDO observations of an X class flare on 2011 September 6, which exhibits an extended EUV late phase. Based on the observational facts found in this flare, we propose a specific scenario, three-stage magnetic reconnection, to explain the production of the EUV late phase.", "pages": [ 1 ] }, { "title": "2. OBSERVATIONS AND DATA ANALYSIS", "content": "We use data from the three instruments onboard SDO . The EUV Variability Experiment (EVE; Woods et al. 2012) measures full-disk integrated EUV irradiance from 0.1 to 105 nm with 0.1 nm spectral resolution and 10 s temporal cadence. The EVE data used in this study are primarily from the MEGS-A channel (Hock et al. 2012a), which covers the 7-37 nm wavelength range with a nearly 100% duty cycle. The Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) provides simultaneous full-disk images of the transition region and corona in 10 channels with 1 . '' 5 spatial resolution and 12 s temporal resolution. In addition, magnetograms from the Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012) are chosen to check the magnetic topology of the flare. The flare under study occurred in NOAA active region (AR) 11283 on 2011 September 6, positioned close to the disk center. It is an X2.1 class flare that started at 22:12 UT, peaked around 22:21 UT, and ended at 22:24 UT, as revealed by the GOES 0.1-0.8 nm light curve in Figure 1. To study the thermal evolution of the flare, in Figure 1 we also plot the time profiles of the backgroundsubtracted irradiance in six EVE spectral lines, which cover temperatures from log T ∼ 4 . 9 to log T ∼ 7. The flare exhibits the all four characteristics pointed out in Woods et al. (2011). First, the cold chromospheric He II 30.4 nm emission showed an impulsive enhancement and peaked around 22:19 UT, almost coincident with the period. Therefore, compared to the typical EUV late phase as demonstrated in Woods et al. (2011), which is directly preceded by the flare's main phase, in this flare there is another 'intermediate phase' characterized by the moderate peak between the main phase and the late phase. RHESSI 50-100 keV hard X-ray (HXR) emission peak, as well as the spikes in the emissions from Fe IX to Fe XVI . This implies that the chromosphere, as responding to impact of the non-thermal electrons accelerated during the flare's impulsive phase, was instantly heated to log T ∼ 6 . 4 (Milligan et al. 2012; Chamberlin et al. 2012). Second, the hot coronal Fe XX/XXIII 13.3 nm emission closely resembled the GOES SXR time series and peaked around 22:21 UT, which is believed to be a result of the chromospheric evaporation caused by the initial heating. The cooler emissions then peaked sequentially with decreasing temperatures within a short period of 70 s, revealing a fast cooling rate over 1 × 10 5 K s -1 . Third, the cool coronal Fe IX 17.1 nm and moderately warm coronal Fe XIV/XII 21.1 nm emissions decreased after the peak of the main phase and turned into a coronal dimming. A coronal mass ejection (CME) was observed being associated with the flare, and the coronal dimming is most likely to reflect the mass drainage during the CME launch (e.g., Aschwanden et al. 2009). Fourth, as the dimming in cool coronal emissions developed and other emissions returned to the pre-flare level, the warm coronal Fe XVI 33.5 nm emission showed another small enhancement that started from ∼ 23:00 UT and lasted until the first few hours of 2011 September 7. It should correspond to the EUV late phase as defined in Woods et al. (2011), which will be further validated by the aftermentioned AIA imaging observations. Note that in this late phase there was also likely a very weak increase in the moderately hot coronal Fe XVIII 9.39 nm emission. Besides these four features, we find that additional moderate enhancements in the EVE 30.4, 33.5, and 9.39 nm emissions were seen to start from ∼ 22:33 UT and peak around 22:40 UT. The EVE 13.3 nm emission and the GOES SXR flux also showed a hump during this Spatially-resolved AIA observations of the flare evolution in six coronal channels are presented in the online animation. Figure 2 shows some snapshots of the animation in the AIA 33.5 and 17.1 nm channels, respectively, as well as a light-of-sight (LOS) magnetogram of AR 11283 taken at 22:14 UT by HMI. Three main polarities are identified and labeled as P1, N1, and P2, among which P2 is a parasitic positive polarity embedded in the negative polarity N1. The flare first occurred along the southeastern part of the polarity inversion line (PIL) between N1 and P2, behaving as a sigmoid-to-fluxrope eruption pattern (cf, Liu et al. 2010), which is best seen in AIA 13.1 and 9.4 nm. Note that prior to the event a semi-circular filament was observed to lie along the PIL, the southern part of which has been successfully modeled as a flux rope (FR) by Feng et al. (2013) and Jiang et al. (2013) by using different nonlinear force-free field (NLFFF) extrapolation methods. From ∼ 22:18 UT the FR started to rise rapidly, with the flare ribbons becoming more elongated along the PIL and the outer one eventually turning into a circular ribbon (Figures 2(a) and (e)). Following the brightening of the main flare ribbons, a remote flare ribbon appeared in P1, over a distance of 80 '' east of the main flare region. The situation is very similar to those studied in Masson et al. (2009) and Wang & Liu (2012), indicating the presence of a 3D null-point magnetic topology, which is also expected from the LOS magnetogram. Afterwards, the region on the eastern side of the remote ribbon quickly turned into a coronal dimming that persisted during the whole evolution of the flare. This first stage eruption was followed by a second stage eruption, which was seen as the continuous ejection of cold material from the northern part of the filament starting from ∼ 22:33 UT (Figures 2(b) and (f)). This moderate ejection produced new post-flare loops mainly on its southern side rather than centered on the filament itself. In AIA 13.1 and 9.4 nm (see online animation), there appeared a second set of longer but less prominent loops connecting the eastern side of the filament (in N1) and the abovementioned remote flare ribbon (in P1), which were not visible in the cooler coronal channels at that moment. The second stage eruption lasted about 30 minutes. As the post-flare loops in the main flare region cooled down, longer post-flare loops connecting N1 and P1 became more and more dominant in particular in the warm and cool coronal channels. The appearance of these loops coincided with the enhancement in the EVE Fe XVI 33.5 nm emission starting from ∼ 23:00 UT, and the loops were spatially distinct from the original post-flare loops, further supporting the identification of an EUV late phase for this third stage evolution. These late phase loops were first localized at the same location of the second set of loops previously only seen in the hotter coronal channels during the second stage (Figures 2(c) and (g)), later became more spatially spread (Figure 2(h)). As noted in Woods et al. (2011), the late phase loops are very diffuse X (arcsecs) X (arcsecs) X (arcsecs) X (arcsecs) in morphology in the warm coronal AIA 33.5 nm channel, but quite distinct in the cool coronal channels such as AIA 17.1 nm. It is worth noting that the remote flare ribbon brightened again and showed a long-lasting eastward (dimming-ward) expansion starting from the second stage eruption when long loops were found to anchor on it. This is an indicator of progressive magnetic reconnection according to the standard flare model. In Figure 2(f) we place a slice roughly perpendicular to the remote ribbon to trace its movement in AIA 17.1 nm. Because of the low brightness of the remote ribbon, we simply use linear fit to track the ribbon expansion. As revealed in Figure 3, the expansion velocity of the remote ribbon turned from 4.5 km s -1 to 0.76 km s -1 , coinciding with the transition from the second stage to the third stage, and also consistent with the evolution of the EVE 33.5 nm emission. The expansion velocities are considerably lower than those in some typical two-ribbon flares, which are several tens of km s -1 (e.g., Asai et al. 2004; Miklenic et al. 2007), implying less and less ener- getic magnetic reconnections during the second and third stages with much lower reconnection rates than the main flare reconnection. The AIA base difference images in Figure 4 highlight the flare's three-stage evolution more clearly. To determine the contribution of different parts of the AR to the flare emission, we identify three regions, which are indicated by the color boxes in the AIA 33.5 nm images, and calculate the light curve by summing the count rate over all pixels in each region. The red box represents the main flare region, the blue one outlines the outer loop region, which is believed to account for the EUV late phase, and the black one surrounds the full AR. We plot the background-subtracted AIA light curves in 33.5 nm from the three regions in Figure 4(g), and compare the AIA 33.5 nm full AR light curve with the EVE 33.5 nm profile in Figure 4(h). First, the general similarity between the two profiles in Figure 4(h) suggests that the change in the EVE full-disk integrated irradiance comes mainly from the AR. Actually, there was no other major activity on the visible disk during the period of this event. Second, as expected, emission from the main flare region dominates in the main phase and the intermediate phase, while the outer loop region is responsible for the late phase. Note that in the main and intermediate phases there were also some intensity increases from the outer loop region, which we attribute to the brightening of the remote flare ribbon and CCD-bleed interference stripes extending from the intensively flaring site. To study the thermal evolution the EUV late phase loops, we select a sub-region defined by the white parallelogram in Figure 4(f), which includes the most prominent late phase loops seen in AIA 33.5 nm, and calculate the intensity variability in six AIA coronal channels for this region. As revealed in Figure 4(i), the EUV late phase can be decomposed into many episodes; each episode is characterized by a cooling process, with the emissions peaking sequentially with decreasing temperatures. Furthermore, combined with the imaging observations, it is found that these multiple peaks appear to be related to different, but adjacent, late phase loops cooling into the corresponding emission temperature ranges at different times, in particular in the cool coronal channels. This indicates the presence of a continuous, but fragmented both temporally and spatially, energy injection during the EUV late phase, challenging the conclusion of Liu et al. (2013) that the appearance of the late phase loops is mainly a cooling-effect rather than the result of a later energy injection. We pick up four most prominent episodes labeled as L1-L4, and list the time of peak emission seen in six AIA channels for each episode in Table 1. It is seen that in L2 the AIA 9.4 nm peak follows the AIA 19.3 nm peak, and all detectable peaks in AIA 13.1 nm are delayed from the corresponding AIA 17.1 nm peaks. This behavior can be expected from the AIA temperature response functions (TRFs) calculated using the latest CHIANTI atomic database (Landi et al. 2013), which also show a second cool coronal peak at log T ∼ 6 . 0 for AIA 9.4 nm and log T ∼ 5 . 8 for AIA 13.1 nm, and suggests the warm corona nature of the late phase loops. Between L2 and L3 the variabilities in AIA 9.4 and 33.5 nm showed opposite patterns, further suggesting that during this period the AIA 9.4 nm channel is mainly sensitive to cool late phase loops that cool from warm coronal temperatures. L1 is a little bit special, in which the appearance of the late phase loops in AIA 33.5 nm (Figure 4(f)) is a combined effect of the cooling from the hotter AIA 9.4 nm loops (Figure 4(c)) and the late phase energy injection. The cooling rates in these four episodes are 0.3-1.2 × 10 4 K s -1 , orders of magnitude lower than that in the main phase derived from the EVE observations.", "pages": [ 1, 2, 3, 4 ] }, { "title": "3. DISCUSSION AND CONCLUSION", "content": "The 2011 September 6 X2.1 flare conforms to the criteria for an EUV late phase flare defined in Woods et al. (2011). To our knowledge, this may be the first report of the EUV late phase of a flare above the X class. The ratio of the late phase peak to the main phase peak is only 9.4%, significantly lower than those in Woods et al. (2011). We attribute this low ratio to the intense energy injection during the impulsive phase, which causes a strong main phase peak of ∼ 100% above the background. According to previous studies, the multipolar magnetic fields may be a necessary condition for the production of an EUV late phase (e.g., Hock et al. 2012b; Liu et al. 2013). We propose a three-stage magnetic reconnection scenario to explain the flare evolution under such a magnetic topology. The main flare reconnection between N1 and P2 is triggered by the tether-cutting mechanism (Moore et al. 2001), as evidenced by the sigmoid-to-flux-rope eruption pattern. The appearance of the remote flare ribbon may suggest a secondary null-point reconnection (Masson et al. 2009; Reid et al. 2012). A magnetic nullpoint was indeed found by Jiang et al. (2013), but the outer spine field lines surrounding the null-point extend to the northwest rather than the eastern remote ribbon. Nevertheless, large-scale overlying field lines that connect P1 and N1 do exist. When the FR erupts, it strongly reconnects with and stretches these overlying field lines, causing the brightening of the remote ribbon and deep dimming at the far side of the remote ribbon. This FRdriven reconnection should belong to the breakout reconnection (Antiochos et al. 1999). The eruption of the FR largely removes the overlying magnetic confinement. Therefore, the northern part of the filament starts to rise as a sequence of the torus instability (Kliem & Torok 2006). The rising filament also drives the breakout reconnection but in a gentle manner, producing two sets of side-lobe post-flare loops (Figure 4(b)). Since the southern side-lobe loops located in the main flare region are more closer to the reconnection site, they are more prominent than the northeastern ones, responsible for the flare's intermediate phase. As the filament-driven breakout reconnection turns the lower overlying field lines into side lobes, new reconnection occurs between the higher overlying field lines that are previously stretched. The process is similar to the main flare reconnection, but is much less energetic because of the weaker magnetic fields that are involved. Hence the reconnection only produces moderately heated warm EUV late phase loops (Figure 4(f)). These loops cool as they retract, sequentially brightening in cooler coronal channels. Meanwhile, the whole AR gradually recovers to its pre-flare state. In a large-scale extent, the on-going reconnection can occur at different sites and with different rates, making the energy injection rather fragmented. The slow cooling rates for the late phase loops can be expected from their long length (cf. Cargill et al. 1995). The three-stage magnetic reconnections show a casual relationship between each other, and the EUV late phase is mainly a production of the third stage reconnection. It is believed that some very early late phase loops (see Figure 4(b)) have already been produced during the second stage breakout reconnection (Hock et al. 2012b). The loops are not visible in cooler coronal channels just because they are still relatively hot at that moment. The diffuse morphology of the late phase loops seen in AIA 33.5 nm images is also reflected from the more smooth AIA 33.5 nm profile, as compared with the less smooth cooler emission profiles shown in Figure 4(i). Woods et al. (2011) attribute this phenomenon to the relatively slow cooling rate at warm coronal temperatures compared to that at cooler coronal temperatures. However, it seems not to be the case for the current event as revealed in Table 1. We present an alternative explanation in terms of the AIA TRFs. The AIA 33.5 nm TRF is quite flat from the peak with decreasing temperatures. It means that as a warm late phase loop cools its visibility in AIA 33.5 nm does not change much; therefore, there could be many overlapping loops visible at the same time. Nevertheless, the TRFs for the cooler coronal channels (AIA 21.1, 19.3, and 17.1 nm) have a sharp peak. In these channels, a small change of the loop temperature around the TRF peak will significantly change the loop intensity, resulting in the sharp loop morphology and the large fluctuation of the intensity profile. We are grateful to the anonymous referee for his/her insightful comments. This work is supported by NSFC (11103009, 10933003, 11203014, and 11078004), and by 973 project of China (2011CB811402). SDO is a mission of NASA's Living With a Star (LWS) program.", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485 Asai, A., Yokoyama, T., Shimojo, M., et al. 2004, ApJ, 611, 557 Aschwanden, M. J., Nitta, N. V., Wuelser, J.-P., et al. 2009, ApJ, 706, 376 Chamberlin, P. C., Milligan, R. O., & Woods, T. N. 2012, Sol. Phys., 279, 23 Feng, L., Wiegelmann, T., Su, Y., et al. 2013, ApJ, 765, 37 Friedman, H. 1963, ARA&A, 1, 59 Hock, R. A., Woods, T. N., Klimchuk, J. A., Eparvier, F. G., & Jones, A. R. 2012b, ArXiv: 1202.4819 Jiang, C., Feng, X., Wu, S. T., & Hu, Q. 2013, ApJ, 771, L30 Kliem, B., & Torok, T. 2006, Physical Review Letters, 96, 255002 Landi, E., Young, P. R., Dere, K. P., Del Zanna, G., & Mason, H. E. 2013, ApJ, 763, 86 Masson, S., Pariat, E., Aulanier, G., & Schrijver, C. J. 2009, ApJ, 700, 559 6 Moore, R. L., Sterling, A. C., Hudson, H. S., & Lemen, J. R. 2001, ApJ, 552, 833 Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3 Reid, H. A. S., Vilmer, N., Aulanier, G., & Pariat, E. 2012, A&A, 547, A52 Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207 Wang, H., & Liu, C. 2012, ApJ, 760, 101 Woods, T. N., Hock, R., Eparvier, F., et al. 2011, ApJ, 739, 59 Woods, T. N., Eparvier, F. G., Hock, R., et al. 2012, Sol. Phys., 275, 115", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "The Vela supernova remnant is the closest supernova remnant to Earth containing an active pulsar, the Vela pulsar (PSR B0833 -45). This pulsar is the archetype of the middle-aged pulsar class and powers a bright pulsar wind nebula (PWN), Vela -X, spanning a region of 2 · × 3 · south of the pulsar and observed in the radio, X-ray and very high energy γ -ray domains. The detection of the Vela -X PWN by the Fermi Large Area Telescope (LAT) was reported in the first year of the mission. Subsequently, we have re-investigated this complex region and performed a detailed morphological and spectral analysis of this source using 4 years of Fermi -LAT observations. This study lowers the threshold for morphological analysis of the nebula from 0.8 GeV to 0.3 GeV, allowing inspection of distinct energy bands by the LAT for the first time. We describe the recent results obtained on this PWN and discuss the origin of the newly detected spatial features. Subject headings: Gammarays: general - ISM: individual objects: Vela-X - pulsars: general pulsars: individual (Vela, PSR J0835 -4510)", "pages": [ 1 ] }, { "title": "THE VELA -X PULSAR WIND NEBULA REVISITED WITH 4 YEARS OF FERMI LARGE AREA TELESCOPE OBSERVATIONS", "content": "M.-H. GRONDIN 1,2,3 , R. W. ROMANI 4 , M. LEMOINE-GOUMARD 5,6 , L. GUILLEMOT 7 A. K. HARDING 8 , T. REPOSEUR 5 , Draft version November 6, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The supernova remnant (SNR) G263.9-3.3 (aka the Vela SNR) is the closest composite SNR to Earth containing an active pulsar, the Vela pulsar (PSR B0833 -45) and is therefore studied in great detail across the electromagnetic spectrum. Located at a distance of only D = 290 pc (Caraveo et al. 2001; Dodson et al. 2003), the Vela pulsar has a characteristic age of τ c = 11 kyr, a spin period of P = 89 ms and a spin-down power of ˙ E = 7 × 10 36 erg s -1 . First discovered as a radio loud pulsar (Large et al. 1968), its pulsations were successively detected in high energy (HE) γ -rays (Thompson 1975), optical (Wallace et al. 1977) and X-rays ( Ogelman et al. 1993). Recent γ -ray observations by the Fermi Large Area Telescope (LAT) have confirmed its detection above 20 MeV and enabled a more detailed study of its γ -ray properties than possible with the previous missions SAS-II, COS-B and CGROEGRET (Kanbach et al. 1994, and references therein). These observations reveal a magnetospheric emission over 80% of the pulsar period, and a strong and complex phase dependence of the γ -ray spectrum, in particular in the peaks of the light curve (Abdo et al. 2009a, 2010a). The 8 · -diameter Vela SNR is also known to host several regions of non-thermal and diffuse radio emission labelled Vela -X, Vela -Y and Vela -Z (Rishbeth 1958). The brightest one ( ∼ 1000 Jy), Vela -X , spans a region of 2 · × 3 · (referred to as the ' halo ') surrounding the Vela pulsar and shows a filamentary structure. In particular, the brightest radio filament has an extent of 45 ' × 12 ' and is located south of the pulsar. The flat spectrum of Vela -X with respect to Vela -Y and Vela -Z and its high degree of radio polarization have led to strong presumptions that Vela -Xis the pulsar wind nebula (PWN) associated with the energetic and middleaged Vela pulsar (Weiler & Panagia 1980). The rotational energy of the pulsar is dissipated through a magnetized wind of relativistic particles. A PWN forms at the termination shock resulting from the interaction between the relativistic wind and the surrounding material, e.g. the supernova ejecta (Gaensler & Slane 2006). Following its radio discovery, the Vela -X region has been intensively observed at every wavelength. X-ray observations by ROSAT have unveiled a diffuse and nebular emission (with an extent of 1.5 ' × 0.5 ' ) coincident with the bright radio filament and referred to as the ' cocoon ' (Markwardt & Ogelman 1995). The Vela -X region has been significantly detected up to 0.4 MeV by OSSE with a spectrum consistent with the E -1 . 7 spectrum seen between optical and X-rays (de Jager et al. 1996). High resolution Chandra observations have enabled the detection of bright and compact non-thermal X-ray emission composed of two toroidal arcs (17 '' and 30 '' away from the pulsar) and a 4 ' -long collimated 'jet'-like structure (Helfand et al. 2001). Finally, very high energy (VHE) γ -ray observations by the H.E.S.S. telescopes have revealed bright emission spatially coincident with the cocoon, whose spectrum peaks at ∼ 10 TeV (Aharonian et al. 2006). This detection has confirmed the non-thermal nature of the cocoon; however, the relativistic particle population responsible for the X-ray and TeV emission can hardly account for the halo structure observed in radio. At this time, several scenarios have been proposed to reconcile multi-wavelength data. Horns et al. (2006) proposed a hadronic model, in which the VHE γ -ray emission is explained by proton-proton interactions inside the cocoon followed by neutral pion decays. In parallel, de Jager et al. (2008) suggested the existence of two electron populations in Vela -X: a young population that produces the narrow cocoon seen in X-rays and at VHE, and a relic one responsi- le for the extended halo observed in radio. According to this model, significant emission from the halo should be detectable in the Fermi -LAT energy range. An alternative scenario was recently proposed by Hinton et al. (2011), which explains the observations by diffusive escape of particles in the extended halo structure. The first HE γ -ray detection of the Vela -X PWN by the Fermi -LAT was reported in the first year of the mission. The source is significantly extended (with an extension of σ Disk = 0.88 · ± 0.12 · assuming a uniform disk hypothesis) and its spectrum is well reproduced with a simple power law having a soft index ( Γ ∼ 2.41 ± 0.09 stat ± 0.15 syst ) in the 0.2 20 GeV energy range (Abdo et al. 2010b). The detection of Vela -Xin the 0.1 - 3 GeV energy range was also reported by the AGILE Collaboration (Pellizzoni et al. 2010). Abramowski et al. (2012) recently reported the detection of faint TeV emission spatially coincident with the radio halo, in addition to the bright emission already reported and matching the X-ray emission. This new result challenges the simple interpretation of a young electron population being responsible for the X-ray and VHE emission. We have re-investigated this complex region in HE γ -rays and performed a detailed morphological and spectral analysis of this source using 4 years of Fermi -LAT observations. The energy range for morphological analysis is extended down to 0.3 GeV allowing the first study of energy-dependent morphology by the LAT. In this paper, we report the results of this analysis and discuss the main implications of the new spectrally-resolved spatial information in the context of the theoretical models described above. In particular, we discuss the possible interpretation of the energy-dependent morphology brought to light with 4 years of Fermi -LAT observations.", "pages": [ 1, 2 ] }, { "title": "2. LAT DESCRIPTION AND OBSERVATIONS", "content": "The LAT is a γ -ray telescope that detects photons by conversion into electron-positron pairs and operates in the energy range from 20 MeV to greater than 300 GeV. Details of the instrument and data processing are given in Atwood et al. (2009). The on-orbit calibration is described in Abdo et al. (2009b) and Ackermann et al. (2012). The following analysis was performed using 48 months of data collected starting 2008 August 4, and extending until 2012 August 4 within a 15 · × 15 · region around the position of the Vela pulsar. Only γ -rays in the from the Pass 7 'Source' class were selected from this sample. This class corresponds to a good compromise between the number of selected photons and the background rate. We have used the P7SOURCE V6 Instrument Response Functions (IRFs) to perform the following analyses. We excluded photons with zenith angles greater than 100 · to reduce contamination from secondary γ -rays originating in the Earth's atmosphere (Abdo et al. 2009b).", "pages": [ 2 ] }, { "title": "3. TIMING ANALYSIS OF THE PULSAR PSR J0835-4510", "content": "The Vela pulsar is the brightest steady point source in the γ -ray sky, with pulsed photons observed up to 25 GeV, and is located within the Vela -X PWN. Previous analyses of its γ -ray properties using Fermi -LAT observations have shown that magnetospheric emission is observed over 80% of the pulsar period (Abdo et al. 2010a). The detailed study of Vela -X requires working in the offpulse window of the Vela pulsar light curve (i.e. 20% of the pulsar period) to avoid contamination from the pulsed emission. Because the Vela pulsar exhibits substantial timing ir- regularities, phase assignment generally requires a contemporary ephemeris. To perform the following analysis, γ -ray photons were phase-folded using an accurate timing solution derived from Fermi -LAT observations. The Vela pulsar is extremely bright in γ -rays, and its continuous observations by the LAT since the beginning of the mission enables us to directly construct regular times of arrival (TOAs) that are then used to generate a precise pulsar ephemeris (Ray et al. 2011). The Vela pulsar experienced a glitch, i.e. a large jump in rotational frequency, near MJD 55428. To avoid any contamination from the Vela pulsar in the analysis of its PWN, data between MJD 55407 and MJD 55429 were excluded from the dataset, and two timing solutions (pre- and post-glitch) were used to phase-fold the γ -ray photons. The pre-glitch ephemeris was built using 198 TOAs covering the period from the beginning of the science phase of the Fermi mission (2008 August 04) to the glitch, while the post-glitch timing solution was built using 197 TOAs from the glitch to 2012 August 04. For both timing solutions, we fit the γ -ray TOAs to the pulsar rotation frequency and first five derivatives. The fit further includes 10 harmonically related sinusoids, using the FITWAVES option in the TEMPO2 package (Hobbs et al. 2006), to flatten the timing noise. The post-fit rms is 91.3 µ s and 97.7 µ s (i.e. 0.1% of the pulsar phase) for the pre- and post-glitch ephemeris respectively. These timing solutions will be made available through the Fermi Science Support Center (FSSC) 9 . We define phase 0 for the model based on the fiducial point from the radio timing observations, which is the peak of the radio pulse at 1.4 GHz. Pulse phases were assigned to the LAT data using the Fermi plug-in provided by the LAT team and distributed with TEMPO2. Only γ -ray photons in the 0.8 - 1.0 pulse phase interval, corresponding to the off-pulse window, were selected and used for the spectral and morphological analysis presented in the following sections. Figure 1 shows the γ -ray light curve of the Vela pulsar obtained in the 0.1 - 300 GeV energy range using events in a 1 · radius around the position of the Vela pulsar and the definition of the off-pulse window (blue dashed lines). It is worth noting that this phase interval was chosen to be narrower than the one used in the previous Fermi -LAT analysis to avoid any contamination from the Vela pulsar, which was estimated to be ∼ 6%in the 0.7 - 0.8 phase interval (Abdo et al. 2010b).", "pages": [ 2, 3 ] }, { "title": "4. ANALYSIS OF THE VELA -X PWN", "content": "The spatial and spectral analysis of the γ -ray data was performed using two different tools, gtlike and pointlike . gtlike is a maximum-likelihood method (Mattox et al. 1996) implemented in the Science Tools distributed by the FSSC. pointlike is an alternate binned likelihood technique, optimized for characterizing the extension of a source (unlike gtlike ), that has been extensively tested against gtlike (Kerr 2011; Lande et al. 2012). These tools fit a model of the region, including sources, residual cosmic-ray, extragalactic and Galactic backgrounds, to the data. In the following analysis, the Galactic diffuse emission is modeled using the standard model gal 2yearp7v6 v0.fits . The residual charged particles and extragalactic radiation are described by a single isotropic component with a spectral shape described by the file iso p7v6source.txt . The models and their detailed description are released by the LAT Collaboration 10 . Sources within 10 · of the Vela pulsar are extracted from the Second Fermi -LAT Catalog (Nolan et al. 2012) and used in the likelihood fit. The nearby bright and extended SNRs Puppis A and Vela Jr are modeled with their best-fit models, i.e. a uniform disk of radius 0.38 · for Puppis A (Hewitt et al. 2012) and the template of the TeV emission as seen with H.E.S.S. for Vela Jr (Tanaka et al. 2011). The spectral parameters of sources closer than 3 · to Vela -X are left free, while the parameters of all other sources are fixed at the values from Nolan et al. (2012). Due to the longer integration time of our analysis (48 months vs. 24 months in the catalog) and the overwhelming brightness of the Vela pulsar in the full phase interval, the appearance of additional sources in our region of interest is expected. These sources, denoted with the identifiers BckgA and BckgB, were also considered in the analysis of SNR Puppis A and were fit at the following positions : BckgA at α (J2000) = 125 . 77 · , δ (J2000) = -42 . 17 · with a 68% error radius of 0.06 · ; BckgB at α (J2000) = 128 . 14 · , δ (J2000) = -43 . 39 · with a 68% error radius of 0.05 · . More details on these sources are available in Hewitt et al. (2012).", "pages": [ 3 ] }, { "title": "4.1. Morphology", "content": "Previous analysis of the Vela -X PWN using 11 months of Fermi -LAT data have shown that the source is significantly extended above 0.8 GeV, with an extension of σ Disk = 0.88 · ± 0.12 · assuming a uniform disk (hereafter labelled ' Disk 11 m '; Abdo et al. 2010b). The increasing statistics and the improvement of the IRFs with respect to Abdo et al. (2010b) allow a more detailed study of the source and the use of a lower energy threshold of 0.3 GeV. To study the morphology of an extended source, a major requirement is to have the best possible angular resolution. Consequently, we restrict the LAT data set to front events only, i.e. events which convert in the thin layers of the tracker, which benefit from higher angular resolution 11 (Atwood et al. 2009). Figure 2 presents the Fermi -LAT Test Statistic (TS) map of γ -ray emission around the Vela -X PWN above 0.3 GeV using front events only. The TS is defined as twice the difference between the likelihood L 1 obtained by fitting a source model plus the background model to the data, and the likelihood L 0 obtained by fitting the background model only : TS = 2( logL 1 -logL 0 ). This skymap contains the TS value for a point source at each map location, thus giving a measure of the statistical significance for the detection of a γ -ray source in excess of the background. The diffuse Galactic and isotropic emission, as well as nearby sources are included in the background model and subtracted from the map. We used pointlike to measure the source extension using five different spatial hypotheses: a point source, a uniform disk hypothesis, a Gaussian distribution, an elliptical Gaussian distribution and an elliptical disk (Lande et al. 2012) assuming a power-law spectrum. The results of the extension fits and the improvement of the TS when using spatially extended models are summarized in the first half of Table 1, along with the number of additional degrees of freedom with respect to the null hypothesis. The improvement of the likelihood fit between a Gaussian distribution and the point-source hypothesis 12 (difference in TS of 216, which corresponds to an improvement at a ∼ 15 σ level) supports a significantly extended source. The best-fit model in the 0.3 - 100 GeV energy range is obtained for an elliptical Gaussian distribution. This best-fit model represents a 5 σ improvement with respect to a symmetric Gaussian distribution ( ∆ TS = 28 for two additional degrees of freedom), which means that the source is also significantly elongated. The best-fit center of gravity of the emission region is (R.A., Dec.) = (128.40 · ± 0.05 · , -45.40 · ± 0.05 · ). The best-fit width along the major axis is 1.04 · ± 0.09 · , while the bestfit intrinsic width along the minor axis is 0.46 · ± 0.05 · . The major axis of the fitted distribution is at a position angle (P.A.) of 40.3 · ± 4.0 · . We also examined the correlation of the γ -ray emission from Vela -X with multi-wavelength observations of this 12 The formula used to derive the significance of an improvement when comparing two different spatial models with different numbers of degrees of freedom is extracted from Particle Data Group (Beringer et al. 2012). * L 1 and L 0 are defined as the likelihood values corresponding to the fit of the spatial model described in the first column plus the background model and the fit of the background model only (null hypothesis). ** Add. d.o.f. : additional degrees of freedom. Galactic longitude (deg) Galactic longitude (deg) PWN using pointlike . We compared the TS obtained with the best-fit model, i.e. the elliptical Gaussian distribution, with the TS obtained using the templates derived from WMAP (61 GHz radio image, shown by green contours in Figure 2) and H.E.S.S. observations. For each analysis, a power law spectrum was assumed. The resulting TS values, which are equivalent to 2∆( log ( L )) , are summarized in the second half of Table 1. When comparing the results obtained by modeling the Fermi-LAT emission with multi-wavelength templates, using the H.E.S.S. template significantly decreases the value of the likelihood with respect to the WMAP template, as noted in the first publication reporting the Fermi -LAT detection of the Vela -X PWN (Abdo et al. 2010b). However, we still observe a good correlation between the radio and the Fermi -LAT observations. We also divided the radio template into two halves, as indicated in Figure 2, to look for an energy-dependent morphological behavior. The split radio model provides an improvement at ∼ 3.6 σ and ∼ 6.5 σ levels with respect to the single radio template and the Southern radio wing model respectively, and is also confirmed by the spectral analysis (see Section 4.2). The multi-wavelength templates and the analytical models cannot be compared directly since the models are not nested. In the following analysis we decided to use the best geomet- rical morphology implemented in pointlike , namely the elliptical Gaussian distribution. Figure 3 presents the Fermi -LAT TS maps of γ -ray emission around the Vela -X PWN in two energy bands (0.3 1 GeV, 1 - 100.0 GeV) using front events only. Radio and TeV contours have been overlaid for comparison. We attempted to characterize the energy-dependent shape of the PWN by estimating the source extension in these energy intervals. The centroids and extensions in the different energy ranges are summarized in Table 2. From Figure 3 we note that the emission in the 'Northern wing' (defined with respect to the Galactic coordinates) of the radio emission is bright in the lower energy band and becomes faint above 1 GeV, which might be an indication of a softer spectrum than the 'Southern wing'. It is worth noting that we report here for the first time the detection of γ -ray emission from the Northern wing of the Vela -XPWN. This detection is clearly visible in the TS map presented in Figure 4, in which the Southern radio wing was included in the background model. Table 1 shows that the loglikelihood of the fit is significantly improved by using the split radio templates instead of the Southern radio wing only. The discovery was enabled by the low energy threshold (0.3 GeV) now considered in this analysis. In addition, the extension and position of the Southern wing are in full agreement with the results of the morphological fit performed above 0.8 GeV and reported in the first Fermi -LAT paper on Vela -X(Abdo et al. 2010b).", "pages": [ 3, 4, 5 ] }, { "title": "4.2. Spectrum", "content": "The following spectral analyses are performed with gtlike using front and back events between 0.2 and 100 GeV. We used the best morphological model from Table 1, i.e. the elliptical Gaussian distribution, to represent the γ -ray emission observed by the LAT, as discussed in Section 4.1. Assuming this spatial shape, the γ -ray source observed by the LAT is detected with a TS of 940 above 0.2 GeV. The spectrum of Vela -X above 0.2 GeV is presented in Figure 5. It is well described by a smoothly broken power law : where Γ 1 = 1.83 ± 0.07 ± 0.27, Γ 2 = 2.88 ± 0.21 ± 0.06 are the spectral indices below and above the break energy E b = 2.1 ± 0.5 ± 0.5 GeV. The parameter β is fixed to the value 0.2 as in standard Fermi -LAT analyses (e.g. Buehler et al. 2012). The first error is statistical, while the second represents our estimate of systematic effects as discussed below. The integrated flux renormalized to the total phase above 0.2 GeV is (1.83 ± 0.08 ± 0.25) × 10 -7 cm -2 s -1 . This spectral model is favored over the simple power law and an exponential cut-off power law at 6.6 σ and 2.7 σ levels respectively. This is in agreement with results obtained independently using pointlike . Similar results are obtained with the radio template, as can be seen in Table 3. The Fermi -LAT spectral points shown in Figure 5 were obtained by dividing the 0.2 - 100 GeV range into 10 logarithmically-spaced energy bins and performing a maximum likelihood spectral analysis to estimate the photon flux in each interval, assuming a power-law shape with fixed photon index Γ = 2 for the source. The normalizations of the diffuse Galactic and isotropic emission were left free in each energy bin. A 99.73% C.L. upper limit is computed when the statistical significance is lower than 3 σ . Bins at the highest energies corresponding to upper limits were combined. Four different systematic uncertainties can affect the LAT flux estimation : uncertainties in the Galactic diffuse background, in the morphology of the LAT source, in the effective area and in the energy dispersion. The fourth one is relatively small ( ≤ 10%) and has been neglected in this study. The main systematic at low energy is due to the uncertainty in the Galactic diffuse emission since Vela -X is located only ∼ 3 · from the Galactic plane. Different versions of the Galactic diffuse emission, generated by GALPROP (Strong et al. 2004), were used to estimate this error. The observed γ -ray intensity of nearby source-free regions on the Galactic plane is compared AN ELLIPTICAL GAUSSIAN DISTRIBUTION. with the intensity expected from the Galactic diffuse models. We adopted the strategy described in Abdo et al. (2010c) to estimate the expected intensity of the Galactic diffuse emission for different models. The difference, namely the local departure from the best-fit diffuse model, is found to be ≤ 6 %. By changing the normalization of the Galactic diffuse model artificially by ± 6 %, we estimate the systematic error on the integrated flux and on the spectral index. The second systematic is related to the morphology of the LAT source. The fact that we do not know the true γ -ray morphology introduces another source of error that becomes significant when the size of the source is larger than the PSF. Different spatial shapes have been used to estimate this systematic error: a disk, a Gaussian distribution and the radio template. The third uncertainty, common to every source analyzed with the LAT data, is due to the uncertainties in the effective area. This systematic is estimated by using modified instrument response functions (IRFs) whose effective area bracket that of our nominal IRF. These 'biased' IRFs are defined by envelopes above and below the nominal dependence of the effective area with energy by linearly connecting differences of (10%, 5%, 10%) at log(E) of (2, 2.75, 4) respectively. We combine these various errors in quadrature to obtain our best estimate of the total systematic error at each energy and propagate them through to the fit model parameters. for each of the spatial templates described in Table 1. Using the elliptical Gaussian distribution and the smoothly broken power law and assuming a distance of D = 290 pc, the γ -ray luminosity of Vela -X above 0.2 GeV is L γ ≈ 2.4 × 10 33 ( D/ 290 pc) 2 erg s -1 , yielding a γ -ray efficiency of η = L γ / ˙ E = 0.03 % of the spin-down power of the Vela pulsar. We attempted to characterize the energy-dependent morphology of the Vela -X PWN by fitting the spectra associated with each of the split radio templates with independent spectral models. The results are summarized in Table 3. The Northern wing is well modeled with a simple power law of index 2.25 ± 0.07 ± 0.20 while the Southern wing is better described by a smoothly broken power law. The corresponding spectral parameters are the following : Γ 1 = 1.81 ± 0.10 ± 0.24 and Γ 2 = 2.90 ± 0.25 ± 0.07 , with a break energy of E b = 2.1 ± 0.5 ± 0.6 GeV. Because of a potential interdependence of the wing spectral fits, the Northern parameters were obtained after 2 iterations. In a first step, both wings were fitted simultaneously. The Northern wing being much fainter than the Southern one, the spectral parameters of the Northern wing were re-adjusted in the second step, using fixed parameters for the Southern wing. Both iterations yield consistent results within statistical errors. However, the fit and spectral points obtained in the second step for the Northern wing are much more in agreement with each other. It is worth noting that the flux of the Northern wing is approximately half of the one in the Southern wing. However the Northern wing is located closer to the Galactic plane (i.e. in a region with a larger contribution from the Galactic diffuse background), which renders the emission from this wing less than half as significant as than the Southern wing. The improvement for the split radio model with respect to the single radio template is at ∼ 4 σ level, which is consistent with the fact that the integral fluxes of the two radio wings are significantly different (see Table 3). Figure 6 (left and right) presents the spectra of the two regions modeled with the two split templates. Interestingly, this analysis shows that below ∼ 2 GeV, the Northern wing has a softer spectrum by an index of ∼ 0.5 with respect to the Southern wing, confirming the first indications given by the TS maps (see Figure 3). However, it should be noted that the steep spectrum of the Northern wing is mainly constrained by the upper limits at high energy. In this context, the likelihood of the fit is improved by only 2.5 σ when using a free power-law model instead of a broken power-law with fixed energy break and spectral indices (frozen at the values obtained for the Southern wing). More statistics are therefore needed to confirm spectral differences between the Northern and Southern regions. The careful reader may note that the best spectral fit of 4 years of Fermi -LAT data (this paper) is obtained with a smoothly broken power law, while the fit of the 11 months of data yielded a simple power law of index Γ ∼ 2.4 ± 0.1 and a weaker flux as presented in Abdo et al. (2010b). These differences arise from the three main improvements (described below) made in this new analysis. First, a larger data set now enables us to spatially model the Vela -X γ -ray emission with an elliptical Gaussian distribution, i.e. a more elaborate morphology than the' Disk 11 m ' model considered in the previous publication. The smaller extension of the Disk 11 m with respect to the elliptical Gaussian distribution above 0.2 GeV therefore yields a fainter flux integrated over 0.2 GeV. For comparison, the spectral parameters obtained by fitting the 4-year dataset with a power law and a smoothly broken power law using the Disk 11 m spatial model are included in Table 3. Secondly, the increased statistics now allow a significant detection of a spectral break at ∼ 2.0 GeV in the γ -ray domain, which was not possible with only 11 months of data. Using Disk 11 m , the smoothly broken power law is preferred to the simple power law at 5.3 σ level. Finally, using the Disk 11 m model, the harder spectrum obtained with the 4-year dataset (spectral index of Γ = 2.24 ± 0.04 for a power law, see the first row labelled ' Disk 11 m ' in Table 3) with respect to the 11-month dataset (which yielded a spectral index of Γ = 2.4 ± 0.1) presented in Abdo et al. (2010b) likely arises from the slight contamination of the 11month dataset by the Vela pulsar at low energies (below 1 GeV), which was estimated to be ∼ 6% of the Vela -X flux. Defining the off-pulse window as 20% of the pulsar phase (instead of 30% in Abdo et al. (2010b)) ensures that we do not suffer contamination from the Vela pulsar in the new analysis.", "pages": [ 5, 6, 7 ] }, { "title": "4.3. Multi-wavelength data", "content": "Spectral measurements at different frequencies may help to better understand the origin of the emission observed from a source via the modeling of its spectral energy distribution. Considering the strong connection between the radio domain and the GeV energy range emphasized in Abdo et al. (2010b), we examined in particular the data obtained in radio in this complex region. Seven-year all-sky data of the Wilkinson Microwave Anisotropy Probe (WMAP) were used to extract the spectrum of the Vela -X PWN at high radio frequencies. Five bands were analyzed, with effective central frequencies of 23, 33, 41, 61 and 94 GHz (Jarosik et al. 2011). Figure 7 represents the composite radio sky map in the Vela -X field of", "pages": [ 7 ] }, { "title": "5.1. Constraining the magnetic field in the halo extended region", "content": "view. This image also shows evidence for a contaminating high frequency component superimposed on the PWN, especially on the Western side of the Northern wing, for Galactic coordinates (l, b) ∼ (265.8 · , -2.5 · ). We extracted the radio spectral measurements using the regions delimited with the white ellipses and red circles for the source and background estimation respectively. The radio spectral points obtained within the Southern and Northern regions in the five frequencies covered by WMAP are summarized in Table 4 and represented in Figure 8. When excluding the highest frequency spectral point (which may be contaminated as shown in Figure 7), the fluxes in the two regions are well modeled with power laws S i ∝ ν -α i (with i = S, N for the South and North respectively) of indices α S = 0.76 ± 0.15 and α N = 0.41 ± 0.08. Given the contamination noted above, this radio slope difference between the two wings may not be significant. Additional data, e.g. from Planck, and especially measurements above 40 GHz, are required to confirm this result. To better constrain the physical parameters related to the PWN and its environment, we will use the above described spectral measurements in the following section. A low-frequency (0.4 GHz) spectral point is extracted from Haslam et al. (1982) and will be used as an upper limit on the flux in each wing of the radio emission. Determining the mechanism responsible for γ -ray emission is crucial in order to measure the underlying relativistic particle population accelerated in a PWN. This new analysis confirms the excellent correlation between the GeV and the radio morphologies, showing that the γ -ray emission extends well outside the narrow cocoon detected in X-rays and VHE. In a first step, we attempted to reproduce the multiwavelength spectral energy distributions of the Southern and Northern wings of Vela -X without taking into account the emissions in the cocoon (detected in X-rays and TeV) which could be produced by a separate electron population, as explained in Abdo et al. (2010b). The objective is to constrain the total energy injected in the form of electrons in the halo as well as the mean value of the magnetic field in this extended region. For this purpose, we used a one-zone model similar to the one described in Grondin et al. (2011). The hadronic scenario, according to which the VHE emission is produced by proton-proton interactions and neutral pion decays, was proposed by Horns et al. (2006) but seems to be disfavored because of the large particle density required with respect to the density derived from X-ray observations (LaMassa et al. 2008). Therefore, it will be disregarded in the following. The spectral differences between the Northern and the Southern wing being marginal with the current statistics of the WMAP and Fermi -LAT data, we did not try to reproduce this effect in the following scenario. In addition, due to the contamination visible at high frequencies in the WMAP data, we did not try to fit the spectral point at 94 GHz. We assume the same leptonic spectrum injected in each region. The WMAP spectral index of ∼ 0 . 4 in the Southern region (which is the less contaminated) requires a particle index of γ ∼ 0 . 4 × 2 + 1 ≈ 1 . 8 , kept fixed in our model (see Table 5). In both regions, the leptonic spectrum injected shows an energy cut-off at the highest energy which is fitted and constrained by the observational data. Electrons suffer energy losses due to ionization, bremsstrahlung, synchrotron processes and inverse Compton (IC) scattering. Escape outside the halo is also taken into account assuming Bohm diffusion. The modification of the electron spectral distribution due to such losses is determined following Aharonian et al. (1997). The electron population is evolved over the estimated lifetime of the pulsar (11 kyr). We fix here the pulsar braking index to the canonical value of n = 3.0. The magnetic field and spin-down power of the pulsar are assumed to remain constant throughout the age of the system and do not depend on the size of the PWN. Neither the interaction of the reverse shock nor the diffusion of the leptons within the PWN are modeled, since there are not enough observables to sufficiently constrain their corresponding parameters. In this context, this phenomenological model is used to reproduce the multi-wavelength data assuming that the Vela pulsar is the only source of energy. The IC photon field includes the cosmic microwave background (CMB), far infrared from the dust (IR; temperature of 25 K, density of 0.44 eV cm -3 ) and starlight (Optical; temperature of 7500 K, density of 0.44 eV cm -3 ), reasonable for the locale of Vela -X(de Jager et al. 2008). We assume a distance of D = 290 pc and a size for each region (i.e. the Northern and Southern wings) of 10 pc. All in all, our simple one-zone model has four free parameters adjusted to reproduce the photon spectral energy distri- bution as seen in radio (Haslam et al. 1982, this work) and γ -rays (this work) in both regions: the magnetic field B in the halo, the exponential cut-off energy (which are both assumed to be the same in the Northern and Southern regions) and the fraction η of the pulsar spin-down power injected to particles in each region. The best fit, obtained by minimizing the χ 2 statistic between model and data points, is shown in Figure 9 (top and bottom). As can be seen in this figure, a simple power-law injection model can reasonably well reproduce the multi-wavelength data of the two wings. The bestfit parameters are summarized in Table 6. Model fitting is achieved by minimizing the χ 2 between model and data using the simplex method described in Nelder & Mead (1965). This algorithm is included within the ROOT framework provided by the CERN 13 . For each ensemble of N variable parameters we evolve the system over the pulsar lifetime and calculate the χ 2 between model curves and flux data points. The simplex routine subsequently varies the parameters of interest to minimize the fit statistic. We estimate parameter errors by using MINOS, which is designed to calculate the correct errors in all cases, especially when there are non-linearities. The theory behind the method is described in Eadie et al. (1971). It is worth noting that ∼ 26% and ∼ 13% of the total energy injected by the pulsar (which represents 100% of the spin-down power injected in particles accounting for all losses during the lifetime of the system) is required to power the radio to γ -ray emission from the Southern and Northern wings respectively. In addition, the total energy injected into leptons and the magnetic field derived are both in very good agreement with previous estimates (de Jager et al. 2008; Aharonian et al. 2006; Abdo et al. 2010b). The synchrotron/IC peak ratio of the cocoon implies a magnetic field of 4 µ G with very small uncertainty, which can be compared to our value of ∼ 5 µ G in the halo. Since no data are available to trace the synchrotron peak and better constrain the magnetic field in the halo, we cannot exclude similar values in the halo and in the cocoon.", "pages": [ 8, 9 ] }, { "title": "5.2. Rapid diffusion of electrons ?", "content": "The radio emission, arising from synchrotron radiation, traces the magnetic field distribution. On the other hand, in a leptonic scenario, the HE emission is produced via IC scattering and directly traces the underlying relativistic electron distribution. The new Fermi -LAT results together with the correlation between the radio and γ -ray data now provide direct evidence that low energy electrons are present in the extended halo. The recent H.E.S.S. detection of TeV emission coincident with the extended radio halo (Abramowski et al. 2012) is additional evidence that electrons are present in this large structure. In addition to that, the above simple modeling provides further evidence that the magnetic fields in the halo and cocoon regions are not strongly different. In that case, the real puzzle is to understand the origin of the electrons present in the extended radio halo since significant emission is now detected by Fermi -LAT and H.E.S.S. up to ∼ 1 · from the Vela pulsar, i.e ∼ 10 pc from the powering pulsar. A potential scenario was first proposed by Van Etten & Romani (2011) to explain the large size of the PWN HESS J1825 -137: rapid diffusion of high energy particles with τ esc ∼ 90( R/ 10 pc) 2 (E e / 100 TeV) -1 year (where R and E e are the radius of the PWN and the energy of the injected electrons respectively) which is 1000 times faster than standard Bohm diffusion. This is in contradiction with the common assumption of toroidal magnetic fields with strong magnetic confinement. The authors argue that turbulence and mixing caused by the passage of the reverse shock might provide the necessary disruption to the magnetic field structure to allow particles to diffuse far more rapidly. More recently, this fast diffusion was invoked for Vela -X by Hinton et al. (2011) to interpret the steep Fermi -LAT spectrum and the absence of > 100 GeV photons in the extended radio halo. Their best fit to the data yields a factor of 2000 enhancement over Bohm diffusion. In such a scenario, the γ -ray flux observed by Fermi -LAT at the outer realm of the Vela -X extended nebula would be produced by high energy electrons that were injected when the pulsar was much younger. However, their model does not produce any TeV emission at large distance from the pulsar since high energy electrons escape too fast. A way to solve this issue would be to inject two populations of electrons (as suggested earlier by Abdo et al. 2010b) and decrease the diffusion time so that 10 TeV photons can still be visible up to 1 · (10 pc) from the powering pulsar in a magnetic field B of 5 µ G. Since the standard Bohm diffusion time scale is τ diff ∼ 34( R/ 1 pc) 2 (E e / 10 TeV) -1 (B / 10 µ G) kyr (Zhang et al. 2008), a diffusion only ∼ 20 times faster than Bohm diffusion would be needed. It should be noted that this scenario should lead to spectral differences between the inner and the outer regions of the PWN due to radiative cooling of the electrons during their propagation, which is in contradiction with the recent H.E.S.S. results (Abramowski et al. 2012). Energy-dependent diffusive escape and stochastic re-acceleration in the radio halo could explain the absence of spectral variations in the TeV regime but such complex modeling is out of the scope of our paper. Another possibility would be that these electrons do not come from the Vela pulsar and are directly accelerated in this extended structure through stochastic acceleration due to turbulent magnetic fields in the outer PWN flow or in the surrounding SNR plasma, since there is no evidence of shocks in this region. Such 2 nd -order Fermi acceleration accounts well for the radio emission from supernova remnants (Scott & Chevalier 1984). It provides an excess of accelerated electrons that will radiate through synchrotron and IC radiation as seen in radio by WMAP and γ -rays by Fermi -LAT. The maximum energy to which electrons can be accelerated by such mechanism depends highly on the level of turbulence, which is unknown. To reproduce the Fermi -LAT spectrum, a reasonable maximum energy of ∼ 140 GeV is required. Obviously, in this case, the radio/ Fermi halo would not be linked with the X-ray/TeV cocoon. The TeV extended halo, if related to the radio structure, would need a second component of accelerated electrons to reproduce the peaked γ -ray spectrum. Unfortunately, the comparison of the radio, GeV and TeV emissions is limited by the differences in resolution and sensitivity of the instruments involved. In particular, the current multi-wavelength observations do not allow conclusions to be drawn about whether the TeV and radio emissions arise from the exact same location.", "pages": [ 9 ] }, { "title": "6. CONCLUSION", "content": "Using four years of Fermi -LAT observations and a lower energy threshold for the morphological analysis than previously, we report for the first time the detection of γ -ray emission from the Northern wing of the Vela -XPWN.Thebest-fit geometrical morphological model in the 0.3 - 100 GeV energy range is obtained for an elliptical Gaussian distribution. We also report the detection of a significant energy break at E b = 2.1 ± 0.5 GeV in the Fermi -LAT spectrum as well as a marginal spectral difference between the Northern and the Southern wings. WMAP data have also been used to characterize the synchrotron emission in the two wings of the radio halo. However, the WMAP image shows evidence for a contaminating high frequency component superimposed on the PWN, especially on the Northern wing, and the radio slope difference observed between the two wings may not be significant. Further observations to characterize the radio and γ -ray spectra are required and will help understand the origin of the excess of low energy electrons detected in these two wavelengths. High frequency radio observations are clearly needed to spatially resolve the radio emission, especially above 60 GHz. Increased sensitivity with the continued observations by Fermi -LAT will also enable any spectral differences between the Northern and Southern regions to be firmly established. In addition, observations with H.E.S.S.-II will provide a better overlap with Fermi -LAT and therefore a direct link to verify if the TeV signal is related to the extended structure detected by Fermi . In addition, deeper observations with high sensitivity instruments such as XMM -Newton will help to better constrain the spectra and their potential spatial variations in the cocoon and halo in the X-ray domain. Despite a large sample of multi-wavelength data, Vela -X remains an excellent case to study and the future observations in radio and γ -rays will obviously provide new clues to understand the acceleration mechanisms taking place in this complex object, as well as new surprises.", "pages": [ 9, 10 ] }, { "title": "Acknowledgements", "content": "The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat 'a l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucl'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d' ' Etudes Spatiales in France. MHG acknowledges support from the Alexander von Humboldt Foundation. MHG wishes to thank Felix A. Aharonian and Patrick O. Slane for helpful comments and discussions.", "pages": [ 10 ] }, { "title": "REFERENCES", "content": "Abdo, A. et al., 2009a, ApJ, 696, 1084 Abdo, A. et al., 2009b, Phys. Rev. D, 80, 122004 Abdo, A. et al., 2010a, ApJ, 713, 154 Abdo, A. et al., 2010b, ApJ, 713, 146 Abdo, A., et al., 2010c, ApJ, 714, Abdo, A. et al., 2010c, ApJ, 722, 1303 Abramowski, A. et. al., 2012, A&A, 548, A38 Ackermann, M. et al., 2012, ApJS, 203, 4 Aharonian, F. A., Atoyan, A. M. & Kifune, T., 1997, MNRAS, 291, 162 Aharonian, F. A. et al., 2006, A&A, 448, L43 Alvarez, H., et al. 2001, A&A, 372, 636 Atwood, W. et al., 2009, ApJ, 697, 2, 1071 Beringer, J. et al. (Particle Data Group), 2012, Phys. Rev. D86, 010001 Buehler, R. et al. 2012, ApJ, 749, 26 Caraveo, P. A., De Luca A., Mignani R. P. & Bignami, G. F. 2001, ApJ, 561, 930 de Jager, O. C., Harding, A. K., Strickman, M. S., ApJ, 460, 729 de Jager, O. C., Slane, P. O., LaMassa S. M., 2008, ApJL, 689, L125 Dodson, R., et al. 2003, ApJ, 596, 1137 Eadie, W. T., Drijard, D., James, F., Roos, M. and Sadoulet, B. , 1971, Statistical Methods in Experimental Physics, North-Holland, 1971, pp. 204-205 Fang, J. & Zhang, L., 2010, A&A, 515, 20 Gaensler, B. M. & Slane, P. O., 2006, ARA&A, 44, 17 Grondin, M.-H. et al, 2011, ApJ, 738, 42 Haslam, C. G. T., Salter, C. J., Stoffel, H. & Wilson, W. E. 1982, A&AS, 47,1 Hewitt, J., Grondin, M.-H., Lemoine-Goumard, M., Reposeur, T. et al, 2012, ApJ, 759, 89 Hinton, J., Funk, S., Parsons, R. D. & Ohm, S., 2011, ApJL, 743, L7 Helfand, D. J., Gotthelf, E. V., & Halpern, J. P. 2001, ApJ, 556, 380 Hobbs, G. B., Edwards, R. T., & Manchester, R. N., 2006, MNRAS, 369, 655 Horns, D., Aharonian, F., Santangelo, A., Hoffmann, A. I. D., & Masterson, C. 2006, A&A, 451, L51 Jarosik, N., et al. 2011, ApJS, 192, 14 Kanbach, G. et al., 1994, A&A, 289, 855 Kerr, M. 2011, PhD Thesis, arXiv:1101.6072v LaMassa, S. M., Slane, P. O., & De Jager, O. C. 2008, ApJ, 689, L121 Lande, J. et al., 2012, ApJ, 756, 5 Large, M. I., Vaughan, A. E., and Mills, B. Y., 1968, Nature, 220, 340 Markwardt, C. B., & ¨ Ogelman, H. 1995, Nature, 375, 40 Mattox, J. R. et al., 1996, ApJ, 461, 396 Nelder, J. A. & Mead, R., 1965, Comput. J., 7, 308 Nolan, P. L. et al. 2012, ApJS, 199, 31 ¨ Ogelman, H., Finley, J. P. and Zimmerman, H. U., 1993, Nature, 361, 136 Pellizzoni, A. et al., 2010, Science, 327, 663 Ray, P. S., et al., 2011, ApJS, 194, 17 Reynolds, S. P., Gaensler, B. M. & Bocchino, F., 2012, Space Science Reviews, 166, 231 Rishbeth, H., 1958, Australian Journal of Physics, 11, 550 Scott, J. S., Chevalier, R. A., 1975, Astrophys. J. Lett. 197, L5 Slane, P.O. et al., 2010, ApJ, 720, 266 Slane, P.O., et al., 2012, ApJ, 749, 131 Spitkovsky, A., 2008, ApJL, 682, L5 Strong, A. W., Moskalenko, I. V., & Reimer. O. 2004, ApJ, 613, 962 Tanaka, T. et al., 2011, ApJL, 740, L51 Thompson, D. J., Fichtel, C. E., Kniffen, D. A., ¨ Ogelman, H. B., 1975, ApJ, 200, L79 Van Etten, A., & Romani, R. W., 2011, ApJ, 742, 62 Wallace, P. T. et al., 1977, Nature, 266, 692 Weiler, K. W., Pangia, N., 1980, A&A, 90, 269 Zhang, L, Chen, S. B., & Fang, J., 2008, ApJ, 676, 1210 BEST-FIT PARAMETERS FOR THE SOUTHERN AND NORTHERN WINGS.", "pages": [ 10, 12 ] } ]

[ { "title": "Paul F. Goldsmith", "content": "Jet Propulsion Laboratory, California Institute of Technology Received ; accepted", "pages": [ 1 ] }, { "title": "1. Abstract", "content": "We use UV measurements of interstellar CO towards nearby stars to calculate the density in the diffuse molecular clouds containing the molecules responsible for the observed absorption. Chemical models and recent calculations of the excitation rate coefficients indicate that the regions in which CO is found have hydrogen predominantly in molecular form. We carry out statistical equilibrium calculations using CO-H 2 collision rates to solve for the H 2 density in the observed sources without including effects of radiative trapping. We have assumed kinetic temperatures of 50 K and 100 K, finding this choice to make relatively little difference to the lowest transition. For the sources having T ex 10 only, for which we could determine upper and lower density limits, we find < n ( H 2 ) > = 49 cm -3 . While we can find a consistent density range for a good fraction of the sources having either two or three values of the excitation temperature, there is a suggestion that the higherJ transitions are sampling clouds or regions within diffuse molecular cloud material that have higher densities than the material sampled by the J = 1-0 transition. The assumed kinetic temperature and derived H 2 density are anticorrelated when the J = 2-1 transition data, the J = 3-2 transition data, or both are included. For sources with either two or three values of the excitation temperature, we find average values of the midpoint of the density range that is consistent with all of the observations equal to 68 cm -3 for T k = 100 K and 92 cm -3 for T k = 50 K. The data for this set of sources imply that diffuse molecular clouds are characterized by an average thermal pressure between 4600 and 6800 Kcm -3 . Keywords: ISM: molecules - radio lines: ISM", "pages": [ 2 ] }, { "title": "2. Introduction", "content": "Diffuse clouds have been studied over a broad range of wavelengths encompassing radio observations of the 21 cm H i line to UV observations of H 2 and other molecules. They have been been found to encompass a wide range of densities, temperatures, and column densities. For low column densities, the gas is essentially atomic (H 0 ) and ionic (C + ), but as the column density and extinction increase, molecules (starting with H 2 ) gradually become dominant, and the term 'diffuse molecular cloud' (Snow & McCall 2006) is appropriate. Ground-based observations of strong millimeter continuum sources (Liszt & Lucas 1998) and UV observations of early-type stars (e.g. Sheffer et al. 2008) have both allowed observations of rotational transitions of the CO molecule. The UV observations are particularly powerful in that they allow simultaneous measurements of multiple transitions, which are sensitive to the column density of the different CO rotational levels in the cloud in the foreground of the star. The column density of H 2 can also be determined, which allowing determination of the abundances of a number of different species and also isotope ratios as a function of column density (Sheffer et al. 2007). The density of diffuse molecular clouds is an important parameter for analyzing emission in various tracers as well as determining a number of critical cloud properties such as their thermal pressure. One important tracer of several phases of the interstellar medium including diffuse clouds is the fine structure transition of ionized carbon ([C ii ]). Several groups using the Herschel Satellite (Pilbratt et al. 2010) have carried out extensive observations of this submillimeter ( λ = 158 µ m) transition. Among these, a large-scale survey of the Milky Way has been attempting to apportion the observed [CII] emission among the different phases of the interstellar medium (Langer et al. 2010; Pineda et al. 2013). The [C ii ] emission from the diffuse cloud component of the interstellar medium will almost certainly be subthermal given that the critical density for the [C ii ] fine structure line is glyph[similarequal] 2000 - 6000 cm -3 (Goldsmith et al. 2012). Since the [C ii ] transition is optically thin or in the effectively optically thin limit (Goldsmith et al. 2012), the inferred column density of ionized carbon in the diffuse interstellar medium will vary inversely as the density in the clouds responsible for the [CII] emission. The cooling and thermal balance are also sensitively dependent on the density, so that understanding the structure of diffuse clouds and their role in the formation of denser clouds and star formation requires knowledge of the density in the diffuse interstellar medium. In this paper we use the relative populations of the lower CO rotational levels to determine the density in the diffuse clouds along the line of sight to early-type stars observed in the UV. Data on sixty four sources were taken from Sheffer et al. (2008). We have supplemented these data with observational results on eight distinct sources observed by Sonnentrucker et al. (2007), who also present data obtained by Lambert et al. (1994) and Federman et al. (2003) for three sources. Two additional, distinct sources were observed by Burgh, France, & McCandliss (2007). In Section 3 we discuss the transformation of the Sheffer et al. (2008) data to standard excitation temperatures that characterize successive rotational transitions, and in Section 4 derive the uncertainties in the excitation temperatures resulting from their column density measurements. In section 5 we discuss the possibility of radiative excitation, and conclude that it is unlikely to play a significant role. We focus on collisional excitation of CO, concluding that collisions with H 2 molecules are dominant in the clouds of interest. Section 6 gives the results for different categories of diffuse clouds defined by which CO transitions have been observed. In Section 7 we discuss and summarize our results. The Appendix gives an explanation of long-standing apparently anomalous results for the excitation temperature in the low-density limit from multilevel statistical equilibrium calculations that can, in fact, be understood in terms of the allowed collisions and the spontaneous decay rates.", "pages": [ 3, 4, 5 ] }, { "title": "3. Excitation Temperatures", "content": "The excitation temperature, T ex , is defined by the relative local densities in two different energy levels, or (having a clear physical meaning if conditions are uniform along the line of sight) by the relative column densities, N , of the two levels of a given species. Denoting the upper and lower levels by u and l , respectively, and their statistical weights by g u and g l , the relationship is where ∆ E ul is the energy difference between the upper and the lower level. The excitation temperature can be defined between any pair of levels, but it is of greatest utility for two levels connected by a radiative transition that can be observed. Sheffer et al. (2008) use UV absorption measurements to determine the column densities in a number of the lowest transitions of the carbon monoxide (CO) molecule, and define the excitation temperatures of the excited rotational levels ( J = 1, 2, 3) relative to the ground state, J = 0. The lowest excitation temperature thus defined corresponds to the CO J = 1 - 0 transition at 115.3 GHz. The excitation temperatures related to column densities of the J = 2 and J = 3 levels relative to J = 0 do not correspond to observable transitions. It is convenient for density determinations to deal with pairs of levels connected by a radiative transition, so that the collision rate directly competes with an allowed radiative processes. The results tabulated by Sheffer et al. (2008) can easily be transformed into the desired excitation temperatures through the following relationships, in which T 01 , T 02 , and T 03 are the excitation temperatures of the indicated pairs of levels determined by Sheffer et al. (2008), and T ex 01 , T ex 21 , and T ex 32 are the excitation temperatures for radiatively-connected pairs of levels. We define for each transition the equivalent temperature, T ∗ ul = ∆ E ul /k , The transformed results for the stars observed by Sheffer et al. (2008) are given in Table 1, along with the molecular hydrogen column density determined for each line of sight. In two cases, the H 2 column density was estimated by Sheffer et al. (2008) from the column densities of CO and CH, and these values are singled out by a note in the Table. For four lines of sight, Sheffer et al. (2008) did not include N(H 2 ), but values for these were found in the literature and values with associated references are given in column 5 of Table 1. The data in Table 6 of Lambert et al. (1994), Table 1 of Burgh, France, & McCandliss (2007), and Table 12 of Sonnentrucker et al. (2007) are presented in the form of excitation temperatures between adjacent rotational levels and so can be used directly. These data are presented in Table 2, along with references to the original observational papers.", "pages": [ 5, 6 ] }, { "title": "4. Uncertainties", "content": "In deriving excitation temperatures, T ex from column densities N , we use the usual relationship given in equation 1, which leads to the expression for the excitation temperature Taking the partial derivatives with respect to upper and lower level column densities, we find Defining the rms uncertainties as σ T ex , σ N l , and σ N u , respectively, and combining the fractional uncertainties as the sum of the squared uncertainties in the lower and upper level column densities gives us For the UV absorption data of interest, the excitation temperatures are on the order of 0.6 to 0.8 times T ∗ (e.g. 4 K for the J = 1-0 transition having T ∗ = 5.5 K). It is thus reasonable to take T ex /T ∗ glyph[similarequal] 0.7, which gives us Sheffer et al. (2008) give only the uncertainty in the total column density of CO. While it is not clear exactly how the uncertainty in the total column density is related to the fractional uncertainty in the column density of a single level, we simply assume that the fractional uncertainty in an individual column density is equal to the total CO column density uncertainty, given as 20% by Sheffer et al. (2008). Then the fractional uncertainty in the excitation temperature is glyph[similarequal] 0.7 √ 2 times 20%. It thus seems that a reasonably generous 1 σ value is σ T ex /T ex = 0.2. The observations taken from other papers (Table 2) explicitly include uncertainties in individual excitation temperatures. As seen in that Table, these vary considerably from source to source, but are of the same order as given by the above analysis.", "pages": [ 6, 7 ] }, { "title": "5.1. Non-Collisional Excitation", "content": "The excitation of CO can, in principle, be affected by radiative processes following its formation. The unshielded photodissociation rate of 12 CO in a radiation field having the standard Draine value (Draine 1978) is k i 0 = 2 × 10 -10 s -1 (UMIST 2012). For the H 2 and CO column densities of the clouds in this sample, the shielding factor is glyph[similarequal] 0.5 (see Van Dishoeck & Black 1988, Table 5), and thus the CO photodissociation rate within the cloud, which we take equal to the formation rate, is on the order of 10 -10 s -1 . The characteristic time scale is thus glyph[similarequal] 300 yr. The vibrational decay rate is enormously faster, with A ( v = 1 -v = 0) = 30.6 s -1 (Chandra, Maheshwari, & Sharma 1996). Thus, any CO molecule formed will very rapidly decay to the ground vibrational state. The spontaneous decay rates for the rotational transitions are many orders of magnitude slower, ranging from A 10 = 7.2 × 10 -8 s -1 to A 32 = 2.5 × 10 -6 s -1 for the transitions considered here (CDMS). The collision rates necessary to achieve the observed subthermal excitation of CO (see Section 6) are glyph[similarequal] 10 -8 s -1 or 100 times the formation timescale. Thus with all CO molecules being in the ground vibrational state, the collision rate that determines the rotational level populations will be much more rapid than formation/destruction rate, and it is reasonable that the effect of a post-formation cascade (as can affect the population of the levels of H 2 ) will be unimportant. Wannier et al. (1997) suggested that the radiation from a nearby, dense molecular cloud could be sufficient to provide the observed excitation of CO in a diffuse molecular cloud. This requires that the two clouds have the same velocity and that the solid angle of the cloud providing the radiative pumping be large enough to make the radiative excitation rate comparable to the spontaneous decay rate of the transition observed. Sonnentrucker et al. (2007) pointed out that a critical test of this model follows from the fact that the pumping cloud, while optically thick in 12 CO would almost certainly be optically thin in 13 CO. The result would be a much lower radiative pumping rate for 13 CO than for 12 CO, and the excitation temperatures of the rare isotopologue would thus be significantly smaller. Sonnentrucker et al. (2007) conclude that for 7 sight lines (including some observed by others) T ex 10 ( 12 CO) is, within the uncertainties, equal to that of 13 CO. An additional consideration is that the excitation temperature T ex J,J -1 ( 12 CO) increases with increasing J ; this increase is predicted by the collisional excitation model discussed in the Appendix. For 13 CO, there are only two sources with excitation temperatures determined for more than one transition. Both of these, HD147933 (Lambert et al. 1994) and HD24534 (Sonnentrucker et al. 2007), show this behavior. Given the constraints imposed by the limited signal to noise ratio, it is difficult to be definitive, but we agree with Sonnentrucker et al. (2007) that radiative excitation by nearby clouds does not play a major role in determining the excitation temperature of the lower rotational transitions of CO and that excitation is primarily by collisions. Zsarg'o & Federman (2003) similarly concluded that optical pumping is generally unimportant for excitation of CI in diffuse clouds.", "pages": [ 8, 9 ] }, { "title": "5.2. Collisional Excitation of CO in Diffuse Molecular Clouds", "content": "Analyzing the excitation of CO and determining the density of diffuse clouds is linked to their structure. Possibly important collision partners in diffuse clouds are electrons, atomic hydrogen (H 0 ) and molecular hydrogen (H 2 ). In these clouds, carbon is largely in ionized form (see, for example Figure 1) so the fractional abundance of electrons glyph[similarequal] 10 -4 throughout diffuse molecular clouds. Crawford & Dalgarno (1971) calculated the cross sections for excitation of lowJ transitions of CO due to collisions with electrons. Their results are reproduced by the general, but more approximate treatment of Dickinson & Richards (1975), who calculate excitation rate coefficients and fit a quite convenient general formula. Application to the low-dipole moment CO molecule, yields characteristic deexcitation rate coefficients between 0.4 and 0.5 × 10 -8 cm 3 s -1 , for kinetic temperatures between 50 K and 150 K, and for J upper between 1 and 6. These are approximately a factor of 100 larger than those for collisions with atomic or molecular hydrogen (see discussion in Section 5.5). However, this is not a sufficient factor to compensate for the much lower fractional abundance of electrons, and in consequence electrons should not be a significant source of collisional excitation for CO in diffuse clouds. The more detailed discussion in Section 5.5 thus considers only excitation by collisions with H 0 and H 2 . Note that this situation is quite different than that for high-dipole moment molecules such as HCN (Dickinson et al. 1977), since the coefficients for electron excitation scale approximately as µ 2 . Thus for HCN or CN, electron excitation rate coefficients will be 10 4 to 10 5 times larger than those for collisions with atoms or molecules.", "pages": [ 9, 10 ] }, { "title": "5.3. Cloud Structure, the H 0 to H 2 Transition, and Excitation Analysis", "content": "We are left with atomic and molecular hydrogen as being significant for collisional excitation of CO in diffuse molecular clouds. The distribution and abundance of each varies through a cloud due to the competition between formation and photodissociation; the latter is mediated by self-shielding. The processes determining the transformation between H 0 and H 2 are well-treated by the Meudon PDR code (Le Petit et al. 2006). We have carried out a number of runs modeling slabs exposed to the interstellar radiation field on both sides, with a uniform density defined by n (H) = n (H 0 ) + 2 n (H 2 ). The critical results are summarized in Table 3. We include the molecular fraction defined by f (H 2 ) = 2 n (H 2 )/(2 n (H 2 ) + n (H 0 )), defined in the central portion of the cloud, and also integrated through the entire cloud, which we denote F (H 2 ) following Snow & McCall (2006). For clouds having extinction exceeding a few tenths of a magnitude, a large fraction of hydrogen is in molecular form. What is particularly important to note is that the H 2 fraction in the centers of the slab is high, generally ≥ 0.75, and in some relevant cases, > 0.9. From the data on color excess presented by Rachford et al. (2002) and Sheffer et al. (2008) we can determine the total hydrogen column density N (H) and the integrated hydrogen fraction for some of the sources observed here. The values for most sources are glyph[similarequal] 0.5, confirming that, for the sources here, a large fraction of the hydrogen will be in molecular form. This is consistent with the information presented in Table 2 of Burgh, France, & McCandliss (2007) showing that F (H 2 ) ≥ 0.24 for 8 of the 9 sources with N (H 2 ) > 10 20 cm -2 . The one exception, HD102065, has a reasonable density range determined with a single CO transition (Table 4), but an enhanced UV field could result in the low integrated molecular fraction, F (H 2 ) = 0.1 (Burgh, France, & McCandliss 2007). F (H 2 ) is not obviously correlated with N (H), suggesting that other characteristics such as cloud density and environment play an important role in determining the balance between atomic and molecular hydrogen. This situation is illustrated by the cloud model results shown in Figure 1. The H 2 density exceeds that of H 0 for visual extinctions ≥ 0.03 mag, and in the central region of the cloud, n (H 2 ) glyph[similarequal] 50 cm -3 , which is quite similar to the average value determined below in Section 6. The kinetic temperature varies between glyph[similarequal] 50 K and glyph[similarequal] 100 K throughout the cloud, also in good agreement with observations (e.g. Table 6 of Sheffer et al. 2008). While this treatment does not consider all combinations of extreme conditions, it is reasonable to conclude that diffuse molecular clouds have a largely molecular hydrogen core, surrounded by a region in which the hydrogen is primarily atomic. The size of the molecular core, the peak H 2 fraction, and the integrated H 2 fraction all increase with increasing cloud density, and decrease as the strength of the interstellar radiation field increases. It is thus plausible that in diffuse clouds with visual extinction of a few tenths to glyph[similarequal] 1 mag, such as most of those observed in the above-cited papers, the density of H 2 is a factor 2 to 10 times larger than that of H 0 . This is consistent with the properties suggested for 'transitional clouds' studied in HI self-absorption by Kavars et al. (2005). A consideration when comparing these models with observations is the question of multiple clouds along the line of sight. Welty & Hobbs (2001) find that their extremely high spectral resolution (ground-based) observations require multiple, relatively narrow velocity components in order to obtain good fits to their observed K I line profiles. Such resolution is not available for UV observations of CO, but the observed cloud parameters may, in fact, refer to the sum of a number of individual components. The major effect of multiple clouds is that the extinction in each individual component cloud is smaller than the total line of sight extinction. The clouds being considered here have (measured) H 2 column densities between 1 and 6 × 10 20 cm -2 . This alone corresponds to extinctions between 0.1 and 0.6 mag. If we assume a nominal integrated H 2 fraction F (H 2 ) = 0.6, N (H 0 ) = 1.33 N (H 2 ), and the atomic hydrogen column density envelope raises the total extinction through the cloud to glyph[similarequal] 0.2 - 1.0 mag. If we have, for example, three equal component clouds along the line of sight, each has extinction between 0.07 and 0.33 mag. In conditions of standard radiation field intensity and n (H) ≥ 100 cm -3 , the peak H 2 fraction will reach 0.5 for the lowest column density clouds, and will be close to unity for those having the highest column densities. Thus, even the presence of a modest number of components along the line of sight will not change the basic picture of a constituent cloud having a primarily molecular H 2 core surrounded by a H 0 envelope.", "pages": [ 10, 11, 12 ] }, { "title": "5.4. CO Chemistry", "content": "Another factor is the chemistry of CO. While it is not appropriate to go into this in much detail here, a short review is important to appreciate where in the observed clouds the CO being observed is actually located. In clouds where hydrogen is atomic, the only route to form CO starts with the radiative association reaction of C + and H 0 . This reaction is extremely slow, but the CH + that forms yields (through reaction with O) a slow rate of CO production, and combined with relatively unattenuated photodissociation, results in a low fractional abundance of CO. In clouds with the hydrogen in the form of H 2 , a similar radiative association reaction between C + (which will still be the dominant form of carbon due to its lower ionization potential) and H 2 can take place, but it is glyph[similarequal] 40 times faster than that with H 0 , leading to somewhat higher CO fractional abundances. A second path is the chemical reaction C + + H 2 → CH + + H. However, this reaction is endothemic by 4640 K, and thus is extremely slow at normal cloud temperatures. The above pathways lead to a fractional abundance of CO in diffuse molecular regions glyph[similarequal] 3 × 10 -8 , similar to that seen in Figure 1, but significantly below that observed for diffuse molecular clouds being considered here ( 〈 N(CO)/N(H 2 〉 = 3 × 10 -7 ; Federman et al. (1980), Sheffer et al. (2008)). Elitzur & Watson (1978) suggested that presence of shock heating would significantly raise the temperature of the molecular gas and enhance the abundance of CH + . This would also have the effect of increasing the abundance of CO. This could resolve the discrepancy between model and observations, but has the undesirable consequence of copiously producing OH via the reaction O + H 2 → OH + H, which is endothermic by 3260 K. The predicted OH fractional abundance exceeds that observed by a large factor. In order to exploit the rapid reaction between C + and H 2 at high temperatures without overproducing OH, Federman et al. (1996) suggested that Alfv'en waves could heat diffuse clouds and the outer portions of larger molecular clouds with the special effect of raising the temperature of the ions and not that of the neutrals. Thus, CH + , and CO abundances could be enhanced without overproducing OH. This 'superthermal' chemistry was supported by observations of various species by Zsarg'o & Federman (2003) and has subsequently been incorporated into different models, notably that of Visser, van Dishoeck, & Black (2009), that successfully reproduce the run of CO vs H 2 column densities. An alternative explanation that explains the abundances of a number of species in diffuse molecular clouds is heating in regions of turbulent dissipation, discussed by Godard, Falgarone, & Pineau des Forˆets (2009). What is essential for the present discussion is that these models are entirely dependent on having molecular hydrogen as the starting point. In contrast, no models starting with atomic hydrogen can achieve fractional abundances of CO close to those observed. Thus, the chemistry of CO strongly suggests that we are tracing a species confined to the portion of the cloud in which hydrogen is largely in the form of H 2 . Note that the Meudon PDR code does not include superthermal chemistry so that the CO fractional abundances predicted (e.g. Figure 1) are significantly below those derived from observations.", "pages": [ 14, 15 ] }, { "title": "5.5. Collision Rate Coefficients", "content": "Due to the importance of CO in the dense interstellar medium in which hydrogen is almost entirely molecular, cross sections and excitation rates for collisions between CO and H 2 have been calculated by several different groups starting more than three decades ago (Green & Thaddeus 1976; Flower 2001; Wernli et al. 2006; Yang et al. 2010). The earliest calculations treated H 2 molecules as He atoms (with a simple scaling for their different mass), which is correct only for the lowest, spherically symmetric level (J = 0) of para-H 2 . This calculation is thus strictly speaking not applicable to ortho-H 2 , which is expected to have comparable or even greater abundance than the para-H 2 spin modification. While the details of the potential surfaces and the quantum calculations have evolved, the results are all quite similar. Figure 2 includes the more detailed calculations that treat ortho-H 2 and para-H 2 as separate species. As seen in Figure 2, there is very little difference between the deexcitation rate coefficients for ortho- and for para- H 2 . To model the excitation of CO by collisions with H 2 in diffuse clouds, we use the most recent results of Yang et al. (2010) as given in the LAMDA database. We have adopted an ortho-to-para H 2 ratio (OPR) equal to 3, but varying the OPR from 1 to 3 does not make a significant difference in the excitation temperature for a given total H 2 density. The results for the three lowest CO transitions for three kinetic temperatures representative of the temperatures measured for this sample of diffuse molecular cloud sight lines are shown in Figure 3. We see that the excitation temperatures of all three transitions increase monotonically with the H 2 density throughout this range. For densities below glyph[similarequal] 50 cm -3 , the excitation temperatures T ex JJ -1 increase as J increases. This apparently surprising behavior is discussed in detail in the Appendix. All transitions are assumed to be optically thin for these calculations. The situation for collisions between hydrogen atoms and CO molecules is less satisfactory. H 0 -CO collisions were analyzed by Chu & Dalgarno (1975), who found cross sections for H 0 -CO collisions comparable to those for H 2 -CO collisions for small ∆ J which are numerically the largest. Green & Thaddeus (1976) carried out calculations for H 0 colliding with CO, with quite different results. First, the magnitude of the collision rate coefficients are an order of magnitude smaller (1-2 × 10 -11 cm 3 s -1 compared to 10 -10 cm 3 s -1 ). Second, the Green & Thaddeus (1976) rate coefficients are very small for | ∆J | ≥ 3 compared to those for | ∆J | = 1 or 2. A more recent calculation (Balakrishnan, Yan, & Dalgarno 2002) suggested cross sections for H 0 -CO collisions a factor glyph[similarequal] 25 larger in total magnitude (compared to Green & Thaddeus (1976)), and with large-∆ J transitions enhanced by up to 1000. However, a subsequent reconsideration by Shepler et al. (2007) of the interaction potential employed suggested that the results of Balakrishnan, Yan, & Dalgarno (2002) are erroneous 1 . This is confirmed by a very recent paper by Yang et al. (2013), which finds results for H 0 -CO collisions very similar to those of Green & Thaddeus (1976). In Figure 4 we compare the excitation temperatures of the three lowest CO rotational transitions produced by collisions with H 0 and H 2 (based on the Green & Thaddeus (1976) rate coefficients for collisions with H 0 and the Yang et al. (2010) rate coefficients for collisions with H 2 ). The form of the excitation temperature curves are the same for both types of colliders, but shifted by a factor of glyph[similarequal] 5 for the two lowest transitions and a factor glyph[similarequal] 25 for the J = 3-2 transition. The relative ordering of the T ex is highly sensitive to the density of colliding particles. One contributor to this is the fact that the J = 1-0 transition is heading towards a population inversion with equal upper and lower level populations and resulting infinite excitation temperature. From this we see that for H 2 fraction f (H 2 ) ≥ 0.3, collisions with H 2 will be dominant, and for f (H 2 ) ≥ 0.5, the H 0 is unimportant for collisional excitation. Based on the modeling of diffuse molecular clouds in terms of the distribution of H 2 and H 0 as a function of optical depth, the CO chemistry, and the relative magnitudes of the collision rate coefficients for collisions with H 2 and H 0 as well as the lack of convincing evidence for any non-collisional excitation mechanism, it appears very likely that collisions with H 2 molecules are the dominant source of collisional excitation of CO molecules. Consequently, from the observed excitation temperatures we should be able to derive the H 2 density in diffuse molecular clouds. Any result is, of course, subject to the caveat of being an average of the regions with CO along the line of the sight, including multiple clouds, if present.", "pages": [ 15, 16, 19, 20 ] }, { "title": "6. Density Determination", "content": "We divide the analysis into two parts. The first is for the sources for which there is only data for the lowest transition. For these sources with only a single value of the excitation temperature to fit, we can determine an upper limit to the density or a range of densities, depending on the value of T ex 10 . The second part is for sources with multiple transitions, for which we must also consider the consistency between the results from the different transitions observed. For the Sheffer et al. (2008) data, we consider the uncertainty in the value of T ex to be that given in Section 4, namely σ T ex /T ex = 0.2, while for the other data we use the uncertainties provided.", "pages": [ 20 ] }, { "title": "6.1. Optical Depth", "content": "The optical depth of the J = 1-0 CO transition can be written where N 12 is the column density of CO in units of 10 12 cm -2 , δv kms is the FWHM line width in kms -1 , and f J =1 is the fraction of the molecules in the upper ( J = 1) state. Under the subthermal conditions encountered here, it is not correct to assume that all transitions have the same excitation temperature or to adopt the usual expression for the partition function, Q = KT/hB 0 . The higherJ transitions have lower optical depths for the densities in the range of interest for these diffuse molecular clouds. The excitation-dependent terms in the above equation vary among the clouds studied here, but their product is not far from unity. While the lines are not spectrally resolved, high-resolution ground-based observations of species such as CH and CH + as templates suggest FWHM line widths glyph[similarequal] 3 km s -1 . Thus, τ (1 , 0) glyph[similarequal] 10 -4 N 12 . Even if the entire line of sight CO column density is incorporated into a single cloud, the optical depth for almost all clouds included here is considerably less than unity, with the highest column density cloud (having N 12 = 10 4 ) just reaching this limit. If the column density is divided among several clouds that each subtends only a small solid angle as seen by the others, the radiative trapping will be reduced accordingly. We thus thus do not consider trapping to be a significant contributor to the CO excitation for the clouds considered here, although this will not be the case, for example, for translucent clouds with larger CO column densities.", "pages": [ 20, 22 ] }, { "title": "6.2. Kinetic Temperatures", "content": "The kinetic temperature shows considerable variation among diffuse molecular clouds. Savage et al. (1977) employed UV observations of H 2 in the J = 0 and J = 1 rotational levels, and with the assumption that the relative population of these ground rotational levels of para- and ortho-H 2 reflects the kinetic temperature, found that clouds with N (H 2 ) greater than 10 18 cm -2 have kinetic temperatures between 45 and 128 K, with an average values for 61 stars of 77 ± 17 (rms) K. Rachford et al. (2002) used a similar technique, finding a slightly lower mean value, with < T k > = 68 K, and a variance of 15 K, although there were three lines of sight having T k > 94 K. Sheffer et al. (2008), again using the same technique, find the average value of the excitation temperature of J =1 relative to J = 0, < T 01 (H 2 ) > = 77 ± 17 K for 56 lines of sight. This should be a good measure of the kinetic temperature. The range of T k determined by HI absorption and emission studies (Heiles & Troland 2003) of the Cold Neutral Medium extends to somewhat lower temperatures, but the column density-weighted peak kinetic temperature is 70 K. The range 50 K ≤ T k ≤ 100 K thus largely covers the measured range of kinetic temperatures determined for the diffuse molecular clouds considered here.", "pages": [ 22, 23 ] }, { "title": "6.3. Sources with J = 1-0 Observations Only", "content": "Of the 76 sources, 44 are in this category. The results are given in Table 4, for which we adopt T k = 100 K. For the sources for which T ex 10 together with the statistical uncertainties define a range of allowed densities, we give the minimum and maximum H 2 densities, n min and n max . Given the statistical uncertainty in the excitation temperature, and the T ex 10 vs n (H 2 ) curve seen in the lower panel of Figure 3, we consider that we have only an upper limit on the density of a source having T ex 10 ≤ 3.5 K. We denote this maximum density n max , and there is no entry for the minimum density n min . As is immediately seen in Figure 3, the dependence of the excitation temperature on the kinetic temperature for n (H 2 ) ≤ 100 cm -3 is much smaller for the J = 1-0 transition than for the higher transitions. For most of the density range of interest, the change in log( n (H 2 )) required to achieve a particular T ex is no more than 0.1 dex for kinetic temperature changing from 100 K to 50 K, and less than that for the kinetic temperature changing from 100 K to 150 K. The H 2 density required to achieve a given excitation temperature increases as the kinetic temperature decreases due to the reduced excitation rates; the J = 3 level is 33 K above the ground state. We adopt a kinetic temperature of 100 K for analysis of the J = 1 - 0 only clouds. The modest sensitivity to kinetic temperature indicates that our lack of knowledge of the kinetic temperature in a given cloud or the likely variation in the kinetic temperature throughout a single cloud will not produce a significant error in the derived value of the H 2 density compared to that resulting from the uncertainty in the excitation temperature arising from the imprecisely known column densities. Of the 44 sources with only T ex 10 data, 30 have only upper limits on n (H 2 ) and 14 have both lower and upper limits. The values of n max for the former are relatively modest, all below 200 cm -3 , with the average value of log ( n max ) equal to 1.57, corresponding to < n max > glyph[similarequal] 37 cm -3 . This category includes, but is not restricted to, clouds having the lowest H 2 column densities. For each source the logarithm of the midpoint density, log ( n mid ) is the average of the logarithms of n max and n min . For the 14 sources with upper and lower limits, the average value of n min is 22 cm -3 , and of n max is 105 cm -3 . The average of the midpoint values of log ( n ( H 2 )) is 1.69 corresponding to < n mid > = 49 cm -3 . These sources thus represent a population of diffuse clouds having relatively low densities. Thermal balance calculations indicate that these low density diffuse clouds will have relative high kinetic temperatures, thus justifying our adoption of 100 K for the nominal value of T k .", "pages": [ 23, 24 ] }, { "title": "6.4. Sources with Observations of Two or Three Transitions", "content": "Our sample includes 18 sources with two, and 14 sources with data for three transitions. The results for these 32 sources are given in Table 5. For each source we give the minimum and maximum density for each transition as discussed above, for kinetic temperature equal to 100 K. A dash indicates that there was no excitation temperature for that transition. In the last two columns we give the range of densities that satisfies all of the data available, if such a consistent range exists. Sources for which there is an upper limit only for a given transition have no entry in the appropriate n min column. For sources with data for more than one transition, there is the possibility of no density simultaneously yielding the different excitation temperatures even when the errors are included. For 18 sources, we find a range of densities consistent with all transitions observed. For 7 sources for which there was no formal consistent solution, an additional 0.1 dex in density allows a consistent density or density range to be found. These combined densities are indicated by an (s) by the derived density or density range. The absence of an entry in both of the final two columns indicates there was no density consistent with the excitation temperatures for that source; there are 7 sources in this category. We have 18 sources with data on T ex 10 and T ex 21 . Of these, 11 have a density range or upper limit consistent with the measurements of both transitions, while 3 additional sources are in this category if the additional 0.1 dex uncertainty is allowed. For the 12 sources with consistent density ranges, we find < n min > = 22 cm -3 , and < n max > = 70 cm -3 . The average value of the midpoint densities is < log ( n mid ) > = 1.63 corresponding to < n mid > = 43 cm -3 . This is slightly lower than, but certainly consistent with, the value obtained for the sources for which we have T ex 10 data only. This suggests that the two lowest CO transitions are not probing very different regions within or among diffuse molecular clouds along the line of sight. We have 14 sources with data on excitation temperatures for 3 transitions. Since the three different excitation temperatures are differently sensitive to density, these sources are the most demanding in terms of defining a single characteristic density responsible for the entirety of the emission. Of these 14 sources, 7 have H 2 density ranges consistent with all three transitions, with 4 additional sources included if we allow the additional 0.1 dex in density added to range for each transition. For the 11 sources with consistent density range, we find < n min > = 75 cm -3 , < n max > = 118 cm -3 , and < n mid > = 94 cm -3 . These values are somewhat higher than for the two previous categories, which suggests that inclusion of the J = 3 - 2 transition does tend to select out clouds or regions within clouds having somewhat higher densities. Given the uncertainties, the values of < n mid > of 49, 42, and 94 cm -3 can be taken together to define and average density < n mid > = 60 cm -3 for the 36 sources with H 2 density ranges, again assuming a kinetic temperature T k = 100 K. Of the 32 sources with multiple transition data, we obtain a consistent density ranges for 9 (50%) of those with the two lowest transitions, and 7 (50%) of those with three transitions. If we include the stretch sources, these numbers go up to 12 (67%) and 10 (71%). Thus, the inclusion of sources with 3 as compared to 2 transitions leaves the fraction of sources for which a consistent density range can be found essentially unchanged. Of the 7 sources with no consistent density solution, 5 can be characterized as having the J = 1-0 transition implying too-low density (compared to J = 2-1 (4 sources) or to both J = 2-1 and 3-2 (1 source)). The 2 remaining sources are characterized by having J = 2-1 transition implying a density range higher than that indicated by the J = 1-0 and 3-2 transitions. While the J = 3-2 transition data are suggestive of higher densities, it is not obvious that a density gradient or multiple density components affect level populations in a way that prevents obtaining a single density solution. In fact, the simple combination of two densities generally produces a solution that is simply an intermediate value. This is illustrated in Figure 5, in which we have combined two different clouds having densities of 10 cm -3 and 100 cm -3 with the low density component (cloud 1) having a fraction between 0 and 1 of the total CO column density. We assume that both clouds have the same kinetic temperature. Since all lines are optically thin, it is straightforward to calculate the excitation temperatures that would be derived from the relative column densities. The result is that the variation in the three excitation temperatures produced by varying the relative amount of high and low density cloud material mimics quite closely the variation in the excitation temperatures produced by a single cloud having density between that of the lower density and the higher density cloud.", "pages": [ 24, 25, 26 ] }, { "title": "6.5. Average Density and Thermal Pressure", "content": "The values for the H 2 density of each source have been found in terms of maximum and minimum values of log(n(H 2 )) that are consistent with the data including errors predominantly due to statistical uncertainties. The sources having 2 or 3 values of kinetic temperature are likely to be the most valuable for assessing the effect of kinetic temperature changes since the different transitions have upper levels significantly higher than for the J = 1-0 transition, although there are relatively fewer of the multiple-transition sources. Figure 3 shows that while T ex 10 is relatively insensitive to T k , the higher transitions show increasing sensitivity, as expected for the larger level separation (equivalent to 33 K for the J =3-2 transition). The collision rates increase monotonically with kinetic temperature in the range of interest, and thus the density required to obtain a given kinetic temperature is lower for a higher value of T k . In Figure 7 we show graphically the range of densities for each of the excitation temperatures in six sources in this category. As anticipated, the allowed densities are shifted to higher values for the lower kinetic temperature. This applies to the individual transitions as well as for the allowed ranges for the combined set of three transitions. For four of the six sources, the allowed density range for the combined set of transitions is substantial. However, for HD148937, the combined transition density range is very narrow, only 0.1 dex. For HD147683 there is nominally no density consistent with all three excitation temperatures, but log ( n (H 2 )) is within 0.1 dex of the upper limit from the J = 1-0 transition and an equal amount from the lower limit of the J = 2-1 transition for T k = 100 K, and similarly log ( n (H 2 )) = 2.5 for T k = 50 K. There is no obvious pattern from changing the kinetic temperature other than the shift to slightly higher densities for 50 K compared to 100 K kinetic temperature. It therefore does not seem possible to use the available data to put tighter constraints on the kinetic temperature; measurements of higherJ transitions would be required to do this. We can find the average value of the midpoint density for several different groupings of our sources, and the results are given in Table 6. We see that < n mid > is essentially the same for the 14 sources for which we have only T ex 10 and the 12 sources for which we have values for T ex 10 and T ex 21 . For both categories, < n mid > glyph[similarequal] 45 cm -3 , at a kinetic temperature of 100 K. The minimum, maximum, and midpoint densities are all greater when we consider 3 rather than 2 excitation temperatures, as discussed above. If we include the 23 sources with 2 or 3 excitation temperatures the average value of the midpoint density is < n mid > = 68 cm -3 , compared to 42 cm -3 for two excitation temperatures, and 94 cm -3 for three excitation temperatures, all for T k = 100 K. For a lower kinetic temperature of 50 K, we obtain somewhat higher densities. For the sources with data on two excitation temperatures, < n min > = 32 cm -3 , < n mid > = 67 cm -3 , and < n max > = 143 cm -3 . For the sources with data on three excitation temperatures, < n min > = 104 cm -3 , < n mid > = 135 cm -3 , and < n max > = 176 cm -3 . All of these results are in line with previous determinations of densities of diffuse clouds. There do seem to be clear variations among the sources included in this study, with some sources having n(H 2 ) only a few tens cm -3 (HD23478 and HD24398), while HD147888 has a density at least a factor of 10 higher. The thermal pressure suggested by these results is moderately large. The anticorrelation between assumed T k and derived n(H 2 ) suggests that a thermal pressure derived by taking their product is reasonably robust against errors in the kinetic temperature. Using the midpoint densities for the sources with two or three values of excitation temperature as the largest statistical sample with reasonable sensitivity to kinetic temperature, we find for T k = 100 K, p/k = 6800 Kcm -3 , and for T k = 50 K, p/k = 4600 Kcm -3 . Further taking the average of these two yields a thermal pressure p/k = 6700 Kcm -3 . This value is noticeably above the median value determined from UV absorption studies of CI by Jenkins & Tripp (2001), 2240 Kcm -3 , but within the range of the sources studied similarly by Jenkins (2002), 10 3 Kcm -3 ≤ p/k ≤ 10 4 Kcm -3 . A more comprehensive C I study of 89 stars by Jenkins & Tripp (2011) finds a lognormal pressure distribution with < log ( p/k ) > = 3.58, corresponding to < p/k > = 3800 Kcm -3 . It is possible that while the densities found here from CO are still quite modest, the regions may be the envelopes of molecular clouds, which are characterized by significantly higher thermal pressure than for diffuse molecular clouds (Wolfire, Hollenbach, & McKee 2010).", "pages": [ 26, 28, 29, 30 ] }, { "title": "6.6. Correlation Between Volume Density and Column Density", "content": "The present data allow us to examine whether there is a correlation between volume density and column density for this sample of diffuse clouds. We have 14 sources with density ranges from T ex 10 alone and molecular hydrogen column densities. These are plotted with diamond (black) symbols in Figure 6. We also have 16 sources from our sample with density ranges determined by excitation temperatures from 2 (9 sources) and 3 (7 sources) transitions; these are plotted with square (red) symbols. For the T ex 10 only sources, there is no significant correlation of volume and column density. The data for the multiple-transition sources suggests a weak correlation, with a linear best fit n (H 2 ) rising from 10 cm -3 to 100 cm -3 as N (H 2 ) increases from 10 20 cm -2 to 10 21 cm -2 . It is clear, however, that a linear relationship is not consistent with the data for HD 147888, which has a density glyph[similarequal] 4 higher than the general trend. These data suggest that there is at least a component of diffuse molecular clouds for which volume density and column density are correlated.", "pages": [ 30 ] }, { "title": "7. Discussion and Summary", "content": "We have used the UV CO absorption data of Sheffer et al. (2008) and published data on other sources, together with statistical equilibrium calculations, to determine the volume density in diffuse interstellar molecular clouds. We have a total of 76 sources, of which 44 have T ex 10 data only, 18 sources having T ex 10 and T ex 21 data , and 14 sources having T ex 10 , T ex 21 , and T ex 32 data. It does not appear likely that non-collisional processes play a major role in excitation of CO in diffuse clouds. Collisional excitation is expected to result primarily from collisions with H 2 molecules as the H 0 to H 2 transition occurs at substantially lower values of column density than does the C + -C o -CO transition. Recent calculations confirm that excitation rate coefficients for CO-H 2 collisions are significant larger than for CO-H 0 collisions. The CO and H 2 column densities of the sources indicate that the fractional abundance of CO is several orders of magnitude below its asymptotic value in well-shielded regions, and also that the CO rotational transitions are optically thin, with the sources having the largest CO column densities reaching τ glyph[similarequal] 1. We have assumed a kinetic temperature of 100 K as representative for the more diffuse clouds, but also discuss the effect of T k a factor of 2 lower, especially for analysis of sources having data on higher J transitions. For 30 of the sources having only T ex 10 data, we obtain only upper limit to n (H 2 ) for which the average value is < log ( n max ) > = 1.57, corresponding to < n max > = 37 cm -3 . For the remaining 14 T ex 10 -only sources we find a range of H 2 densities that is consistent with the value of the excitation temperature and its estimated uncertainty, and thus determine n min as well as n max . For these sources we find < n min > = 22 cm -3 and < n max > = 105 cm -3 . Defining log ( n mid ) as the average of log ( n max ) and log ( n min ) for each source, the average midpoint density for these 14 sources is given by < n mid > = 49 cm -3 . Of the 18 sources having T ex 21 and T ex 10 data, 14 yield a consistent density range or upper limit. For the 12 sources with density ranges, < n mid > = 42 cm -3 for T k = 100 K and 67 cm -3 for T k = 50 K. Of the 14 sources having T ex 10 , T ex 21 , and T ex 32 data, 11 yield density ranges that are consistent for all three transitions, yielding < n mid > = 94 cm -3 for T k = 100 K and 135 cm -3 for T k = 50 K. Taking the sources with either two or three values of the excitation temperature, we find < n mid > = 68 cm -3 for T k = 100 K and 92 cm -3 for T k = 50 K. Thus, while there are undoubtedly some selection biases, it appears that this sample of diffuse molecular clouds, having H 2 column densities between few × 10 20 and glyph[similarequal] 10 21 cm -2 is reasonably characterized by a density between 50 and 100 cm -3 . The clouds in this sample clearly do not all have the same volume density, with the extreme cases being a factor of a few below and above the range given here. The anticorrelation between derived density and the assumed kinetic temperature allows plausible determination of the internal thermal pressure of these clouds, which is found to be relatively large with p/k in the range 4600 to 6800 cm -3 K. As we are analyzing clouds in which hydrogen is largely molecular, but in which the fractional abundance of CO is so small that this species would be extremely difficult to detect in emission, the present results help characterize the 'CO-Dark Molecular Component' of the interstellar medium. We thank Drs. N. Balakrishnan and L. Wiesenfeld for very helpful information about collision rate coefficients and potential energy surfaces. We thank Nicolas Flagey and Jorge Pineda for useful discussions about dealing with the uncertainties in the column densities of CO and the rotational excitation temperatures, and Bill Langer for clarifying a number of points and a careful reading of the manuscript. The anonymous referee also made significant contributions by pointing out particular aspects of UV studies of diffuse clouds that would otherwise have been missed, and by carefully checking of the data presented here. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.", "pages": [ 30, 33, 34 ] }, { "title": "8.1. INTRODUCTION", "content": "This appendix addresses the issue that the excitation temperature of the levels of simple molecules and atoms in the limit of very low densities does not asymptotically approach the temperature of the background radiation field. This applies to rigid rotor molecules such as CO, and also to simple atomic systems such as C I and O I . Using the RADEX program (Van der Tak et al. 2007) to analyze CO excitation by collisions with ortho-H 2 molecules for a kinetic temperature of 100 K, background temperature of 2.7 K, and H 2 density of 0.01 cm -3 yields the results shown in Table 7. All transitions are optically thin, and since collisional deexcitation rate coefficients are glyph[similarequal] few × 10 -11 cm 3 s -1 , all transitions should be highly subthermal, given that the A -coefficient for the lowest transition is 7.2 × 10 -8 s -1 . This result, that the excitation temperatures seem unreasonably large and increase with increasing J (albeit not perfectly monotonically), is found in the output of all statistical equilibrium programs examined. It thus does not seem to be an artifact of the calculation, but rather is a property of the solutions of the rate equations in the low density limit. While this may seem to be a curiosity, it is important if one has level populations derived from UV absorption, for example, and one wishes to solve for densities that result in highly subthermal excitation. This is suggested by the FUSE and HST observations of CO of Sheffer et al. (2008) and others that are discussed in this paper.", "pages": [ 35 ] }, { "title": "8.2.1. Definitions", "content": "In order to gain some insight into the behavior of the excitation temperatures, we can use a three level model with two transitions to capture the essence of the multilevel CO problem. This is a complete representation of the situation for atomic C I and O I fine structure systems and a very good approximation for CO at low densities. We denote the levels 1, 2, and 3 (not to be confused with rotational quantum numbers), their energies as E 1 , E 2 , and E 3 , and the downwards spontaneous rates and collision rates as A 21 , A 32 , C 21 , C 32 , and C 31 . The energies of the three levels lead to equivalent temperatures for the three transitions kT 21 = E 2 -E 1 , kT 32 = E 3 -E 2 , and kT 31 = E 3 -E 1 , where k is Boltzmann's constant. The background radiation field is assumed to be a blackbody at temperature T bg producing energy density U ( T bg ). The downwards stimulated rates are B 21 U and B 32 U , where the B 's are the stimulated radiative rate coefficients and U is understood to be evaluated at the frequency of the transition in question. The upwards stimulated rates are related to the downwards rates through the statistical weights g 1 , g 2 , and g 3 and detailed balance, giving g 1 B 12 U = g 2 B 21 U and g 2 B 23 U = g 3 B 32 U . The collision rates are the product of the collision rate coefficients and the colliding partner density. Thus for collisions with H 2 , The upwards rates are related to the downwards rates through detailed balance and the kinetic temperature T k through g 1 C 12 = g 2 C 21 exp( -T ∗ 21 /T k ), g 1 C 13 = g 3 C 31 exp( -T ∗ 31 /T k ), and g 2 C 23 = g 3 C 32 exp( -T ∗ 32 /T k ). The level population per statistical weight defines the excitation temperature through equation 1. With these definitions, the rate equations for the level populations n 1 , n 2 , and and which yield T ex = T bg for both transitions.", "pages": [ 36, 37 ] }, { "title": "8.2.4. Low Density Limit with No Background Radiation", "content": "With the above expressions we can examine the low density limit, together with the effect of varying the background radiation temperature. We first consider the no-background n 3 lead to the following equations for the ratios of the column densities of adjacent levels (connected by radiative transitions): and", "pages": [ 37 ] }, { "title": "8.2.2. High Density Limit", "content": "In this limit with C glyph[greatermuch] A , BU , we find that equations 11 and 12 yield N 2 /N 1 = ( g 2 /g 1 ) e ( -T ∗ 21 /T k ) and N 3 /N 2 = ( g 3 /g 2 ) e ( -T ∗ 32 /T k ) , respectively. This is exactly as expected in the thermalized limit when collisions dominate.", "pages": [ 37 ] }, { "title": "8.2.3. Zero Collision Rate Limit", "content": "In this limit limit ( T bg = 0). In this case, dropping collisional terms where they compete directly with a spontaneous rate, we find and If the ∆ J = 2 collision rate coefficients are zero, equation 16 reduces to In this case of purely 'dipole-like' collisions, we also find We thus see that the excitation temperature of each transition approaches zero as the collision rate approaches zero. If collisions between levels 1 and 3 are allowed, then in the limit of very low collision rate, we find This obviously has an entirely different behavior than that of the lower transition given by equation 15. The excitation temperature of the upper transition approaches an asymptotic limit since the first fraction is a constant determined by the molecular radiative rates, and the second fraction is a constant determined by the relative collision rates. For the latter, in the limit of zero background, the excitation temperature for the upper transition is independent of the density , and is given by In the low density limit (with no background), the excitation temperature for the lower transition is which does depend on the collision partner density through the proportionality of the collision rates and the density (equation 10). To restate the obvious, the excitation temperature of the upper (level 3 - level 2) transition does not approach zero even if the collision rate is arbitrarily small. This is because there is not a simple competition between collisional and radiative processes. This is in contrast with the lower (level 2 - level 1) transition, for which the excitation temperature does approach zero for low collision rate. This reflects the fact that in the limit of very infrequent collisions, level 2 is populated exclusively by collisions from level 1 (which has most of the population) and depopulated by radiative decay back to level 1. It is thus evident that the excitation temperature of a particular transition in a multilevel system can behave in the apparently counterintuitive way of having T ex not approach zero as the collision rate approaches this value. We next give some examples of three-level systems, and will extend the discussion to systems with more than 3 levels in Section 8.3", "pages": [ 37, 38, 39 ] }, { "title": "8.2.5. Examples of Different Systems and Comparisons with Numerical Calculations", "content": "8.2.6. CO We consider the lowest three rotational levels of CO to illustrate the preceding analytic results. The rate coefficients for collisions with para-H 2 from Yang et al. (2010) at a kinetic temperature of 100 K are R 12 = 9 . 7 × 10 -11 cm 3 s -1 and R 13 = 1 . 4 × 10 -10 cm 3 s -1 . Since the collision rates and rate coefficients are proportional (equation 10), this gives ( R 12 + R 13 ) /R 13 = 1.7, which yields (for no background radiation) T ex 32 = 3 . 35 K. The collisional excitation rates for ortho-H 2 - CO collisions from Flower (2001) and Wernli et al. (2006) as extrapolated in the LAMDA database (home.strw.leidenuniv.nl/ moldata/) at a kinetic temperature of 100 K are R 12 = 2 . 65 × 10 -10 cm 3 s -1 and R 13 = 2 . 63 × 10 -10 cm 3 s -1 . This gives ( R 12 + R 13 ) /R 13 = 2.1 and an excitation temperature (for no background radiation) T ex 32 = 3 . 2 K. The value from the full multilevel RADEX calculation is 3.6 K. The difference is due to the effect of the higher levels, discussed in Section 8.3. T ex 32 is essentially constant for H 2 densities up to 100 cm -3 , at which point it begins to rise due to the collision rate becoming comparable to the spontaneous decay rate. The excitation temperature of the lower transition, as expected, varies continuously as a function of the H 2 density. This behavior is shown in Figure 8. The excitation temperature of the lower transition is rising sharply as n (H 2 ) approaches 100 cm -3 , because with the relatively large rate for ∆ J = 2 collisions, we can have a situation in which level 3 ( J = 2) is populated by collisions from level 1 ( J = 0). The radiative decays to level 2 ( J = 1) add to the direct collisional population of that level and result in level 2 ( J = 1) becoming overpopulated relative to level 1( J = 0) as seen in Goldsmith (1972). As the H 2 density increases, the negative excitation temperatures characteristic of the population inversion are preceded by very high positive values of the excitation temperature of the lowest transition, T ex 21 . The behavior of T ex 32 is largely independent of the choice of collision partner or which calculation of the collision rate coefficients is adopted. The Green & Thaddeus (1976) rate coefficients for CO-H 2 collisions give ( R 12 + R 13 ) /R 13 = 1.61 and T ex 32 = 3.4 K, almost identical to the results from Yang et al. (2010), although the values for the individual coefficients are slightly larger. Green & Thaddeus (1976) also give the results for collisions with H and He atoms, which give T ex 32 = 3.3 K and 3.47 K, respectively, almost identical to the values produced by collisions with H 2 molecules. The three fine structure levels of C I make this system a highly appropriate test of this behavior for low collision rates. The ground state (level 1) is 3 P 0 , the first excited state (level 2 at E/k = 23.65 K above the grounds state) is 3 P 1 , and the second excited state (level 3 at 62.51 K above the ground state) is 3 P 2 . Adopting the deexcitation rate coefficients of Schroder et al. (1991) for collisions with ortho-H 2 , we find for a kinetic temperature of 100 K that R 12 = 1 . 68 × 10 -10 cm 3 s -1 and R 13 = 1 . 85 × 10 -10 cm 3 s -1 . With A 32 = 2 . 65 × 10 -7 s -1 and A 21 = 7 . 9 × 10 -8 s -1 , we find from equation 20 that T ex 32 = 16.7 K. The excitation temperature of the lower transition from equation 21 is 3.65 K for a hydrogen density of 1 cm -3 and no background radiation. The value for the upper transition agrees within a few tenths K with that from RADEX, and that of the lower transition agrees within 0.1 K.", "pages": [ 39, 40, 41 ] }, { "title": "8.2.8. Effect of Background Radiation", "content": "The ratio of the downwards stimulated emission rate due to the background radiation field to the spontaneous decay rate is given by where T bg is temperature of the background radiation field, which we assume to be a blackbody. Let us consider the situation in which C ul glyph[lessmuch] A ul with no background radiation field ( T bg = 0). If we consider increasing the background temperature, we will reach a point at which B ul U = C ul . This occurs when The required background temperature thus depends on how much smaller the collision rate is than the spontaneous rate. For this value of background temperature, we should expect the excitation temperature to approach the background temperature since spontaneous and stimulated rates are both comparable to or greater than the collision rate. The results for the 3 level model for CO are shown in Figure 9, for a kinetic temperature of 100 K and a H 2 density of 1 cm -3 . At this density, A 21 /C 21 = 2.1 × 10 3 and A 32 /C 32 = 1.1 × 10 4 . Equation 23 gives T bg ' = 0.7 K for the lower transition and 1.2 K for the higher transition, both of which are in reasonable agreement with Figure 9. Since T ex for the lower transition is less than T bg ' , the excitation temperature increases as T bg increases. As discussed previously, the excitation temperature of the higher transition is relatively large with no background present, and so it initially drops as T bg increases, before joining the T ex = T bg curve. For this density and the two lower transitions, T bg ' is significantly smaller than the background temperature required to make B ul U = A ul , which is just T bg '' = T ∗ /ln (2) = 1 . 44 T ∗ . We can see from the excitation temperatures given in Table 7 that the stimulated rate produced by a background temperature equal to 2.7 K is sufficient to bring the excitation temperatures of two lowest transitions close to equilibrium with the the background temperature. For the higher transitions, the blackbody radiation function falls off sufficiently rapidly that the background becomes insignificant, and the rise of the excitation temperature as one moves up the ladder is essentially the same as from that with no background present at all.", "pages": [ 41, 42 ] }, { "title": "8.3. Systems with More than Three Levels", "content": "Analytic solution of the level populations in systems having many levels is in general tedious. The exception is for dipole-like collisions for which only adjacent levels are coupled. In this situation the ratio of column densities of any pair of adjacent levels can be written analogous to equation 17 and 18, but in which the background radiation can be included. If collisions connect non-adjacent levels, one typically resorts to numerical solutions based on matrix inversion. In the low collision rate limit, the population (for no background radiation) will be limited to the lowest level, since the collisional excitation rate is by assumption less than any spontaneous decay rate. In the case of low but nonzero collision rate, the population will be restricted to the lowest few levels. This will also be the case if there is a background radiation field that produces a stimulated rate comparable to the collision rate for only the lowest few transitions. For a molecule with simple rotor structure, the analytic solution for no background radiation yields an equation similar to equation 19, but with some modifications due to the collisions that change the rotational quantum number by a range of integers. We can write the population ratio of pair of levels u and l in the absence of any background radiation as where l -1 indicates the level below the lower level of the pair in question, and k max is the highest level that is connected by collisions to level 1 or the highest level included in the calculation. The limits on the summation in the numerator reflect the fact that collisions from the lowest level to levels above the upper level of the pair of interest all decay radiatively much faster than any collisional process, and thus effectively populate the upper level of the pair. The sum in the denominator yields the total rate of collisions that populate both members of the pair of levels of interest. Equation 25 can be used to obtain the expression for the excitation temperature where the P ul term reflects the collisional population rates and can be written The fractional population of the higher levels will be very small in this limit, but as seen from Table 7, the excitation temperatures are well-defined. It may seem surprising that the higher levels in this case are populated by collisions directly from the lowest level (or levels). We can verify this numerically, and for simplicity set the background radiation temperature to zero. We use the collision rate coefficients for ortho-H 2 CO collisions from Flower (2001) and Wernli et al. (2006) as extrapolated in the LAMDA database (home.strw.leidenuniv.nl/ moldata/), and for definitiveness consider CO rotational levels 20 and 19. The collisional deexcitation rate coefficients are R 20 1 = 7.5 × 10 -17 cm 3 s -1 and R 20 19 = 1.1 × 10 -10 cm 3 s -1 . At a kinetic temperature of 100 K, the excitation rates are R 1 20 = 2.7 × 10 -20 cm 3 s -1 and R 19 20 = 3.9 × 10 -11 cm 3 s -1 . The ratio of the rates of population of level 20 from level 1 to that from level 19 is given by which in the present case is equal to 2 × 10 8 . It is thus clear that collisions that transfer population from the lowest level (or few lowest levels) to high-lying levels are the dominant excitation mechanism for the higher rotational levels of CO in the low density limit. Returning to the issue of the expected excitation temperatures for high-J transitions, we must evaluate equation 26. There are two factors that must be considered to estimate the sum of the collisions to the levels above the upper level of the transition of interest. First, the collisional deexcitation rates decrease as ∆ J increases. Second, the upwards rates (from the ground state) are further reduced by the increasing upper level energy, even for a moderately high kinetic temperature of 100 K. The result is that the rate to the lower level of the transition is significantly larger than that to the upper and to higher-lying levels. If we consider the transition between levels 10 and 9 ( J = 9-8) with ortho-H 2 collisions (as discussed above) at 100K, we find P 10 9 = 3.16, which results in T ex 10 9 = 30.7 K. This agrees (fortuitously) well with the 30.6 K Radex result given in Table 7. For the transition between levels 18 and 17 ( J = 17 - 16), P 18 17 = 5.5, which results in T ex 18 17 = 48.4 K. This compares to the RADEX result T ex 18 17 = 49.6 K (Table 7). Given we included excitation only up to level 20, the agreement is very satisfactory.", "pages": [ 42, 43, 44, 45 ] }, { "title": "8.4. Conclusions", "content": "We have analyzed the initially surprising behavior of the excitation of the CO rotational ladder under conditions of very low density, for which the excitation temperature increases steadily as one moves from lower to higher levels. The same effect is generally observed for rigid rotors, for simple atomic fine structure systems, as well as for molecules with more complex term schemes. This behavior can be understood by considering the limit in which collisional deexcitation can be ignored. Radiative decay then makes the population of all levels other than the ground state quite small. A (rare) collision from the ground state to an excited state is followed by a radiative cascade. The populations of the upper and lower levels of a transition are determined by the collisions into the respective levels, plus the radiative cascade from higher levels. The result is level populations that depend essentially only on the relative magnitudes of the A-coefficients for decay into and out of the lower level of the transition of interest. In consequence, the excitation temperature is proportional to the equivalent temperature T ∗ = hf/k of the transition. The impact on the lower levels of CO is modest because the stimulated transition rate from the cosmic microwave background radiation is sufficient to make the excitation temperature equal to the background temperature. The same is not true for the higher levels, for which the background is unimportant.", "pages": [ 45, 46 ] }, { "title": "REFERENCES", "content": "Balakrishnan, N., Yan, M., & Dalgarno, A. 2002, ApJ, 568, 443 Burgh, E.B., France, K., & McCandliss, S.R. 2007, ApJ, 658, 446 Chandra, S., Maheshwari, V.U., & Sharma, A.K. 1996, A&AS, 117, 557 Chu, S.-I. & Dalgarno, A. 1975, Proc. R. Soc. Lond. A., 342, 191 Cologne Database for Molecular Spectroscopy (CDMS) www.astro.uni- koeln.de/cdms/catalog Crawford, O.H. & Dalgarno, A. 1971, J. Phys. B., 4, 494 Dickinson, A.S. & Richards, D. 1975, J. Phys. B, 8, 2846 Dickinson, A.S., Phillips, T.G., Goldsmith, P.F., et al. 1977, A&A, 54, 645 Draine, B.T. 1978, ApJS, 36, 595 Elitzur, M., and Watson, W.D. 1978, ApJ, 222, L141 Federman, S.R., Glassgold, A.E., Jenkins, E.B., & Shaya, E.J. 1980, ApJ, 242, 545 Federman, S.R., Rawlings, J.M.C., Taylor, S.D., & Williams, D.A. 1996, MNRAS, 279, L41 Federman, S.R., Lambert, D.L., Sheffer, Y., et al. 2003, ApJ591, 986 Green, S. & Thaddeus, P. 1976, ApJ, 205, 766 Flower, D.R. 2001, J. Phys. B., 34, 2731 Godard, B., Falgarone, E., & Pineau des Forˆets, G. 2009, A&A, 495, 847 Goldsmith, P.F. 1972, ApJ, 176, 597 Goldsmith, P.F., Langer, W.D., Pineda, J.L., & Velusamy, T. 2012, ApJS, 203, 13 Heiles, C. & Troland, T.H. 2003, ApJ, 586, 1067 Jenkins, E.B., Drake, J.F., Morton, D.C. 1973, ApJ, 181, L122 Jenkins, E.B. & Tripp, T.M. 2001, ApJS, 137, 297 Jenkins, E.B. 2002, ApJ, 580, 938 Jenkins, E.B. & Tripp, T.M. 2011, ApJ, 734, 65 Kavars, D.W., Dickey, J.M., McClure-Griffiths, N.M., et al. 2005, ApJ, 626, 887 Lambert, D.L., Sheffer, Y., Gilliland, R.L., & Federman, S.R., 1994, ApJ, 420, 756 Langer, W. D., Velusamy, T., Pineda, J. L., et al. 2010, A&A, 521, L17 Le Petit, F., Nehem'e, C., Le Bourlot, J., & Roueff, E. 2006, ApJS, 164, 506 Liszt, H.S. & Lucas, R. 1998, A&A, 339, 561 Liszt, H.S. 2006, A&A, 458, 507 Pilbratt, G. L., Riedinger, J. R., Passvogel, T., et al. 2010, A&A, 518, L1 Pineda, J.L., Langer, W.D., Velusamy, T., & Goldsmith, P.F. 2013, A&A, 554, A103 Rachford, B.L., Snow, T.P., Tumlinson, J., et al. 2002, ApJ, 577, 221 Savage, B.D., Bohlin, R.C., Drake, J.F., & Budich, W. 1977, ApJ, 216, 291 Schroder, K., Staemmler, V., Smith, M.D., Flower, D.R., & Jaquet, R. 1991, J. Phys. B: At. Mol. Opt. Phys., 24, 2487 Sheffer, Y., Rogers, M., Federman, S.R., et al. 2007, ApJ, 667, 1002 Sheffer, Y., Rogers, M., Federman, S.R. et al. 2008, ApJ, 687, 1075 Shepler, B.C., Yang, B.H., Kumar, T.J.D., et al. 2007, A&A, 475, L15 Snow, T.P. & McCall, B.J. 2006, ARA&A, 44, 367 Sonnentrucker, P., Welty, D.E., Thorburn, J.A., & York, D.G. 2007, ApJS, 168, 58 UMIST RATE2012 www.udfa.net Van Dishoeck, E.F. & Black, J.H. 1988, ApJ, 334, 771 Van der Tak, F.F.S., Black, J.H., Schoier, F.L., Jansen, D.J., van Dishoeck, E.F. 2007, A&A468, 627 Visser, R., van Dishoeck, E.F., & Black, J.H. 2009, A&A, 503, 323 Wannier, P.G., Penpraes, B.E., & Andersson, B.-G. 1997, ApJ, 487, L65 Welty, D.E. & Hobbs, L.M. 2001, ApJS, 133, 345 Wernli, M., Valiron, P., Faure, A., et al. 2006, A&A, 446, 367 Wolfire, M, Hollenbach, D., & McKee, C.F. 2010, ApJ, 716, 1191 Yang, B., Stancil, P.C., Balakrishnan, N., & Forrey, R.C. 2010, ApJ, 718, 1062 Yang, B., Stancil, P.C., Balakrishnan, N., et al. 2013, arXiv1305.2376v1 Zsarg'o, J. & Federman, S.R. 2003, ApJ, 589, 319 1 Excitation temperatures derived from data given in Sheffer et al. (2008), using equations 2 - 4 in the present paper. 2 N(H 2 ) from Rachford et al. (2002). 3 Estimates of N(H 2 ) by Sheffer et al. (2008) based on correlations with column densities of other species. 4 N(H 2 ) from Sheffer et al. (2007). b Burgh, France, & McCandliss (2007) a n (H) = n (H 0 ) + 2 n (H 2 )", "pages": [ 49, 50, 51, 54, 55, 56 ] } ]

[ { "title": "ABSTRACT", "content": "Gamma-ray bursts (GRBs) and GeV-TeV selected radio loud Active Galactic Nuclei (AGNs) are compared based on our systematic modeling of the observed spectral energy distributions of a sample of AGNs with a single-zone leptonic model. We show that the correlation between the jet power ( P jet ) and the prompt gamma-ray luminosity ( L jet ) of GRBs is consistent, within the uncertainties, with the correlation between jet power and the synchrotron peak luminosity ( L s , jet ) of flat spectrum radio quasars (FSRQs). Their radiation efficiencies ( ε ) are also comparable ( > 10% for most sources), which increase with the bolometric jet luminosity ( L bol , jet ) for FSRQs and with the L jet for GRBs with similar power-law indices. BL Lacs do not follow the P jet -L s , jet relation of FSRQs. They have lower ε and L bol , jet values than FSRQs, and a tentative L bol , jet -ε relation is also found, with a power-law index being different from that of the FSRQs. The magnetization parameters ( σ ) of FSRQs are averagely larger than that of BL Lacs. They are anti-correlated with ε for the FSRQs, but positive correlated with ε for the BL Lacs. GeV Narrow-line Seyfert 1 galaxies potentially share similar properties with FSRQs. Based on the analogy between GRBs and FSRQs, we suggest that the prompt gamma-ray emission of GRBs is likely produced by synchrotron process in a magnetized jet with high radiation efficiency, similar to FSRQs. The jets of BL Lacs, on the other hand, are less efficient and are likely more matter dominated. Subject headings: galaxies: jets-BL Lacertae objects: general-quasars: general-gamma-ray burst: general-methods: statistical", "pages": [ 1 ] }, { "title": "RADIATION MECHANISM AND JET COMPOSITION OF GAMMA-RAY BURSTS AND GEV-TEV SELECTED RADIO LOUD ACTIVE GALACTIC NUCLEI", "content": "Jin Zhang 1 , En-Wei Liang 2,1 , Xiao-Na Sun 2 , Bing Zhang 3 , Ye Lu 1 , Shuang-Nan Zhang 1,4 Draft version October 17, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The emission of gamma-ray bursts (GRBs) and radio loud active galactic nuclei (AGNs) is generally believed to be produced in relativistic jets powered by their central black holes (BHs). GRBs are the most luminous transients in the universe. Their progenitors are thought to be either core collapses of massive stars (Woosley 1993; Paczy'nski 1998) or mergers of two compact objects (Eichler et al. 1989). Such a catastrophic event likely gives birth to a new-born stellar-mass BH, which accretes mass from a torus. An ultra-relativistic jet is launched from the central engine, which dissipates its kinetic or magnetic energy at a large radius to power the gamma-rays observed. Radio-loud AGNs are believed to be powered by a super-massive rotating BH, which launches a mildly relativistic jet (Urry & Padovani 1995). Most confirmed extragalactic GeV-TeV sources are blazars, a sub-sample of radio-loud AGNs. They are divided into flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs), based to whether or not strong emission line features are observed. The broad band spectral energy distributions (SEDs) of blazars usually show two bumps, which are generally explained as synchrotron radiation and inverse Comp- 3 Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA ton (IC) scattering of a same population of relativistic electrons, respectively. The seed photons for the IC process can be from the synchrotron radiation itself (i.e., the synchrotron self-Compton [SSC] model, Maraschi et al. 1992; Ghisellini et al. 1996; Zhang et al. 2012b), or from an external radiation field (e.g., the external Compton [EC] model, Dermer et al. 1992). Narrow-Line Seyfert 1 Galaxies (NLS1s) were also identified as a new class of GeV gamma-ray AGNs by the Fermi/LAT (Abdo et al. 2009a). Their broad band SEDs are similar to that of FSRQs, which are also well explained by this synchrotron+IC leptonic jet model (Abdo et al. 2009a). In contrast to blazars whose radiation mechanisms (especially for the low-frequency component) are well understood, the radiation mechanism of GRB prompt gammaray emission is still not identified. This prompt emission spectrum is usually well described by a smooth broken power-law function called the 'Band function' (Band et al. 1993), with a spectral peak typically in the range of 100s of keV. The origin of this spectrum is under intense debate. Although synchrotron emission has been proposed as the leading model (e.g., M'esz'aros et al. 1994; Wang et al. 2009; Daigne et al. 2011; Zhang & Yan 2011), other mechanisms including Comptonization of quasi-thermal photons from the fireball photosphere (e.g., Rees & M'esz'aros 2005; Pe'er et al. 2006; Giannios 2008; Beloborodov 2010) and synchrotron selfCompton (e.g., Racusin et al. 2008; Kumar & Panaitescu 2008) have been also suggested. Even though photosphere emission is believed to dominate the spectrum in a small fraction of GRBs such as GRB 090902B (Abdo et al. 2009b; Ryde et al. 2010; Zhang et al. 2011) and probably GRB 090510 (Ackermann et al. 2010), the Band component in most GRBs may not be the modified photosphere emission. In particular, Zhang et al. (2012a) show that the peak energy of the Band component of GRB 110721A is beyond the 'death line' in the L iso -E p diagram for the photosphere models, where L iso is the isotropic prompt gamma-ray luminosity and E p is the peak energy of the νf ν spectrum, suggesting that at least some GRBs have a dominant emission component above the photosphere in the optically thin region. Since the SSC mechanism suffers some other criticisms (e.g., Piran et al. 2009; Resmi & Zhang 2012), synchrotron radiation is a natural candidate (see also Veres et al. 2012). Because both a quasi-thermal photosphere emission component and a non-thermal component are predicted to co-exist in the standard fireball shock model (M'esz'aros & Rees 2000; Pe'er et al. 2006), the non-detection of the photosphere component (or in some cases a weak component) would point towards a magnetically dominated jet in GRBs (Zhang & Pe'er 2009; Daigne & Mochkovitch 2002; Zhang et al. 2011). Such a scenario is favored in the Fermi era, although further independent support is needed to make a stronger case. It has long been speculated that the physics in different BH jet systems may be essentially the same (Mirabel 2004; Zhang 2007). A comparative study between the properties of AGNs and GRBs may shed light on the nature of these jets in different scales. By comparing the radio-to-optical spectral properties of optically bright GRB afterglows and blazars, Wang & Wei (2011) suggested that GRB afterglows share a similar emission process with high frequency-peaked BL Lac objects. Wu et al. (2011) found that GRBs and blazars may have a similar intrinsic synchrotron luminosity in the co-moving frame, which is Doppler-boosted by different degrees in different systems, with GRBs having a higher Doopler boosting factor ( δ ) than blazars. Lately, Nemmen et al. (2012) found that the energy dissipation efficiency of BH powered jets is similar over ten orders of magnitude in jet power, establishing a physical connection between AGNs and GRBs. This letter compares the synchrotron radiation of AGNs with the prompt gamma-rays of GRBs and study their radiation efficiency ( ε ) and jet composition of AGNs and GRBs. We make use of our detailed modeling results for blazars and NLS1s to infer more physical parameters. The data and sample are presented in § 2. A comparison of GRBs and AGNs in the jet power-luminosity plane is presented in § 3. Jet radiation efficiency and magnetization degree ( σ ) are presented in § 4. Conclusions are presented in § 5.", "pages": [ 1, 2 ] }, { "title": "2. SAMPLES AND DATA", "content": "Our AGN sample includes 19 BL Lacs, 23 FSRQs, and four NLS1s. They are GeV-TeV sources and 51 well-sampled broad-band SEDs are available for these sources. The SEDs are taken from Zhang et al. (2012b) 5 and Abdo et al. (2009a, 2010). We fit the SEDs with a synchrotron+SSC model for BL Lacs. The details of our SED fits for the BL Lacs can be found in Zhang et al. (2012b). For NLS1s and FSRQs, a synchrotron+SSC+EC model is used to fit the SEDs. The external photon field of the EC process is taken as emission from the broad-line regions (BLRs) of FSRQs and NLS1s. The radiation from the BLRs is taken as a black body spectrum, with energy density measured in the comoving frame as U ' BLR = 3 . 76 × 10 -2 Γ 2 erg/cm 3 (Ghisellini et al. 2010), where Γ is the bulk Lorentz factor of the radiating region. Most AGNs in our sample are blazars. Since for blazars we are likely looking at the jet within the 1 / Γ cone, and that the probability is the highest at the rim of the cone, we take δ ∼ Γ in all the calculations, where δ is the beaming factor. The radiation region is taken as a homogeneous sphere with radius R , which is given by R = cδ ∆ t . The minimum variability timescale is taken as ∆ t = 12 hr for the FSRQs and NLS1s. The electron distribution is taken as a broken power law, which is characterized by a normalization term ( N 0 ), a break energy γ b and indices ( p 1 and p 2 ) in the range [ γ min , γ max ]. These parameters are preliminarily derived from the observed SEDs and are further refined in our SED modeling (Zhang et al. 2012b). N 0 and γ b depend on the synchrotron peak frequency ( ν s ) and peak flux ( ν s f ν s ) as well as other model parameters. We let ν s and ν s f ν s as free parameters in stead of N 0 and γ b . γ max is poorly constrained, but it does not significantly affect our results. We fix it at a large value. Therefore, the free parameter set of our SED modeling is { B , δ, ν s , ν s f ν s , γ min } , where B is the magnetic field strength of the radiating region. We randomly generate a parameter set in broad spaces and measure the consistency between the model result and data with a probability p ∝ e -χ 2 r / 2 , where χ 2 r is the reduced χ 2 . The center values and 1 σ uncertainties of these parameters are derived from Gaussian fits to the profiles of the p distributions. Since we set γ min /greaterorequalslant 2, the p distributions of γ min for some sources have a cutoff at γ min = 2. The γ min of NLS1s are poorly constrained in 1 σ confidence level. Taking PKS 1454-354 as an example, Figure 1 shows our best SED fits and the probability distributions of the parameters. Our results for the FSRQs and NLS1s are reported in Table 1. The observed SEDs with our best model fitting parameters are presented online 6 . We adopt the conventional assumption that the jet power is carried by electrons and protons with one-to-one ratio, magnetic fields, and radiation, i.e., P jet = ∑ i P i , where P i = πR 2 Γ 2 cU ' i and U ' i ( i = e , p , B , r) is the energy density associated with each ingredient measured in the co-moving frame (Ghisellini et al. 2010). Note that the radiation power P r should be a part of P jet before radiation, since P e is only the power carried by the electrons after radiation. It is estimated with the bolometric luminosity L bol , i.e., P r = πR 2 Γ 2 cU ' r = L obs Γ 2 / 4 δ 4 ≈ L bol / 4 δ 2 . The radiation efficiency and the magnetization parameter of the jets can be calculated by respectively. The jet luminosities (bolometric L bol , jet , synchrotron peak L s , jet , and IC peak L c , jet ) are derived with L w , jet = f b L w , iso , where 'w' stands for 'bol', 's', or 'c', and f b = 1 -cos θ ≈ θ 2 / 2 ≈ 1 / 2Γ 2 is the relativistic beaming correction factor. Our results are reported in Table 2. The GRB sample includes 52 typical GRBs reported by Nemmen et al. (2012), excluding two low-luminosity GRBs (980425 and 060218) since they may have a different origin from most high-luminosity long GRBs (Liang et al. 2007). Their isotropic gamma-ray luminosity, jet luminosity, and jet power are calculated with L iso = E γ, iso (1 + z ) /T 90 , L jet = f b L iso , and P jet = f b ( E k , iso + E γ, iso )(1 + z ) /T 90 , respectively, where T 90 is the duration and E γ, iso is the equivalent isotropic gamma-ray energy of GRBs, f b is the beaming factor estimated with the jet opening angles derived from the observed break in their afterglow lightcurves, and E k , iso is the isotropic kinetic energy of GRB fireballs estimated from the X-ray luminosity during the afterglow phase using the standard afterglow model. The GRB radiation efficiency is defined as ε = E γ, iso / ( E γ, iso + E k , iso ) (Zhang et al. 2007).", "pages": [ 2, 3 ] }, { "title": "3. GRB AND AGN SEQUENCE IN THE LUMINOSITY - JET POWER PLANE", "content": "We first plot P jet as a function of L s , iso , L s , jet , L bol , jet , and L c , jet for AGNs in Figure 2. We make the best linear fits to the data with the minimum χ 2 technique by considering the errors in both X and Y axes, if available, for the FSRQs and BL Lacs, respectively. The derived P jet -L relations are marked in Figure 2 7 . Tight correlations between P jet and luminosities are observed for FSRQs, but not for BL Lacs. We do not make correlation analysis for NLS1s because we have only four NLS1s in our sample. Since the prompt emission mechanism of GRBs is not identified, we add the GRBs to these L -P jet planes by plotting their P jet against L iso or L jet . Our best linear fit results are shown in Figure 2. Interestingly, the L s , jet -P jet relation of FSRQs is well consistent with the L jet -P jet relation of GRBs. We have log P jet = (14 . 2 ± 3 . 6) + (0 . 72 ± 0 . 08) log L s , jet with r = 0 . 75 and p = 4 . 3 × 10 -5 for the FSRQs, and log P jet = (14 . 1 ± 2 . 8)+(0 . 73 ± 0 . 06) log L jet with r = 0 . 84 and p = 4 . 7 × 10 -15 for the GRBs. We make best linear fits to the combined FSRQ and GRB sample and obtain P jet ∝ L 0 . 79 ± 0 . 01 s , jet , with a Pearson correlation coefficient of r = 0 . 98 (chance probability p ∼ 0) and a dispersion of 0.44 dex. NLS1s are roughly in the 3 σ confidence band of this relation, but most BL Lacs are out of this band. This relation spans ten orders of magnitude in luminosity with a small dispersion, indicating that both FSRQs and GRBs form a well sequence. Although tight L s , iso -P jet , L c , jet -P jet and L bol , jet -P jet relations are also found for the FSRQs, their intercepts are significantly different from the L jet -P jet relation of the GRBs. These results suggest that the physical properties between FSRQs and GRBs are likely similar and the dominant radiation mechanism of GRBs may be the synchrotron radiation of relativistic electrons in the jets.", "pages": [ 3 ] }, { "title": "4. RADIATION EFFICIENCY AND JET COMPOSITION", "content": "The standard internal shock model of GRBs predicts ε ∼ 5% (e.g., Kumar 1999; Panaitescu et al. 1999). Some GRBs satisfy such a constraint, but most of them do not(Zhang et al. 2007; see also Fan & Piran 2006). Dissipative photosphere emission (Lazzati et al. 2011) and internal-collision-induced magnetic reconnection and turbulence (ICMART) in a Poynting-fluxdominated wind (Zhang & Yan 2011) have been suggested to achieve high radiation powers of GRBs. These two scenarios invoke distinctly different jet composition. While the photosphere model invokes a hot, matterdominated fireball, the ICMART model invokes a magnetically dominated emitter in the emission region. Figure 3(a) shows ε as a function of L bol , jet for AGNs and ε as a function of L jet for GRBs. The ε of both GRB and FSRQ jets are comparable, with a large fraction being greater than 10%, and they increase with L bol , jet or L jet with similar power-indices. Lloyd-Ronning & Zhang (2004) reported a similar correlation between ε and E γ, iso for GRBs. Our best linear fits yield ε ∝ L 0 . 41 ± 0 . 05 jet with r = 0 . 4 and p = 0 . 003 for the GRBs and ε ∝ L 0 . 35 ± 0 . 05 bol , jet with r = 0 . 78 and p = 1 . 1 × 10 -5 for the FSRQs. The slopes of both FSRQs and GRBs are consistent within error bars. The BL Lacs have a lower efficiency (normally 0 . 03% ∼ 10%) than FSRQs and GRBs. A large fraction of the BL Lacs are out of the 3 σ confidence band of the ε -L bol , jet relation for the FSRQs. A weak L bol , jet -ε correlation is also found for the BL Lacs alone, which is ε ∝ L 0 . 82 ± 0 . 10 bol , jet with r = 0 . 5 and p = 0 . 013. The index is different from that of the FSRQs and GRBs. We further study the jet composition of AGNs with our modeling results. Even though current GRB modeling does not allow us to constrain the magnetization parameter σ , this can be done for the AGN sample. In Figure 3(b), we plot ε against σ for all AGNs in our sample. The FSRQs tend to have higher ε and σ values than the BL Lacs. Their σ values are close to or exceeding unity. An anti-correlation between ε and σ is found for the FSRQs, i.e., ε ∝ σ -0 . 32 ± 0 . 09 with r = -0 . 63 and p = 0 . 001. This is due to P B dominates P jet for FSRQs, and an anti-correlation between ε and σ may be expected from Eqs (1) and (2). NLS1s show similar trend, but systematically have a lower ε than FSRQs. BL Lacs have lower ε and σ values, and a weak correlation is found, i.e., ε ∝ σ 0 . 57 ± 0 . 09 with r = 0 . 55 and p = 0 . 005. The dramatic difference of the ε -σ correlations between FSRQs and BL Lacs may further signal the different jet properties of two kinds of sources. The FSRQ jets are likely highly magnetized and the BL Lac jets are less radiation efficiency and matter dominated. Since GRBs have a similar radiation efficiency and efficiency-luminosity dependence as FSRQs, one may suggest that the jet properties of GRBs are analogous to FSRQs. This supports the idea that GRB emission is due to magnetic dissipation in a highly magnetized jet (e.g. Zhang & Yan 2011).", "pages": [ 3 ] }, { "title": "5. CONCLUSIONS AND DISCUSSION", "content": "We have presented a comparative study of GRBs and radio loud AGNs, making use of our systematical SED modeling results for a GeV-TeV selected sample of AGNs. We show that the P jet -L s, jet relation of FSRQs is consistent with the P jet -L jet relation of GRBs. The radiation efficiencies of both FSRQs and GRBs are comparable and even increase with L bol , jet with a similar power-law index. BL Lacs typically have a lower ε and L bol , jet than FSRQs, and a tentative L bol , jet -ε relation is found with a different slope from that of the FSRQs. An anti-correlation between ε and σ is found for FSRQs, but this correlation is positive for the BL Lacs. Based on the analogy between GRBs and FSRQs, we suggest that GRBs are likely produced by synchrotron process in a magnetized jet with high radiation efficiency. The jets of NLS1s potentially share similar properties with FSRQs. The jets of BL Lacs, on the other hand, are low radiation efficiency and likely matter dominated. Compared with AGNs, GRBs are less understood. There is a list of open questions in GRB physics, including jet composition, energy dissipation and radiation mechanisms (e.g., Zhang 2011). Our comparative study between GRBs and AGNs shed new light on some of these open questions of GRBs. For example, the clear jet power - luminosity correlation suggests that the dominant radiation mechanism of GRB prompt emission is similar to the low energy peak of SEDs for blazars, namely, synchrotron radiation (see also an independent study of Uhm & Zhang 2013). The close analogy between GRBs and FSRQs in their radiation efficiency luminosity dependence and the fact that FSRQs have a moderate to high magnetization parameter σ suggest that GRB emission is likely from energy dissipation in a highly magnetized jet (e.g., Zhang & Yan 2011). The high radiation efficiency argument alone may not disfavor the photosphere model of GRBs. However, when combining the luminosity - jet power correlation as presented in Figure 2, the magnetic dissipation model is further favored since it invokes synchrotron radiation as the dominant radiation mechanism. All these are also consistent with Roming et al. (2006), who discovered that some GRBs with high radiation efficiency tend to have tight early UVOT upper limits, which could be caused by suppression of the reverse shock emission in a magnetized jet (e.g., Zhang & Kobayashi 2005; Mimica et al. 2009). Above conclusions are based on the leptonic model of radiation for the AGNs and comparative analysis between the AGNs and GRBs. The derivation of P jet of AGNs is essential for our analysis. The estimate of P e and P p , especially P p are significantly affected by the γ min values, which are simply taken as unity in previous work (e.g., Ghisellini et al. 2010). As shown in Table 1, we find that the typical γ min value is 45, which lowers both P e and P p , with more drastic decrease of P p , as compared with Ghisellini et al. (2010). As a result, both ε and σ derived in our paper are systematically higher than those derived in Ghisellini et al. (2010). Note that we assumed one cool proton for one relativistic electron in jet power calculations. In case of that the jet power is carried by positron-electron pairs, magnetic field, and radiation, but no protons, the main results for FSRQs hold and the conclusions of our analysis are still valid, but no significant luminosity-radiation efficiency and magnetization parameter-radiation efficiency correlations are found for BL Lacs. We thank helpful discussion with G. Ghisellini, JianYan Wei, Xue-Feng Wu, and Zi-Gao Dai. This work is supported by the National Basic Research Program (973 Programme) of China (Grant 2009CB824800), the National Natural Science Foundation of China (Grants 11078008, 11025313, 11133002, 10725313), Guangxi Science Foundation (2013GXNSFFA019001, 2011GXNSFB018063, 2010GXNSFC013011). BZ acknowledges support from NSF (AST-0908362).", "pages": [ 3, 4 ] }, { "title": "REFERENCES", "content": "/s32 /s32", "pages": [ 5 ] }, { "title": "Model parameters of SED fits for the FSRQs and NLS1s in our sample", "content": "/s32 /s32 /s32 Zhang, B., & Pe'er, A. 2009, ApJ, 700, L65 Zhang, B., & Yan, H. 2011, ApJ, 726, 90 Zhang, J., Liang, E.-W., Zhang, S.-N., & Bai, J. M. 2012b, ApJ, 752, 157 Zhang, S. N. 2007, Highlights of Astronomy, 14, 41 a The data of BL Lacs are taken from Zhang et al. (2012b). The source names marked with 'H' or 'L' are for a high or low state of the sources as defined in Zhang et al. (2012b). /s32 /s32 /s32 /s32 /s32 /s32", "pages": [ 5, 6, 7 ] } ]

[ { "title": "ABSTRACT", "content": "High-frequency quasi-periodic oscillations (QPOs) appear in general-relativistic magnetohydrodynamicsimulations of magnetically choked accretion flows around rapidly rotating black holes (BHs). We perform polarized radiative transfer calculations with ASTRORAY code to explore the manifestations of these QPOs for SgrA*. We construct a simulation-based model of a radiatively inefficient accretion flow and find model parameters by fitting the mean polarized source spectrum. The simulated QPOs have a total sub-mm flux amplitude up to 5% and a linearly polarized flux amplitude up to 2%. The oscillations reach high levels of significance 10 -30 σ and high quality factors Q ≈ 5. The oscillation period T ≈ 100 M ≈ 35 min corresponds to the rotation period of the BH magnetosphere that produces a trailing spiral in resolved disk images. The total flux signal is significant over noise for all tested frequencies 87 GHz, 230 GHz, and 857 GHz and inclination angles 10 · , 37 · , and 80 · . The non-detection in the 230 GHz Sub-Millimeter Array light curve is consistent with a low signal level and a low sampling rate. The presence of sub-mm QPOs in SgrA* will be better tested with the Atacama Large Millimeter Array. Subject headings: accretion, accretion disks - black hole physics - Galaxy: center - instabilities - magnetohydrodynamics (MHD) - radiative transfer", "pages": [ 1 ] }, { "title": "SUBMILLIMETER QUASI-PERIODIC OSCILLATIONS IN MAGNETICALLY CHOKED ACCRETION FLOW MODELS OF SGRA*", "content": "ROMAN V. SHCHERBAKOV 1,2,3 , JONATHAN C. MCKINNEY 2,4 Draft version October 16, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Quasi-periodic oscillations (QPOs) in the emission from black hole (BH) accretion disks and jets are found in systems with both stellar mass BHs (Remillard & McClintock 2006) and supermassive BHs (SMBHs) (Gierli'nski et al. 2008; Reis et al. 2012). The high-frequency QPOs (HFQPOs) with a period T about the orbital period at the innermost stable circular orbit (ISCO) potentially probe the region close to the event horizon, offering a chance to test accretion and jet theories in the strong gravity regime. There have been multiple claims of the HFQPOs from the SMBH SgrA* in the Milky Way center, which then provides a unique opportunity to study the HFQPOs up-close. Henceforth, we set the speed of light and gravitational constant to unity ( c = 1 and G = 1), such that 1 M = 21 s for SgrA* with the BH mass M BH = 4 . 3 × 10 6 M /circledot (Ghez et al. 2008; Gillessen et al. 2009). A HFQPO period commonly claimed for SgrA* is T = 17min = 48 M . This was suggested in the K band during a flare (Genzel et al. 2003), in 7 mm data by Yusef-Zadeh et al. (2011), and in images obtained with very long baseline interferometry (VLBI) by Miyoshi et al. (2011). Periods of T = 28min = 79 M (Genzel et al. 2003), T =23min= 65 M , and T = 45min = 127 M (Trippe et al. 2007; Hamaus et al. 2009) were reported in the IR observations of flares. However, Do et al. (2009) analyzed a long K band light curve and found no statistically significant power spectrum density excess. A statistically significant longer period of T = 2 . 5 -3hrs ≈ 470 M was reported by Mauerhan et al. (2005) in 3 mm data. Miyoshi et al. (2011) claimed a range of periods from 17 min [email protected] 1 http://astroman.org Department of Astronomy, University of Maryland, College Park, MD20742, USA to 56 min due to the spirals with multiple arms, but did not compute significance. Physical origins of the HFQPOs are highly debated. Models are often based on an ISCO orbital frequency, an epicyclic frequency, and frequencies of various pressure and gravity modes (e.g., Kato 2001, 2004; Remillard & McClintock 2006; Wagoner 2008). The underlying physics involves beat oscillations (van der Klis 2000), resonances between flow modes and normal frequencies in general relativity (GR) (Abramowicz & Klu'zniak 2001), trapped oscillations (Nowak & Wagoner 1991), parametric resonances (Abramowicz et al. 2003), and disk magnetospheric oscillations (Li & Narayan 2004). The analytic methods were followed by blind QPO searches in magnetohydrodynamic (MHD) simulations. Non-GR 2D MHD simulations exhibited spiral patterns of Rossby waves detectable in simulated light curves (Tagger & Melia 2006; Falanga et al. 2007). Non-GR 3D MHD simulations of thick accretion disks by Chan et al. (2009) developed the QPOs with a period T = 39 M 5 in a simulated X-ray light curve. The 3D GRMHD simulations and radiative transfer by Schnittman et al. (2006) revealed weak transient QPOs. Similar simulations and radiative transfer by Dolence et al. (2009, 2012) showed a spiral structure producing oscillations with periods T = 6 -9min = 17 -25 M in simulated NIR and X-ray light curves. The GRMHD simulations of tilted disks produce tentative QPOs in dynamical quantities with T ≈ 170 M = 1 hr (Henisey et al. 2012). Such MHD simulations start with a weak magnetic field, which is amplified by the magneto-rotational instability (MRI) that generates incoherent turbulence. However, when magnetized gas falls onto a BH, the disk becomes saturated with more magnetic flux than the MRI can generate (McKinney et al. 2012). The 3D GRMHD simulations of radiatively inefficient accretion flows (RIAFs, as applicable to SgrA*) with ordered magnetic flux were performed by McKinney et al. (2012). The resultant magnetically choked accretion flow (MCAF) has a BH magnetosphere that significantly affects the sub-Keplerian equatorial inflow. The simulations showed high-quality disk-BH magnetospheric QPOs in dynamical quantities with an m =1(one-arm)toroidal mode and a rotating inflow pattern in the equatorial plane. We quantify the QPO signal and its statistical significance in simulated SgrA* light curves based on the GRMHD simulations of MCAFs. We do a targeted search for the known QPO period T ≈ 100 M . In Section 2 we describe the 3D GRMHD simulations and the application to SgrA*. We perform GR polarized radiative transfer calculations with ASTRORAY code, fit the SgrA* mean polarized spectrum, and find the best-fitting model parameters. In Section 3 we describe timing analysis. We study the light curves of the bestfitting model viewed at different inclination angles θ . We find statistically significant QPOs in total and some linearly polarized (LP) fluxes. We image a correspondent equatorial plane spiral wave. In Section 4 we compare our simulated QPOs with previous work and discuss the observability in SgrA*.", "pages": [ 1, 2 ] }, { "title": "2.1. GRMHD Simulations", "content": "The initial gas reservoir is a hydrostatic torus (Gammie et al. 2003), within which magnetic field loops are inserted. The MRI action on the initial field leads to MHD turbulent accretion that eventually causes magnetic flux to saturate near the BH (McKinney et al. 2012). We focus on a simulation with a dimensionless spin a ∗ = 0 . 9375, which is close to a ∗ ≈ 0 . 9 favored in simulation-based modeling of the SgrA* spectrum and the emitting region size (Mo'scibrodzka et al. 2009; Dexter et al. 2010; Shcherbakov et al. 2012). The simulation is performed in spherical coordinates ( r , θ, φ ) with resolution Nr × N θ × N φ = 272 × 128 × 256. It reached a quasi-steady state by time t = 8 , 000 M and ran till t = 28 , 000 M . In steady state near the BH event horizon, the sub-Keplerian inflow is balanced against the BH magnetosphere resulting in vertical compression of the disk. The BH magnetosphere and disk exhibit the QPOs in dynamical quantities such as the magnetic field energy density. Atoroidal wobbling mode with m = 1 is eminent in the jet polar region and disk plane. It was identified with pattern rotation of the BH magnetospheric region pierced by the infalling matter streams. The streams form due to magnetic RayleighTaylor instabilities (e.g. Stone & Gardiner 2007). The pattern rotates with an angular frequency Ω F ≈ 0 . 2 Ω H , where Ω H = a ∗ / (2 rH ) is the BH angular frequency and rH = (1 + √ 1 -a 2 ∗ ) M is the horizon radius. The angular frequency Ω F is close to the rotation frequency ≈ 0 . 27 Ω H of the field lines attached to the BH at the equatorial plane in a paraboloidal magnetospheric solution (Blandford & Znajek 1977).", "pages": [ 2 ] }, { "title": "2.2. SgrA* Accretion Flow Model", "content": "We use this MCAF 3D GRMHD simulation to model the SgrA* accretion flow. We follow Shcherbakov et al. (2012) to define the electron temperature and extrapolate quantities to outer radii r > 50 M . A power-law extension of density to r > 50 M is n ∝ r -β , while the proton temperature is continued as Tp ∝ r -1 . The magnetic field strength is extended as b ∝ √ nTp ∝ r ( -1 -β ) / 2 to preserve a constant local ratio of magnetic field energy to thermal energy. The slope β is found by connecting the known density at r = 3 × 10 5 M to the density in the inner region (Shcherbakov & Baganoff 2010). Correct simultaneous evolution of the simulations and the radiation field is considered, despite the radiative transfer is conducted in post-processing. Wefocus on the accretion disk as the source of SgrA* emission and will consider jet emission (e.g., Falcke et al. 2004) in future studies. The simulated matter density is artificial near the polar axis, because matter is injected there to avoid an exceedingly high local ratio of magnetic energy to rest-mass energy that is difficult for GRMHD codes to evolve. The injected material does not change flow dynamics because it is energetically negligible. Nevertheless, a small amount of hot matter in the polar region can shine brightly as revealed by Mo'scibrodzka et al. (2009). The matter densities are zeroed out in a bipolar cone with an opening angle θ = 26 · . If that artificial matter was not removed, then none of our models would be consistent with the observed image size at 230 GHz (Doeleman et al. 2008) and the observed polarized SgrA* spectrum. The radiative transfer is performed with our ASTRORAY code (Shcherbakov & Huang 2011; Shcherbakov et al. 2012). We compute radiation over a quasi-steady simulation period between t = 8 , 000 M and t = 28 , 000 M . Following the previous work, we fit the total flux of SgrA* at 87 -857 GHz, the LP fraction at 87 GHz, 230 GHz, and 345 GHz, and the circular polarization (CP) fraction at 230 GHz and 345 GHz. We vary the heating constant C , which determines the electron temperature Te close to the BH, the accretion rate ˙ M , and the inclination angle θ . Fitting the mean SgrA* spectrum with the mean simulated spectrum we reach χ 2 / dof = 1 . 55 for dof = 9, which is a better agreement than in our prior work based on weakly magnetized simulations (Penna et al. 2010; Shcherbakov et al. 2012). The correspondent values of parameters are Te = 3 . 2 × 10 10 K at 6 M distance from the center, ˙ M =1 . 0 × 10 -8 M /circledot yr -1 , and θ =37 · . We then perform a timing analysis of the light curves from a number of models.", "pages": [ 2 ] }, { "title": "3.1. Oscillations in Light Curves and Images", "content": "Let us first demonstrate the oscillations in the light curves. In Figure 1 we show the light curves at times t = 25 , 500 M -26 , 100 M for the best-fitting model with the inclination angle θ = 37 · . The light curves are computed for three frequencies with different optical depth: radiation at 87 GHz is optically thick, the optical depth at 230 GHz is about τ ∼ 1, while radiation is optically thin at 857 GHz. The total flux (top panel) shows regular oscillations with the amplitude ∆ F ≈ 0 . 05 Jy at 87 GHz and ∆ F ≈ 0 . 15 Jy at 230 GHz. Fluctuations at 857 GHz with the amplitude ∆ F ≈ 0 . 2 Jy are less regular. The LP fraction fluctuates at 2% level at all three frequencies, which translates into the relative variations of up to 50% and the absolute LP flux variations ∆ F ≈ 0 . 06 Jy. The LP and CP fractions and the electric vector position angle (EVPA) exhibit substantial variations over long timescales at 87 GHz. The variations of the EVPA at 230 GHz and 857 GHz are about 5 · -10 · . The CP fraction oscillates by 0 . 3% at 87 GHz and by 0 . 15% at the higher frequencies. The absolute CP flux variations are ∆ F ≈ 0 . 005 Jy at 230 GHz. In Figure 2 we show how the amplitude of oscillations depends on the inclination angle θ at 230 GHz. Shown are the light curves for the best-fitting inclination angle θ = 37 · , for almost face-on θ = 10 · , and for almost edge-on θ = 80 · . The total flux exhibits the same oscillation amplitude ∆ F ≈ 0 . 15 Jy independent of θ . The edge-on and the best-fitting cases produce comparable variations of the LP fraction, while cancelations of the polarized fluxes emitted across the flow lower both the mean and the fluctuation amplitude of the LP fraction in the face-on case. Correspondingly, the EVPA fluctuates dramatically in the face-on case. The CP fraction oscillates at 0 . 5% level in the edge-on case, while the other cases exhibit oscillation amplitude 0 . 15%. The face-on accretion flow images with θ = 10 · are shown in Figure 3: the time series of the total intensity images in the top row and of the LP intensity images in the bottom row. The total intensity images show a clear one-arm spiral rotating with a period T ≈ 100 M . The LP intensity spiral is spatially offset from the total intensity spiral, as the region of the brightest total intensity exhibits the strongest LP cancelations. The total intensity spiral looks similar to that in Dolence et al. (2012), despite different angular velocities. Note that we report the intensity images, while Dolence et al. (2012) showed the images of the dynamical quantities.", "pages": [ 2, 3 ] }, { "title": "3.2. Statistical Analysis", "content": "Let us quantify significance of the QPOs. Following Papadakis & Lawrence (1993) we start with an autocovari- ance where k =0 , ± 1 ∆ t , ..., ± ( N -1) ∆ t and xt is the sample of simulated fluxes normalized to have its mean ¯ x equal unity. Then we compute a periodogram In our analysis ∆ t = 4 M , which appears large enough to avoid aliasing at periods T > 50 M . All periodograms are logsmoothed Papadakis & Lawrence (1993) to 0 . 08dex as a compromise between stronger random noise and larger smearing of the QPO peaks. The determination of statistical significance of the QPOs involves comparison of the simulated periodogram with the random noise periodograms. We follow the procedure in Timmer & Koenig (1995) for random noise generation. We employ a log-smoothed to 3 . 0dex periodogram of the simulated light curve as the underlying non-QPO periodogram. This approximation produces a smooth curve comparable to fits of the non-QPO power spectrum with a power-law of a broken power-law. We draw the random noise Fourier transform from a normalized Gaussian distribution and perform the inverse Fourier transform to generate the noise light curves. We then compute noise autocovariances and periodograms. We find a 3 σ significance curve based on 2 , 592 random noise samples. We do not correct for the blind search, since the periodograms are binned, and we target the QPOs with a period T ≈ 100 M . The periodograms with their 3 σ significance curves are depicted in the top six panels of Figure 4 for several frequencies and inclination angles. The top curves in each panel are for almost face-on inclination θ = 10 · , the middle curves are for θ = 37 · , and the bottom curves are for almost edge-on θ = 80 · . We report the statistical significance levels and quality factors Q = T QPO / FWHM. The total flux and the LP fraction periodograms exhibit peaks significant to 5 -30 σ with Q = 2 -5 at the period T ≈ 100 M for most studied frequencies and inclination angles. Our method allows for the maximum measurable quality of Q = 8 -11. A face-on disk shows weaker QPOs, which are not significant at 857 GHz. The LP fraction oscillations are weak for a face-on disk for all frequencies due to random cancelations of the LP. The total flux and the LP fraction at 230 GHz show the strongest oscillations, which further encourages SgrA* observations at 1 . 3 mm wavelength. We also detect marginally significant LP fraction oscillations with a period T = 1000 M ≈ 4 hr. The bottom left panel of Figure 4 shows the analysis for the different approximations to the non-QPO periodogram: the powerlaw fit, the broken power-law fit, and the log-smoothed to 3 . 0dex source periodogram. The bottom right panel of Figure 4 shows the analysis for the different azimuthal viewing angles φ = 0deg, φ = 120deg, and φ = 240deg. The QPO peaks in these six cases stay prominent despite the significance level (number of sigmas) varies by 50%. The significant QPO peaks among the cases presented in Figure 4 (top panels) stay significant, when we switch to the broken power-law fits to the non-QPO periodograms. We characterize presence of the oscillations and stability of the oscillation period by a spectrogram in Figure 5. The spectrogram indicates that most of the time oscillations with a period T = 90 -100 M are present. However, no oscillations occur around time t =22 , 000 M , when the accretion rate peaks due to weaker magnetic field.", "pages": [ 3, 4 ] }, { "title": "4.1. Summary and Comparison to Previous Work", "content": "Here we report the QPOs in the simulated SgrA* light curves for models based on the state-of-the-art 3D GRMHD simulations of the magnetically choked RIAFs. The minimization procedure produces a fit with χ 2 / dof = 1 . 55 for dof = 9 to the mean polarized sub-mm source spectrum. The correspondent simulated total flux light curve shows regular oscillations with the period T ≈ 100 M ≈ 35 min and the amplitude ∆ F ≈ 0 . 15 Jy at 230 GHz. Less regular fluctuations with ∆ F ≈ 0 . 2 Jy are seen at 857 GHz. Weaker oscillations with ∆ F ≈ 0 . 05 Jy are seen at 87 GHz, which probes the optically thick emission from ∼ 10 M radius. The LP fraction exhibits periodic modulations at 50% relative level, but the absolute LP flux amplitude is only about ∆ FLP ≈ 0 . 06 Jy. The QPOs are significant above 3 σ in the total flux light curves for all tested inclination angles 10 · , 37 · , and 80 · and frequencies 87 GHz, 230 GHz, and 857 GHz, while the LP fraction shows less prominent QPOs at 87 GHz and in a face-on case. Our main T ≈ 35 min period is longer than the claimed SgrA* observed period 17 -20 min, while the simulated periods T = 6 -9 min in Dolence et al. (2012) are shorter. Their simulation has the same BH spin a ∗ =0 . 9375 as does our simulation, but the simulation by Dolence et al. (2012) reaches a relatively weak BH horizon magnetic flux and produces a thinner disk with height-to-radius ratio of H / R ∼ 0 . 2. The resultant MRI-dominated accretion flow has a Keplerian rotation and shorter QPO periods. They did not identify their QPO mechanism, although they noted their turbulence is unresolved (Shiokawa et al. 2012) and this might lead to artificial QPOs (Henisey et al. 2009). Our MCAF model has a subKeplerian rotation and the QPOs driven by the interaction of the disk with the rotating BH magnetosphere (Li & Narayan 2004) that leads to longer periods. Our relatively thick disk with H / R ∼ 0 . 6 is expected for a RIAF, and our simulations resolve well the disk turbulence (McKinney et al. 2012) sug- the QPOs are robust. Based upon these works, SgrA* QPOs might be explained by a ∗ =0 . 9375 with an intermediate gas rotation rate, magnetization, or H / R . Flow cooling, whose marginal importance for SgrA* was suggested by Drappeau et al. (2013), can self-consistently choose the disk thickness H / R in simulations. In MCAFs, the steady-state BH horizon magnetic flux has a positive correlation with H / R (McKinney et al. 2012), so cooling can lead to more Keplerian rotation and a weaker magnetosphere, and then the QPO period from MCAFs could be comparable to claimed for SgrA*.", "pages": [ 4, 5 ] }, { "title": "4.2. Observing QPOs in SgrA*", "content": "We showed that the QPOs, though highly significant, have a maximum sub-mm amplitude of 5% or ∆ F ∼ 0 . 15 Jy. Low sensitivity and low sampling rate of current sub-mm instruments might prohibit observational detection of such oscillations (Marrone 2006). The SMA achieves 5% accuracy and samples every 10 min at 1 . 3 mm with a correspondent 20 min Nyquist period (Marrone et al. 2008). A weak signal with T ∼ 30 min period is readily masked by aliasing and noise in the SMA data. The total flux QPO amplitude ∆ F ≈ 0 . 15 Jy is larger than the LP flux amplitude ∆ F ≈ 0 . 06 Jy. However, the observational error of the total flux can also be larger. The SMA measures the LP flux to the leakage level of ∼ 0 . 3% = 9 mJy, while the total flux is measured to ∼ 0 . 7% = 20 mJy due to calibration uncertainties (Marrone et al. 2007). Then it is about equally difficult to detect the total flux oscillations and the LP flux oscillations. The ALMA gives more hope in detecting SgrA* sub-mm QPOs. It covers a wide frequency range 84 -720 GHz, has a collecting area of ∼ 7 × 10 3 m 2 about 30 times that of the SMA, and can sample every few minutes (Brown et al. 2004). The ALMA observations of SgrA* will have a flux error under 0 . 05 Jy, which is enough to reveal the predicted oscillations were they present on an observation night. As our modeling indicates, the QPOs are absent when the magnetic field is weak due to destruction by magnetic field reversals. The future implementation of the Event Horizon Telescope may allow to measure the QPOs in the source size variations (Doeleman et al. 2009). The anticipated brighter state of SgrA* (Mo'scibrodzka et al. 2012) after the cloud infall may alter our predictions: at a constant ν the flux increase is compensated by the lower fractional level of the QPOs due to the higher optical depth.", "pages": [ 5, 6 ] }, { "title": "5. ACKNOWLEDGEMENTS", "content": "The authors thank Chris Reynolds, Jim Moran, and the anonymous referee for comments. This work was supported by NASA Hubble Fellowship grant HST-HF51298.01 (RVS) and NSF/XSEDE resources provided by NICS (Kraken/Nautilus) under the awards TG-AST080025N (JCM) and PHY120005 (RVS/JCM).", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Abramowicz, M. A., Karas, V., Kluzniak, W., Lee, W. H., & Rebusco, P. 2003, PASJ, 55, 467 Abramowicz, M. A., & Klu'zniak, W. 2001, A&A, 374, L19 Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433 Brown, R. L., Wild, W., & Cunningham, C. 2004, Advances in Space Research, 34, 555 Chan, C.-k., Liu, S., Fryer, C. L., Psaltis, D., Özel, F., Roc kefeller, G., & Melia, F. 2009, ApJ, 701, 521 Dexter, J., Agol, E., Fragile, P. C., & McKinney, J. C. 2010, ApJ, 717, 1092 Do, T., Ghez, A. M., Morris, M. R., Yelda, S., Meyer, L., Lu, J. R., Hornstein, S. D., & Matthews, K. 2009, ApJ, 691, 1021 Doeleman, S. S., Fish, V. L., Broderick, A. E., Loeb, A., & Rogers, A. E. E. 2009, ApJ, 695, 59 Doeleman, S. S., et al. 2008, Nature, 455, 78 Dolence, J. C., Gammie, C. F., Mo'scibrodzka, M., & Leung, P. K. 2009, ApJS, 184, 387 Dolence, J. C., Gammie, C. F., Shiokawa, H., & Noble, S. C. 2012, ApJ, 746, L10 Drappeau, S., Dibi, S., Dexter, J., Markoff, S., & Fragile, P. C. 2013, MNRAS, 431, 2872 Falanga, M., Melia, F., Tagger, M., Goldwurm, A., & Bélanger, G. 2007, ApJ, 662, L15 Falcke, H., Körding, E., & Markoff, S. 2004, A&A, 414, 895 Gammie, C. F., McKinney, J. C., & Tóth, G. 2003, ApJ, 589, 444 Genzel, R., Schödel, R., Ott, T., Eckart, A., Alexander, T., Lacombe, F., Rouan, D., & Aschenbach, B. 2003, Nature, 425, 934 Ghez, A. M., et al. 2008, ApJ, 689, 1044 Gierli'nski, M., Middleton, M., Ward, M., & Done, C. 2008, Nature, 455, 369 Gillessen, S., Eisenhauer, F., Trippe, S., Alexander, T., Genzel, R., Martins, F., & Ott, T. 2009, ApJ, 692, 1075 Hamaus, N., Paumard, T., Müller, T., Gillessen, S., Eisenhauer, F., Trippe, S., & Genzel, R. 2009, ApJ, 692, 902 Henisey, K. B., Blaes, O. M., & Fragile, P. C. 2012, ApJ, 761, 18 Henisey, K. B., Blaes, O. M., Fragile, P. C., & Ferreira, B. T. 2009, ApJ, 706, 705 Kato, S. 2001, PASJ, 53, 1 Kato, Y. 2004, PASJ, 56, 931 Li, L.-X., & Narayan, R. 2004, ApJ, 601, 414 Marrone, D. P. 2006, PhD thesis, Harvard University Marrone, D. P., Moran, J. M., Zhao, J.-H., & Rao, R. 2007, ApJ, 654, L57 Marrone, D. P., et al. 2008, ApJ, 682, 373 Mauerhan, J. C., Morris, M., Walter, F., & Baganoff, F. K. 2005, ApJ, 623, L25 McKinney, J. C., Tchekhovskoy, A., & Blandford, R. D. 2012, MNRAS, 423, 3083 Miyoshi, M., Shen, Z.-Q., Oyama, T., Takahashi, R., & Kato, Y. 2011, PASJ, 63, 1093 Mo'scibrodzka, M., Gammie, C. F., Dolence, J. C., Shiokawa, H., & Leung, P. K. 2009, ApJ, 706, 497 Mo'scibrodzka, M., Shiokawa, H., Gammie, C. F., & Dolence, J. C. 2012, ApJ, 752, L1 Nowak, M. A., & Wagoner, R. V. 1991, ApJ, 378, 656 Papadakis, I. E., & Lawrence, A. 1993, MNRAS, 261, 612 Penna, R. F., McKinney, J. C., Narayan, R., Tchekhovskoy, A., Shafee, R., &McClintock, J. E. 2010, MNRAS, 408, 752 Reis, R. C., Miller, J. M., Reynolds, M. T., Gültekin, K., Maitra, D., King, A. L., & Strohmayer, T. E. 2012, Science, 337, 949 Remillard, R. A., & McClintock, J. E. 2006, Ann. Rev. Astron. Astr., 44, 49 Schnittman, J. D., Krolik, J. H., & Hawley, J. F. 2006, ApJ, 651, 1031 Shcherbakov, R. V., & Baganoff, F. K. 2010, ApJ, 716, 504 Shcherbakov, R. V., & Huang, L. 2011, MNRAS, 410, 1052 Shcherbakov, R. V., Penna, R. F., & McKinney, J. C. 2012, ApJ, 755, 133 Shiokawa, H., Dolence, J. C., Gammie, C. F., & Noble, S. C. 2012, ApJ, 744, 187 Stone, J. M., & Gardiner, T. 2007, ApJ, 671, 1726 Tagger, M., & Melia, F. 2006, ApJ, 636, L33 Timmer, J., & Koenig, M. 1995, A&A, 300, 707 Trippe, S., Paumard, T., Ott, T., Gillessen, S., Eisenhauer, F., Martins, F., & Genzel, R. 2007, MNRAS, 375, 764 van der Klis, M. 2000, ARA&A, 38, 717 Yusef-Zadeh, F., Wardle, M., Miller-Jones, J. C. A., Roberts, D. A., Grosso, N., & Porquet, D. 2011, ApJ, 729, 44", "pages": [ 6, 7 ] } ]

[ { "title": "ABSTRACT", "content": "Based on several magnetic nonpotentiality parameters obtained from the vector photospheric active region magnetograms obtained with the Solar Magnetic Field Telescope at the Huairou Solar Observing Station over two solar cycles, a machine learning model has been constructed to predict the occurrence of flares in the corresponding active region within a certain time window. The Support Vector Classifier, a widely used general classifier, is applied to build and test the prediction models. Several classical verification measures are adopted to assess the quality of the predictions. We investigate different flare levels within various time windows, and thus it is possible to estimate the rough classes and erupting times of flares for particular active regions. Several combinations of predictors have been tested in the experiments. The True Skill Statistics are higher than 0.36 in 97% of cases and the Heidke Skill Scores range from 0.23 to 0.48. The predictors derived from longitudinal magnetic fields do perform well, however they are less sensitive in predicting large flares. Employing the nonpotentiality predictors from vector fields improves the performance of predicting large flares of magnitude ≥ M5.0 and ≥ X1.0. Subject headings: methods: statistical - Sun: activity - Sun: flares - Sun: photosphere - Sun: surface magnetism", "pages": [ 1 ] }, { "title": "Magnetic Nonpotentiality in Photospheric Active Regions as a Predictor of Solar Flares", "content": "Xiao YANG 1 , 2 , GangHua LIN 1 , HongQi ZHANG 1 , and XinJie MAO 1 , 3 [email protected]", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Solar flares are sudden processes that release tremendous energy in a short period of time in the solar atmosphere. They lead to transient heating of local regions and the dramatic enhancement of electromagnetic radiation and high-energy particle ejection. Some large eruptions toward the Earth have an impact on normal human activities. It is worthwhile to make short-term predictions of solar flares to reduce losses. For a long period of time, solar physicists have been trying to understand the physics of flares, in order to make predictions by simulating the evolutions of magnetic fields in the solar atmosphere and by obtaining information from the solar interior. At present, however, it seems relatively feasible to make predictions based on the statistical relationships between solar eruptions and the evolution of other solar phenomena. Some authors predict flares based on morphological parameters or remote information from different sources (e.g., Gallagher et al. 2002; Qahwaji & Colak 2007; Li et al. 2007; Colak & Qahwaji 2009; Bloomfield et al. 2012). Such predictors require manual intervention before entering the prediction process, and therefore are not suitable for automatic operations. There are some other flare-prediction studies adopting the measures deduced from longitudinal magnetic fields (e.g., Georgoulis & Rust 2007; Yu et al. 2009; Song et al. 2009; Mason & Hoeksema 2010; Yuan et al. 2010; Ahmed et al. 2013). The accumulation of magnetic nonpotential energy is of importance for solar eruptions. Mason & Hoeksema (2010) mentioned the importance of the vector-field data to obtain the most promising flare-predictive magnetic parameters. Leka and Barnes (2007) contributed a great amount to the exploration of the differences of magnetic-field properties between flare-imminent and flare-quiet active regions, however, the number or the time spans of their samples were quite restricted. Lacking long-term consistent observations of vector magnetic fields, the magnetic nonpotentiality was rarely used in solar flare predictions. The vector magnetograms obtained at the Huairou Solar Observing Station over more than 20 yr make the experiments possible. Yang et al. (2012, hereafter Paper I) have calculated the statistical relations between magnetic nonpotentiality and solar flares. By means of the prediction experiments described in this Letter, we can predict the occurrence of flares in particular active regions based on their magnetic properties alone, and also can estimate the starting time and eruption magnitude of the flares. In addition, several classical verification measures of dichotomous predictions are discussed to call for more serious concerns on the verification issue (Doswell et al. 1990). The Heidke Skill Scores (HSS) and the True Skill Statistics (TSS; see Section 3.2) of our 100 group experiments are in the ranges 0.23-0.48 and 0.320.82, respectively. Our results show that the nonpotentiality predictors improve the performance of predicting more powerful flares.", "pages": [ 1, 2 ] }, { "title": "2.1. Data and Preprocessing", "content": "We use the observational data of photospheric active region vector magnetograms obtained by the Solar Magnetic Field Telescope (SMFT; Ai & Hu 1986) at the Huairou Solar Observing Station, National Astronomical Observatories of China. SMFT is a 35 cm aperture vector magnetograph with a tunable birefringent filter. The working spectral line for the vector magnetograms is Fe I λ 5324.19, which is a strong and broad line with an equivalent width of about 0.334 ˚ A and a Land'e factor of 1.5 (Ai et al. 1982). The data employed are selected from all the vector magnetograms during the period from 1988 to 2008 subject to the following criteria: (1) the active regions are located within 30 · from the solar disk center, and (2) only one magnetogram is used for each active region in one observation day. The final data set, which is also used in Paper I, consists of 2173 photospheric vector magnetograms involving 1106 active regions. The detailed descriptions of the data and their distributions during the two solar cycles are in Paper I, as well as the calibration for the vector magnetograms and the determination of the 180 · ambiguity of the transverse field. The records of soft X-ray flares are available from NOAA's National Geophysical Data Center. 1", "pages": [ 2, 3 ] }, { "title": "2.2. Magnetic Nonpotentiality Parameters as Predictors", "content": "The magnetic nonpotentiality parameters as predictors, the inputs for the prediction model, are the mean planar magnetic shear angle ∆ φ , mean shear angle of the vector magnetic field ∆ ψ , mean absolute vertical current density | J z | , mean absolute current helicity density | h c | , absolute averaged twist force-free factor | α av | , mean free magnetic energy density ρ free , effective distance of the longitudinal magnetic field d E , longitudinal-field weighted effective distance d Em (Paper I), mean horizontal gradient of the longitudinal field ∇ h B z , maximum horizontal gradient ( ∇ h B z ) m , length of strong-gradient ( > 0.05 G km -1 ) inversion lines L gnl , and mean density of longitudinal magnetic energy dissipation ε ( B z ) (Cui et al. 2006; Jing et al. 2006). All of the above measures are macroscopic and averaged quantities, which indicate the magnetic nonpotentiality or magnetic complexity of a whole active region. In the calculations, each magnetogram is represented as ( x i , y i ), where x i ∈ R n is the predictor array and y i ∈ { 1 , -1 } is the class label of the magnetogram ( y i = 1 for flaring instances and y i = -1 for non-flaring ones, according to the labeling scheme stated in Section 3.1). We have tried five combinations of predictors:", "pages": [ 3 ] }, { "title": "2.3. Prediction Method: Support Vector Classification", "content": "Predicting whether or not an active region will flare within a certain time interval can be transformed into a classification problem. The support vector machine (SVM) first introduced by Vapnik (Boser et al. 1992; Cortes & Vapnik 1995; Vapnik 1995) is now a widely applied statistical learning theory used to solve classification and regression problems. In recent years, SVM has been applied to the field of astronomy (e.g., Zhang & Zhao 2003; Wo'zniak et al. 2004; Wadadekar 2005; Gao et al. 2008; Beaumont et al. 2011; Peng et al. 2012) including solar physics (e.g., Qu et al. 2003; Qahwaji & Colak 2007; Li et al. 2007; Al-Omari et al. 2010; Labrosse et al. 2010; Alipour et al. 2012). A machine learning system for classification is able to learn and construct a model (from the existing training data with definite category labels) which can classify the training data and predict upcoming ones whose categories are unknown. The maximum margin principle and the kernel function are the two core concepts of the SVM. By solving an optimization problem, the SVM classifier is obtained as an optimal separating hyperplane w · x + b = 0 that separates the two-class data with the maximum distance. When in a linearly non-separable case, a kernel function is employed, then the training vectors x i are mapped into a higher-dimensional feature space in which the data can be linearly separated. The primal optimization problem can be written as C > 0 is the penalty parameter for the sum of slack variables ξ i . 1 2 ‖ w ‖ 2 corresponds to the distance maximization of the two classes. Taking the reciprocal, the square, and the factor 1/2 are for mathematical convenience. φ ( x i ) denote the training vectors in the higher-dimensional space after employing the kernel function. The kernel function is denoted by K ( x i , x j ), and the corresponding dual optimization problem, which is easier to solve, is /negationslash where α i are the Lagrange multipliers. Then the coefficients α i ∗ for the optimal hyperplane are solved from the dual problem. The training vectors x i with α ∗ i = 0 are the support vectors that contribute to the final discriminant function where w ∗ and b ∗ are the corresponding solutions of the primal problem ( b ∗ = y j -∑ l i =1 y i α ∗ i K ( x i , x j ) taking any 0 < α ∗ j < C ). The plus and minus signs of f ( x ) indicate the two different classes. There are a few commonly used kernels like the polynomial kernel, Gaussian radial basis kernel, sigmoid kernel, etc. After trying several kernels in the calculations, we accept the Gaussian radial basis kernel, the mathematical expression of which is K ( x i , x j ) = exp( -‖ x i -x j ‖ 2 /σ 2 ), where σ is a kernel parameter. The SVM software LIBSVM (Chang & Lin 2011) is used in our experiments. Note that this classification or prediction model is based on statistical relations with no obvious physical meanings; nevertheless, the physical parameters closely related to solar flares must make positive effects to the performance of the model. This is exactly why we adopt magnetic nonpotentiality and complexity parameters as predictors.", "pages": [ 3, 4, 5 ] }, { "title": "3.1. Experiment Design", "content": "According to whether the active regions produce flares exceeding a specified class within a certain time window, every magnetogram is labeled as positive (flaring) or negative (non-flaring). The 'time window' in this Letter begins at the observing time of each magnetogram. We set the flaring magnitude thresholds to C1.0, C5.0, M1.0, M5.0, and X1.0, and the time windows 6, 12, 24, and 48 hr. The positive-negative sample ratios are different for different combinations of flaring thresholds and time windows (see Table 1). In each labeled set, we divide the whole set into training and testing subsets, then train the training subset to obtain the classifier and test the rest to evaluate the performance of the classifier. k -fold cross-validation is used for avoiding overfitting. The full set is randomly divided into k subsets with approximately equal size, ( k -1) of which are for training and the remaining is for testing. Training and testing are repeated k times. Each subset is tested exactly once. We take k = 10 for most sets, and k = 5 for the sets whose flaring samples are less than 50. The positive-negative sample ratios of both training and testing subsets are maintained consistent with that of the original set.", "pages": [ 5, 6 ] }, { "title": "3.2. Performance Assessment for Predictions", "content": "The counts of successes and failures obtained from previous dichotomous prediction constitute a 2 × 2 contingency table (confusion matrix in machine learning), as shown in Table 2. The verification measures assessing the prediction performance are derived from the statistics in the table. For simplicity, we use the notations ( x , y , z , w ) to name the four elements of the contingency table. x is the number of the positive events predicted positive (True Positive or Hit), y the number of the positive events predicted negative (False Negative or Miss), z the number of the negative events predicted positive (False Positive or False Alarm), and w the number of the negative events predicted negative (True Negative or Correct Rejection). x and w make a positive impact on the prediction assessment while y and z do the opposite. From Table 2, we can directly obtain eight ratios of the elements with their associated marginal sums: POD 2 = x/ ( x + y ), FOH 3 = x/ ( x + z ), FAR 4 = z/ ( x + z ), POFD 5 = z/ ( z + w ), FOM 6 = y/ ( x + y ), DFR 7 = y/ ( y + w ), PON 8 = w/ ( z + w ), and FOCN 9 = w/ ( y + w ) (cf. Doswell et al. 1990). POD, FOH, PON, and FOCN, in which the numerator is x or w , are hoped to be higher, and the other four are expected to be lower. Other verification measures are also available such as F 1 -measure, HSS, TSS, Critical Success Index (CSI), Gilbert Skill Score (GSS), and Clayton Skill Score (CSS), a summary of which is shown in Table 3. The perfect prediction, which is difficult to achieve in practice, corresponds to these verification measures reaching their upper bounds of 1. Though it has been more than a century since the 'Finley affair' (see Murphy 1996) inducing hot discussions, the study on this seemingly simple 2 × 2 problem remains ongoing (Stephenson 2000). In this work, we only consider the classical verification measures which are more intuitional to utilize in practical operations. The percentage of correct predictions ( x + w ) /N (referred as ACC hereafter) is the simplest but often misleading measure to assess the prediction, especially when one side, event or non-event, is overwhelming. ACC, CSI, and F 1 do not exclude the correct numbers based on the stochastic prediction. The so-called skill scores indicate the relative accuracy of a prediction to some standard reference predictions. The generic form of skill score is where S is a particular measure of accuracy, S ref a reference, and S perfect the perfect prediction. A no-skill prediction scores 0, a positive score shows a better prediction than the reference, and the perfect prediction scores 1. HSS is a skill score from ACC comparing with the random prediction. GSS is the skill-modified CSI, subtracting the expected correct predictions due to chance from x . F 1 is the harmonic mean of POD and FOH, and HSS happens to be the harmonic mean of skill-modified POD and skill-modified FOH (POD s and RS s in Schaefer 1990). The skill-modified ones are always lower than the original ones. These verification measures are related to each other through the connections of x , y , z , and w . A common property of HSS, GSS, TSS, and CSS is that they all have the factor ( x · w -y · z ) in their numerators. This factor becomes zero in the random prediction, and thus these four skill scores all have the value 0, indicating no skill. In the constant prediction (all positive predictions, y = w = 0; or all negative, x = z = 0), this factor is also zero; CSS is meaningless in this case. The values of CSI, F 1 , and ACC in random situations depend on the ratio of events to non-events. Another common property of the above four skill scores is that they are all fair to both events and non-events. Considering non-events as focus, swapping x with w and y with z simultaneously, they remain unchanged; this is not the case with CSI or F 1 . Keeping the numerators of the above four skill scores exactly the same, the differences of their denominators are: GSS is usually less than TSS and HSS, except when y = z = 0 (i.e., in the perfect prediction). There is no definite magnitude relation between TSS and CSS, or between HSS and TSS. HSS is less than TSS if w is overwhelmingly dominant ( w /greatermuch x and usually z > y in optimizing TSS). There is little difference between HSS and TSS if w is not dominant. Detailed introductions to the contingency table and forecast verification can be found in Wilks (2006).", "pages": [ 6, 7, 8 ] }, { "title": "3.3. Experiment Results and More Comments on Verification", "content": "It is nearly impossible to optimize all the verification measures simultaneously (Manzato 2005; see also the results of Bloomfield et al. 2012). Accordingly, we compute the geometric mean of several verification measures (POD, FOH, TSS, HSS, GSS, CSI, F 1 , and 3 √ POD · FOH · FOCN) which we are more concerned about. A grid search process is carried out to obtain a relatively better pair of ( C,σ 2 ) for the final SVM classifier. A12's results 10 are shown in Table 5, in which each value with its error is the arithmetical mean of the specific verification measure in k times testing. The percentage of non-events ( N 0 /N ) is given at the end of each row for reference. F 1 is always higher than CSI, except when x = 0 or y = z = 0. In rare event situations, HSS is close to F 1 , so HSS is likely higher than CSI. In our results, there are only two cases with HSS lower than CSI (C1.0, 48 hr; C1.0, 24 hr). These are the top two cases whose positive samples are in a larger proportion compared with other cases and w is not extremely dominant. The predictors derived from longitudinal magnetic fields (L05) perform somewhat better than those mainly involving transverse components (V06, V08) in predicting flares of ≥ C1.0 and ≥ C5.0. However, the superiority diminishes in predicting more powerful flares. For instance, in the case of ≥ M5.0 or ≥ X1.0 flares, the performance of longitudinal predictors becomes worse than that of other predictor combinations. It seems that the predictors from longitudinal fields are less sensitive in predicting large flares. Overall, there is an improvement in the prediction employing various measures derived from vector magnetic fields (A10, A12). /negationslash HSS and TSS are often discussed and applied in forecast verification (e.g., Woodcock 1976; Doswell et al. 1990; Manzato 2005). HSS = TSS when y = z ; HSS ≡ TSS when N 1 = N 0 . Bloomfield et al. (2012) proposed using TSS instead of HSS as a standard to reliably compare flare forecasts. However, no single scalar measure can cover all the information of the prediction results. Even the unbiased TSS, which is independent of the event frequency, fails to effectively deal with rare event predictions (Doswell et al. 1990). TSS approaches POD in rare event situations, so both w and z contribute little to the results. Experientially, z rises if x 's proportion is increased. The bias (( x + z ) / ( x + y ) = 1) may be unintentionally introduced in optimizing a verification measure (Manzato 2005). Pursuing higher POD or TSS will cause higher FAR and lower FOH. Fewer misses cost more false alarms, but 'crying wolf' may be undesirable. Moreover, the same TSS does not mean the same prediction performance. For instance, Table 4 lists some examples from Woodcock (1976). The prediction P1 has POD = 75% and PON = 50%, and P2 has POD = 50% and PON = 75%. TSS remains the same in the two cases and two predictions, but the results are indeed different. Therefore, only one measure might mislead the prediction verification, and multiple verification measures are probably acceptable. This point of view is as well mentioned in Schaefer (1990), Doswell et al. (1990), Marzban (1998), etc. We believe that, since each data set may have its own intrinsic properties, it is inappropriate to compare different predictions on different trial samples.", "pages": [ 8, 9 ] }, { "title": "4. CONCLUSIONS AND DISCUSSIONS", "content": "Based on the long-term reliable observations of the photospheric vector magnetic fields by SMFT, we adopt some nonpotentiality measures which are not available from observations of only line-of-sight magnetic fields to study the prediction of solar flares. Real-time processing and no manual intervention are two advantages of our prediction system. The data for the input of the prediction model are obtained by local observations, and the key measures as predictors are available without manual operations. From our experiments, the combinations of magnetic measures derived from longitudinal fields perform well in the flare prediction, however, they may be less sensitive than the measures from vector fields in predicting large flares. The information of transverse fields makes a limited contribution to the prediction of low magnitude flares, but it does improve the prediction for large flares such as ≥ M5.0 and ≥ X1.0 ones. Thus, it is reasonable to include transverse field components in flare predictions. To avoid misleading the optimization work or misusing the results from a single verification measure, prediction results should be assessed carefully. It is helpful to consider multiple verification measures. A step like k -fold cross-validation is necessary for improving the generalization capability of the prediction models. The intrinsic properties of various data sets may make a specific tool perform rather differently, and hence, it is then significant to make comparisons in the same data environment. Some researchers have begun to use vector magnetograms from the Helioseismic Magnetic Imager (HMI) on board the Solar Dynamics Observatory to predict solar flares. Yet, the prediction methods founded on statistical information are restricted by the finite time span of HMI data at present. Results of statistical predictions depend on both the historical data set and prediction method employed. There is still a long way to go for the prediction of solar activities employing the exquisite HMI data. The authors are very grateful to the anonymous referees for encouraging comments and beneficial suggestions that improved the manuscript. We are indebted to the HSOS staff and the GOES team for the data they produced. X. Y. acknowledges helpful discussions with Dr. Shangbin Yang. This work is supported by the National Natural Science Foundation of China (60940030, 10921303, 41174153, 10903015, 11003025, 11103037, 11103038, 11203036, and 11221063), the Young Re- searcher Grant of National Astronomical Observatories of Chinese Academy of Sciences (CAS), the Knowledge Innovation Program of CAS (KJCX2-EW-T07), the National Basic Research Program of MOST (2011CB811401), and the Key Laboratory of Solar Activity of CAS.", "pages": [ 9, 10 ] }, { "title": "REFERENCES", "content": "Ahmed, O. W., Qahwaji, R., Colak, T., et al. 2013, SoPh, 283, 157 Ai, G. X., & Hu, Y. F. 1986, PBeiO, 8, 1 Ai, G. X., Li, W., & Zhang, H. Q. 1982, ChA&A, 6, 129 Al-Omari, M., Qahwaji, R., Colak, T., & Ipson, S. 2010, SoPh, 262, 511 Alipour, N., Safari, H., & Innes, D. E. 2012, ApJ, 746, 12 Beaumont, C. N., Williams, J. P., & Goodman, A. A. 2011, ApJ, 741, 14 Bloomfield, D. S., Higgins, P. A., McAteer, R. T. J., & Gallagher, P. T. 2012, ApJL, 747, L41 Boser, B. E., Guyon, I. M., & Vapnik, V. N. 1992, in Proceedings of the Fifth Annual Workshop on Computational Learning Theory, COLT '92 (New York: ACM), 144 Chang, C.-C., & Lin, C.-J. 2011, ACM Transactions on Intelligent Systems and Technology, 2, 27:1, Software available at http://www.csie.ntu.edu.tw/ ~ cjlin/libsvm Chinchor, N. 1992, in Proceedings of the 4th Conference on Message Understanding, MUC4 '92 (Stroudsburg, PA: Association for Computational Linguistics), 22 Clayton, H. H. 1934, BAMS, 15, 279 Colak, T., & Qahwaji, R. 2009, SpWea, 7, 6001 Cortes, C., & Vapnik, V. 1995, Mach. Learn., 20, 273 Cui, Y. M., Li, R., Zhang, L. Y., He, Y. L., & Wang, H. N. 2006, SoPh, 237, 45 Donaldson, R. J., Dyer, R. M., & Kraus, M. J. 1975, in Ninth Conference on Severe Local Storms (Norman, OK: Amer. Meteor. Soc.), 321 Doolittle, M. H. 1888, Bull. Philosophical Soc. Washington, Vol. 10, 83 and 94 Doswell, C. A., III, Davies-Jones, R., & Keller, D. L. 1990, WtFor, 5, 576 Gallagher, P. T., Moon, Y.-J., & Wang, H. M. 2002, SoPh, 209, 171 Gao, D., Zhang, Y.-X., & Zhao, Y.-H. 2008, MNRAS, 386, 1417 Georgoulis, M. K., & Rust, D. M. 2007, ApJL, 661, L109 Gilbert, G. K. 1884, American Meteorological Journal, 1, 166 Hanssen, A. W., & Kuipers, W. J. A. 1965, Mededeelingen en Verhandelingen, 81, 2 Heidke, P. 1926, Geogr. Ann. Stockh., 8, 301 Jing, J., Song, H., Abramenko, V., Tan, C., & Wang, H. 2006, ApJ, 644, 1273 Labrosse, N., Dalla, S., & Marshall, S. 2010, SoPh, 262, 449 Leka, K. D., & Barnes, G. 2007, ApJ, 656, 1173 Li, R., Wang, H. N., He, H., Cui, Y. M., & Du, Z. L. 2007, ChJAA, 7, 441 Manzato, A. 2005, WtFor, 20, 918 Marzban, C. 1998, WtFor, 13, 753 Mason, J. P., & Hoeksema, J. T. 2010, ApJ, 723, 634 Murphy, A. H. 1996, WtFor, 11, 3 Peirce, C. S. 1884, Sci, 4, 453 Peng, N. B., Zhang, Y. X., Zhao, Y. H., & Wu, X. B. 2012, MNRAS, 425, 2599 Qahwaji, R., & Colak, T. 2007, SoPh, 241, 195 Qu, M., Shih, F. Y., Jing, J., & Wang, H. 2003, SoPh, 217, 157 Schaefer, J. T. 1990, WtFor, 5, 570 Song, H., Tan, C. Y., Jing, J., et al. 2009, SoPh, 254, 101 Stephenson, D. B. 2000, WtFor, 15, 221 Van Rijsbergen, C. 1979, Information Retrieval (London: Butterworths), http://www.dcs.gla.ac.uk/Keith/Preface.html Vapnik, V. N. 1995, The Nature of Statistical Learning Theory (New York: Springer) Wadadekar, Y. 2005, PASP, 117, 79 Wandishin, M. S., & Brooks, H. E. 2002, MeApp, 9, 455 Wilks, D. S. 2006, Statistical Methods in the Atmospheric Sciences, (2nd ed.; Burlington, MA: Elsevier Academic Press), Chapter 7 s 8 7 5 . 0 0 1 0 . 0 ± 2 4 7 . 0 8 1 0 . 0 ± 2 1 3 . 0 0 2 0 . 0 ± 9 7 6 . 0 0 1 0 . 0 ± 1 6 7 . 0 8 1 0 . 0 ± 6 0 3 . 0 1 2 0 . 0 ± 1 8 7 . 0 6 0 0 . 0 ± 6 8 7 . 0 3 1 0 . 0 ± 5 7 2 . 0 6 1 0 . 0 ± 8 5 8 . 0 9 0 0 . 0 ± 6 2 8 . 0 8 1 0 . 0 ± 5 3 2 . 0 4 2 0 . 0 ± 3 0 8 . 0 8 0 0 . 0 ± 5 2 8 . 0 1 2 0 . 0 ± 3 1 3 . 0 5 2 0 . 0 ± 0 0 4 5 4 6 6 8 . 0 8 0 0 . 0 ± 7 4 8 . 0 9 1 0 . 0 ± 2 8 2 . 0 2 2 0 . 0 ± 7 3 4 . 0 4 7 1 9 . 0 6 0 0 . 0 ± 6 9 8 . 0 9 2 0 . 0 ± 9 3 2 . 0 5 3 0 . 0 ± 9 7 3 . 0 1 4 5 9 . 0 6 0 0 . 0 ± 8 9 8 . 0 9 2 0 . 0 ± 7 8 1 . 0 1 4 0 . 0 ± 6 0 3 . 0 2 4 8 8 . 0 7 0 0 . 0 ± 0 6 8 . 0 1 2 0 . 0 ± 4 8 2 . 0 6 2 0 . 0 ± 8 3 4 . 0 4 3 2 9 . 0 4 0 0 . 0 ± 7 0 9 . 0 3 2 0 . 0 ± 3 7 2 . 0 0 3 0 . 0 ± 4 2 4 . 0 6 6 5 9 . 0 4 0 0 . 0 ± 4 3 9 . 0 4 2 0 . 0 ± 0 4 2 . 0 1 3 0 . 0 ± 2 8 3 . 0 2 3 7 9 . 0 4 0 0 . 0 ± 9 3 9 . 0 8 2 0 . 0 ± 3 7 1 . 0 0 4 0 . 0 ± 6 8 2 . 0 0 7 6 9 . 0 4 0 0 . 0 ± 2 4 9 . 0 0 2 0 . 0 ± 7 4 2 . 0 6 2 0 . 0 ± 3 9 3 . 0 1 2 8 9 . 0 3 0 0 . 0 ± 0 8 9 . 0 5 7 0 . 0 ± 4 2 2 . 0 4 9 0 . 0 ± 3 4 3 . 0 5 0 9 9 . 0 2 0 0 . 0 ± 2 8 9 . 0 6 5 0 . 0 ± 7 0 2 . 0 6 7 0 . 0 ± 9 2 3 . 0 7 4 9 9 . 0 2 0 0 . 0 ± 5 8 9 . 0 2 4 0 . 0 ± 8 9 1 . 0 9 5 0 . 0 ± 2 2 3 . 0 9 1 8 9 . 0 5 0 0 . 0 ± 6 7 9 . 0 3 7 0 . 0 ± 6 1 3 . 0 4 8 0 . 0 ± 2 6 4 . 0 1 2 8 9 . 0 3 0 0 . 0 ± 3 8 9 . 0 1 6 0 . 0 ± 6 5 2 . 0 9 7 0 . 0 ± 2 9 3 . 0 5 0 9 9 . 0 2 0 0 . 0 ± 5 8 9 . 0 0 1 0 . 0 ± 0 1 2 . 0 3 1 0 . 0 ± 6 4 3 . 0 3 6 9 9 . 0 3 0 0 . 0 ± 7 8 9 . 0 6 5 0 . 0 ± 4 6 1 . 0 4 8 0 . 0 ± 5 6 2 . 0 6 e R n o i t a c fi i r e V : 5 e l b a T w o d n i W e m i T l e v e L e r a l F n o i t c i d e r P N / 0 N C C A S S G r h 8 4 r h 4 2 r h 2 1 0 . 1 C /greaterorequalslant r h 6 r h 8 4 r h 4 2 r h 2 1 0 . 5 C /greaterorequalslant r h 6 r h 8 4 r h 4 2 r h 2 1 0 . 1 M /greaterorequalslant r h 6 r h 8 4 r h 4 2 r h 2 1 0 . 5 M /greaterorequalslant r h 6 r h 8 4 r h 4 2 r h 2 1 0 . 1 X /greaterorequalslant r h 6", "pages": [ 10, 11, 14 ] } ]

[ { "title": "ABSTRACT", "content": "We present new UV observations for NGC 288, taken with the WFC3 detector on board the Hubble Space Telescope , and combine them with existing optical data from the archive to explore the multiple-population phenomenon in this globular cluster (GC). The WFC3's UV filters have demonstrated an uncanny ability to distinguish multiple populations along all photometric sequences in GCs, thanks to their exquisite sensitivity to the atmospheric changes that are tell-tale signs of second-generation enrichment. Optical filters, on the other hand, are more sensitive to stellar-structure changes related to helium enhancement. By combining both UV and optical data we can measure helium variation. We quantify this enhancement for NGC 288 and find that its variation is typical of what we have come to expect in other clusters. Subject headings: stars: Population II - globular clusters individual: NGC 288 1 Based on observations with the NASA/ESA Hubble Space Telescope , obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.", "pages": [ 1 ] }, { "title": "Multi-wavelength Hubble Space Telescope photometry of stellar populations in NGC 288. 1", "content": "G. Piotto 2 , 3 , A. P. Milone 4 , 5 , 6 , A. F. Marino 4 , L. R. Bedin 3 , J. Anderson 7 , H. Jerjen 4 , A. Bellini 7 , S. Cassisi 8", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Recent observations with the Hubble Space Telescope (HST) have shown that colormagnitude diagrams (CMDs) of globular clusters (GCs) are very different from our classical expectations of razor-thin sequences characteristic of single, old populations of stars. In particular, HST near-UV data has shown that most, if not all, GCs host multiple stellar populations, as evidenced by two or more intertwined sequences in the CMD that we can trace from the main sequence (MS), through the sub-giant branch (SGB), up the red-giant branch (RGB) and even along the horizontal branch (HB). ¿From studies of several clusters, we have found that the different sequences can vary their color separation or even invert their relative colors, depending on the photometric-band combinations. These sequences correspond to stellar populations that have different abundances of light elements and helium. A comparison of the photometry with synthetic spectra can provide unique opportunities to estimate the helium content among the stellar populations (e.g. Milone et al. 2012a), even at the level of faint MS stars, which are unreachable by spectroscopic investigations. The cluster analysed in this paper, NGC 288, is already known to host two populations of stars characterized by difference in light-element abundance (e.g. Shetrone & Keane 2000, Kayser et al. 2008, Smith & Langland-Shula 2009, Carretta et al. 2009, Pancino et al. 2010). The RGB of NGC288 is bimodal, when observed in appropriate ultraviolet filters, and each RGB is populated by stars with different abundance of sodium and oxygen (Lee et al. 2009, Roh et al. 2011, Monelli et al. 2013). In this paper, we combine new HST observations with archival data to investigate the evolutionary path of the multiple populations in NGC 288 along the MS, SGB, and RGB. By exploring a wide wavelength region, ranging from the ultraviolet ( ∼ 2750 ˚ A) to the near infrared ( ∼ 8140 ˚ A), we will estimate the helium difference between the two main populations.", "pages": [ 2 ] }, { "title": "2. Data and Data Reduction", "content": "To get the broadest possible perspective on NGC 288's multiple populations, we consolidated photometry from a large number of HST images taken with the Wide Field Channel of the Advanced Camera for Surveys (ACS/WFC) and the ultraviolet/visible channel of the Wide-Field Camera 3 (WFC3/UVIS). Table 1 gives a list of the data sets we used. Most HST data are from the archive, with the exception of the proprietary images from GO-12605 (PI: Piotto), which were specifically taken for this project and are crucial for its success. Photometric and astrometric measurements of ACS/WFC exposures were obtained with the software program described by Anderson et al. (2008). This routine produces a catalog of stars over the field of view by analyzing an entire set of images simultaneously. It measures stellar fluxes independently in each exposure by means of a spatially variable point-spreadfunction model (see Anderson & King 2006), along with a spatially-constant perturbation of the PSF to account for the effects of focus variations. The photometry has been calibrated as in Bedin et al. (2005), using the encircled energy and zero points of Sirianni et al. (2005). The WFC3/UVIS images were reduced as described in Bellini et al. (2010), with img2xym UVIS 09 × 10 , a software routine that is adapted from img2xym WFI (Anderson et al. 2006). Astrometry and photometry were corrected for pixel area and geometric distortion as in Bellini & Bedin (2009), and Bellini, Anderson & Bedin (2011). There are a few filters for which filter-specific distortion solutions are not yet available. For these filters (F395N, F467M, and F547M), we applied the solution for the closest available filter. This introduces small (0.05 pixel) errors in astrometry and negligible errors in photometry. Since the main results of this paper require high-precision photometry, we limited our analysis to the sub-sample of stars well measured. The software routine provides several quality indexes that can be used as diagnostics of the reliability of photometric measurements: i) the rms of the individual position measurements about their mean, after they have been measured in different exposures and transformed into a common reference frame ( rms X and rms Y ), ii) o , the ratio between the estimated flux of the star in a 0.5 arcsec aperture and the flux from neighbor stars that has spilled over into the same aperture), and iii) q , the residuals to the PSF fit for each star (see Anderson et al. 2008 for details). To select the high-quality sub-sample of stars we followed the approach described by Milone et al. (2009, Sect. 2.1). Photometry has been corrected for differential reddening by means of a procedure that has been adopted for several other projects and is described in detail in Milone et al. (2012b). Briefly, we define the fiducial MS for the cluster and then identify for each star a set of neighbors and determine from them their median offset relative to the fiducial sequence; this systematic color and magnitude offset, measured along the reddening line, is our estimate of the local differential-reddening value.", "pages": [ 2, 3 ] }, { "title": "3. The color-magnitude diagram", "content": "A visual inspection of the CMDs that we obtain from the data sets listed in Tab. 2 indicates that the multiple populations along the MS, the RGB, and the SGB are best identified in the m F275W versus m F275W -m F336W and the m F275W versus m F336W -m F438W CMDs shown in Fig. 1. Panels (c) and (d) of the figure show a zoomed-in region around the MS and the RGB, and reveal, for the first time, that both the cluster MS and RGB are split into two sequences. Each sequence approximately contains the same number of stars. In the following, we will use for the two RGBs and MSs of NGC 288 the same nomenclature as previously adopted in our previous works for the cases of 47 Tuc (Milone et al. 2012a), NGC6397 (Milone et al. 2012c), and NGC6752 (Milone et al. 2013). In these papers we demonstrated that, in the m F275W versus m F275W -m F336W CMD, the blue- and the red-RGB stars are the progeny of blue- and red-MS stars, respectively. Here, for analogy, we indicate as MSa and RGBa the MS and RGB sequence with redder m F275W -m F336W colors, while the bluer MS and RGB are named MSb and RGBb, respectively. The double SGB is highlighted in Fig. 1e. The two SGBs are well separated in color (by ∼ 0.05 mag) in the interval -0 . 35 < m F336W -m F438W < -0 . 15, and then merge together at m F336W -m F438W ∼ -0.15, with the faint SGB evolving into RGBa.", "pages": [ 3, 4 ] }, { "title": "3.1. Population ratio", "content": "In order to measure the fraction of stars in each MS, we followed the procedure illustrated in Fig. 2, again using techniques developed in previous studies (e.g. Piotto et al. 2007, 2012). The left panel shows the m F275W vs. m F275W -m F336W CMD of Fig. 1a, zoomed in around the MS region, in the interval 20.65 < m F275W < 23.2, where the bimodal distribution is most evident. The MS ridge line is marked in red. To determine it, we started by selecting a sample of MS stars by means of a hand-drawn, first-guess ridge line. We calculated the median color and the median magnitude of MS stars in bins that were 0.3 magnitude tall. We then interpolated these median points with a spline, and did an iterated sigma-clipping of the 'verticalized' MS (middle panel). In order to obtain the 'verticalized' MS of the middle panel, we subtracted from each star the color of the fiducial line at the same F275W magnitude level, obtaining a ∆( m F275W -m F336W ) value. The right panels of Fig. 2 show the histograms of the distribution of ∆ ( m F275W -m F336W ) for six F275W magnitude intervals. Finally, in each magnitude interval, we fit the histogram with a pair of Gaussians, colored green (for the redder peak) and magenta (for the bluer peak). Hereafter, these colors will be consistently used to distinguish the MSa and MSb populations and their post-MS progeny. From the areas under the Gaussians we estimate that 54 ± 3% of the stars belong to the MSa and 46 ± 3% to MSb. The errors were computed from the rms of the values obtained for the six intervals. In the WFC3/UVIS field of view, which includes the central part of the cluster with radial distance smaller than ∼ 1.2 core radii, the two MSs have almost the same number of stars in each magnitude interval, within the statistical uncertainties. In order to extend the study of stellar populations to the RGB and determine the fraction of RGBa and RGBb stars, in Fig. 3 we show the m F336W -m F438W versus m F275W -m F336W two-color diagram, where the RGB of NGC288 is clearly split into two sequences. Here, we analyze only RGB stars with m F606W < 17 . 85. Note that stars are selected on the basis of their F606W magnitude (in order to avoid any bias introduced to the strong luminosity difference between RGBa and RGBb stars in the ultraviolet filters). The red line is the hand-drawn fiducial line for the RGB. It separates RGBa stars (on the bottom-left side) from RGBb stars (on the upper-right side). We subtracted from the m F336W -m F438W color of each star the corresponding color of the fiducial line, obtaining a ∆( m F336W -m F438W ) index. The 'verticalized' m F275W -m F336W versus ∆( m F336W -m F438W ) diagram is plotted in panel (b) of Fig. 3, while panel (c) shows the histogram of the ∆( m F336W -m F438W ) distribution. The histogram is fitted with the sum of two Gaussians, again colored in green and magenta as above. From the area under the Gaussians we calculated that RGBa stars include 57 ± 5% RGB stars, with the remaining 43 ± 5% stars populating the RGBb. In this case, we simply associated a Poisson error to the fraction of stars in each population. Within one sigma uncertainty, these are the same fractions as for the MSa and MSb stars. From the weighted mean of the values obtained from the MS and RGB analysis, we obtain that population 'a' contains 55 ± 3% and population 'b' the 45 ± 3% of the total number of stars in the central region analyzed in this paper. Our previous studies of 47 Tuc and NGC6397 have demonstrated that any two-color diagram made from the combination of a near-ultraviolet filter (such as F225W or F275W), the F336W filter, and a blue filter (such as F390W, F435W, or F438W) is particularly efficient at disentangling stellar populations with different ligh-element abundances (Milone et al. 2012a, c). These photometric shifts can be interpreted in the light of spectroscopic observations. Carretta et al. (2009) have analyzed GIRAFFE spectra of ∼ 130 stars, twenty-five of which are in common with the HST dataset of this paper. The spectroscopic targets are represented with large circles in Fig. 3 and are colored green and magenta according to their membership in the RGBa or the RGBb. The Na-O anticorrelation from Carretta and collaborators is reproduced in panel (d), while stars for which oxygen-abundance measurements are not available are arbitrarily plotted at the flagged value of [O/Fe]=0.85. The histogram distributions of [Na/Fe] for RGBa (green histogram) and RGBb stars (magenta histogram) are shown in panel (e). Similar to what is observed in the other GCs studied with a similar approach, we find that population 'a' stars are Na-poor and O-rich, in contrast to population 'b' stars, which are depleted in oxygen and enhanced in sodium.", "pages": [ 4, 6, 8 ] }, { "title": "3.2. A multiwavelength analysis of the double MS", "content": "By combining archive and proprietary data, we have access to eleven different photometric bands to build CMDs for NGC 288. We used the UV and blue photometry displayed in Figs. 1 to select the members of population 'a' and 'b', and then plotted their positions in the CMDs obtained with all possible color combinations. UV photometry has proven to be essential to separate the two populations, because of its sensitivity to light-element variations (Marino et al. 2008). On the other hand, optical CMDs are sensitive to He content and allow us to use the color separation of the CMD sequences (MS and RGB) to estimate their average helium difference. In particular, as shown by Sbordone et al. (2011), filters redder than F435W are marginally affected by differences in C N O abundances, while they are sensitive to the helium content of the two MSs. Once we have selected the members of the two populations using the UV color-color diagrams, the optical photometry allows us to estimate the He content. Helium is extremely difficult to measure by spectroscopy in GC stars. Our procedure below is adapted from that in Milone et al. (2013). Fig. 4 shows the fiducial ridge lines for the MSa and MSb stars in the CMDs constructed with m F814W vs. m X -m F814W (where X= F275W, F336W, F395N, F435W, F438W, F467M, F547M, F606W, F625W, or F658N). A visual inspection reveals that MSa is generally redder than MSb, with the only exception of the m F814W vs. m F336W -m F814W baseline. The separation of the two sequences increases for larger color baselines in the remaining CMDs, in close analogy with what has been observed in the cases of ω Cen, NGC6397, 47Tuc, and NGC6752. Finally, we quantified the MS separation by measuring the color difference between MSa and MSb fiducials at a reference magnitude m cut F814W . We repeated this procedure for m cut F814W = 19 . 35 , 19 . 55 , 19 . 75 , 19 . 95 , 20 . 15. As an example, we show the color differences for the case of m cut F814W = 19 . 75 in Fig. 5 . We followed the same procedure as used for the MS to analyze the color separation of RGBa and RGBb. Due to the relatively small number of RGB stars, we calculated the distance between the two RGB fiducials for the two values of m cut F814W = 17 . 25 and 16 . 75. In close analogy to the color behavior of the two MSs, RGBb is typically bluer than the RGBa, with the exception of CMDs based on the m F336W -m F814W color. In the other filters the color distance from the RGBa of the RGBb increases with the color baseline. Results are illustrated in the right panel of Fig. 5 for m cut F814W = 16 . 75. Fig. 5 with synthetic photometry predictions. We used BaSTI isochrones (Pietrinferni et al. 2004, 2009) to calculate the surface temperature ( T eff ) and gravity (log g ) at different m F814W = m cut F814W for two MS populations with helium abundances as listed in Table 2. Table 2 also gives the resulting ( T eff ) and gravity (log g ). In our calculation, we assumed E ( B -V ) = 0 . 03 and ( m -M ) V = 14 . 84 (Harris 1996, 2010 edition). We used the average [O/Fe] values for population 'a' and population 'b' stars derived by the measurements in Carretta et al. (2009): [O/Fe]=0.2 and [O/Fe]= -0.0 for first- and second-generation stars, respectively. Since neither carbon nor nitrogen abundance estimates were available for NGC288, we arbitrarily assumed that the MSa has solar N and C, while MSb stars has a carbon depletion of 0.15 dex ([C/Fe]= -0 . 15) and enhanced in nitrogen by 0.7 dex ([N/Fe]=+0 . 7). To avoid the possibility that the adopted (and uncertain) values of [N/Fe] and [C/Fe] could affect our conclusions, we estimate helium only from filters redder than F435W. We assumed for the MSa primordial helium content (Y=0.246) and assumed for the MSb different helium abundances, with Y ranging from 0.246 to 0.300 in steps of ∆Y=0.001. We used the ATLAS12 program (Kurucz 2005, Castelli 2005, Sbordone et al. 2007) to account for the adopted chemical composition and performed spectral synthesis from ∼ 2,000 ˚ A to ∼ 10,000 ˚ A by using the SYNTHE code (Kurucz2005). Synthetic spectra have been integrated over the transmission curves of the appropriated filters, and, for each value of Y of our grid, we calculated the color difference m X -m F814W . The best fit between models and observations was determined by means of chi-square minimization. The helium difference corresponding to the best-fit models are listed in Table 2 for each adopted m cut F814W value. From the average mean we obtain that population 'b' is helium-enhanced by ∆Y=0.013 ± 0.001, where the error is calculated from the agreement of the independent measurements. Results are shown in Fig. 5 for the case of m cut F814W = 19 . 75, and m cut F814W = 16 . 75. Models well match the data for the visual filters, while the agreement is poorer for the ultraviolet points, as expected since these baselines are very sensitive to C and N variations, and these abundances are not constrained by spectroscopy. A spectroscopic measure of the C and N for the two populations is clearly needed. The C and N abundance differences between population 'a' and population 'b' stars that we arbitrarily adopted for NGC 288 are similar to those measured between first and second-generation stars of the GC M 4 and listed in Tab. 6 by Marino et al. (2008, see also Ivans et al. 1999, Villanova & Geisler 2011). In order to investigate the impact of our choice of C and N abundances on the inferred helium difference, we repeated the same procedure above by assuming that population 'b' stars are nitrogen enhanced by ∆[N/H]=1.0 dex and carbon depleted by ∆[C/H]= -0.5 dex with respect to population 'a' stars. In this case, the resulting ∆Y is consistent with our previous estimate within 0.001 dex, indicating that the conclusions of this paper are not significantly affected by the choice of C and N.", "pages": [ 8, 10, 11 ] }, { "title": "4. Summary", "content": "We used multi-band HST photometry covering a wide range in wavelength to study the multiple stellar populations in NGC 288. Once again, UV photometry has proven essential to allow us to separate distinct stellar populations. For the first time, our photometry shows that this cluster's MS splits into two branches, and we find that this duality is repeated along the SGB and the RGB, similar to what has been observed in other GCs. We calculated theoretical stellar atmospheres for main-sequence stars, assuming different chemical composition mixtures, and compared the predicted colors through the HST filters with our observed colors. The observed color differences between the double MS and RGB of NGC288 are consistent with two populations with different helium and light-element content. In particular, population 'a', which contains slightly more than half of the stars in NGC 288, corresponds to the first stellar generation with primordial He, and O-rich/Na-poor stars, while population 'b' is made of stars enriched in He by ∆ Y = 0 . 013 ± 0 . 001 (internal error) and Na, but depleted in O. High-precision HST photometry allows us to estimate the He content difference at an accuracy beyond reach of spectroscopy. APM and HJ acknowledge the financial support from the Australian Research Council through Discovery Project grant DP120100475. SC is grateful for financial support from PRIN-INAF 2011 'Multiple Populations in Globular Clusters: their role in the Galaxy assembly' (PI: E. Carretta). Support for this work has been provided by the IAC (grant 310394), and the Education and Science Ministry of Spain (grants AYA2007-3E3506, and AYA2010-16717). GP acknowledges partial support by the Universit'a degli Studi di Padova CPDA101477 grant. JA and AB acknowledge support from STSCI grant GO-12605", "pages": [ 11, 13 ] }, { "title": "REFERENCES", "content": "Anderson, J., Bedin, L. R., Piotto, G., Yadav, R. S., & Bellini, A. 2006, A&A, 454, 1029 Anderson, J., & King, I. R. 2006, Instrument Science Report ACS 2006-01, 34 pages, 1 Anderson, J., et al. 2008, AJ, 135, 2055 Bedin, L. R., Cassisi, S., Castelli, F., Piotto, G., Anderson, J., Salaris, M., Momany, Y., & Pietrinferni, A. 2005, MNRAS, 357, 1038 Bellini, A., & Bedin, L. R. 2009, PASP, 121, 1419 Bellini, A., Bedin, L. R., Piotto, G., Milone, A. P., Marino, A. F., & Villanova, S. 2010, AJ, 140, 631 Bellini, A., Anderson, J., & Bedin, L. R. 2011, PASP, 123, 622 Sbordone, L., Salaris, M., Weiss, A., & Cassisi, S. 2011, A&A, 534, A9", "pages": [ 13, 14 ] } ]

[ { "title": "ABSTRACT", "content": "Short-hard gamma-ray bursts (GRBs) are widely believed to be produced by the merger of two binary compact objects, specifically by two neutron stars or by a neutron star orbiting a black hole. According to the Li-Paczynski kilonova model, the merger would launch sub-relativistic ejecta and a near-infrared/optical transient would then occur, lasting up to days, which is powered by the radioactive decay of heavy elements synthesized in the ejecta. The detection of a late bump using the Hubble Space Telescope ( HST ) in the near-infrared afterglow light curve of the short-hard GRB 130603B is indeed consistent with such a model. However, as shown in this Letter, the limited HST near-infrared lightcurve behavior can also be interpreted as the synchrotron radiation of the external shock driven by a wide mildly relativistic outflow. In such a scenario, the radio emission is expected to peak with a flux of ∼ 100 µ Jy, which is detectable for current radio arrays. Hence, the radio afterglow data can provide complementary evidence on the nature of the bump in GRB 130603B. It is worth noting that good spectroscopy during the bump phase in short-hard bursts can test validity of either model above, analogous to spectroscopy of broad-lined Type Ic supernova in long-soft GRBs. Subject headings: Gamma rays: general - radiation mechanisms: non-thermal", "pages": [ 1 ] }, { "title": "IS THE LATE NEAR-INFRARED BUMP IN SHORT-HARD GRB 130603B DUE TO THE LI-PACZYNSKI KILONOVA?", "content": "Zhi-Ping Jin 1 , Dong Xu 2 , Yi-Zhong Fan 1 , Xue-Feng Wu 3 , and Da-Ming Wei 1", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "GRB 130603B triggered the Burst Alert Telescope (BAT) on board the Swift satellite at 15:49:14 UT on 2013 June 3 (Melandri et al. 2013). It had a T 90 duration of 0 . 18 ± 0 . 02 s in the 15-350 keV band (Barthelmy et al. 2013) and the BAT light curve reveals no trace of extended emission at the ∼ 0 . 005 counts det -1 s -1 level (Norris et al. 2013). The spectral lag analysis reveals no significant delay of the high and low energy photons (Norris et al. 2013). All these facts together render GRB 130603B a prototypical short-hard gamma-ray burst (GRB; de Ugarte Postigo et al. 2013; Bromberg et al. 2013). GRB 130603B is the first short GRB with absorption spectroscopy (de Ugarte Postigo et al. 2013). The other remarkable discovery made in GRB 130603B is an infrared bump appearing at t ∼ 9 days after the burst (Tanvir et al. 2013; Berger et al. 2013), which has been interpreted as the infrared/optical transient powered by the radioactive decay of heavy elements synthesized in the sub-relativistic ejecta launched by either the neutron star binary merger or the neutron star-black hole merger, i.e., the Li-Paczynski kilonova (e.g., Li & Paczynski 1998; Rosswog 2005; Metzger et al. 2010; Goriely et al. 2011; Kasen et al. 2013; Barnes & Kasen 2013; Tanaka & Hotokezaka 2013; Grossman et al. 2013). If the Li-Paczynski kilonova origin is established, the neutron star binary merger or the neutron star-black hole merger origin of some (if not all) short GRBs, a model proposed in 1990s (Eichler et al. 1989; Narayan et al. 1992), will be confirmed. Hence, some short and long GRBs do have very different physical origin, as widely speculated (Narayan et al. 2001). In view of the fundamental importance of such a kind of interpretation, it is necessary to check whether other possibilities exist, and if they do, how these possibilities can be further constrained. That is the main purpose of this letter.", "pages": [ 1 ] }, { "title": "2. THE DIFFICULTY OF INTERPRETING THE LATE INFRARED BUMP AS THE REGULAR AFTERGLOW", "content": "Tanvir et al. (2013) suggested that there are two reasons against the regular afterglow origin of the late infrared bump of GRB 130603B. One is that the optical afterglow lightcurve of GRB 130603B drops with time more quickly than t -2 for t > 10 hours after the trigger of the burst. The near-infrared flux, on the other hand, is in excess of the same extrapolated power law (see Figure 2 of Tanvir et al. 2013). The other is the significant color evolution of the transient, defined as the difference between the magnitudes in each filter, which evolves from R 606 -H 160 ≈ 1 . 7 ± 0 . 15 mag at about 14 hr to greater than R 606 -H 160 ≈ 2 . 5 mag at ∼ 9 days. Below we discuss the afterglow model extensively and demonstrate that in the regular fireball afterglow model no color evolution should be present in the time interval of 0 . 6 -9 days, which is in support of Tanvir et al. (2013)'s argument. As found in de Ugarte Postigo et al. (2013), assuming a spectral break ν -0 . 5 between the optical and Xray bands as that expected in the standard afterglow model (Piran 1999), the near-infrared to X-ray spectral energy distribution (SED) of GRB 130603B 8.5 hr after the burst onset can be nicely fitted by an extinction of A V = 0 . 86 mag and a Small Magellanic Cloud (SMC) extinction law. The SED is well fitted with a spectral break at ν break ≈ 10 16 Hz and the lower and high energy spectral indexes are α O = -0 . 65 ± 0 . 09 and α X = -1 . 15 ± 0 . 11, respectively. Using the optical afterglow data at t ∼ 0 . 6 day (Tanvir et al. 2013) and the public X-ray afterglow data, we obtain a very similar SED but ν break shifts to ∼ 6 × 10 15 Hz (see Figure 1). The optical and X-ray spectra suggest that the break frequency is the so-called cooling frequency ν c in the fireball afterglow model (Piran 1999). For the burst born in stellar wind, ν c ∝ t 1 / 2 , i.e., the later the observation, the higher the cooling frequency. The SEDs at t ∼ 0 . 35 day and 0 . 6 day are not in support of such a tendency. Instead they are in agreement with the case of that ν c ∝ t -1 / 2 for the burst born in the ISM-like medium. Hence, ν c ∼ 10 16 ( t/ 0 . 35 day) -1 / 2 ∼ 2 × 10 15 Hz at t ∼ 9 days, which is still well above the optical band. Such results holds as long as the ejecta sideways expansion is unimportant. If the sideways expansion is important, in both wind- and ISM-like medium models, ν c ∝ t 0 . All these facts together rule out the presence of a significant color evolution of the infrared/optical afterglow emission in the time interval of 0 . 6 -9 day. Therefore the observed soft infrared/optical emission at t ∼ 9 days should have a different physical origin. Tanvir et al. (2013) and Berger et al. (2013) have interpreted the infrared bump as the Li-Paczynski kilonova. This kind of interpretation is extremely attractive. However, the Hubble Space Telescope ( HST ) data is very rare and other scenarios should also been investigated.", "pages": [ 1, 2 ] }, { "title": "3. THE SECOND-COMPONENT JET MODEL FOR THE INFRARED BUMP OF GRB 130603B?", "content": "Two component jet model has been adopted to interpret some peculiar afterglow emission of both long and short GRBs (for the former, see, e.g., Berger et al. 2003; Huang et al. 2004; Racusin et al. 2008; for the latter, see Jin et al. 2007). In such a model, the narrow energetic core produce prompt γ -ray emission and then the early bright afterglow emission while the much wider but less energetic ejecta component will emerge at a late time, depending on its bulk Lorentz factor. The infrared bump likely peaks at t ∼ 9 days after the burst onset, then the initial Lorentz factor of the ejecta should satisfy where E k , w is the kinetic energy of the mildly relativistic outflow component and n is the number density of the circum-burst medium (for simplicity, below we just discuss the ISM-like medium that is favored by the SEDs). Note that here and throughout the text the convenience Q x = Q/ 10 x has been adopted except for specific notations. We also point out that a mildly relativistic outflow component is not unexpected. For example, in both the double neutron star merger scenario the neutron starblack hole merger scenario, a wide but mildly relativistic outflow surrounding the ultra-relativistic GRB ejecta may be formed as a result of the interaction of the outflow with the surrounding material (e.g., Aloy et al. 2005). After the merger of the double neutron stars, a supramassive/stable magnetar rather than a black hole may be formed (e.g., Gao & Fan 2006; Zhang 2013; Giacomazzo & Perna 2013). The wind of the magnetar that possibly suffers from significant kinetic energy loss via gravitational wave radiation (Fan et al. 2013) may be able to accelerate the material ejected from the double neutron star merger to a mildly relativistic velocity as well (Fan & Xu 2006; Gao et al. 2013). The cooling Lorentz factor of the external forward shock electrons can be estimated as ν c ≈ 10 16 Hz E -1 / 2 k , 51 /epsilon1 -3 / 2 B , -2 n -1 0 ( t/ 1 day) -1 / 2 (1 + z ) -1 / 2 (Piran 1999), where /epsilon1 B is the fraction of shock energy given to the magnetic field and z = 0 . 356 is the redshift of GRB 130603B (de Ugarte Postigo et al. 2013). For the narrow and wider ejecta components, the number density of the medium should be the same and the initial kinetic energy is expected to be different and usually we have E k , n > E k , w . As mentioned above, for the narrow ejecta component ν c , n ∼ 10 16 Hz at t ∼ 0 . 35 day, hence To interpret the identified softness of near-infrared bump (i.e., ∆( R 606 -H 160 ) ≈ 0 . 8 ± 0 . 15 mag), the synchrotron radiation spectrum of the second-component ejecta should be softer than that of the early ( t ∼ 0 . 6 day) afterglow by a factor of ν -0 . 75 ± 0 . 14 . The required power-law distribution index of the electrons accelerated by the wide-component ejecta is p w ∼ 2 . 8 ± 0 . 3 as long as ν c , w < ν F606W at t ≥ 9 days (where p n ∼ 2 . 3 has been adopted), for which we need It is unclear why the narrow and wide outflow components have different /epsilon1 B (possibly also /epsilon1 e and/or p ). However, we note that the best-fitted microphysical parameters of GRBs differ from burst to burst (Panaitescu & Kumar 2001) and no universal values have been obtained. Moreover, in the modeling of the afterglow emission of some GRBs within the two-component jet scenario, the best-fitted microphysical parameters are found to be different for the narrow and wide components (e.g., Jin et al. 2007; Racusin et al. 2008). Hence, we suggest that the request /epsilon1 B , w > /epsilon1 B , n is reasonable and possible. Simultaneous with the ∼ 0 . 2 µ Jy infrared emission, the radio radiation flux is expected to be for p w ∼ 2 . 5 and ν radio ∼ 10 10 Hz. This is because both the typical synchrotron radiation frequency ν m , w ≈ 4 × 10 9 Hz E 1 / 2 k , w , 50 /epsilon1 1 / 2 B , w , -1 /epsilon1 2 e , w , -1 ( t/ 9 day) -3 / 2 and the synchrotron self-absorption frequency ν a , w ≈ 5 . 7 × 10 9 Hz E 0 . 35 k , w , 50 /epsilon1 0 . 35 B , w , -1 n 0 . 3 0 /epsilon1 0 . 46 e , w , -1 ( t/ 9 day) -0 . 73 are below ∼ 10 10 Hz. Note that even for p w ∼ 2 . 3, we have F ν radio ∼ 100 µ Jy. Such a flux is bright enough to be reliably detected by some radio arrays (for example, the Karl G. Jansky Very Large Array) in performance. The non-detection will in turn impose a tight constraint on the external forward shock radiation origin of the infrared bump. To better show our idea, we calculate the flux numerically. The code used here has been developed in Fan & Piran (2006) and Zhang et al. (2006). The dynamical evolution of the outflow is calculated using the formulae in Huang et al. (2000), which can be used to describe the dynamical evolution of the outflow for both the relativistic and non-relativistic phases. The energy distribution of the shock-accelerated electrons is calculated by solving the continuity equation with the power-law source function Q = Kγ -p w e , normalized by a local injection rate (Moderski et al. 2000). The cooling of the electrons due to both synchrotron and inverse Compton has been taken into account. In Figure 2, we have presented one numerical example which can fit the limited HST data of GRB 130603B. The physical parameters adopted in the fit are as follows: /epsilon1 e , w = 0 . 1, /epsilon1 B , w = 0 . 1, p w = 2 . 5, n = 1 . 0 cm -3 , E k , w = 4 × 10 50 erg, the initial Lorentz factor of the outflow Γ 0 = 3 . 0, and the half-opening angle is assumed to be θ j = 1 . 0. As one can see both the temporal and spectral properties of the infrared bump of GRB 130603B can be reproduced. The radio afterglow emission is so bright ( ∼ 200 µ Jy) that can be well detected by Karl G. Jansky Very Large Array-like telescopes. The non-detection would impose a tight constraint on the mildly relativistic outflow model.", "pages": [ 2, 3 ] }, { "title": "4. DISCUSSION", "content": "Our understanding of short GRBs, a kind of γ -ray outbursts with a duration less than two seconds (Kouveliotou et al. 1993), has been revolutionized due to the successful performance of the Swift satellite. For a good fraction of short GRBs, the binaryneutron-star or the black hole-neutron star merger model (Eichler et al. 1989; Narayan et al. 1992) has been supported by their host galaxy properties and by the nonassociation with bright supernovae. A smoking-gun signature, i.e., a supernova-like infrared/optical transient powered by the radioactive decay of heavy elements synthesized in the ejecta launched by either the neutron star binary merger or the neutron star-black hole merger (i.e., the Li-Paczynski kilonova), however, has not yet been unambiguously identified. The best candidate of such a smoking-gun signature is likely the in- /s32 frared bump detected at t ∼ 9 days after the onset of GRB 130603B (Tanvir et al. 2013; Berger et al. 2013). If the Li-Paczynski kilonova origin is confirmed, the neutron star binary merger or alternatively the neutron starblack hole merger origin of some (if not all) short GRBs will be established. Hence, some short and long GRBs do have very different physical origin (Narayan et al. 2001). In view of the fundamental importance, it is necessary to check whether or not other possible physical origins of the infrared bump of GRB 130603B exist. As shown in this Letter the HST near-infrared data is very limited and can be interpreted as the synchrotron radiation of the external shock driven by a wide mildly relativistic outflow. Interestingly, a wide mildly relativistic outflow associated with the ultra-relativistic GRB ejecta has been 'observed' in the numerical simulation (e.g., Aloy et al. 2005) or 'predicted' in the magnetar central engine model of short GRBs (e.g., Fan & Xu 2006; Gao et al. 2013). In the mildly relativistic outflow model, the radio emission is expected to peak at t ∼ 10 6 s (see Fig.2) with a flux ∼ 100 µ Jy, which is detectable for some radio arrays in performance. While the synchrotron radio radiation powered by the kilonova outflow is expected to peak at t ∼ 8 × 10 8 ( V nova / 0 . 1 c ) -5 / 3 E 1 / 3 nova , 51 n -1 / 3 0 s, where E nova ( V nova ) is the kinetic energy (velocity) of the kilonova outflow and c is the speed of light. Hence, the radio afterglow data can provide complementary evidence on the nature of the near-infrared bump in GRB 130603B or similar events in the future. It is worth noting that good spectroscopy during the bump phase in short-hard bursts can test validity of either model above, analogous to spectroscopy of broad-lined Type Ic supernova in long-soft GRBs. If the mildly relativistic outflow model has been confirmed by (future) observations, the near-infrared bump like that detected in GRB 130603B is still valuable for those interested in searching for the electromagnetic counterparts of the merger of two neutron stars or a neutron star and a black hole since such a kind of signal is expected to be almost isotropic due to its low bulk Lorentz factor. As shown in the numerical simulation (Aloy et al. 2005), the mildly relativistic outflow may /s32 be common, thus the observational prospect can not be ignored.", "pages": [ 3, 4 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This work was supported in part by the 973 Programme of China under grants 2009CB824800 and 2013CB837000, the National Natural Science of China under grants 11073057, 11103084 and 11273063, and the Foundation for Distinguished Young Scholars of Jiangsu Province, China (No. BK2012047). Y.Z.F. and X.F.W. are also supported by the 100 Talents programme of Chinese Academy of Sciences. D.X. acknowledges support from the ERC-StG grant EGGS-278202 and IDA.", "pages": [ 4 ] }, { "title": "REFERENCES", "content": "Huang, Y. F., Gou, L. J., Dai, Z. G., & Lu, T., 2000, ApJ, 543, 90 Huang, Y. F., Wu, X. F., Dai, Z. G., Ma, H. T., & Lu, T. 2004, ApJ, 605, 300 Jin, Z. P., Yan, T., Fan, Y. Z., & Wei, D. M. 2007, ApJ, 656, L57 Kasen, D., Badnell, N. R. & Barnes, J. 2013, ApJ, 774, 25 Kouveliotou C., Meegan, C. A., Fishman, G. J., et al. 1993,", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "We calculate the structure and short-term evolution of a gamma ray burst central engine in the form of a turbulent torus accreting onto a stellar mass black hole. Our models apply to the short gamma ray burst events, in which a remnant torus forms after the neutron star-black hole or a double neutron star merger and is subsequently accreted. We study the 2-dimensional, relativistic models and concentrate on the effects of black hole and flow parameters as well as the neutrino cooling. We compare the resulting structure and neutrino emission to the results of our previous 1-dimensional simulations. We find that the neutrino cooled torus launches a powerful mass outflow, which contributes to the total neutrino luminosity and mass loss from the system. The neutrino luminosity may exceed the Blandford-Znajek luminosity of the polar jets and the subsequent annihillation of neutrino-antineutrino pairs will provide an additional source of power to the GRB emission. Subject headings: accretion, accretion disks; black hole physics; magnetohydrodynamics (MHD); neutrinos; relativistic processes; gamma ray burst:general", "pages": [ 1 ] }, { "title": "ACCRETION AND OUTFLOW FROM A MAGNETIZED, NEUTRINO COOLED TORUS AROUND THE GAMMA RAY BURST CENTRAL ENGINE", "content": "Agnieszka Janiuk 1 , Patryk Mioduszewski 1 1 Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland and Monika Moscibrodzka 2 2 Department of Physics, University of Nevada Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA Draft version June 16, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Gamma Ray Bursts (GRB), known for about forty years (Klebesadel et al. 1973) are extremely energetic transient events, visible from the most distant parts of the Universe. They last from a fraction of a second up to a few hundreds of seconds and are isotropic, nonrecurrent sources of gamma ray radiation (10 keV - 20 MeV). Short gamma ray bursts were distinguished in the KONUS data by (Mazets & Golentskii 1981) and further two distinct classes of events, long and short, were found by (Kouveliotou et al. 1993). The energetics of these events points to a cosmic explosion as a source of the burst, associated with the compact objects such as black holes and neutron stars. The short timescales and high Lorentz factors of the gamma ray emitting jets are most likely produced in the process of accretion of rotating gas on the hyper-Eddington rates that proceeds onto a newly born stellar mass black hole. The key properties of such a scenario are therefore deep gravitational potential of the black hole and significant amount of the angular momentum that supports the rotating torus. Accretion of magnetized torus onto a black hole with a range of spin parameters was studied by De Villers et al. (2003); McKinney & Gammie (2004) and applied to the long gamma ray bursts (Nagataki 2009). The relativistic simulations of accretion flows with an ideal gas equation of state were studied e.g., by Hawley & Krolik (2006) and McKinney & Blandford (2009) and recently more sophisticated models with a realistic EOS were proposed by Barkov & Komissarov (2008, 2010) and Barkov (2008); Barkov & Baushev (2011). This central engine gives rise to the most powerful jets (see e.g. the reviews by Zhang & Meszaros (2004); Piran (2005); Gehlers et al. (2009); Metzger (2010), Gehlers et al. (2009), Metzger (2010)). Despite the existence of still unsolved problems, such as the composition of the outflows, the emission mechanisms creating the gamma rays, or the form of energy that dominates the jet (i.e. kinetic or Poynting flux), the jets themselves are believed to be powered by accretion and rotation of the central black hole. In this process, the strong large-scale magnetic fields play a key role in transporting the energy to the jets (McKinney 2006; Tchekhovskoy et al. 2008; Dexter et al. 2012). In addition to the magneto-rotational mechanism of energy extraction, the annihilation of neutrinoantineutrino pairs, emitted from the accreting torus, may provide some energy reservoir available in the polar regions to support jets. The neutrinos are produced in central engines of both short and long GRBs, the latter being modeled in the frame of the collapsing massive star scenario (Woosley 1993; Paczynski 1998). The recent numerical simulations of the 'hypernovae' aimed to capture the effects of both MHD and neutrino transport in the supernova explosion modeling (Burrows et al. 2007), using a flux-limited neutrino diffusion scheme in the Newtonian dynamics. The general relativistic simulations by Shibata et al. (2007) on the other hand, consider the neutrino cooling of the accreting torus around the black hole and capture the neutrino-trapping effect in a qualitative way. In this work, we study the central engine, composed of a stellar mass, rotating black hole and accreting torus that has formed from the remnant matter at the base of the GRB jet. We start from an axially symmetric, configuration of matter filling the equipotential surfaces around a Kerr black hole (Fishbone & Moncrief 1976; Abramowicz et al. 1978), assuming an initial poloidal magnetic field. The MHD turbulence amplifies the field and leads to the transport of angular momentum within the torus. In the dynamical calculations, we use a realistic equation of state while we account for the neutrino cooling (Yuan 2005; Janiuk et al. 2007). We study the evolution and physical properties of such an engine, its neutrino luminosity and production of a wind and outflow from the polar regions. Our calculations are 2-D and relativistic, therefore this work is a generalization of the model presented in Janiuk et al. (2007); Janiuk & Yuan (2010), where a simpler steady-state, 1 dimensional model of a torus around a rotating black hole was analyzed, using approximate correction factors to the pseudo-Newtonian potential that allowed to mimic Kerr metric. The microphysics however is currently described using the EOS from that work and neutrino cooling is incorporated into the HARM scheme via the cooling function. The total pressure invoked to compute the cooling is contributed by the free and degenerate nuclei, electron-positron pairs, helium, radiation and partially trapped neutrinos. This allows us to compute the optical depths for neutrino absorption and scattering and the neutrino emissivities in the optically thin or thick plasma. The article is organized as follows. In § 2, we describe our model, the initial conditions, the dynamical evolution of the system and the assumed chemical composition as well as the processes responsible for energy losses via neutrino cooling. In § 3, we present the results, describing the effects of (i) black hole mass (ii) its spin (iii) torus mass, and (iv) magnetic field strength. We also discuss the effect of neutrino cooling on the torus structure, in comparison with the reference model with such cooling neglected. Finally, we compare our results with the 1-D simulations of the vertically averaged torus, emphasizing the effects of 2-dimensional computations. We discuss the results in § 4.", "pages": [ 1, 2 ] }, { "title": "2. MODEL OF THE HYPERACCRETING DISK", "content": "The model computations are based on the axisymmetric, general relativistic MHD code HARM-2D , described by Gammie et al. (2003) and Noble et al. (2006). The nuclear equation of state is discussed in detail in Janiuk et al. (2007). The goal of our calculations is to investigate the overall structure of a magnetized, turbulent accretion disk in which nuclear reactions take place and the gas looses energy via neutrino cooling, and in particular to expand our previous 1-dimensional models based on α viscosity, to the case of 2-D GRMHD.", "pages": [ 2 ] }, { "title": "2.1. Initial conditions and dynamical model", "content": "We start the numerical calculations from the equilibrium model of a thick torus around a spinning black hole as introduced by Fishbone & Moncrief (1976) and Abramowicz et al. (1978). The parameters of the model are the central black hole mass, M BH = 3 -10 M /circledot , the dimensionless spin of the black hole, a = 0 . 8 -0 . 98, and the total mass of the surrounding gas, M torus = 0 . 1 -2 . 5 M /circledot (see Table 1 for the list of models). We seed the torus with a poloidal magnetic field (magnetic field lines follow the constant density surfaces); the strength of the initial magnetic field is normalized by the gas to magnetic pressure ratio at the pressure maximum of the ini- tial structure of the disk ( β = P gas /P mag = 5 -100). In the dynamical calculations, we use P = ( γ -1) u equation of state with the adiabatic index γ = 4 / 3. To follow the evolution of the gas dynamics near a black hole we use a numerical MHD code HARM-2D . The numerical code is designed to solve magnetohydrodynamic equations in the stationary metric around a black hole. In this work, we modify the MHD code to account for the chemical composition of the nuclear matter accreting onto black hole in the GRB environment (described in more detail in § 2.2). At each time moment of the simulation we calculate the gas nuclear composition assuming the balance of nuclear equilibrium reactions. This gives us expected neutrino cooling rates which we incorporate into the code. After each time step of the dynamical evolution the total internal energy of gas is reduced by Q ν ∆ t factor using an explicit method with n -sub-cycles. The procedure for calculating the neutrino cooling takes into account the change of the gas internal energy in the comoving frame, which is a correct relativistic approach. We do not account for the neutrino transfer though, and the effects like the gravitational redshift are neglected. Our models have numerical resolution of the grid 256x256 points in r and θ directions (see also Sect. 3.1.2). The grid is logarithmic in radius and condensed in polar direction towards the equatorial plane, as in Gammie et al. (2003).", "pages": [ 2 ] }, { "title": "2.2. Chemical composition and neutrino cooling", "content": "We assume that the neutrino emitting plasma consists of protons, electron-positron pairs, neutrons and helium nuclei. The gas is in beta equilibrium, so that the ratio of protons to neutrons satisfies the balance between forward and backward nuclear reactions. Neutrinos are formed in the URCA process, electron positron pair annihilation, nucleon -nucleon bremsstrahlung, plasmon decay. These reactions are: and and For a given temperature and density, the neutrino cooling rate is calculated from the balance between the above reactions, supplemented with the conditions of the conservation of the baryon number and charge neutrality (Yuan 2005; see also Kohri & Mineshige 2002, Chen & Beloborodov 2007, Janiuk al. 2007). We assume that the cooling proceeds via electron, muon and tau neutrinos in the plasma opaque to their absorption and scattering. The URCA process and plasmon decay produce the electron neutrinos only, while the other processes produce neutrinos of all flavors. The emissivities of these processes are and the two latter being iterated numerically (the full set of Equations is given in the Appendix of Janiuk et al. 2007). Here ρ 10 is the baryon density in the units of 10 10 g/cm 3 and T 11 is temperature in the units of 10 11 K. The emissivities are given in the units of [erg cm -3 s -1 ]. We neglect here the term of neutrino cooling by photodissociation of helium nuclei, since at the temperatures and densities obtained in the presented models, this term will be practically equal to zero. The plasma can be opaque to neutrinos, so we use the optical depths, given by the equations derived in Di Matteo et al. 2002 : where absorption of the electron neutrinos is determined by and for the muon and tau neutrinos is given by We also account for the neutrino scattering and the scattering optical depth is given by: where C s , p = (4( C V -1) 2 +5 α 2 ) / 24, C s , n = (1+5 α 2 ) / 24, C V = 1 / 2 + 2 sin 2 θ C , with α = 1 . 25 and sin 2 θ C = 0 . 23 (Yuan 2005; Reddy et al. 1998). The neutrino cooling rate is finally given by (13) and the neutrino luminosity emitted by the plasma is where dV is the unit volume in the Kerr geometry. The optical depths for absorption and scattering are calculated approximately by assuming the disk vertical thickness equal to the pressure scale-height, H = c s / Ω K , where c s is the speed of sound and Ω K = c 3 GM BH ( a + r 3 / 2 ) -1 is the Keplerian frequency (see e.g. Lopez-Camara et al. 2009). The resulting thickness is roughly proportional to a fraction of the disk radius and the typical ratios are H/r ∼ 0 . 3 -0 . 5. We do not account for the neutrino heating in the jets via the annihilation process, because of large uncertainties in the internal energy computations in the jet. The neutrino cooling is limited to the torus and wind only, via the density and temperature ranges for which the cooling is operating (10 6 -10 13 g cm -3 and 10 7 -10 12 K, respectively). Therefore the jets are not shown in the neutrino cooling maps.", "pages": [ 2, 3 ] }, { "title": "3. RESULTS", "content": "3.1. Effect of the BH parameters and torus mass on the ˙ M and neutrino luminosity We studied the models with the black hole mass of M BH = 3 M /circledot or M BH = 10 M /circledot , and the torus mass was assumed equal to about 0.1, 0.3, 0.7, 1.0 or 2.6 M /circledot (Table 1). In Figure 1, we show the time evolution of the mass accretion rate onto black hole, for the two values of torus and black hole mass. The average accretion rate onto black hole is not changing much with the black hole spin and is about 0.3-1.0 M /circledot s -1 for most SBH models. The accretion rate for the first 2-3 milliseconds is very small, and then grows to about 0.2-0.5 M /circledot s -1 and starts varying. During such flares, it exceeds momentarily 2-5 M /circledot s -1 . These flares are however very short in duration. The mean accretion rate in our models does not exceed 1 M s -1 . /circledot The magnitude of the flares depends on the black hole spin, and largest is for a = 0 . 8 in the small disk models (SBH). The amplitude of flares is by a factor of ∼ 2 -3 larger for the black hole mass of 10 M /circledot (LBH). In the LBH models, the case with a = 0 . 9 shows higher flares at the early evolution, while the a = 0 . 8 model is flaring in the late times. After the time of about t = 3000 M , the accretion rate decreases, the flaring ceases and a rather stable value below ˙ M /lessorsimilar 0 . 3 M /circledot s -1 is reached. The late time activity ceases because of the decay of magnetic turbulence characteristic for axisymmetric models. In Figures 2 and 3 we show the maps of the torus structure calculated in the 2-D model for the black hole mass M BH = 3 M /circledot and 10 M /circledot , and torus mass of 0 . 1 M /circledot and 1 . 0 M /circledot , respectively (models SBH3 and LBH3 in Table 1). The snapshots, taken at the end of the simulation for time t = 2000 GM BH /c 3 , present the baryon density ρ , gas temperature T and magnetic β parameter overplotted with magnetic field lines, as well as the neutrino cooling. The neutrino luminosity evolution with time is shown in Figure 4 (models with small and large black hole mass). For the black hole mass of 3 M /circledot and torus of 0.1 M /circledot , the initial neutrino luminosity calculated using Eq. 14, is about 10 52 erg s -1 . Then the luminosity gradually grows to over 10 53 erg s -1 and peaks at time t = 0 . 01 s, which is equal to about 660 M. For the black hole mass of 10 M /circledot and more massive torus of 1.0 M /circledot , the total luminosity is higher and at maximum reaches values almost 10 54 erg s -1 , at about t = 0 . 04 s (equal to about 800 M). At the end of the simulation, the neutrino luminosity is about 2 × 10 53 in this model and depends mostly on the ratio between the torus and black hole mass. The exact values of L ν at the end of the simulation are given in Table 1, for a range of parameters. The neutrinos are emitted from the torus as well as from the hot, rarefied wind. The luminosity of this wind gives substantial contribution to the total luminosity and it is about 8-13 % for SBH models, and 10-15 % for LBH models, anticorrelating with the black hole spin. This fraction was estimated geometrically, i.e. the wind luminosity was calculated by integrating the emissivities over the volume above and below 30 · from the mid-plane. The luminosity of the densest parts of the torus, on the other hand, which can be estimated e.g. by weighing the total emissivity by the plasma density, is not more than 10 48 -10 49 erg s -1 , because the opacity for neutrino absorption and scattering in this regions reaches τ ∼ 0 . 1. The velocity field maps at the end of the simulation, for M BH = 3 M /circledot and M BH = 10 M /circledot are shown in Figure 5. The figures show results of the models with highest β = 100 at time t=4000 M, so that we could obtain clear polar jets. In the first case, the torus is turbulent, the wind outflow occurs, but most of material is swept back from the outermost regions and finally accretes onto black hole. Some fraction of gas is lost via the hot winds at moderate latitudes. In the second model, the disk winds are sweeping the gas out from the system, both in the equatorial plane and at higher latitudes. We identified the regions of the wind in the computation domain by defining three conditions that must be satified simultaneously: (i) the radial velocity of the plasma is positive (ii) the denisty is smaller than 10 9 g cm 3 and (iii) the gas pressure is dominant, β > 0 . 1. The two latter conditions are somewhat arbitrary but they are necessary to distinguish the wind from the turbulent dense torus and from the magnetized jets. The winds are located approximately at radii above 10 R g and latitudes between about 30 · -60 · and 120 · -150 · . The velocity in the wind is 0.005 - 0.18 of the velocity of light (models SBH) and 0.002 - 0.06 (models LBH). In the first case, it is on the order of the escape velocity, while in the second case the winds are bound by the black hole gravity (cf., e.g., McKinney (2006), who found the winds with half opening angles of θ = 16 -45 · and mildly relativistic velocities). Such large-scale circulations can be determined in the simulations with a much larger radial domain (e.g. Narayan et al. (2012); McKinney, Tchekhovskoy & Blandford (2012)). The effect of the wind is the mass loss from the system. We estimated quantitatively the evolution of the mass during the simulation. The total mass removed from the torus as a function of time, calculated by integrating the density over the total volume, differs significantly from the total mass accreted onto the black hole (i.e. the time integrated mass accretion rate through the inner boundary, subtracted from the initial mass). For models with M BH = 3 M /circledot , the denser and cooler torus, with smaller gas pressure to magnetic pressure ratio, launches a wind and about 50% of mass is lost through wind, while the rest is accreted onto black hole. However, for the black hole of 10 M /circledot , after the wind is launched, it takes away about 75% of mass from the system. In other words, the average mass loss rate in the winds is either equal to or larger (in particular, in LBH models, it may be even 3 times larger) than the accretion rate onto the black hole. The results are weakly sensitive to the black hole spin value. The physical conditions in the winds are different from those in the torus. The densities are a few orders of magnitude smaller, between 5 × 10 6 and 10 9 g cm -3 , while the temperatures in the wind are very high, in the range 7 × 10 9 -5 × 10 10 K (in general, the winds in models LBH are slightly hotter and less dense than in SBH). Such high temperatures, above the treshold for electron-positron pair production, T = m e c 2 ≈ 5 × 10 9 K, are the key condition for neutrino emission processes. The neutrino cooling is then efficient and only weakly depends on density. In the clumps with ρ > ∼ 10 8 g cm -3 , the nuclear processes lead to neutrino production, while the optical depths for their absorption are very small. The hot, rarefied, transient polar jets appear as well on both sides of the black hole, as seen in Figure 5 as well as in the maps in Figs. 2 and 3. The limitation of our model is only that here we do not study the neutrino emission in these jets. In this Section, we show the results of the models where the thickness of the torus is given by the pressure scaleheight at the equator. This is about 0.3 times the radius. We also tested the approximate condition for the disk thickness being a fraction of the radius, H ∼ 0 . 5 r . We verified that the disk thickness parametrization of neutrino cooling does not affect much the accretion rate onto black hole neither the total luminosity.", "pages": [ 3, 4 ] }, { "title": "3.1.1. Optically thin and thick tori", "content": "We find no clear neutrinosphere in the models where the torus to the black hole mass ratio is small and the accretion rate is below ∼ 1 M /circledot s -1 . In these models, the torus and wind are both optically thin to neutrinos and radiate efficiently. The optical depths due to the scattering and absorption of neutrinos, calculated in the equatorial plane, are shown in Figure 6. As shown in the top and middle panels of the Figure, τ tot ≈ 0 . 15 in the innermost parts of the torus at the equator, for the model with black hole mass M BH = 3 M /circledot and disk mass of 0 . 1 M /circledot (i.e., SBH3 and LBH3). Above the equator, the optical depths are much smaller. Also, the model with back hole mass M BH = 10 M /circledot and disk mass of 1 . 0 M /circledot gives small neutrino optical depths, up to about 0.05. The flow is optically thin to neutrinos for the magnetic field parameter β = 50 as well as β = 5. Therefore the neutrino pressure is much less than both the gas and magnetic pressures. In the bottom panel of the Figure 6, we show the results from the model SBH8, where the torus mass was assumed 1 . 0 M /circledot and the black hole mass was M BH = 3 M /circledot . The accretion rate onto the black hole was in this case larger than 1 . 0 M /circledot s -1 and the optical thicknesses to the neutrino absorption and scattering were larger than unity within the inner 3 gravitational radii in the torus equatorial plane. The neutrino luminosity of the plasma is affected by the opacities. However, the neutrino trapping effect that was clearly present in the 1-D models, is now rather subtle and plays a role in the densest, equatorial regions of the torus. In Figure 7 we plot the neutrino luminosity weighted by the plasma density, i.e. < L ν > ρ = ∫ Q ν ρdV / ∫ ρdV . We see, that after the initial conditions of the simulation are relaxed, about 0.01 s for the lack hole mass M BH = 3 M /circledot , the luminosity of the more massive torus drops below the value obtained for the less massive one, optically thin to neutrinos. Still, the total neutrino luminosity of the system is dominated by the optically thin wind, and the total L ν of the more massive torus is large (e.g. at t end it is equal to 9 × 10 52 and 4 × 10 53 erg s -1 respectively; see Table 1). In Figure 4, we show the total neutrino luminosity (i.e. the disk and wind luminosity), in the models with different BH and disk mass. These models are optically thin. In Figure 7, we show the luminosity weighted by the density, which represents the densest parts of the disk, where the optical depths could be larger than 1. The meaning of Fig. 7 is therefore to compare the optically thick and thin models, which have luminosities slightly different due to neutrino absorption. Still, the luminosities are on the same order of magnitude, after the initial conditions are relaxed. The differences in the initial conditions leading to the luminosity differences are mainly due to a larger size and mass of the disk in the compared models, determined by the initial location of the pressure maximum. After the torus redistributes itself and matter accretes through the black hole horizon, the initial conditions are relaxed.", "pages": [ 4, 5, 6 ] }, { "title": "3.1.2. Resolution tests", "content": "As a standard resolution, we use 256 × 256 zones in r and θ . For numerical test, we also checked two other resolutions, for the model SBH3. The lowest resolution model was with 128 × 128 zones and highest resolution was with 512 × 512 zones. We found the increase of total neutrino luminosity with resolution at late times of the evolution, up to a factor of 2 between the two extreme cases. The time averaged neutrino luminosity is equal to 4 . 74 × 10 52 , 1 . 04 × 10 53 and 6 . 55 × 10 52 erg s -1 , for the low, medium and high resolution models, respectively. Also, the relaxation from initial conditions is reached earlier for the largest resolution. For the disk structure, the increase of resolution results in a slight temperature increase and density rise in the inner regions of the torus, because the magneto-rotational turbulence is better resolved and accretion rate is increased. The time dependence of accretion rate onto black hole is finest for highest resolution models. The peaks in the accretion rate are higher, occur earlier during the evolution and continue to the end of simulation. Still, we conclude that it is justified to keep the moderate resolution as the basic one, as it satisfies the balance between accuracy and computation time.", "pages": [ 6, 7 ] }, { "title": "3.2. Effects of the black hole spin", "content": "We ran our small and large black hole simulations with three values of the black hole spin parameters, a = 0 . 98, a = 0 . 9, and a = 0 . 8. The value of black hole spin is qualitatively not very significant for the average properties of the torus. For the lower spins, the torus is slightly hotter and less magnetized, with the neutrino emissivity being smaller both in the torus and in the wind. The flaring activity, shown in the Figure 1 and discussed above, is stronger for smaller black hole spins at late times, and the accretion rate onto black hole occasionally reaches 3-4 or even 5-6 M /circledot s -1 , depending on the black hole to torus mass ratio. The fast spinning black holes launch powerful and steady polar jets. However, tha values of the Blandford-Znajek luminosity as given in Table 1, do not differ significantly for our spins (a=0.8-0.98). These results should be further verified by the 3-dimensional simulations with a range of grid resolutions. The mean accretion rate onto the black hole decreases with black hole spin, as given in Table 1. The result is therefore the same as in De Villers et al. (2003), regardless of the neutrino cooling included.", "pages": [ 7 ] }, { "title": "3.3. Effect of the magnetic field", "content": "The magnetic field in our simulations was parametrized with initial conditions of β = P gas /P mag of a fixed value with a maximum at the pressure maximum radius and zero everywhere outside of the torus. The mean value of β , integrated over the total volume, was at t = 0 infinite due to such initial conditions, but at the end of the simulation converged to the value assumed for the torus. The mean β weighted by the density was always a bit larger than the total volume integrated beta due to the dominating gas pressure in the disk. Changing the magnetic field normalization β affects somewhat the resulting structure of the torus. The torus density increases with β : the maximum density at the equatorial plane for the torus around a 3 M /circledot black hole with β init = 50 is ρ max ≈ 1 . 5 × 10 12 g cm -3 , for β init = 10 it is 3 . 5 × 10 11 g cm -3 , and for β init = 5 it is 1 . 5 × 10 11 g cm -3 (all results are for t = 0 . 03 s of the torus evolution; the models we compare are SBH3, SBH4 and SBH5). Similar trend in density is found for other torus to black hole mass ratios. The temperature of the torus is roughly similar for all the β values we tested and T max ≈ 1 . 2 × 10 11 K, however the jets are cold only for the highest β . The latter might be affected by numerical effects, so we do not analyze the jets structure here. For the largest β init we tested, the contrast between the highly magnetized polar jets and weakly magnetized disk is most pronounced. For smaller β , we have a region of mildly magnetized flow in the intermediate latitudes. The speed of evolution of the disk also depends on β and the shortest relaxation time is for the model with smallest β init , because the viscous time scale is small in this case. On the other hand, the large β means that the magnetic field is weak and therefore the action of magnetic dynamo most quickly dies out. Also, the accretion rate on average is larger for small β , i.e. the accretion rate correlates with the viscosity, the same as in a standard accretion disk. We compared the accretion rates for several values of β parameter. We noticed that the flares are higher when β decreases, so for the most magnetized plasma we studied, the accretion rate can reach even 10 M s -1 . /circledot In Figure 8 we show the neutrino luminosity for β = 100. The general evolution of the luminosity does not depend on β , so the maximal neutrino luminosity is reached at time ∼ 0 . 01, and then L ν slowly decreases. The value of the maximum luminosity exceeds 2 × 10 53 erg s -1 . This value does not depend significantly on β parameter and the differences (see Table 1) should be attributed mainly to numerical uncertainties (see Section 3.1.2). The Figure 8 shows the simulation up to time 4000 M (model SBHlb). for the initial configuration, estimated as the ratio between the total thermal energy and neutrino luminosity, is in this model equal to 0.12 s, while in the models SBH4 and SBH5 ( beta = 10 and β = 5), it is equal to τ ν ≈ 0 . 05 -0 . 07 s.", "pages": [ 7 ] }, { "title": "3.4. Comparison to the models without neutrino cooling", "content": "The torus around the spinning black hole at hyperEddington rates is cooled by neutrinos and in the 1-D simulations the neutrino cooling effects were studied e.g., by Janiuk et al. (2007); Chen & Beloborodov (2007). To quantify the effect of neutrino cooling in 2D MHD simulations, we ran a test model with no cooling assumed. In Figure 9 with a thin dashed line we plot the accretion rate as a function of time for an exemplary model without neutrino cooling. The average accretion rate onto black hole is lower in these models than in the cooled models, for the same black hole spin and magnetic field. Decreasing the β parameter, i.e. increasing the viscosity, results in the increase of the accretion rate, similarly to the α -disks. The density of the disk in the models without cooling is smaller in the equatorial plane, the disk being less compact (i.e., less dense and geometrically thicker) and hotter than in the neutrino-cooled disks. The disk with- out cooling is also more magnetized i.e. the ratio of gas to magnetic pressure, β , is on average smaller in the disk. This is because the pressure decreases with smaller density, albeit the higher temperatures in the plasma. The distribution of gas to magnetic pressure in the equatorial plane is shown in Figure 10. The maps of the density, temperature and magnetic field are shown in Figure 11. Also, the thickness of the torus, measured by the pressure scale height at the equator, is larger in case of no neutrino cooling, as shown in the example in Figure 12. The ratio of H/r is about 0.3-0.5 in the model without neutrino cooling, and it is 0.1-0.3 in the cooled disk (initial approximation of H = 0 . 5 r was used to compute the neutrino opacities). To sum up, the mass accretion rate remains similar, but the structure of the disk changes, compared to the torus evolving with no neutrino cooling: the disk is geometrically thinner and more magnetized.", "pages": [ 7, 8 ] }, { "title": "3.5. Comparison with 1-dimensional models", "content": "In this section, we quantify the effects of 2-dimensional GR MHD approach with respect to the simplified 1-D neutrino cooled torus model (Janiuk et al. 2007) and compare the 1D and 2D models. The 1-D model is parametrized by the black hole mass, spin and α viscosity. To compare its results with the relaxed model in 2-D simulations, we set these parameters to 3 M /circledot , 0.98 and 0.1, respectively, which corresponds to the SBH5 2-D model in the Table 1. The accretion rate is taken equal to 0 . 17 M /circledot s -1 which is the mean acretion rate computed after evolving the 2-D model. The structure of the disk in our 1-D model is calculated assuming the zero-torque boundary condition at the marginally stable circular orbit. Its location is dependent on the black hole spin, according to the formulae by Bardeen (1970) (see Janiuk & Yuan (2010) Eq. (17)). This condition is used for standard α -disks and does not apply in the MHD simulations. The total mass of the torus, calculated up to 50 r g , is computed from integration of the converged surface density profile. The resulting value is of the same order as that assumed in the 2-D calculations by defining the location of the pressure maximum, the difference being mainly due to lower density in the inner ∼ 6 R g of the 2-D model equatorial plane. The viscosity in the 1-D simulations was parametrized by means of the Shakura & Sunyaev (1973) α constant. In the 2-D model, the viscosity is due to the magnetic turbulence, as parametrized with an initial value of β inside the torus and infinite outside it, and then depending on the location and evolving in time. The angular momentum is transported outwards due to magneto-rotational turbulence. In consequence, no constant value of viscosity is obtained, but after the initial conditions imposed by β init = P gas /P mag are relaxed, the system slowly converges to a value β = u ( γ -1) 1 / 2 B 2 , which approximately corresponds to α via the relation α ≈ 1 / (2 β ). This approximate relation might be verified with a 3-D model of the magneto-rotational instability with Maxwell and Reynolds stresses computed directly. The 2-dimensional structure of the torus is basically consistent with the results of 1-D models. The results are shown in Figure 13. The equatorial density profiles have the same average slopes and normalisations are within the same order of magnitude, up to 20 r g , however they differ due to the types of boundary conditions. The temperature profiles have the same slopes in 1-D and 2-D equatorial plane. Their relative normalisations differ only slightly and they depend mostly on α value. We note that in the 2-D models the temperature is more sensitive to resolution, as the MHD turbulence is better resolved. The neutrino cooling profiles in the 1-D and 2-D models are similar within 2 orders of magnitude. At inner parts of the tori the boundary conditions are different, and at outer parts the neutrino emissivity in 2-D model decreases due to drop in density and temperature. Close to the inner edge of the torus, the emissivity in the 2-D model strongly varies, because of the magnetic turbulence and thermal flickering, which was not accounted for in the 1-D model.", "pages": [ 8, 9 ] }, { "title": "4. SUMMARY AND DISCUSSION", "content": "We calculated the structure and short-term evolution of a gamma ray burst central engine in the form of a turbulent torus accreting onto a black hole. We studied the models with a range of value of the black hole spin, its mass to the torus mass ratio and magnetization. We found that (i) in the 2-dimensional computations, the neutrino cooling changes the torus structure, making it denser, geometrically thinner and less magnetized; (ii) the total neutrino luminosity reaches 10 53 -10 54 erg s -1 , for the torus to black hole mass ratio 0.03-0.1, and the time of its peak anticorrelates with the black hole spin; (iii) at the end of the simulation, t ∼ 0 . 03 or t ∼ 0 . 1 s for smaller or larger black hole, the neutrino luminosity is about 10 52 -10 53 erg s -1 , increasing with black hole spin; this is by 1-2 orders of magnitude larger than the Blandford-Znajek luminosity of the jets computed in our models; (iv) the neutrino cooled torus launches a fast, rarefied wind that is responsible for a powerful mass outflow, correlated with the torus to black hole mass ratio; (v) the contribution of the wind to the total neutrino luminosity is on the order of 10% and correlates with its mass; (vi) the density and temperature profiles in the equatorial plane of the 2-dimensional MHD torus are well reproduced by the vertically averaged profiles calculated in the 1-dimensional α -disk model, however in the latter case the torus is cooler by a factor of 1.5-2; (vii) the neutrino cooling rates are similar for the inner ∼ 20 -30 R g in the 1D and 2D calculations. The structure of the central engine we modeled is relevant for any gamma ray burst, the free parameters being mainly the black hole spin and initial magnetic field strength. Without neutrino cooling, all the results scale with the black hole mass and the assumed mass and size of the initial torus. Here we have shown only the short timescale calculations, with no extra inflow of matter to the outer edge of the disk, which would be relevant for the subclass of long GRBs central engines. The internal structure of the torus should not depend on that, as supported e.g. by the recent observations by Swift showing that flares in both short and long GRBs are likely produced by the same intrinsic mechanism (Margutti et al. 2011). In the short GRB models, during the evolution of the post-merger disks the rings of material of a mass between 0.01 and 0.1 M /circledot can fall back from the eccentric orbits. In this way, the neutrino luminosity may brighten a few times on a timescale of > 1 second (Lee et al. 2009). Mass fallback from the stellar envelope material is also a key feature of the collapsar model for the long GRBs. The mass of the torus assumed in most of our models is about 0.1-1.0 M /circledot , when the black hole mass is fixed at 3 or 10 M /circledot . A more massive torus, which can form in the center of a massive star as a 'collapsar' central engine, would result in accreting a substantial amount of mass and angular momentum onto the black hole. Therefore the evolution of the black hole mass and spin should consistently be taken into account, as shown e.g. by (Janiuk, Moderski & Proga 2008). This is currently neglected in our calculations, and we focus on the torus much less massive than the accreting black hole, M torus /M BH ≤ 0 . 25. This is still relevant for the compact binary merger scenario. The initial conditions used in our models, similarly to other simulations, is based on the equlibrium torus solution and embedded magnetic field of a specified topology and stregnth. The recently simulated mergers of hypermassive neutron stars (e.g. Shibata et al. (2011)) follow the evolution of matter and electromagnetic energy ejection during several tens of milliseconds and show that already at this stage the toroidal magnetic field component is developed and relativistic outflows occur. Then, it is expected that the neutron star will eventually collapse to a black hole, after a substantial loss of the angular momentum due to the gravitational wave emission, and the transient torus with a lifetime of about 100 milliseconds will power the GRB engine. Our simulation covers this last stage of the event; obviously conditions for initial magnetic field are mostly artificial at t=0. However, the toroidal field forms in our computations really quickly, i.e. after one orbit, and the evolution of the neutrino luminosity and flares should match then the outcome of the former compact object merger. The black hole-neutron stars merger simulations (for a review see Shibata & Taniguchi (2011)) lead mostly to the formation of a massive black hole with a remnant disk of less than 10 % of the total inital mass of the binary. Its density depends on the initial mass ratio and primary BH spin, as well as on the neutron star's EOS. The final BH spin is determined mostly by its initial value. Overall, the coalescence of high mass ratio binaries with a ≤ 0 . 75 is a promissing channel for a short GRB progenitor, forming a massive disk plus BH system. Our simulations are aimed to realize this scenario. More detailed studies of the dynamical evolution of the post-merger system, with initial conditions based on the direct output of the merger simulations rather than the quasi-steadystate torus, are planned for our future work (see e.g. by Schwab et al. (2012) for the post-merger evolution of binary white dwarfs). The distribution of the compact binaries from the population synthesis models shows two peaks: double black holes constitute about two-thirds of the popula- tion, while the double neutron star binaries are about 28% (Belczynski et al. 2010). The remaining pairs can contain a low mass black hole and a neutron star system. However, as recently computed by Dominik et al. (2011; in preparation), the most compact binary pairs contain a neutron star and a black hole of mass 7-13 M /circledot . The details of the mass distribution depend on the evolutionary scenario (presence of the common envelope phase) and are sensitive to the assumed metallicity. Therefore, a plausible short GRB scenario may involve a 3 M /circledot black hole with a small disk, as well as a black hole of M BH = 10 M . /circledot The luminosity of the torus is comparable to that obtained from relativistic hydrodynamical simulations (Jaroszynski 1993, 1996; Birkl et al. 2007). Also, the relativistic MHD simulations by Shibata et al. (2007) reported the neutrino luminosity on the order of L ν ∼ 10 54 erg s -1 , depending on black hole spin ( a ≤ 0 . 9) and torus mass. To compute the electromagnetic luminosity of the observed GRBs, one needs to consider the efficiency of neutrino-antineutrino annihilation process, as well as swallowing of some fraction of neutrinos by the black hole due to the curvature effects. Most of the neutrinos are formed within 10 R g . The luminosity obtained in our simulation will lead to the annihilation luminosity on the order of L ν ¯ ν ≈ a few times 10 50 erg s -1 (Zalamea & Beloborodov 2011), providing an additional energy reservoir to power the GRB jet. This is on the same order of magnitude as the BlandfordZnajek luminosity in the polar jets. The jet power can be calculated from our models by integrating the electromagnetic energy flux on the black hole horizon over the surface area (McKinney & Gammie 2004). Depending on black hole spin it reaches the values in the range of L BZ ∼ 4 × 10 50 -3 × 10 52 erg s -1 , consistently with other estimates (Lee et al. 2000; Komissarov & Barkov 2009). For the same black hole spin and magnetic β parameter, the models with neutrino cooling give about a factor of two smaller L BZ than the non-cooled models. Our results show that the disks around larger mass black holes are in general less dense and cooler, for the same black hole spin and accretion rate. They are however brighter in neutrinos, as their peak luminosity scales directly with mass. The wind outflows launched form the surface of the accreting torus are driven by magnetic pressure which can also halt the accretion rate onto black hole. The wind is bright in neutrinos, giving an additional contribution to the total luminosity of the system. The general relativistic simulations that ignore the radiative (and neutrino) cooling have recently been discussed e.g. in ref McKinney, Tchekhovskoy & Blandford (2012). They discuss various topologies and stregths of initial magnetic field and confirm that the value of initial β parameter affects the final, or time-averaged, viscosity. The latter might be to some extent verified by the observations of accreting X-ray sources (see King et al. (2007)), to help determine on whether the α scales with only magnetic or the total pressure. We note that in our simulations the limitations of assumed axisymmetry in the model do not allow to fully constrain effective α . The simulations presented in Krolik et al. (2005) show the existence of the polar jet outflows. The authors do not discuss massive winds, as they concentrate mostly on the accretion disk properties. However, McKinney (2006) reports on the existence of winds with moderately relativistic velocities (Γ ∼ 1 . 5) and half opening angles of 16-45 · . The results shown in this work are obtained with a detailed neutrino cooling description in which we have incorporated the chemical composition of nuclear matter where the reactions lead to the neutrino production (Janiuk et al. 2007). The simulations discussed in Dibi et al. (2012) include the radiative cooling for low luminosities and accretion rates, appropriate for the case of radiativily inefficient flows in AGN. The scale height of the disk in their results is affected by the radiative cooling by a factor of 30-50 per cent, however the density and thickness of the inner torus might still be partly affected by the initial conditions assumed in these simulations. Qualitatively, our results are similar to theirs, as the neutrino cooling also leads to the denser and thinner torus inside 10-15 gravitational radii. The 'bump' outside that radius, seen in the final snapshots from our simulations, may partly also be affected by initial conditions. However, the difference may also arise because of a stronger radial dependence of neutrino cooling than it is in the case of photon cooling. Similarly to Dibi et al. (2012), our dynamical model uses a simplified version of EOS. We note that the electrons are degenerate near the disk equatorial plane between the BH horizon and r ≈ 20 R g , e.g. in the model SBH2. In this small region, the dynamical computations with γ = 4 / 3 might not be suitable to describe the degenerate electrons (see Barkov & Komissarov (2008, 2010)). To model degenerate gas one could introduce a new equation of state (e.g. P = P ( ρ 0 ) ρ 1 0 /n where ρ 0 is the density of the electrons and n is a politropic index (see Paschalidis et al. 2011; Malone et al. 1975). The latter however is a mayor change of the numerical scheme since the matter is composed of also partially degenerate and nondegenerate electrons, protons, helium nuclei and neutrons which can still be described by perfect gas law. Moreover, to account for the pressure of photons and neutrinos one would need to follow the evolution of radiation and neutrino energy-momentum tensor coupled to the evolution of matter. Sill, in our present model the energy carried out from the system by the neutrinos does not depend on the EOS used in the interior of the disk and most of the energy is generated in the disk wind. Of course, it is possible that the change of the EOS would influence the wind strength, structure and neutrino luminosity. It would be interesting to explore the wind launching mechanism in this case and we plan to study this in future work.", "pages": [ 9, 10, 11 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank Chris Belczynski, Michal Dominik, Bozena Czerny and Marek Sikora for helpful discussions. We also thank the anonymous referee for insightful comments. This research was supported in part by grant NN 203 512638 from the Polish Ministry of Science and Higher Education.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "King A. R., Pringle J. E., Livio M., 2007, MNRAS , Klebesadel, R.W., Strong, I. B., Olson, R.A., 1973, ApJL , Kohri, K., Mineshige, S., 2002, ApJ , Komissarov S.S., Barkov M.V., 2009, MNRAS , Kouveliotou, C., et al., 1993, ApJL , 413 , 101 Lopez-Camara D., Lee W.H., Ramirez-Ruiz E., 2009, ApJ , 804 577 , 311 397 , 1153 376 , 1740 182 , 85 692 ,", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "Following our recent paper (Petrosian & Chen 2010), we have developed an inversion method to determine the basic characteristics of the particle acceleration mechanism directly and nonparametrically from observations under the leaky box framework. In the above paper, we demonstrated this method for obtaining the energy dependence of the escape time. Here, by converting the Fokker-Planck equation to its integral form, we derive the energy dependences of the energy diffusion coefficient and direct acceleration rate for stochastic acceleration in terms of the accelerated and escaping particle spectra. Combining the regularized inversion method of Piana et al. (2007) and our procedure, we relate the acceleration characteristics in solar flares directly to the count visibility data from RHESSI . We determine the timescales for electron escape, pitch angle scattering, energy diffusion, and direct acceleration at the loop top acceleration region for two intense solar flares based on the regularized electron flux spectral images. The X3.9 class event shows dramatically different energy dependences for the acceleration and scattering timescales, while the M2.1 class event shows a milder difference. The M2.1 class event could be consistent with the stochastic acceleration model with a very steep turbulence spectrum. A likely explanation of the X3.9 class event could be that the escape of electrons from the acceleration region is not governed by a random walk process, but instead is affected by magnetic mirroring, in which the scattering time is proportional to the escape time and has an energy dependence similar to the energy diffusion time. Subject headings: acceleration of particles - Sun: flares - Sun: X-rays, gamma rays", "pages": [ 1 ] }, { "title": "DETERMINATION OF STOCHASTIC ACCELERATION MODEL CHARACTERISTICS IN SOLAR FLARES", "content": "Qingrong Chen and Vah'e Petrosian Department of Physics, Stanford University, Stanford, CA 94305, USA Draft version June 19, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Solar flares are a complex multiscale phenomenon powered by the explosive energy release from non-potential magnetic fields through reconnection in the solar corona. The total energy released in a large flare can reach up to ∼ 10 32 -10 33 erg within ∼ 10 2 -10 3 s and ∼ 10-50% of this energy goes into acceleration of electrons and ions to relativistic energies in the impulsive phase (Lin & Hudson 1976; Lin et al. 2003; Emslie et al. 2012). In particular, the suprathermal electrons produce hard X-ray (HXR) emission up to a few hundred keV through the well understood bremsstrahlung process (Lin 1974; Dennis 1988; Krucker et al. 2008b; Holman et al. 2011). HXR observations in the past two decades from the Yohkoh /Hard X-ray Telescope and the Reuven Ramaty High Energy Solar Spectroscopic Imager ( RHESSI ; Lin et al. 2002; Hurford et al. 2002) have significantly advanced our understanding of electron acceleration in solar flares. Detection of distinct coronal HXR sources located near the top of the flare loop in addition to the commonly seen footpoint (FP) sources (e.g., Masuda et al. 1994; Aschwanden 2002; Petrosian et al. 2002; Battaglia & Benz 2006; Krucker et al. 2008a; Ishikawa et al. 2011; Chen & Petrosian 2012; Sim˜oes & Kontar 2013) has revealed that the primary electron acceleration takes place in the corona with an intimate relation to the energy release process by magnetic reconnection. More recently RHESSI further observed a second coronal X-ray source located above the loop top (LT) source. The higher energy emission of the two coronal sources is found to be closer to each other (e.g., Sui & Holman 2003; Sui et al. 2004; Liu et al. 2008; Chen & Petrosian 2012; Liu et al. 2013). This property, complemented with the extreme ultraviolet (EUV) observations of the context (e.g., Wang et al. 2007), further suggests that electron acceleration occurs most likely in the reconnection outflow regions, rather than in the current sheet (Holman 2012; Liu et al. 2013). Several different acceleration mechanisms may operate in the outflow regions as a result of the explosive energy release by reconnection, involving either the kinetic effects of small amplitude electromagnetic fluctuations or the spatial-temporal variations of the large scale magnetic fields. Among these mechanisms, the model of stochastic acceleration (SA), also known as the second-order Fermi process (Fermi 1949), has achieved considerable success in interpreting the high energy features of solar flares (e.g., Miller et al. 1997; Petrosian 2012). Resonant interactions of particles with a broad spectrum of plasma waves or turbulence in the corona, presumably excited by the large scale outflows from the reconnection region, lead to momentum diffusion and pitch angle scattering of particles (Sturrock 1966; Tsytovich 1966, 1977; Tversko ˇ i 1967, 1968). Several variants of the SA mechanism have been applied to acceleration of electrons and ions in solar flares (e.g., Melrose 1974; Barbosa 1979; Ramaty 1979; Ryan & Lee 1991; Hamilton & Petrosian 1992; Steinacker & Miller 1992; Miller & Roberts 1995; Miller et al. 1996; Park et al. 1997; Petrosian & Liu 2004; Emslie et al. 2004; Liu et al. 2006; Grigis & Benz 2006; Bykov & Fleishman 2009; Bian et al. 2012; Fleishman & Toptygin 2013). Resonant pitch angle scattering, a necessary prerequisite for efficient acceleration (e.g., Tversko ˇ i 1967; Miller 1997; Melrose 2009), increases the time electrons stay at the LT acceleration region (e.g., Petrosian & Donaghy 1999), before they escape to the thick target FPs of the flare loop. This enhances the HXR radiation at the coronal LT region and naturally explains the aforementioned HXR morphological structure associated with the flare loop. Particle spectra resulting from SA by turbulence are generally described by the so-called leaky box version of the Fokker-Planck kinetic equation (e.g., Ramaty 1979; Steinacker & Miller 1992; Park & Petrosian 1995; Petrosian & Liu 2004). The accelerated and escaping electron spectra and the resulting bremsstrahlung HXR spectra at the LT and FPs are found to be sensitive to the turbulence spectrum and the background plasma properties (Petrosian & Donaghy 1999; Petrosian & Liu 2004). There have been continued efforts to constrain the wave-particle interaction coefficients and the property of turbulence from solar flare HXR (and γ -ray) observations, mainly through a parametric forward fitting procedure (Hamilton & Petrosian 1992; Park et al. 1997; Liu et al. 2009). Although there has been systematic theoretical modeling of the SA mechanism in attempt to explain the spectral features of RHESSI HXR observations (Petrosian & Liu 2004; Grigis & Benz 2006), the spatially resolved imaging spectroscopic data from RHESSI have been rarely under direct quantitative comparison to constrain the SA model characteristics. By taking advantage of the recently developed electron flux spectral images (Piana et al. 2007) via regularized inversion from the RHESSI count visibility data, Petrosian & Chen (2010) initiated direct determination of the SA model characteristics from the radiating electron flux spectra at the LT and FP regions. We have derived the energy dependences of the escape time and pitch angle scattering time. In this paper, by fully utilizing the leaky box Fokker-Planck equation describing the acceleration process, we further derive the energy dependence of the energy diffusion coefficient, which also gives the direct acceleration rate by turbulence, directly and non-parametrically from the spatially resolved electron spectra in solar flares. This provides a complete determination of all unknown SA model quantities in the Fokker-Planck equation. In the next section, we present the equations describing the particle acceleration, transport, and radiation processes. In Section 3, we show the general determination of the escape time and how the inversion of the Fokker-Planck kinetic equation leads to the energy diffusion coefficient in terms of purely observable quantities. In Section 4, we apply the formulas to two RHESSI solar flares to determine the SA model characteristics based on the electron flux images. In the final section we give a brief summary and discuss implications of the results for acceleration and transport of electrons in solar flares.", "pages": [ 1, 2 ] }, { "title": "2. ACCELERATION, TRANSPORT, AND RADIATION", "content": "In this paper, we are interested in the spatially averaged characteristics of the mechanism accelerating the background thermal particles. For a homogeneous acceleration region these would give the actual values. In application to solar flares, the acceleration region with volume V , cross section A , and size L = V/ A , which we assume to be consisted of a fully ionized hydrogen plasma with background number density n LT , would be embedded at the apex of the flare loop. We define a free stream- ing time across the acceleration region as τ cross = L/v . We assume that this region contains a certain level of turbulence to scatter and accelerate particles.", "pages": [ 2 ] }, { "title": "2.1. Leaky Box Acceleration Model", "content": "The very complex details of particle diffusion in the momentum space due to wave-particle interactions are most commonly illuminated by the quasilinear theory (Kennel & Engelmann 1966; Schlickeiser 1989, and references therein), through the momentum and pitch angle diffusion coefficients, namely D pp , D pµ , and D µµ . However, acceleration by turbulence (and some other mechanisms, e.g., shocks) requires a pitch angle scattering time ( τ scat ∼ 1 /D µµ ) that is much shorter than other timescales. As a result, particles rapidly attain a nearly isotropic distribution, and instead of free streaming, they diffuse out of the accelerator via a random walk process. By translating the spatial diffusion into an escape term from the accelerator, and transforming from the momentum space to the energy domain, the evolution of the particle distribution function N ( E,t ), averaged over the pitch angle and integrated over the physical space, is conventionally described by the leaky box model. We use the following slightly modified variant of the leaky box Fokker-Planck equation 1 (Park & Petrosian 1996; Petrosian 2012), which is more convenient for our purpose here, where D EE and A ( E ) are the energy diffusion coefficient and the acceleration rate (due to turbulence and all other interactions), respectively, ˙ E L ( E ) is the energy loss rate, and ˙ Q ( E ) and N ( E ) /T esc ( E ) are the rates of injection of seed particles and escape of the accelerated particles, respectively. The energy diffusion coefficient by turbulence is If turbulence is the only agent of acceleration, then the acceleration rate is (Petrosian 2012) where γ is the Lorentz factor. The escape time is related to the spatial diffusion of particles along the magnetic field lines, which depends on the pitch angle scattering time τ scat as (e.g., Schlickeiser 1989; Steinacker & Miller 1 From the standard form of the Fokker-Planck formalism (Chandrasekhar 1943), ∂ ∂t N = ∂ 2 ∂E 2 [ D EE N ] -∂ ∂E [ A d ( E ) N ], where D EE ≡ 〈 (∆ E ) 2 2∆ t 〉 and A d ( E ) ≡ 〈 ∆ E ∆ t 〉 = A ( E ) + dD EE dE for SA only, it is easy to show that the total energy of the accelerated particles E ( t ) = ∫ ∞ 0 EN ( E,t ) dE varies with time as d dt E = ∫ ∞ 0 A d ( E ) NdE . Thus, it is A d ( E ), rather than A ( E ), that gives the actual energy gain rate or direct acceleration rate (Tsytovich 1966, 1977; Ramaty 1979). Insertion of A d ( E ) into the above equation yields a form of Equation (1), the steady state of which is a first-order (instead of second-order) ordinary differential equation for D EE . 1992; Petrosian 2012) The above relation is valid when the scattering time is much shorter than the crossing time. We further add τ cross to the escape time, which extends its validity to the opposite case and assures that the escape time is longer than the crossing time. For further discussions about the above equations, see Petrosian & Liu (2004) and Petrosian (2012). The effect of the geometry of the large scale magnetic fields on the escape time will be discussed in Section 5. For solar flare X-ray radiating electrons below a few MeV, the energy loss rate ˙ E L is dominated by Coulomb collisions with the background electrons, where n is the background electron density, r 0 is the classical electron radius with 4 πr 2 0 = 10 -24 cm 2 , and ln Λ is the Coulomb logarithm taken to be 20 for solar flare conditions. In most astrophysical systems, in particular in solar flares, the dynamic timescale is generally much longer than the acceleration and other timescales, then it is justified to treat the steady state leaky box equation. Solution of this equation provides the spectrum and the escape rate of the accelerated particles, N ( E ) and N ( E ) /T esc , respectively, or equivalently, the accelerated and escaping flux spectra (in units of particles cm -2 s -1 keV -1 ), From the above particle spectra, we can obtain the escape time as and from Equations (5 and 4), we can obtain the pitch angle scattering time τ scat and the averaged pitch angle diffusion rate 〈 D µµ 〉 .", "pages": [ 2, 3 ] }, { "title": "2.2. Particle Transport", "content": "The escape rate N ( E ) /T esc serves as the seed source ˙ Q tr for the subsequent transport of particles outside the acceleration region. If the particles lose all their energy in the transport region, i.e., we are dealing with a thick target process, then the volume integrated particle spectrum is governed by the steady state transport kinetic equation (Longair 1992), where ˙ E tr L is the energy loss rate at the thick target transport region. Then solution of this equation gives rise to the effective thick target radiating particle spectrum (Longair 1992; Johns & Lin 1992), For Coulomb collisions in solar flares, the energy loss rate ˙ E tr L should be evaluated with the mean density n tr from the loop legs to FPs.", "pages": [ 3 ] }, { "title": "2.3. Bremsstrahlung HXR Radiation", "content": "In solar flares, the accelerated and escaping electrons produce bremsstrahlung HXR emission along the flare loop, for which the angle-averaged differential photon flux (in units of photons s -1 keV -1 ) is written as a linear Volterra integral equation of the first kind, where σ ( /epsilon1, E ) is the angle-averaged bremsstrahlung cross section. The quantity X ( E ) represents the integration over the volume of interest of the electron flux spectrum F ( E,s ) multiplied with the background proton density n ( s ), where A ( s ) is the cross section of the loop along the magnetic field lines. In what follows, we refer to X ( E ) as the volume integrated radiating electron flux spectrum. Thus at the LT acceleration region, The transport of the escaping electrons from the loop legs to FPs is described by the classical thick target model (Brown 1971; Syrovat-Skii & Shmeleva 1972; Petrosian 1973). The radiating electron flux spectrum integrated over the whole thick target, but mainly at the FPs, produced by the escaping electrons is given by (e.g., Park et al. 1997) where N tr eff is given by Equation (10). It should be noted that as a result of the density dependence of the Coulomb energy loss rate, the thick target radiating electron spectrum X FP and consequently the bremsstrahlung photon spectrum J FP are independent of the thick target density profile (e.g., Syrovat-Skii & Shmeleva 1972; Park et al. 1997). As explained below, the volume integrated radiating electron flux spectra X LT and X FP can be obtained directly and non-parametrically from RHESSI data.", "pages": [ 3 ] }, { "title": "3. DETERMINATION OF MODEL QUANTITIES", "content": "The two unknown diffusion coefficients in the SA model are D ( p ) and D µµ , or equivalently, D EE and T esc , which we aim to determine from observations. In comparison, one has relatively good knowledge or estimate of the energy loss rate ˙ E L and the source term ˙ Q , which depend primarily on the background medium properties. Therefore, given the accelerated and escaping particle flux spectra F acc and F esc from observations, in particular, X LT and X FP from solar flares, we can in principle determine the two unknown model quantities.", "pages": [ 3 ] }, { "title": "3.1. Escape Time", "content": "Equation (15) for the escape time as As already indicated above, one can in general determine the first unknown quantity, namely, the escape time, simply from the ratio between F acc and F esc (Equation 8), or alternatively from the ratio between N ( E ) and N ( E ) /T esc . For solar flare bremsstrahlung, on the other hand, we deal with a thick target transport process and the escaping electrons produce an effective radiating spectrum N tr eff . By differentiating Equation (10), we then determine the escape time as (see also Petrosian & Chen 2010) Note that for Coulomb collisions in a cold target, we have d ln ˙ E tr L d ln E = -1 γ 2 + γ /similarequal -1 2 at the non-relativistic limit.", "pages": [ 4 ] }, { "title": "3.2. Energy Diffusion Coefficient", "content": "Now we are left with the second unknown quantity, namely, the energy diffusion coefficient, and it turns out that the derivation for D EE is very simple. By using the relation between A ( E ) and D EE (Equation 3), we rewrite the steady state leaky box equation as below, Integration of the above equation from E to ∞ gives For particle energies far above the injection energy, acceleration results in N/T esc /greatermuch ˙ Q , so that ˙ Q can be ignored from the above equation. Therefore, given the escape time T esc as determined above, we can derive the formula for D EE once again purely in terms of observables. This formula can be further simplified for the thick target transport model. The integral inside the square brackets is related to the effective thick target radiating spectrum for the escaping particles (Equation 10). Thus we express D EE as In summary, by differentiating the effective thick target radiating spectrum due to the escaping particles and converting the Fokker-Planck equation to the integral form, we can express the unknown model quantities, the escape time T esc and the energy diffusion coefficient D EE , purely in terms of observables with minimal assumptions.", "pages": [ 4 ] }, { "title": "3.3. Solar Flare Radiating Electron Spectra", "content": "For Coulomb collisional energy loss in solar flares, we have ˙ E tr L / ˙ E L = n tr /n LT and the following relation, Thus, in terms of the volume integrated radiating electron flux spectra X LT and X FP in solar flares, we rewrite and Equation (18) for the energy diffusion coefficient as 2 where τ L = E/ ˙ E L is the energy loss time at the LT acceleration region (with density n LT ). We further define the energy diffusion time due to turbulence as τ diff = E 2 / 2 D EE and direct acceleration time as τ acc = E/A d (see Footnote 1).", "pages": [ 4 ] }, { "title": "3.4. Interplay between Timescales", "content": "The shape of the accelerated electron spectrum is a result of the interplay between the competing processes involved in the leaky box Fokker-Planck Equation (1). Conversely, we can gain some insight into these physical processes from the electron spectra as we have shown above. Both T esc and D EE primarily depend on the ratio X LT /X FP . By eliminating X LT /X FP from Equations (20 and 21), we relate the timescales for the physical processes as below, where η LT = -d ln X LT d ln E + 2 γ γ +1 and η FP = -d ln X FP d ln E + 2 γ 2 + γ . If the (non-relativistic) X-ray radiating electron spectra X LT and X FP in solar flares are nearly power laws, then both η LT and η FP vary very slowly with energy. We now consider two extreme cases. On the one hand, if X LT /X FP /lessmuch 1, which is applicable to most flare observations, then we have η FP T esc /lessmuch τ L and roughly τ acc ∼ τ diff /similarequal ( η LT η FP / 2) T esc . On the other hand, if X LT /X FP /greatermuch 1, which is representative for a few very rare events with an extremely bright LT source, then we have η FP T esc /greatermuch τ L and τ acc ∼ τ diff /similarequal ( η LT / 2) τ L .", "pages": [ 4 ] }, { "title": "4. APPLICATIONS TO RHESSI OBSERVATIONS", "content": "The volume integrated radiating electron flux spectra X LT and X FP have been generally inferred from the HXR spectra of the LT and FP sources using the Volterra integral Equation (11), which is an ill-posed inverse problem and there are no unique solutions. This is commonly carried out by a forward fitting procedure (e.g., Holman et al. 2003), but there have also been attempts to determine these electron flux spectra by the inversion of this equation (Brown et al. 2006). Several methods, such as analytic solution (Brown 1971), matrix inversion (Johns & Lin 1992), and regularized inversion (Piana et al. 2003; Kontar et al. 2005) have been used for this task. Here we use the more recent and direct procedure described below.", "pages": [ 4 ] }, { "title": "4.1. Regularized Electron Imaging Spectroscopy", "content": "Piana et al. (2007) noted that the most fundamental product of the temporal modulation from RHESSI as . a Fourier imager is the count visibilities, the Fourier components of the source spatial distribution, which are related via essentially the same Volterra Equation (11) to the electron flux visibilities, the Fourier components of the so-called electron flux spectral images. By the same regularized inversion method as mentioned above, Piana et al. (2007) first inverted the electron flux visibility spectrum from the count visibility spectrum. This requires knowledge of the bremsstrahlung cross section and the detector response function. Then by applying visibility-based imaging algorithms to the these visibilities, they reconstructed the images of the mean radiating electron flux multiplied by the column depth N ( x, y ) along the line-of-sight, 3 namely, X ( x, y ; E ) = N ( x, y ) F ( x, y ; E ), where x and y are the spatial coordinates. From these electron flux images over a sequence of energy bins, one can then extract the volume integrated radiating electron flux spectra X ( E ) = ∫ X ( x, y, E ) dxdy for spatially separated LT and FP sources of solar flares. With availability of this regularized 'electron' imaging spectroscopy, one can now better constrain the acceleration and transport processes in solar flares (e.g., Prato et al. 2009; Petrosian & Chen 2010; Torre et al. 2012; Guo et al. 2013; Massone & Piana 2013; Codispoti et al. 2013). Torre et al. (2012), assuming a spectrum of accelerated electrons, used a similar integration of the transport equation to determine the energy loss rate along the flare loop. As can be explicitly seen from Equations (20 and 21), simultaneous detection of both the LT and FP sources in solar flares over a wide energy range is essential to determine the SA model characteristics as a function of electron energy. For this purpose, we have carried out a systematic search of high energy events (Chen & Petrosian 2009), for which both the LT and FP emission during the impulsive phase is imaged by RHESSI . We have found a few such events close to the solar limb with the HXR emission detected above 50 keV from both the LT and FP sources. We reconstruct the regularized electron flux images using the MEM NJIT algorithm (Schmahl et al. 2007). In the data analysis performed below, the electron flux spectra X LT and X FP are extracted from the electron images using the Object Spectral Executive (OSPEX; Smith et al. 2002) package of the Solar SoftWare. The fittings in this paper are implemented using a non-linear least squares fitting program, MPFIT, based on the Levenberg-Marquardt algorithm (Markwardt 2009; Mor'e 1977). We apply the above data analysis procedure to the GOES X3.9 class solar flare on 2003 November 3 and the GOES M2.1 class flare on 2005 September 8 and determine the SA model characteristics. In Table 1, we list the basic information of these two flares, and the 3 More exactly, the electron flux spectral images represent a 2 N ( x, y ) F ( x, y ; E ) / 10 50 , where x and y are in units of arcsec and a = 7 . 25 × 10 7 cm arcsec -1 . From these images, the sum of the pixel intensities within one region of interest, after multiplication by the square of the pixel size (in units of arcsec), yields the volume integrated radiating electron flux spectra X ( E ) defined in Equation (12) for that region, in units of 10 50 electrons cm -2 s -1 keV -1 . In the current paper, we have corrected our misinterpretation of the observed electron flux spectra by up to a constant as made in the upper panel of Figure 2 in Petrosian & Chen (2010). Basic information and power law indices of the electron flux spectra ( ∝ E -δ ) and the SA model quantities and timescales ( ∝ E s ) in two RHESSI flares. Note . -a The mean value of the broken power law fitting. Note . -b To calculate τ acc , we approximate the logarithmic derivative of D EE with its power law fitting. power law indices for the radiating electron flux spectra and the SA model quantities and related timescales.", "pages": [ 4, 5 ] }, { "title": "4.2. The 2003 November 3 Event", "content": "The 2003 November 3 solar flare of X3.9 class (Solar Object Locator: SOL2003-11-03T09:43) is an intense solar eruptive event close to the west solar limb. The unusually bright HXR emission from the coronal LT source, detectable up to 100-150 keV along with two FP sources by RHESSI (Chen & Petrosian 2012), makes this event particularly suitable for our purpose to determine the SA model characteristics. Figure 1 shows the regularized electron flux spectral images. The LT and FP sources are clearly visible up to 250 keV, about twice the highest photon energy for the LT source. We then extract the volume integrated radiating electron flux spectra X ( E ) above 34 keV from the LT source and the two FP sources (Figure 2, left panel). The LT spectrum can be fitted by a power law with an index ∼ 3.0, while the flatter FP spectrum can be better fitted by a broken power law with the indices ∼ 2.1 and ∼ 2.8 below and above the break energy ∼ 91 keV, respectively. From analysis of the X-ray images (Chen & Petrosian 2012), we obtain the density at the LT acceleration region to be n LT ∼ 5 × 10 10 cm -3 with the LT size to be L ∼ 10 9 cm. In Figure 2 (right panel), we plot the electron escape time and the energy diffusion time as calculated from the above spatially resolved spectra X LT and X FP (left panel). Except for the leftmost data point at the lowest energy, the escape time T esc is clearly much longer than the crossing time τ cross and is ∼ 5-15 times shorter than the energy loss time τ L . The escape time shows an overall trend increasing with energy, and can be fitted with a power law, T esc ∝ E 0 . 8 . From the escape time, we calculate the pitch angle scattering time, which decreases with energy as τ scat ∝ E -1 . 8 . This implies a pitch angle diffusion rate of D µµ ∝ E 1 . 8 . Here we attribute the above scattering time purely to turbulence. Contribution from Coulomb collisions will be at the scale of the energy loss time and therefore negligible for electrons above 34 keV in this event. The energy diffusion time varies as τ diff ∝ E 1 . 1 and is about half the energy loss time. Thus we have the energy diffusion coefficient D EE ∝ E 0 . 9 . The direct acceleration time τ acc is very close to the energy diffusion time. It is obvious that the energy diffusion time and pitch angle scattering time have very different energy dependences in this event.", "pages": [ 5, 7 ] }, { "title": "4.3. The 2005 September 8 Event", "content": "The 2005 September 8 solar flare (SOL2005-0908T16:49) is an M2.1 class event occurring at the southeast quadrant of the Sun near the limb. As seen from the RHESSI HXR images and the Transition Region and Coronal Explorer ( TRACE ) 171 ˚ A EUV images, the flare consists of two interacting loops, with their northern loop legs visually overlapped along the line-of-sight. Furthermore, the coronal LT source appears higher at altitude with increasing HXR energy (Chen & Petrosian 2009). Here we model the acceleration region associated with the two loops as a single leaky box for the whole flare. We take the density and size of this single accelerator to be 2 × 10 10 cm -3 and 1 . 5 × 10 9 cm, respectively. Figure 3 displays the electron flux images up to 130 keV, in which two flare loops can be clearly resolved. We extract the radiating electron flux spectra at the LT and FP sources summed over the two loops. As shown in Figure 4 (left panel), both the LT and FP spectra can be well fitted by a power law, with the indices ∼ 4.8 and ∼ 3.5, respectively, the difference of which is larger than that in the 2003 November 3 flare. Due to the relatively softer LT source in this event, all the model timescales are flatter than those in the 2003 November 3 flare. As in Figure 4 (right panel), the escape time can be fitted with a power law, T esc ∝ E 0 . 2 . The scattering time varies as τ scat ∝ E -0 . 9 , and the pitch angle diffusion rate as D µµ ∝ E 0 . 9 . The energy diffusion time and acceleration time can be fitted with a similar power law, τ diff ∝ τ acc ∝ E 0 . 5 . The energy diffusion coefficient is found to be D EE ∝ E 1 . 5 . Again, the energy dependences for the energy diffusion time and the pitch angle scattering time are very different, but now the difference is smaller than that in the 2003 November 3 flare.", "pages": [ 7 ] }, { "title": "4.4. Numerical Verification", "content": "For the above two events, we also use the power law forms of the escape time T esc and the energy diffusion coefficient D EE determined directly from observations as input to the steady state leaky box Fokker-Planck Equation (1), and solve for the electron spectra N ( E ) numerically using the Chang-Cooper finite difference scheme (Chang & Cooper 1970; Park & Petrosian 1996). We then calculate the effective thick target radiating spectra for the escaping particles. As shown in Figures 2 and 4, these numerical model spectra in general agree very well with the observed spectra from RHESSI . This is a mere self-consistency check justifying our procedure.", "pages": [ 7 ] }, { "title": "5. SUMMARY AND DISCUSSIONS", "content": "Following our earlier paper (Petrosian & Chen 2010), we have developed a new method for the determination of the energy dependences of basic characteristics of the SA mechanism. The particle spectrum is determined by three such characteristics, but only two of them, namely the momentum and pitch angle diffusion coefficients ( D pp and D µµ ), play a major role. As is well known, for a nearly isotropic particle distribution at a homogeneous acceleration region, the diffusion equation in the momentum space reduces to the so-called leaky box FokkerPlanck equation, including the energy diffusion coefficient and direct acceleration rate ( D EE and A d ( E ) related to D pp ) and escape time ( T esc related to D µµ ). We show that from the observed spectra of the accelerated and escaping particles, we can determine these two unknowns of the SA mechanism by inversion of the particle acceleration and transport equations and thus gain insight into the properties of the required plasma waves or turbulence. It should be noted that, in contrast to the usual forward fitting method, this is a non-parametric method of relating the acceleration coefficients directly to observables. In this paper we have applied the above procedure to acceleration of electrons in solar flares based on RHESSI HXR observations, assuming mainly SA by turbulence with negligible contribution to acceleration from electric fields or shocks. 4 Here we also employ the regularized inversion procedure of Piana et al. (2007) to produce the electron flux spectral images from the RHESSI count visibilities, thus relating the acceleration model coefficients directly and non-parametrically to the raw RHESSI data. We have applied our method to two intense flares observed by RHESSI on 2003 November 3 (X3.9 class) and 2005 September 8 (M2.1 class). The results from both events exhibit some interesting behaviors, as summarized below. energy and is roughly parallel to the escape time for both flares (except in the few low energy bins). The same is true for the direct acceleration time. This behavior is what is expected when the FP sources are stronger than the LT source ( X FP > X LT ). Clearly knowledge thus gained from observations can then be used to directly compare with theoretical model predictions. It should, however, be emphasized that the derivation of the electron escape time does not involve assumptions about any specific acceleration mechanisms and thus it may impose the most severe constraint on the theoretical models.", "pages": [ 7, 8 ] }, { "title": "5.1. Comparison with SA Modeling", "content": "We now discuss whether the above results, specifically the discordant energy dependences of the acceleration and scattering time, can be reconciled with the predictions of the SA model. This is a test of the above Assumption II while keeping Assumption I. As described in the quasilinear theory, the momentum and pitch angle diffusion coefficients are related to the turbulence energy density spectrum W ( k ) ∝ k -q , where k is the wave number, plasma dispersion relation ω ( k ), and resonance conditions (e.g., Schlickeiser 1989; Dung & Petrosian 1994). In the relativistic range, we expect similar energy dependences for the momentum diffusion and pitch angle scattering timescales, τ diff ∼ ( c/v A ) 2 τ scat ∝ E 2 -q , where v A is the Alfv'en speed. In the non-relativistic regime of interest here, the relation becomes more complex. Most past works in the literature modeling the SA mechanism in solar flares assume plasma waves propagating parallel to the large scale magnetic fields (e.g., Steinacker & Miller 1992; Dung & Petrosian 1994; Pryadko & Petrosian 1997, 1998; Petrosian & Liu 2004). As shown in Figure 5, there is a considerable variation of the model timescales at low energies, especially for the energy diffusion (or direct acceleration) time. The scattering time from the models increases or is nearly constant with energy at the non-relativistic regime. It generally obeys an approximate relation, τ scat ∝ E (3 -q ) / 6 (Petrosian & Liu 2004), so that a steep turbulence spectrum with q > 3 is needed for a scattering time that decreases with energy. Thus, for the two flares studied above, we need q > 14 and q > 8, respectively. However, for such steep turbulence spectra, the diffusion and acceleration timescales will most likely decrease with energy. Petrosian & Liu (2004) provided another approximate expression, τ diff ∝ E (7 -q ) / 6 , which is valid in a limited non-relativistic range. This would disagree with our results under the random walk approximation. We therefore conclude that the observational results from the intense X3.9 class flare on 2003 November 3 is not consistent with the SA model predictions as presented above. While the results from the weaker M2.1 class flare on 2005 September 8 with weaker and softer LT emission have less severe disagreement with the SA model if electrons are interacting with a steep portion of the turbulence spectrum, possibly in the damping range beyond the inertial range 5 (Petrosian & Chen 2010; Petrosian 2012). However, a steep turbulence spectrum will require more energy in the turbulence unless its spectral range of the steep part is narrow. Obliquely propagating waves will have different energy dependences for these timescales, but limited information on perpendicularly propagating waves seems to give similar energy dependences for the momentum and pitch angle diffusion timescales (Pryadko & Petrosian 1999). We should emphasize that the two intense events on 2003 November 3 and 2005 September 8 are not representative of typical flares. As mentioned above, the escape time and energy diffusion time are primarily determined by the ratio X LT /X FP . The discrepancy between the above observational results and the SA model predictions is related to the relatively bright LT source with a flat spectrum. On the contrary, imaging spectroscopic observations have indicated that for most flares, the LT source is much weaker and has a steeper spectrum (e.g., Petrosian et al. 2002; Liu 2006; Shao & Huang 2009). Furthermore, spectral studies of the over-the-limb solar flares with their FP sources occulted from Yohkoh (Tomczak 2001, 2009) and from RHESSI (Krucker & Lin 2008; Saint-Hilaire et al. 2008) found that their spectral index on average is larger than that from the disk flares by 1.5 and 2, respectively. Thus, for those more common flare events, we can expect flatter energy dependences for the escape time, energy diffusion time, and scattering time as determined from observations, which would be closer to the SA model predictions.", "pages": [ 8, 9 ] }, { "title": "5.2. Shock Acceleration", "content": "Addition of acceleration by a shock does not seem to help in this regard. If a standing shock exists at the acceleration region with a high speed u sh , it may dominate the acceleration rate, with an acceleration time τ sh ∝ v 2 τ scat (Petrosian 2012). If the escape of particles is again diffusive in nature ( T esc ∝ 1 /v 2 τ scat ), then we expect a simple inverse relation between the escape time and shock acceleration time, T esc ∝ 1 /τ sh . On the other hand, if shock acceleration is dominant, then from integration of the Fokker-Planck equation (with D EE = 0), we obtain Therefore the acceleration rate by a shock, which now depends only on the pitch angle diffusion rate, is similar to the energy diffusion coefficient derived above (Equation 17). We can then express the shock acceleration timescale in terms of observables as τ sh = E/A sh ( E ) = τ L (1+ X FP /X LT ) -1 . For the case X LT < X FP , our observations indicate that roughly τ sh ∝ T esc (Equation 22), which is in direct disagreement with the above inverse relation expected from the shock acceleration model. Thus for both SA and shock acceleration, we encounter contradiction with the above Assumption I that the long escape time is due to the random walk approximation expected in the strong diffusion limit ( τ scat /lessmuch τ cross ). Alternatively, a long escape time may arise in a magnetic mirror geometry as we discuss next (e.g., Chen & Petrosian 2012).", "pages": [ 9, 10 ] }, { "title": "5.3. Weak Diffusion and Magnetic Mirroring", "content": "Assumption I involves the use of Equation (5), which in the strong diffusion limit gives the random walk relation, but in the weak diffusion limit ( τ scat /greatermuch τ cross ) gives T esc → τ cross . However, in addition to plasma waves or turbulence, magnetic reconnection may restructure the the large scale magnetic fields into a configuration that can also trap and accelerate particles in the LT region (e.g., Somov & Kosugi 1997; Karlick'y & Kosugi 2004; Minoshima et al. 2011; Grady et al. 2012). The newly reconnected, cusp-shaped magnetic field lines relax and shrink to the underlying closed loops and may form a magnetic mirror geometry in the corona. A cuspshaped geometry is often seen from soft X-ray and EUV images and coronal HXR sources in many events have been found to be located near such a structure (e.g., Sun et al. 2012; Liu et al. 2013). If the magnetic field lines converge from the center of the LT acceleration region to where the particles escape into the loop legs, then the escape time will be affected. In the strong diffusion limit we would still expect a random walk process. While in the weak diffusion limit, instead of T esc → τ cross , we expect an escape time T esc ∝ τ scat , which is the time needed to scatter particles into the loss cone (Kennel 1969; Melrose & Brown 1976). The proportionality constant depends on several factors but primarily on the pitch angle distribution and the mirroring ratio η m ∼ B L /B 0 , the ratio of the magnetic field intensity from the boundary to the center of the acceleration region. As well known from the conservation of the magnetic moment (the first adiabatic invariant), only particles with a pitch angle smaller than the mirroring angle, or a pitch angle cosine | µ | > µ cr = √ 1 -1 /η m , can escape from this magnetic trap and penetrate to the loop FPs. In absence of scattering, the rest of the particles will be trapped in the mirror. But when there is scattering, the particles with high pitch angles will be scattered into the loss cone, perhaps after several bounces back and forth between the mirroring points. For example, for an isotropic pitch angle distribution, we can obtain an average escape time as which for strong convergence ( µ cr → 1) would give T esc ∼ τ scat . Malyshkin & Kulsrud (2001) showed that for electrons injected into the trap with µ = 0, one has a proportionality constant of ln η m . For strong convergence, they also identified an intermediate range 1 / 2 η m /lessmuch τ scat /τ cross /lessmuch 2 η m , where T esc ∼ 2 η m τ cross . In summary, in a converging field geometry, we expect the escape time first decreases with an increasing scattering time, but instead of becoming equal to the crossing time, it then reaches a minimum and begins to increase linearly with the scattering time when the latter exceeds the crossing time. Thus, a long escape time can arise not only from a short scattering time in the strong diffusion scenario, but also from a long scattering time in a converging magnetic field configuration. Some earliest mechanisms that were proposed to produce a distinct coronal HXR source (e.g., Leach 1984; Fletcher & Martens 1998) basically adopted the second scenario with Coulomb collisions as the scattering agent for the suprathermal electrons. In addition, an observed escape time that increases with energy requires a scattering time that also increases with energy. For scattering due to Coulomb collisions in a fully ionized hydrogen plasma in the non-relativistic limit, the electron pitch angle scattering time follows τ Coul scat ∼ τ Coul L ∝ E 3 / 2 (Trubnikov 1965; Melrose & Brown 1976; Bai 1982; Aschwanden 2002). Therefore, if Coulomb scattering is the agent that scatters electrons into the loss cone and cause their escape, the escape time will be comparable to the energy loss time and scales with energy as T esc ∝ E 3 / 2 . This is in disagreement with the above observational results from both events. Even if the scattering time is comparable to the crossing time, then we would be in the intermediate regime and the escape time would vary like the crossing time as E -1 / 2 , which also disagrees with observations. On the other hand, if scattering is dominated by waveparticle interactions, then because the above observations gives roughly similar energy dependences for the energy diffusion time and escape time, this would mean similar dependences for the scattering time and energy diffusion time. This would be in better agreement with theoretical expectations in the SA model by turbulence. But for the shock acceleration model, our observations imply a relation T esc ∝ τ sh , and in a converging field configuration, we expect T esc ∼ τ scat . While the shock acceleration model predicts τ sh ∝ v 2 τ scat (Section 5.2). We therefore conclude that this very first nonparametric determination of the SA model characteristics directly from the observed data could be reconciled with stochastic acceleration by turbulence, if the LT acceleration region is surrounded by a cusp-shaped magnetic geometry with a relatively large mirroring ratio. On the other hand, our results do not seem to be consistent with what is expected in acceleration in a standing shock at the LT region of the flare, regardless of whether there is strong field convergence. The exact treatment of the acceleration and transport of electrons in a more realistic geometry cannot be treated by the leaky box model and requires inclusion of the kinematic effects of a magnetic mirror in a dynamic flare environment that is more complicated and will be treated in future works. This work was supported by NASA grants NNX10AC06G and NNX13AF79G. Q.C. thanks the discussions and help from Anna Massone with the electron flux spectral images and Kim Tolbert with OSPEX. We thank the referee for constructive comments. RHESSI is a NASA small explorer mission. Facilities: RHESSI .", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "Aschwanden, M. J. 2002, Space Sci. Rev., 101, 1 Fleishman, G. D., & Toptygin, I. N. 2013, MNRAS, 429, 2515 Fletcher, L., & Martens, P. C. H. 1998, ApJ, 505, 418 Kontar, E. P., Emslie, A. G., Piana, M., Massone, A. M., & Brown, J. C. 2005, Sol. Phys., 226, 317 Massone, A. M., & Piana, M. 2013, Sol. Phys., 283, 177 Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., & Ogawara, Y. C. McKee, & C. Max, 135 Torre, G., Pinamonti, N., Emslie, A. G., et al. 2012, ApJ, 751, 129 Trubnikov, B. A. 1965, Reviews of Plasma Physics, 1, 105", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "We report Submillimeter Array (SMA) observations of the 880 µ m dust continuum emission for four dust-obscured galaxies (DOGs) in the local universe. Two DOGs are clearly detected with S ν (880 µ m) = 10 -13 mJy and S/N > 5, but the other two are not detected with 3 σ upper limits of S ν (880 µ m) = 5 -9 mJy. Including an additional two local DOGs with submillimeter data from the literature, we determine the dust masses and temperatures for six local DOGs. The infrared luminosities and dust masses for these DOGs are in the range 1 . 2 -4 . 9 × 10 11 ( L /circledot ) and 4 -14 × 10 7 ( M /circledot ), respectively. The dust temperatures derived from a two-component modified blackbody function are 23 -26 K and 60 -124 K for the cold and warm dust components, respectively. Comparison of local DOGs with other infrared luminous galaxies with submillimeter detections shows that the dust temperatures and masses do not differ significantly among these objects. Thus, as argued previously, local DOGs are not a distinctive population among dusty galaxies, but simply represent the high-end tail of the dust obscuration distribution. Subject headings: galaxies: active - galaxies: evolution - galaxies: formation - galaxies: starburst infrared: galaxies - submillimeter: galaxies", "pages": [ 1 ] }, { "title": "DUST PROPERTIES OF LOCAL DUST-OBSCURED GALAXIES WITH THE SUBMILLIMETER ARRAY", "content": "Ho Seong Hwang, Sean M. Andrews, and Margaret J. Geller Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138, USA Last updated: August 21, 2021", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Recent studies suggest that the cosmic star formation density peaks around z = 2, and then decreases by an order of magnitude towards z = 0 (e.g., Magnelli et al. 2013; Behroozi et al. 2013). Interestingly, over the last 11 billion years, this cosmic star formation density is dominated by infrared luminous galaxies rather than ultraviolet (UV) luminous galaxies (Takeuchi et al. 2005; Reddy et al. 2008, 2012; Bouwens et al. 2010; Heinis et al. 2013; Burgarella et al. 2013). Therefore, studying highz dusty galaxies is critical for understanding the change in the star formation activity of galaxies with cosmic time (Elbaz et al. 2011; Lutz et al. 2011; Oliver et al. 2012). Among many methods for identifying highz dusty galaxies, a simple optical/mid-infrared color criterion with ( R -[24]) ≥ 14 (mag in Vega, or S ν (24 µ m)/ S ν ( R ) ≥ 982) is very efficient in selecting z ∼ 2 star-forming galaxies with large dust obscuration: dust-obscured galaxies (DOGs, Dey et al. 2008; Fiore et al. 2008; Penner et al. 2012; Hwang et al. 2012). These DOGs seem responsible for 10-30% of the total star formation rate density of the universe at z = 1 . 5 -2 . 5 (Calanog et al. 2013). These objects are divided into two groups depending on the shape of their spectral energy distributions (SEDs) at rest-frame near- and midinfrared wavelengths: 'bump' and 'power-law' DOGs (Dey et al. 2008). The SEDs of bump DOGs show a restframe 1.6 µ m stellar bump, resulting from the minimum opacity of the H -ion in the atmospheres of cool stars (John 1988). In contrast, the power-law DOGs have a rising continuum with weak polycyclic aromatic hydrocarbon (PAH) emission, probably resulting from the hot dust component heated by active galactic nucleus (AGN) (Houck et al. 2005; Desai et al. 2009). Numerical simulations suggest that the DOGs are a diverse population ranging from intense gas-rich galaxy mergers to secularly evolving star-forming disk galaxies (Narayanan et al. 2010). However, because of their extreme distances, it is difficult to fully understand the nature of these extremely dusty galaxies. To study the physical properties of DOGs in detail (e.g., morphology, SED, dust mass and temperature), we focus on the rare local analogs of DOGs discovered recently (Hwang & Geller 2013, hereafter HG13). Thanks to their proximity and the wealth of multiwavelength data, the local DOGs are a useful testbed for studying what makes a DOG a DOG and for improving the understanding of the nature of their highz siblings. Using the Wide-field Infrared Survey Explorer ( WISE ; Wright et al. 2010) and Galaxy Evolution Explorer ( GALEX ; Martin et al. 2005) data, we identified 47 DOGs at 0 . 05 < z < 0 . 08 with large flux density ratios between mid-infrared ( WISE 12 µ m) and nearUV ( GALEX 0.22 µ m) bands 2 [i.e., S ν (12 µ m)/ S ν (0.22 µ m) ≥ 892] in the Sloan Digital Sky Survey (SDSS, York et al. 2000) data release 7 (DR7, Abazajian et al. 2009). The observational data for local and highz DOGs suggest a common underlying physical origin of the two populations; both seem to represent the high-end tail of the dust obscuration distribution resulting from various physical mechanisms rather than a unique phase of galaxy evolution (HG13). The current multiwavelength data for local DOGs mostly cover only λ ≤ 100 µ m from the Infrared Astronomical Satellite ( IRAS ; Neugebauer et al. 1984); there are only five DOGs with AKARI 140 µ m data (Murakami et al. 2007). There are no useful data on 2 We first used AKARI 9 µ m and GALEX NUV data, roughly equivalent to the R -band (0.65 µ m) and Spitzer 24 µ m data originally used for selecting z ∼ 2 DOGs (Dey et al. 2008). We then used WISE 12 µ m data instead of AKARI 9 µ m to increase the sample size (see HG13 for details). the 'Rayleigh-Jeans' side of the infrared SED peak; these data are essential for deriving dust temperatures and dust masses for these galaxies (Hwang et al. 2010; Dale et al. 2012; Symeonidis et al. 2013). Quantifying the dust properties is important because the combination of dust and stellar properties gives better constraints on the nature of these heavily obscured galaxies. We can also directly compare these quantities with model predictions (Narayanan et al. 2010). The comparison of these local DOGs with other dusty galaxies can establish a possible evolutionary link among them. We thus conducted Submillimeter Array (SMA; Ho et al. 2004) observations of the 880 µ m continuum emission for four bright local DOGs to derive the physical parameters of their dust content. We report the results from this pilot survey. Section 2 describes the sample and the details of the SMA observations and data reduction. We derive the physical parameters of the dust content in local DOGs, and compare them with other submillimeter detected, infrared luminous galaxies in Section 3. We discuss and summarize the results in Section 4. Throughout, we adopt flat ΛCDM cosmological parameters: H 0 = 70 km s -1 Mpc -1 , Ω Λ = 0 . 7 and Ω m = 0 . 3.", "pages": [ 1, 2 ] }, { "title": "2.1. Sample", "content": "HG13 identified 47 local DOGs with S ν (12 µ m) > 20 mJy at 0 . 05 < z < 0 . 08 in the SDSS DR7. These DOGs have extreme flux density ratios between midinfrared and UV bands with S ν (12 µ m)/ S ν (0.22 µ m) ≥ 892. The infrared luminosities of the DOGs are in the range 3 × 10 10 < L IR / L /circledot < 7 × 10 11 with a median L IR of 2 . 1 × 10 11 (L /circledot ). These infrared luminosities are based on an SED fit to the photometric data at 6 µ m < λ ≤ 140 µ m with the SED templates and fitting routine of Mullaney et al. (2011), DECOMPIR 3 . From these SED fits, we computed the expected 880 µ m flux densities for the 47 DOGs, and selected the four DOGs with the largest, predicted flux densities at 880 µ m for SMA observation (see the target list in Table 1).", "pages": [ 2 ] }, { "title": "2.2. Observations and Data Reduction", "content": "Four local DOGs were observed in the compact configuration (8-70 m baselines) of the 8-element Submillimeter Array (SMA; Ho et al. 2004) interferometer at Mauna Kea, Hawaii in early 2013 (see Table 1 for an observing journal). The SMA dual-sideband receivers were tuned to a local oscillator (LO) frequency of 342 GHz (877 µ m), and the correlator was configured to process 2 × 2 GHz (intermediate frequency) IF bands per sideband centered ± 4-8 GHz from the LO, divided into 48 spectral 'chunks' that each contained 64 individual 1.6875 MHz channels. In each track, observations of two target DOGs were interleaved with nearby quasars on a 15 minute cycle. Additional observations of 3C 84, 3C 279, Uranus, and Titan were made for calibration purposes when the science targets were at low elevations. Observing conditions were good, with precipitable water vapor levels at 1.5-2.0 mm and stable phase behavior. The raw visibilities were reduced with the MIR software package. The bandpass response was calibrated with observations of 3C 84 and 3C 279, and the antennabased complex gains were determined by frequent observations of a nearby quasar: 0854+201 for LDOG-07, J1310+323 for LDOG-26, J1635+381 for LDOG-39, and L1549+026 for LDOG-41. The absolute amplitude scale was set based on observations of Uranus and Titan, and should have a systematic uncertainty of ∼ 10% or less. After calibration, the individual spectral channels for each sideband and IF band were combined into a composite wideband continuum visibility set. Those data were then Fourier inverted assuming natural weighting, deconvolved with the CLEAN algorithm, and then restored with a synthesized beam (with a FWHM of roughly 2 . '' 2 × 1 . '' 8). The imaging and deconvolution procedures were conducted with the MIRIAD software package. We show the resulting 880 µ m continuum aperture synthesis images for the four local DOG targets in Figure 1; we also show SDSS ur and WISE 3.4/22 µ m cutout images. None of the DOGs are resolved in the 880 µ m synthesis images. The SMA synthesis maps for two DOGs in the top panels (LDOG -07 and LDOG -25) show clear detections with S ν (880 µ m) = 10 -13 mJy and S/N > 5. However, the other two DOGs in the bottom panels (LDOG -39 and LDOG -41) are not visible in the synthesis maps. They are are not detected with 3 σ upper limits of S ν (880 µ m) = 5 -9 mJy. We list the four target DOGs in Table 1 with the SMA observation log and the measured 880 µ m flux densities.", "pages": [ 2 ] }, { "title": "3.1. Determination of Physical Parameters of Dust Content in Local DOGs", "content": "We first compute the infrared luminosities of the DOGs using the SED templates and fitting routine of Mullaney et al. (2011). This routine decomposes the observed SED of a galaxy into two components (i.e., a hostgalaxy and an AGN). Therefore, we can also measure the contribution of (buried) AGN to the total infrared lumi- nosity of a galaxy. This method is the same as in HG13, but we have additional submillimeter data to constrain the fit. The SED fit with the Mullaney et al. (2011) routine does not provide the dust temperatures and masses for the galaxies. We thus fit the observational data again using a modified blackbody function with two (warm and cold) dust components (Dunne & Eales 2001; Vlahakis et al. 2005; Willmer et al. 2009): where A w and A c are the relative contributions of warm and cold dust components, T w and T c are dust temperatures, B ( ν , T ) is the Planck function, and β is the dust emissivity index. We examined two values of β (i.e., 1.5 and 2.0), and found that β = 2 . 0 generally provides better fits. Therefore, we use β = 2 . 0 for the fit, consistent with Vlahakis et al. (2005) and Willmer et al. (2009). We then compute the dust mass from the observed flux density (Hildebrand 1983), defined by where k d is the dust mass opacity coefficient, D L is the luminosity distance, and S ν obs is the observed flux density 4 with S ν obs = S ν obs ,w + S ν obs ,c . We adopt k rest d = 0 . 383 cm 2 g -1 at 850 µ m from Draine (2003). We use k rest d at 850 µ m rather than at 880 µ m to be consistent with the comparison sample of galaxies (see Section 3.2.1). Note that the k d value is usually very uncertain; it can change by a factor of 2 (e.g., k 850 d = 0 . 77 cm 2 g -1 in James et al. 2002). Therefore, the resulting dust mass can also change depending on the k d value adopted. For S ν obs , we use the flux densities expected from the modified blackbody fit at 850(1+ z ) µ m. We use the photometric data at 20 µ m < λ ≤ 880 µ m for both SED fits. We also compile the farinfrared/submillimeter data in the literature, and include them for the fit (e.g., Herschel 100-500 µ m data for LDOG -07 from the H-ATLAS program; Rigby et al. 2011; Pilbratt et al. 2010). Figure 2 shows the photometric data for the DOGs along with the best-fit SEDs for infrared luminosities 4 Note that if we use S ν obs derived from equation (1) instead of the observed flux density, equation (2) can be simply expressed as D 2 L ν β rest ( A w + A c ) / k rest d . (left panels) and for dust temperatures and masses (right panels). The SEDs for two DOGs in the middle panels (e-h) are not well constrained because the SMA flux densities are upper limits, not used for the SED fit. We thus flag the derived quantities for these DOGs with lower and upper limits depending on parameters in the following Figures and Tables. Table 2 lists the infrared luminosities, dust temperatures of the warm and cold components, total dust masses, and dust mass ratios between warm and cold components of the four DOGs. We compute the uncertainty in each parameter by randomly selecting flux densities at each band within the associated error distribution (assumed to be Gaussian) and then refitting. We compiled the far-infrared/submillimeter data in the literature, and found two more DOGs with existing submillimeter data. LDOG -08 in Figure 2(i-j) has Herschel 100 -500 µ m data from the H-ATLAS program (Rigby et al. 2011), and LDOG -35 in Figure 2(k-l) has SCUBA 850 µ m data from Ant'on et al. (2004). We show these two DOGs in Figure 2 and Table 2, and include in our analysis.", "pages": [ 2, 3, 4, 5 ] }, { "title": "3.2. Comparison of Dust Content between Local DOGs and Infrared Luminous Galaxies with Submillimeter Detection", "content": "To see whether the local DOGs with submillimeter detection are a population distinct from other submillimeter detected, infrared luminous galaxies, we compare the dust parameters between the two populations. Because of the small number of local DOGs with submillimeter data and because of the inhomogenous selection criteria for submillimeter detected, infrared luminous galaxies (see next Section), we simply examine their relative distribution in several parameter spaces.", "pages": [ 5 ] }, { "title": "3.2.1. Local Infrared Luminous Galaxies with Submillimeter Detection", "content": "Among many studies based on submillimeter observations of local galaxies (e.g., Willmer et al. 2009; Clements et al. 2010; Dale et al. 2012), we select a comparison sample including only the galaxies with infrared luminosities similar to the local DOGs (i.e., 10 11 /lessorsimilar L IR / L /circledot /lessorsimilar 10 12 ) and with submillimeter data at λ ≥ 850 µ m. We first use the galaxies in the SCUBA local universe galaxy survey (SLUGS; Dunne et al. 2000; Dunne & Eales 2001). Among 104 galaxies with SCUBA 850 µ m data in the survey, we select 63 galaxies at z > 0 . 01 with available mid- and far-infrared data. The lower redshift limit removes very nearby, extended galaxies that could be resolved in the mid-infrared. For 48 out of 63 galaxies, we use WISE , IRAS , and AKARI data at 3.4-160 µ m from the SDSS galaxy catalog with multiwavelength data compiled in HG13. For the remaining 15 galaxies, we adopt the mid- and far-infrared data from the Great Observatories All-sky LIRG Survey (GOALS; Armus et al. 2009); Spitzer and IRAS data at 3.6-160 µ m in U et al. (2012). There could be some potential DOG candidates in this GOALS sample, not covered in HG13 (i.e., SDSS). We do identify eight potential DOG candidates with S Spitzer 8 µ m /S GALEX 0 . 22 µ m > 982 in the GOALS sample, and do not include them in the comparison sample 5 . We do not include them in the DOG sample either, because the selection criteria (e.g., observed bands, mid-infrared flux density limits) are not exactly the same as HG13. We also use the luminous infrared galaxies with SMA 880 µ m data in Wilson et al. (2008). Among 15 galaxies in the paper, we include seven systems that do not overlap with the SLUGS sample and that do not have two distinct interacting galaxies. Their mid- and far-infrared flux densities are again adopted from HG13 (six galaxies) and GOALS (one galaxy). In summary, there are 62 galaxies with submillimeter, mid- and far-infrared data at z > 0 . 01 for comparison with the local DOGs. We apply the same SED fitting routines of Section 3.1 to these galaxies to derive physical parameters including infrared luminosity, dust mass and temperature. Using this sample, we first confirm that our measurements agree well with previous measurements: the dust masses in Dunne et al. (2000), Dunne & Eales (2001) and Willmer et al. (2009), and the dust temperatures in Willmer et al. (2009). Moreover, we note that there are recent sophisticated models that provide several dust parameters simultaneously from the SED fit (e.g., Draine & Li 2007; da Cunha et al. 2008; see also Walcher et al. 2011 for a review). Because of the small number of bands in the far-infrared/submillimeter regimes for the DOGs, we restrict our analysis to simple models (e.g., two-component modified blackbody function) rather than sophisticated ones that require many observational data points. Our simple approach works well. For example, the dust masses derived in this study for the galaxies in Willmer et al. (2009) show excellent agreement with those based on the Draine & Li (2007) models. We apply the same fitting routine both to local DOGs and to other infrared luminous galaxies with submillimeter detections. Thus, the comparison between the two suffers no bias resulting from different SED fitting methods.", "pages": [ 5 ] }, { "title": "3.2.2. Comparisons of Dust Temperature and Mass", "content": "Figure 3 displays several parameters related to the dust temperature as a function of total infrared luminosity. The top panels show the temperature T cold of the cold dust component. The cold dust temperature is in a very narrow range both for DOGs (circles) and for other infrared luminous galaxies (squares). Remarkably, T cold does not change much with infrared luminosity. The temperature of the warm dust component, T warm , in the middle panel also does not depend on infrared luminosity, consistent with previous studies (Dunne & Eales 2001). However, it shows a large dispersion from 45 K to 125 K. The dust temperatures of all the DOGs except the one with T warm ∼ 125 K are well mixed with those of other infrared luminous galax- s with similar infrared luminosities. To examine the behavior of the warm dust component in galaxies, we plot the observed flux density ratio, S ν (22 µ m)/ S ν (60 µ m), in the bottom panel. For comparison, we also plot the contours and gray dots indicating the distribution of IRAS 60 µ m detected SDSS galaxies at z > 0 . 01 regardless of submillimeter detection. The panel shows that the four DOGs with small AGN contribution (purple circles) are indistinguishable from other infrared luminous galaxies (squares). Two DOGs and two infrared luminous galaxies with a large AGN contribution (green and cyan symbols) have larger flux density ratios than other galaxies, consistent with expectation (de Grijp et al. 1985; Veilleux et al. 2009; Lee et al. 2012). In Figure 4, we plot several parameters related to the dust mass as a function of infrared luminosity. The top panels show the dust mass ratios between the cold and warm components. Again the DOGs do not differ from other infrared luminous galaxies. We run a Kolmogorov-Smirnov (K-S) test to determine whether the DOGs (circles) and other infrared luminous galaxies (squares) are drawn from the same distribution. The K-S test cannot reject the hypothesis that the ratio distributions of the two samples are extracted from the same parent population. If we run the K-S test for the galaxies in the same infrared luminosity range (i.e., 1 . 2 × 10 11 < L IR , total /L /circledot < 4 . 7 × 10 11 ), the conclusion does not change. The middle panels show the total dust mass, M dust = M cold + M warm . The dust mass roughly correlates with infrared luminosity, consistent with previous studies (Dunne & Eales 2001; Magdis et al. 2012). The Spearman correlation coefficient ( ρ s ) is 0.51 and the probability of obtaining the correlation by chance is < 0.01%, confirming the correlation between the two. The dust masses of the DOGs are indistinguishable from other infrared luminous galaxies. The K-S test also confirms this impression. The bottom panels display the ratios between dust masses ( M dust ) and stellar masses ( M star ). We use the stellar mass estimates in the MPA/JHU DR7 value-added galaxy catalog 6 . These estimates are based on the fit to SDSS five-band photometry with the Bruzual & Charlot (2003) models (see also Kauffmann et al. 2003). We convert the stellar masses in the catalog that are based on the Kroupa initial mass function (IMF; Kroupa 2001) to those with a Salpeter IMF (Salpeter 1955) by dividing them by 0.7 (Elbaz et al. 2007). The stellar masses in this catalog are not available for all the galaxies in this study. If we use the stellar masses derived from WISE 3.4 µ m luminosities to increase the sample size (Hwang et al. 2012), the conclusions do not change. The ratios, M dust / M star , for the majority of the galaxy samples are between 10 -4 and 10 -2 , consistent with previous results for star-forming galaxies in the local universe (Santini et al. 2010; Dunne et al. 2011; Skibba et al. 2011). The ratios for the DOGs except the AGN-dominated outlier again do not differ from other infrared luminous galaxies.", "pages": [ 5, 6, 7 ] }, { "title": "4. DISCUSSION AND SUMMARY", "content": "We conducted SMA observations of four local analogs of DOGs to measure the physical parameters of their dust content. Two DOGs are clearly detected at 880 µ m with S ν (880 µ m) = 10 -13 mJy and S/N > 5; the other two are not detected with 3 σ upper limits of S ν (880 µ m) = 5 -9 mJy. In addition to these four DOGs, we compiled submillimeter data for additional two DOGs from the literature. Thus, we determine the dust temperatures and masses for a total of six local DOGs. The comparison of these DOGs with other infrared luminous galaxies with submillimeter detection indicates no significant difference in dust parameters between the two populations. Previous studies suggest that there are two types of DOGs for both local and highz DOGs: star formation (SF)- and AGN-dominated ones in their near- and mid-infrared SEDs (Dey et al. 2008; HG13). The reason for the extreme flux density ratios between midinfrared and UV bands in SF-dominated DOGs mainly results from abnormal faintness in the UV rather than extreme brightness in the mid-infrared (Penner et al. 2012; HG13). This conclusion also applies to AGN-dominated DOGs, but the large mid-infrared fluxes from the AGN dust also contribute to the extreme flux density ratios. For the six local DOGs, the dust masses and temperatures are similar to those of other submillimeter detected, infrared luminous galaxies with similar infrared luminosities. Thus, the DOGs are not a distinctive population among dusty galaxies. In other words, the main reason they are selected as DOGs is not an extremely large dust content, but simply results from a large dust obscuration along the light of sight. This conclusion explains the significant fraction of local DOGs with highly inclined disks (see Figure 6 in HG13; see also Kartaltepe et al. 2012 for disk-dominated, highz DOGs). Merging processes also change the dust geometry to favor large dust obscuration (Penner et al. 2012). The DOGs with large AGN contribution clearly contain a hot dust component with T /greaterorsimilar 80 K (see middle panels in Figure 3). Although the galaxies with a large AGN contribution tend to be selected as DOGs because of their large mid-infrared fluxes (see bottom left panel in Figure 9 of HG13), not all infrared luminous galaxies with large AGN contribution are selected as DOGs. One interesting feature of the AGN-dominated DOGs is that their cold temperatures are similar to those of SFdominated DOGs and other infrared luminous galaxies (see top panels in Figure 3). This result is consistent with recent conclusions that the effect of AGN on starforming galaxies does not appear on the 'Rayleigh-Jeans' side of the infrared SED peak (i.e., cold components), but only appears on the 'Wien' side (i.e., warm components) (Hatziminaoglou et al. 2010; Kirkpatrick et al. 2012). There are several studies on the dust temperatures and masses for highz DOGs (Bussmann et al. 2009; Melbourne et al. 2012; Sajina et al. 2012; Wu et al. 2012). Because of the different SED fitting methods and because of the small number of farinfrared/submillimeter data for highz DOGs, a direct comparison of dust parameters between local and highz DOGs is not very meaningful. Moreover, the infrared luminosity range for highz DOGs does not overlap with local DOGs. A rough comparison of the dust temperatures based on currently available data (see Figure 13 in HG13) suggests that the dust temperatures for the majority of highz DOGs are similar to or lower than for local DOGs even though the infrared luminosities of highz DOGs are much higher than for local DOGs. There are also some hot DOGs at high redshift with dust temperatures much higher than for local DOGs (Wu et al. 2012). Farinfrared and submillimeter data for a larger number of DOGs in both low and high redshifts with similar infrared luminosities will be useful for a thorough comparison between the two populations. This study clearly shows the importance of submillimeter data in understanding the dust content of local DOGs. We plan to extend this study to a larger sample of local DOGs with the Caltech Submillimeter Observatory (G.-H. Lee et al., in preparation). Facility: SMA We thank the anonymous referee for his/her useful comments that improved the manuscript. We also thank Jong Chul Lee, Gwang-Ho Lee and Jubee Sohn for useful discussions. H.S.H. acknowledges the Smithsonian Institution for the support of his post-doctoral fellowship. The Smithsonian Institution also supports the research of MJG. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.", "pages": [ 7 ] }, { "title": "REFERENCES", "content": "Abazajian, K. N., Adelman-McCarthy, J. K., Agueros, M. A., et al. 2009, ApJS, 182, 543 Armus, L., Mazzarella, J. M., Evans, A. S., et al. 2009, PASP, Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2010, ApJ, 709, L133 Bussmann, R. S., Dey, A., Borys, C., et al. 2009, ApJ, 705, 184 Calanog, J. A., Wardlow, J., Fu, H., et al. 2013, ApJ, in press (arXiv:1304.4593) Clements, D. L., Dunne, L., & Eales, S. 2010, MNRAS, 403, 274 da Cunha, E., Charlot, S., & Elbaz, D. 2008, MNRAS, 388, 1595 Dale, D. A., Aniano, G., Engelbracht, C. W., et al. 2012, ApJ, 745, 95 Desai, V., Soifer, B. T., Dey, A., et al. 2009, ApJ, 700, 1190 Dey, A., Soifer, B. T., Desai, V., et al. 2008, ApJ, 677, 943 Dunne, L., Eales, S., Edmunds, M., et al. 2000, MNRAS, 315, 115 Dunne, L., & Eales, S. A. 2001, MNRAS, 327, 697 Fiore, F., Grazian, A., Santini, P., et al. 2008, ApJ, 672, 94 Hatziminaoglou, E., Omont, A., Stevens, J. A., et al. 2010, A&A, 518, L33 Hwang, H. S., Geller, M. J., Kurtz, M. J., Dell'Antonio, I. P., & Fabricant, D. G. 2012, ApJ, 758, 25 Hwang, H. S., Elbaz, D., Magdis, G., et al. 2010, MNRAS, 409, 75 James, A., Dunne, L., Eales, S., & Edmunds, M. G. 2002, MNRAS, 335, 753 John, T. L. 1988, A&A, 193, 189 Kauffmann, G., Heckman, T. M., White, S. D. M., et al. 2003, MNRAS, 341, 33 Kroupa, P. 2001, MNRAS, 322, 231 Lutz, D., Poglitsch, A., Altieri, B., et al. 2011, A&A, 532, A90 Magdis, G. E., Daddi, E., B'ethermin, M., et al. 2012, ApJ, 760, 6 Magnelli, B., Popesso, P., Berta, S., et al. 2013, A&A, 553, A132 Martin, D. C., Fanson, J., Schiminovich, D., et al. 2005, ApJ, 619, L1 Melbourne, J., Soifer, B. T., Desai, V., et al. 2012, AJ, 143, 125 Mullaney, J. R., Alexander, D. M., Goulding, A. D., & Hickox, Takeuchi, T. T., Buat, V., & Burgarella, D. 2005, A&A, 440, L17 U, V., Sanders, D. B., Mazzarella, J. M., et al. 2012, ApJS, 203, 9 Veilleux, S., Rupke, D. S. N., Kim, D., et al. 2009, ApJS, 182, 628 Vlahakis, C., Dunne, L., & Eales, S. 2005, MNRAS, 364, 1253 Walcher, J., Groves, B., Budav'ari, T., & Dale, D. 2011, Ap&SS, 331, 1 Willmer, C. N. A., Rieke, G. H., Le Floc'h, E., et al. 2009, AJ, 138, 146", "pages": [ 7, 8 ] } ]

[ { "title": "ABSTRACT", "content": "The interaction of TeV gamma rays from distant blazars with the extragalactic background light produces relativistic electron-positron pair beams by the photon-photon annihilation process. Using the linear instability analysis in the kinetic limit, which properly accounts for the longitudinal and the small but finite perpendicular momentum spread in the pair momentum distribution function, the growth rate of parallel propagating electrostatic oscillations in the intergalactic medium is calculated. Contrary to the claims of Miniati and Elyiv (2013) we find that neither the longitudinal nor the perpendicular spread in the relativistic pair distribution function do significantly affect the electrostatic growth rates. The maximum kinetic growth rate for no perpendicular spread is even about an order of magnitude greater than the corresponding reactive maximum growth rate. The reduction factors to the maximum growth rate due to the finite perpendicular spread in the pair distribution function are tiny, and always less than 10 -4 . We confirm the earlier conclusions by Broderick et al. (2012) and us, that the created pair beam distribution function is quickly unstable in the unmagnetized intergalactic medium. Therefore, there is no need to require the existence of small intergalactic magnetic fields to scatter the produced pairs, so that the explanation (made by several authors) of the FERMI non-detection of the inverse Compton scattered GeV gamma rays by a finite deflecting intergalactic magnetic field is not necessary. In particular, the various derived lower bounds for the intergalactic magnetic fields are invalid due to the pair beam instability argument. Subject headings: cosmology: diffuse radiation - cosmic rays - gamma rays: theory - instabilities - plasmas", "pages": [ 1 ] }, { "title": "PLASMA EFFECTS ON FAST PAIR BEAMS II. REACTIVE VERSUS KINETIC INSTABILITY OF PARALLEL ELECTROSTATIC WAVES", "content": "R. SCHLICKEISER 1 , 2 , S. KRAKAU 1 , M. SUPSAR 1 1 Institut fur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universitat Bochum, D-44780 Bochum, Germany 2 Research Department Plasmas with Complex Interactions, Ruhr-Universitat Bochum, D-44780 Bochum, Germany Draft version June 24, 2021", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The new generation of air Cherenkov TeV γ -ray telescopes (HESS, MAGIC, VERITAS) have detected about 30 cosmological blazars with strong TeV photon emission: the most distant ones are 3C279 (redshift zr = 0 . 536), 3C66A ( zr = 0 . 444) and PKS 1510-089 ( zr = 0 . 361). Any of these more distant than zr = 0 . 16 produces energetic e ± particle beams in double photon collisions with the extragalactic background light (EBL). These pairs with typical Lorentz factors γ = 10 6 Γ 6 are expected to inverse Compton (IC) scatter on the cosmic microwave background (CMB) radiation, on a typical length scale lIC ∼ 0 . 75 Γ -1 6 Mpc, thus producing gamma-rays with energy of order 100 GeV, which have not been detected by the FERMI satellite. Given the still relatively short distance lIC , both pair production and IC emission occur primarily in cosmic voids of the intergalactic medium (IGM), which fill most of cosmic volume. It has been argued that the inverse Compton scattered gamma-rays then are still energetic enough for further pair-production interactions giving rise to a full electromagnetic cascade as in vacuum. However, the pair-beam is subject to two-stream-like instabilities of both electrostatic and electromagnetic nature (Broderick et al. 2012, Schlickeiser et al. 2012a). In this case the electromagnetic pair cascade does not contribute to the multiGeV flux, as most of the pair beam energy is transferred to the IGM with important consequences for its thermal history. Moreover, there is no need to require the existence of small intergalactic magnetic fields to scatter the produced pairs, so that the explanation of the FERMI non-detection of the inverse Compton scattered GeV gamma rays by a finite de- [email protected], [email protected], [email protected] flecting intergalactic magnetic field (Neronov and Vovk 2010, Tavecchio et al. 2011, Dolag et al. 2011, Taylor et al. 2012, Dermer et al. 2011, Takahashi et al. 2012, Vovk et al. 2012) is not necessary. In their instability analysis Schlickeiser et al. (2012a hereafter referred to as paper I) and Broderick et al. (2012) have approximated the pair parallel momentum distribution function g ( x ) = δ ( x -xc ) by a sharp delta-function, where x = p ‖ / ( mec ) denotes the parallel pair momentum p ‖ in units of mec = 5 . 11 · 10 5 eV/ c ( c : speed of light), which is commonly referred to as reactive linear instability analysis. This approximation has been recently criticized by Miniati and Elyiv (2013), who noted that the finite momentum spread of the pair distribution function (referred to as kinetic instability study) will significantly reduce the maximum electrostatic growth rate to a level that the full electromagnetic pair cascade as in vacuum is not modified. The study of Cairns (1989), based on nonrelativistic kinetic plasma equations, indicated that the kinetic/reactive instability character depends strongly on the plasma beam and plasma background parameters, such as beam density nb , beam speed β 1 c and background particle density Ne and temperature Te . Severe differences between reactive and kinetic instability rates occur particularly for beam to background particle density ratios exceeding nb / Ne > 10 -5 . However, as argued below, in our case of pair beams in the IGM medium this ratio is of order nb / Ne /similarequal 10 -15 , much below the critical value 10 -5 , so that we are in a regime where reactive and kinetic instability studies should not differ significantly according to Cairns (1989). However, as noted the work of Cairns (1989) is based on nonrelativistic kinetic plasma equations. It is the purpose of this work to investigate the claim of Miniati and Elyiv (2013) for parallel propagating electrostatic fluctuations using the correct relativistic kinetic plasma equations. Relativistic kinetic instability studies are notoriously difficult and complicated due to plasma particle velocities close to the speed of light. Therefore extreme care is necessary in order to include all relevant relativistic effects. We therefore will repeat in detail the linear instability analysis in the kinetic limit using the realistic pair momentum distribution function. For mathematical simplicity we will restrict our analysis to parallel wave vector orientations with respect to the direction of the TeV gamma rays generating the relativistic pairs. In our analysis we will also use a more realistic modelling of the fully-ionized IGM plasma as isotropic thermal distributions.", "pages": [ 1, 2 ] }, { "title": "2.1. Intergalactic medium", "content": "The unmagnetized IGM consists of protons and electrons of density Ne = 10 -7 N 7 cm -3 . Any neutral atoms or molecules do not participate in the electromagnetic interaction with the pairs. In paper I we have modelled the IGM plasma with the cold isotropic particle distribution functions ( a = e , p ) where p ‖ and p ⊥ denote the momentum components parallel and perpendicular to the incoming γ -ray direction in the photon-photon collisions, respectively. Here we take into account the finite temperature Ta of the IGM plasma particles, adopting the isotropic Maxwellian distribution function β a = √ 2 kbTa / ( mac 2 ) is the thermal IGM velocity in units of the speed of light. Photoionization models of the IGM (Hui and Gnedin 1997, Hui and Haiman 2003) indicate nonrelativistic electron temperatures Te = 10 4 T 4 K, implying very small values of β e = 1 . 8 · 10 -3 T 1 / 2 4 /lessmuch 1 and large values of µe /greatermuch 1. If we scale the proton temperature Tp = χ Te , we obtain β p = √ χξβ e with the electron-proton mass ratio ξ = me / mp = 1 / 1836. For proton to electron temperature ratios χ /lessmuch ξ -1 = 1836 we find that β p /lessmuch β e .", "pages": [ 2 ] }, { "title": "2.2. Intergalactic pairs from photon-photon annihilation", "content": "Schlickeiser et al. (2012b) analytically calculated the pair production spectrum from a power law distribution of the gamma-ray beam up to the maximum energy M (all energies in units of mec 2 ), interacting with the isotropically soft photon Wien differential energy distribution N ( k 0 ) ∝ k 2 0 exp ( -k 0 / Θ ) representing the EBL with Θ /similarequal 2 · 10 -7 corresponding to 0.1 eV. They found that the pair production spectrum is highly beamed into the direction of the initial gamma-ray photons, so that a highly anisotropic, ultrarelativistic velocity distribution of the pairs results. With respect to the parallel momentum x = p ‖ / ( mec ) the pair momentum distribution function is strongly peaked at Mc = Θ -1 for the case of effective pair production M /greatermuch Mc . The differential parallel momentum spectrum of the generated pairs can be well approximated as with the step function H ( x ) = [ 1 +( x / | x | )] / 2, and the two characteristic normalized momenta where τ 0 = σ T N 0 R , with the total number density of EBL photons N 0 /similarequal 1 cm -3 , denotes the traversed optical depth of gamma rays. Both characteristic momenta xb > xc /greatermuch 1 are very large compared to unity as Mc /similarequal 2 · 10 6 . As noted in Schlickeiser et al. (2012b) the analytical approximation (3) agrees rather well with the numerically calculated production spectrum using the code of Elyiv et al. (2009). The parallel momentum spectrum of pairs (3) exhibits a strong peak at xc , is exponentially reduced ∝ exp ( -xc / x ) at smaller momenta, and exhibits a broken power law at higher momenta (see Fig. 7 in Schlickeiser et al. 2012b). During this analysis here we will simplify the parallel momentum spectrum (3) slightly to the form where we keep the essential features of the spectrum (3), namely the exponential reduction below xc , and the powerlaw behavior at high parallel momentum values. But instead of allowing for the broken power-law behavior above and below xb , we represent this part only as a single power law with spectral index s = p -( 1 / 2 ) . As we will see later, this simplification only affects the damping rate of plasma fluctuations, whereas the growth rate is caused by the exponential reduction below xc . The associated pair phase space density is then given by with the normalization factor A 0 determined by the total beam density In paper I we have ignored any finite spread of the pair distribution function in perpendicular momentum p ⊥ , i.e. Here we will allow for such a perpendicular spread by adopting with finite values of b . The special form (9) of the perpendicular momentum distribution function is chosen because of the limit which can be readily proven by inspecting with an arbitrary function W ( p ⊥ ) the expression Using the Taylor expansion of the function W near p ⊥ = 0 readily yields Therefore, in the limit b = 0 the broadened perpendicular distribution function (9) reduces to the distribution function (8) with no perpendicular spread. Using the phase space density (6) with Eqs. (5) and (9) in the normalization condition (7) then yields where Γ ( a ) is the gamma function and U ( a , b , z ) denotes the confluent hypergeometric function of the second kind. Its argument xc is very large, so that we have approximated U ( s -1 , s , xc ) /similarequal x 1 -s c for values of s > 1. Therefore the normalization factor has to be Now we estimate the value of the maximum normalized perpendicular momentum b . With extensive Monte Carlo simulations Miniati and Elyiv (2013) determined the maximum angular spread of the beamed pairs to ∆φ = 10 -5 in agreement with the kinematic estimate (see Eq. (5) of Miniati and Elyiv (2013)) where we use the invariant maximum center of mass energy square s 0 = E γΘ / mec 2 . This maximum angular spread determines so that with Eq. (4) which for τ 0 = 10 3 τ 3 is well below unity. The maximum perpendicular momenta of the generated pair distribution are less than 40 keV/c.", "pages": [ 2, 3 ] }, { "title": "2.3. Reactive instability results", "content": "As noted before, in paper I we approximated the parallel pair distribution function (11) by a sharp delta-function mecg ( x ) = δ ( x -xc ) and ignored any finite spread i.e. G ( p ⊥ ) = δ ( p ⊥ ) . Moreover, we modelled the unmagnetized IGMas a fully-ionized cold electron-proton plasma. In agreement with the earlier reactive instability study of Broderick et al. (2012), we found that very quickly oblique (at propagation angle θ ) electrostatic fluctuations are excited. The growth rate ( ℑω ) max and the real part of the frequency ( ℜω ) max at maximum growth are given by and respectively, with the electron plasma frequency ω p , e = 17 . 8 N 1 / 2 7 Hz. Note that we have corrected a mistake in paper I in the numerical factor in the growth rate (12). nb = 10 -22 n 22 cm -3 represent typical pair densities in cosmic voids, xc = 10 6 xc , 6 and with β 1 = xc / √ 1 + x 2 c . The maximum growth rate occurs at the oblique angle θ E = 39 . 2 degrees and provides as shortest electrostatic growth time Even, if nonlinear plasma effects are taken into account, we concluded in paper I that most of the pair beam energy is dissipated generating electrostatic plasma turbulence, which prevents the development of a full electromagnetic pair cascade as in vacuum. For later comparison we note that for parallel wave vector orientations θ = 0 Eq. (14) reduce to implying for the real and imaginary frequency parts at maximum growth (19) - (20) and With Dirac's formula", "pages": [ 3, 4 ] }, { "title": "3. ELECTROSTATIC DISPERSION RELATION", "content": "The dispersion relation of weakly damped or amplified ( | γ | /lessmuch ω R ) parallel electrostatic fluctuations with wavenumber k and freuency ω = ω R + ı γ in an unmagnetized plasma with gyrotropic distribution functions is given by (Schlickeiser 2010) The dispersion function Λ ( k , ω ) is symmetric Λ ( ω , -k ) = Λ ( ω , k ) with respect to the wavenumber k , so that it suffices to discuss positive values of k > 0. Inserting the distribution functions (2), (6) and (9), using nonrelativistic values of β a /lessmuch 1, then provides where Z ' ( t ) denotes the first derivative of the plasma dispersion function (Fried and Conte (1961); Schlickeiser and Yoon (2012, Appendix A)) with complex argument as z = ω / ( kc ) = R + ıI with R = ω R / ( kc ) and I = γ / ( kc ) . For weakly damped/amplified fluctuations we use the approximations We notice that the imaginary part is the same in both approximations. The expression with q = p ⊥ / ( mec ) , represents the pair beam contribution to the electrostatic dispersion relation. The first x -integral in Eq. (29) vanishes because g ( 0 ) = g ( ∞ ) = 0 leaving where P denotes the principal value, we obtain for the limit with Because of the small factor ( 2 nb / Ne ) /lessmuch 1 we ignore the contribution of the real principal part of Eq. (32) to the dispersion relation (27), but keep the imaginary part with the result The last integral has a nonvanishing value provided that x 0 ( R , q ) ∈ [ 0 , ∞ ] , which requires subluminal real phase speed ( | R | ≤ 1). where we have introduced the integral the normalized wavenumber and the normalization constant (15). Separating the dispersion function into real and imaginary parts Λ = ℜΛ + ı ℑΛ we find and We emphasize that the real part of the dispersion function (37) is symmetric in R , so that it suffices to discuss positive values of R > 0. It remains to calculate with the parallel pair beam distribution (5) so that the integral (35) becomes J ( b ) = where we introduce With property (13) we obtain for no perpendicular spread In Appendix A we derive approximations of the integral (40), valid for values of b ≤ b 0, where b 0 = 7 . 2 · 10 -2 , according to the estimate (18), is significantly smaller than unity. In terms of the value (42) at b = 0 we obtain with where the correction function with can be expressed in terms of Dawson's integral F ( X ) (see definition (103)). If the correction function (45) is smaller than unity, the perpendicular spread will reduce the growth rate γ 0 of fluctuations. If the correction function (45) is greater than unity, it will enhance the growth rate γ 0; each case compared to the case of no perpendicular spread b = 0.", "pages": [ 4, 5 ] }, { "title": "3.1. General kinetic instability analysis", "content": "For weakly damped or amplified ( | γ | /lessmuch ω R ) fluctuations the real and imaginary phase speed (or frequency) parts of the fluctuations are given by (Schlickeiser 2002, p. 263) and respectively, where R = ω R / ( kc ) = ω R / ( ω p , e κ ) . We then find that is given by the difference of the growth rate γ p ( κ ) from the anisotropic relativistic pair distribution and the positively counted Landau damping rate γ L ( κ ) from the thermal IGM plasma with and 3.2. Electrostatic modes In Appendix B we show that the dispersion relation (47) provides two collective electrostatic modes: Langmuir oscillations and ion sound waves. The Langmuir oscillations with the dispersion relation occur at normalized wavenumbers κ L ≤ κ /lessmuch β -1 e , where Eq. (52) corresponds to the dispersion relation of Langmuir oscillations (see Appendix B). Likewise, the ion sound waves with the dispersion relation only exists for values of χ /lessmuch 1 or Tp /lessmuch Te at wavenumbers κ /lessmuch ( χ 1 / 2 β e ) -1 = 43 / β p . Because there are no indications for such large differences in the proton to electron temperature in the IGM, we will not consider ion sound waves in the following.", "pages": [ 5, 6 ] }, { "title": "4. KINETIC INSTABILITY ANALYSIS OF LANGMUIR OSCILLATIONS FOR NO PERPENDICULAR SPREAD", "content": "We start with the case of no perpendicular spread b = 0 in the relativistic pair distribution function. We use Eq. (42) to find for the growth rate (50) γ p ( κ , b = 0 ) which is positive for values of A > s corresponding to given the very large value of xc (see Eq. (4). As long as R ≤ 1 -ε with the pair parallel momentum distribution provides a positive growth rate γ b . At wavenumbers κ L < κ /lessmuch β -1 e the dispersion relation (118) of Langmuir oscillations readily yields because Langmuir oscillations occur at phase speeds R /greatermuch β e . Inserted into Eqs. (56) and (51) the growth rate as a function of the variable (41) becomes", "pages": [ 6 ] }, { "title": "4.1. Growth rate", "content": "In Fig. 1 we plot the growth rate γ p ( b = 0 ) / ω p , e for the case of no angular spread b = 0 as a function of the normalized wavenumber κ for xc = 10 6 , and different values of the spectral index s = 1 . 5 , 2 , 2 . 5. Because of the large value of xc = 10 6 , all growth rates peak in an extremely narrow range of wavenumber values. First, it can be seen that the weak amplification condition γ p /lessmuch ω R ≤ ω p , e is well satisfied at all values of κ . Secondly, the growth rate γ p exhibits a pronounced maximum.", "pages": [ 6 ] }, { "title": "4.2. Maximum growth rate", "content": "The function C ( A , s ) , defined in Eq. (61), is plotted in Fig. 2 for three values of s = 1 . 5 , 2 , 2 . 5. It has one zero at AN ( s ) = s , is negative for smaller A < s , and positive for larger A > s , in agreement with Eq. (57). Extrema are located at values of A satisfying For values of 1 < s ≤ 2 the function C ( A , s ) attains its maximum value at For the special case s = 2 we find AN ( 2 ) = 2 and A 0 ( 2 ) = 3 and the maximum value For values of s > 2 the function C ( A , s ) has a negative minimum at with and the constant whereas the Landau damping rate is The variable (41) as a function of the normalized wavenumber reads corresponding to · - · and a positive maximum at It is straightforward to show that the location of the maximum A 0 ( s ) < AN ( s ) is always above the location of the zero AN ( s ) , in agreement with Fig. 2. In Table 1 we calculate the locations A 0 ( s ) and values of C max ( s ) for different values of s . For ease of exposition we continue with the simplest case s = 2. From Eq. (60) we then obtain for the maximum kinetic growth rate which occurs at A 0 = 3, corresponding to values of K 0 ( R ) = xc / 3 and values of slightly below unity. In Fig. 3 we show the growth rate from Fig.1 now as a function of the variable A . We note that the location of the maximum and the zero in the case s = 2 agree exactly with the analytical values. With the dispersion relation (52) and the definition (53) we find for the corresponding wavenumber Maximum growth occurs at frequencies in perfect agreement with the reactive result (24). Moreover, the maximum growth rate (71) is given by which is about an order of magnitude larger than the maximum reactive growth rate (25). Apparently, the spread in parallel momentum of the pair distribution function does not reduce the maximum growth rate of parallel Langmuir oscillations, in disagreement with the result of Miniati and Elyiv (2013). At the same values of R 0 and κ 0, because of the exponential factor, the Landau damping rate (63) of Langmuir oscillations is negligibly small 5. KINETIC INSTABILITY ANALYSIS OF LANGMUIR OSCILLATIONS FOR FINITE PERPENDICULAR SPREAD With the correction function (45) for finite perpendicular spreads below the limit b 0, the growth rate in this case is simply related to the growth rate γ p ( b = 0 ) . The growth rate γ p ( b ) with finite spread as compared to the growth rate γ p ( b = 0 ) with no finite spread is enhanced (reduced) if the correction function (45) is greater (smaller) than unity. The correction function (45) reads with the function We noted before that the growth rate γ p ( b = 0 ) is positive only for values of A > s , so we restrict our analysis to this range. For A > s the function (79) is positive for all values of A > s > 1. With A = s + t the function (79) reads with t ∈ ( 0 , ∞ ] . The function is strictly decreasing, as is always negative. No extreme values occur in the interval ( 0 , ∞ ] . For later use we note that the condition h ( A , s ) = 1 / 2 leads to the equation which is identical to Eq. (66), determining the maximum growth rate γ max p ( b = 0 ) through the function C ( A , s ) . Hence, at the maximum A 0 ( s ) the function for all values of s . Moreover, for larger values of A > A 0 ( s ) , the function h ( A , s ) < 1 / 2.", "pages": [ 6, 7, 8 ] }, { "title": "5.1. Correction function for the maximum growth rate", "content": "The maximum growth rate γ max p ( b = 0 ) occurs at A 0 ( s ) listed in Table 1. For values of b < 0 . 1 the variable (44) is smaller than unity for all values of b < 0 . 1, because for s ≤ 4 we calculated A 0 ( s ) ≤ A 0 ( 4 ) = 5 . 56. We therefore use the series expansion (106) for Dawson's integral in Eq. (78) to find With the value (83) the quadratic terms vanishes and we obtain the correction For the maximum value of b = 0 . 1, we calculate the reduction factor B ( A 0 ( s ) , b , s ) -1 for different values of s . The results are listed in Table 1. As can be seen, the reduction factors due to the finite spread in the pair distribution function are tiny, always less than ( -10 -4 ) . Contrary to the statement of Miniati and Elyiv (2013) we find that the finite perpendicular spread does not significantly reduce the maximum growth rate.", "pages": [ 8 ] }, { "title": "5.2. General behavior of the correction function", "content": "Dawson's integral satisfies the linear differential equation so that the first derivative of the correction function (78) is given by The extreme value of the correction function B ( XE ) occurs at XE given by the solution of the transcendental equation Inserting this condition into Eq. (78) we obtain for the extreme value of the correction function We recall that for values of A > A 0 ( s ) , corresponding to XE > b √ A 0 ( s ) / 2, the function h < 1 / 2. The first and second derivative of function (90) are given by and The function BE has a single maximum at X 2 E =( 1 -2 h ) / 2 h given by For given b , Eq. (90) corresponds to the extreme value where we inserted the function h ( AE , s ) from Eq. (79). Even without knowing the value AE , we can draw some interesting conclusions from Eq. (94). For values of s /lessmuch AE /lessmuch ( 2 / b 2 ) the function (94) approaches producing at most a tiny correction over a wide range of s /lessmuch Ae /lessmuch 200 in agreement with our earlier discussion of the maximum growth rate.", "pages": [ 8 ] }, { "title": "6. SUMMARY AND CONCLUSIONS", "content": "The interaction of TeV gamma rays from distant blazars with the extragalactic background light produces relativistic electron-positron pair beams by the photon-photon annihilation process. The created pair beam distribution is unstable to linear two-stream instabilities of both electrostatic and electromagnetic nature in the unmagnetized intergalactic medium. Based on a linear reactive instability analysis Broderick et al. (2012) and Schlickeiser et al. (2012) have concluded that the created pair beam distribution function is quickly unstable to the excitation of electrostatic oscillations in the unmagnetized intergalactic medium, so that the generation of inverseCompton scattered GeV gamma-ray photons by the pair beam is significantly suppressed. Because most of the pair kinetic energy is transferred to electrostatic fluctuations, less kinetic pair energy is available for inverse Compton interactions with the microwave background radiation fields. Therefore, there is no need to require the existence of small intergalactic magnetic fields to scatter the produced pairs, so that the explanation (made by several authors) of the FERMI non-detection of the inverse Compton scattered GeV gamma rays by a finite deflecting intergalactic magnetic field is not necessary. In particular, the various derived lower bounds for the intergalactic magnetic fields are invalid due to the pair beam instability argument. Miniati and Elyiv (2013) have argued that the more appropriate linear kinetic instability analysis, accounting for the longitudinal and the small but finite perpendicular momentum spread in the pair momentum distribution function, significantly reduces the growth rate of electrostatic oscillations by orders of magnitude compared to the linear reactive instability analysis, concluding that the pair beam instability does not modify the pair cascade as in vacuum. We therefore have repeated the linear instability analysis in the kinetic limit for parallel propagating electrostatic oscillations using the realistic pair distribution function with longitudinal and perpendicular spread. Contrary to the claims of Miniati and Elyiv (2013) we find that neither the longitudinal nor the perpendicular spread in the relativistic pair distribution function do significantly affect the electrostatic growth rates. The maximum kinetic growth rate for no perpendicular spread is even about an order of magnitude greater than the corresponding reactive maximum growth rate. The reduction factors to the maximum growth rate due to the finite perpendicular spread in the pair distribution function are tiny, and always less than 10 -4 . We confirm the earlier conclusions by Broderick et al. (2012) and Schlickeiser et al. (2012a), that the created pair beam distribution function is quickly unstable in the unmagnetized intergalactic medium. As our analysis has shown, relativistic kinetic instability studies are notoriously difficult and complicated due to plasma particle velocities close to the speed of light. Therefore extreme care is necessary in order to include all relevant relativistic effects, as done in the present study. We gratefully acknowledge partial support of this work by the Mercator Research Center Ruhr (MERCUR) through grant Pr-2012-0008, and the German Ministry for Education and Research (BMBF) through Verbundforschung Astroteilchenphysik grant 05A11PCA.", "pages": [ 9 ] }, { "title": "7. APPENDIX A: APPROXIMATIONS OF THE INTEGRAL J ( B )", "content": "We introduce so that according to Eqs. (40) The substitution q = tan t provides with Because b is significantly smaller than unity, we approximate so that with arctan b /similarequal b We restrict our analysis to values of β e ≤ R /lessmuch R 0, where R 0 denotes the real phase speed (72), where the maximum growth rate γ max p ( b = 0 ) for no angular spread occurs (see Sect. 4.2). In this case the variable (41) is always larger than 3. The main contribution to the integral (101) is then indeed provided by small values of q /lessmuch 1, so that the approximation (100) is justified. The integral (101) can be expressed in terms of Dawson's integral (Abramowitz and Stegun 1972, Ch. 7.1; Lebedev 1972, Ch. 2.3), the error function of imaginary argument, as with Dawson's integral (103) has a maximum Fm = 0 . 541 at xm = 0 . 924, the series expansion and the asymptotic expansion In Figures 4 and 5 we compare the numerically evaluated exact integral (99) with its approximation (104) for p = 2 and two values of A = 3 and A = 100. In both cases the agreement is excellent for values of b < 0 . 1. According to Eqs. (96) and (98) we obtain the approximations The small argument expansion (106) readily yields which agrees with Eq. (42), so that the correction function (45) becomes with 8. APPENDIX B: COLLECTIVE ELECTROSTATIC MODES Eq. (47) together with the real part of the dispersion relation (37) reads (a) In the case of phase speeds larger than β e , In order to use the asymptotic expansions (28) for protonelectron temperature ratios χ < ξ -1 = 1836 we have to consider three cases: both arguments of the Z ' -function are large compared to unity, so that we may use the asymptotic expansion (b) In the case of intermediate phase speeds, we use the expansion (114) in the third term of Eq. (112) and the asymptotic expansion for small arguments in the second term of Eq. (112). (c) In the case of very small phase speeds, we use the expansion (116) in the second and third term of Eq. (112). We consider each case in turn. Here we readily obtain for Eq. (112) 8.1. Large phase speed R /greatermuch β e yielding the dispersion relation with the solution The requirement R /greatermuch β e implies the wavenumber restriction β 2 e κ 2 /lessmuch 2 . 5. Likewise, the subluminality requirement R < 1 demands In this wavenumber range the solution (119) reduces to corresponding to Langmuir oscillations for 2 -1 / 2 β e ≤ k λ De /lessmuch 1 with the electron Debye length λ De = β ec / √ 2 ω p , e . 8.2. Intermediate phase speed √ χξβ e = β p /lessmuch R /lessmuch β e In this case we derive for Eq. (112) yielding the dispersion relation with the two formal solutions The first solution violates the restriction R 2 /lessmuch β 2 e , leaving as only solution This ion sound wave solution has to fulfill the second restriction R 2 /greatermuch χξβ 2 e , corresponding to the condition which is only possible for values of χ /lessmuch 1 or Tp /lessmuch Te . In this case the solution (128) holds for wavenumbers κ 2 β 2 e /lessmuch χ -1 . Therefore the ion sound wave solution only exists for Tp /lessmuch Te at wavenumbers ( λ Dek ) 2 /lessmuch ( 2 χ ) -1 with frequencies 8.3. Very small phase speed R /lessmuch √ χξβ e = β p In this case we derive for Eq. (112) yielding the dispersion relation The very small phase speed requirement R 2 /lessmuch χξβ 2 e corresponds to which cannot be fulfilled. Therefore no electrostatic mode with very small phase speeds exists.", "pages": [ 9, 10, 11 ] }, { "title": "REFERENCES", "content": "Abramowitz, M., Stegun, I. A., 1972, Handbook of Mathematical Functions, NBS, Washington Broderick, A. E., Chang, P., Pfrommer, C., 2012, ApJ 732, 22 Cairns, I. H., 1989, Phys. Fluids B 1, 204 Dermer, C. D., Cavadini, M., Razzaque, S., Finke, J. D., Chiang, J., Lott, B., 2011, ApJ 733, L21 Dolag, K., Kachelriess, M., Ostrapchenko, S., Tomas, R., 2011, ApJ 727, L4 Elyiv, A., Neronov, A., Semikoz, D. V., 2009, Phys. Rev. D 80, 023010 Fried, B. D., Conte, S. D., 1961, The Plasma Dispersion Function, Academic Press, New York Hui,L., Gnedin, N. Y., 1997, MNRAS 292, 27 Hui,L., Haiman, Z., 2003, ApJ 596, 9 Lebedev, N. N., 1972, Special Functions and their applications, Dover, New York Miniati, F., Elyiv, A., 2013, ApJ 770, 54 Neronov, A., Vovk, I., 2010, Science 328, 73 Schlickeiser, R., 2002, Cosmic Ray Astrophysics, Springer, Berlin Schlickeiser, R., 2010, Phys. Plasmas 17, 112105 Schlickeiser, R., Elyiv, A., Ibscher, D., Miniati, F., 2012b, ApJ 758, 101 Schlickeiser, R., Ibscher, D., Supsar, M., 2012a, ApJ 758, 102 Schlickeiser, R., Yoon, P. H., 2012, Phys. Plasmas 19, 022105 Takahashi, K., Mori, M., Ichiki, K., Inoue, S., Takami, H., 2012, ApJ 744, L42 Tavecchio, F., Ghisellini, G., Bonnoli, G., Foschini, L., 2011, MNRAS, 414, 3566 Taylor, A. M., Vovk, I., Neronov, A., 2011, A & A 529, A144 Vovk, I., Taylor, A. M., Semikoz, D. V., Neronov, A., 2012, ApJ 747, L14", "pages": [ 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "An investigation of the steep, high-frequency (i.e., ν ∼ 12GHz) radio spectra among a sample of 31 local infrared-bright starburst galaxies is carried out in light of their HST -based merger classifications. Radio data covering as many as 10 individual bands allows for spectral indices to be measured over three frequency bins between 0 . 15 -32 . 5GHz. Sources having the flattest spectral indices measured at ∼ 2 and 4 GHz, arising from large free-free optical depths among the densest starbursts, appear to be in ongoing through post-stage mergers. The spectral indices measured at higher frequencies (i.e., ∼ 12GHz) are steepest for sources associated with ongoing mergers in which their nuclei are distinct, but either share a common stellar envelope and/or exhibit tidal tails. These results hold after excluding potential AGN based on their low 6.2 µ m PAH EQWs. Consequently, the low-, mid-, and high-frequency spectral indices each appear to be sensitive to the exact merger stage. It is additionally shown that ongoing mergers, whose progenitors are still separated and share a common envelope and/or exhibit tidal tails, also exhibit excess radio emission relative to what is expected given the far-infrared/radio correlation, suggesting that there may be a significant amount of radio emission that is not associated with ongoing star formation. The combination of these observations, along with high-resolution radio morphologies, leads to a picture in which the steep high-frequency radio spectral indices and excess radio emission arises from radio continuum bridges and tidal tails that are not associated with star formation, similar to what is observed for so-called 'taffy' galaxies. This scenario may also explain the seemingly low far-infrared/radio ratios measured for many highz submillimeter galaxies, a number of which are merger-driven starbursts. Subject headings: galaxies:active - galaxies:starbursts - infrared:galaxies - radio continuum:galaxies stars:formation", "pages": [ 1 ] }, { "title": "THE ROLE OF MERGER STAGE ON GALAXY RADIO SPECTRA IN LOCAL INFRARED-BRIGHT STARBURST GALAXIES", "content": "Eric J. Murphy Infrared Processing and Analysis Center, California Institute of Technology, MC 220-6, Pasadena CA, 91125, USA; [email protected] Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA Submitted to ApJ June 7, 2013; Accepted August 15, 2013", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Encoded in the radio spectra of star-forming galaxies, which are typically well characterized by a power law ( S ν ∝ ν -α ), lies information on the thermal and non-thermal energetic processes powering them. Both thermal and non-thermal emission processes are typically associated with massive star formation, underlying the basis for the well known far-infrared/radio correlation (de Jong et al. 1985; Helou et al. 1985; Condon 1992). Far-infrared emission arises from re-radiated UV/optical photons that heat dust grains surrounding massive starforming regions. The young, massive O/B stars in such regions, whose lifetimes are /lessorsimilar 10Myr, produce ionizing radiation that is proportional to the amount of freefree emission. Stars more massive than /greaterorsimilar 8 M /circledot end their lives as supernovae (SNe), whose remnants (SNRs) are thought to be the primary accelerators of cosmicray (CR) electrons, which emit synchrotron emission as they propagate through a galaxy's magnetized interstellar medium (ISM). The non-thermal emission typically dominates the freefree emission at frequencies /lessorsimilar 30GHz (Condon 1992), having a relatively steep spectrum (i.e., α ≈ 0 . 83; Niklas et al. 1997). Thermal bremsstrahlung (free-free) emission, on the other hand, has a much flatter spectrum ( α ≈ 0 . 1), making it difficult to separate this component from the non-thermal emission. Thus, at frequencies /greaterorsimilar 30GHz, where the thermal fraction starts to become large, radio observations should become robust measures for the ongoing star formation rate in galaxies (Murphy et al. 2011b, 2012). However, this has been shown to not necessarily be the case for a number of local luminous infrared galaxies (LIRGs), whose infrared (IR; 8 -1000 µ m) luminosities exceed L IR /greaterorsimilar 10 11 L /circledot . In a number of these infrared-bright starbursts, their high-frequency (i.e., /greaterorsimilar 10 GHz) radio spectra are much steeper than expected for an increased thermal fraction, and in some cases, even show possible evidence for spectral steepening (Clemens et al. 2008, 2010; Leroy et al. 2011). Understanding the physical underpinnings driving this behavior can greatly help with the interpretation of radio observations for higher redshift starbursts, which is important given that infrared-luminous galaxies appear to be much more common in the early universe and dominate the star formation rate density in the redshift range spanning 1 /lessorsimilar z /lessorsimilar 3, being an order of magnitude larger than today (e.g., Chary & Elbaz 2001; Le Floc'h et al. 2005; Caputi et al. 2007; Murphy et al. 2011a; Magnelli et al. 2013). In the local universe, it is well-known that LIRGs and ultraluminous LIRGs (ULIRGs; L IR /greaterorsimilar 10 12 L /circledot ) appear to be undergoing an intense starburst phase. Within these systems are compact star-forming regions that have been been triggered predominantly through major mergers (see e.g., Armus et al. 1987, 1988, 1989, 1990; Sanders et al. 1988a,b; Murphy et al. 1996; Veilleux et al. 1995, 1997, 2002). Major mergers have the ability to significantly complicate the interpretation of observed radio properties when individual systems are not resolved, the classic case being the so-called 'taffy' galaxies (Condon et al. 1993, 2002). When unresolved, the systems appear to have nearly a factor of ∼ 2 more radio continuum emission relative to what is expected giving the far-infrared/radio correlation, as well as unusually steep (i.e., α ≈ 1 . 0) radio spectra. However, when resolved, it is clearly found that the integrated radio properties are driven by radio continuum emission that forms a bridge connecting the galaxy pairs; the radio continuum emission in the bridges are characterized by a steep spectrum and contain roughly an equal amount of emission as the individual galaxies, which have radio spectral indices and far-infrared/radio ratios typical of normal spiral galaxies (Condon et al. 1993, 2002). In this paper, an explanation is presented for occur- rences of steep radio spectra observed in local ULIRGs within the context of there merger stage. The paper is organized as follows. In § 2 the sample and data used in the analysis are presented. In § 3 the major results are described, and then discussed in § 4 to piece together a self consistent picture describing the radio properties for this sample of local starbursts. Finally, the main conclusions are summarized in § 5.", "pages": [ 1, 2 ] }, { "title": "2. DATA AND ANALYSIS", "content": "The galaxy sample being analyzed here is drawn from sources in the IRAS revised Bright Galaxies Sample (Soifer et al. 1989; Sanders et al. 2003) having 60 µ m flux densities larger than 5.24 Jy and far-infrared (FIR; 42 . 5 -122 . 5 µ m) luminosities ≥ 10 11 . 25 L /circledot . The 40 systems that meet these criteria were originally imaged by Condon et al. (1991) at 8.4 GHz with 0 . '' 25 resolution. For 31 of these sources, Clemens et al. (2008) presented additional new very large array (VLA) 22.5GHz data (obtained in D-configuration), along with new 8.4 GHz data for 7 objects (obtained in C-configuration) and archival observations to increase the radio spectral coverage of each system. Additional lower frequency data at 244 and 610MHz were obtained and presented in Clemens et al. (2010), and for 14 and 13 sources existing 6.0 and 32.5 GHz measurements, respectively from Leroy et al. (2011) are used. The present analysis focuses on these 31 galaxies (see Table 1) having well sampled radio spectra between 0.15 and 32.5 GHz (typically 6 bands with as many as 10 including some combination of data observed at 0.151, 0.244, 0.365, 0.610, 1.4, 4.8, 6.0, 8.4, 15.0, 22.5, and 32.5 GHz). While these radio data span over an order of magnitude in frequency and have varying spatial resolutions, missing short-spacing data will not significantly affect the results given that the highest frequency radio data should be sensitive to /greaterorsimilar 1 ' angular scales, much larger than the typical emitting regions for this sample of infrared-bright galaxies. For NGC3690, UGC08387, UGC08696, and IRASF15163+4255, the 22.5 GHz flux densities of Clemens et al. (2008) are significantly lower than the 32.5 GHz flux densities presented in Leroy et al. (2011). These discrepancies may be associated with the difficulties of measuring flux densities through the pressure-broadened 22.3GHz line of atmospheric water vapor, and therefore the 22.5 GHz flux densities for these 4 sources are not included in the present analysis. Additionally, the 15 GHz flux density UGC 08387, which was reduced from archival VLA data by Clemens et al. (2008), is found to be discrepant with the combination of lower frequency radio data and the 32.5 GHz flux density of Leroy et al. (2011), and is therefore not used. For IRAS F15163+4255, the 4.8 GHz flux density measured by Gregory & Condon (1991) using the 91 m telescope in Green Bank is found to be significantly discrepant with the 6.0 GHz flux density of Leroy et al. (2011), and is dropped from the analysis as well. IRAS -based far-infrared fluxes and logarithmic farinfrared-to-radio ratios at 1.4 and 8.4 GHz, each of which were reported in Clemens et al. (2008), are also used in the analysis. The far-infrared flux, F FIR , is estimated using IRAS 60 and 100 µ m flux densities and the relation given by Helou et al. (1985) such that, Logarithmic F FIR /radio ratios at frequency ν are also defined using the relation given by Helou et al. (1985) such that Each of these values is given in Table 1 for each source. The mid-infrared spectral properties were collected by the Spitzer Infrared Spectrograph (IRS; Houck et al. 2004) as part of the Great Observatories All-Sky LIRG Survey (GOALS Armus et al. 2009), and are taken from Stierwalt et al. (2013). For 2 sources, IRS observations missed the nucleus of the galaxy (IRAS F01417+1651, IRASF03359+1523). In the analysis 6.2 µ m polycyclic aromatic hydrocarbon (PAH) equivalent widths (EQWs) and 9.7 µ m silicate strengths (see Stierwalt et al. 2013) are used. Similar to Murphy et al. (2013), the measured 6.2 µ m PAH EQWs is used as an active galactic nuclei (AGN) discriminant. Sources hosting AGN typically have very small PAH EQWs (e.g., Genzel et al. 1998; Armus et al. 2007). Specifically, starburst dominated systems appear to have 6.2 µ m PAH EQWs that are /greaterorsimilar 0 . 54 µ m (Brandl et al. 2006), while AGN have 6.2 µ m PAH EQWs are /lessorsimilar 0 . 27 µ m. In this paper it is assumed that all sources having PAH EQWs ≥ 0 . 27 µ m are primarily powered by star formation. Given that the PAH EQW is used as an AGN discriminant, we exclude both sources missing IRS data in the present analysis when focusing on star formation dominated systems. The silicate strength at 9.7 µ m is defined as s 9 . 7 µ m = log( f 9 . 7 µ m /C 9 . 7 µ m ), where f 9 . 7 µ m is the measured flux density at the central wavelength of the absorption feature and C 9 . 7 µ m is the expected continuum level in the absence of the absorption feature. For two sources (UGC04881 and NGC3690), IR luminosity-weighted averages of individual IRS measurements from each nuclei are used. This analysis additionally makes use of merger classifications based on available Hubble Space Telescope ( HST ) imaging for 29 galaxies taken as part of an HST /ACS survey of the GOALS sample (Kim et al. 2013, A.S. Evans et al. 2013, in preparation). The merger classifications were assigned using both HST optical [i.e., Advance Camera for Surveys (ACS) B - (F435W) and I -band (F814W)] and near-infrared [i.e., Near Infrared Camera and Multi-Object Spectrometer (NICMOS) H -band (F160W)] imaging. The individual mergers stages are classified by an integer value ranging from 0 to 6, and are described in detail by Haan et al. (2011). Briefly, the numerical classifications are defined in the following way: 0 = non-merger, 1 = pre-merger, 2 = ongoing merger with separable progenitor galaxies, 3 = ongoing merger with progenitors sharing a common envelope, 4 = ongoing merger with double nuclei plus tidal tail, 5 = post-merger with single nucleus plus prominent tail, and 6 = post-merger with single nucleus and a disturbed morphology. For two sources, merger classifications are not available. HST data were not taken for NGC 6286, and the HST /ACS data saturated for the observations of UGC08058 (Mrk231). Each of these properties is given in Table 1.", "pages": [ 2, 3, 4 ] }, { "title": "3. RESULTS", "content": "A number of the infrared-bright galaxies being investigated here are known to have high-frequency (i.e., /greaterorsimilar 10 GHz) radio spectra that are much steeper than expected given that the thermal fraction should increase with frequency and work to flatten the spectra (e.g., Clemens et al. 2008, 2010; Leroy et al. 2011). In the following section, correlations between various galaxy properties that may help to explain such steep, highfrequency radio spectral indices are investigated after removing potential AGN from the sample using the midinfrared spectroscopic data. A number of physical considerations have already been put forth and discussed by Clemens et al. (2008), including those used to explain the high frequency turnover in the starburst galaxy NGC1569 by Lisenfeld et al. (2004), and we refer the reader to these papers. To summarize, these physical processes include synchrotron aging, stochastic events such as radio hypernovae, rapid temporal variations in the star formation rate, and the escape of low energy CRs through convective transport. Each scenario has been found to be unsatisfactory for this sample of local starburst galaxies in a large part due to the extremely short (i.e., ∼ 10 4 yr; Condon et al. 1991) radiative lifetimes estimated for CR electrons in such systems, which requires a near continuous injection of particles.", "pages": [ 4, 5 ] }, { "title": "3.1. Radio Spectral Curvature", "content": "The radio spectra of each source are broken up into low-, mid-, and high-frequency bins, so the spectral curvature of each source can be crudely measured. The low frequency bin is defined as ν < 5 GHz, the mid-frequency bin spans 1 GHz < ν < 10 GHz, and the high-frequency bin is defined as ν > 4GHz. This allows the radio spectral index to be calculated using typically 3 or more data points in each frequency bin (see exact numbers in Table 2). The spectral index is estimated by an ordinary least squares fit over each frequency bin range, weighted by the photometric errors. Each spectral index is given in Table 2 along with uncertainties from the fitting. Additionally given in Table 2 is the average frequency, weighted by the signal-to-noise of each photometric data point, over which the radio spectral index was calculated. In the top panel of Figure 1, the median spectral indices of each bin are plotted as a function of the median frequency over which the spectral indices were calculated. The median frequencies of the low-, mid-, and high-frequency bins are ≈ 2, 4, and 12 GHz, respectively. The horizontal error bar indicates the standard deviation in the average frequencies, while the vertical error bar is the error on the average spectral index. Average spectral indices are shown for the entire sample, as well as for stabursting and AGN-dominated systems, as defined by their 6.2 µ m PAH EQWs (see § 2). While a rather clear distinction between the average radio spectral indices measured between 1.49 and 8.44 GHz (essentially the α mid presented here) was pointed out by Murphy et al. (2013), whereby AGNdominated sources typically have a much flatter radio spectral index compared to starburst-dominated sources, the high-frequency radio spectral indices are statistically indistinguishable between both AGN- and starburstdominated systems. Using the 6.2 µ m PAH EQW to split the sources up into starburst and AGN-dominated systems, the median high-frequency radio spectral index for starburst dominated systems is 0.78, with a standard deviation of 0.16. For the AGN-dominated systems, the median spectral index is 0.71, albeit with a slightly larger scatter of 0.20. Thus, the steep spectral indices at ∼ 12 GHz do not seem to be the result of radio emission associated with an AGN. In the bottom panel of Figure 1 the average spectral indices in each frequency bin are plotted for only those sources whose energetics are thought to be dominated by star formation based on having a 6.2 µ m PAH EQW ≥ 0 . 27 µ m. The over plotted lines correspond to expectations based on 4 different radio spectra that are shown in Figure 2 to illustrate the change in radio spectral index as a function of frequency based on the increased thermal fraction towards higher frequencies for a normal star-forming galaxy. The radio-to-infrared models are based on the physical description given in Murphy (2009), where the total infrared (8 -1000 µ m) flux is set to F IR = 10 -9 Wm -2 and the source is assumed to follow the far-infrared/radio correlation. The radio spectra are constructed following the prescription given in Murphy (2009), resulting in a non-thermal spectral index of ≈ 0.83 with 1.4 GHz thermal radio fractions of f 1 . 4GHz T ≈ 5 (dashed line), 10 (dotted line), and 15% (dot-dashed line). These 1.4 GHz thermal fractions correspond to 30 GHz thermal fractions of f 30GHz T ≈ 33, 51, and 63%, respectively. As pointed out by Clemens et al. (2008), and also seen by Leroy et al. (2011) using new 33 GHz VLA data, there is a clear trend of spectral flattening/steepening towards lower/higher frequencies among this sample of local, merger-driven starburst galaxies, which is clearly discrepant from the expectations of typical radio spectra for star-forming galaxies. However, by taking the model having a 1.4 GHz thermal fraction of 5%, modified by free-free absorption where the free-free optical depth is τ ff = ( ν/ν b ) -2 . 1 and becomes unity at ν b = 1GHz, the model largely fits the observed trend (solid line).", "pages": [ 5 ] }, { "title": "3.2. The High-Frequency Indices Compared to the Compactness of the Sources", "content": "Recently, it has been shown that the flattening of the radio spectrum measured between 1.4 and 8.4 GHz (i.e., essentially the spectral index in the mid-frequency bin presented here) among this sample of local infraredbright starbursts increases with increasing 9.7 µ msilicate optical depth, 8.44GHz brightness temperature, and decreasing size of the radio source even after removing potential AGN (Murphy et al. 2013), supporting the idea that compact starbursts show spectral flattening as the result of increased free-free absorption (Condon et al. 1991). In Figure 3 the strength of the 9.7 µ m silicate feature is plotted against the high-frequency radio spectral index indicating the absence of any such trend. Even after removing sources identified as harboring AGN due to their small 6.2 µ m PAH EQW, a trend is still not found. This indicates that the compactness of the starburst likely does not affect the steepness of the highfrequency radio spectrum.", "pages": [ 5, 6 ] }, { "title": "3.3. The Radio Spectral Indices as a Function of Merger Stage", "content": "As already shown by Condon et al. (1991), there is a general trend among this sample of local starbursts in which more compact (i.e., higher surface brightness) sources tend to have flatter radio spectral indices, consistent with compact sources becoming optically thick at low (i.e., ν /lessorsimilar 5GHz) radio frequencies. Another way one can illustrate this is by plotting the radio spectral index as a function of merger stage, which is done for the low-, mid-, and high-frequency radio spectral indices in Figure 4. The top, middle, and bottom panels plot radio spectral indices at frequencies of ∼ 2, 4, and 12 GHz, respectively, against merger stage. The horizontal lines in each panel corresponds to the expected spectral index at that frequency given the model radio spectra shown in Figure 2. Only two models are shown; a normal starforming galaxy having a 1.4 GHz thermal fraction of 10%, and a star-forming galaxy with an original 1.4 GHz thermal fraction of 5% except that the free-free optical depth becomes unity at 1 GHz. In the top and middle panels, it seems that the flattest spectrum sources falling below what is expected for a normal star-forming galaxy, as indicated by the dotted line, are only found in ongoing and post-mergers with strongly disturbed morphologies. The only outlier appears to be the non-merger IRAS F10173+0828, which has a flat spectrum at all frequency bins and hosts an OH megamaser (Mirabel & Sanders 1987) along with a highly compact radio core (Lonsdale et al. 1993). This finding is robust even after removal of potential AGN as indicated by low 6.2 µ m PAH EQWs. However, this trend does not persist when the high-frequency spectral indices, measured at ∼ 12GHz, are plotted against merger sequence in the bottom panel. After removing potential AGN, the steepest spectrum sources, falling well above what is expected for a normal star-forming galaxy as indicated by the dotted line, are only found among those sources categorized as ongoing mergers in which galaxy nuclei are distinct, but share a common envelope and/or exhibit tidal tails as observed in their stellar light. In contrast, pre- and post-stage mergers do not seem to exhibit such steep spectral indices. Thus, the low-, mid-, and high-frequency spectral indices each appear to be sensitive to the exact stage of the merger.", "pages": [ 6, 7 ] }, { "title": "3.4. Far-Infrared-to-Radio Ratios as a Function of Merger Stage", "content": "In Figure 4 it has been shown that sources having the steepest spectra appear to be associated with systems classified as ongoing mergers in which the progenitors are separable and share a common envelope or display strong tidal features. To determine whether merger stage affects the total radio continuum emission, rather than just the spectral index, the difference between the nominal and observed logarithmic F FIR /radio ratios at 1.4 and 8.4 GHz are plotted in the top and bottom panels of Figure 5, respectively, against merger stage. The difference between the nominal and observed F FIR /radio ratios are defined as, such that a positive value indicates excess radio emission per unit far-infrared emission, and q 0 ,ν is the nominal F FIR /radio ratio for local star-forming systems. Under the assumption that the far-infrared emission is a good measure for the total star formation rate in each system, a large value of δq ν indicates excess radio emission per unit star formation rate. At 1.4 GHz, q 0 , 1 . 4GHz ≈ 2 . 34dex with a scatter of 0.26 dex among ∼ 1800 galaxies spanning nearly 5 orders of magnitude in luminosity (e.g. Yun et al. 2001), among other properties (e.g. Hubble type, far-infrared color, and F FIR /optical ratio). Assuming a typical radio spectrum, with a spectral index of α ≈ 0 . 8 (Condon 1992), this translates into a nominal F FIR to 8.4 GHz flux density ratio of q 0 , 8 . 4GHz ≈ 2 . 94dex. In the top panel of Figure 5, it can be shown that galaxies having the largest 1.4 GHz excesses appear to be ongoing mergers that share a common envelope before and after removal of potential AGN. However, there is a good deal of scatter in this figure, and there are a number of sources for which significant excesses of farinfrared emission are observed relative to the measured 1.4 GHz flux densities. Since the q 1 . 4GHz ratios are clearly affected by free-free absorption at low frequencies in the most compact starbursts (Condon et al. 1991), it may be better to see if such trend persists using emission at a higher frequency, where free-free absorption is considered to be largely negligible, such as 8.4 GHz. This is shown in the bottom panel of Figure 5, where δq 8 . 4GHz is plotted against merger stage and the exact same behavior is found, albeit with a much smaller dispersion per merger sequence bin. Excluding potential AGN, there appears to be a trend in which sources classified as ongoing mergers that share a common envelope have excess radio emission per unit star formation rate relative to both early and post-stage mergers, which exhibit nominal F FIR /radio ratios. Thus, similar to radio spectral indices, the far-infrared-to-radio ratios appear to be sensitive to merger stage.", "pages": [ 7 ] }, { "title": "4. DISCUSSION", "content": "As previously stated, a number of physical scenarios to explain the steep radio spectral indices among this sample of local starburst galaxies have been proposed in the literature, particularly by the original study of Clemens et al. (2008). While simple arguments to produce steep spectra via increased synchrotron and inverse Compton losses appear to fall short, given that the most compact starbursts are not the sources that exhibit the steepest spectra, an alternative explanation is proposed. It has been know for some time that, when caught in the act, merging systems can create bridges of synchrotron emission that stretch between the progenitor galaxies, whose brightness contours resemble stretching strands of 'taffy' (Condon et al. 1993). The two most well-known cases of 'taffy' galaxies are the face-on colliding systems UGC12914/5 (Condon et al. 1993) and UGC 813/6 (Condon et al. 2002). The space between the spiral galaxy pairs UGC12915/5 and UGC813/6 are filled with synchrotron emission, implying that they must contain both relativistic electrons and magnetic fields. The magnetic field of the bridge is presumably stripped from the merging spiral galaxy disks as they interpenetrated during a recent collision, however there is some debate as to whether the relativistic electrons in the bridge have escaped the merging spirals, as originally suggested by Condon et al. (1993), or are rather a new population of relativistic electrons that have been diffusively accelerated in a shock associated with a gas dynamical interaction during the merger (Lisenfeld & Volk 2010). As discussed below, if such a scenario is also responsible for the properties of the starbursting mergers investigated here, the latter explanation appears more appealing. Other prominent features of the 'taffy 'systems include: (1) An exceptionally steep spectrum located at the center of the radio bridges (i.e., α ≈ 1 . 3, which is roughly ∆ α = 0 . 5 steeper than the spectral indices of the galaxy disks). (2) A ratio of F FIR /radio emission for the entire system that is significantly lower than that of normal star-forming galaxies, but typical for the individual spiral galaxy disks. (3) A significant fraction of H i stripped by the collisions that resides between the spiral disks. (4) A significant amount of molecular gas in the bridge (Braine et al. 2003) whose physical conditions are comparable to those in the diffuse clouds of the Galaxy (Zhu et al. 2007). (5) A small amount of warm (i.e., 5 -17 µ m; Jarrett et al. 1999) and cold (i.e., 450 and 850 µ m; Zhu et al. 2007) dust located in the bridge, indicating a low amount on ongoing star formation. (6) Rotational lines of H 2 emission that dominate the midinfrared spectrum and appear strongest near the center of the bridge (Peterson et al. 2012).", "pages": [ 7, 8 ] }, { "title": "4.1. 'Taffy'-Like Starbursts", "content": "Among the merging starbursts investigated here, those having the steepest high-frequency ( ν ∼ 12GHz) radio spectra, and significant excesses of radio emission relative to their observed far-infrared emission, appear to be galaxies in an ongoing merger and either share a common stellar envelope or have significant tails. Both of these radio characteristics are also found in 'taffy' systems, whose steep spectra and excess radio emission per unit star formation rate arises from the radio continuum emission in the bridge connecting the merging galaxies. Given that simple physical arguments based on the escape and radiative cooling times of CR electrons to explain these properties among local compact starburst fail, a taffy-like merger scenario appears to be an appealing way to explain the low F FIR /radio ratios and steep high-frequency spectra. In Figure 6, radio continuum maps are shown for 6 galaxies that largely span the full merging sequence and do not appear to harbor buried AGN based on their measured 6.2 µ m PAH EQWs. Each system has been highlighted as a star in Figures 4 and 5. Mrk331 is a pre-merger, exhibiting a symmetric 1.4 GHz radio continuum morphology, as well as a typical radio spectrum and F FIR /radio ratio. UGC04881 is characterized as an ongoing merger with separable progenitor galaxies. IC1623 is optically classified as an ongoing merger whose progenitors share a common envelope, which is consistent with its 1.4 and 8.4 GHz radio morphologies. It has a high-frequency radio spectral index of ≈ 0.95, and a factor of 1.6 excess radio emission relative to what is expected given the far-infrared/radio correlation. The excess radio emission appears easily explained by summing the diffuse synchrotron emission that is not associated with the colliding star-forming galaxy disks. IRASF15163+4255 is classified (by the HST data) as an ongoing merger with two nuclei and a prominent tidal tail. This source has very weak 1.4 and 8.4 GHz radio continuum emission that appears to form a bridge between the two progenitor galaxies, keeping it right on the far-infrared/radio correlation. This source also exhibits a somewhat steep high frequency radio spectral index of ≈ 0.84. Like IRAS F15163+4255, UGC 08387 is also classified as an ongoing merger harboring two nuclei and a prominent tidal tail. Given that the progenitor galaxies are not separated by a large angular extent, the high (0 . '' 25) 8.4 GHz radio continuum map of Condon et al. (1991) is necessary to resolve both components. Similar to IC 1623, the 8.4 GHz morphology is consistent with its optical merger classification, as there are tidal tails of synchrotron emission. UGC 08387 has a high-frequency radio spectral index of ≈ 0 . 70 and 40% excess radio emission at 8.4 GHz than expected giving its far-infrared flux. Finally, the post-stage, ultra-compact, merger IRAS F01364+1042 is shown, also requiring the use of the high-resolution 8.4 GHz radio map in which their appears to be a single source having seemingly undisturbed radio continuum contours. This system has an extremely flat low- and mid-frequency radio spectral index, presumably due to free-free absorption occurring in the compact starbursts, but a rather normal highfrequency spectral index. Consequently, its q 1 . 4GHz value is significantly larger than average, unlike its rather typical q 8 . 4GHz value. the merger process. A similar situation is observed by plotting the 1.4 to 8.4 GHz (3 . '' 1 resolution) spectral index contours on the I -band (F814W) HST /ACS image of IRASF15163+4255 in Figure 8. While the northern and southern galaxy nuclei appear to have significantly different spectral indices, peaking around α 8 . 4GHz 1 . 4GHz ≈ 0 . 6 and α 8 . 4GHz 1 . 4GHz ≈ 0 . 8 , respectively, there is a strong steepening of the indices towards the emission connecting the two galaxy nuclei peaking at α 8 . 4GHz 1 . 4GHz ≈ 1 . 1 at the midpoint between the interacting disks. Explaining the steep radio spectral indices and lower than average F FIR /radio ratios as the result of ongoing mergers seems to be fairly well supported by the data. To date, it has been hard to explain the steep highfrequency spectra in this sample of starbursts given that the rapid ( ∼ 10 4 yr) synchrotron and inverse Compton cooling times render diffusion and escape losses to be negligible, and thus cannot work to steepen the spectra. In fact, escape as an explanation is in disagreement with the data, given that the loss of CR electrons would effectively lower the total synchrotron power, whereas galaxies with steep high-frequency spectral indices can have low F FIR /radio ratios. By creating a magnetized medium through the dynamical interaction of the merging galaxy pairs, similar to that of the taffy systems, provides a location where the CR electrons can radiatively cool without the continuous injection and acceleration of new particles. Given the rapid cooling times due to radiative losses in the compact starbursts embedded in the merging disks, Focusing in on IC 1623, 1.4 to 8.4 GHz (4 . '' 7 resolution) spectral index contours are overlaid on an I -band (F814W) HST /ACS image in Figure 7. The radio spectral indices are found to be rather typical relative to normal star-forming galaxies for both merging galaxy disks (i.e., α 8 . 4GHz 1 . 4GHz ∼ 0 . 7). The eastern galaxy disk hosts the most deeply embedded star formation, and the radio spectral index clearly flattens right on the peak of the starburst. Similar to the 'taffy' systems, the radio spectral index is found to steepen significantly along the radio continuum bridge connecting the galaxy pair, peaking roughly at the midpoint between the two galaxy disks with α 8 . 4GHz 1 . 4GHz ≈ 0 . 92. Furthermore, like the 'taffy' systems, high-resolution 12 CO observations show a significant amount of molecular gas located in the overlap regions connecting the two galaxy nuclei (Yun et al. 1994; Iono et al. 2004), thus providing material for the galactic magnetic fields to be anchored as they stretch during ing times of CR electrons that may populate a bridge, assuming the magnetic field stays in equipartition with the disk it is being pulled out of, and that the radiation field and ISM density leads to negligible inverse Compton and bremsstrahlung cooling, respectively. Thus, it appears more likely that charged particles in such regions have been diffusively accelerated via shocks associated with the available mechanical energy of the merger (e.g., Lisenfeld & Volk 2010). The speed of the galaxy collisions are a few ∼ 100kms -1 (e.g., G. Privon et al. 2013, in preparation), and therefore super-Alfv'enic for typical ISM Alfv'en speeds a few ∼ 10kms -1 . The Alfv'enic Mach number is much larger than unity, at the order M 2 A ∼ O (100) /greatermuch 1, allowing for shocks that can diffusively accelerate CR particles. This is similar to the Alfv'enic Mach numbers within the Sedov-Taylor expansion phase of a SNRs. it seems unlikely that the relativistic electrons associated with synchrotron bridges and/or tidal tails were injected and accelerated in the starburst itself, and have propagated to such distances. As an example, the distance to the midpoint between the two 1.4 GHz 'hot spots' in the merging disks of IC 1623 is ≈ 5 '' , which projects to a linear distance of ≈ 2kpc at a distance of 85.5 kpc. This distance is actually smaller than the ≈ 15 '' (6 kpc) separation between the two galaxy nuclei. Following the description for propagation and physical processes responsible for CR electron energy losses given in Murphy (2009), it can be shown that the time for 1.4 GHz emitting electrons to reach this midpoint is significantly longer than their radiative lifetime. Assuming random walk diffusion, characterized by an energy-dependent diffusion coefficient, the propagation time is ∼ 2 . 6 × 10 7 yr. The radiative lifetime for CR electrons emitted within a 2 . '' 5 radius of the western galaxy, which likely includes the bulk 1.4 GHz emission associated with star formation in that disk, is ∼ 2 . 9 × 10 5 yr, nearly two orders of magnitude shorter. Additionally, the time that the merger is expected to spend in either of these classifications is no more than a few times ∼ 10 7 yr (Haan et al. 2011), which is roughly an order of magnitude longer than the synchrotron cool- This explanation for diffusive acceleration via shocks associated with the merger is additionally supported by integral field spectroscopic observations of infraredbright mergers. For the case of IC 1623, Rich et al. (2011) have shown that optical emission line diagnostic ratios indicate the presence of widespread shock excitation induced by ongoing merger activity. The energy associated with the shocks in the interacting regions between the two galaxy disks is estimated to be ∼ 4 × 10 42 erg s -1 based on the amount of H α line emission having widths /greaterorsimilar 100kms -1 in the interacting region (i.e., 80 × L H α where L H α ∼ 5 × 10 40 erg s -1 ; Rich et al. 2011, 2010). This value is nearly a factor of ∼ 1 . 5 × 10 3 times larger than the total 8.4 GHz luminosity of the source ( νL 8 . 4GHz ∼ 3 × 10 39 erg s -1 ), suggesting that, for a proton-to-electron ratio of ∼ 100 in this GeV energy range, ∼ 4% of the total mechanical luminosity from the shock needs to go into accelerating CRs to explain a factor of ∼ 2 extra radio emission. This is much less than the typical 10 -30% efficiency of particle acceleration in SNRs (e.g., Berezhko & Volk 1997; Kang & Jones 2005; Caprioli et al. 2010).", "pages": [ 8, 9, 10 ] }, { "title": "4.2. The Case for a Steeper Injection Spectrum in Dense Starbursts", "content": "While it is argued here that the most likely explanation for both the steep high-frequency radio spectra and 'excess' radio emission in this sample of local starbursts arises from synchrotron bridges and tails associated with the stage of the merger, another explanation that has not been currently explored is a systematic change (steepening) in the injection spectrum in this sample of starbursts. Naively, such an scenario does not seem that implausible given the ISM conditions in dense starbursts. The efficiency in which CRs are accelerated in SNRs plays a significant role in determining the synchrotron emissivity from galaxies. One could imagine a scenario in which the adiabatic phase of supernovae is halted as it expands into the ambient medium, thus reducing the amount of energy lost to adiabatic expansion leaving 'extra' energy that could be used in the acceleration of CRs, thereby increasing the total synchrotron emission per unit star formation rate relative to normal galaxies. For example, the modeling of Dorfi (1991, 2000) shows that the total acceleration efficiency for CRs increases from ∼ 0.15 E SN to ∼ 0.25 E SN , where E SN = 10 51 erg is the total explosion energy of the SNe, when the external ISM density is increased from ∼ 1 cm -3 to ∼ 10 cm -3 . Additionally, an increase in the injection efficiency can work to steepen the CR injection spectrum (Caprioli 2012). The ISM densities of starbursts are typically much larger (e.g., n ISM ∼ 10 4 cm -3 ), thus the evolution of SNRs will likely be different and may work to increase the synchrotron emissivity in such galaxies through an increased efficiency in particle acceleration, as well as result in steeper radio spectra due to a steeper initial injection spectra. However, given that the densest starbursts, which exhibit the flattest low- and mid-frequency spectral indices are associated with late/post-stage mergers, and also have typical high-frequency radio spectral indices as well as normal F FIR /radio flux density ratios, this explanation seems less likely. This additionally suggests that the excess radio emission in starbursts as a result of increased secondary electrons may not be a likely explanation (e.g., Murphy 2009; Lacki et al. 2010), since the densest starbursts (i.e., post mergers) provide the environment for which secondary production should be the most efficient. For example, in a dense starburst, having a much larger ISM density, the cross-section for collisions between CR nuclei and the interstellar gas is increased, thus increasing the number of e ± 's for a fixed primary nuclei/electron ratio, which may actually dominate the diffuse synchrotron emission. Yet, the densest, post-merger starbursts do not show evidence for excess radio emission per unit star formation rate like those systems in which the progenitors are still clearly separated and exhibit a non-negligible amount of diffuse radio emission in features such bridges and tails.", "pages": [ 10, 11 ] }, { "title": "4.3. An Explanation for Low FIR/Radio Ratios in High-z SMGs?", "content": "A significant fraction of submillimeter galaxies (SMGs) detected at redshifts between 2 /lessorsimilar z /lessorsimilar 4 similarly show excess (i.e., a factor of /greaterorsimilar 3) radio emission relative to their total far-infrared emission and a nominal F FIR /radio ratio (e.g., Kov'acs et al. 2006; Valiante et al. 2007; Capak et al. 2008; Murphy et al. 2009; Daddi et al. 2009b,a; Coppin et al. 2009; Knudsen et al. 2010; Smolˇci'c et al. 2011). Although, it is worth pointing out that this is not true for all samples of SMGs at these redshifts (e.g., Chapman et al. 2010). AGN provide a likely explanation for the excess radio emission in such sources given that a number are detected in hard (2 . 0 -8 . 0keV) X-rays (Alexander et al. 2005), however, it is possible that those sources without evidence for AGN may exhibit excess radio emission associated with being involved in an ongoing merger. At such cosmological distances, it is currently unclear what fraction of SMGs are major mergers rather than isolated disks, but the morphologies for a number of resolved sources seem to suggest major mergers that are driving intense bursts of star formation (e.g., Chapman et al. 2003; Hodge et al. 2013). With the Atacama Large Millimeter/submillimeter Array (ALMA) now online, resolving these dusty starbursts into individual components is becoming easier (e.g., Hezaveh et al. 2013; Hodge et al. 2013). For instance, the lensed starforming galaxy SPT-S053816-5030.8, at a redshift of z = 2 . 783, has a radio spectrum that appears to flat- ten ( α ≈ 0 . 18) towards low frequencies, while having a rather steep ( α ≈ 0 . 76) spectrum at high frequencies (Aravena et al. 2013), similar to what is seen among the local compact starbursts investigated here. While the estimated (rest-frame) q 1 . 4GHz value for this source is slightly larger than the local average value, consistent with expectations given the spectral flattening at lower frequencies assuming optically-thick free-free emission, the q 8 . 3GHz value is ≈ 0.36 dex smaller than the local average value, implying excess radio emissions per unit star formation rate. New imaging at 350GHz using ALMA along with lens modeling (Hezaveh et al. 2013) suggests that this SMG is in fact composed of two galaxies, one of which is a compact source that dominates the far-infrared emission. Both the radio continuum properties and 350GHz morphology suggests that the source is consistent with being powered by merger-driven star formation as observed in local (U)LIRGs. Thus, at least for this SMG, excess nonthermal radio emission associated with the merger may provide a natural explanation for its steep radio spectral index at high frequencies and presumably excess radio emission per unit star formation rate relative to normal star-forming galaxies rather than requiring the need to invoke cosmic conspiracies (e.g., Lacki & Thompson 2010). Having better multifrequency radio data for a large number of these high-redshift SMGs, to see if their spectra is also steep at high frequencies, may help to explain exactly why this population of starbursting galaxies appear to have significantly more radio emission per unit star formation rate than expected.", "pages": [ 11 ] }, { "title": "5. CONCLUSIONS", "content": "An examination of the radio continuum properties for a sample of local infrared-bright starbursts has been investigated against their merger classification. This was done to shed light on the curious nature of the radio spectra among such sources, specifically those having steeper than expected radio spectra observed at frequencies ∼ 12GHz as pointed out by Clemens et al. (2008). The main conclusions from this investigation can be summarized as follows: steep high-frequency radio spectral indices and excess radio emission arises from radio continuum bridges and tidal tails in which a new population of relativistic electrons have been accelerated and can radiatively cool producing a steep spectrum. Such a scenario is consistent with high-resolution radio morphologies of the sources as a function of merger stage, as well as the radio spectral index map for the merging galaxy pairs IC 1623 and IRASF15163+4255. We thank the anonymous referee for useful comments that helped to significantly improve the content and presentation of this paper. E.J.M. thanks S. Stierwalt, S. Haan, J. A. Rich, G. C. Privon, L. Armus, L. Barcos, A.K. Leroy, and P.A. Appleton, for useful discussions, M. Clemens for providing reduced X-band maps, and D.C. Kim and A.S. Evans for providing their reduced HST images. E.J.M. is also grateful to J.J. Condon for giving the paper a careful reading and providing useful comments. E.J.M. acknowledges the hospitality of the Aspen Center for Physics, which is supported by the National Science Foundation Grant No. PHY-1066293. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.", "pages": [ 11, 12 ] }, { "title": "REFERENCES", "content": "Alexander, D. M., Bauer, F. E., Chapman, S. C., et al. 2005, ApJ, 632, 736 Aravena, M., Murphy, E. J., Aguirre, J. E., et al. 2013, ArXiv e-prints Armus, L., Heckman, T., & Miley, G. 1987, AJ, 94, 831 Armus, L., Heckman, T. M., & Miley, G. K. 1988, ApJ, 326, L45 -. 1989, ApJ, 347, 727 -. 1990, ApJ, 364, 471 Armus, L., Charmandaris, V., Bernard-Salas, J., et al. 2007, ApJ, 656, 148 Armus, L., Mazzarella, J. M., Evans, A. S., et al. 2009, PASP, 121, 559 Berezhko, E. G., & Volk, H. J. 1997, Astroparticle Physics, 7, 183 Braine, J., Davoust, E., Zhu, M., et al. 2003, A&A, 408, L13 Brandl, B. R., Bernard-Salas, J., Spoon, H. W. W., et al. 2006, ApJ, 653, 1129 Capak, P., Carilli, C. L., Lee, N., et al. 2008, ApJ, 681, L53 Dorfi, E. A. 1991, A&A, 251, 597 -. 2000, Ap&SS, 272, 227 Murphy, E. J. 2009, ApJ, 706, 482 Valiante, E., Lutz, D., Sturm, E., et al. 2007, ApJ, 660, 1060 Veilleux, S., Kim, D.-C., & Sanders, D. B. 2002, ApJS, 143, 315 Veilleux, S., Kim, D.-C., Sanders, D. B., Mazzarella, J. M., & Soifer, B. T. 1995, ApJS, 98, 171 Veilleux, S., Sanders, D. B., & Kim, D.-C. 1997, ApJ, 484, 92 Yun, M. S., Reddy, N. A., & Condon, J. J. 2001, ApJ, 554, 803 Yun, M. S., Scoville, N. Z., & Knop, R. A. 1994, ApJ, 430, L109 Zhu, M., Gao, Y., Seaquist, E. R., & Dunne, L. 2007, AJ, 134, 118", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "We suggest a new picture of supermassive black hole (SMBH) growth in galaxy centers. Momentumdriven feedback from an accreting hole gives significant orbital energy but little angular momentum to the surrounding gas. Once central accretion drops, the feedback weakens and swept-up gas falls back towards the SMBH on near-parabolic orbits. These intersect near the black hole with partially opposed specific angular momenta, causing further infall and ultimately the formation of a smallscale accretion disk. The feeding rates into the disk typically exceed Eddington by factors of a few, growing the hole on the Salpeter timescale and stimulating further feedback. Natural consequences of this picture include (i) the formation and maintenance of a roughly toroidal distribution of obscuring matter near the hole; (ii) random orientations of successive accretion disk episodes; (iii) the possibility of rapid SMBH growth; (iv) tidal disruption of stars and close binaries formed from infalling gas, resulting in visible flares and ejection of hypervelocity stars; (v) super-solar abundances of the matter accreting on to the SMBH; and (vi) a lower central dark-matter density, and hence annihilation signal, than adiabatic SMBH growth implies. We also suggest a simple sub-grid recipe for implementing this process in numerical simulations. Subject headings: accretion, accretion disks - black hole physics - galaxies: evolution - quasars: general", "pages": [ 1 ] }, { "title": "BLACK HOLE FORAGING: FEEDBACK DRIVES FEEDING", "content": "Walter Dehnen 1 and Andrew King Draft version August 20, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The relation between supermassive black holes (SMBHs) and their host galaxies is a major theme of current astrophysics. The scaling relations (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Haring & Rix 2004) between the SMBH mass M and the velocity dispersion σ and mass M bulge of the host spheroid strongly suggest that the hole's enormous binding energy affects the host in important ways. A credible picture of this process is gradually emerging (e.g. Silk & Rees 1998; Fabian 1999; King 2003, 2005; Zubovas & King 2012). But we are still far from a deterministic theory of SMBH-galaxy coevolution, because we have no cogent picture of how the host affects the hole, i.e. of what causes SMBH mass growth. We know that this must largely occur through accretion of gas: the Soltan (1982) relation implies that mass growth produces electromagnetic radiation with accretion efficiency η /similarequal 0 . 1 × rest-mass energy, at least at low redshifts. This rules out dark-matter accretion as a major contributor, and direct accretion of stars through tidal disruption is inefficient (Frank & Rees 1976). Because all gas has angular momentum, accretion on to the hole at the smallest scales must be through an accretion disk. But these scales must indeed be small: the viscous timescale approaches a Hubble time at scales of only a few times 0 . 1pc if the accreting gas can cool, so that the disk aspect ratio H/R /lessmuch 1 (e.g. King & Pringle 2006, 2007; α /lessorsimilar 1 is the standard Shakura & Sunyaev (1973) viscosity parameter). However, if M disk /M /greaterorsimilar ( H/R ) ∼ 0 . 003, the 1 Theoretical Astrophysics Group, University of Leicester, Leicester LE1 7RH, U.K.; [email protected], [email protected] disk is self-gravitating and forms stars instead of accreting. Therefore, for efficient black-hole growth, /greaterorsimilar 10 2 -3 individual accretion events are required, each of which contributes only a small fraction of M and lasts /lessorsimilar 10 6 yr (King et al. 2008), implying that the accretion disks have radii R disk /lessorsimilar 0 . 003pc. Yet the gas that the hole must eventually accrete, which can be of order 10 8 -9 M /circledot , must occupy a far larger region R gas ∼ 10 -100pc. So the missing element in current treatments is a connection between these scales, telling us how gas falls from a region of size R gas to make a succession of disks at scales ∼ R disk . In numerical simulations of galaxy evolution, Bondi (1952) accretion is a popular choice, but has several critical drawbacks. Two of these are crucial. The first is that in reality all gas has significant angular momentum, and so cannot fall in radially, in the way envisaged for Bondi accretion. Angular momentum is the main barrier to accretion. However since R gas /lessorsimilar scale height of the ISM, the cold gas in this region is probably not in large-scale rotation, i.e. has a distribution of (partly) opposing angular momenta with a small net angular momentum. Therefore, a way of cancelling these opposing angular momenta would greatly enhance accretion. A second serious problem in using the Bondi formula is its implication that gas falls towards the black hole because of the destabilizing influence of its gravity. But the hole's mass is so small compared to that of even a small region of the galaxy that this is implausible. As we remarked above, the property of the hole which is highly significant for the galaxy is not its mass M , but its binding energy ηc 2 M , where η /similarequal 0 . 1. In mass terms, the hole is typically only one part in about 10 -3 of the galaxy bulge stellar mass M bulge (Haring & Rix 2004). But for binding energies the situation is reversed: a hole of mass 10 8 M /circledot has ηc 2 M ∼ 10 61 erg, while the bulge binding energy is ∼ σ 2 M bulge ∼ 10 58 erg for a typical velocity 1 dispersion σ /similarequal 200 kms -1 (this disparity is even bigger for smaller SMBH if these follow the scaling relations). This suggests that the cause of black hole accretion ultimately involves its effects on the galaxy, i.e. feedback. We already know quite a lot about black-hole feedback in galaxies, and how it produces the SMBH-galaxy scaling relations. What is important for our purposes here is that the feedback is carried by quasi-spherical winds driven by radiation pressure; these are detected via blueshifted X-ray iron absorption lines (e.g. Pounds et al. 2003a,b; Tombesi et al. 2010, 2011). The winds have momentum scalars ˙ M out v /similarequal L Edd /c , where L Edd is the Eddington luminosity of the hole, ˙ M out is the wind outflow rate, and v ∼ ηc its velocity (King & Pounds 2003). The winds interact with the host galaxy by shocking against its interstellar gas, giving initial postshock temperatures ∼ 10 10 K. While the SMBH is growing, these shocks lie close to the hole. Here the much cooler ( ∼ 10 7 K) radiation field produced by accretion removes most of the shock energy through the inverse Compton effect (King 2003). So only the wind ram pressure, i.e. the momentum rate L Edd /c mentioned above, is communicated to the host ISM (these are called 'momentum-driven' flows). This thrust can push the host ISM only modestly outwards, and is apparently unable to prevent the hole from growing. But once the hole mass reaches the M -σ scaling relation, i.e. with f g the local gas fraction, the wind shocks are able to move far away from the hole (King 2003, 2005), beyond the critical radius R cool ∼ 0 . 5kpc where the radiation field of the accreting black hole becomes too dilute to cool the shocked wind. This now expands adiabatically ('energy-driven' flow), sweeping the host ISM before it at high speed ( ∼ 1000kms -1 ) and largely clearing the galaxy bulge of gas (Zubovas & King 2012). This terminates black hole growth, leaving the hole near the mass (2).", "pages": [ 1, 2 ] }, { "title": "2. FEEDBACK CAUSES FEEDING", "content": "This sequence shows that the growth of the supermassive black hole towards the M -σ relation is characterized by quasispherical momentum-driven outflow episodes which push the interstellar gas out, but do not unbind it. This changes the dynamical state of the ISM in two important ways. First, the SMBH driven wind does not transfer angular momentum to the gas, but increases its gravitational energy. This results in a decrease of the typical pericentric radius of the gas. Second, gas with differing angular momenta is pushed together, leading to (partial) cancellation. When a black-hole accretion episode ends, the outward thrust supporting the gas against gravity drops, and it must fall back from the radius R shell of the swept-up region. Clearly, this infall is unlikely to be spherically symmetric. Instead, individual clumps or high-density regions fall on ballistic orbits. Because of the cancellation of angular momentum and the increase of gravitational energy during the outflow phase, these orbits are highly eccentric with pericenters much closer to the hole than the radii from which the gas was originally swept up dur- ing the wind feedback phase. On such eccentric orbits, any gas cloud is likely to be tidally stretched, forming a stream, in particular near pericenter. We now estimate the resulting density of clouds/streams on such orbits. Consider a population of clouds/streams orbiting with the same peri- and apocentric radii R ± , and hence with the same orbital energy and specific angular momentum Here, Φ ± ≡ Φ( R ± ), where Φ( R ) = -GMR -1 +Φ bulge ( R ) is the total gravitational potential. Neglecting collisions and internal shocks, the phase-space density of clouds/streams is conserved and simply the product of delta functions in E and L 2 . Integrating it over all velocities yields the spatial density with where m is the total gas mass. We identify the apocenter with the radius of the initially swept-up shell, R + = R shell , and numerically evaluate C and the mass m fixed at R + =2 R inf with R inf ≡ GM/σ 2 (6) the radius of the hole's sphere of influence. Because eccentric orbits have a long residence time near apocenter, the density is maximal there and most of the gas is now further from the hole than before, in a kind of thick shell near R shell . However, the infalling gas creates a second density maximum near pericenter, where the clouds/streams tend to collide with probability ∝ ρ 2 and with significant relative velocity. Near apocentre, on the other hand, collisions are not only less likely (because the orbiting clouds simply return near to their initial position, avoiding each other) but also have modest relative velocities and thus do not lead to cancelation of angular momentum. These high-impact-velocity collisions near pericentre (which are neglected in Fig. 1) lead to accretion-disk formation because the gas loses energy much faster than angular momentum, a process familiar from accretion in close binary systems. The colliding gas must shock and lose much of its orbital energy to cooling. In addition, the collisions may cancel some, potentially most, of the angular momentum, creating a cascade of ever smaller but less eccentric orbits. Ultimately, gas on the innermost orbits circularises and forms a disk. If more gas penetrates to this radius, the disk is destroyed but quickly replaced by an even smaller one. Moreover, any misalignment of the disk angular momentum with the black hole spin results in disk tearing, when angular-momentum cancellation leads to a further reduction of the inner disk radius by a factor 10 -100 (Nixon et al. 2012). This whole process is rather complex and chaotic, but certainly has the potential to transfer some of the gas from R gas ∼ 10 -100pc into an accretion disk at R disk ∼ 0 . 001 -0 . 01pc, where standard viscosity-driven accretion physics takes over the mass transport, and feeds the SMBH on a timescale of ∼ 10 6 yr. 3. THE FEEDING RATE The fundamental feature of our picture is that once central accretion (and hence feedback) slows, gas is no longer supported against gravity. This suggests that during the chaotic infall phase, gas feeds a small-scale accretion disk around the SMBH at some fraction of the dynamical infall rate ˙ M feed /lessorsimilar ˙ M dyn /similarequal f g σ 3 G . (7) For an SMBH with mass M close to M σ , this exceeds the Eddington accretion rate ˙ M Edd by factors ∼ 10 -100 at most (King 2007). This feeding rate should characterise the rapid growth phases for the SMBH. For gas fractions /greaterorsimilar 0 . 1 it implies disk feeding at rates a few times ˙ M Edd . This is likely to result in the following scenario (cf. King & Pringle 2006; 2007). The outer parts of the disk may become self-gravitating and form stars, while the remaining gas flows inwards under the disk viscosity at slightly superEddington rates. This leads to SMBH accretion at about ˙ M Edd , and similar mass outflow rates, with momentum scalars ˙ M out v /similarequal L Edd /c (King & Pounds 2003). This fits self-consistently with the feedback needed to give the observed M -σ scaling relation (King 2003, 2005). Once central accretion stops, the SMBH should be quiescent for the sum of the infall timescale R + /σ and the viscous timescale (1). In general infall is more rapid, so the controlling timescale is probably viscous and depends critically on the radius R disk at which the chaotic infall process places the disk. We note that in our picture, both the precise value of the mass feeding rate and its duty cycle are determined by essentially stochastic processes. This makes it difficult to go beyond the simple estimates given here either analytically or numerically. We return to this problem in the last section. 4. BLACK HOLE OBSCURATION We expect this same mechanism to produce the putative accretion 'torus' at radii larger than R disk . This structure is postulated (Antonucci & Miller 1985; Antonucci 1993) to cover a large solid angle, obscuring the hole along many lines of sight, and so accounting for the populations of unobscured (Type I) and obscured (Type II) active galactic nuclei. The main problem in understanding the torus in physical terms is that it must consist of cool material, which by its nature cannot form a vertically extended disk or torus. However, a large solid angle is natural if much of this obscuring gas is not yet settled into a disk, but still falling in on a range of orbits of very different inclinations. The column density Σ= ∫ ρ R . of a population of gas clouds/streams with total mass m and common apo- and pericentric radii is Σ ∼ m 2 π ( R -+ R + ) √ R -R + . (8) (using equation (4) with Φ bulge =0). This diverges for small pericentric radii R -, so the black hole must be obscured either completely or, more probably, for many lines of sight and/or extended periods of time. In fact, the obscuring matter may not be in form of a torus at all but merely a collection of clouds/streams orbiting the hole on eccentric orbits. Whatever the geometry of the obscuring matter, our model renders the standard geometrical explanation for AGN unification (Antonucci 1993) time-dependent, since the orientation of that matter changes randomly over time and because we expect cyclicly recurring inflow phases. This is in line with observational evidence of occasional changes between Seyfert types (e.g. Alloin et al. 1985; Shappee et al. 2013). 5. THE CENTRAL BUBBLE Our discussion so far has not specified the physical scale R shell where the momentum-driven outflows are typically halted. Our feeding mechanism works independently of this scale, but it may set the duty cycle and orientation of the individual accretion disk episodes. We note that King & Pounds (2013) have recently suggested that radiation pressure from the central active nucleus tends to create a shell of gas at a characteristic radius R tr ∼ 50( σ/ 200kms -1 ) 2 pc, at which the gas becomes transparent to the radiation from the accretion disk. This is larger than the radius (6) of the sphere of influence by a factor M σ /M and the shell's mass is com- arable with the final mass M σ of the hole. In this picture, momentum-driven outflows must be halted here, as their inertia is of course far smaller. This means that R shell /similarequal R tr . This idea agrees with observations of warm absorbers, which can be interpreted as arrested momentum-driven outflows. 6. DISCUSSION We have suggested that black hole feeding is ultimately caused by feedback. By elongating the gas orbits and promoting collisions, this causes cancellation of opposed specific gas angular momenta, allowing accretion disks to form at small distances from the black hole, where they can feed the hole on time scales close to Salpeter (1964). This is different from a situation where the gas is initially pressure supported, when cooling and collisions of the resulting condensations can lead to turbulent infall (Gaspari et al. 2013). Our picture explains a number of other aspects. As we have shown above, a near-toroidal topology for obscuring gas is a natural result. It is also clear that the orientation of the accretion structure (disk + 'torus') cannot be constant over time, but must be essentially random. This is just the situation envisaged in the picture of chaotic accretion suggested by King & Pringle (2006, 2007), which results in relatively low black hole spins. This implies rapid mass growth and low gravitational-wave recoil velocities for merging black holes. The impact of the black hole wind on the gas which ultimately falls in may cause some of it to form stars, and this can also happen in the collisions during gas infall. Of course, any gas converted to and/or heated by stars is prevented from participating in the black hole feeding. However, at each feeding cycle only a small fraction of the gas within R shell is required to reach R disk , and only gas locked in stellar remnants and dwarfs is ultimately prevented from accreting. Because angular momentum has been largely cancelled, such newly formed stars fall in on near-parabolic orbits. This has several consequences. First, stars coming too close to the hole create visible tidal disruption events (Rees 1988); second, tidal dissociation of close binaries produces hypervelocity stars (Hills 1988); finally, massive stars which escape these fates inject metal-enriched gas into their surroundings. In any plausible picture most of this gas remains near to the hole, and could undergo repeated star formation. This may be the origin of the high chemical enrichment observed in AGN spectra (Shields 1976; Baldwin & Netzer 1978; Hamann & Ferland 1992; Ferland et al. 1996; Dietrich et al. 1999, 2003a,b; Arav et al. 2007). We note that the idea of feedback-stimulated feeding opens the possibility of runaway growth: the black hole forages for its own food, and grows still faster. Given an abundant food supply (i.e. f g /greaterorsimilar 0 . 1) this growth is stopped only as the hole reaches the limiting M -σ mass and drives all the food away. A runaway SMBH like this would of course have a tendency to grow at the Eddington rate for most of its (short) feeding frenzy. This may explain very massive SMBH observed at high redshifts (e.g. Barth et al. 2003; Willott et al. 2003; Fan et al. 2003; Mortlock et al. 2011). Here the close proximity of all galaxies means that many are likely to be gas-rich (i.e. f g /greaterorsimilar 0 . 1) because of mergers, so runaways are favored. One interesting aspect of the proposed mechanism is the mutual dependence of feeding and feedback on each other. Clearly, this whole process must be started by some initial accretion which was not triggered by feedback, but by sufficient gas coming within /lessorsimilar 0 . 001pc of the infant hole. Such an event could be triggered by a galactic merger, but must be relatively rare. This implies that early SMBH formation may be somewhat random, but more likely in frequently perturbed/merging galaxies. Conversely, if the SMBH's neighbourhood at R /lessorsimilar R gas acquires some net rotation, for example, during a merger, then the distribution of angular momenta is unlikely to allow for angular-momentum cancellation. In such a situation, the SMBH suffers from starvation. Despite sitting tantalisingly close to its food, it cannot reach it nor bring it down easily. However, if the rotating gas can cool, it will form a disk (and possibly stars), clearing most of the space and opening the possibility for re-starting the feeding cycle. Also, our proposed feeding mechanism will not work efficiently if the feedback is dominated by a collimated jet rather than wide-angle outflows. This is obvious if the impact shocks are efficiently cooled (momentum-driven flow) as the jet simply carves a narrow hole in the gas it impacts. If the shocks do not cool (energy-driven), their effect is wider but still unlikely to cause feeding in the way described here. Finally, we note that the episodic in- and outflows of a fraction f g ∼ 0 . 1 of matter at velocities well above σ entail abrupt changes in the gravitational potential in the inner R shell ∼ 10 -100pc. Therefore, the growth of the SMBH is not an adiabatic process for the dynamics of collisionless matter on these scales. Instead, the abrupt variations in the potential redistribute the orbital actions. This renders the central dark-matter density smaller than current estimates (by e.g. Young 1980; Quinlan et al. 1995) based on the adiabatic assumption, though possibly still larger than in absence of a SMBH. This implies a significant reduction in the expected darkmatter annihilation signal from SMBH hosting galaxy centers. 7. A SUBGRID RECIPE We have suggested that feeding of supermassive black holes may in many cases be stimulated by feedback. A practical question is how one might implement this process in simulations of galaxy formation which cannot resolve the hole's sphere of influence, let alone the dynamics and cooling of infall and outflow, and instead must use a subgrid recipe. Clearly, any Bondi-like subgrid recipe adapted to account for the angular momentum of the gas at /greaterorsimilar R gas cannot adequately describe these dynamics. Instead a completely different approach is required. We have seen that feedback-induced feeding generally occurs at a fraction of the dynamical infall rate (7) when it operates. This is generally slightly super-Eddington (for f g /greaterorsimilar 0 . 1). This in turn makes the SMBH grow at about the Eddington rate, and rejects the remainder of the mass in a wind, which is what causes the feedback. If M tes SMBH mass growth. Given the discussion above, a suitable subgrid recipe is as follows. Grow M from surrounding gas at the rate ˙ M =min { /epsilon1 ˙ M dyn , ˙ M Edd } (see equation 7) with /epsilon1 ∼ 0 . 1. If M ACKNOWLEDGMENTS We thank Ken Pounds, Chris Nixon, and Peter Hague for helpful conversations. Theoretical astrophysics in Leicester is supported by an STFC Consolidated Grant. REFERENCES Alloin D., Pelat D., Phillips M., Whittle M., 1985, ApJ, 288, 205 Antonucci R., 1993, ARA&A, 31, 473 Antonucci R. R. J., Miller J. S., 1985, ApJ, 297, 621 Arav N., Gabel J. R., Korista K. T., et al., 2007, ApJ, 658, 829 Baldwin J. A., Netzer H., 1978, ApJ, 226, 1 Barth A. J., Martini P., Nelson C. H., Ho L. C., 2003, ApJ, 594, L95 Bondi H., 1952, MNRAS, 112, 195 Dietrich M., Appenzeller I., Hamann F., et al., 2003a, A&A, 398, 891 Dietrich M., Appenzeller I., Wagner S. J., et al., 1999, A&A, 352, L1 Dietrich M., Hamann F., Shields J. C., et al., 2003b, ApJ, 589, 722 Fabian A. C., 1999, MNRAS, 308, L39 Fan X., Strauss M. A., Schneider D. P., et al., 2003, AJ, 125, 1649 Ferland G. J., Baldwin J. A., Korista K. T., et al., 1996, ApJ, 461, 683 Ferrarese L., Merritt D., 2000, ApJ, 539, L9 Frank J., Rees M. J., 1976, MNRAS, 176, 633 Gaspari M., Ruszkowski M., Oh S. P., 2013, MNRAS, 432, 3401 Gebhardt K., Bender R., Bower G., et al., 2000, ApJ, 539, L13 Hamann F., Ferland G., 1992, ApJ, 391, L53 Haring N., Rix H.-W., 2004, ApJ, 604, L89 Hills J. G., 1988, Nature, 331, 687 King A., 2003, ApJ, 596, L27 King A., 2005, ApJ, 635, L121 King A. R., 2007, in IAU Symposium 238, ed. V. Karas, G. Matt, 31 King A. R., Pounds K. A., 2003, MNRAS, 345, 657 King A. R., Pounds K. A., 2013, submitted to MNRAS King A. R., Pringle J. E., 2006, MNRAS, 373, L90 King A. R., Pringle J. E., 2007, MNRAS, 377, L25 King A. R., Pringle J. E., Hofmann J. A., 2008, MNRAS, 385, 1621 Mortlock D. J., Warren S. J., Venemans B. P., et al., 2011, Nature, 474, 616 Nixon C., King A., Price D., Frank J., 2012, ApJ, 757, L24 Pounds K. A., King A. R., Page K. L., O'Brien P. T., 2003a, MNRAS, 346, 1025 Pounds K. A., Reeves J. N., King A. R., Page K. L., O'Brien P. T., Turner M. J. L., 2003b, MNRAS, 345, 705 Quinlan G. D., Hernquist L., Sigurdsson S., 1995, ApJ, 440, 554 Rees M. J., 1988, Nature, 333, 523 Salpeter E. E., 1964, ApJ, 140, 796 Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Shappee B. J., Grupe D., Mathur S., et al., 2013, The Astronomer's Telegram, 5059, 1 Shields G. A., 1976, ApJ, 204, 330 Silk J., Rees M. J., 1998, A&A, 331, L1 Soltan A., 1982, MNRAS, 200, 115 Tombesi F., Sambruna R. M., Reeves J. N., et al., 2010, ApJ, 719, 700 Tombesi F., Sambruna R. M., Reeves J. N., Reynolds C. S., Braito V., 2011, MNRAS, 418, L89 Willott C. J., McLure R. J., Jarvis M. J., 2003, ApJ, 587, L15 Young P., 1980, ApJ, 242, 1232 Zubovas K., King A., 2012, ApJ, 745, L34 fixed at R + =2 R inf with the radius of the hole's sphere of influence. Because eccentric orbits have a long residence time near apocenter, the density is maximal there and most of the gas is now further from the hole than before, in a kind of thick shell near R shell . However, the infalling gas creates a second density maximum near pericenter, where the clouds/streams tend to collide with probability ∝ ρ 2 and with significant relative velocity. Near apocentre, on the other hand, collisions are not only less likely (because the orbiting clouds simply return near to their initial position, avoiding each other) but also have modest relative velocities and thus do not lead to cancelation of angular momentum. These high-impact-velocity collisions near pericentre (which are neglected in Fig. 1) lead to accretion-disk formation because the gas loses energy much faster than angular momentum, a process familiar from accretion in close binary systems. The colliding gas must shock and lose much of its orbital energy to cooling. In addition, the collisions may cancel some, potentially most, of the angular momentum, creating a cascade of ever smaller but less eccentric orbits. Ultimately, gas on the innermost orbits circularises and forms a disk. If more gas penetrates to this radius, the disk is destroyed but quickly replaced by an even smaller one. Moreover, any misalignment of the disk angular momentum with the black hole spin results in disk tearing, when angular-momentum cancellation leads to a further reduction of the inner disk radius by a factor 10 -100 (Nixon et al. 2012). This whole process is rather complex and chaotic, but certainly has the potential to transfer some of the gas from R gas ∼ 10 -100pc into an accretion disk at R disk ∼ 0 . 001 -0 . 01pc, where standard viscosity-driven accretion physics takes over the mass transport, and feeds the SMBH on a timescale of ∼ 10 6 yr.", "pages": [ 2, 3 ] }, { "title": "3. THE FEEDING RATE", "content": "The fundamental feature of our picture is that once central accretion (and hence feedback) slows, gas is no longer supported against gravity. This suggests that during the chaotic infall phase, gas feeds a small-scale accretion disk around the SMBH at some fraction of the dynamical infall rate For an SMBH with mass M close to M σ , this exceeds the Eddington accretion rate ˙ M Edd by factors ∼ 10 -100 at most (King 2007). This feeding rate should characterise the rapid growth phases for the SMBH. For gas fractions /greaterorsimilar 0 . 1 it implies disk feeding at rates a few times ˙ M Edd . This is likely to result in the following scenario (cf. King & Pringle 2006; 2007). The outer parts of the disk may become self-gravitating and form stars, while the remaining gas flows inwards under the disk viscosity at slightly superEddington rates. This leads to SMBH accretion at about ˙ M Edd , and similar mass outflow rates, with momentum scalars ˙ M out v /similarequal L Edd /c (King & Pounds 2003). This fits self-consistently with the feedback needed to give the observed M -σ scaling relation (King 2003, 2005). Once central accretion stops, the SMBH should be quiescent for the sum of the infall timescale R + /σ and the viscous timescale (1). In general infall is more rapid, so the controlling timescale is probably viscous and depends critically on the radius R disk at which the chaotic infall process places the disk. We note that in our picture, both the precise value of the mass feeding rate and its duty cycle are determined by essentially stochastic processes. This makes it difficult to go beyond the simple estimates given here either analytically or numerically. We return to this problem in the last section.", "pages": [ 3 ] }, { "title": "4. BLACK HOLE OBSCURATION", "content": "We expect this same mechanism to produce the putative accretion 'torus' at radii larger than R disk . This structure is postulated (Antonucci & Miller 1985; Antonucci 1993) to cover a large solid angle, obscuring the hole along many lines of sight, and so accounting for the populations of unobscured (Type I) and obscured (Type II) active galactic nuclei. The main problem in understanding the torus in physical terms is that it must consist of cool material, which by its nature cannot form a vertically extended disk or torus. However, a large solid angle is natural if much of this obscuring gas is not yet settled into a disk, but still falling in on a range of orbits of very different inclinations. The column density Σ= ∫ ρ R . of a population of gas clouds/streams with total mass m and common apo- and pericentric radii is (using equation (4) with Φ bulge =0). This diverges for small pericentric radii R -, so the black hole must be obscured either completely or, more probably, for many lines of sight and/or extended periods of time. In fact, the obscuring matter may not be in form of a torus at all but merely a collection of clouds/streams orbiting the hole on eccentric orbits. Whatever the geometry of the obscuring matter, our model renders the standard geometrical explanation for AGN unification (Antonucci 1993) time-dependent, since the orientation of that matter changes randomly over time and because we expect cyclicly recurring inflow phases. This is in line with observational evidence of occasional changes between Seyfert types (e.g. Alloin et al. 1985; Shappee et al. 2013).", "pages": [ 3 ] }, { "title": "5. THE CENTRAL BUBBLE", "content": "Our discussion so far has not specified the physical scale R shell where the momentum-driven outflows are typically halted. Our feeding mechanism works independently of this scale, but it may set the duty cycle and orientation of the individual accretion disk episodes. We note that King & Pounds (2013) have recently suggested that radiation pressure from the central active nucleus tends to create a shell of gas at a characteristic radius R tr ∼ 50( σ/ 200kms -1 ) 2 pc, at which the gas becomes transparent to the radiation from the accretion disk. This is larger than the radius (6) of the sphere of influence by a factor M σ /M and the shell's mass is com- arable with the final mass M σ of the hole. In this picture, momentum-driven outflows must be halted here, as their inertia is of course far smaller. This means that R shell /similarequal R tr . This idea agrees with observations of warm absorbers, which can be interpreted as arrested momentum-driven outflows.", "pages": [ 3, 4 ] }, { "title": "6. DISCUSSION", "content": "We have suggested that black hole feeding is ultimately caused by feedback. By elongating the gas orbits and promoting collisions, this causes cancellation of opposed specific gas angular momenta, allowing accretion disks to form at small distances from the black hole, where they can feed the hole on time scales close to Salpeter (1964). This is different from a situation where the gas is initially pressure supported, when cooling and collisions of the resulting condensations can lead to turbulent infall (Gaspari et al. 2013). Our picture explains a number of other aspects. As we have shown above, a near-toroidal topology for obscuring gas is a natural result. It is also clear that the orientation of the accretion structure (disk + 'torus') cannot be constant over time, but must be essentially random. This is just the situation envisaged in the picture of chaotic accretion suggested by King & Pringle (2006, 2007), which results in relatively low black hole spins. This implies rapid mass growth and low gravitational-wave recoil velocities for merging black holes. The impact of the black hole wind on the gas which ultimately falls in may cause some of it to form stars, and this can also happen in the collisions during gas infall. Of course, any gas converted to and/or heated by stars is prevented from participating in the black hole feeding. However, at each feeding cycle only a small fraction of the gas within R shell is required to reach R disk , and only gas locked in stellar remnants and dwarfs is ultimately prevented from accreting. Because angular momentum has been largely cancelled, such newly formed stars fall in on near-parabolic orbits. This has several consequences. First, stars coming too close to the hole create visible tidal disruption events (Rees 1988); second, tidal dissociation of close binaries produces hypervelocity stars (Hills 1988); finally, massive stars which escape these fates inject metal-enriched gas into their surroundings. In any plausible picture most of this gas remains near to the hole, and could undergo repeated star formation. This may be the origin of the high chemical enrichment observed in AGN spectra (Shields 1976; Baldwin & Netzer 1978; Hamann & Ferland 1992; Ferland et al. 1996; Dietrich et al. 1999, 2003a,b; Arav et al. 2007). We note that the idea of feedback-stimulated feeding opens the possibility of runaway growth: the black hole forages for its own food, and grows still faster. Given an abundant food supply (i.e. f g /greaterorsimilar 0 . 1) this growth is stopped only as the hole reaches the limiting M -σ mass and drives all the food away. A runaway SMBH like this would of course have a tendency to grow at the Eddington rate for most of its (short) feeding frenzy. This may explain very massive SMBH observed at high redshifts (e.g. Barth et al. 2003; Willott et al. 2003; Fan et al. 2003; Mortlock et al. 2011). Here the close proximity of all galaxies means that many are likely to be gas-rich (i.e. f g /greaterorsimilar 0 . 1) because of mergers, so runaways are favored. One interesting aspect of the proposed mechanism is the mutual dependence of feeding and feedback on each other. Clearly, this whole process must be started by some initial accretion which was not triggered by feedback, but by sufficient gas coming within /lessorsimilar 0 . 001pc of the infant hole. Such an event could be triggered by a galactic merger, but must be relatively rare. This implies that early SMBH formation may be somewhat random, but more likely in frequently perturbed/merging galaxies. Conversely, if the SMBH's neighbourhood at R /lessorsimilar R gas acquires some net rotation, for example, during a merger, then the distribution of angular momenta is unlikely to allow for angular-momentum cancellation. In such a situation, the SMBH suffers from starvation. Despite sitting tantalisingly close to its food, it cannot reach it nor bring it down easily. However, if the rotating gas can cool, it will form a disk (and possibly stars), clearing most of the space and opening the possibility for re-starting the feeding cycle. Also, our proposed feeding mechanism will not work efficiently if the feedback is dominated by a collimated jet rather than wide-angle outflows. This is obvious if the impact shocks are efficiently cooled (momentum-driven flow) as the jet simply carves a narrow hole in the gas it impacts. If the shocks do not cool (energy-driven), their effect is wider but still unlikely to cause feeding in the way described here. Finally, we note that the episodic in- and outflows of a fraction f g ∼ 0 . 1 of matter at velocities well above σ entail abrupt changes in the gravitational potential in the inner R shell ∼ 10 -100pc. Therefore, the growth of the SMBH is not an adiabatic process for the dynamics of collisionless matter on these scales. Instead, the abrupt variations in the potential redistribute the orbital actions. This renders the central dark-matter density smaller than current estimates (by e.g. Young 1980; Quinlan et al. 1995) based on the adiabatic assumption, though possibly still larger than in absence of a SMBH. This implies a significant reduction in the expected darkmatter annihilation signal from SMBH hosting galaxy centers.", "pages": [ 4 ] }, { "title": "7. A SUBGRID RECIPE", "content": "We have suggested that feeding of supermassive black holes may in many cases be stimulated by feedback. A practical question is how one might implement this process in simulations of galaxy formation which cannot resolve the hole's sphere of influence, let alone the dynamics and cooling of infall and outflow, and instead must use a subgrid recipe. Clearly, any Bondi-like subgrid recipe adapted to account for the angular momentum of the gas at /greaterorsimilar R gas cannot adequately describe these dynamics. Instead a completely different approach is required. We have seen that feedback-induced feeding generally occurs at a fraction of the dynamical infall rate (7) when it operates. This is generally slightly super-Eddington (for f g /greaterorsimilar 0 . 1). This in turn makes the SMBH grow at about the Eddington rate, and rejects the remainder of the mass in a wind, which is what causes the feedback. If M tes SMBH mass growth. Given the discussion above, a suitable subgrid recipe is as follows. Grow M from surrounding gas at the rate ˙ M =min { /epsilon1 ˙ M dyn , ˙ M Edd } (see equation 7) with /epsilon1 ∼ 0 . 1. If M ACKNOWLEDGMENTS We thank Ken Pounds, Chris Nixon, and Peter Hague for helpful conversations. Theoretical astrophysics in Leicester is supported by an STFC Consolidated Grant. REFERENCES Alloin D., Pelat D., Phillips M., Whittle M., 1985, ApJ, 288, 205 Antonucci R., 1993, ARA&A, 31, 473 Antonucci R. R. J., Miller J. S., 1985, ApJ, 297, 621 Arav N., Gabel J. R., Korista K. T., et al., 2007, ApJ, 658, 829 Baldwin J. A., Netzer H., 1978, ApJ, 226, 1 Barth A. J., Martini P., Nelson C. H., Ho L. C., 2003, ApJ, 594, L95 Bondi H., 1952, MNRAS, 112, 195 Dietrich M., Appenzeller I., Hamann F., et al., 2003a, A&A, 398, 891 Dietrich M., Appenzeller I., Wagner S. J., et al., 1999, A&A, 352, L1 Dietrich M., Hamann F., Shields J. C., et al., 2003b, ApJ, 589, 722 Fabian A. C., 1999, MNRAS, 308, L39 Fan X., Strauss M. A., Schneider D. P., et al., 2003, AJ, 125, 1649 Ferland G. J., Baldwin J. A., Korista K. T., et al., 1996, ApJ, 461, 683 Ferrarese L., Merritt D., 2000, ApJ, 539, L9 Frank J., Rees M. J., 1976, MNRAS, 176, 633 Gaspari M., Ruszkowski M., Oh S. P., 2013, MNRAS, 432, 3401 Gebhardt K., Bender R., Bower G., et al., 2000, ApJ, 539, L13 Hamann F., Ferland G., 1992, ApJ, 391, L53 Haring N., Rix H.-W., 2004, ApJ, 604, L89 Hills J. G., 1988, Nature, 331, 687 King A., 2003, ApJ, 596, L27 King A., 2005, ApJ, 635, L121 King A. R., 2007, in IAU Symposium 238, ed. V. Karas, G. Matt, 31 King A. R., Pounds K. A., 2003, MNRAS, 345, 657 King A. R., Pounds K. A., 2013, submitted to MNRAS King A. R., Pringle J. E., 2006, MNRAS, 373, L90 King A. R., Pringle J. E., 2007, MNRAS, 377, L25 King A. R., Pringle J. E., Hofmann J. A., 2008, MNRAS, 385, 1621 Mortlock D. J., Warren S. J., Venemans B. P., et al., 2011, Nature, 474, 616 Nixon C., King A., Price D., Frank J., 2012, ApJ, 757, L24 Pounds K. A., King A. R., Page K. L., O'Brien P. T., 2003a, MNRAS, 346, 1025 Pounds K. A., Reeves J. N., King A. R., Page K. L., O'Brien P. T., Turner M. J. L., 2003b, MNRAS, 345, 705 Quinlan G. D., Hernquist L., Sigurdsson S., 1995, ApJ, 440, 554 Rees M. J., 1988, Nature, 333, 523 Salpeter E. E., 1964, ApJ, 140, 796 Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Shappee B. J., Grupe D., Mathur S., et al., 2013, The Astronomer's Telegram, 5059, 1 Shields G. A., 1976, ApJ, 204, 330 Silk J., Rees M. J., 1998, A&A, 331, L1 Soltan A., 1982, MNRAS, 200, 115 Tombesi F., Sambruna R. M., Reeves J. N., et al., 2010, ApJ, 719, 700 Tombesi F., Sambruna R. M., Reeves J. N., Reynolds C. S., Braito V., 2011, MNRAS, 418, L89 Willott C. J., McLure R. J., Jarvis M. J., 2003, ApJ, 587, L15 Young P., 1980, ApJ, 242, 1232 Zubovas K., King A., 2012, ApJ, 745, L34 tes SMBH mass growth. Given the discussion above, a suitable subgrid recipe is as follows. Grow M from surrounding gas at the rate ˙ M =min { /epsilon1 ˙ M dyn , ˙ M Edd } (see equation 7) with /epsilon1 ∼ 0 . 1. If M ACKNOWLEDGMENTS We thank Ken Pounds, Chris Nixon, and Peter Hague for helpful conversations. Theoretical astrophysics in Leicester is supported by an STFC Consolidated Grant. REFERENCES Alloin D., Pelat D., Phillips M., Whittle M., 1985, ApJ, 288, 205 Antonucci R., 1993, ARA&A, 31, 473 Antonucci R. R. J., Miller J. S., 1985, ApJ, 297, 621 Arav N., Gabel J. R., Korista K. T., et al., 2007, ApJ, 658, 829 Baldwin J. A., Netzer H., 1978, ApJ, 226, 1 Barth A. J., Martini P., Nelson C. H., Ho L. C., 2003, ApJ, 594, L95 Bondi H., 1952, MNRAS, 112, 195 Dietrich M., Appenzeller I., Hamann F., et al., 2003a, A&A, 398, 891 Dietrich M., Appenzeller I., Wagner S. J., et al., 1999, A&A, 352, L1 Dietrich M., Hamann F., Shields J. C., et al., 2003b, ApJ, 589, 722 Fabian A. C., 1999, MNRAS, 308, L39 Fan X., Strauss M. A., Schneider D. P., et al., 2003, AJ, 125, 1649 Ferland G. J., Baldwin J. A., Korista K. T., et al., 1996, ApJ, 461, 683 Ferrarese L., Merritt D., 2000, ApJ, 539, L9 Frank J., Rees M. J., 1976, MNRAS, 176, 633 Gaspari M., Ruszkowski M., Oh S. P., 2013, MNRAS, 432, 3401 Gebhardt K., Bender R., Bower G., et al., 2000, ApJ, 539, L13 Hamann F., Ferland G., 1992, ApJ, 391, L53 Haring N., Rix H.-W., 2004, ApJ, 604, L89 Hills J. G., 1988, Nature, 331, 687 King A., 2003, ApJ, 596, L27 King A., 2005, ApJ, 635, L121 King A. R., 2007, in IAU Symposium 238, ed. V. Karas, G. Matt, 31 King A. R., Pounds K. A., 2003, MNRAS, 345, 657 King A. R., Pounds K. A., 2013, submitted to MNRAS King A. R., Pringle J. E., 2006, MNRAS, 373, L90 King A. R., Pringle J. E., 2007, MNRAS, 377, L25 King A. R., Pringle J. E., Hofmann J. A., 2008, MNRAS, 385, 1621 Mortlock D. J., Warren S. J., Venemans B. P., et al., 2011, Nature, 474, 616 Nixon C., King A., Price D., Frank J., 2012, ApJ, 757, L24 Pounds K. A., King A. R., Page K. L., O'Brien P. T., 2003a, MNRAS, 346, 1025 Pounds K. A., Reeves J. N., King A. R., Page K. L., O'Brien P. T., Turner M. J. L., 2003b, MNRAS, 345, 705 Quinlan G. D., Hernquist L., Sigurdsson S., 1995, ApJ, 440, 554 Rees M. J., 1988, Nature, 333, 523 Salpeter E. E., 1964, ApJ, 140, 796 Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Shappee B. J., Grupe D., Mathur S., et al., 2013, The Astronomer's Telegram, 5059, 1 Shields G. A., 1976, ApJ, 204, 330 Silk J., Rees M. J., 1998, A&A, 331, L1 Soltan A., 1982, MNRAS, 200, 115 Tombesi F., Sambruna R. M., Reeves J. N., et al., 2010, ApJ, 719, 700 Tombesi F., Sambruna R. M., Reeves J. N., Reynolds C. S., Braito V., 2011, MNRAS, 418, L89 Willott C. J., McLure R. J., Jarvis M. J., 2003, ApJ, 587, L15 Young P., 1980, ApJ, 242, 1232 Zubovas K., King A., 2012, ApJ, 745, L34", "pages": [ 4, 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank Ken Pounds, Chris Nixon, and Peter Hague for helpful conversations. Theoretical astrophysics in Leicester is supported by an STFC Consolidated Grant.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Alloin D., Pelat D., Phillips M., Whittle M., 1985, ApJ, 288, 205 Antonucci R., 1993, ARA&A, 31, 473 Gebhardt K., Bender R., Bower G., et al., 2000, ApJ, 539, L13 Hamann F., Ferland G., 1992, ApJ, 391, L53 Astronomer's Telegram, 5059, 1", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "We present the results of our detailed timing studies of an anomalous X-ray pulsar, 1RXS J170849.0400910, using Rossi X-ray Timing Explorer ( RXTE ) observations spanning over ∼ 6 yr from 2005 until the end of RXTE mission. We constructed the long-term spin characteristics of the source and investigated time and energy dependence of pulse profile and pulsed count rates. We find that pulse profile and pulsed count rates in the 2 -10 keV band do not show any significant variations in ∼ 6 yr. 1RXS J170849.0-400910 has been the most frequently glitching anomalous X-ray pulsar: three spin-up glitches and three candidate glitches were observed prior to 2005. Our extensive search for glitches later in the timeline resulted in no unambiguous glitches though we identified two glitch candidates (with ∆ ν/ν ∼ 10 -6 ) in two data gaps: a strong candidate around MJD 55532 and another one around MJD 54819, which is slightly less robust. We discuss our results in the context of pulsar glitch models and expectancy of glitches within the vortex unpinning model. Subject headings: pulsars: individual (AXP 1RXS J170849.0-400910 ) -stars: neutron -X-rays: stars", "pages": [ 1 ] }, { "title": "LONG-TERM TIMING AND GLITCH CHARACTERISTICS OF ANOMALOUS X-RAY PULSAR 1RXS J170849.0-400910", "content": "Sinem S¸as¸maz Mus¸ 1 , Ersin Go˘gus¸ 1 Draft version March 19, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Glitches, sudden jumps in the rotation frequency of neutron stars, are the unique events that provide invaluable information on the internal structure of extremely compact stars. Originally detected from rotation powered neutron stars (see e.g., Richards & Comella 1969; Radhakrishnan & Manchester 1969), glitches are generically not associated to changes in the radiative behavior of the source. (but see, Weltevrede et al. 2011). Therefore, the proposed glitch models involve dynamical variations in the neutron star interior instead of an external torque mechanism. The size of the glitch typically reflects the underlying internal dynamics of the neutron star: small-size glitches (∆ ν / ν ∼ 10 -9 , aka. Crab -like glitches) are explained by the decrease of the moment of inertia of the pulsar (Ruderman 1969; Baym & Pines 1971) and large-size glitches (∆ ν / ν ∼ 10 -6 , aka. Vela -like glitches) are described as the angular momentum transfer from inner crust neutron superfluid to the crust by the sudden unpinning of the vortices that are pinned to the inner crust nuclei (Anderson & Itoh 1975; Pines et al. 1980). Anomalous X-ray Pulsars (AXPs) are slowly rotating (P ∼ 2 -12 s) neutron stars with persistent emission being significantly in excess of their inferred rotational energy loss rate. So far, there has been no evidence of binary signature in AXPs. They are young systems ( ∼ 10 4 yr) as inferred from their characteristic spin-down ages (P/2 ˙ P ), and also supported by their location on the plane of Milky Way, and the association of at least five AXPs with their supernova remnants. Almost all AXPs emitted short duration, energetic bursts in X-rays (see, e.g., Gavriil et al. 2002; Kaspi et al. 2003 and for a recent review Rea & Esposito 2011). Their surface dipole magnetic field strengths inferred from their periods and [email protected] 1 Sabancı University, Faculty of Engineering and Natural Sciences, Orhanlı -Tuzla, 34956 Istanbul Turkey spin-down rates are on the order of 10 14 -10 15 G, which is much higher than that of conventional magnetic field strengths of pulsars. The decay of their extremely strong magnetic fields is proposed as the source of energy for their persistent X-ray emission and burst activity (Thompson & Duncan 1995, 1996; Thompson et al. 2002). Recently, observational evidence of dipole field decay was reported by Dall'Osso et al. (2012). Glitch activity from an AXP was first seen in 1RXS J170849.0-400910 (Kaspi et al. 2000). Thanks to almost continuous spin monitoring of AXPs with RXTE for more than a decade, sudden spin frequency jumps have now been observed from six AXPs (see, e.g., Kaspi et al. 2003; Dall'Osso et al. 2003; Woods et al. 2004; Morii et al. 2005; Israel et al. 2007a,b; Dib et al. 2008, 2009; Gavriil et al. 2011). Fractional glitch amplitudes (∆ ν / ν ) of these events range from 10 -8 to 10 -5 (Dib et al. 2009; ˙ I¸cdem et al. 2012) and fractional postglitch change in spin-down rates (∆ ˙ ν /˙ ν ) are between -0.1 and 1 (Kaspi et al. 2003; Dib et al. 2009). Glitches from AXPs somehow resemble those from radio pulsars, but contain some peculiar distinctive features in their recovery behavior and associated radiative characteristics (Woods et al. 2004; Morii et al. 2005; Dib et al. 2008, 2009; Gavriil et al. 2011). AXP 1E 2259+586 went into an outburst in conjunction with a glitch (Kaspi et al. 2003; Woods et al. 2004). AXP 1E 1048.1-5937 has shown X-ray burst correlated with a glitch event (Dib et al. 2009). During the burst active phase of AXP 4U 0142+61 between 2006 and 2007, six short bursts and a glitch with a long recovery time were observed (Gavriil et al. 2011). AXP 1E 1841-045 has exhibited bursts and glitches, but not coincidentally (Dib et al. 2008; Zhu & Kaspi 2010; Kumar & Safi-Harb 2010; Lin et al. 2011). Israel et al. (2007b) reported a burst and an extremely large glitch (∆ ν / ν ∼ 6 × 10 -5 ) from CXOU J164710.2-455216, but the possibility of such a glitch was ruled out by Woods et al. (2011). How- he latter team point out that a glitch with the size of usual AXP glitches may indeed have occurred. 1RXS J170849.0-400910 has been the most frequently glitching AXP (Kaspi et al. 2000, 2003; Dall'Osso et al. 2003; Israel et al. 2007a; Dib et al. 2008), but it has not shown any bursts or remarkable flux variability related to the glitch epochs. It is still unclear whether glitches are always associated with radiative enhancements. Recently, Pons & Rea (2012) suggested that in the context of the starquake model, glitches observed in the bright sources can be related to the radiative enhancements but due to the bright quiescent state of these sources and fast decay of the enhancements, these events can be observed as small changes in the luminosity or only detected in faint sources. 1RXS J170849.0-400910 is an AXP with a spin period of ∼ 11 s. After the discovery of its spin period (Sugizaki et al. 1997), it has been monitored with RXTE for ∼ 13.8 yr. Analyzing the first ∼ 1.4 yr of data Israel et al. (1999) and Kaspi et al. (1999) have concluded that the source is a stable rotator. The continued monitoring has been essential in detecting three unambiguous glitches and three glitch candidates without any significant pulse profile variations (Kaspi et al. 2000, 2003; Dall'Osso et al. 2003; Israel et al. 2007a; Dib et al. 2008). There appears to be a correlation between intensity and spectral hardness: the Xray spectrum gets softer(harder) while the X-ray flux decreases(increases), possibly in relation with glitches (Rea et al. 2005; Campana et al. 2007; Rea et al. 2007; Israel et al. 2007a). Gotz et al. (2007) reported the same correlation in the hard X-rays using INTEGRAL /ISGRI data. However, den Hartog et al. (2008) claimed that they did not find the reported variability in their analysis. Thompson et al. (2002) proposed that external magnetic field can twist and untwist. Twisting and untwisting of the external magnetic field can lead to cracks and unpin the vortices for the glitches (Thompson & Duncan 1996; Dall'Osso et al. 2003). Such twist/untwist of the magnetic field with a period of ∼ 5 -10 yr has been suggested as an explanation for the observed correlations (Rea et al. 2005; Campana et al. 2007). Here, we report on the analysis of long-term RXTE observations of 1RXS J170849.0-400910 spanning ∼ 6 yr. In § 2 we describe RXTE observations that we used in our analysis. We present long-term timing characteristics of the source in § 3.1. In § 3.2 & § 3.3 we constructed the pulse profiles, calculated pulsed count rates and examined their variability both in time and energy. We present the results of our extensive search for glitches in § 3.4. Finally, in § 4 we discuss our results in the context of glitch models and expectancy of glitches in the vortex unpinning model.", "pages": [ 1, 2 ] }, { "title": "2. RXTE OBSERVATIONS", "content": "1RXS J170849.0-400910 has been almost regularly monitored with RXTE in 528 pointings since the beginning of 1998. Phase connected timing behavior of the source was investigated by Dib et al. (2008) using the RXTE data collected between 1998 January 12 and 2006 October 7, Dall'Osso et al. (2003) using data from 1998 January 13 to 2002 May 29, and Israel et al. (2007a) using from 2003 January 5 to 2006 June 3. Here we ana- lyzed RXTE data collected in 280 pointings between 2005 September 25 and 2011 November 17 with the Proportional Counter Array (PCA). Note that the first 49 pointings in our sample were also used by Dib et al. (2008). We included them in order to maintain the continuity in the timing characteristics of 1RXS J170849.0-400910. Exposure times of individual RXTE observations ranged between 0.25 ks (in one observation) and 2.5 ks, with a mean exposure time of 1.9 ks (see Figure 1 for a distribution of exposure times). For our timing analysis, we used data collected with all operating Proportional Counter Units (PCUs) in GoodXenon mode that provides a fine time resolution of 1 µ s.", "pages": [ 2 ] }, { "title": "3.1. Phase Coherent Timing", "content": "We selected events in the 2 -6 keV energy range from the top Xenon layer of each PCU in order to maximize the signal-to-noise ratio, as done also by Dib et al. (2008). All event arrival times were converted to the solar system barycenter and binned into light curves of 31.25 ms time resolution. We inspected each light curve for bursts and discarded the time intervals with the instrumental rate jumps. We merged observations together if the time gap between them was less than 0.1 days. The first set of observations (i.e., segment 0 in Table 1) which includes 49 observations from Dib et al. (2008) were folded initially with the spin ephemeris given by Dib et al. (2008) and later by maintaining the phase coherence. We then cross-correlated the folded pulse profiles with a high signal-to-noise template pulse profile generated from a subset of observations and determined the phase shifts of observations with respect to the template. We fitted phase shifts with whose coefficients yield the spin frequency, and its higher order time derivatives, if required. In Table 1 we list the best fit spin frequency and frequency derivatives to the specified time intervals, obtained using also listed number of time of arrivals (TOAs). In Figure 2 (a) we present the spin frequency evolution of 1RXS J170849.0-400910, and in (b) phase residuals after subtraction of the best fit phase model given in Table 1. We obtained frequency derivatives by fitting a second order polynomial to the sub-intervals of about 2.5 months long data and present them in Figure 2 (c).", "pages": [ 2, 3 ] }, { "title": "3.2. Pulse Profile Evolution", "content": "We investigated long term pulse profile evolution of the source both in energy and time. For the pulse profile analysis, we excluded data collected with PCU0 and the data of PCU1 for the observations after 2006 December 25 due to the loss of their propane layers (therefore, having elevated background levels). We obtained the pulse profiles with 32 phase bins by folding the data in six energy bands with the appropriate phase connected spin ephemeris given in Table 1. The energy intervals investigated are 2 -10 keV, 2 -4 keV, 4 -6 keV, 6 -8 keV, 8 -12 keV and 12 -30 keV. In order to account for the different number of operating PCUs, we normalized the rates of each bin with the number of active PCUs. Finally, we subtracted the DC level and divided by the maximum rate of each profile. In Figures 3 and 4, we present the normalized pulse profiles for the six segments given in Table 1 in six energy bands and their evolution in time. The 2 -10 keV pulse profiles of 1RXS J170849.0400910 are characterized by a broad structure formed by the superposition of two features: the main peak near the pulse phase, φ ∼ 0.55 and a weaker shoulder around φ ∼ 0.85. Pulse profiles of the two lowest energy bands exhibit an additional shoulder (near phase ∼ 0.35) in the 55203 -55516 epoch (Segment 4), which is not clearly seen in any other epochs. Pulse profiles in the 2 -4 keV band consist of the main peak in all epochs, while the shoulder feature ( φ ∼ 0.85) is either weak or non-existent. The shoulder appears in the 4 -6 keV band, and becomes more dominant above 6 keV. Pulse profiles above 8 keV contain only the shoulder feature. Note the fact that the duty cycle of the pulse profiles drops with increasing energy. The dominance of the secondary peak (shoulder) with the increase in photon energy was also reported in den Hartog et al. (2008) by using INTEGRAL , XMM -Newton and earlier RXTE observations. We calculated the Fourier Powers (FPs) for a quantitative measure of the pulse profile variations. First we computed the Fourier transform of each profile and calculated the powers in the first six harmonics as FP k = 2(a 2 k + b 2 k )/( σ 2 a k + σ 2 b k ). Here a k and b k are the coefficients in the Fourier series, and σ a k and σ b k are the uncertainties in the coefficients a k and b k , respectively. Second, we corrected the powers for the binning using equation 2.19 of van der Klis (1989) and calculated upper and lower limits to the FPs by using the method described in Groth (1975) (and also in Vaughan et al. 1994). Finally, we normalized the FPs by the total power. We show in Figure 5, the time evolution of the normalized harmonic powers in the first three Fourier harmonics. We find that the FPs remain fairly constant in time in all investigated energy intervals.", "pages": [ 3 ] }, { "title": "3.3. Pulsed Count Rates", "content": "PCA is not an imaging instrument; it collects all events originating within about 1 o (FWHM) field centered near the position of 1RXS J170849.0-400910. Therefore, we cannot construct a precise X-ray light curve of the source using PCA observations since the accurate determination of X-ray background with the PCA is not possible. Nevertheless, we can trace the behavior of the pulsed Xray emission of 1RXS J170849.0-400910 since there is no other pulsed X-ray source with exactly the same pulse period in the vicinity. X-rays originating from the other sources in the field of view (even the pulsed ones) are averaged out after folding the data with the spin frequency of 1RXS J170849.0-400910 and remain within the DC level. For these reasons, we calculated the rms pulsed count rates of the source using where R i are the count rates in each phase bin, ∆R i are their uncertainties, R ave is their average and N is the number of phase bins. Note that this is a background exempt representation of pulsed intensity of the source. In Figure 2(d) we present the time variation of rms pulsed count rates in the 2 -10 keV energy range. Here, each pulsed intensity value is an average of about 1 month of data accumulation. We find that the rms pulsed count rate in the 2 -10 keV band does not show any significant variation. Figure 6 presents the pulsed count rates as a function of energy (in other words, rough energy spectra of the pulsed X-ray emission from 1RXS J170849.0-400910). Power law fits to these rough energy spectra yield a general trend from a more steep shape to a more shallow one as time progresses.", "pages": [ 3 ] }, { "title": "3.4. Search for Glitches", "content": "There is no explicit glitch detected in our data sample as it can be seen from the fit results to the phase drifts in Table 1. To investigate whether there are any small amplitude variations in phase drifts (i.e. frequency jumps), we fitted phase shifts using the MPFITFUN 2 (Markwardt 2009) procedure which performs Levenberg-Marquardt least-squares fit with the corresponding phases of a glitch model containing a jump in every ∼ 0.1 day and a linear decay, as follows: where ν 0 ( t ) is the preglitch frequency evolution, ∆ ν is the frequency jump, ∆ ˙ ν is the change of the frequency derivative after the glitch and t g is the epoch of the glitch. First we applied this methodology to a previously published glitch in 2005 June and a candidate glitch in 2005 September. We detected the frequency jumps (∆ ν ) and glitch epochs in agreement with the published values (Israel et al. 2007a; Dib et al. 2008). We then carried out the glitch search in all six epochs listed in Table 1 as follows: For each epoch, we analyzed the fit results on the ∆ ν versus the reduced χ 2 plane and identified the set of parameters corresponding to the lowest reduced χ 2 value. We then computed rms fluctuations of phase residuals using the possible glitch parameters and compared them with those obtained using the polynomial fit results listed in Table 1. We find that rms phase residual fluctuations with respect to the glitch model fits do not indicate any improvement in the fit quality compared to the polynomial fits (Figure 7). Moreover, the largest glitch amplitude (∆ ν ) obtained is about 3 × 10 -8 Hz in segments 0, 1 and 5 which could well be due to random fluctuations of phases, as can be seen in Figure 7. Consecutive RXTE observations were typically performed at 7 -10 day time intervals. Due to Sun con- straints, there were five longer gaps of ∼ 50 days in our data set. In order to assess the probability for the detection of a glitch that might have occurred during these longer gaps, we adopted the detectability criterion defined as (Alpar & Ho 1983; Alpar & Baykal 1994): where δν and δ ˙ ν specify the total error on the spin frequency and frequency derivative determined on both ends of the gap, ∆t denotes the duration of the gap, and ∆ ν is the change in spin frequency due to a putative glitch. Equation 5 implies that ∆ ν has to be much bigger than maximum phase error accumulated across the gap in order to identify it as a possible glitch event. We calculated the total phase error for each gap adopting the timing solutions on both sides of the gaps, and present these results in Table 2. We then applied the glitch search methodology to ∼ 250 day long data segments centered around each gap (gap segment), and evaluated minimum χ 2 searches as explained above. Best-fit timing solutions are listed in Table 2. Among all gaps, only glitch amplitudes in gap segment 3 (54687 -54913) and gap segment 5 (55406 -55666) satisfy condition 5. In particular, the glitch amplitude in gap segment 5 is ∼ 7 times larger than the noise criterion which makes it a rather strong candidate for a possible glitch event. The putative glitch identified in gap segment 3 has an amplitude ∼ 4 times larger than the corresponding minimum noise criterion. The amplitudes of estimated glitch events in gap segments 1, 2 and 4 possess large errors. The rms fluctuations of phase residuals in gap segments are similar in gap segments 1, 2, 3, and 5, while they are much larger in gap segment 4. Note that glitch amplitude in gap segment 4 is affected from an outlier phase measurement (see Figure 8), without which the glitch amplitude becomes even less significant. We, therefore, identified two glitch candidates; a strong case in the gap segment 5, and another one in gap segment 3 which is slightly less robust. We discuss their implications below.", "pages": [ 3, 4, 7 ] }, { "title": "4. DISCUSSION AND CONCLUSIONS", "content": "We performed detailed long term timing studies of 1RXS J170849.0-400910 spanning ∼ 6 yr. Together with the earlier extensive study of the source by Dib et al. (2008), our investigation considers the entire database of RXTE observations of 1RXS J170849.0-400910. In our long-term timing investigations, it was possible to describe the phase shifts with a second order polyno- al in only one interval (Segment 4 in Table 1), while all other parts required higher order terms. These results are similar to what has been obtained by Dib et al. (2008), Archibald et al. (2008), and Israel et al. (2007a), confirming the fact that 1RXS J170849.0-400910 is indeed a noisy pulsar. The pulse profile of 1RXS J170849.0-400910 in the 2 -10 keV band does not show any significant variations over the last ∼ 6 yr, maintaining its general pulse structure as in the earlier epochs. A minor structure (described as a shoulder above) in the pulse profile below 4 keV becomes stronger with energy and dominates the pulse profiles above 8 keV, as also noted by den Hartog et al. (2008) regarding earlier observations of the source. We also find no significant changes in the rms pulsed count rates (i.e., a measure of the pulsed flux) in the 2 -10 keV range. In these respects, 1RXS J170849.0400910 exhibits an almost stable pulsed X-ray emission behavior. We constructed a coarse energy spectrum of the rms pulsed count rates for each observation segment and found that it becomes gradually harder with time, as indicated by a shallowing power law index. As a result of ∼ 14 yr of RXTE observations, three glitches with two different recovery characteristics were unveiled unambiguously, and three candidate glitches were suggested in the time baseline between 1999 and 2005. Such a glitching behavior of 1RXS J170849.0400910 made this system one of the most frequently glitching pulsars (Israel et al. 2007a; Dall'Osso et al. 2003; Dib et al. 2008). It is important to report the fact that, we do not find any unambiguous glitches in the time interval between 2006 and 2011. However, glitch search in the gaps yielded a strong candidate in gap 5 with glitch amplitude ∼ 10 -7 which is ∼ 7 times larger than the noise in this gap and on the order of largest glitches observed from this source. We identified another candidate in gap segment 3, although it is slightly less robust. Glitches are generally explained by models involving the neutron star crust, superfluid component of the inner crust or core superfluid and starquakes. The superfluid vortex unpinning model involves the crust and inner crust superfluid (Anderson & Itoh 1975; Alpar et al. 1984a). In this model vortices formed by superfluid are pinned to the neutron-rich nuclei. While the crust spins down due to the electromagnetic torques, a rotational lag between the superfluid component and the crust builds up. When a critical value of rotational lag ( δ Ω ≡ Ω s -Ω c , where Ω s and Ω c are the superfluid's and crust's rotational rate, respectively) is reached, vortices suddenly unpin, resulting in transfer of angular momentum to the crust, i.e., glitch. This lag also determines the glitch occurrence time interval. This model is successful in explaining large glitches (∆ ν / ν ∼ 10 -6 ), such as those observed from the Vela pulsar with an occurrence time interval of ∼ 2 yr (Alpar et al. 1981, 1984b). Another class of models invokes starquakes, which are triggered by the cracking of the solid neutron star when growing internal stresses strain the crust beyond its yield point (Ruderman 1969, 1976, 1991; Baym & Pines 1971). This critical strain can be reached due to several mechanisms: the star spin down causes a progressive decrease of the equilibrium oblateness of the crust (Ruderman 1969, 1991; Franco et al. 2000); variations of the core magnetic field, due to the motion of core superfluid vortices cou- a Values in parenthesis are the uncertainties in the last digits of their associated measurements pled to it (Srinivasan et al. 1990; Ruderman et al. 1998); and, in strongly magnetized neutron stars, the rapid diffusion of the core magnetic field (or the 'turbulent' evolution of the crustal field) provides an alternative channel to produce crustal fractures (Thompson & Duncan 1996; Rheinhardt & Geppert 2002). Dall'Osso et al. (2003), based on the different recovery characteristics of the glitches of 1RXS J170849.0-400910, proposed that they can be explained by a magnetically-driven starquake model since they intrinsically involve local processes and a higher degree of complexity. In order to discriminate between different possible models, Alpar & Baykal (1994) following Alpar & Ho (1983), investigated the global properties of large pulsar glitches using a sample of 430 pulsars, excluding the Vela pulsar. As these sources are not continuously monitored due to limited telescope times or other observational constraints, there are unavoidable data gaps in between successive pointings. This case puts a serious constraint on the detectability of a glitch if it occurs in a data gap of a pulsar with noisy timing behavior. They introduced a noise criterion (see Eqn. 5) for significantly detecting frequency jumps in the observational gaps. Therefore, they restricted their analysis to the 19 pulsar glitches with ∆ ν / ν > 10 -7 . They estimated the physical parameters, e.g., inter-glitch time for the vortex unpinning model and the glitch size for the core-quake model. The parameters of the former model were estimated with two different assumptions for unpinning: First, the critical glitch parameter is taken as δ Ω which is a representative of the number of vortices that is unpinned at the time of the glitch. Second, this parameter is taken as fractional density of the unpinned vortices that is proportional to δ Ω / Ω, as the density of vortices ∝ Ω. They also assumed that the probability of observing n glitches is given by Poisson statistics. Glitch size estimation from the core-quake model is far bigger than the glitch amplitudes of the Vela pulsar and sample mean. Thus, their work statistically excluded the core-quake model. They also compared the parameter estimates of the vortex un- pinning model with those of glitches from Vela and other pulsars, and concluded that the vortex unpinning model with a constant fractional vortex density ( 〈 δ Ω / Ω 〉 ) is the most compatible model and can represent an invariant for glitches. To test the glitch expectancy within the vortex unpinning model for 1RXS J170849.0-400910 glitches, we applied the same statistical glitch expectancy analysis (see Equation 11 of Alpar & Baykal 1994) and estimated the expected number of glitches using ∼ 14 yr of RXTE observations. We calculated the critical fractional vortex density of the vortex unpinning model by using the time span between 1998 January and 2005 November, which contains three glitches and three glitch candidates (Dib et al. 2008; Dall'Osso et al. 2003; Israel et al. 2007a). For a single pulsar, ˙ ν / ν value is not expected to fluctuate between observations. However, this is not the case for 1RXS J170849.0-400910 as it changes between -1.87 × 10 -12 s -1 and -1.31 × 10 -12 s -1 with an average value of -1.66 × 10 -12 s -1 , which further implies the noisy timing characteristics of the source. Therefore, we performed our calculations for all these three values. First we included the observational gaps into the total time span which, by the chosen noise criterion, restricts our analysis to large glitches with ∆ ν / ν on the order of 10 -6 . Using ˙ ν / ν values and observed number of glitches with ∆ ν / ν ∼ 10 -6 (i.e., n = 2), we obtain the upper, lower and average values for critical parameter value of the vortex unpinning model. We note an important fact here that a glitch candidate (i.e., near candidate glitch 2 in Dib et al. (2008)) was reported by Israel et al. (2007a) with a fractional amplitude of 1.2 × 10 -6 . If the latter report is correct, the number of large glitches in the 1998 -2005 interval would be 3 (i.e, n = 3) which changes the critical parameter. Finally, we excluded all data gaps except the ones with glitches reported in them, and the ones that satisfied the noise criterion in our analysis, and we considered all reported glitches with ∆ ν / ν /greaterorsimilar 10 -7 (i.e., n = 6) and calculated the critical parameters for this case as well. In Table 3 we list the values of the critical parameter for each of the above-mentioned cases and their corresponding expected number of glitches in the time intervals between 1998 -2005, 1998 -2011, and 2006 -2011. As expected, the average value of the critical parameter yields the observed number of glitches in the 1998 -2005 interval. We find that the total number of expected glitches with fractional amplitudes of /greaterorsimilar 10 -6 (n = 2 in Table 3) varies between 3.2 and 4.6 if the time baseline spans untill the end of the RXTE coverage of the source in 2011 November. The number of glitches in the 2006 -2011 time range, where we found a strong candidate, were expected to range from 1.4 to 2.0. We then repeated the above procedure, this time excluding all data gaps except the ones with reported candidate glitches. In this case, the noise criterion allows consideration of all glitches with ∆ ν / ν ∼ 10 -7 (i.e., n = 6), and we re-calculated the critical parameters (see Table 3). Glitch expectancy analysis within the context of vortex unpinning model suggests that 1RXS J170849.0400910 might have had, on average, two large glitches in 6 yr, corresponding to the interval of 2006 -2011 (Table 3). The two significant glitch candidates we identified in gap segments are, therefore, important, since they comprise the observed number to match with the expectancy of the vortex creep model. As far as only glitch statistics is concerned, this case implies that the mechanism leading to the observed glitches in 1RXS J170849.0-400910 is internal. However, where particular glitch characteristics were concerned (e.g., discrepancies in glitch recovery), the vortex unpinning model is argued to be not sufficient (Dall'Osso et al. 2003). 1RXS J170849.0-400910 is the only member of the magnetar family that has not exhibited energetic X-ray bursts. Almost all other AXPs, that have experienced timing glitches, emitted energetic bursts either in conjunction with (e.g., 1E 2259+586, Woods et al. (2004)) or contemporaneous to their glitches. It is, therefore, suggestive that a common mechanism might be responsible for both glitches and bursts. The dipole magnetic field strength of 1RXS J170849.0-400910 as inferred from its spin period and spin-down rate is about 4.6 × 10 14 G, that is strong enough to produce significant deformation in the neutron star crust and eventually lead to the release of energy via bursts (Thompson & Duncan 1995). Nevertheless, the condition on 1RXS J170849.0400910 has not given rise to any observable bursts, even though it has experienced the largest number of glitches among all magnetars. While a common mechanism could reproduce coincident energetic bursts and glitches in general, it might be generating glitches but not detectable enhancements and bursts in 1RXS J170849.0-400910, possibly due to this source having slightly lower crust shear modulus, so that the release of less energy can still produce breaks in the crust. The energetic bursts, however, are not accounted for within the context of the vortex unpinning model which appears to be favored for this source in our statistical investigations. Recently Eichler & Shaisultanov (2010) suggested that vortices can be unpinned mechanically via oscillations rather than by a sudden heat release. According to their estimation, the relative velocity between the crust and superfluid, which is generated by the mechanical energy release at the depths below 100 m, can exceed the critical velocity lag and unpin the vortices. In order to explain the radiatively silent glitches seen in some AXPs (as in the case of 1RXS J170849.0-400910) they proposed that mechanically triggered glitch event might not be accompanied by a long-term X-ray brightening since a glitch can be triggered by a less energy release. In this picture, the origin of X-ray brightening is also through mechanical energy release and these flux enhancements are expected to be accompanied with glitch events. This scenario can be diagnosed through the exact timing of glitches with radiative enhancements (Eichler & Shaisultanov 2010). We would like to thank M. Ali Alpar and the anonymous referee for helpful comments. SS¸M acknowledges support through the national graduate fellowship program of the Scientific and Technological Research Council of Turkey (T UB ˙ ITAK).", "pages": [ 7, 8, 9, 10 ] }, { "title": "REFERENCES", "content": "Alpar. M. A., Anderson, P.W., Pines, D., & Shaham, J. 1981, ApJ, 249, L29 Alpar, M. A. & Ho, C. 1983, MNRAS, 204, 655 Alpar. M. A., Anderson, P.W., Pines, D., & Shaham, J. 1984a, ApJ, 276, 325 Alpar. M. A., Anderson, P. W., Pines, D., & Shaham, J. 1984b, ApJ, 278, 791 Alpar, M. A., & Baykal, A. 1994, MNRAS, 269, 849 Anderson, P. W., & Itoh, N. 1975, Nature, 256, 25 Archibald, A. M., Dib, R., Livingstone, M. A., & Kaspi, V. M. 2008, in AIP Conf. Ser. 983, 40 Years of Pulsars: Millisecond Pulsars, Magnetars and More, eds. C. Bassa, Z. Bassa, A. Cumming & V. M. Kaspi (Melville, NY: AIP), 265 Baym, G., & Pines, D. 1971, Ann.Phys., 66, 816 Campana, S., Rea, N., Israel, G. L., Turolla, R., & Zane, S. 2007, A&A, 463, 1047 Dall'Osso, S., Israel, G. L., Stella, L., Possenti, A., & Perozzi, E. 2003, ApJ, 599, 485 Dall'Osso, S., Granot, J., & Piran, T. 2012, MNRAS, 422, 2878 Cummings, J., Perna, R., & Stella, L. 2007b, ApJ, 664, 448 Kaspi, V. M., Chakrabarty, D., & Steinberger, J. 1999, ApJ, 525, L33 Kumar, H. S., & Safi-Harb, S. 2010, ApJ, 725, L191 Lin, L., et al. 2011, ApJ, 740, L16 Markwardt, C.B. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 411, Astronomical Data Analysis Software and Systems XVIII, ed. D.A. Bohlender, D. Durand & P. Dowler, 251 Morii, M., Kawai, N., & Shibazaki, N. 2005, ApJ, 622, 544", "pages": [ 10, 11 ] } ]

[ { "title": "ABSTRACT", "content": "Near-future data from ESA's Gaia mission will provide precise, full phase-space information for hundreds of millions of stars out to heliocentric distances of ∼ 10 kpc. This 'horizon' for full phasespace measurements is imposed by the Gaia parallax errors degrading to worse than 10%, and could be significantly extended by an accurate distance indicator. Recent work has demonstrated how Spitzer observations of RR Lyrae stars can be used to make distance estimates accurate to 2%, effectively extending the Gaia , precise-data horizon by a factor of ten in distance and a factor of 1000 in volume. This Letter presents one approach to exploit data of such accuracy to measure the Galactic potential using small samples of stars associated with debris from satellite destruction. The method is tested with synthetic observations of 100 stars from the end point of a simulation of satellite destruction: the shape, orientation, and depth of the potential used in the simulation are recovered to within a few percent. The success of this simple test with such a small sample in a single debris stream suggests that constraints from multiple streams could be combined to examine the Galaxy's dark matter halo in even more detail - a truly unique opportunity that is enabled by the combination of Spitzer and Gaia with our intimate perspective on our own Galaxy. Subject headings: Galaxy: structure - Galaxy: halo - cosmology: dark matter", "pages": [ 1 ] }, { "title": "SPITZER, GAIA, AND THE POTENTIAL OF THE MILKY WAY", "content": "Adrian M. Price-Whelan 1,2 Kathryn V. Johnston 1 Accepted to ApJ Letters", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The existence of vast halos of unseen dark matter surrounding each galaxy has long been proposed to explain the surprisingly large motions of the baryonic matter that we can see (e.g., Rubin & Ford 1970). Dark-matter-only simulations of structure formation lead us to expect that these dark matter halos should have density distributions that are described by a universal radial profile (Navarro et al. 1996) with a variety of triaxial shapes (Jing & Suto 2002). The inclusion of baryons in the simulations tends to soften the triaxiality of the dark matter in the inner regions of the halo (e.g., as the disk forms, Bailin et al. 2005) and can alter the radial profile through a combination of adiabatic contraction and energetic feedback (e.g. Pontzen & Governato 2012). Hence, measurements of the shape, orientation, radial profile, and extent of dark matter halos provides information about the formation of these vast structures, as well as the messy baryonic processes that continue to shape them. The Milky Way is the best candidate for such a detailed study of a dark matter halo since we can resolve large samples of stellar tracers. Thousands of blue horizontal branch stars selected from the Sloan Digital Sky Survey (SDSS) have been used to probe the Milky Way mass out to tens of kpc (SDSS, see Deason et al. 2012a; Kafle et al. 2012), and estimates with combined tracers extend to 150kpc (Deason et al. 2012b). This approach assumes that the tracers represent a random sampling of phase-mixed orbits drawn from a smooth distribution function, however large area surveys have revealed the existence of large-scale spatial inhomogeneities in the form of giant stellar streams (Newberg et al. 2002; Majewski et al. 2003; Belokurov et al. 2006), 1 Department of Astronomy, Columbia University, 550 W 120th St., New York, NY 027, USA 2 [email protected] demonstrating that a significant fraction of the stellar halo is neither randomly sampled nor is fully phasemixed. A complimentary approach to measuring the mass distribution is to instead take advantage of the non -random nature of the Galaxy's stellar distribution and utilize the knowledge that stars in streams were once all part of the same object. Such approaches can require orders of magnitude fewer tracers than a randomly sampled population to achieve comparable accuracy. One method is to simply fit orbits to observations of streams (e.g., Koposov et al. 2010). However, the assumption that debris traces a single orbit is actually incorrect (see Johnston 1998; Helmi & White 1999) and changes in orbital properties along debris streams can lead to systematic biases in measurements of the Galactic potential (Eyre & Binney 2009; Varghese et al. 2011). Sanders & Binney (2013a) recently demonstrated that this bias is equally problematic for the very thinnest, coldest streams, whose observed properties may be indistinguishable from those of the parent orbit (e.g. such as the globular cluster, GD1 - see Koposov et al. 2010), as for the much more extended and hotter streams (e.g. such as debris from the Sagittarius dwarf galaxy - see Majewski et al. 2003) where offsets from a single orbit are clearly apparent. One way to address these biases is to run self-consistent N-body simulations of satellite destruction in a variety of potentials with the aim of simultaneously constraining both the properties of the satellite and the Milky Way. Many studies of the Sagittarius debris system (hereafter Sgr) have adopted this approach, with the most recent work attempting to place constraints on the triaxiality and orientation of the dark matter halo (Law & Majewski 2010). The promise of near-future data sets including full phase-space information has also inspired other approaches. Binney (2008) and Pe˜narrubia et al. (2012) demonstrate that the distribution of energy and entropy in debris, respectively, will be minimized only for a correct assumption of the form of the Galactic potential. Sanders & Binney (2013b) examine the distribution of debris in action-angle co-ordinates and show that stars stripped from the same disrupted object must lie along a single line in angle-frequency space, providing a constraint that can be used as a potential measure. In this Letter we re-examine and update a complimentary approach to using tidal debris as a potential measure (originally proposed by Johnston et al. 1999b) in the context of current and near-future observational capabilities, and apply it to a simulation of the Sgr debris system. In Section 2 we outline the observational prospects and Sgr properties that motivated this re-examination. In Section 3 we present the updated potential measure and test it with synthetic observations of simulated Sgr debris. In Section 4 we highlight the advantages and shortcomings of this method. We conclude in Section 5.", "pages": [ 1, 2 ] }, { "title": "2. CONTEXT AND MOTIVATION", "content": "The method presented in Section 3 takes advantage of three distinct developments: (i) the demonstration of a technique for deriving distances to individual RR Lyrae stars with 2% accuracies (Section 2.1); (ii) the prospect of proper motion measurements of the same stars with ∼ 10 µ as/yr precision (Section 2.2); and (iii) the tracing of debris associated with Sgr around the entire Galaxy (Section 2.3)", "pages": [ 2 ] }, { "title": "2.1. Spitzer and 2% distance errors to RR Lyrae in the halo", "content": "There is a long tradition for using RR Lyrae stars in the Galaxy to study structure (e.g. Shapley 1918), substructure (e.g. Sesar et al. 2010), and distances to satellite galaxies (e.g. Clementini et al. 2003). However, studies of RR Lyrae at optical wavelengths are limited by both metallicity effects on the intrinsic brightness of these stars and variable extinction along the line of sight. Moreover, systematic differences between instruments make it difficult to tie observations across the sky to a common scale. At longer wavelengths, RR Lyrae promise tighter constraints on distances. Madore & Freedman (2012) have recently shown, using five stars with trigonometric parallaxes measured by Hubble (Benedict et al. 2011), that the dispersion in the mid-IR Period-Luminosity (PL) relation (first mapped by Longmore et al. 1986) at wavelengths measurable by NASA's Spitzer mission is ∼ 0.03 mag. This implies that it is possible to use Spitzer to determine distances that are good to 2% for individual RR Lyrae stars out to ∼ 60 kpc ( Spitzer 's limit for detecting and measuring RR Lyrae). For comparison, distance measurements of Blue Horizontal Branch stars typically achieve ∼ 10-15% uncertainties (if appropriate color measurements are available, e.g., Deason et al. 2012b).", "pages": [ 2 ] }, { "title": "2.2. Gaia and the age of astrometry", "content": "The Gaia satellite (Perryman et al. 2001) is an astrometric mission which aims to measure the positions of billions of stars with 10-100 µ as accuracies. Combined with expected proper motion accuracies, this will enable full six-dimensional phase-space maps of the Galaxy with 10 < 10% distance errors for heliocentric distances of up to ∼ 6 kpc for RR Lyrae stars. Figure 1 shows the Gaia end-of-mission distance and tangential velocity error estimates for RR Lyrae. Within 2 kpc, Gaia will measure distances to these stars with better than 2% accuracy - RR Lyrae in this volume can be used to test and calibrate the Spitzer PL relation described above. Beyond the 2 kpc threshold, the midIR PL relation for RR Lyrae will provide better distance measurements. The combination of Spitzer and Gaia data will extend the 'horizon' of where precise, six-dimensional phasespace maps of the Galaxy are possible from < 10 kpc to 60 kpc. This enormous increase in volume will greatly refine data on debris systems in the halo.", "pages": [ 2 ] }, { "title": "2.3. The Sagittarius debris system", "content": "Sgr was discovered serendipitously during a radial velocity survey of the Galactic bulge (Ibata et al. 1994). Signatures of extensive stellar streams associated with Sgr have since been mapped across the sky in carbon stars (Totten & Irwin 1998), M giants selected from 2MASS (Majewski et al. 2003), main sequence turnoff stars from SDSS (Belokurov et al. 2006), and RR Lyrae in the Catalina Sky Survey (Drake et al. 2013). Sgr stream data has inspired a rich set of models (e.g., Johnston et al. 1999a; Fellhauer et al. 2006). Most recently, Law & Majewski (2010, hereafter LM10) combined all the (then) current data on the Sgr debris to constrain both a model of its evolution and the potential in which it orbits. (Note that new observational work by Belokurov et al. (2013) suggest that the trailing tail of Sgr debris does not match the LM10 model.) Figure 2 shows particle positions from the final time-step of the LM10 N-body simulation of dwarf satellite disruption along the expected Sgr orbit in the best-fitting Milky Way halo model. The simulation was run in a three-component potential, with a triaxial, logarithmic halo model of the form where C 1 , C 2 , and C 3 are combinations of the x and y axis ratios ( q 1 , q 2 ) and orientation of the halo with respect to the baryonic disk ( φ ): Acomparison of simulations and data enabled LM10 to make an assessment of the three-dimensional mass distribution of the Milky Way's dark matter halo through constraints on the potential parameters v halo , q 1 , q z , and φ . Combined Spitzer and Gaia measurements of distances and proper motions of RR Lyrae in the Sgr debris will open up new avenues for potential constraints. Figure 1 shows that a 2% distance error is smaller than the distance range in the stream (top panel). Similarly, Gaia proper motion error estimates correspond to tangential velocity errors less than the velocity dispersion for much of the stream (bottom panel). The next section outlines a new method to take advantage of this information.", "pages": [ 2, 3 ] }, { "title": "3. DESCRIPTION AND TEST OF OUR ALGORITHM", "content": "With access to 6D information for stars in a tidal stream, each star becomes a powerful potential measure by exploiting the fact that the stars must have come from the same progenitor: if the orbits of the stars and progenitor are integrated backwards in a a potential that accurately models the Milky Way, the stars should recombine with the progenitor (imagine watching satellite destruction in 'rewind'). If the potential is incorrect, the orbits of the stars will diverge from that of the progenitor and thus will not be recaptured by the satellite system (Figure 3). This approach was originally proposed by Johnston et al. (1999b) and was tested on the proposed characteristics of the Space Interferometry Mission (Unwin et al. 2008). Below we present an updated version of the algorithm: the promise of 2% distances to RR Lyrae stars (see Section 2.1) enables a direct measurement (rather than approximate estimate, as previously assumed) of the position of a star within its debris structure. The test statistic that quantifies how well stars recombine with the satellite has also been rigorously redefined.", "pages": [ 3 ] }, { "title": "3.1. The algorithm: Rewinder", "content": "Quantifying this method requires a sample of stars with known full space kinematics ( x i , v i ) | t =0 (e.g., measurements of all position and velocity components for these stars today at t = 0), the orbital parameters for the progenitor system ( x p , v p ) | t =0 , and a functional form for the potential, Φ( θ ). For a given set of potential parameters, θ , the orbits of the stars and progenitor are integrated backwards for several Gigayears. At each timestep t j , for each particle i , a set of normalized, relative phasespace coordinates are computed where ( x , v ) i and ( x , v ) p are the phase-space coordinates for the particles and progenitor, respectively. These definitions require an estimate of the mass of the satellite, m sat , which, combined with the orbital radius of the satellite, R , and the computed enclosed mass of the potential within R , M enc , sets the instantaneous tidal radius and escape velocity, These quantities are computed at each time step to take the time dependence into account, neglecting massloss from the satellite. Qualitatively, when the distance in this normalized six-dimensional space, D ps ,i = √ | q i | 2 + | p i | 2 glyph[lessorsimilar] 2, the star is likely recaptured by the satellite (in the absence of errors, we find that ∼ 90% of the initially bound particles come within this limit when integrating all orbits backwards). Johnston et al. (1999b) imposed a similar condition as a hard boundary and maximized the number of recaptured particles in a given backwards-integration. What follows is a description of an updated procedure with a statisticallymotivated choice for an objective function. For each star, i , the phase-space distance, D ps , is computed at each timestep t j , and the vector with the minimum phase-space distance is stored Thus, the matrix A ik contains these minimum phasespace distance vectors for each star, where k ∈ [1 , 6]. Intuitively, the variance of the distribution of minimum phase-space vectors will be larger for orbits integrated in an incorrect potential relative to the distribution computed from the 'true' orbital history of the stars: in an incorrect potential, the orbits of the stars relative to the orbit of the progenitor spread out in phase space. Thus, the generalized variance of the distribution - computed for a given set of potential parameters, θ - is a natural choice for the scalar objective function, f ( θ ), used in constraining the potential of the Milky Way", "pages": [ 3, 4 ] }, { "title": "3.2. Application to Simulated Data", "content": "The LM10 simulation data (see Section 2.3) is a perfect test-bed for evaluating the effectiveness of this method. We start by extracting both particle data and the satellite orbital parameters from the present-day snapshot of the simulation data. 3 We then 'observe' a sample of 100 stars from the first leading and trailing wraps of the stream. The radial velocity and distance errors are drawn from Gaussians ( ε RV ∼ N ( µ = 0 , σ = 10 km / s) and ε D ∼ N (0 , 0 . 02 × D )) and the proper motion errors are computed from the expected Gaia error curve. 4 The generalized variance defines a convex function over which we optimize four of the six logarithmic potential parameters: v circ , φ , q 1 , and q z ( q 2 and R c are degenerate with combinations of the other parameters). Figure 4 shows one-dimensional slices of the objective function produced by varying each of the potential parameters by ± 10% around the true values and holding all others fixed. -4 In anticipation of extending the above method to include a true likelihood function, we use a parallelized Markov Chain Monte Carlo (MCMC) algorithm (Foreman-Mackey et al. 2013) to sample from our ob- 3 www.astro.virginia.edu/ srm4n/Sgr/data.html ~ jective function. 5 We use the median value of the converged sample distribution as a point estimate for the potential parameters. To assess the uncertainty in the derived halo parameters, we sample 100 stars 100 times and estimate the potential parameters with each resampling. Figure 5 shows the recovered parameters for each sample and demonstrates the power of this method: a moderately sized sample of RR Lyrae alone places strong constraints on the shape and mass of the Galaxy's dark matter halo. From the covariance matrix derived from the distribution of points in Figure 5, we find the mean recovered parameters and one-sigma deviations to be q 1 = 1 . 36 ± 0 . 02, q z = 1 . 36 ± 0 . 03, φ = 96 . 0 ± 1 . 5 degrees, and v halo = 123 . 2 ± 1 . 6 km/s.", "pages": [ 4, 5 ] }, { "title": "4. DISCUSSION, STRENGTHS, AND LIMITATIONS", "content": "The strengths of this method stem from its simplicity: it requires only a rough estimate of the satellite mass m sat combined with backwards integration of orbits. Rewinder does not assume that stream stars follow a single orbit and instead relies on the fact that each star is on a different orbit. There are also no assumptions made about of the internal distribution of satellite stars. Thus, Rewinder is applicable to any debris that is known to come from a single object and not restricted to the coldest tidal streams. In principle, it could also be applied to the vast stellar debris clouds that have been discovered (e.g., the Triangulum-Andromeda and HerculesAquila clouds; Rocha-Pinto et al. 2004; Belokurov et al. 2006), or even stars that have only associations in orbital properties and do not form a coherent spatial structure (e.g. Helmi & White 1999). The method trivially extends to combining constraints from multiple debris systems at once by simply integrating all debris from several satellites simultaneously, with D ps defined appropriately for each star. It is also important to characterize the the limitations of this method. Firstly, the measurement errors for RR Lyraes associated with the very coldest streams (e.g., 5 Though MCMC is typically not an efficient optimization tool, in this case objective function is both noisy and expensive to compute. The stochasticity and easy parallelization of the algorithm outperforms other optimizers on this problem. the globular clusters Pal5 and GD1; Odenkirchen et al. 2002; Koposov et al. 2010) will likely be too large to resolve the minute differences in orbital properties between the debris and satellite. Second, the present prescription neglects orbital evolution (e.g., dynamical friction) and scattering of stream stars due to the potential of the satellite. Preliminary simulations (to be fully explored in forthcoming work) suggest that these two points can be neglected for satellite masses between ∼ 10 7 and ∼ 10 9 M glyph[circledot] . Lastly, the current version of the algorithm relies on knowledge of the current position and velocity of the parent satellite, which may not be available (e.g., the Orphan Stream; Belokurov et al. 2007).", "pages": [ 5 ] }, { "title": "5. CONCLUSIONS AND MOTIVATION FOR FUTURE WORK", "content": "This paper presents an algorithm for measuring the Galactic potential that anticipates combined data from the Spitzer and Gaia satellite missions which promise precise, full phase-space measurements of RR Lyrae stars in the halo of our Galaxy. When applied to a sample of 100 stars (with realistic observational errors) drawn from the Law & Majewski (2010) N-body simulation of the destruction of the Sgr dwarf satellite, Rewinder recovers the depth, shape, and orientation of the dark matter potential to within a few percent. While the tests presented in this paper are very simple, the accuracy of potential recovery promised by such a small sample of stars provides strong motivation for further theoretical work to: 1) develop a robust generative model that utilizes the concepts demonstrated by Rewinder ; 2) investigate the power of using multiple debris structures; and 3) examine how Rewinder might work with less accurate measurements or missing dimensions. Our results also motivate an observational campaign with Spitzer to survey RR Lyrae stars in debris structures around the Milky Way to get precise distances to combine with near-future Gaia velocity data. If just 100 stars in a single stellar stream allow us to study the depth, shape, and orientation of the Milky Way potential, larger samples in multiple structures (e.g., the Orphan Stream; Sesar et al. 2013) offer the prospect of assessing these quantities as a function of Galactocentric radius. Tracing the mass in a dark matter halo with this level of detail is impossible for any other galaxy in the Universe. We thank Barry Madore for providing the inspiration for this work. Thanks to David Hogg for statistical advice. Thanks also to Steve Majewski, David Law, and David Nidever. We also thank the anonymous referee for useful suggestions. APW is supported by a National Science Foundation Graduate Research Fellowship under Grant No. 1144155. This work was supported in part by the National Science Foundation under Grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics.", "pages": [ 5, 6 ] }, { "title": "REFERENCES", "content": "Bailin, J., Kawata, D., Gibson, B. K., et al. 2005, ApJ, 627, L17 Belokurov, V., Zucker, D. B., Evans, N. W., et al. 2006, ApJ, 642, L137 Binney, J. 2008, MNRAS, 386, L47 Deason, A. J., Belokurov, V., Evans, N. W., & An, J. 2012a, MNRAS, 424, L44 Deason, A. J., Belokurov, V., Evans, N. W., et al. 2012b, MNRAS, 425, 2840 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194 Ivezi'c, ˇ Z., Sesar, B., Juri'c, M., et al. 2008, ApJ, 684, 287", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "The Census of High- and Medium-mass Protostars (CHaMP) is the first largescale (280 · < l < 300 · , -4 · < b < 2 · ), unbiased, sub-parsec resolution survey of Galactic molecular clumps and their embedded stars. Barnes et al. (2011) presented the source catalog of ∼ 300 clumps based on HCO + (1-0) emission, used to estimate masses M . Here we use archival mid-infrared to mm continuum data to construct spectral energy distributions. Fitting two-temperature greybody models, we derive bolometric luminosities, L . We find the clumps have 10 /lessorsimilar L/L /circledot /lessorsimilar 10 6 . 5 and 0 . 1 /lessorsimilar L/M/ [ L /circledot /M /circledot ] /lessorsimilar 10 3 , consistent with a clump population spanning a range of instantaneous star formation efficiencies from 0 to ∼ 50%. We thus expect L/M to be a useful, strongly-varying indicator of clump evolution during the star cluster formation process. We find correlations of the ratio of warm to cold component fluxes and of cold component temperature with L/M . We also find a near linear relation between L/M and Spitzer -IRAC specific intensity (surface brightness), which may thus also be useful as a star formation efficiency indicator. The lower bound of the clump L/M distribution suggests the star formation efficiency per free-fall time is /epsilon1 ff < 0 . 2. We do not find strong correlations of L/M with mass surface density, velocity dispersion or virial parameter. We find a linear relation between L and L HCO + (1 -0) , although with large scatter for any given individual clump. Fitting together with extragalactic systems, the linear relation still holds, extending over 10 orders of magnitude in luminosity. The complete nature of the CHaMP survey over a several kiloparsecscale region allows us to derive a measurement at an intermediate scale bridging those of individual clumps and whole galaxies. Subject headings: stars: formation - stars: pre-main-sequence - ISM: dust - surveys", "pages": [ 2, 3 ] }, { "title": "The Galactic Census of High- and Medium-mass Protostars. II. Luminosities and Evolutionary States of a Complete Sample of Dense Gas Clumps", "content": "Bo Ma 1 , Jonathan C. Tan 1 , 2 , Peter J. Barnes 1 Department of Astronomy, University of Florida, FL, 32611, USA Department of Physics, University of Florida, FL, 32611, USA Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Stars form from the gravitational collapse of the densest regions of giant molecular clouds (GMCs). In particular star clusters, likely the dominant mode of star formation (Lada & Lada 2003; Gutermuth et al. 2009), are born from ∼ parsec-scale gas clumps within GMCs. However, many open questions remain (see, e.g. McKee & Ostriker 2007; Tan et al. 2012; Hennebelle & Falgarone 2012). How are GMCs formed out of the diffuse interstellar medium? Why does star formation occur in only a small fraction of the available gas in GMCs? What is the star formation rate and efficiency over the GMC lifetime and what processes control this? What is the timescale of star cluster formation: is it fast (Elmegreen 2000, 2007) or slow (Tan et al. 2006) with respect to the free-fall time? What processes control the evolution and overall star formation efficiency of a star-forming clump? To help address some of these open questions, Barnes et al. (2011, hereafter Paper I) have designed a multi-wavelength survey, the Census of High- and Medium-mass Protostars (CHaMP). Starting in the 3mm band, the aim of CHaMP has been to map a complete sample of molecular gas structures in a 20 · × 6 · region in the Galactic plane (280 · < l < 300 · , -4 · < b < +2 · ), and to measure their associated star formation activity from the near- to far-IR. Using the 4m Nanten telescope, this region was first surveyed in the J=1-0 transitions of 12 CO, 13 CO, C 18 O and HCO + (Yonekura et al. 2005). This sequence of species traces progressively higher densities and the mapping was carried out in this order so as to identify all the locations of dense gas, without having to map the entire region in the tracers of the densest gas. So 13 CO was only observed where the 12 CO integrated intensity was above 10 K km s -1 , and C 18 O and HCO + were observed where 13 CO was brighter than 5 K km s -1 . Then a follow-up campaign was begun to map the dense gas regions found in the Nanten survey. The follow-up is conducted in a number of 3mm molecular transitions with the 22m Mopra telescope at much higher sensitivity and angular resolution than Nanten telescope (Paper I). This observing strategy distinguishes the CHaMP survey from all other Galactic plane surveys of dense gas. In Paper I, maps of the CHaMP regions in HCO + (1-0) line emission observed by the Mopra telescope were presented. A total of 303 massive molecular clumps were identified. This sample has the following properties: integrated line intensities 1-30 K km s -1 , linewidths 1-9 km s -1 , FWHM sizes 0.2-2 pc, mean mass surface densities Σ ∼ 0 . 01 to ∼ 1 g cm -2 and masses ∼ 10 to ∼ 10 4 M /circledot . In this paper we use archival infrared and millimeter data to investigate the SEDs and luminosities of these HCO + clumps, with the goal being to characterize their evolutionary state with respect to star cluster formation. The paper is organized as follows. § 2 describes the IR and mm data used in this study. § 3 describes our methods of estimating clump fluxes. § 4 presents our results, including the clump masses, bolometric fluxes, bolometric luminosities, luminosity-to-mass ratios, warm and hot component fluxes, and cold component temperatures and bolometric temperatures. In particular, we examine the correlation of various potential tracers of embedded stellar content with the luminosity-to-mass ratio, and then emphasize the use of this ratio as an evolutionary indicator for star cluster formation. § 5 presents further discussion, including searches for potential correlation of luminosity-to-mass ratio with clump mass surface density and virial parameter. It also discusses the luminosity versus HCO + line luminosity relation from clumps to whole galaxies. § 6 summarizes our conclusions.", "pages": [ 4, 5 ] }, { "title": "2. Infrared and Millimeter Observational Data", "content": "The first goal of this paper is to measure fluxes at various wavelengths coming from the CHaMP clumps. Here we describe the main observational datasets that we use to derive these fluxes.", "pages": [ 5 ] }, { "title": "2.1. MSX", "content": "The Midcourse Space Experiment (MSX) was launched in April 1996. It conducted a Galactic plane survey (0 · < l < 360 · , | b | < 5 · ), which covers all the CHaMP clumps. The four MSX band wavelengths are centered at 8.28, 12.13, 14.65 and 21.3 µ m. The best image resolution is ∼ 18 '' in the 8 . 28 µ m band, with positional accuracy of about 2 '' . The instrumentation and survey are described by Egan & Price (1996). Calibrated images of the Galactic plane were obtained from the online MSX image server at the IPAC website at: http://irsa.ipac.caltech.edu/data/MSX/. For simplicity, we assume conservative common absolute flux uncertainties of 20% for all the IR data ( MSX, IRAS, Spitzer IRAC ), similar to that estimated for IRAS (M. Cohen, private comm.).", "pages": [ 6 ] }, { "title": "2.2. IRAS", "content": "The Infrared Astronomical Satellite (IRAS) performed an all sky survey at 12, 25, 60 and 100 µ m. The nominal resolution is about 4 ' at 60 µ m. High Resolution Image Restoration (HIRES) uses the Maximum Correlation Method (MCM, Aumann et al. 1990) to produce higher resolution images, better than 1 ' at 60 µ m. Sources chosen for processing with HIRES were processed at all four IRAS bands with 20 iterations. The pixel size was set to 15 '' with a 1 · field centered on the target. The absolute fluxes of the IRAS data are expected to be accurate to about 20%.", "pages": [ 6 ] }, { "title": "2.3. Spitzer IRAC", "content": "The Spitzer InfraRed Array Camera (IRAC) is a four-channel camera that provides simultaneous 5 . 2 ' × 5 . 2 ' images at 3.6, 4.5, 5.8, and 8 µ m with a pixel size of 1 . 2 '' × 1 . 2 '' and angular resolution of about 2 '' at 8 µ m. We searched the Spitzer archive at http://irsa.ipac.caltech.edu/applications/Spitzer/SHA/ for IRAC data near the positions of our HCO + clumps. We found IRAC data for 284 out of our 303 clumps. Most of these data are from two large survey programs: PID 189 (Churchwell, E., 'The SIRTF Galactic Plane Survey') and PID 40791 (Majewski, S., 'Galactic Structure and Star Formation in Vela-Carina'). We used the post basic calibration data to estimate the fluxes of these clumps, which we assume has a 20% uncertainty.", "pages": [ 6, 7 ] }, { "title": "2.4. Millimeter data", "content": "Hill et al. (2005) carried out a 1.2-mm continuum emission survey toward 131 star-forming complexes using the Swedish ESO Submillimetre Telescope (SEST) IMaging Bolometer Array (SIMBA). SIMBA is a 37-channel hexagonal bolometer array operating at a central frequency of 250 GHz (1.2mm), with a bandwidth of 50 GHz. It has a half power beam width of 24 '' for a single element, and the separation between elements on the sky is 44 arcsec. Hill et al. list the 1.2-mm flux for 404 sources, 15 of which are in our sample.", "pages": [ 7 ] }, { "title": "3.1. Definition of Clump Angular Area and HCO + Masses", "content": "Paper I presented maps of the CHaMP region in HCO + (1-0) line emission using the 22m Mopra telescope, identifying 303 massive molecular clumps. Elliptical clump sizes were defined based on 2D Gaussian fitting for each HCO + clump. The ellipse size quoted in columns 9 and 10 of Table 4 of Paper I is the FWHM angular size of the major and minor axes of the Gaussian fit. Clump masses, M , were evaluated based on integrating the derived column density distribution over the full area of the Gaussian profile ( M col listed in column 9 of their Table 5). Note the derivation of mass surface densities and masses from the observed HCO + (1-0) intensity depends on: (1) in the view of one of our team (JCT), the conversion of observed HCO + (1-0) line intensity to total HCO + column density is assumed to have an uncertainty ∼ 30%; in the view of another (PJB), there is no identifiable reason for this assumption, since the analysis in Paper I showed that there is no such uncertainty, beyond the points mentioned next. (2) the abundance of HCO + ( X HCO + ≡ n HCO + /n H2 = 1 . 0 × 10 -9 was adopted in Paper I, being a median value from a number of observational and astrochemical studies). The uncertainty in this mean abundance is itself uncertain: in this paper we will assume a factor of 2 uncertainty, i.e. a range of 0.5 to 2 . 0 × 10 -9 for the mean abundance. In addition, clump to clump variations in X HCO + are expected: we will assume a dispersion of a factor of 2. There may be a number of effects that lead to systematic variation of X HCO + with environmental conditions. For example, we have recently found (Barnes et al. 2013) that the HCO + abundance may be enhanced in the vicinity of ionizing radiation from massive stars, with a possibly lower X HCO + in the majority of darker, more quiescent clumps. If confirmed, this particular effect would tend to have the effect of increasing the masses quoted here for the more quiescent clumps, but decreasing the masses for the minority of vigorously star-forming clumps. Future work to improve the calibration of HCO + -derived masses is needed. (3) the distance to the sources (the clumps' median distance uncertainty is estimated in Paper I to be 20% based mostly on classical distance estimates to GMC complexes and assuming association of clumps with a particular GMC complex. Here we use a slightly larger, more conservative value of 30% for the absolute distance uncertainty [see also Paper I for a more extensive discussion of distance estimates], i.e. leading to ∼ 60% uncertainties in M ). Combining these uncertainties, we conclude that the absolute mass estimate of any particular clump may be uncertain by as much as a factor of ∼ 3. To measure the continuum fluxes at various wavelengths coming from the CHaMP HCO + clumps, we define the clump size as two times larger than the FWHM ellipse derived in Paper I, i.e. its radial extent is equal to 1 FWHM at a given position angle. For a 2D Gaussian flux distribution as assumed in Paper I, the area inside this ellipse encloses 93.75% of the total flux. Thus with this definition of clump size we expect to enclose close to 100% of the total HCO + flux measured in Paper I, and presumably close to 100% of the continuum flux associated with each clump. The clumps are highly clustered in space so that on a scale of two times the FWHM ellipse the majority of them, ∼ 70% of the sample, suffer from overlap with a neighboring clump ( ∼ 30% overlap on the scale of one times the FWHM ellipse). While the original clump definition from Paper I also used their velocity space information, sometimes nearby clumps also overlap in velocity to some extent. We have developed an approximate method to estimate the fluxes of these clumps where there are image pixels belonging to more than one ellipse. We first calculate the angular distance, normalized by the size of the ellipse, from the overlapped pixel to the center of each clump. This normalized angular distance is defined as r norm = (( d x /a ) 2 +( d y /b ) 2 ) 1 / 2 , where d x and d y are the angular distance from the overlapped pixel to the minor and major axis of each ellipse, and a and b are the angular sizes of the major and minor axis of the ellipse. Then the flux of each overlapped pixel is assigned to its nearest ellipse according to this normalized angular distance.", "pages": [ 7, 8, 9 ] }, { "title": "3.2. Clump and Background Flux Measurements", "content": "The MSX and IRAS data exist for all 303 CHaMP clumps and these form the basis for our spectral energy distribution measurements. We describe here the method we use to derive the fluxes from the clumps based on these imaging data. We then describe how we utilize the mid-infrared IRAC data and the mm data where it is available. Using the coordinates, sizes and geometries of the HCO + sources, fluxes were deduced first by directly integrating over the images, this total flux being expressed as F ν, tot . However, we expect that some fraction of this flux can come from foreground and background sources along the line of sight that are not associated with the clump. For simplicity we refer to this foreground and background emission as the 'background flux', F ν,b . We evaluate F ν,b as the median pixel value in the region between the clump ellipse (as defined here) and an ellipse twice as large (i.e. four times the FWHM size of Paper I), excluding areas that are part of other clumps. In the end, we derived two fluxes: without and with background subtracted, which are F ν, tot and F ν = F ν, tot -F ν,b , respectively. The error of the fluxes are estimated from the combination of two terms. The first is the uncertainty in the absolute flux from the particular telescope. The data used here are generally assumed to be accurate to about 20%. The second term is from the background subtraction. Since in the Galactic plane it is often difficult to estimate the background emission, we treat the background level as an error term in our flux error estimation. So the fractional error is (0 . 2 2 +(F ν, b / F ν, tot ) 2 ) 1 / 2 . In the following, we have carried out the analysis for both flux estimates, F ν, tot and F ν . Next we use a two-temperature grey body model to fit the spectral energy distribution (SED) in order to estimate the bolometric fluxes, F tot (no background subtracted) and F (background subtracted), (calculated by integrating over the fitted SED and assuming negligible flux escapes in the near-IR and shorter wavelengths) and temperatures of the clumps, following the method of Hunter et al. (2000) and Fa'undez et al. (2004). Each temperature component of the grey body model is described by: where B ν ( T ) = (2 hν 3 /c 2 ) / [exp( hν/kT ) -1] is the Planck function for the black body flux density (where c is the speed of light, h is the Planck constant and k is the Boltzmann constant), Ω is the angular size of the source, and T is the temperature. The dependence of the optical depth, τ ν , with frequency, ν , is given by: where β is the emissivity index and ν 0 is the turnover frequency. In this fitting procedure, for the colder component (subscript 'c'), we explored parameter values in the ranges T c = 10 -50 K and β c = 1 . 0 -2 . 5. These values of β c are those expected from laboratory experiments and observational results (see Schnee et al. 2010 and references therein). Also, Fa'undez et al. (2004) found β c to be in this range for their sample of sources. We find ν 0 to generally be in the range 3 -30 THz. For the warmer component (subscript 'w'), T w was allowed to have values in the range 100 -300 K, while β w was fixed to 1 following Hunter et al. (2000) and Fa'undez et al. (2004). The choice of β w = 1 is motivated both by theoretical calculations and by observational evidence (Whittet 1992, pp. 201-203). The angular size of the colder component, Ω c was set equal to the angular size of the clump, including accounting for reduction due to overlap with other clumps. For the warmer component the angular size, Ω w , is derived from the best fitting result, always being smaller than the angular size of the clump. The values of T c are not particularly well constrained by the IRAS data, which extend to a longest wavelength of only 100 µ m. For those 15 sources where we do have mm fluxes reported from SEST-SIMBA, we examine how the two-temperature grey body model fit changes when we do make use of the mm flux. Note, for the mm fluxes, not having access to estimates of F ν,b , we assume that background subtraction makes a negligible difference, i.e. F ν,b /lessmuch F ν . In Fig. 1 we present the SED and model fits of BYF 73 (G286.2+0.2), which is one of the more massive and actively star-forming clumps in the sample (Barnes et al. 2010), as one example to show the effect of the mm flux measurement. The results from only the MSX and IRAS data are: T c = 35 . 2 , 33 . 2 K, β c = 1 . 45 , 1 . 52 and ν 0 = 101 , 31 . 1 THz, without and with background subtraction, respectively. Adding in the mm flux we now derive T c = 32 . 8 , 30 . 4 K, β c = 1 . 82 , 1 . 78 and ν 0 = 15 . 4 , 11 . 1 THz for these same cases. The bolometric flux, obtained by integrating over the model spectrum, changes from (1 . 32 , 1 . 22) × 10 -7 erg -1 s -1 cm -2 , without and with background subtraction respectively, to (1 . 30 , 1 . 20) × 10 -7 erg -1 s -1 cm -2 when the mm flux is utilized. The fitting results for all 15 sources with mm flux measurements using the no background subtraction method are summarized in Table 1. These results show that T c typically changes by /lessorsimilar 5 K after including the mm flux. 1 The mean value changes by about 10%. We find β c changes from 2 . 0 ± 0 . 33 to 1 . 85 ± 0 . 41 after utilizing the mm flux. Most importantly, we find the bolometric fluxes, F tot and F , typically change by /lessorsimilar 10% after including the mm flux. Thus we conclude that the lack of longer wavelength data for the main sample only introduces a modest uncertainty of ∼ 10% in F tot and F . Note, however, that the limited FIR/sub-mm coverage of the SEDs, even with the SEST-SIMBA data, prevent accurate measurement of T c , ν 0 and β . This situation will be improved with forthcoming data from the Herschel Hi-GAL survey (Molinari et al. 2010). In this paper, we choose not to use the IRAC data for our fiducial SED fitting (although we do examine certain correlations of clump properties with the flux in the IRAC bands). The IRAC data are not available for about 10% of the CHaMP clumps (generally those furthest from the midplane) and we wish to maintain the same procedure for all the clumps in the sample. Furthermore, the bolometric luminosity is dominated by the colder component, even for clumps with the most active star formation (see, e.g. Fig. 1). For BYF 73, when we compare SED fitting (no background subtraction) with just MSX+IRAS a using MSX and IRAS data b using MSX, IRAS & mm data to that with IRAC+MSX+IRAS we see: T c changes from 35 . 1 K to 35 . 0 K, T w changes from 215 K to 230 K and F changes from 1 . 323 × 10 -7 erg -1 s -1 cm -2 to 1 . 328 × 10 -7 erg -1 s -1 cm -2 .", "pages": [ 9, 10, 11, 12, 14, 15 ] }, { "title": "4.1. HCO + Masses", "content": "In this paper we set the clump mass, M , equal to that derived from analysis of HCO + (1-0) emission, M col (listed in column 9 of Table 5, Paper I). The distribution of these masses is presented in Fig. 2a. The masses range from ∼ 10 -10 4 M /circledot , with mean of 723 M /circledot and median of 427 M /circledot . The clump masses and other clump properties are also listed in Table 2. Additional, secondary clump properties are listed in Table 3. As discussed above, uncertainties in absolute clump mass are likely to be at the level of about a factor of 4, mainly due to uncertainties in HCO + abundance. We expect relative clump masses are somewhat better determined, especially since a large fraction of the CHaMP clumps are in the Carina sprial arm, with about half in the same η Carinae giant molecular association at a common distance of ∼ 2 . 5 kpc.", "pages": [ 15 ] }, { "title": "4.2. Bolometric Fluxes", "content": "The bolometric flux distributions without, F tot and with, F , background subtraction are presented in Fig. 2b. The mean 10 σ sensitivity of the 4 IRAS bands are 0.7, 0.65, 0.85 and 3.0 Jy, which correspond to a bolometric flux of about 3 × 10 -10 erg -1 s -1 cm -2 for a source with a typical angular size of 60 '' . This limit is also shown in Fig. 2b. We see that F tot can be detected at better than 10 σ for nearly all of the CHaMP clumps. We assume the uncertainty in F tot is about 20% from the absolute flux calibration of the IR observations and about 10% from the two temperature greybody model fitting, i.e. adding in quadrature to about 22%. For the faintest clumps, the total flux from the direction of the clump, F tot , can be similar to that of the background (i.e. the region surrounding the clump). The background subtracted flux, F ν , can thus be very small (or even formally negative) at a particular wavelength. The uncertainty assigned to F ν is of order the same level as the background. For deriving bolometric fluxes, the flux at 100 µ m is typically most important. Thus we flag those clumps that have a 100 µ m background flux that is > 0 . 75 times the clump flux, and consider these values of F , L and L/M to be highly uncertain, i.e. /greaterorsimilar 100% uncertainties.", "pages": [ 15, 17 ] }, { "title": "4.3. Bolometric Luminosities", "content": "Given the clump distances from Paper I and our derived bolometric fluxes, we calculate the bolometric luminosities L tot and L (without and with background subtraction, respectively). The distributions of L tot and L are shown in Fig. 2c. Adopting a typical distance uncertainty of 30% as explained in § 3.1, we then estimate an uncertainty in L tot of about 64%. L has somewhat greater uncertainty due to background flux estimation, and again we flag those sources where we expect this source of error dominates. The mean luminosities are 〈 L tot 〉 = 5 . 2 × 10 4 L /circledot and 〈 L 〉 = 4 . 2 × 10 4 L /circledot . For reference, this is about the luminosity of a 20 M /circledot zero age main sequence (ZAMS) star (Schaller et al. 1992). The median values of L tot and L are 1 . 06 × 10 4 L /circledot and 6 . 2 × 10 3 L /circledot , respectively: half of the sample are lower in luminosity than a single 12 M /circledot ZAMS star. Note that the previous surveys of dust emission toward massive star forming regions by Mueller et al. (2002) and Fa'undez et al. (2004) found 〈 L tot 〉 = 2 . 5 × 10 5 L /circledot and 2 . 3 × 10 5 L /circledot , respectively. These values are much larger that those of the CHaMP clumps. We attribute this difference as being due to the different selection criteria of the samples: CHaMP is a complete sample of dense gas independent of star formation activity, while these other surveys were selected based on (massive) star formation indicators.", "pages": [ 17, 18 ] }, { "title": "4.4. Luminosity-to-mass ratios", "content": "During the evolution of star-forming clumps, i.e. the formation of star clusters, the gas mass will decrease due to incorporation into stars and dispersal by feedback, causing the luminosity-to-mass ratio to increase. So L/M should be an evolutionary indicator of the star cluster formation process. The distribution of L/M is shown in Fig. 2d. Three dotted vertical lines at L/M = 0 . 078 , 0 . 77 , 3 . 9 L /circledot /M /circledot are used to show the values expected of clouds with dust temperatures of T = 10 , 15 , 20 K, which can be achieved in starless clumps via external heating, as evidenced by temperature measurements of Infrared Dark Clouds (e.g. Pillai et al. 2006). These values are calculated via where the latter evaluation is based on integrating the opacities of the Ossenkopf & Henning (1994) moderately-coagulated thin ice mantle dust model (and adopting a gas-to-dust mass ratio of 155) for clouds with 0 . 01 < Σ / g cm -2 < 1 and 10 < T/ K < 20 (there is a modest dependence of L/M on Σ 0 . 02 , which we ignore, normalizing the numerical factor of eq. (3) to Σ = 0 . 03 g cm -2 , typical of the CHaMP clump sample). Values of L/M ∼ 1 L /circledot /M /circledot are thus expected to define the lower end of the L/M distribution, as is observed. To understand the upper end of the observed distribution, consider a clump with an equal mass of gas and stars that are on the zero age main sequence (ZAMS). For a Salpeter IMF down to 0.1 M /circledot , this will have L/M ∼ 600 L /circledot /M /circledot (Leitherer et al. 1999; Tan & McKee 2002). Other IMFs typically considered for Galactic star-forming regions give similar numbers to within about a factor of two. This value is close to the upper end of the distribution of L/M shown in Fig. 2d. Note that as the gas mass goes to very small values, L/M should rise far above 600 L /circledot /M /circledot . However, in this case a smaller fraction of the bolometric luminosity will be re-radiated in the MIR and FIR, and so would be missed by our analysis. Also such 'revealed' clusters with small amounts of dense gas would not tend to be objects in the CHaMP sample, which is complete only on the basis of emission of dense gas tracers. To investigate the relation between bolometric luminosity and gas mass (i.e. how luminosity depends on mass), we also show the correlation between L tot and M in Fig. 3a and the correlation between L and M in Fig. 3b. The best-fit power law results (e.g., following methodology of Kelly 2007) are as follows: with Spearman rank correlation coefficient r s = 0 . 54 and probability for a chance correlation p s /lessmuch 10 -4 (formally p s = 1 . 2 × 10 -24 , but this value depends sensitively on the assumed shape of the tails of the distribution functions, which are not well-defined for real datasets) for the no background subtraction method and with r s = 0 . 55 and p s /lessmuch 10 -4 (formally p s = 7 . 2 × 10 -25 ; note the open symbols in Fig. 3b have larger uncertainties, explaining the asymmetric distribution of points about the best fit relation) for the background subtraction method. Both show significant positive correlations. The more massive the clump is, the more luminous it tends to be. The mean, median and standard deviation of log( L tot /M/ [ L /circledot /M /circledot ]) are 1.34, 1.43 and 0.77 respectively for non-background subtraction method. For the background subtraction method, the mean, median and standard deviation of log( L/M/ [ L /circledot /M /circledot ]) are 1.06, 1.25 and 0.97 respectively. Molinari et al. (2008) have studied the SEDs of 42 potentially massive individual young stellar objects (YSOs). By fitting the SEDs with YSOs models they obtained the bolometric luminosity and envelope mass, M env . They presented the L bol -M env diagram as a tool to diagnose the pre-MS evolution of massive YSOs. For their sample, the mean, median and standard deviation of log( L/M ) are 1.91, 1.77 and 0.66 respectively. This illustrates the different nature of their sample: objects that are already forming massive stars and with much higher values of L/M . However, we caution that systematic differences could also arise because of the different methods being used to derive masses (i.e. HCO + versus mm flux-based masses). Similarly, Mueller et al. (2002), Beuther et al. (2002) and Fa'undez et al. (2004) reported mean values of log( L/M ) as 2 . 04 ± 0 . 34, 1 . 18 ± 0 . 34 and 1 . 75 ± 0 . 38. Note here that in Beuther et al. (2002) they have used opacity from Hildebrand (1983), which is 4.9 times smaller than the opacity from Ossenkopf & Henning (1994) used in Mueller et al. (2002) and Fa'undez et al. (2004). So the mass derived in Beuther et al. (2002) would be 4.9 times smaller and their mean log( L/M ) will be 1 . 87 ± 0 . 34 if they adopt the opacity from Ossenkopf & Henning (1994).", "pages": [ 18, 19, 21 ] }, { "title": "4.5. The Warm Component", "content": "From the two temperature fitting process, we derived the total, F w , tot , and backgroundsubtracted, F w , flux for the warm component. The distributions of F w , tot and F w are shown in Fig. 4a. The correlation of F w , tot with F tot is shown in Fig. 4b, and that of F w with F in Fig. 4c. These both show significant correlations. We derive a best-fit power law fit for the and The Spearman rank correlation coefficients (see Fig. 4) indicates a positive correlation exists in both cases. Our findings support the idea that as stars gradually form in molecular clumps and the luminosity-to-mass ratio increases, a larger fraction of the bolometric flux will emerge at shorter wavelengths. The specific functional form of this correlation is a constraint on radiative transfer models of star cluster formation. dependence of F w , tot on F tot , finding For the background subtracted case, which we consider to be the most accurate measure of the intrinsic properties of the clumps, we try two different constrained fits, finding: The distributions of F w , tot /F tot and F w /F are shown in Fig. 4d. The warm component flux generally accounts for 10% -30% of the total flux, so F w and F are not independent, which can contribute to these correlations. To investigate if there are any systematic trends associated with the warm component during star cluster formation as measured by the clump luminosity to mass ratio, we show the correlation of F w , tot /F tot versus L tot /M in Fig. 4e and F w /F versus L/M in Fig. 4f. The power law fit results of this positive correlation are as follows:", "pages": [ 21, 23 ] }, { "title": "4.6. The Hot (IRAC Band) Component", "content": "We now search for any correlation of the IRAC band flux, which extends from ∼ 3 -9 µ m, with the bolometric flux and the luminosity to mass. These relatively short wavelengths are more sensitive to hot dust directly heated by embedded young stars. We first measure the total IRAC band flux using a simple trapezoidal rule integration in the four IRAC bands, without background subtraction, F IRAC , tot , and then subtract the background to derive F IRAC . The distributions of F IRAC , tot and F IRAC are shown in Fig. 5a. The correlation of F IRAC , tot with F tot is shown in Fig. 5b, and that of F IRAC with F in Fig. 5c. These both show highly significant correlations. The power law fit results of these two correlations are as follows: and, trying two constrained fits, The distributions of F IRAC , tot /F tot and F IRAC /F are shown in Fig. 5d. The IRAC component flux generally accounts for ∼ 1% -10% of the total flux, so F IRAC and F are essentially independent, unlike for F w (above). To investigate if there are any systematic trends associated with the IRAC (hot) component during star cluster formation as measured by the clump luminosity to mass ratio, we show the correlation of F IRAC , tot /F tot versus L tot /M in Fig. 5e and F IRAC /F versus L/M in Fig. 5f. The best-fit power law relations are as follows: The Spearman rank correlation coefficient of F IRAC , tot /F tot versus L tot /M is negative. We expect this is due to the fact that F tot and L tot are correlated, while F IRAC , tot is often dominated by 'background' (i.e. both background and foreground, i.e. unrelated) emission. Attempting a power law fit for F IRAC /F versus L/M , we find but with r s = -0 . 14 and p s = 0 . 05, indicating there is not significant correlation. So there is no evidence for an increase in the relative importance of the hot component as cluster evolution (as measured by L/M ) proceeds. As the luminosity input into the clump rises, a fairly constant fraction emerges in the IRAC bands. Again, this result can provide a constraint on theoretical models of star cluster formation. In order to more directly probe the evolution of IRAC-traced hot dust emission and its possible correlation with luminosity to mass ratio, we also calculated the IRAC band specific intensity (surface brightness) without, I IRAC , tot and with, I IRAC background subtraction (Fig. 6). Note that both the specific intensities and the luminosity to mass ratios are essentially independent of distance uncertainties. The best-fit relations are as follows: with r s = 0 . 57 and p s /lessmuch 10 -4 (formally p s = 10 -13 ), and, trying two constrained fits, The former has r s = 0 . 66 and p s /lessmuch 10 -4 (formally p s = 10 -19 ). Thus the IRAC band specific intensity, which is essentially independent of L/M (since only a very small fraction of L emerges at these wavelengths) and more directly traces embedded stellar populations, has a significant correlation with L/M , thus validating the use of L/M as an evolutionary indicator of star cluster formation. The specific functional form of the correlation is a constraint on radiative transfer models of star cluster formation. The near linear relation of I IRAC with L/M (although with large scatter, which may be expected from IMF sampling) suggests that I IRAC also has a near linear dependence on embedded stellar content relative to gas mass, i.e. the instantaneous star formation efficiency, /epsilon1 ' ≡ M ∗ /M , which, note, is normalized by the gas mass. (We define /epsilon1 ≡ M ∗ / ( M ∗ + M ), which becomes similar to /epsilon1 ' when /epsilon1 ' /lessmuch 1.) Thus, for a Salpeter IMF down to 0.1 M /circledot (see § 4.4), where we have used the numerical result of the constrained linear fit (eq. 18). This may be a useful relation for estimating star formation efficiencies of statistical samples of star-forming clumps (at least those with similar densities to local Galactic clumps), when only IRAC data are available and a background subtraction can be performed.", "pages": [ 24, 26, 28 ] }, { "title": "4.7. Cold Component Dust Temperature and Bolometric Temperature", "content": "We now search for any dependence of the cold component dust temperature, T c , tot (based on total fluxes with no background subtracted) and T c (based on fluxes after background subtraction), with the luminosity to mass ratio. We note that the available data for the clumps generally are limited at long wavelengths to the IRAS 100 µ m data and so our accuracy for estimating T c is limited to about ± 5 K (see § 3.2). The distributions of T c , tot and T c are shown in Fig. 7a. The mean values are 〈 T c , tot 〉 = 33 ± 5 K and 〈 T c 〉 = 33 ± 7 K. These results are similar to those derived in other surveys, such as: 〈 T 〉 = 29 ± 9 K in the large sample of Mueller et al. (2002); 〈 T 〉 = 45 ± 11 K in the large sample of Sridharan et al. (2002); 〈 T 〉 = 32 ± 5 K in the sample of Molinari et al. (2000); 〈 T 〉 = 35 ± 6 K in the sample of Hunter et al. (2000); 〈 T 〉 = 30K in the sample of Molinari et al. (2008); and 〈 T 〉 = 32K in the sample of F'aundez et al. (2004). The correlation of T c, tot with L tot /M is shown in Fig. 7b and that of T c with L/M in Fig. 7c. We see clear positive correlations are present - the temperature rises as L/M increases. We find best-fit relations: with r s = 0 . 81 and negligible value of p s , and with r s = 0 . 65 and negligible value of p s . 'Bolometric temperature', T bol , has been proposed as a measure of the evolutionary development of a young stellar object (YSO) (Ladd et al. 1991; Myers & Ladd 1993; Myers et al. 1998). It is the temperature of a blackbody having the same weighted mean frequency as the observed SED. As the envelopes in YSO systems are dispersed, their bolometric temperatures will rise. This is because the FIR emission decreases while the NIR and MIR emission increases. We calculated the bolometric temperature for our molecular clumps following Myers & Ladd (1993): where 〈 ν 〉 ≡ ∫ ∞ 0 νF ν dν/ ∫ ∞ 0 F ν dν is the flux weighted mean frequency. The coefficient of 〈 ν 〉 in eq. (22) is chosen so that a blackbody emitter at temperature T has T bol = T . The distributions of T bol , tot (based on total fluxes with no background subtraction) and T bol (based on fluxes after background subtraction) are shown in Fig. 7d. These have mean values 92 ± 18 K and 113 ± 44 K, respectively. For comparison, Mueller et al. (2002) find a mean value of 78 ± 21 K for their sample. The correlation of T bol , tot with L tot /M is shown in Fig. 7e and that of T bol with L/M in Fig. 7f. We do not find significant correlations, since the best-fit relations are with r s = -0 . 15 and p s = 0 . 06, and with r s = -0 . 15 and p s = 0 . 06. We suspect the lack of significant correlation is because the uncertainties in deriving T bol are relatively large compared to the expected size of any trend for T bol to increase during star cluster formation. This is in contrast to the measures F w /F and I IRAC , which show clear changes by about a factor of 10 or more as L/M increases.", "pages": [ 28, 30, 31 ] }, { "title": "5.1. Dependence of L and L/M on Mass Surface Density, Σ", "content": "Consider a clump that forms stars at a fixed efficiency per free-fall time, /epsilon1 ff . The overall accretion rate to stars is where we have normalized to a value of /epsilon1 ff estimated by Krumholz & Tan (2007). Then the accretion luminosity is Here f acc is the fraction of the accretion power that is radiated. While for individual protostars we expect f acc ∼ 0 . 5 because of the mechanical luminosity of protostellar outflows, in early-stage star-forming clumps much of the outflow kinetic energy is likely to be liberated via radiative shocks and thus contribute to the total clump luminosity. Thus we adopt f acc = 1 as a fiducial value. In the above equation, ¯ m ∗ is the mean protostellar mass, weighted by accretion energy release. For a Salpeter IMF from 0.1 to 120 M /circledot , the mean stellar mass is 0.353 M /circledot , while the mean gravitational energy is 2 . 06 GM 2 /circledot / ¯ r ∗ , assuming ¯ r ∗ is independent of m ∗ (discussed below). For accretion near the end of individual star formation, this implies ¯ m ∗ /similarequal 1 . 4 M /circledot , however the typical unit of accretion energy release will be when the protostar has 2 -1 / 2 of its final mass. Thus we estimate ¯ m ∗ /similarequal 1 M /circledot as a typical fiducial value in eq. (26). The protostellar evolution models of Tan & McKee (2002), developed for protostars forming with accretion rates appropriate for cores fragmenting from a clump with Σ /similarequal 1 g cm -2 (see also Stahler 1988; Palla & Stahler 1992; Nakano et al. 2000; McKee & Tan 2003), indicate that the sizes of all protostars are close to ∼ 3 to 4 R /circledot when their masses are /lessorsimilar 1 M /circledot . After this the size increases along the deuterium core burning sequence, reaching about 6 R /circledot by the time the protostars have 1 . 5 M /circledot . After this, sizes stay relatively constant until m ∗ ∼ 5 M /circledot . Given these relatively modest changes in r ∗ with m ∗ , we adopt a fiducial value of ¯ r ∗ = 4 R /circledot in eq. (26). We can now use eqs. (26) and (3) to estimate minimum values of L/M for star-forming clumps. We have where T is the dust temperature expected from ambient heating of starless clumps. Note that because of internal stellar luminosities that will contribute in addition to L acc , L min /M provides only a lower bound on the distribution of L/M of star-forming clumps. In Fig. 8a and b, we plot the dependence of L tot and L with Σ. Note Σ, like M , is based on the HCO + observations and analysis. We estimate Σ as M/ 2 divided by the projected area of the FWHM ellipse of Paper I. This will give a value of Σ for the typical mass element in the clump. We find best-fit relations: with r s = 0 . 33 and p s /lessmuch 10 -4 (formally p s = 6 × 10 -9 ), and with r s = 0 . 34 and p s /lessmuch 10 -4 (formally p s = 2 × 10 -9 ). In Fig. 8c and d, we plot the dependence of L tot /M and L/M with Σ. We do not find any evidence for a correlation, since the best-fit relations are with r s = 0 . 07 and p s = 0 . 26, and with r s = -0 . 12 and p s = 0 . 14. One caveat of the above results is that L/M and Σ are inversely correlated via M , and this may be making it more difficult to discern any rise of L/M with Σ. We note that high Σ clumps, e.g. with Σ > 0 . 1 g cm -2 all have L/M /greaterorsimilar 4 L /circledot /M /circledot . We also considered our other 'good' cluster evolution indicators, F w /F , I IRAC and T c and their dependence on Σ. However, we did not find any significant correlations of these properties with Σ. In Fig. 8d we also show the predictions of eq. (28) for clumps with M = 10 3 M /circledot , T = 10 , 15 , 20 K forming stars at fixed /epsilon1 ff = 0 . 002 , 0 . 02 , 0 . 2. Models with T ∼ 10 -15 K appear to define the lower boundary of the populated region of the observed L/M versus Σ parameter space, but obtaining precise constraints on /epsilon1 ff is difficult because of the sensitivity of L/M to the adopted temperature. The models with high values of /epsilon1 ff = 0 . 2, even with T = 10 K appear to exceed the observed L/M of a significant number of the clumps, thus we tentatively conclude that /epsilon1 ff < 0 . 2. This analysis will be improved once FIR data become available allowing individual clump temperatures to be accurately measured from their spectral energy distributions. The implications of the detailed distribution of L/M of the clump population and its implication for star cluster formation theories will be examined in a future paper.", "pages": [ 31, 33, 34, 35 ] }, { "title": "5.2. Dependence of L and L/M with Velocity Dispersion and Virial Parameter", "content": "In Fig. 9a and b, we explore the dependence of L tot and L on the 1D velocity dispersion, σ , (as measured from HCO + (1-0) in Paper I). We find best-fit relations: with r s = 0 . 28 and p s /lessmuch 10 -4 (formally p s = 6 . 8 × 10 -6 ), and with r s = 0 . 26 and p s /lessorsimilar 10 -4 (formally p s = 3 . 2 × 10 -5 ). We expect that σ correlates with M for clumps that are self-gravitating. Since L correlates with M , this can explain the observed, weaker correlation of L with σ . Similarly, in Fig. 9c and d we show the dependence of L tot /M and L/M with σ . We do not find significant correlations, since the best-fit relations are: with r s = -0 . 05 and p s = 0 . 41, and with r s = -0 . 04 and p s = 0 . 46. Thus there is no apparent correlation of these variables. If star clusters were built-up hierarchically from a merger of smaller clumps, one might have expected to see increasing L/M with σ . The virial parameter, α vir ≡ 5 σ 2 R/ ( GM ) (Bertoldi & McKee 1992), is proportional to the ratio of a clump's kinetic and gravitational energies. In Fig. 10a and b we show the dependence of L tot and L with α vir . We find best-fit relations: with r s = -0 . 24 and p s = 1 × 10 -4 , and with r s = -0 . 27 and p s /lessorsimilar 10 -4 (formally p s = 2 × 10 -5 ). These (only moderately) significant correlations may be explained by the fact that smaller virial parameters indicate more gravitationally bound systems, which should be more prone to star formation. However, these relations may alternatively be driven by the fact that more massive clumps tend to have smaller virial parameters (Bertoldi & McKee 1992; Paper I) and that luminosity correlates with mass ( § 4.4). This second explanation appears to be supported by the following results. In Fig. 10c and d we show the dependence of L tot /M and L/M with α vir (note, these are equivalent of correlating L tot and L with σ 2 R ). We do not find significant correlations since the best-fit relations are: with r s = 0 . 06 and p s = 0 . 36, and with r s = 0 . 009 and p s = 0 . 89. So these data do not reveal any correlation of cluster evolutionary stage (as measured by L/M ) with degree of gravitational boundedness. Note that the absolute values of α vir appear relatively high, e.g. compared to the somewhat larger 13 CO clouds and clumps analyzed by Roman-Duval et al. (2010), which have ¯ α vir ∼ 1 (see also Tan et al. 2013). As discussed above ( § 3.1), potential systematic uncertainties, especially in the measurement of mass via an assumed HCO + abundance, may be causing an overestimation of α vir , but these uncertainties are not expected to lead to a median value of the HCO + clump sample that is close to unity. Thus the dynamics of the HCO + clumps may be dominated by surface pressure, rather than by their self-gravity (see also Paper I). This is consistent with the fact that most of the HCO + clumps have relatively low L/M and low star formation activity, so we may expect them to have values of α vir in the range ∼ 1 - 30, similar to results found by Bertoldi & McKee (1992). However, it is interesting that we do not see a trend of decreasing α vir with increasing L/M . Possible explanations are: (1) the uncertainties in α vir (which depends on M , R and σ 2 ) and L/M are large enough to wash-out any correlation that is present; (2) the importance of self-gravity, as measured at the HCO + clump scale, does not grow during star cluster formation. Improved mass, luminosity and velocity dispersion measurements are needed to investigate this issue further.", "pages": [ 35, 38, 39 ] }, { "title": "5.3. Dependence of L with HCO + (1-0) line luminosity", "content": "Gao & Solomon (2004) found a tight linear correlation between the infrared luminosity (hereafter we refer to this as the bolometric luminosity, L ) and the amount of dense gas as traced by the luminosity of HCN in both normal galaxies and starburst galaxies. This may suggest that the star formation rate (thought to be proportional to L , at least in starbursts) simply scales with the mass of dense gas. Similarly, Juneau et al. (2009) found an index of 0 . 99 ± 0 . 26 in their study of the relation between the bolometric luminosity and HCO + line luminosity in a sample of 34 nearby galaxies. On the much smaller scales of clumps, the luminosity should not be such a good measure of SFR (Krumholz & Tan 2007), rather tracing embedded stellar content. Still, by surveying a sample of massive dense star formation clumps in CS(7-6), CS(2-1), HCN(1-0) and HCN(3-2), Wu et al. (2005, 2010) have extended the relation of L -L HCN(1 -0) proposed by Gao & Solomon (2004) down to L ∼ 10 4 . 5 L /circledot (see Fig. 11). The CHaMP survey provides a way to connect these scales, by being a complete census of dense gas and thus star formation activity over a several kpc 2 region of the Galaxy. The CHaMP clumps span the full range of evolution of these sources that will be averaged over in extragalactic observations. In addition, by its improved sensitivity, the CHaMP survey allows us to extend the bolometric luminosity versus dense gas line luminosity relation down to much smaller values of source bolometric luminosity. In Fig. 11 we also plot the CHaMP sources. We fit a power-law relation between L and L HCO + (1 -0) (because of the uncertainties in background subtracted luminosities, we only fit to those sources with L > 10 1 . 5 L /circledot ). Only fitting to the CHaMP clumps (via a least-squares fit in log L ) yields: Similarly, a fit to both the CHaMP sample and the extragalactic HCO + (1-0) of Graci'a-Carpio et al. (2006) yields: Finally a fit to the total CHaMP data point and the extragalactic sample yields: This last fit is expected to be the most accurate for extending current extragalactic results down to lower luminosities. Our results suggest that the L -L HCO + (1 -0) relation in clumps (when averaged over a complete sample) is almost the same as that found when averaging over whole galaxies.", "pages": [ 39, 40, 42 ] }, { "title": "6. Summary", "content": "A total of 303 dense gas clumps have been detected using the HCO + (1 -0) line in the CHaMP survey (Paper I). In this paper we have derived the SED for these clumps using Spitzer, MSX and IRAS data. By fitting a two-temperature grey-body model to the SED, we have derived the colder component temperature, colder component flux, warmer component temperature, warmer component flux, bolometric temperature and bolometric flux of these dense clumps. Adopting clump distances and HCO + -derived masses from Paper I, we have calculated the bolometric luminosities and luminosity-to-mass ratios. These dense clumps typically have masses ∼ 700 M /circledot , luminosities ∼ 5 × 10 4 L /circledot and luminosity-to-mass ratios ∼ 70 L /circledot /M /circledot . During the evolution of star-forming clumps, i.e. the formation of star clusters, the luminosity will increase and the gas mass will decrease due to incorporation into stars and dispersal by feedback, causing the luminosity-to-mass ratio to increase. So L/M should be a good evolutionary indicator of the star cluster formation process. The observed range of L/M from ∼ 0 . 1 L /circledot /M /circledot to ∼ 1000 L /circledot /M /circledot corresponds to that expected for evolution from starless clumps to those with near equal mass of stars and gas. The fraction of the warmer component flux in the bolometric flux, F w /F has a positive correlation with the luminosity-to-mass ratio, supporting the idea that as stars form in molecular clumps and L/M increases, a larger fraction of the bolometric flux will come out at shorter wavelengths. We also find that the colder component dust temperature, T c , has a positive correlation with L/M : the bulk of the clump material appears to be getting warmer as star cluster formation proceeds. However, we caution that our measurements of T c are relatively poor (they will be improved with the acquisition of Herschel observations in this region of the Galaxy). We also find a highly significant correlation of specific intensity in the Spitzer -IRAC bands (3-8 µ m), I IRAC with L/M . This has the potential to be a useful evolutionary indicator for the star cluster formation process. We investigated the dependence of L/M with mass surface density, Σ, velocity dispersion, σ and virial parameter, α vir . The lower limit of the distribution of L/M with Σ is consistent with a model for accretion luminosity powered by accretion rates that are a few percent of the global clump free-fall collapse rate. We do not see strong trends of L/M with Σ and, if present, real effects may be masked by the intrinsic correlation of these variables via M . Similarly, we do not find strong correlations between L/M and σ or α vir . The bolometric luminosity has a nearly linear correlation with the dense gas mass as traced by HCO + (1 -0) line luminosity, and this relation holds for over 10 orders of magnitude from molecular clumps in the Milky Way to infrared ultraluminous infrared galaxies. Our results have extended the previously observed relation of Wu et al. (2010) (via HCN(1 -0) line observation) down to much lower luminosity clumps. The complete nature of our sample also gives a measurement at intermediate scales ( ∼ several kpc 2 ) that connects the individual clump results with the extragalactic results, which are averages over clump populations. JCT acknowledges support from NSF CAREER grant AST-0645412; NASA Astrophysics Theory and Fundamental Physics grant ATP09-0094; NASA Astrophysics Data Analysis Program ADAP10-0110. PJB thanks Lisa Torlina and George Papadopoulos at the University of Sydney for their work on an earlier version of this project. Facilities: IRAS, MSX, Spitzer (IRAC), Mopra (MOPS)", "pages": [ 42, 43 ] }, { "title": "REFERENCES", "content": "Aumann, H. H., Fowler, J. W., & Melnyk, M. 1990, AJ, 99, 1674 Barnes, P. J., Yonekura, Y., Ryder, S. D., et al. 2010, MNRAS, 402, 73 Barnes, P. J., Yonekura, Y., Fukui, Y., et al. 2011, ApJS, 196, 12 Barnes, P. J., Ryder, S. D., O'Dougherty, S. N., et al. 2013, MNRAS, 432, 2231 Beltr'an, M. T., Brand, J., Cesaroni, R., Fontani, F., Pezzuto, S., Testi, L., & Molinari, S. 2006, A&A, 447, 221 Bertoldi, F. & McKee, C. F. 1992, ApJ, 395, 140 Beuther, H., Schilke, P., Menten, K. M., Motte, F., Sridharan, T. K., & Wyrowski, F. 2002, ApJ, 566, 945 Bronfman, L., Nyman, L.-A., & May, J. 1996, A&AS, 115, 81 Egan, M. P., & Price, S. D. 1996, AJ, 112, 2862 Elmegreen, B. G. 2000, ApJ, 530, 277 Elmegreen, B. G. 2007, ApJ, 668, 1064 Fa'undez, S., Bronfman, L., Garay, G., Chini, R., Nyman, L.-˚ A., & May, J. 2004, A&A, 426, 97 Gao, Y., & Solomon, P. M. 2004, ApJ, 606, 271 Graci'a-Carpio, J., Garc'ıa-Burillo, S., Planesas, P., & Colina, L. 2006, ApJ, 640, L135 Gutermuth, R. A., Megeath, S. T., Myers, P. C., et al. 2009, ApJS, 184, 18 Hennebelle, P., & Falgarone, E. 2012, arXiv:1211.0637 Hildebrand, R. H. 1983, QJRAS, 24, 267 Hill, T., Burton, M. G., Minier, V., et al. 2005, MNRAS, 363, 405 Hunter, T. R., Churchwell, E., Watson, C., Cox, P., Benford, D. J., & Roelfsema, P. R. 2000, AJ, 119, 2711 Juneau, S., Narayanan, D. T., Moustakas, J., Shirley, Y. L., Bussmann, R. S., Kennicutt, R. C., & Vanden Bout, P. A. 2009, ApJ, 707, 1217 Kelly, B. C. 2007, ApJ, 665, 1489 Krumholz, M. R., & Tan, J. C. 2007, ApJ, 654, 304 Lada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57 Ladd, E. F., Adams, F. C., Casey, S., Davidson, J. A., Fuller, G. A., Harper, D. A., Myers, P. C., & Padman, R. 1991, ApJ, 366, 203 Leitherer, C. et al. 1999, ApJS, 123, 3 Li, A., & Draine, B. T. 2001, ApJ, 554, 778 McKee, C. F., & Tan, J. C. 2003, ApJ, 585, 850 McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565 Molinari, S., Brand, J., Cesaroni, R., & Palla, F. 2000, A&A, 355, 617 Molinari, S., Pezzuto, S., Cesaroni, R., Brand, J., Faustini, F., & Testi, L. 2008, A&A, 481, 345 Molinari, S. et al. 2010, A&A, 518, 100 Mueller, K. E., Shirley, Y. L., Evans, N. J., II, & Jacobson, H. R. 2002, ApJS, 143, 469 Myers, P. C., & Ladd, E. F. 1993, ApJ, 413, L47 Myers, P. C. et al. 1998, ApJ, 492, 703 Nakano, T., Hasegawa, T., Morino, J.-I., & Yamashita, T. 2000, ApJ, 534, 976 Ossenkopf, V., & Henning, T. 1994, A&A, 291, 943 Palla, F., & Stahler, S. W. 1992, ApJ, 392, 667 Pillai, T., Wyrowski, F., Carey, S. J., & Menten, K. M. 2006, A&A, 450, 569 Preibisch, T., Ossenkopf, V., Yorke, H. W., & Henning, T. 1993, A&A, 279, 577 Reid, M. et al. 2009, ApJ, 700, 137 Roman-Duval, J., Jackson, J. M., Heyer, M., Rathborne, J., & Simon, R. 2010, ApJ, 723, 492 Sanders, D. B., & Mirabel, I. F. 1996, ARA&A, 34, 749 Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269 Schnee, S., Enoch, M., Noriega-Crespo, A., et al. 2010, ApJ, 708, 127 Sridharan, T. K., Beuther, H., Schilke, P., Menten, K. M., & Wyrowski, F. 2002, ApJ, 566, 931 Stahler, S. W. 1988, ApJ, 332, 804 Tan, J. C. & McKee, C. F. 2002, in ASP Conf. Ser. 267, Hot Star Workshop III: The Earliest Phases of Massive Star Birth, ed. P. Crowther (San Francisco: ASP), 267 Tan, J. C., Krumholz, M. R., & McKee, C. F. 2006, ApJ, 641, L121 Tan J. C., Shaske S. N., Van Loo S., 2013, IAUS, 292, 19 Whittet, D. C. B. 1992, Dust in the galactic environment Institute of Physics Publishing, 306 p., Wu, J., Evans, N. J., II, Gao, Y., Solomon, P. M., Shirley, Y. L., & Vanden Bout, P. A. 2005, ApJ, 635, L173 Wu, J., Evans, N. J., Shirley, Y. L., & Knez, C. 2010, ApJS, 188, 313 Yonekura, Y., Asayama, S., Kimura, K., et al. 2005, ApJ, 634, 476 + O 7 .3 0 5 6 .7 0 2 5 .1 1 1 . ly n o e lin b ila P m r a r i b v α d n a v σ . le b a t is h t f o t a m r o f e h t w o h s o t s ie r t n e 4 t s r fi e h t w o h s ly n o e W a ) ) 0 - 1 ( + O C H L ( g lo r i v α t , C A 2 m 0 .6 8 .7 2 - 0 .5 8 - 8 .6 8 - 8 .6 7 - 5 .6 7 - a 5 5 .7 3 .2 3 - 7 .4 9 - 5 .9 9 - 0 .8 8 - 4 .9 8 - b 5 5 .9 1 .2 3 - 8 .1 9 - 4 .4 9 - 9 .5 8 - 7 .5 8 - c 5 3 .0 6 .0 2 - 1 .8 7 - 4 .8 7 - 8 .4 7 - 2 .4 7 - d 5 H C g ( I I R g / s / c -2 -2 -2 -2 e f r o lo ) C A R I I ( g lo ) t o t , C A R I F ( g lo ) C A R I F ( g lo ) t o t , w F ( g lo ) w F ( g lo F Y B r e r s / 2 m c / s / g r e 2 m c / s / g r e 2 m c / s / g r e 2 m c / s / g r e 2 m c / s / g r e . o N e h t f o s ie t r e p o r p l a ic s y h p y r a d n o c e S . 3 le b a T 2 c p s / m k K 3 .4 1 - 58 -", "pages": [ 44, 45, 46, 47, 58 ] } ]

[ { "title": "ABSTRACT", "content": "We report on observations of the supernova remnant Cassiopeia A in the energy range from 100 MeV to 100 GeV using 44 months of observations from the Large Area Telescope on board the Fermi Gamma-ray Space Telescope . We perform a detailed spectral analysis of this source and report on a low-energy break in the spectrum at 1 . 72 +1 . 35 -0 . 89 GeV. By comparing the results with models for the γ -ray emission, we find that hadronic emission is preferred for the GeV energy range. Subject headings: gamma-rays: general, ISM: supernova remnants, supernovae: individual (Cassiopeia A), Acceleration of particles, radiation mechanisms: nonthermal", "pages": [ 1 ] }, { "title": "Fermi -LAT Detection of a Break in the Gamma-Ray Spectrum of the Supernova Remnant Cassiopeia A", "content": "Y. Yuan 1 , 2 , S. Funk 1 , 3 , G. J'ohannesson 4 , J. Lande 1 , 5 , L. Tibaldo 1 , Y. Uchiyama 6 , 7", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "With an age of ∼ 350 years, the supernova remnant (SNR) Cassiopeia A (Cas A) is one of the youngest objects of this class in our Galaxy. It is also one of the best studied objects with both thermal and non-thermal broad-band emission ranging from radio through X-ray all the way to GeV and TeV gamma rays. It is the brightest radio source in the sky outside of our solar system (Baars et al. 1977) and is located at a distance of 3.4 -0 . 1 kpc (Reed et al. 14 1995). Non-thermal emission tracing the acceleration of particles to relativistic energies 15 has been detected in both the forward and reverse shocks (see e.g. Gotthelf et al. 2001; 16 Hughes et al. 2000; Helder & Vink 2008; Maeda et al. 2009), in particular seen through high17 angular resolution X-ray studies. Fast variability and small filaments seen in these X-ray 18 observations also suggest rather large magnetic fields of 0.1-0.3 mG in the shock region of Cas 19 A (Patnaude & Fesen 2007, 2009; Uchiyama & Aharonian 2008). The observed brightness 20 variations might, however, also be produced by local enhancements of the turbulent magnetic 21 field (Bykov et al. 2008). 22 Gamma-ray observations further corroborate the existence of non-thermal particles in 23 the shell of Cas A. The SNR was first detected at TeV energies with the HEGRA telescope 24 system (Aharonian et al. 2001), later confirmed by MAGIC (Albert et al. 2007) and VERI25 TAS (Acciari et al. 2010), and subsequently detected at lower (GeV) energies with the Large 26 Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope ( Fermi ) (Paper I, 27 Abdo et al. 2010a). Those observations revealed a rather modest gamma-ray flux, compared 28 to the synchrotron radio through X-ray emission, further strengthening the argument for a 29 rather high magnetic field. The field can hardly be significantly less than 100 µ G(Abdo et al. 30 2010a), consistent with earlier studies (see e.g. Vink & Laming 2003; Parizot et al. 2006). It 31 should be stressed that the magnetic field is likely to be non-uniform. This was originally 32 proposed by Atoyan et al. (2000) who suggested greatly amplified magnetic fields of up to 1 33 mGin compact filaments. Because both the photon and matter densities in the shock regions 34 are rather high, these gamma-ray studies also suggested that the non-thermal electron (and 35 proton) densities are somewhat low, compared to estimates of the explosion energy (only a 36 few percent). The centroids for the GeV to TeV emission seem to be shifted towards the west37 ern region of the remnant where nonthermal X-ray emission is also brightest (Helder & Vink 38 39 However, given the gamma-ray data published so far it was not possible to unam40 biguously determine the particle population responsible for the bulk of the emission, in 41 particular to distinguish between gamma rays produced through the bremsstrahlung and in42 verse Compton (IC) leptonic processes and the neutral pion decay hadronic process. Lower43 energy gamma rays (below 1 GeV) hold the key to distinguishing between these scenarios, 44 since a sharp low-energy roll-over in the spectrum of hadronically-produced gamma rays 45 is expected (Stecker 1971). Continuous observations of Cas A with the Fermi -LAT have 46 provided us a better opportunity to investigate the gamma-ray emission in the /lessorsimilar 1 GeV 47 range. 48 The telescope has been in routine scientific operation since 2008 August 4. With its wide 50 field of view of 2.4 sr, the LAT observes the whole sky every ∼ 3 hours. More details about 51 the LAT instrument and its operation can be found in Atwood et al. (2009). In addition, 52 the data reduction process and instrument response functions recently have been improved 53 based on two years of in-flight data (so-called Pass7v6, Ackermann et al. 2012b). According 54 to the updated instrument performance, the point-spread function of the LAT gives a 68% 55 containment angle of < 6 · radius at 100 MeV and < 0 . · 3 at > 10 GeV for normal incidence 56 photons in P7SOURCE class. The sensitivity of the LAT for a point source with a power 57 law photon spectrum of index 2 and a location similar to Cas A is ∼ 9 × 10 -9 ph cm -2 s -1 for 58 a 5 σ detection above 100 MeV after 44 months of sky survey. Our analysis takes advantage 59 60 63", "pages": [ 1, 2, 3 ] }, { "title": "2. Analysis Method", "content": "We analyzed Fermi -LAT observations of Cas A using data collected from 2008 August 64 4 to 2012 April 18 (Mission elapsed time 239557565.63 - 356436692.23, about 44 months of 65 data). The analysis was performed in the energy range 100 MeV-100 GeV using the LAT Sci66 ence Tools 1 as well as an independent tool pointlike . In particular, we used the maximum67 likelihood fitting packages pointlike to fit the position and test for significant spatial exten68 sion of Cas A, then with the updated localization result we used gtlike to fit the spectrum 69 of the source. Our analysis procedure is very similar to that of the second LAT source cat70 alog (2FGL, Nolan et al. 2012). When analyzing the data, we used the P7SOURCE class 71 event selection and P7 V6 instrument response functions (IRFs, Ackermann et al. 2012b). 72 In order to reduce contamination from gamma rays produced in the Earth's limb, we ex73 cluded events with reconstructed zenith angle greater than 100 · , and selected times when 74 the rocking angle was less than 52 · . 75 Emission produced by the interactions of cosmic rays with interstellar gas and radiation 76 fields substantially contributes to the gamma-ray intensities measured by the LAT near the 77 Galactic plane. We accounted for it using the standard diffuse model used in the 2FGL 78 analysis. We also included the standard isotropic template accounting for the isotropic 79 gamma-ray background and residual cosmic-ray contamination. 2 In addition, we modeled 80 as background sources all nearby 2FGL sources: in pointlike we used a circular region of 81 interest (ROI) with a radius of 15 · centered on Cas A; in gtlike we used a square region of 82 interest with a size of 20 · × 20 · aligned with Galactic coordinates, using a spatial binning of 83 0 . · 125 × 0 . · 125. We adopt the same parameterizations as 2FGL for these sources, while left 84 free the spectral parameters of 5 2FGL sources that were either nearby or had a significant 85 residual when assuming the 2FGL values: 2FGL J2333.3+6237, 2FGL J2257.5+6222c, 2FGL 86 J2239.8+5825, 2FGL J2238.4+5902, 2FGL J2229.0+6114. In addition, we added 4 sources 87 89 90", "pages": [ 3, 4 ] }, { "title": "3.1. Spatial Analysis", "content": "Because of the wide and energy-dependent point-spread function of the LAT, nearby 91 sources must be carefully modeled to avoid bias during a spectral analysis. Therefore, before 92 analyzing Cas A, we performed a dedicated search for nearby point-like sources not included 93 in the 2FGL catalog. We did so by adding sources in the background model at the positions of 94 significant residual test statistic (TS, which follows the same definition as that in Nolan et al. 95 2012) until the residual TS < 25 within the entire pointlike ROI. Table 1 lists the four 96 significant new sources found in this study. We have not found any counterparts for the new 97 sources yet. 98 Figure 1 shows a count map above 800 MeV of the region surrounding Cas A. The 99 relatively bright source coincident with the SNR Cas A has a TS value of ∼ 600. First, we 100 used pointlike to fit the position of this source and test for any possible spatial extension. 101 The best fit position of the source, in Galactic coordinates, is l, b = 111 . · 74 , -2 . · 12, with a 102 statistical uncertainty of 0 . · 01 (68% containment). To account for the systematic error in 103 the position of Cas A, we added 0 . · 005 in quadrature as was adopted for the 2FGL analysis 104 (Nolan et al. 2012). 105 106 107 108 109 This location is only 0 . · 02 away from the central compact object (CCO) (Pavlov & Luna 2009), as shown in Figure 2. This confirms that the GeV source is most likely the γ -ray counterpart of the Cas A SNR. Following the method described in Lande et al. (2012), we used a disk spatial model to fit the extension of Cas A. We found that the emission was not significantly spatially extended (TS ext = 0 . 1) and has an extension upper limit of 0 . · 1 at 110 95% confidence level. Note that this upper limit is larger than the shell of Cas A. 111 112", "pages": [ 4, 5 ] }, { "title": "3.2. Spectral Analysis", "content": "We performed a spectral analysis of Cas A in the energy range from 100 MeV to 100 GeV 113 using gtlike . We first fit Cas A with a power-law spectral model and found an integral flux 114 of (6 . 17 ± 0 . 43 stat ) × 10 -11 erg cm -2 s -1 in the energy range from 100 MeV to 100 GeV and 115 a photon index of Γ = 1 . 80 ± 0 . 04 stat . The results are consistent with the previous analysis 116 of Abdo et al. (2010a). 117 We then tested for a break in the spectrum of Cas A by fitting the spectrum with a smoothly-broken power-law spectral model Here, N 0 is the prefactor; E 0 is a fixed energy scale (taken to be 1 GeV); E b is the break 118 energy; Γ 1 and Γ 2 are the photon indices before and after the break, respectively; β is a 119 small, fixed parameter that describes the smoothness of the transition at the break (taken 120 to be 0.1). 121 We tested for the significance of this spectral feature using a likelihood ratio test: where L is the Poisson likelihood of observing the given data assuming the best-fit model. 122 We obtained TS break = 48 . 2, indicating that the break is significant. The resulting spectral 123 parameters are quoted in Table 2. 124 We then computed a spectral energy distribution (SED) in 8 bins per energy decade by 125 fitting the flux of Cas A independently in each energy bin (the lowest 6 bins were combined 126 into 3 bins). The SED of Cas A, along with the all-energy spectral fit, is plotted in Figure 3. 127 Statistical upper limits are shown in energy bins where TS of the flux is less than 4. These 128 upper limits are calculated at 95% confidence level using a Bayesian method (e.g., Helene 129 1983). 130 131 132 133 134", "pages": [ 5, 6 ] }, { "title": "3.3. Systematic Errors", "content": "We estimated the systematic errors on the spectrum of Cas A due to uncertainty in our model of the Galactic diffuse emission and due to uncertainty in our knowledge of the IRFs of the LAT. To probe the uncertainties due to the modeling of Galactic diffuse emission we use a se135 ries of alternative models (de Palma et al. 2013). These models differ from the standard one 136 in the sense that de Palma et al. 1) adopt different gamma-ray emissivities for the interstellar 137 gas, different gas column densities, and use a different approach for incorporating spatially 138 extended residuals; 2) vary a select number of important input parameters of the model 139 (Ackermann et al. 2012a): the H i spin temperature, the cosmic-ray source distribution, and 140 height of the cosmic-ray propagation halo; 3) allow more freedom in the fit by separately 141 scaling components of the model in four Galactocentric rings. Although these models do 142 not span the complete uncertainty of the systematics involved with Galactic diffuse emission 143 modeling, they were selected to probe the most important systematic uncertainties. 144 145 146 147 148 149 150 151 At low energy ( < 1 GeV), our uncertainty in the modeling of the Galactic diffuse emission leads to significant uncertainty in the spectral analysis of Cas A, because the integrated intensity of the diffuse emission on the scale of the energy dependent point spread function of the LAT becomes comparable with the flux of the source. By examining the residual maps after fitting, we found that the standard diffuse model overshoots the data for a region ∼ 2 · from Cas A (Figure 4), and this can lead to underestimated upper limits in the SED calculation. This overestimation of diffuse count is most likely due to uncertainty in modeling the 152 gamma-ray emission from the molecular complex associated with NGC 7538 and Cas A in the 153 Perseus arm (e.g., Abdo et al. 2010b). The alternative diffuse models provide a qualitatively 154 better fit of this region when the normalization of each Galactocentic ring was left free, since 155 the increased degrees of freedom allow us to better scale the Galactic diffuse model for this 156 specific region. The improvement can be seen in Figure 4 which shows a residual map with 157 the standard diffuse model and an improved residual map with one of the alternative diffuse 158 models. 159 160 161 162 163 164 165 Even though there is significant systematic uncertainty in the spectral model of Cas A at lower energies, TS break was greater than 20 using all of the alternative diffuse models and is therefore robust against this systematic uncertainty. We estimated the systematic error due to uncertainty in the IRFs using the method described in Ackermann et al. (2012b). Following this method, we set the pivot in the bracketing IRFs at 2 GeV, near the spectral peak in our SED. Again, we found the spectral 169 break to be robust against uncertainty in IRFs. 166 The systematic errors on the estimated spectral parameters due to both systematic 167 uncertainties are included in Table 2. 168", "pages": [ 6, 7 ] }, { "title": "4. Discussion", "content": "In Figure 3, the new spectral data points measured with the Fermi -LAT are overlaid 170 with those from Paper I. The newly-measured spectrum is consistent with the previous 171 result, except that most of the new data points lie slightly above the old measurement. This 172 is likely due to the changed event classifications and improved IRFs of the LAT as well as 173 Note. - The spectral and spatial parameters of the new sources found in the region surrounding Cas A. l and b are the Galactic longitude and latitude of the source and TS is the significance of the detection of the source (in the energy range from 100 MeV to 100 GeV). The sources were modeled with a power-law spectral model and the flux is computed from 100 MeV to 100 GeV. Note. - Spectral fit of Cas A assuming a smoothly-broken power-law spectral model. Energy flux is quoted from 100 MeV to 100 GeV. ∆ stat is the statistical error; ∆ sys,diffuse is the estimated systematic error due to uncertainties in modeling the Galactic diffuse emission; ∆ sys,IRFs is the estimated systematic error due to uncertainty in our knowledge of the IRFs of the LAT. ∆ sys is derived by adding the two components of systematic errors in quadrature. updated background models. In Paper I, we argued that the GeV-TeV gamma rays detected 174 from Cas A can be interpreted in terms of either a leptonic or a hadronic model. In these 175 models, cosmic-ray electrons and protons (and ions) are accelerated in Cas A and produce 176 the gamma-ray emission. In what follows, we revisit the gamma-ray emission models and 177 then discuss the new LAT spectrum. 178 The synchrotron X-ray filaments found at the locations of outer shock waves indicate 179 efficient acceleration of cosmic-ray electrons at the forward shocks (Hughes et al. 2000; 180 Gotthelf et al. 2001; Vink & Laming 2003; Bamba et al. 2005; Patnaude & Fesen 2009). 181 Moreover, X-ray studies with Chandra suggest that electron acceleration to multi-TeV ener182 gies also takes place at the reverse shock propagating inside the supernova ejecta (Uchiyama & Aharonian 183 2008; Helder & Vink 2008). The detections of TeV gamma rays with HEGRA (Aharonian et al. 184 2001), MAGIC (Albert et al. 2007) and VERITAS (Acciari et al. 2010), established the ac185 celeration of multi-TeV particles in the remnant. Because of the small radius of 2 . 5 ' of Cas A, 186 these experiments lacked the angular resolution to determine the spatial distribution of the 187 gamma rays and the sites of particle acceleration. 188 It is widely considered that diffusive shock acceleration (DSA: see e.g., Malkov & O'C Drury 189 2001, for a review) operating at the forward shocks is responsible for the energization 190 of the cosmic-ray particles. Most DSA models, which provide predictions of gamma-ray 191 spectra of SNRs, focus on the acceleration at the forward shock (e.g., Ellison et al. 2010; 192 Morlino & Caprioli 2012). Recently, newly-developed non-linear DSA models have included 193 the effects of acceleration of particles at reverse shocks and their subsequent transport 194 (Zirakashvili & Ptuskin 2012). Zirakashvili et al. (2013) have demonstrated that about 50% 195 of the gamma-ray flux at 1 TeV from Cas A can be contributed by the reverse-shocked 196 medium. Although the nonthermal X-ray filaments and knots in the reverse-shock region 197 are interesting sites of particle acceleration (Uchiyama & Aharonian 2008), we assume that 198 the gamma-ray emission comes predominantly from the forward shock region. Note that 199 our discussion on leptonic versus hadronic emission would not be greatly affected by this as200 sumption, because we allow for parameter space that is relevant also for the reverse-shocked 201 regions. 202 The gamma-ray emission models are constrained by the gas and radiation density and 203 by the magnetic field in the gamma-ray production region. We assume the simplest model 204 where cosmic rays are distributed uniformly in the shell of the remnant. The fluxes of 205 bremsstrahlung and π 0 -decay gamma-ray emission scale linearly with the average gas density 206 ( ∝ ¯ n ). Likewise the IC flux is proportional to the radiation energy density ( ∝ U ph ) as long as 207 IC scattering is in the Thomson regime. The synchrotron flux scales as ∝ B ( s +1) / 2 for a fixed 208 density of electrons with a power-law index of s . The magnetic field only indirectly affects 209 the gamma-ray flux by determining the amount of relativistic electrons that are required to 210 produce the observed synchrotron radio emission. This in turn can be used to calculate the 211 bremsstrahlung and IC fluxes. Therefore the gamma-ray flux constrains the magnetic field 212 in the shell (Cowsik & Sarkar 1980). 213 The outer shock waves are currently propagating into a dense circumstellar wind. The 214 density behind the blastwave is estimated as n H /similarequal 10 cm -3 from the measured hydrodynam215 ical quantities such as shock velocities (Laming & Hwang 2003). The radiation field for IC 216 scattering is dominated by far infrared (FIR) emission from the shock-heated ejecta, char217 acterized by a temperature of 100 K and an energy density of ∼ 2 eV cm -3 (Mezger et al. 218 1986). Using the gas and infrared densities, which are well constrained from the multiwave219 length data, it was shown in Paper I that bremsstrahlung by relativistic electrons dominates 220 the leptonic component below ∼ 1 GeV, and IC/FIR becomes comparable to bremsstrahlung 221 above 10 GeV, for the assumed electron acceleration spectrum Q e ( E ) ∝ E -2 . 34 exp( -E/E m ) 222 with E m = 40 TeV (Vink & Laming 2003). The power-law index was set to match the 223 radio-infrared spectral index of α = 0 . 67 (Rho et al. 2003), since both the GeV gamma-ray 224 emission and the radio synchrotron emission sample similar electron energies. We note that 225 the IC scattering of FIR exceeds IC of cosmic microwave background by a factor of ∼ 3 at 226 10 GeV. 227 Figure 5 compares the leptonic model presented in Paper I with our new LAT mea228 surement. The magnetic field B = 0 . 1 mG used in the leptonic model is consistent with 229 B = 0 . 08-0 . 16 mG estimated by Vink & Laming (2003) who interpreted the width of a syn230 chrotron X-ray filament as the synchrotron cooling length. The field is somewhat lower than 231 B /similarequal 0 . 3 mG estimated by Parizot et al. (2006) who took into account a projection effect. 232 Unlike the TeV band where the electrons responsible for the gamma-ray emission suffer from 233 severe synchrotron losses, the gamma-ray spectral shape near 1 GeV does not depend on 234 the magnetic field. This can be seen, for example, in Araya & Cui (2010) who employed 235 different magnetic field strengths (by a factor of 6) between two radiation zones. 236 Also shown in Figure 5 is the hadronic model presented in Paper I. To achieve a 237 better match with the new measurement, the normalization of the model spectrum is in238 creased by 27% from Paper I. The model was calculated for a proton spectrum of Q p ( p ) ∝ 239 p -2 . 1 exp( -p/p m ) with an exponential cutoff at cp m = 10 TeV, where p denotes momentum of 240 accelerated protons. The total proton content amounts to W p ( > 10 MeV c -1 ) /similarequal 4 × 10 49 erg, 241 which is less than 2% of the estimated explosion kinetic energy of E sn = 2 × 10 51 erg 242 (Laming & Hwang 2003; Hwang & Laming 2003) 3 . 243 Paper I already showed that the leptonic model cannot fit the turnover well at low 244 energies because the bremsstrahlung component that is dominant over IC below 1 GeV has 245 a steep spectrum. Note that the spectral shape of the bremsstrahlung component copies 246 the electron spectrum with spectral index s = 2 . 34, which in turn is determined from the 247 radio-infrared spectral index of α = 0 . 67 (Rho et al. 2003). If we use a steeper power law 248 for the electron energy distribution based on a global spectral index of α = 0 . 77 in the 249 radio wavelengths (Baars et al. 1977) or a spectral shape with curvature that reproduces 250 the hardening ( α = 0 . 77 → 0 . 67) in the integrated spectrum, the discrepancies between the 251 bremsstrahlung model and the Fermi -LAT data become even larger. Araya & Cui (2010), 252 who reported the results of Fermi -LAT analysis of Cas A independently, also showed that the 253 electron bremsstrahlung with such a steep electron index could not explain the Fermi -LAT 254 spectrum. However, uncertainties in the Galactic diffuse emission at low energies prevented 255 a definitive conclusion regarding the inconsistency between the bremsstrahlung model and 256 the gamma-ray data. In this paper, a more detailed investigation of these uncertainties at 257 low energy now confirms the hadronic origin of the GeV γ -ray emission from Cas A. The 258 new LAT spectrum can be described by a broken power law with a second power-law index 259 of Γ 2 = 2 . 17 ± 0 . 09. A comparison between the LAT spectrum and the TeV γ -ray spectra 260 suggests that additional steepening between the LAT and the TeV bands is necessary. Indeed, 261 the TeV γ -ray spectra measured with HEGRA, MAGIC, and VERITAS are consistent with 262 a power law with a photon index of Γ TeV = 2 . 5 ± 0 . 4 stat ± 0 . 1 sys , Γ TeV = 2 . 3 ± 0 . 2 stat ± 0 . 2 sys , 263 and Γ TeV = 2 . 61 ± 0 . 24 stat ± 0 . 2 sys , respectively, which are somewhat steeper than the second 264 index Γ 2 = 2 . 17 ± 0 . 09 of the LAT spectrum. However, given the relatively large statistical 265 uncertainties of the TeV γ -ray fluxes, we refrain from solidifying the presence of the cutoff. 266 If confirmed, efficient acceleration of particles to PeV energies in Cas A is questioned. 267 The Fermi -LAT results on two historical SNRs, Tycho's SNR (Giordano et al. 2012) 268 and Cas A, support hadronic scenarios for these objects. Tycho's SNR is the remnant of a 269 Type Ia supernova, while Cas A is that of a core-collapse SN (specifically Type IIb). This in270 dicates that both Type Ia and core-collapse SNRs can convert a substantial fraction of their 271 kinetic expansion energies into cosmic-ray energies, and makes SNRs energetically favorable 272 candidates for the origin of Galactic cosmic rays. Recently, direct spectral signatures of 273 the π 0 -decay emission have been found in two middle-aged SNRs interacting with molecular 274 clouds: W44 and IC 443 (Ackermann et al. 2013; Giuliani et al. 2011). Although spectro275 scopic evidence for the π 0 -decay emission from Cas A is not as strong as these two cases, our 276 ejecta is similar to that in the forward shock region. Therefore, the total proton content estimated here can be interpreted roughly as a sum of the cosmic-ray contents in the forward shock region and that in the reverse-shocked ejecta. results presented in this paper demonstrate the importance of the gamma-ray measurements 277 of SNRs below 1 GeV. 278 The Fermi -LAT Collaboration acknowledges generous ongoing support from a number 279 of agencies and institutes that have supported both the development and the operation of the 280 LAT as well as scientific data analysis. These include the National Aeronautics and Space 281 Administration and the Department of Energy in the United States, the Commissariat 'a 282 l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National 283 de Physique Nucl'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana 284 and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, 285 Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization 286 (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallen287 berg Foundation, the Swedish Research Council and the Swedish National Space Board in 288 Sweden. 289 Additional support for science analysis during the operations phase is gratefully acknowl290 edged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d' ' Etudes 291 Spatiales in France. 292 293", "pages": [ 7, 8, 12, 13, 14, 15, 16 ] } ]

[ { "title": "ABSTRACT", "content": "We have monitored the BL Lac object S5 0716+714 in five intermediate optical wavebands from 2004 September to 2011 April. Here we present the data that include 8661 measurements. It represents one of the largest databases obtained for an object at optical domain. A simple analysis of the data indicates that the object was active in most time, and intraday variability was frequently observed. In total, the object varied by 2.614 magnitudes in the i band. Strong bluer-when-brighter chromatism was observed on long, intermediate, and short timescales. Subject headings: BL lacertae Object: individual (S5 0716+714) - galaxies: active galaxies: photometry", "pages": [ 2 ] }, { "title": "Seven-Year Multi-Color Optical Monitoring of BL Lacertae Object S5 0716+714", "content": "Yan Dai Department of Astronomy, Beijing Normal University, Beijing 100875, China Department for Popularization of Astronomy, Beijing Planetarium, 138 Xizhimenwai Street, Beijing 100044, China Jianghua Wu, Zong-Hong Zhu Department of Astronomy, Beijing Normal University, Beijing 100875, China [email protected] Xu Zhou, Jun Ma Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100012, China Qirong Yuan Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210046, China and Lingzhi Wang Department of Astronomy, Beijing Normal University, Beijing 100875, China Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Blazars constitute the most variable subclass of active galactic nuclei (AGNs). Depending on whether or not showing strong emission lines in spectra. Blazar is divided into flat-spectrum radio quasars (FSRQs) and BL Lacertae (BL Lac) objects. BL Lac objects are characterized by non-thermal continuum emission across the whole electromagnetic spectrum with absent or weak emission and absorption lines (Stickel et al. 1993), variable and high polarization (Angel & Stockman 1980; Impey & Neugebaur 1988; Gabuzda et al. 1989), large amplitude and rapid variability at all wavelengths from radio to gamma rays (Ravasio et al. 2002; Bottcher et al. 2003), and superluminal motion of radio components (Denn et al. 2000). S5 0716+714 is a distant BL Lac. In 1979, it was discovered in a survey for sources with a 5 GHz flux greater than 1 Jy (Kuhr et al. 1981). Because of the featureless spectrum and its strong optical polarization, it was identified as a BL Lacertae object by Biermann et al. (1981). The redshift of S5 0716+714 was uncertain until Wagner et al. (1996) estimated a value bigger than 0.3. Afterwards, Nilsson et al. (2008) acquired a deep i-band image of this object and derived a redshift of 0 . 31 ± 0 . 08 by using the host galaxy as a standard candle. Most recently, Danforth et al. (2012) set an upper bound of z < 0 . 304 with a confidence level of 90% for this object. This source is one of the most studied BL Lac objects, because it has high brightness and strong variability. The optical duty cycle of S5 0716+714 is nearly unity, indicating that the source is always in an active state in the visible (Wagner & Witzel 1995). Strong bluer-when-brighter correlations were found for both internight and intranight variations (Wu et al. 2005, 2007, 2012; Poon et al. 2009; Hao et al. 2010; Chandra et al. 2011). This source has been intensively monitored by a number of authors. During a 4-week period of continuous monitoring, the source displayed in both optical and radio regimes a transition between states of fast and slow variability with a change of the typical variability timescale from about 1 to about 7 days (Quirrenbach et al. 1991). Wagner et al. (1996) investigated the rapid variations of this object in the radio, optical, ultraviolet, and X-ray regimes and found that it always keeps high amplitude change on the timescale of a few days. Sagar et al. (1999) showed an average V -R color of this BL Lac to be ∼ 0.4 mag in their one month long BVRI optical monitoring campaign in 1994. Raiteri et al. (2003) reported that the long-term optical brightness variations of this source appear to have a characteristic timescale of 3.3yr and four major optical outbursts were observed at the beginning of 1995, in late 1997, at the end of 2000, and in fall 2001. In particular, an exceptional brightening of 2.3 mag in 9 days was detected in the R band on 2000 October 30. Color analysis on the optical light curves reveals only a weak general correlation between the color index and the source brightness. Recently, Poon et al. (2009) monitored the BL Lac object S5 0716+714 in the optical band during 2008 October and December and 2009 February with a best temporal resolution of about 5 minutes in the BVRI bands. Typical timescales of microvariability range from 2 to 8 hr. The overall V -R color index ranges from 0.37 to 0.59. Strong bluer-when-brighter chromatism was found on internight timescales. The overall variability amplitude decreases with decreasing frequency. We have monitored S5 0716+714 since 2004. Here we present the data during the period from 2004 to 2011. A simple analysis is performed and the results are described. This paper is organized as follows. The Observation and data analysis is described in Section 2. Section 3 presents the light curves. Section 4 shows the comparison result of i -data from us and R -data from other authors. The relation of color and magnitude is described in Section 5. The conclusions are given in Section 6.", "pages": [ 3, 4 ] }, { "title": "2. Observations and data analysis", "content": "Our optical monitoring program of S5 0716+714 was carried out with the 60/90 cm Schmidt telescope located at the Xinglong Station of the National Astronomical Observatories of China (NAOC). Prior to 2006, a Ford Aerospace 2048 × 2048 CCD camera was mounted at its main focus. The CCD has a pixel size of 15 µ m, and its field of view is 58' × 58', resulting in a resolution of 1 . '' 7 pixel -1 . At the beginning of 2006, the 2k CCD was replaced by a new 4096 × 4096 CCD. The field of view is now 96' × 96', resulting in a resolution of 1 . '' 3 pixel -1 . The telescope is equipped with 15 intermediate-band filters, covering a wavelength range from 3000 to 10000 ˚ A. This paper includes data from 2004 September 10 to 2011 April 24. Excluding the nights with bad weather and those devoted to other targets, the actual number of nights for S5 0716+714 observations is 332. We used filters in e , i , and m bands to observe in 2004-2006, and then changed to the c , i , and o bands from 2006 December. The central wavelengths of the c , e , i , m , and o bands were 4210, 4920, 6660, 8020, and 9190 ˚ A, respectively. The central wavelength of i band is similar to the R band. With the observational results of stars, the magnitudes in these two bands can be transformed with R = i + 0 . 1 (Zhou et al. 2003). Depending on the weather and seeing conditions, the exposure time of different bands range from 30s to 480s, and the exposures per night of different bands varies between 2 to 43. The data reduction procedure includes bias subtraction, flat-fielding, extraction of instrumental aperture magnitude, and flux calibration. We used differential photometry. For each frame, the instrumental magnitudes of the blazar and four comparison stars (See Fig.1) were extracted at first. The radii of the aperture and the sky annuli were adopted as 3, 7, and 10 pixels, respectively. Then the brightness of the blazar was measured relative to the average brightness of the three reference stars 3, 4, and 5. Star 6 acted as a check star, which has an apparently similar brightness as the blazar (for a reasonable selection of reference and check stars, see Howell et al. 1988). The differential magnitude of star 6 is the difference between the magnitude of star 6 and the average magnitude of star 4 and 5, so as to verify the stable fluxes of the four comparison stars, and to verify the accuracy of our measurements. The c , e , i , m , and o magnitudes of the 4 comparison stars were obtained by observing them and the standard star HD 19945 on a photometric night and are listed in Table 1.", "pages": [ 5, 6 ] }, { "title": "3. Light Curves", "content": "The samples of observational log and results are given in Tables 2-6. The columns are observation date and time in universal time, Julian date, exposure time in second, magnitude and error of S5 0716+714, and differential magnitude of star 6 (its nightly averages were set to zero). Figure 2 shows the light curves of the overall monitoring period in the five bands. The source remained active during the whole monitoring period. The variation amplitudes of e , i , and m bands from 2004 September 10 to 2006 March 29 are 1.200, 1.156, and 1.127 mags, respectively, and the variation amplitudes of c , i , and o bands from 2006 December 6 to 2011 April 24 are 2.763, 2.614, and 2.522 mags, respectively. The amplitude of variation tends to decrease with decreasing frequency. The light curves keep fluctuating during the whole monitoring period. The curves get the extreme bright value on 2004 September 10 (JD 2,453,259), 2007 October 20 (JD 2,454,394), 2008 April 22 (JD 2,454,579), and 2010 September 28 (JD 2,455,468), and get the extreme dark value on 2005 January 29 (JD 2,453,400), 2007 December 16 (JD 2,454,451), and 2011 April 24 (JD 2,455,676). A faintest optical state was recorded on 2007 December 16 (JD 2,454,451), . Nilsson et al. (2008) acquired a deep i-band image of this object at that state and derived a redshift of 0 . 31 ± 0 . 08 by using the host galaxy as a 'standard candle'. The BL Lac object S5 0716+714 is one of the brightest BL Lac objects noted for its microvariability. In order to confirm whether or not the object was variable in one day, a quantitative assessment was carried out. For each of the 233 nights with observational duration longer than 2 hours, we used a chi-square inspection to check whether there is intraday variability (IDV) (Penston & Cannon 1970; Kesteven et al. 1976; de Diego 2010). The chi-square value was compared with the critical value at the 95% confidence level. If the former was greater than the latter, the null hypothesis that there was no variability was rejected. As a result, 138 nights (62% of 233) with IDV were identified. The observed fastest variation of S5 0716+714 varied by 0.117 mags in 1.1 hrs in the c band on 2011 March 5 (JD 2,455,626), as indicated by the bottom line with arrows in Figure 3. The similar work was made by Villata et al. (2000). They found that the source exhibited strong variability with similar trend but different amplitudes in all bands, and noted the monotonic brightness increase in B band for about 130 minutes. The steepest (linear) part has a rising rate of 0.002 mag per minute and a duration of about 45 minutes.", "pages": [ 6, 7 ] }, { "title": "4. Compare with Other Data", "content": "In Section 2, a transforming formula between the i and R magnitudes was mentioned. This formula was derived from the observational result of stars (Zhou et al. 2003). However, the spectral shape of blazars is quite different from that of stars. So the transforming formula may be different for blazars. Therefore, we made a comparison between our i -band data and the R -band data of other authors in order to find an empirical relation between them. Villata et al. (2000) and Poon et al. (2009) have made intensive monitoring on the same object. Their data were adopted and matched in time with ours with a threshold of less than 0.02 days (or 28.8 minutes). As the results, 24 R -i matches were found for Villata et al. and our data, and 97 R -i matches were found for Poon et al. and our data. The average time differences are 0.0038 and 0.0009 days for the 20 and 97 matches, respectively. Two R -i diagrams were plotted in the top and middle panels of Figure 4. Two linear regressions give the transforming formulae as R = (0 . 964 ± 0 . 015) × i + (0 . 279 ± 0 . 187) and R = (0 . 976 ± 0 . 009) × i + (0 . 059 ± 0 . 123), respectively. If all matches are plotted together, as shown in the bottom panel of Figure 4, the linear regression gives the formula as R = (0 . 897 ± 0 . 004) × i +(1 . 127 ± 0 . 048).", "pages": [ 7, 8 ] }, { "title": "5. Relation of Color and Magnitude", "content": "The long-term color behavior of S5 0716+714 was studied based on our data. For the 2004-2006 data, the e -m color was calculated and plotted vs. the e magnitude in the top panel of Figure 5. For the data after 2006 December, the c -o color was calculated and plotted vs. the c magnitude in the central panel of Figure 5. The bottom panels illustrate how the c -o color and c magnitude changed with time. Despite the discontinuity in the top panel and significant scatter in the central panel, there is an overall bluer-when-brighter chromatism in both panels. In order to investigate the color behavior of S5 0716+714 on the intermediate timescale, three episodes in our monitoring are isolated. They are from JDs 2,454,101 to 2,454,115, from JDs 2,454,429 to 2,454,463, and from JDs 2,455,597 to 2,455,629. These three episodes lasted from two weeks to more than one month. In these periods, we have relatively continuous monitoring, and the object showed significant variations. The light curves and the corresponding color-magnitude diagrams are plotted in Figure 6 for the three episodes. There are strong color-magnitude correlations. The correlation coefficients are 0.89, 0.78, and 0.82, respectively. The bluer-when-brighter chromatism on intermediate timescale is found by other authors (e.g., Villata et al. 2000; Wu et al. 2007). On intraday timescales, the object also displays strong bluer-when-brighter chromatism. Some examples are given in Figure 7. Depending on the balance between escape, acceleration, and cooling of the electrons with different energy, either soft (low energy) or hard (high energy) lags are expected (Kirk et al. 1998). This will lead to a loop-like path of the blazars state in a colormagnitude (or spectral index-flux) diagram. The direction of this spectral hysteresis can be either clockwise or anticlockwise. It depends on the relative position between the observing frequency and the peak frequency of the synchrotron component in the SED of the blazar as well as the relative values of the acceleration, cooling and escape timescales (Chiaberge & Ghisellini 1999; Dermer 1999). In our monitoring, we got an inconspicuous loop flare in internight timescale and an obvious loop flare in intranight timescale, which are shown in Figure 8. The upper-left panel is the light curve of JDs 2,454,390 ∼ 2,454,398, which shows a flare in cio bands. The corresponding color-magnitude diagram is displayed in the upper-right panel, in which the numbers denote time sequence. There were not significant variations during these day, so we averaged the data in day to decrease system error. The points spread as diagonal distribution in color-magnitude diagram. The loop flare is far less obvious. The result of intranight timescale, JD 2,455,621, is shown in the bottom of Figure 7. The numbers denote the time sequence. An anticlockwise loop can be seen in the lower-right panel. As Kirk et al. (1998) supposed, if the loop is traced anticlockwise, there might be a flare propagating from lower to higher energy, as particles are gradually accelerated into the radiating window. The frequency of c and o band are 7 . 13 × 10 14 Hz and 3 . 26 × 10 14 Hz. In 1999, Giommi et al. found that the frequency is between 10 14 and 10 15 Hz. Our anticlockwise loop imply that the peak frequency of short band in the SED of the blazar should be higher than × 14 7 . 13 10 Hz.", "pages": [ 8, 9 ] }, { "title": "6. Conclusions", "content": "We have monitored the BL Lac object S5 0716+714 in five intermediate optical wavebands from 2004 September to 2011 April by the 60/90 cm Schmidt telescope located at the Xinglong Station of the National Astronomical Observatories of China (NAOC). We collected 8661 data points with error less than 0.05 mags. It represents one of the largest databases obtained for an object at optical domain and can be used to study both the longand short-term flux and spectral variability of this object. It can also be correlated with the data in the radio, X-ray, or gamma-ray wavelengths in order to investigate the broad-band behavior of this object. A simple analysis of the data indicates that the object was active in most time. The overall amplitudes of e , i , and m bands from 2004 September 10 to 2006 March 29 are 1.200, 1.156, and 1.127 mags, respectively, and the overall amplitudes of c , i , and o bands from 2006 December 6 to 2011 April 24 are 2.763, 2.592, and 2.522 mags, respectively. The amplitude of variation tends to decrease with decreasing frequency. The bluer-when-brighter phenomenon is effectively confirmed on long, intermediate, and short timescales. It is an important support to the shock-in-jet model in which shocks propagate down the relativistic jet, accelerating particles and/or compressing magnetic fields, leading to the observed flux and spectral variability (Marscher & Gear 1985; Qian et al. 1991). There were 138 nights of IDV captured during the whole monitoring period. The authors thank the anonymous referee for constructive suggestions and insightful comments. We thank Villata, M. for kindly sending us the WEBT data on S5 0716+714. This work has been supported by Chinese National Natural Science Foundation grants 11273006, 11173016, and 11073023. WJH is supported by National Basic Research Program of China 973 Program 2013CB834900. ZHZ acknowledges that this work is supported by the Ministry of Science and Technology National Basic Science Program (Project 973) under Grant No.2012CB821804, the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University.", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "Angel, J. R. P., Stockman, H. S. 1980, ARA&A, 18, 321 Biermann, P.L., et al. 1981, ApJ, 247, 53 B¨ottcher, M. et al. 2003, ApJ, 596, 847 Chandra, S., Baliyan, K. S., Ganesh, S., & Joshi, U. C. 2011, ApJ, 731, 118 Chiaberge, M. & Ghisellini, G. 1999, MNRAS, 306, 551 Danforth, C. W., Nalewajko, K., France, K., & Keeney, B. A. 2012, submitted to ApJ, arXiv:1209.3325 Denn, G. R., Mutel, L. R., Marscher, A. P. 2000, ApJS, 129, 61 Dermer, C. D. 1998, ApJ, 501, 157 de Diego, J. A. 2010, AJ, 139, 1269 Fukugita, M., et al. 1996, AJ, 111, 1748 Gabuzda, D. C., Cawthorne, T. V., Roberts, D. H., & Wardle, J. F. C. 1989, ApJ, 347, 701 Hao, J., Wang, B., Jiang, Z., & Dai, B. 2010, RAA, 10, 125 Howell, S. B., Mitchell, K. J., & Warnock, A. 1998, AJ, 95, 247 Kesteven, M. J. L., Bridle, A. H., & Brandie, G. W. 1976, AJ, 81, 919 Impey, C. D., & Neugebaur, G. 1988, AJ, 95, 307 Kirk, J. G., Rieger, F. M., & Mastichiadis, A. 1998, A&A, 333, 452 K¨uhr, H., Witzel, A., Pauliny-Toth, I. I. K., & Nauber, U. 1981, A&AS, 45, 367 Marscher, A. P., & Gear, W. K. 1985, ApJ, 298, 114 Nilsson, K., Pursimo, T., Sillanpaa, A., Takalo, L. O., & Lindfors, E. 2008, A&A, 487, 29 Oke, J. B. & Gunn, J. E. 1983, ApJ, 266, 713 Penston, M. V., & Cannon, R. D. 1970, Royal Greenwich Obs. Bull., 159, 83 Poon, H., Fan, J. H., & Fu, J. N. 2009, ApJS, 185, 511 Qian, S. J., Quirrenbach, A., Witzel, A., Krichbaum, T. P., Hummel, C. A., & Zensus, J. A. 1991, A&A, 241, 15 Quirrenbach, A., et al. 1991, ApJ, 372, 71 Raiteri, C. M., et al. 2003, A&A, 402, 151 Ravasio, M. et al. 2002, A&A, 383, 763 Sagar, R., Gopal-Krishna, Mohan, V., Pandey, A. K., Bhatt, B. C., & Wagner, S. J. 1999, A&AS, 134, 453 Stalin, C. S., Gopal-Krishna, Sagar, R., Wiita, P. J., Mohan, V., & Pandey, A. K. 2006, MNRAS, 366, 1337 Stickel, M., Fried, J. W., & Kuhr, H. 1993, A&AS, 98, 393 Vagnetti, F., Trevese, D., & Nesci, R. 2003, ApJ, 590, 123 Villata, M., et al. 2000, A&A, 363, 108 Wagner, S. & Witzel, A. 1995, ARA&A, 33, 163 Wagner, S. J., et al. 1996, AJ, 111, 2187 Wu, J., et al. 2005, AJ, 129, 1818 Wu, J., et al. 2007, AJ, 133, 1599 Wu, J., et al. 2012, AJ, 143, 108 Yan, H., et al. 2000, PASP, 112, 691 Zhou, X., et al. 2003, A&A, 397, 361 Note. - Table 2 is published in its entirety in the electronic edition of the The Astrophysical Journal Supplement . A portion is shown here for guidance regarding its form and content. Note. - Table 3 is published in its entirety in the electronic edition of the The Astrophysical Journal Supplement . A portion is shown here for guidance regarding its form and content. Note. - Table 4 is published in its entirety in the electronic edition of the The Astrophysical Journal Supplement . A portion is shown here for guidance regarding its form and content. Note. - Table 5 is published in its entirety in the electronic edition of the The Astrophysical Journal Supplement . A portion is shown here for guidance regarding its form and content. Note. - Table 6 is published in its entirety in the electronic edition of the The Astrophysical Journal Supplement . A portion is shown here for guidance regarding its form and content.", "pages": [ 12, 13, 14, 25, 26, 27, 28, 29 ] } ]

[ { "title": "ABSTRACT", "content": "We investigate the S/N of a new incoherent dedispersion algorithm optimized for FPGA-based architectures intended for deployment on ASKAP and other SKA precursors for fast transients surveys. Unlike conventional CPU- and GPUoptimized incoherent dedispersion algorithms, this algorithm has the freedom to maximize the S/N by way of programmable dispersion profiles that enable the inclusion of different numbers of time samples per spectral channel. This allows, for example, more samples to be summed at lower frequencies where intra-channel dispersion smearing is larger, or it could even be used to optimize the dedispersion sum for steep spectrum sources. Our analysis takes into account the intrinsic pulse width, scatter broadening, spectral index and dispersion measure of the signal, and the system's frequency range, spectral and temporal resolution, and number of trial dedispersions. We show that the system achieves better than 80% of the optimal S/N where the temporal resolution and the intra-channel smearing time are smaller than a quarter of the average width of the pulse across the system's frequency band (after including scatter smearing). Coarse temporal resolutions suffer a ∆ t -1 / 2 decay in S/N, and coarse spectral resolutions cause a ∆ ν -1 / 2 decay in S/N, where ∆ t and ∆ ν are the temporal and spectral resolutions of the system, respectively. We show how the system's S/N compares with that of matched filter and boxcar filter detectors. We further present a new algorithm for selecting trial dispersion measures for a survey that maintains a given minimum S/N performance across a range of dispersion measures. Subject headings: methods: observational - surveys - instrumentation: detectors - pulsars: general - radio continuum: general", "pages": [ 1, 2 ] }, { "title": "Performance of a novel fast transients detection system", "content": "Nathan Clarke [email protected] Jean-Pierre Macquart 1 and Cathryn Trott 1 ICRAR/Curtin University, Bentley, WA 6845, Australia", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The dispersive nature of the plasma that pervades interstellar and intergalactic space causes the observed arrival time of impulsive astrophysical radio signals to be strongly frequency dependent. In cold plasmas the dispersive delay is proportional to λ 2 DM, where the dispersion measure, DM, is the line-of-sight electron column density. The effects of dispersion are particularly manifest in searches for pulsars and short-timescale transients at long wavelengths ( λ /greaterorsimilar 0 . 1 m) with sufficient sensitivities to detect objects at large distances. This applies to several current and planned high-sensitivity surveys on next-generation radio telescopes, which are being conducted in the regime in which the effects of interstellar, and potentially intergalactic, dispersion are extreme (e.g. the LOFAR Transients Key Project; Stappers et al. 2011; CRAFT, Macquart 2011; Arecibo PALFA Survey; Cordes et al. 2006; HTRU survey Keith et al. 2010; Burke-Spolaor et al. 2011). The effects of dispersion smearing are in principle fully reversible if the electron column through which the radiation propagated can be determined. However, a number of practical factors prevent complete recovery of the signal to the same strength as an undispersed pulse. For the process of incoherent dedispersion, in which the signal is reconstructed from a filterbank of intensities gridded in time and frequency (Cordes & McLaughlin 2003), there are three primary means by which the S/N is degraded. 1. The finite resolution of the filterbank limits the S/N of the dedispersed signal when there is residual dispersion smearing across the individual filterbank channels (i.e. when the dispersive delay across the bandwidth of the channel exceeds the temporal resolution). 2. Finite computational power limits the number of DM trials that can be searched in a survey, resulting in a loss of sensitivity to events with DMs in between trials. 3. The signal is smeared over a large number of temporal bins, which degrades the signal strength in the presence of system noise (Cordes & McLaughlin 2003). The process of coherent dedispersion (Hankins & Rickett 1975), in which the raw signal voltages recorded from the antenna are convolved with the inverse of the transfer function of the dispersive medium, achieves the optimum S/N recovery of the dispersed signal by eliminating effects 1 and 3. However, for the purposes of conducting blind surveys for oneoff transient events, the data- and compute-intensive nature of coherent dedispersion renders it too slow to be practical with present technology. The technique of incoherent dedispersion offers a viable alternative when processing resources are limited. Incoherent dedispersion is the mainstay of most current pulsar search and transients survey detection algorithms (e.g., Wayth et al. 2011; Ter Veen et al. 2011). A complete understanding of its performance is crucial to understanding the optimal dedispersion strategy when computational resources are finite. For instance, if a real-time detection system can only dedisperse the signal at a fixed number of trial dispersion measures, what is the optimal choice of trial DMs? A related problem is to quantify the effect of a given dedispersion strategy on the completeness statistics of the survey. Though these are old questions, the answers have acquired a renewed urgency because they are needed to inform the design of next generation surveys for impulsive signals (e.g., D'Addario 2010). These questions have been addressed in the past (e.g., Cordes & McLaughlin 2003), but without addressing the degrading effects of implementing boxcar templates as opposed to true matched filters, and only considering a general approach to analysing the effects of temporal and spectral resolution, and DM error. The optimization of blind surveys for pulsars and transients is particularly pressing in the context of SKA time-domain system design, where extreme data rates make offline data storage impractical in many instances, and necessitate real-time processing of the data stream. These factors influence SKA system design and drive backend hardware processing requirements, which can comprise a sizable fraction of the total cost of the instrument. Incoherent dedispersion techniques have been employed for several decades. An early technique, known as the tree algorithm (Taylor 1974), consists of a regular structure of delay and sum elements that transforms an input signal of N frequency channels to N dedispersed output signals, with O ( N log 2 N ) operations. While the tree algorithm is a process-efficient technique and has been popular, particularly in early pulsar surveys, it has some draw-backs that limit its sensitivity: a) it assumes that signal dispersion is linear with frequency, b) the dispersion measures for each of the dedispersed outputs are fixed to linear distributions from 0 (no dispersion) to the DM at which the gradient of the dispersion curve is one temporal bin per spectral channel (thus called the 'Diagonal DM'), and c) each dedispersed output sample is the sum of only one sample from each of the N channels of the dynamic spectrum. Additional processing stages are often employed to mitigate some of these limitations: for example, Manchester et al. (2001) linearize dispersion by inserting artificial ('dummy') channels between the real frequency channels, and then divide the linearized data into smaller groups of adjacent channels, or sub-bands, before dedispersing each sub-band using the tree algorithm; and a broad distribution of trial DMs is achieved by successively summing the data samples in pairs and repeating the dedispersion process. Another algorithm called DART (a Dedisperser of Autocorrelations for Radio Transients) used in the V-FASTR transient detection system for the VLBA (Wayth et al. 2011) arranges samples of the signal's dynamic spectrum into vectors, one vector per frequency channel, with each vector containing a time series of samples of up to several seconds. The vectors are then skewed with delay offsets appropriate to the trial DM, then summed to produce the dedispersed time series for that trial. In many ways the DART algorithm is more flexible than the tree algorithm: It supports an arbitrary number and distribution of trial dispersion measures, and it supports arbitrary dynamic-spectrum dispersion curves, including curves proportional to λ 2 . However, it too sums only one sample from each input channel to produce each dedispersed output sample. A new transients detection system called Tardis is being developed for the Commensal Real-time ASKAP Fast Transients (CRAFT) survey (Macquart et al. 2010). For this system D'Addario (2010) describes a dedisperser that can, for each output sample of a given trial, sum dynamic spectrum samples from multiple temporal bins per spectral channel. Thus, for large DMs where pulse power can be distributed over many temporal bins per spectral channel, additional dynamic spectrum samples can be included in the sum to improve the S/N of the dedispersed output. The Tardis implementation of this system (Clarke et al., in prep.) allows arbitrary sets of dynamic spectrum samples to be selected for the dedispersion sums for each trial. The samples of each set are selected a priori depending on the DM, pulse width and spectral index assumed for the trial. The pulse width can include the signal's intrinsic width and also temporal broadening of the signal due to interstellar scattering. Equal weight is given to all samples in each trial sum. In this paper, we examine the S/N performance of the fast transients detector proposed in D'Addario (2010) and implemented in Tardis, and we present a sample selection algorithm aimed at maximizing the S/N of each dedispersed output signal. While matched filter detectors perform weighted sums of signal samples, with weightings determined by assumed pulse profiles, we show that our new detector yields comparable performance using unweighted sums. In the second part of the paper, we use the new detector to describe how performance is affected by the temporal and spectral resolutions of the system, the magnitude of dispersion and DM error. We use these results as tools with which to decide how to choose the optimal balance of resources for a given system (spectral and temporal resolution, and trial DMs), extending previous work in these areas to form concrete recommendations for system design with dynamic spectrum detectors. In § 2 we define the problem and specify the Tardis dedispersion algorithm mathematically. The S/N reduction associated with finite temporal and spectral resolution is examined in § 3, and in § 4 we examine how the S/N reduces with increasing dispersion measures. In § 5 we compare the performance of the new algorithm with that of time-series and dynamic spectrum matched filters, and the traditional boxcar filter. Then in § 6 we study the residual temporal smearing due to differences between trial DMs and true dispersion measures of signals (i.e. DM errors), how these errors impact the S/N performance, and present a new algorithm for selecting trial DMs to maximize the completeness of fast transients surveys. Our conclusions are outlined in § 7.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2. A dynamic spectrum fast transients detection system", "content": "In this section we examine the S/N performance of the incoherent fast transients detection system outlined in D'Addario (2010) and advance an alternative sample selection algorithm that aims to maximize the S/N performance.", "pages": [ 5 ] }, { "title": "2.1. Dedispersion fundamentals", "content": "Consider a pulse whose intrinsic emitted power per unit bandwidth is of the form, where P 0 has dimensions WHz -1 , α is the spectral index of the pulse and f ( t ) is a dimensionless function that describes the intrinsic pulse profile. P ν is the energy received per unit time per unit bandwidth at a given time t and frequency ν . 1 Interstellar dispersion introduces a delay in the signal arrival time of an amount t d = DM /κν 2 , where the dispersion measure (DM) is the integral of the electron density along the propagation path of the signal, and κ = 2 . 41 × 10 -16 pc.cm -3 .s is a constant (Hankins & Rickett 1975). Furthermore, multipath propagation, or scattering, in the interstellar medium can cause broadening of the temporal width of the signal, and diffractive and refractive scintillation modulations of the signal intensity (Rickett 1990). Scattering is highly dependent on the signal frequency, and on the direction and distance of the source in a manner that strongly correlates with dispersion measure. We model scatter broadening as a convolution in time (denoted by an asterisk) with a general scattering impulse response function, h d ( t ; ν, DM). h d is dimensionless and as temporal smearing due to scattering involves no attenuation in signal power, its area is unity. ( h d approaches the dirac delta function in the limit of no scattering.) Scintillation causes deep (up to 100% of the mean) amplitude modulations in time and frequency. Scintillation time scales are generally too large to be relevant to detecting fast transients. The only instance in which frequency modulation plays an important role is where the decorrelation bandwidth is comparable to the observed bandwidth; larger modulations affect all frequencies within the observed bandwidth equally, and smaller modulations average-out across the band. Optimization of the S/N subject to the effects of scintillation is prohibitive in a computationally limited system, because scintillation is a stochastic process with multitudes of possibilities that compound an already large parameter space. For this reason we choose not to include scintillation in our model. Considering dispersion and temporal smearing due to scattering, our model for the observed power per unit bandwidth is The average power received over temporal and spectral intervals [ t, t +∆ t ] and [ ν, ν +∆ ν ] respectively is In digital systems, the dynamic spectrum of a signal is quantised in frequency and time into discrete samples. If we assume that the time dimension is quantised to a resolution of ∆ t and that frequency is quantised into channels of ∆ ν , then each sample represents the average power within a ∆ t -by-∆ ν cell of the dynamic spectrum, as illustrated in Figure 1. Each sample includes contributions from the signal, i.e. the dispersed pulse, and noise from the sky and the receiver. Thus if sample s represents the average power in the cell [ t s , t s + ∆ t ; ν s , ν s +∆ ν ], then sample s would have a value of ¯ P ( t s , ν s ) + ¯ P N ( t s , ν s ), where the former term is the average power of the pulse within the cell (as modeled in eq. (3)), and the latter term is the average noise power within the cell. The system described by D'Addario (2010) involves summing selected samples of the dynamic spectrum, where samples are selected based on their relative time t and frequency ν , and on the dispersion measure, pulse width and spectral index assumed for the trial. We will consider how to select the samples in the next section. For now, assume that S is the set of samples selected to dedisperse the signal for a given trial. The pulse component (ignoring noise) of the time series output of the dedisperser for that trial can be modeled as For the purposes of detecting astronomical pulses, we aim to maximize the dedispersed signal power relative to statistical variations in the noise power. Our figure of merit is therefore the signal-to-noise ratio (S/N) calculated as a ratio of P dedisp to the noise error (i.e. the standard deviation of the noise). The uncertainty principle implies that the product of the temporal and spectral resolution cannot be less than unity, and in this paper we assume that ∆ ν ∆ t /greatermuch 1 such that the central limit theorem holds and the noise contribution to each sample can be assumed to be normally distributed. To simplify our analysis, we ignore self-noise generated from the signal; self-noise is typically small compared with sky and receiver noise. Using the radiometer equation, the noise error in a cell of bandwidth ∆ ν and interval ∆ t can be modeled as where k is Boltzmann's constant and T sys is the system equivalent noise temperature. Generally, T sys is a frequency dependent parameter that represents the overall noise temperature of the system, including natural radio emissions from the sky and gain fluctuations in the receiver electronics; however, variations in system temperature are often relatively small across the operating bandwidth of the receiver, and for the purposes of the analyses in this paper we assume that T sys is constant with frequency. Since the noise is normally distributed, the average total noise power after summing the samples for a given trial is where N S is the number of samples in set S . The dedispersion process therefore produces a S/N ratio given by:", "pages": [ 5, 6, 7, 8 ] }, { "title": "2.2. Sample selection for maximum Signal-to-Noise Ratio (S/N)", "content": "Assume that we have a set of samples S for dedispersing our signal, and consider the possibility of adding another sample, ς , to our set. If we were to include this sample, then the new dedispersed signal power would be: and the new dedispersed noise error would be: The ratio of the new S/N to the old would then be: /negationslash On average, we improve the overall S/N by adding sample ς to our sum when eq. (10) is greater than unity. That is, when: /negationslash Eq. (11) provides a criterion for adding a new sample ( ς ) to an existing set of samples ( S ) used in the dedispersion sum for a given trial. We use this criterion to select, a priori, sets of dynamic spectrum samples to be summed by the dedisperser for each trial. The ¯ P ( t, ν ) terms, on both sides of the relation, are predicted using eq. (3) and the DM, pulse width and spectral index parameters targeted for the trial. The discrete time offset, n , controls the time at which a dedispersed pulse will appear at the output of the dedisperser relative to the time that the corresponding dispersed pulse arrives at its input, and is therefore chosen to minimize the dedispersion latency and the amount of physical storage required within the dedisperser. The set of samples that maximizes the S/N may not be unique. To achieve the maximum S/N with the fewest samples, we recommend the following procedure: Beginning with an empty set, include a sample that has the highest average signal power (as predicted using eq. (3)), then add successive samples in order of highest average signal power until eq. (11) is no longer satisfied.", "pages": [ 8, 9 ] }, { "title": "3. S/N variation with temporal and spectral resolution", "content": "In this section we look at how the signal-to-noise ratio performance of our fast transients detection system varies with temporal and spectral resolution. We show that systems employing finer resolutions generally achieve better S/N performance than systems employing coarser resolutions, but there is a sweet spot beyond which finer resolutions yield smaller S/N gains. To simplify the analysis we assume that the scatter broadened pulse has a rectangular profile and that rather than using the procedure described in § 2.2 to select samples, set S includes any sample that includes a non-zero component of signal power. That is, S includes any sample, s , for which ¯ P ( t s + n ∆ t, ν s ) > 0. The signal power for each sample is predicted using eq. (3), the DM, intrinsic pulse width, scatter broadening and spectral index parameters targeted for the trial, and an arbitrary discrete time offset, n = n 0 . We have already shown that the S/N can be improved by excluding some samples with small, non-zero amounts of signal power, so the following analysis will use a less than optimal value for the S/N, but this is fine for the purposes of exploring the effects of resolution on the S/N and later in this section we will see how the more rigorous sample selection algorithm improves the S/N. Figure 2 illustrates the profile of the dispersed, rectangular pulse defining the samples of set S . Here we define t A c and t B c to represent the earliest and latest times at which the pulse appears in channel c , respectively. If we define ν c to be the highest frequency within channel c , then we have t A c = DM /κν 2 c , and t B c ≈ t A c + τ ' ( ν c ; DM) + ∆ τ c . Note that the approximation for t B c assumes that τ ' ( ν c ; DM) is approximately constant across the frequency band for channel c , which becomes less accurate with coarser spectral resolutions. 2 The ∆ τ c term is the dispersion smearing time of the signal across channel c , which can be approximated as ∆ τ c ≈ 2 DM∆ ν/κν 3 c . Thus, if all samples containing non-zero signal power are included in S , then the total number of samples in S is If a dispersed pulse matching our prescribed profile is input to this dedisperser, then, by design, the peak S/N will occur at n = n 0 and will be A derivation for eq. (13) is given in Appendix A. It can be shown that for large numbers of frequency channels, i.e. C /greatermuch 1, the term in brackets in eq. (13) converges to a constant, and for the remainder of this analysis we assume that this term has little influence on how the peak S/N varies with spectral resolution. However, note that this assumption does not hold for spectrally steep signals at coarse spectral resolutions, because under these conditions the bracketed term can vary significantly with C . For brevity we define which is the maximum S/N that the system would achieve if it had infinitely fine resolution and if there were no scatter broadening. With scatter broadening, the maximum S/N attenuates with ( τ/τ ' av ) 1 / 2 , where τ ' av is the average scatter broadened width of the pulse across the system bandwidth, β . Thus, we define the maximum scatter broadened S/N as We also define the nominal spectral resolution which is the spectral resolution at which the average intra-channel smearing time (i.e. the average of ∆ τ c over all channels) equals the average scatter broadened pulse width, τ ' av . Note that the term in brackets in eq. (16) is a function of the number of spectral channels, C , and converges to a constant for large C . With these definitions, we now examine the effects of spectral and temporal resolution on the peak S/N using the approximations in equations (12) and (13). We consider nine cases, one for each combination of 'fine', 'nominal' and 'coarse' resolution in frequency and time, and for each case the reductions of equations (12) and (13) are captured in Table 1. In Table 1 we show that the peak S/N converges to the optimal value, SNR ' 0 , at fine temporal and spectral resolutions; drops down to SNR ' 0 / 2 at nominal resolutions; then decays proportional to ∆ t -1 / 2 and ∆ ν -1 / 2 at coarse resolutions. Note that while the condition for nominal spectral resolution is the same across all temporal resolutions, the conditions for nominal temporal resolution vary with the spectral resolution, from ∆ t ≈ τ ' av at fine spectral resolutions, ∆ t ≈ 2 τ ' av at nominal spectral resolutions, to ∆ t ≈ τ ' av ∆ ν/ ∆ ν 0 at coarse spectral resolutions. These conditions are consistent in that each represents the temporal resolution at which the peak S/N is 1 / √ 2 of the value it would be at an infinitesimally fine temporal resolution. In the special case where both the temporal resolution and the average intra-channel smearing are limited to some arbitrary multiple of the pulse width, ∆ t = Av.[∆ τ c ] = mτ ' av , i.e. ∆ t/τ ' av = ∆ ν/ ∆ ν 0 = m , we have For example, for rectangular pulses, S/Ns greater than 82% of optimal (0 . 82 SNR ' 0 ) can be achieved when the temporal resolution and intra-channel smearing are less than a quarter of the averaged scatter broadened pulse width ( m< 0 . 25). √ So far we have used a simple and intuitive means of estimating the S/N. The surface plot in Figure 3 justifies this by showing how the peak S/N varies with the spectral and temporal resolution of our system when employing the sample selection algorithm described in § 2.2. One sees that the curve conforms with the characteristics described in the above analysis. The sample selection algorithm slightly improves the S/N predicted in the above analysis, and in this case the relative improvement in S/N is at a maximum of ∼ 20% at the nominal resolution point. Table 1 and Figure 3 show that significant improvements in S/N performance can be realized by increasing the resolution of a system from coarse to nominal regimes, but also that progressively less S/N improvement can be achieved as the resolution increases beyond nominal. Finer resolutions generally come at the cost of higher data volumes and faster processing, and these costs need to be weighed against the improvement in performance and overall science goals of the survey.", "pages": [ 9, 10, 11, 12 ] }, { "title": "4. S/N variation with dispersion measure", "content": "The S/N performance of an incoherent dedispersion system also depends on the dispersion measure of the trial. This dependence can be seen from the relations in Table 1. By rearranging the relations in the left-hand column of Table 1, the table becomes a description of the S/N for nine regions of DM vs ∆ t space. Thus, for a given spectral resolution, the DM dimension is divided into 'low' (DM /lessmuch DM τ ), 'nominal' (DM ≈ DM τ ) and 'high' (DM /greatermuch DM τ ) dispersion measures, where Similarly, by replacing the normalized ∆ ν axis with a normalized DM axis (DM/DM τ ), the surface plot in Figure 3 serves to illustrate the variation in S/N with dispersion measure. DM τ is the dispersion measure at which the average intra-channel smearing time equals the average width of the scatter broadened pulse. At DMs larger than DM τ , intra-channel smearing losses start to become significant, causing the S/N to decay proportional to DM -1 / 2 . Intra-channel smearing loss is a common problem for all incoherent dedispersion systems (Hankins & Rickett 1975) and such systems often require finer spectral resolutions in order to target larger dispersion measures without significant loss in S/N.", "pages": [ 12, 13, 14 ] }, { "title": "5. Comparison with matched filters", "content": "We now compare the performance of the Tardis dedisperser detector to the performance of (i) detectors commonly employed for fast transient and pulsar detection, and (ii) theoretically optimal detectors. The Tardis dedisperser detector operates on power samples in time and frequency space (the dynamic spectrum), whereas conventional fast transient detectors typically operate on time series data, obtained by averaging the dedispersed dynamic spectrum dataset over spectral channels (e.g., Wayth et al. 2011; Deneva et al. 2009). We consider detectors operating in both the dynamic spectrum (time and frequency samples) and temporal (time samples alone) domains. The matched filter detector (MF) is the optimal linear detector for data with generalized Gaussian noise, and operates on the dataset with a replica of the signal profile. When the signal profile is unknown, approximate templates can be evaluated to optimize detection performance. The major disadvantage of the MF is the need for knowledge of the pulse profile. One method of avoiding this issue is to consider a simple boxcar template (specifically, a unit height rectangular pulse with variable width, typically binned geometrically): this detector will have sub-optimal performance compared with the MF for signals that are not rectangular and/or of differing temporal width compared with the template. The boxcar template has been employed in recent fast transients experiments (e.g., Wayth et al. 2011; Deneva et al. 2009). In addition to comparing the Tardis dedisperser detector with the time series boxcar detector, we also wish to compare its performance with optimal detectors: the matched filter applied in both the temporal (time series dataset), and dynamic spectrum (time and frequency dataset) domains. The former represents the best-case detector when data are averaged over spectral channels, and the latter represents the best performance achievable with dispersed dynamic spectrum data. Throughout we consider white Gaussian noise, although this is not necessary for the method (knowledge of the noise properties is required, however), and assume that the pulse timing is optimal (pulse arrival aligns with the beginning of a sample, and is known: in general, this is determined empirically). We also omit any contribution to the noise power from signal self-noise, because this is typically small compared with the power contribution from the sky and receiver. This simplification allows us to treat yielding, the noise as an additive quantity. We now use the first and second order statistics of the detection test statistic (Kay 1998) to derive expressions for the signal-to-noise ratios for each of these detectors, and discuss the key differences and similarities between them. Time series matched filter (MF): Noting that the matched filter multiplies the data by a replica of the expected pulse profile (yielding a square in the following expression), the signal power and noise error are, respectively: where N t denotes the number of temporal samples, and C is the number of spectral channels. The number of samples in the MF denominator is ∼ √ N t C /greaterorsimilar √ N S , because the Tardis detector does not have to use all of the spectral channels. This can lead to the MF incorporating more noise power than is optimal if one had the full dynamic dataset (the MF presented here is optimal in the time-series domain). Therefore, the performance of the two detectors depends on the nature of the signal being detected, and consequently, on the spectral and temporal resolution of the experiment.", "pages": [ 14, 15 ] }, { "title": "Dynamic spectrum matched filter (DSMF):", "content": "For this detector, all included samples are summed in quadrature. This detector weights each sample according to the expected signal strength, reducing the effective noise contribution to the test statistic. The obvious drawback to implementation is the requirement for full knowledge of the pulse shape. Time series boxcar detector (Box.): /negationslash This expression is similar to that for the Tardis detector, with the major difference that it is forced to include all of the noise power over the spectral channels. For finite temporal and spectral resolution, and DM =0, N S < C N t , and the Tardis detector will always yield improved performance compared with the boxcar MF. Note that this also considers boxcar templates that are optimally matched to the actual signal pulse width: in the general case, when the pulse width is unknown, the boxcar template will not be matched, and the performance will be further degraded. For a perfectly dedispersed pulse (where N S = C N t and ¯ P ( t, ν ) = ¯ P ), it is straightforward to show that all detectors yield the same S/N. It is obvious from these expressions that the Tardis dedisperser detector is a width-optimized boxcar detector in the dynamic spectrum domain (the sample inclusion/exclusion criterion provides the width optimization). Figure 4 displays the detection performance for each detector as a function of the normalized spectral resolution, ∆ ν/ ∆ ν 0 . The two matched filter detectors perform well, with the DSMF performing the best across the range tested, as expected. The Tardis dedisperser detector performs well relative to the time series matched filter. At low ∆ ν/ ∆ ν 0 (high resolution), the time series matched filter performs better. This reflects the Tardis detector's binary choice for either including or excluding samples: while excluding a sample may retain a higher S/N, signal power is nonetheless excluded (rather than being optimally-weighted, as for the time series matched filter). At very poor resolution, the two curves cross: the dedispersed signal is substantially broadened at low resolution, and the additional noise power incorporated into the time series matched filter degrades its performance. The time series boxcar detector has variable performance, depending on how wellmatched the coarse temporal bins are to the underlying signal. We have chosen a single possible realization of its performance. Its performance matches that of the others at high resolution, when the pulse width is matched perfectly to a tested bin width, and the start of the pulse aligns with the start of the bin. The matched filters, for both the time series and dynamic spectrum domains, demonstrate superior performance compared with the Tardis detector, for a wide range of system and signal parameters. Matched filters are, however, difficult to implement in practise, given the need for full knowledge of the signal profile - a major obstacle for fast transients surveys. On the other hand, boxcar filters, implemented in either the time-series or dynamic spectrum domains, are blind to pulse shape, and suffer performance degradation accordingly (note that the Tardis dedisperser detector is further superior to some time-series implementations [e.g., VFASTR], because it does not use a base-2 discretized temporal binning to produce the trial templates). The Tardis dedisperser detector presented here attempts to balance the performance/signal knowledge trade-off, by exploiting the performance advantages of working in the dynamic spectrum domain and with a sample-selection criterion, to offset performance loss due to lack of pulse shape knowledge. In addition, the Tardis dedisperser detector is computationally efficient to implement, requiring only summing of samples (compared with matched filters, which perform weight and sum operations).", "pages": [ 15, 16, 17 ] }, { "title": "6. Survey completeness", "content": "In our analyses so far we have assumed that the dispersion measure of the received signal is known. However, dispersion measures vary with distance from the source and the content of the intervening ISM along the line-of-sight to the source, and when surveying the sky for new sources, the dispersion measure applicable to each received transient is generally unknown. It is therefore necessary for the system to dedisperse the signal using a range of trial dispersion measures and search each dedispersed signal for transient content. Real-time dedispersion and detection processes are compute-intensive, and the computation power increases linearly with the number of trial dispersion measures. As we will see, the number of trials and the distribution of those trials across the range of dispersion measures targeted by the survey are critical design choices; they determine the completeness of the survey in terms of the average S/N performance. In this section we describe how a set of trial dispersion measures can be chosen to maximize the completeness of a fast transients survey.", "pages": [ 18 ] }, { "title": "6.1. Pulse broadening due to DM error", "content": "We begin by illustrating how differences between the dispersion measure assumed for a given trial and the actual dispersion measure of an observed pulse can cause the resulting dedispersed signal to be broadened in time. Temporal broadening due to DM error is a well documented effect (Burns & Clark 1969; Cordes & McLaughlin 2003; D'Addario 2010) and for completeness we review this in the context of the models presented in this paper. /negationslash Assume that the set of samples, S , is chosen to maximize the dedispersed S/N for signals that have a dispersion measure equal to the trial DM, DM trial , and assume that we attempt to dedisperse a signal whose actual dispersion measure, ̂ DM, differs from the trial DM, i.e. ̂ DM = DM trial . The dedispersed S/N would be: The ratio of eq. (7) and eq. (24) gives the 'relative' S/N for a signal whose DM does not equal the trial DM: Using the sample selection criterion outlined in § 2.2 for determining S , and the relation for the relative S/N in eq. (25), the profiles for a series of test pulses, each with differing dispersion measures, are plotted in Figure 5 for a trial DM of 30 pc.cm -3 . These examples are calculated for signals received in the 700 MHz to 1004 MHz frequency band, with 1 MHz channel resolution and square-law detected samples integrated to a temporal resolution of 1 ms. Each dispersed pulse input to the dedisperser is modeled using eq. (3) with a rectangular scatter broadened pulse profile of width 1 ms. The plots show how pulses become increasingly smeared as the differences between the trial and actual DMs increase. The visible asymmetries in the dedispersed pulses, more notable for those with larger absolute DM errors, are a consequence of the natural ν -2 bend in the dispersion curve. If the actual DM of the pulse is less than that of the trial, then the trial will initially intersect the dispersed pulse in the low frequency channels, and the point of intersection will progress to the higher frequency channels as the trial sweeps past the dispersed pulse. Since both the trial and the pulse are more dispersed at lower frequencies, the leading edge of the resulting dedispersed pulse is more extended than its trailing edge. The converse occurs for pulses with DMs larger than that of the trial: as the trial sweeps past the dispersed pulse, the point of intersection moves from high to low frequency channels, causing the trailing edge of the resulting dedispersed pulse to be more extended than its leading edge.", "pages": [ 18, 19, 20 ] }, { "title": "6.2. S/N variation with DM error", "content": "The temporal broadening of a pulse due to the difference between its true dispersion measure and a given trial DM (i.e. the DM error) reduces the S/N of the dedispersed signal for that trial. Cordes & McLaughlin (2003) shows that, in general, temporal broadening reduces the S/N according to where W i and SNR i are the temporal width and S/N of the 'incident' pulse (i.e. before the pulse is broadened), and W b and SNR b are the temporal width and S/N of the broadened pulse. If we consider W i to be the width of the pulse after it has been dedispersed to a perfectly matched trial DM, such that there is no DM error, then W i can be approximated using and if W b is the width of the pulse after it has been dedispersed to an un-matched trial DM, with a DM error of δ DM, then where ∆ t DM residual is the component of the pulse width due to residual dispersion smearing (that which cannot be corrected for by dedispersion); ∆ t δ DM is the component due to the DM error, δ DM; and as defined earlier, ∆ t and τ ' av are the temporal resolution of our dedispersion system and the average scatter broadened width of the pulse, respectively. For coherent dedispersion systems, the residual smearing after dedispersion is essentially zero (i.e. ∆ t DM residual = 0). However, incoherent dedispersion systems can only remove interchannel smearing; the residual (intra-)channel smearing can be approximated as The component of smearing due to DM error, ∆ t δ DM , is equivalent to the smearing of a signal with a dispersion measure equal to δ DM across the entire frequency band, β . This smearing can likewise be approximated as By substituting these approximations back into eq. (26) it follows that where DM diag is known as the 'diagonal DM', i.e. the DM at which the average smearing time across each channel equals the temporal resolution; and DM τ is the DM at which the average smearing time across each channel equals the average width of the scatter broadened pulse. Using the approximation given in eq. (31), Figure 6 illustrates how the S/N attenuates as the DM error increases. The DM error is normalised to a value of 1 C √ DM 2 +DM 2 diag +DM 2 τ , which implies that for sufficiently small dispersion measures, DM /lessmuch DM diag , or DM /lessmuch DM τ , the S/N attenuation for a given DM error is independent of the dispersion measure; whereas for sufficiently large dispersion measures, DM /greatermuch DM diag and DM /greatermuch DM τ , the S/N attenuation for a given DM error is expected to be less for larger dispersion measures. With greater scatter broadening, the normalisation value increases, which means that although the overall S/N (SNR i ) reduces with scatter broadening, DM errors cause less attenuation in the relative S/N.", "pages": [ 20, 21 ] }, { "title": "6.3. Choosing trial dispersion measures", "content": "To maximize the average S/N performance of our detection system across all signals within the DM range of our survey, we aim to choose a set of trial DMs that maintains a limited S/N attenuation between trials. We do so by constraining the relative S/N to some minimum constant value where /epsilon1 is referred to as the DM error factor. It can be shown that this constraint is equivalent to limiting the temporal broadening due to DM error to ∆ t δ DM < /epsilon1 W i . By substituting the approximation for the relative S/N given in eq. (31) into eq. (33) we can show that the DM error needs to be constrained to Therefore, given some arbitrary limit to the reduction in S/N that we are prepared to accept between trial DMs (i.e. ( /epsilon1 2 +1) -1 / 4 ), and noting that we can space our trial DMs at intervals of 2 δ DM, we can choose a set of trial dispersion measures that adhere to this limit as follows: where DM i , ∀ i ∈ [0 , 1 , ..., N -1], are the dispersion measures chosen for our set of N trials, with each successive subscript denoting a successively larger dispersion measure. DM 0 can be set to the minimum dispersion measure in the range to be searched, and each successive trial DM can be calculated from the trial DMs preceeding it using eq. (35). It follows from eq. (35) that where the trial DMs are small, i.e. where DM n /lessmuch DM diag and DM n /lessmuch DM τ , and where scatter broadening is either insignificant or independent of the DM, the DM error ( δ DM) is approximately constant and the trial DMs are approximately uniformly (linearly) spaced, i.e. But where the trial DMs become dominant (i.e. DM j /greatermuch DM diag and DM j /greatermuch DM τ , for some j < n ), the trial DMs become approximately exponential with n , i.e. For coherent dedispersion systems, since there is no channelization and consequently no residual intra-channel smearing, the DM i terms disappear from the right-hand side of eq. (35) and the trial DMs follow a linear spacing where scatter broadening is small or constant with DM, and become more spread-out at higher DMs where scatter broadening becomes significant. Therefore generally more trial DMs are needed for coherent dedispersion systems than for incoherent dedispersion systems. Figure 7 demonstrates how the choice of trial DMs can impact the S/N performance for the Tardis fast transients detection system planned for the CRAFT survey. CRAFT aims to survey the sky for milli-second-scale transients from both galactic and extra-galactic sources by making use of ASKAP's wide (30 degree 2 ) field of view. Given the high luminosities of recently detected extra-galactic fast transients (e.g., Lorimer et al. 2007; Keane et al. 2011), it is reasonable to expect that Tardis may detect sources to redshifts of z /lessorsimilar 3, implying IGM dominated dispersion measures up to ∼ 3000 pc.cm -3 (Inoue 2004). Thus a range from 10 to 3000 pc.cm -3 is targeted for CRAFT, which the Tardis system intends to cover with 442 trials. The plot in Figure 7(a) shows the S/N performance where those 442 trials are distributed using the relation given in eq. (35). The S/N consists of a series of finely spaced peaks and troughs, where at the peaks, the true dispersion measure of the pulse matches a trial DM, and at the troughs, the true dispersion measure falls in the middle of two adjacent trial DMs. Note the relative drop-out, i.e. the ratio of the S/N of a trough to the S/N of its adjacent peaks, is constant across the full range of dispersion measures. The DM error factor in this case is 1.492, giving a S/N drop-out between trial DMs of about 0.75 relative to surrounding peaks. Also note the general DM -1 / 2 attenuation in S/N discussed above in § 4. We compare the plot in Figure 7(a) with the plot in Figure 7(b) where the same number of trials are exponentially distributed across the DM range. Here we see that the exponential distribution packs trials unnecessarily tightly at low dispersion measures, leaving fewer trials available for higher DMs and overall poorer average performance across the entire range. It is well known that dedispersion systems with larger numbers of channels (i.e. finer spectral resolutions) require more trial DMs to achieve the same S/N drop-out between trials (Hankins & Rickett 1975), and this can be seen from the dependence on C in eq. (35). Essentially, systems with coarser spectral resolutions suffer more significant intra-channel smearing, making them less sensitive to DM error than systems with finer spectral resolutions. This is demonstrated in Figure 8 for a putative high radio frequency (21 to 23 GHz) survey for milli-second pulsars at the Galactic Center where dispersion measures as high as ∼ 2000-5000 pc.cm -3 can be expected. For the purposes of this example we consider DMs in the range 50 to 10,000 pc.cm -3 . Two possibilities are plotted: In (a), the band is divided into 256 channels of 7.8125 MHz, and in (b), the band is divided into 32 channels of 62.5 MHz. Each target the same number of trial DMs (128), but to do so, the finer spectral resolution example must suffer a higher DM error factor. This can be seen in the plots as slightly larger drop-outs between trial DMs: In (a), troughs are 93% of the peaks; while in (b), troughs are 98% of the peaks. The underlying cause of this is that at high DMs the system with coarser spectral resolution (b) suffers from significant intra-channel smearing, making it less sensitive to DM error than the finer spectral resolution system. Consequently, for the plot in (b), the trial DMs can be more spread-out at high DMs, and more compact at low DMs, resulting in shallower troughs between trial DMs. On the other hand, the finer spectral resolution system suffers less S/N degradation due to intra-channel smearing and is therefore able to maintain higher overall S/N performance at high dispersion measures. In terms of survey completeness, the finer spectral resolution example is preferable since it has a higher average S/N over the DM range.", "pages": [ 21, 22, 23, 24, 25 ] }, { "title": "7. Conclusions", "content": "In this paper we have examined the signal-to-noise performance of a new incoherent dedispersion algorithm that improves on the performance of traditional algorithms by supporting multiple temporal bins per spectral channel in the sum that forms the dedispersed time series for a given trial. The algorithm has the freedom to include (or exclude) any sample of the dynamic spectrum in its dedispersion sum, thus providing a crude mechanism for matching the profile of a pulse without the computational expense of weighting each sample. Even without sample weights, the new algorithm displays comparable S/N performance to the ideal matched filter (both time-domain and dynamic spectrum matched filters) and improved performance over traditional time-series boxcar filters. Critical parameters affecting S/N performance include the system temperature, frequency range, and spectral and temporal resolutions of the system, the ranges of pulse widths and dispersion measures targeted for the survey, and the number and distribution of trial dispersion measures across the DM range. Given an assumed pulse profile and dispersion measure for a trial, application of the sample selection criterion presented in this paper ensures that the S/N of the dedispersed time series is optimized with a minimal number of samples. The paper has demonstrated that significant improvements in S/N performance can be achieved for moderate increases in resolution when both the temporal resolution and the average intra-channel smearing time are approximately equal to the target pulse width. Progressively less S/N improvement can be achieved as the resolution increases beyond this nominal resolution point, and at coarser resolutions the S/N diminishes with ∆ t -1 / 2 and ∆ ν -1 / 2 . Once a suitable system resolution has been identified, the number and distribution of trial dispersion measures can be determined, and the paper has presented a new trial DM selection algorithm designed to maintain a predefined minimum relative S/N performance across the targeted range of DMs. The authors are grateful to Peter Hall, Larry D'Addario and Stephen Ord for their many and various comments and suggestions. The Centre for All-sky Astrophysics is an Australian Research Council Centre of Excellence, funded by grant CE110001020. The International Centre for Radio Astronomy Research (ICRAR) is a Joint Venture between Curtin University and the University of Western Australia, funded by the State Government of Western Australia and the Joint Venture partners.", "pages": [ 26 ] }, { "title": "A. Peak dedispersed S/N for rectangular pulses", "content": "We derive an expression for the peak dedispersed signal-to-noise ratio for pulses that have rectangular scatter broadened pulse profiles: where f is the intrinsic pulse profile, h d is the impulse response function for scatter broadening, H( t ) is the Heaviside function, τ is the intrinsic width of the pulse, and τ ' ( ν ; DM) is the pulse width after scatter broadening. The fractional term on the right of eq. (A1) accounts for proportional attenuation of the pulse intensity with scatter broadening. For this profile, the average power of the dispersed pulse in the dynamic spectrum bounded by the temporal and spectral limits [ t, t +∆ t ] and [ ν, ν +∆ ν ] is Substituting eq. (A2) into eq. (7) gives an expression for the S/N of the n th sample of the dedispersed time series: where S is the set of N S dynamic spectrum samples chosen for the dedispersion sum. If at n = n 0 the dedisperser receives a pulse whose profile precisely matches the assumed scatter broadened pulse profile modelled in eq. (A1), then the samples of set S will collectively include all of the pulse power and the sum on the right of eq. (A3) will equate to the scatter broadened pulse width, τ ' , leaving Note that the term in brackets converges to a constant for increasing numbers of channels ( C ), and for suitably small channel bandwidths (∆ ν ), can be approximated by", "pages": [ 27 ] }, { "title": "REFERENCES", "content": "Burke-Spolaor, S., Bailes, M., Johnston, S., et al. 2011, MNRAS, 416, 2465 Burns, W. R., & Clark, B. G. 1969, A&A, 2, 280 Cordes, J., & McLaughlin, M. 2003, ApJ, 596, 1142 Cordes, J., Freire, P., Lorimer, D., et al. 2006, ApJ, 637, 446 D'Addario, L. 2010, Searching for Dispersed Transient Pulses with ASKAP, SKA Memo 124 Deneva, J. S., Cordes, J. M., McLaughlin, M. A., et al. 2009, ApJ, 703, 2259 Hankins, T. H., & Rickett, B. J. 1975, in Methods in Computational Physics, ed. B. Alder, S. Fernbach, & M. Rotenberg, Vol. 14, 55-129 Inoue, S. 2004, MNRAS, 348, 999 Kay, S. 1998, Fundamentals of statistical signal processing: detection theory (Prentice-Hall) Keane, E. F., Kramer, M., Lyne, A. G., Stappers, B. W., & McLaughlin, M. A. 2011, MNRAS, 415, 3065 Keith, M., Jameson, A., van Straten, W., et al. 2010, MNRAS, 409, 619 Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777 Macquart, J., Bailes, M., Bhat, N. D. R., et al. 2010, PASA, 27, 272 Macquart, J.-P. 2011, ApJ, 734, 20 Manchester, R., Lyne, A., Camilo, F., et al. 2001, MNRAS, 328, 17 Rickett, B. J. 1990, ARA&A, 28, 561 Stappers, B. W., Hessels, J. W. T., Alexov, A., et al. 2011, A&A, 530, A80 Taylor, J. H. 1974, A&AS, 15, 367 Ter Veen, S., Falcke, H., Fender, R., et al. 2011, in American Institute of Physics Conference Series, Vol. 1357, American Institute of Physics Conference Series, ed. M.Burgay, N.D'Amico, P.Esposito, A.Pellizzoni, & A.Possenti , 331-334 Wayth, R., Brisken, W., Deller, A., et al. 2011, ApJ, 735, 97", "pages": [ 28 ] } ]

[ { "title": "ABSTRACT", "content": "A significant fraction ( ∼ 30%) of the high-energy gamma-ray sources listed in the second Fermi LAT catalog (2FGL) are still of unknown origin, being not yet associated with counterparts at low energies. We recently developed a new association method to identify if there is a γ -ray blazar candidate within the positional uncertainty region of a generic 2FGL source. This method is entirely based on the discovery that blazars have distinct infrared colors with respect to other extragalactic sources found thanks, to the Wide-field Infrared Survey Explorer ( WISE ) all-sky observations. Several improvements have been also performed to increase the efficiency of our method in recognizing γ -ray blazar candidates. In this paper we applied our method to two different samples, the first constituted by the unidentified γ -ray sources (UGSs) while the second by the active galaxies of uncertain type (AGUs), both listed in the 2FGL. We present a catalog of IR counterparts for ∼ 20% of the UGSs investigated. Then, we also compare our results on the associated sources with those present in literature. In addition, we illustrate the extensive archival research carried out to identify the radio, infrared, optical and X-ray counterparts of the WISE selected, γ -ray blazar candidates. Finally, we discuss the future developments of our method based on ground-based follow-up observations. 10", "pages": [ 1, 2 ] }, { "title": "1. Introduction", "content": "Unveiling the nature of the Unidentified Gamma-ray Sources (UGSs) (e.g., Abdo et al. 2009) 11 is one of the biggest challenges in contemporary gamma-ray astronomy. Since the era of the 12 Compton Gamma-ray Observatory many γ -ray objects have not been conclusively associ13 ated with counterparts at other frequencies (Hartman et al. 1999), although various classes 14 have been investigated to understand whether they are likely to be detected at γ -ray energies 15 or not (e.g., Thompson 2008). 16 tifrequency Catalogue of Blazars (ROMA-BZCAT, Massaro et al. 2009; Massaro et al. 2010; 38 Massaro et al. 2011a). 39 radio, infrared, optical and X-ray frequencies to characterize the multifrequency behavior 69 of the γ -ray blazar candidates. We then compare our results on the associated sources 70 with those based on statistical methods developed by Ackermann et al. (2012) in Section 6. 71 Finally, Section 7 is devoted to our conclusions. 72 The most frequent acronyms used in the paper are listed in Table 1. 73 74 75", "pages": [ 2, 3, 4 ] }, { "title": "2.1. The unidentified gamma-ray sources", "content": "Our primary sample of UGSs consists of all the sources for which no counterpart was as76 signed at low energies in the 2FGL or in the 2LAC (Nolan et al. 2012; Ackermann et al. 2011, 77 respectively), for a total of 590 γ -ray objects. 78 We considered and analyzed independently two subsamples of UGSs, distinguishing the 79 299 Fermi sources without any γ -ray analysis flags from the other 291 objects that have a 80 warning in their γ -ray detection. This distinction has been performed because future releases 81 of the Fermi catalogs based on improvements of the Fermi response matrices and revised 82 analyses, could make their detection more reliable, as occurred for a handful of sources 83 flagged in the first Fermi LAT catalog (1FGL, Abdo et al. 2010; Nolan et al. 2012). 84", "pages": [ 4 ] }, { "title": "2.2. The active galaxies of uncertain type", "content": "According to the definition of the 2LAC and 2FGL catalogs, active galaxies of uncertain 86 type (AGUs) are γ -ray emitting sources with at least one of the following criteria: 87 91 92 93 94 95 96 The number of AGUs in the 2FGL, that have been analyzed is 210; excluding γ -ray 97 sources with analysis flags (defined according to both the 2FGL or the 2LAC descriptions). 98 99", "pages": [ 4, 5 ] }, { "title": "3. The Association Procedure", "content": "The complete description of our association procedure together with the estimates of its 100 efficiency and its completeness can be found in D'Abrusco et al. (2013) where we discuss a 101 new and improved version of the association method based on a 3-dimensional parametriza102 tion of the locus occupied by γ -ray emitting blazars with WISE counterparts. Here we 103 provide only an overview. We note that the results of the improved method are in agree104 ment with those of the previous parametrization, thus superseding the previous procedure 105 (Massaro et al. 2011b; Massaro et al. 2012a; Massaro et al. 2012b). 106 The new association procedure was built to improve the efficiency of recognizing γ -ray 107 blazar candidates, to decrease the number of possible contaminants and, at the same time, 108 to determine if a selected γ -ray blazar counterpart is more likely to be a BZB or a BZQ. The 109 main differences between the two association methods reside in the parameter space where 110 the locus has been defined (IR color space for the old version and principal component space 111 for the new one) and in the assignment criteria of the classes for the γ -ray blazar candidates 112 (see D'Abrusco et al. 2013). The new method also takes into account of the correction for 113 Galactic extinction for all the WISE magnitudes 3 according to the Draine (2003) relation. 114 As shown in D'Abrusco et al. (2013), this correction affects only marginally the [3.4]-[4.6] 115 color, in particular at low Galactic latitudes (i.e., | b | < 15 deg). 116 The principal component analysis is designed to reduce the dimensionality of a dataset 117 consisting of usually large number of correlated variables while retaining as much as possible 118 of the variance present in the data in the smallest possible number of orthogonal parame119 ters. This is achieved by transforming the observed parameter into a new set of variables, the 120 principal components. They are ordered so that the first accounts the largest possible vari121 ance of the original dataset and the others in turn have the highest variance possible under 122 the constraint of being orthogonal to the preceding ones (e.g., Pearson 1901; Jolliffe 2002). 123 Thus our new parametrization of the locus in the PC space, where the maximum variance 124 is contained along only one axis, is simpler than any other possible representation in the IR 125 color space. 126 For each γ -ray source we defined a search region : a circular region of radius θ 95 equal 127 to the semi-major axis of the ellipse corresponding to the positional uncertainty region of 128 the Fermi source at 95% level of confidence and centered at the 2FGL position of the γ -ray 129 source (e.g., Nolan et al. 2012). We selected and calculated the IR colors for the WISE 130 sources within the search region detected in all four bands. 131 To compare the infrared colors of generic infrared sources that lie in the search region 132 with those of the γ -ray emitting ones, we developed a 3-dimensional parametrization of 133 the locus in the parameter space of its principal components. The locus was described as 134 a cylinder in the space of the principal components. This choice simplifies and improves 135 the previous description built using irregular quadrilaterals on all the color-color diagrams 136 (Massaro et al. 2012a). Moreover, the cylinder axis is aligned along the first PC axis, which 137 accounts for the larger fraction possible of the variance of the dataset in the IR color space, 138 is the simplest parametrization available. 139 We then assign to each source score value s that is a proxy of the distance between the 140 locus surface and the source location in the 3-dimensional parameter space of the principal 141 components. The values of s allow to to evaluate if the IR colors of a generic source are 142 consistent with those of the known γ -ray emitting blazars. They were weighted taking into 143 account of all the color errors and they are also normalized between 0 and 1. We define 144 three classes (i.e., A, B, C) of reliability for the γ -ray blazar candidates. A generic source is 145 assigned to class A, class B or class C when its score his higher than the threshold values 146 defined by the 90%, 60% and 30% percentiles of the score distributions of all the γ -ray blazars 147 that constitute the locus , respectively. We consider reliable γ -ray blazar candidates only 148 those having the score higher than 70% of their distributions. Thus sources with high values 149 of the score (e.g., > 0.8) are very likely to be blazars and belong to class A, while sources 150 with score values ∼ 0.5 belong to class C and are less probable γ -ray blazars. IR sources 151 that having score values null or extremely low (e.g., ∼ 0.1) were marked as outliers and 152 were not considered as γ -ray blazar candidates (see D'Abrusco et al. 2013, for an extensive 153 explanation on the class definitions). 154 The locus was divided in subregions on the basis of the space density of BZBs and BZQs 155 in the parameter space of its principal components, thereby permitting us to determine if a 156 selected γ -ray blazar candidate is more likely to be a BZB or a BZQ. 157 Finally, we ranked all the WISE sources within each search region and selected as best 158 candidate counterpart for the UGS the one with the highest class; when more than one 159 candidate of the same class was present, we chose the one closest to the γ -ray position as 160 best one. 161", "pages": [ 5, 6 ] }, { "title": "4.1. The unidentified gamma-ray sources", "content": "For the UGSs without γ -ray analysis flags we found 75 γ -ray blazar candidates out of 164 the 299 objects analyzed: 8 sources have 2 candidates, 1 source has 3, and 1 source has 4 165 candidates, while 52 associations are unique. We found 2 γ -ray blazar candidates of class 166 A, 12 of class B and 61 of class C, respectively, in the whole sample of 75 sources; 32 of them 167 are classified as BZB type, 29 as BZQ type and the remaining 14 are still uncertain (see 168 D'Abrusco et al. 2013, for more details). All our γ -ray blazar candidates have a signal-to169 noise ratio systematically larger than 10.9 in the WISE band centered at 12 µ m and larger 170 than ∼ 20 for the 3.4 µ m and 4.6 µ m nominal bands. For all these 75 sources we performed a 171 cross correlation with the major radio, infrared, optical, and X-ray surveys (see Section 5). 172 In the sample of UGSs with γ -ray analysis flags we found 71 γ -ray blazar candidates out 173 of the 291 objects investigated: 6 sources have 2 candidates, 4 sources have 3 candidates, 2 174 sources have 4 and 6 candidates, respectively, while 35 associations are unique. We found 175 8 γ -ray blazar candidates of class A, 20 of class B and 43 of class C, respectively, in the 176 whole sample of 71 sources; 36 of them are classified as BZB type, 22 as BZQ type and the 177 remaining 13 are still uncertain (see D'Abrusco et al. 2013). We also performed the cross 178 correlation with the major radio, infrared, optical, and X-ray databes for these 71 UGSs 179 listed in the Section 5. 180 181", "pages": [ 7 ] }, { "title": "4.2. The active galaxies of uncertain type", "content": "For the AGU sample we found 125 γ -ray blazar candidates out of the 210 sources 182 analyzed: 10 sources have 2 candidates within their search region , while the remaining 183 105 candidates have unique associations. There are 10 γ -ray blazar candidates of class A, 184 39 of class B and 76 of class C, respectively, in the whole sample of 125 sources; 52 out 185 of 125 are classified as BZB type on the basis of the IR colors of blazars of similar type, 186 39 as BZQ type and the remaining 34 are still uncertain (see D'Abrusco et al. 2013, for 187 more details). Eighty-seven sources out of 125 associations correspond to those reported 188 in the 2LAC or in the 2FGL. All our γ -ray blazar candidates have a signal-to-noise ratio 189 systematically larger than 10.9 in the WISE band centered at 12 µ m and larger than ∼ 20 190 for the 3.4 µ m and 4.6 µ m nominal bands. In these case we did not provide any additional 191 radio or X-ray information since it is already present in both the 2LAC and the 2FGL, 192 while a multifrequency investigation has been performed for the remaining 38. Additional 193 IR information for all the AGUs associated will be discussed in Section 5.2. 194", "pages": [ 7 ] }, { "title": "4.3. Comparison with previous associations", "content": "The fraction of sources for which we have been able to find a γ -ray blazar counterpart is 196 about ∼ 15-20% lower than presented in previous analyses of UGSs (Massaro et al. 2012b) 197 and AGUs (Massaro et al. 2012a), respectively. This difference occurs because a more con198 servative approach has been adopted in the new parametrization of the locus . We not limit 199 blazar candidates to those having the scores higher than 30% of the entire distribution of 200 γ -ray emitting blazars (D'Abrusco et al. 2013), rather than 10% as in the previous analy201 sis. These choices made our association method more efficient, so decreasing the number 202 of WISE sources with IR colors similar to those of the γ -ray blazar population. In addition, 203 we now use a search region of radius θ 95 instead of that at 99.9% level of confidence, to be 204 consistent with the associations of the 2FGL and the 2LAC catalogs. All the sources listed 205 in this work as γ -ray blazar candidates were also selected in our previous analysis based on 206 WISE Preliminary data analysis (Massaro et al. 2012b). 207 We note that only three IR WISE sources have the 'contamination and confusion' flag 208 that might indicate a WISE spurious detection of an artifact in all bands (e.g., Cutri et al. 2012). 209 It occurs for WISE J085238.73-575529.4 within the AGUs, WISE J084121.63-355505.9 in the 210 UGS sample, and WISE J125357.07-583322.3 among the UGS with γ -ray analysis flags. The 211 large majority (i.e., ∼ 90%) of the WISE sources considered do not show any WISE analysis 212 flags, with 10% clean in at least two IR bands. 213 Finally, we remark that several γ -ray pulsars have been identified since the release of 214 the 2FGL where they were listed as UGSs. However, we tested these UGSs and we did not 215 find any WISE blazar-like counterpart associable to them. Thus, in agreement with other 216 gamma-ray pulsars listed in the the Public List of LAT-Detected Gamma-Ray Pulsars 4 . 217 218", "pages": [ 8 ] }, { "title": "5. Correlation with existing databases", "content": "We searched in the following major radio,infrared, optical and X-ray surveys as well as 219 in the NASA Extragalactic Database (NED) 5 for possible counterparts within 3 '' .3 of our 220 γ -ray blazar candidates, selected with the WISE association method, to see if additional 221 information could confirm their blazar-like nature. The angular separation of 3 '' .3 from the 222 WISE position was chosen on the basis of the statistical analysis previously performed to 223 assign a WISE counterpart to each ROMA-BZCAT source (D'Abrusco et al. 2013) devel224 oped following the approach described in Maselli et al. (2012a, 2012b). In particular, we 225 found that for all radii larger than 3 '' .3 the increase in the number of IR sources positionally 226 associated with ROMA-BZCAT blazars becomes systematically lower than the increase in 227 number of random associations. This choice of radius results in zero multiple matches. 228 For the radio counterparts we used the NRAO VLA Sky Survey (NVSS; Condon et al. 1998, 229 - N), the VLA Faint Images of the Radio Sky at Twenty-Centimeters (FIRST; Becker et al. 1995; 230 White et al. 1997, - F), the Sydney University Molonglo Sky Survey (SUMSS; Mauch et al. 2003, 231 - S) and the Australia Telescope 20 GHz Survey (AT20G; Murphy et al. 2010, - A); for the 232 infrared we used the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006, - M) since 233 each WISE source is already associated with the closest 2MASS source by the default catalog 234 (see Cutri et al. 2012, for more details). We also marked sources that are variable when 235 having the variability flag higher than 5 in at least one band as in the WISE all-sky catalog 236 (Cutri et al. 2012). Then, we also searched for optical counterparts, with possible spec237 tra available, in the Sloan Digital Sky Survey (SDSS; e.g. Adelman-McCarthy et al. 2008; 238 Paris et al. 2012, - s), in the Six-degree-Field Galaxy Redshift Survey (6dFGS; Jones et al. 2004; 239 Jones et al. 2009, - 6); while for the high energy we looked in the soft X-rays using the 240 ROSAT all-sky survey (RASS; Voges et al. 1999, - X). A deeper X-ray analysis based on 241 the pointed observations present in the XMM-Newton , Chandra , Swift and Suzaku archives 242 will be performed in a forthcoming paper (Paggi et al. 2013). We also considered NED for 243 additional information. 244 We also searched in the USNO-B Catalog (Monet et al. 2003) for the optical coun245 terparts of our γ -ray blazar candidates within 3 '' .3; this cross correlation will be useful to 246 prepare future follow up observations and the complete list of sources together with their 247 optical magnitudes is reported in Appendix. 248 In Table 2 we summarize all the multifrequency information for the UGS samples, 249 without and with the γ -ray analysis flags, respectively, while all the details are given in in 250 Table 3 and Table 4. In Table 5 and Table 6 we report our findings the AGUs. In each table 251 we report the 2FGL source name, together with that of the WISE associated counterpart 252 and a generic one from the surveys cited above. We also report the IR WISE colors, the type 253 and the class of each candidate derived by our association procedure, the notes regarding 254 the multifrequency archival analysis, as the optical classification, and, if known, the redshift. 255 In Table 5 and Table 6, we also indicate if the selected source is the same associated by the 256 2FGL and the 2LAC. Figure 1 shows the 3-dimensional color plot comparing the IR colors 257 of the selected γ -ray blazar candidates with the blazar population that constitutes the locus . 258 260", "pages": [ 8, 9, 11 ] }, { "title": "5.1. Radio counterparts", "content": "In the UGS sample of sources without γ -ray analysis flags, 19 have a counterpart in the 261 NVSS; 7 in the SUMSS and 6 only in the FIRST (5 in common with the previous 19 in 262 the NVSS). In the list of UGSs with γ -ray analysis flags, we found only 4 sources having 263 a radio counterpart in the NVSS, one also detected in the FIRST, but none in the SUMSS 264 or in the AT20G catalogs. In Figure 2 we show the archival NVSS radio image of WISE 265 J134706.89-295842.3, the candidate low-energy counterpart of 2FGLJ1347.0-2956. 266 Within the AGU sample, 12 sources out the 38 new associations proposed have unique 267 counterparts in one of the considered radio survey. Two of them: BZUJ1239+0730 and 268 BZUJ1351-2912, were also classified as Blazars of uncertain type in the ROMA-BZCAT 269 (e.g., Massaro et al. 2011a), while the remaining one are divided as 6 in the NVSS, 1 in the 270 FIRST, 2 in the SUMSS and 1 in the AT20G. 271", "pages": [ 11 ] }, { "title": "5.2. Infrared counterparts", "content": "In the UGS sample of sources without γ -ray analysis flags, there are 43 WISE candidates 273 with counterparts in the 2MASS catalog: 10 out of 75 are variable infrared sources according 274 to the same criterion previously described. 275 The large majority (47 out of 71) of the UGSs, in the sample with γ -ray analysis flags, 276 have counterparts in the 2MASS catalog and 15 out of 71 are variable according to the 277 WISE all-sky catalog. 278 Of the 125 WISE candidates counterparts of the AGUs, 59 are detected in 2MASS, as 279 generally expected for blazars (e.g., Chen et al. 2005). In addition, 25 γ -ray blazar candi280 dates out of 125 have the variability flag in the WISE catalog with a value higher than 5 in 281 at least one band, suggesting that their IR emission is not likely arising from dust. 282 283", "pages": [ 13 ] }, { "title": "5.3. Optical counterparts", "content": "In the sample of UGSs without γ -ray analysis flags, 13 sources have been found with a 284 counterpart in the SDSS, 4 with spectroscopic information (Table 3). Among these 4 sources, 285 two are broad line quasars, promising to be blazar-like sources of BZQ type. One is a Seyfert 286 galaxy: SDSS J015910.05+010514.5 is a contaminant of our association procedure (although 287 our method suggests a better candidate, for 2FGLJ0158.4+0107). The remaining one, NVSS 288 J161543+471126 shows the optical spectrum similar to that of an X-ray Bright, Optically 289 Normal Galaxy (XBONG Comastri et al. 2002). The source SDSS J015836.23+010632.0, 290 another candidate counterpart of 2FGLJ0158.4+0107 is described as a quasar at redshift 291 0.723 in Schneider et al. (2007) and Hu et al. (2008). In addition to these 4 sources, spec292 troscopic information is also available for WISE J230010.16-360159.9 a possible low-energy 293 counterpart of 2FGLJ2300.0-3553, classified as quasar according to Jones et al. (2009). A 294 quasar-like spectrum is then available for SDSS J161434.67+470420.0 candidate counterpart 295 of 2FGLJ1614.8+4703. 296 The search for the optical counterparts for UGSs with γ -ray analysis flags was less suc297 cessful. Only one source has an optical, counterpart: WISE J131552.98-073301.9, associated 298 with 2FGLJ1315.6-0730. This source has a counterpart in both the NVSS and in the FIRST 299 radio survey. According to Bauer et al. (2009), this source is also variable in the optical 300 and it was therefore selected as a blazar candidate. In Figure 3 we show the archival SDSS 301 spectrum of the WISE J161434.67+470420.1 candidate as the low energy counterpart of 302 2FGLJ1614.8+4703. 303 We found only 1 γ -ray blazar candidate in the AGU sample with a counterpart in the 304 SDSS, while 4 of them have a 6dFGS source lying 3 '' .3 from their WISE position. In the 305 case of WISE J033200.72-111456.1 associated with 2FGLJ0332.5-1118, we also found that 306 its 6dFGS optical spectrum appear to be featureless suggesting a BL Lac classification 307 (Jones et al. 2009). The same information has been found for WISE J001920.58-815251.3 308 associated with 2FGLJ0018.8-8154, for which the noisy, featureless 6dFGS optical spectrum 309 points to a BL Lac classification (Jones et al. 2009). 2FGLJ0823.0+4041 and 2FGLJ0858.1310 1952 appear to be associated, both by the 2FGL catalog and our method to broad line 311 quasars. WISE J085805.36-195036.8 associated with 2FGLJ0858.1-1952 is also classified as 312 a quasar at redshift 0.6597 by White et al. (1988). The archival 6dFGS spectrum of WISE 313 J001920.58-815251.3 the candidate low energy counterpart of 2FGLJ0018.8-8154 is available 314 on NED; the absence of features allows us to classify the source as a BZB. 315 316", "pages": [ 13, 14 ] }, { "title": "5.4. X-ray counterparts", "content": "In the UGS sample without γ -ray analysis flags, only 3 objects have X-ray counterparts 317 in the ROSAT all-sky catalog: the Seyfert 1 galaxy SDSS J015910.05+010514.5, the quasar 318 SDSS J161434.67+470420.0 (both described in Section 5.3) and WISE J164619.95+435631.0 319 associated with 2FGLJ1647.0+4351. In addition, SDSS J161434.67+470420.0 is also de320 tected in the Chandra source catalog: CXO J161434.7+470419 as occurs NVSS J161543+471126, 321 alias CXO J161541.2+471111 (Evans et al. 2010). 322 In the UGS list of sources with γ -ray analysis flags, there is only a single object detected 323 in the ROSAT all-sky survey, namely WISE J043947.48+260140.5 uniquely associated with 324 the Fermi source 2FGLJ0440.5+2554c and with a X-ray counterpart also in the Chandra 325 source catalog CXO J043947.5+260140 (Evans et al. 2010). In addition, WISE J060659.94326 061641.5 the unique counterpart of 2FGLJ0607.5-0618c, has the Chandra counterpart CXO 327 J060700.1-061641 (Evans et al. 2010). 328 Finally, in the AGU sample of 38 new γ -ray blazar candidates we found only 1 source in 329 the ROSAT catalog, namely WISE J181037.99+533501.5, associated with the X-ray object 330 1RXS J181038.5+533458 and having a radio counterpart in the NVSS. According to NED, 331 WISE J182352.33+431452.5, associated with 2FGLJ1823.8+4312, is also detected in the 332 X-rays by Chandra : CXO J182352.2+431452 (Massaro et al. 2012d). 333 334", "pages": [ 14, 15 ] }, { "title": "6. Comparison with other methods", "content": "Among the whole sample of 590 UGSs analyzed, 299 without and 291 with γ -ray analysis 335 flag there are 28 sources having at least one γ -ray blazar candidate that were also unidentified 336 in the First Fermi γ -ray LAT catalog (1FGL; Abdo et al. 2010) and they were analyzed 337 using two different statistical approaches: the Classification Tree and the Logistic regression 338 analyses (see Ackermann et al. 2012, and references therein). For these 28 UGSs, analyzed 339 on the basis of the above statistical approaches, we performed a comparison with our results 340 to verify if the 2FGL sources that we associated with a γ -ray blazar candidates have been 341 also classified as AGNs. 342 By comparing the results of our association method with those in Ackermann et al. 343 (2012), we found that 23 out of 28 UGSs that we associate with γ -ray blazar candidates are 344 classified as AGNs, all of them with a probability higher than 66% and 12 of them higher than 345 80% (see Ackermann et al. 2012). Among the remaining 5 sources, 4 have been classified as 346 pulsars, with a very low probability with respect to the whole sample, systematically lower 347 than 56%. In addition, there is one with an ambiguous classification. Consequently, we 348 emphasize that for the subamples where we overlap our results are in good agreement with 349 the classification suggested by Ackermann et al. (2012), consistent with the γ -ray blazar 350 nature of the WISE candidates proposed in our analysis. 351 352", "pages": [ 15 ] }, { "title": "7. Summary and conclusions", "content": "A new association method has been recently developed on the basis of the striking 353 discovery that γ -ray emitting blazars occupy a distinct region in the WISE 3-dimensional 354 Col. (1) 2FGL name. Col. (2) WISE name. Col. (9) Notes: N = NVSS, F = FIRST, S = SUMSS, A=AT20G, M = 2MASS, s = SDSS dr9, 6 = 6dFGS, x = XMM-Newton or Chandra , X = ROSAT; QSO = quasar, Sy = Seyfert, LNR = LINER, BL = BL Lac, XB = X-ray Bright Optically Inactive Galaxies; v = variability in WISE Col. (1) 2FGL name. Col. (2) WISE name. Col. (3) Other name if present in literature and in the following order: ROMA-BZCAT, NVSS, SDSS, AT20G, NED. Cols. (4,5,6) Infrared colors from the WISE all sky catalog corrected for Galactic extinciton. Values in parentheses are 1 σ uncertainties. Col. (7) Type of candidate according to our method: BZB - BZQ - UND (undetermined). color space, well separated from that occupied by other extragalactic and galactic sources 355 (Massaro et al. 2011b; D'Abrusco et al. 2012). According to D'Abrusco et al. (2013) the 356 3-dimensional region occupied by γ -ray emitting blazars is the locus ; its 2-dimensional 357 projection in the [3.4]-[4.6]-[12] µ m parameter space, retains its historical definition of 358 WISE Gamma-ray Strip (Massaro et al. 2011b). Additional improvements, mostly based on 359 the WISE all-sky data release, available since March 2012 (e.g., Cutri et al. 2012), and on 360 a new parametrization of the locus in the parameter space of its principal components have 361 been subsequently developed (D'Abrusco et al. 2013). 362 In this work we describe the results obtained by applying our new association procedure 363 to the search for new γ -ray blazar candidates in the two samples: the unidentified gamma364 ray sources (UGSs), and the active galaxies of uncertain type (AGUs), as listed in the 2FGL 365 (Nolan et al. 2012). 366 We present the complete list of γ -ray blazar candidates found using the WISE observa367 tions. We also perform an extensive archival search to see if the sources associated with our 368 method, show additional blazar-like characteristics; as for example the presence of a radio 369 counterpart and/or of a spectrum that could be featureless as for BZBs or similar to those 370 of broad-line quasars as generally occurs in BZQs. 371 We found 62 γ -ray blazar candidates for the UGS without any γ -ray analysis flag and 372 49 for those with γ -ray analysis flag, out of a total of 590 sources investigated. For the AGUs 373 sample, we confirmed the blazar-like nature of 87 out 210 of AGUs analyzed on the basis of 374 their IR colors. 375 A significant fraction (i.e., ∼ 36%) of the WISE sources associated with our method 376 with UGSs have a radio counterpart, more than 50% are also detected in the 2MASS cat377 alog as generally occurs for blazars, and more than ∼ 10% appear to be variable accord378 ing to the WISE analysis flags (Cutri et al. 2012). Notably, all the sources for which an 379 optical spectrum was available in literature clearly show blazar-like features, being either 380 featureless or having broad emission lines typical of quasars, the only exception being SDSS 381 J015910.05+010514.5, one of the counterparts associated with 2FGLJ0158.4+0107. As gen382 erally expected for γ -ray blazars a handful of the selected candidates are also detected in the 383 X-rays. A deeper investigation of their X-ray counterparts will be addressed in a forthcoming 384 paper (Paggi et al. 2013). All the γ -ray blazar candidates selected with our association pro385 cedure appear to be extragalactic in nature; moreover our selection seems not to be highly 386 contaminated by any class of non-blazar-like sources, as for example obscured quasars or 387 Seyfert galaxies. 388 Our results are in good agreement with those based on different statistical approaches 389 like the Classification Tree and the Logistic regression analyses (Ackermann et al. 2012). In 390 particular, 23 out of 28 UGSs that we associate to a γ -ray blazar candidate are also classified 391 as active galaxies by the above methods at high level of confidence. 392 For UGSs associated with a pulsar in the 2FGL analysis as reported in the Public List 393 of LAT-Detected Gamma-Ray Pulsars (see Section 2.1), we did not find any WISE γ -ray 394 blazar candidate, confirming the reliability of our selection procedure. We provide a list 395 of the UGSs for which we did not find any γ -ray blazar candidates using either the new 396 improved method or the old parametrization (i.e., less conservative), within their positional 397 uncertainty regions at 95% level of confidence. This list of Fermi sources reported in Table 7 398 could be useful for follow up observations aiming at discover new pulsars or to constrain 399 exotic high-energy physics phenomena such as dark matter signatures, or new classes of 400 sources (e.g., Zechlin et al. 2012; Su & Finkbeiner 2012). 401 Finally, we emphasize that additional investigations of different samples of active 402 galactic nuclei, such as Seyfert galaxies, are necessary to study the problem of the con403 tamination of our association method by extragalactic sources with infrared colors similar 404 to those of γ -ray blazars. Moreover extensive ground-based spectroscopic follow up ob405 servations in the optical and in the near IR would be ideal to verify the nature of the 406 selected WISE sources and to estimate the fraction of non-blazar objects, similar to the 407 recent studies performed for the unidentified INTEGRAL sources (e.g., Masetti et al. 2008; 408 Masetti et al. 2009; Masetti et al. 2010; Masetti et al. 2012). 409 Note added to the proofs: The infrared source WISE J182352.33+431452.5, potential 410 counterpart of 2FGL J1823.8+4312, is a possible contaminant of our selections given its 411 optical spectrum typical of an obscured red quasar (D. Stern priv. comm.). 412 We thank the anonymous referee for useful comments that led to improvements in the 413 paper. F. Massaro is grateful to S. Digel and D. Thompson for their helpful discussions and 414 to M. Ajello, E. Ferrara and J. Ballet for their support. The work is supported by the NASA 415 grants NNX12AO97G. R. D'Abrusco gratefully acknowledges the financial support of the US 416 Virtual Astronomical Observatory, which is sponsored by the National Science Foundation 417 and the National Aeronautics and Space Administration. The work by G. Tosti is supported 418 by the ASI/INAF contract I/005/12/0. H. A. Smith acknowledges partial support from 419 NASA/JPL grant RSA 1369566. TOPCAT 6 (Taylor 2005) and SAOImage DS9 were used 420 extensively in this work for the preparation and manipulation of the tabular data and the 421 images. Part of this work is based on archival data, software or on-line services provided by 422 the ASI Science Data Center. This research has made use of data obtained from the High 423 Energy Astrophysics Science Archive Research Center (HEASARC) provided by NASA's 424 Goddard Space Flight Center; the SIMBAD database operated at CDS, Strasbourg, France; 425 the NASA/IPAC Extragalactic Database (NED) operated by the Jet Propulsion Labora426 tory, California Institute of Technology, under contract with the National Aeronautics and 427 Space Administration. Part of this work is based on the NVSS (NRAO VLA Sky Survey); 428 The National Radio Astronomy Observatory is operated by Associated Universities, Inc., 429 under contract with the National Science Foundation. This publication makes use of data 430 products from the Two Micron All Sky Survey, which is a joint project of the University of 431 Massachusetts and the Infrared Processing and Analysis Center/California Institute of Tech432 nology, funded by the National Aeronautics and Space Administration and the National Sci433 ence Foundation. This publication makes use of data products from the Wide-field Infrared 434 Survey Explorer, which is a joint project of the University of California, Los Angeles, and the 435 Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aero436 nautics and Space Administration. Funding for the SDSS and SDSS-II has been provided by 437 the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Founda438 tion, the U.S. Department of Energy, the National Aeronautics and Space Administration, 439 the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding 440 Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by 441 the Astrophysical Research Consortium for the Participating Institutions. The Participating 442 Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, 443 University of Basel, University of Cambridge, Case Western Reserve University, University 444 of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Par445 ticipation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, 446 the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, 447 the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max448 Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), 449 New Mexico State University, Ohio State University, University of Pittsburgh, University of 450 Portsmouth, Princeton University, the United States Naval Observatory, and the University 451 of Washington. 452 453", "pages": [ 15, 16, 18, 21, 22, 23 ] }, { "title": "REFERENCES", "content": "Abdo, A. A., et al. 2009, Astroparticle Physics, 32, 193 454 Abdo, A. A. et al. 2010 ApJS 188 405 455 Mirabal, N. 2009 [arxiv.org/abs/0908.1389v2] 481 Mirabal, N. 2009 ApJ, 701, 129 482 Mirabal, Nieto, D. & Pardo, S. 2010 A&A submitted, [arxiv.org/abs/1007.2644v2] 483 Maselli, A., Massaro, E., Nesci, R., Sclavi, S., Rossi, C., Giommi, P. 2010 A&A, 512A, 74 484 Maselli, A., Cusumano, G., Massaro, E., La Parola, V., Segreto, A., Sbarufatti, B. 2010 485 A&A, 520A, 47 486 Masetti, N. et al. 2008 A&A, 482, 113 487 Masetti, N. et al. 2009 A&A, 495, 121 488 Masetti, N. et al. 2010 A&A, 519A, 96 489 Masetti, N. et al. 2012 A&A, 538A, 123 490 Massaro, E., Giommi, P., Leto, C., Marchegiani, P., Maselli, A., Perri, M., Piranomonte, S., 491 Sclavi, S. 2009 A&A, 495, 691 492 Massaro, E., Giommi, P., Leto, C., Marchegiani, P., Maselli, A., Perri, M., Piranomonte, S., 493 Sclavi, S. 2010 http://arxiv.org/abs/1006.0922 494 Massaro, E., Giommi, P., Leto, C., Marchegiani, P., Maselli, A., Perri, M., Piranomonte, 495 S., 2011 'Multifrequency Catalogue of Blazars (3rd Edition)', ARACNE Editrice, 496 Rome, Italy 497 Massaro, F., D'Abrusco, R., Ajello, M., Grindlay, J. E. & Smith, H. A. 2011b ApJ, 740L, 48 498 Massaro, F., D'Abrusco, R., Tosti, G., Ajello, M., Gasparrini, D., Grindlay, J. E. & Smith, 499 Howard A. 2012a ApJ, 750, 138 500 Massaro, F., D'Abrusco, R., Tosti, G., Ajello, M., Paggi, A., Gasparrini, 2012b ApJ, 752, 61 501 Massaro, F., D'Abrusco, R., Paggi, A., Tosti, G., Gasparrini, D. 2012c ApJ, 750L, 35 502 Massaro, F., Paggi, A., D'Abrusco, R., Tosti, G., Grindlay, J. E., Smith, Howard A., Digel, 503 S. W., Funk, S. 2012d ApJ, 757L, 27 504 Mauch, T., Murphy, T., Buttery, H. J., Curran, J., Hunstead, R. W., Piestrzynski, B., 505 Robertson, J. G., Sadler, E. M. 2003 MNRAS, 342, 1117 506 Monet, D. G. et al. 2003 AJ, 125, 984 507 Murphy, T. et al. 2010 MNRAS, 402, 2403 508 Mukherjee, R. et al., 1997 ApJ, 490, 116 509 Nolan et al. 2012 ApJS, 199, 31 510 Paggi, A., Massaro, F., D'Abrusco, R. et al. 2013 ApJS in prep. 511 Paris, I. et al. 2012 A&A, 548A, 66 512 Pearson, K. 1901 Philosophical Magazine 2, 559. 513 Schneider et al. 2007, AJ, 134, 102 514 Stickel, M., Padovani, P., Urry, C. M., Fried, J. W., Kuehr, H. 1991 ApJ, 374, 431 515 Stocke et al. 1991, ApJS, 76, 813 516 Skrutskie, M. F. et al. 2006, AJ, 131, 1163 517 Su, M. & Finkbeiner, D. P. 2012 ApJ submitted http://arxiv.org/abs/1207.7060v1 518 Urry, C. M., & Padovani, P. 1995, PASP, 107, 803 519 Thompson, D. J. 2008 RPPh, 71k6901 520 Taylor, M. B. 2005, ASP Conf. Ser., 347, 29 521 Voges, W. et al. 1999 A&A, 349, 389 522 White, R. L., Becker, R. H. Helfand, D. J., Gregg, M. D. et al. 1997 ApJ, 475, 479 523 White, G. L. et al. 1988 ApJ, 327, 561 524 Wright, E. L., et al. 2010 AJ, 140, 1868 525 Zechlin, H.-S., Fernandes, M. V., Elsasser, D., Horns, D. 2012 A&A, 538A, 93 526", "pages": [ 23, 25, 26 ] }, { "title": "A. Optical counterparts", "content": "In Tables 8, 9, 10 and 11, we report the magnitudes of the optical counterpart uniquely 528 found within 3 '' .3, for all the γ -ray blazar candidates, selected according to our association 529 procedure. This information permits us to optimize the strategy for the future follow up 530 optical observations needed to clarify the nature of the selected sources and to determine 531 their redshifts via spectroscopy. 532", "pages": [ 27 ] } ]

[ { "title": "L. L. Kitchatinov 1 , 2 ∗", "content": "1 Institute for Solar-Terrestrial Physics, P.O. Box 4026, Irkutsk, 664033 Russia 2 Pulkovo Astronomical Observatory, Pulkovskoe Sh. 65, St. Petersburg, 196140 Russia Abstract. A linear analysis of baroclinic instability in a stellar radiation zone with radial differential rotation is performed. The instability onsets at a very small rotation inhomogeneity, ∆Ω ∼ 10 -3 Ω . There are two families of unstable disturbances corresponding to Rossby waves and internal gravity waves. The instability is dynamical: its growth time of several thousand rotation periods is short compared to the stellar evolution time. A decrease in thermal conductivity amplifies the instability. Unstable disturbances possess kinetic helicity thus indicating the possibility of magnetic field generation by the turbulence resulting from the instability.", "pages": [ 1 ] }, { "title": "DOI: 10.1134/S1063773713080045", "content": "Keywords: stars: rotation - instabilities - waves.", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "Upon arrival on the main sequence, young stars rotate rapidly, with periods of about one day. Solar-type stars spin down with age due to the loss of angular momentum through a stellar wind (Skumanich 1972; Barnes 2003). The braking torque acts on the stellar surface, but the spin-down extends rapidly deep into the convective envelope due to the eddy viscosity existing here. In deeper layers of the radiation zone, the viscosity is low ( ∼ 10 cm 2 /s) and insufficient to smooth out the radial rotation inhomogeneity. Therefore, before the advent of helioseismology, it had been thought very likely that the solar radiation zone rotates much faster than the surface (see, e.g., Dicke 1970). Subsequently, it transpired that the radiation zone rotates nearly uniformly (Shou et al. 1998). In other words, there is a coupling between the convective envelope and deep layers of the radiation zone that is efficient enough to smooth out the rotation inhomogeneity in a short time compared to the Sun's age. Observations of stellar rotation show that the characteristic time of the coupling is < ∼ 10 8 yr (Hartmann & Noyes 1987; Denissenkov et al. 2010). One possible explanation for the smoothing of rotation inhomogeneities in stars is the instability of differential rotation: the turbulence resulting from the instability transports angular momentum in such a way that the rotation approaches uniformity. The difficulty of this explanation stems from the fact that the threshold rotation inhomogeneity for the appearance of hydrodynamic instabilities is not small, q = O (0 . 1) (here, Ω is the angular velocity, r is the radius). One might expect that such instabilities could reduce the rotation inhomogeneity to its threshold value but could not remove it completely. The baroclinic instability that is related to the rotation inhomogeneity indirectly may constitute an exception (Tassoul & Tassoul 1983). In the equilibrium state of a differentially rotating radiation zone, the surfaces of constant pressure and constant density do not coincide. Such a 'baroclinic' equilibrium can be unstable (Spruit & Knobloch 1984). In this paper, we consider the baroclinic instability in a stellar radiation zone with radial differential rotation. As we will see, the instability occurs at a very small rotation inhomogeneity, q /lessmuch 1 . Instabilities are also important for the mixing of chemical species in stars (see, e.g., Pinsonneault 1997). The study of the stability of differential rotation in stellar radiation zones has a long history (Goldreich & Schubert 1967; Acheson 1978; Spruit & Knobloch 1984; Korycansky 1991). This paper differs in that we consider the stability against global disturbances. The horizontal disturbance scale is not assumed to be small compared to the stellar radius. At the same time, a stable stratification of the radiation zone rules out mixing on a large radial scale. Therefore, the radial disturbance scale is assumed to be small. Such an approach was applied to analyze the stability of latitudinal differential rotation (Charbonneau et al. 1999; Gilman et al. 2007; Kitchatinov 2010) in connection with the problem of the solar tachocline. It showed that the horizontally-global modes are actually the dominant ones. As we will see, the same is true for the baroclinic instability. The most unstable disturbances correspond to global Rossby waves ( r -modes) and internal gravity waves ( g -modes), which grow exponentially with time in the presence of a radial rotation inhomogeneity. Therefore, the baroclinic instability may be considered as the loss of stability by a differentially rotating star with respect to the excitation of r - and g -modes of global oscillations. /negationslash Both dominant modes possess kinetic helicity, u · ( ∇ × u ) = 0 . Helicity is indicative the ability of a flow to generate magnetic fields (see, e.g., Vainshtein et al. 1980). Here, our instability analysis joins with another possible explanation for the uniform rotation of the solar radiation zone, the magnetic field effect.", "pages": [ 1, 2, 3 ] }, { "title": "Background Equilibrium and the Origin of Instability", "content": "When analyzing the stability, we will assume the initial equilibrium state to be stationary and cylinder symmetric about the rotation axis. We consider the hydrodynamic stability, i.e., the magnetic field is disregarded. We will proceed from the stationary hydrodynamic equation where ψ is the gravitational potential, the standard notation is used, the influence of viscosity on the global flow is neglected. The main flow component in the radiation zone is rotation, where the usual spherical coordinates ( r, θ, φ ) are used and e φ is the azimuthal unit vector. We assume the rotation to be sufficiently slow, Ω 2 /lessmuch GM/R 3 , for the deviation of stratification from spherical symmetry to be small. The stratification in stellar radiation zones is stable, i.e., the specific entropy s = c v ln( P ) -c p ln( ρ ) , increases with radius r , The buoyancy forces counteract the radial displacements. Therefore, the meridional circulation is small and the main flow component is rotation (3). The characteristic time of meridional circulation in the radiation zone exceeds the Sun's age (Tassoul 1982). Nevertheless, the most important force balance condition follows from the equation for a meridional flow. This condition can be derived by calculating the azimuthal component of the curl of the equation of motion (2). This gives where ∂/∂z = cos θ∂/∂r -r -1 sin θ∂/∂θ is the spatial derivative along the rotation axis, the subscript φ denotes the azimuthal component of the vector. The centrifugal force is conservative only if the angular velocity does not vary with distance z from the equatorial plane. The left part of Eq. (5) allows for the non-conservative part of the centrifugal force, which by itself produces a vortical meridional flow. In a stellar radiation zone, this non-conservative force is balanced by the buoyancy force included in the right part of Eq. (5). In the case of z -dependent differential rotation, the equilibrium is baroclinic: the surfaces of constant pressure and constant density do not coincide. One might expect such an equilibrium to be unstable. This can be seen after the following transformations of the right part of Eq. (5): where g ∗ = -∇ ψ + r sin θ Ω e φ × Ω is the 'effective' gravity. It can be seen from Eq. (6) that the isobaric and isentropic surfaces do not coincide either. Figure 1 explains why an instability is possible in this situation (Shibahashi 1980). For displacements in the narrow cone between the isobaric and isentropic surfaces, the gravitational forces increase the energy of the fluid particles being displaced. The relatively light particles with a positive entropy (temperature) perturbation are displaced opposite to the gravity, while the colder and relatively dense particles are displaced in the direction of gravity. One might expect the disturbances with such displacements to be amplified due to the release of (gravitational) energy of the equilibrium state. Remarkably, the instability arises from the buoyancy forces that usually exhibit a stabilizing effect in stellar radiation zones. It can be seen from Fig. 1 that not the deviation of stratification from spherical symmetry related to rotation but the baroclinicity caused by the rotation inhomogeneity is responsible for the instability. For simplicity, we will neglect the deviation of the pressure distribution from spherical symmetry but will take into account the latitudinal entropy inhomogeneity. For the special case of rotation dependent only on the radius, from Eqs. (5) and (6) we find where q is the rotation inhomogeneity parameter (1).", "pages": [ 3, 4, 5 ] }, { "title": "Linear Stability Equations", "content": "The main approximations and methods of deriving the equations for small disturbances were discussed in detail previously (Kitchatinov 2008; Kitchatinov & Rüdiger 2008). This paper differs only in allowance for the deviation of stratification from barotropy. Therefore, the equations of the linear stability problem will be written without repeating their derivation. We repeat, however, the main approximations and assumptions used in deriving these equations. The initial equilibrium state does not depend on time and longitude. Therefore, the dependence of the disturbances on longitude and time in the linear stability problem can be written as exp(i mφ -i ωt ) , where m is the azimuthal wave number. A positive imaginary part of the eigenvalue, /Ifractur ( ω ) > 0 , means an instability. Stable stratification of the radiation zone prevents mixing on large radial scales. Therefore, the radial scale of disturbances is assumed to be small and the stability analysis is local in radius: perturbations of the velocity, u , and entropy, s ' , depend on radius as exp(i kr ) with kr /greatermuch 1 . At the same time, the mixing in horizontal directions encounters no counteraction and the stability analysis is global in these directions. As we will see, the most unstable disturbances actually have large horizontal scales. We use the incompressibility approximation, div u = 0 . It is justified for disturbances whose wavelength in the radial direction is small compared to the pressure scale height. The magnetic fields are disregarded. The angular velocity is assumed to be dependent on radius only but not on latitude. The equations are written for the scalar potentials P u and T u of the the poloidal and toroidal components of the velocity perturbations: (Chandrasekhar 1961), where is the angular part of the Laplacian. We use non-dimensional variables. The physical quantities can be restored from the normalized perturbations of entropy ( S ) and the poloidal ( V ) and the toroidal ( W ) flow potentials using Eq. (8) and the relations The equation for the entropy perturbations is where ˆ ω = ω/ Ω -m is the dimensionless eigenvalue in the co-rotating frame of reference, µ = cos θ , ˆ λ and Q are the two basic parameters controlling the influence of fluid stratification and differential rotation The finite diffusion is taken into account in the parameters where χ and ν are the thermal diffusivity and viscosity, respectively. The complete system consists of three equations. In addition to Eq. (11) for the entropy perturbations, it includes the equations for the poloidal flow, and the toroidal flow, The eigenvalue problem for the system of equations (11), (14), and (15) was solved numerically. The independent variables were expanded in a series of the associated Legendre polynomials, for example, and similarly for W and V . This leads to a system of linear algebraic equations for the expansion amplitudes S l , W l and V l . The number of equations in the system is not about 3 K but a factor of 2 smaller, because the complete system splits into two independent subsystems governing the eigenmodes symmetric and antisymmetric relative to the equator. Most of the calculations were performed for the following values of dissipation parameters of Eq. (13), typical of the upper part of the solar radiation zone (Kitchatinov & Rüdiger 2008). In the cases where we used other values, this is stipulated.", "pages": [ 5, 6, 7 ] }, { "title": "Symmetry Properties", "content": "Two types of equatorial symmetry are possible: symmetric modes for which S ( µ ) = S ( -µ ) , V ( µ ) = V ( -µ ) and W ( µ ) = -W ( -µ ) , and antisymmetric modes with symmetric W and antisymmetric S and V . For the symmetric and antisymmetric modes, we will use the notations S m and A m , respectively, where m is the azimuthal wave number. These notations correspond to the symmetry relative to the mirror-reflection about the equatorial plane. For example, for the S m -modes, u r and u φ are symmetric relative to the equator, while u θ is antisymmetric. A more significant property of the system of equations (11), (14), and (15) consists in its symmetry relative to the transformation where the asterisk denotes complex conjugation. This means that if the mode with some m is unstable at a certain rotation inhomogeneity q , then at a rotation inhomogeneity of opposite sense ( -q ) there is an unstable mode with the same growth rate and azimuthal wave number -m . Therefore, it will suffice to consider the stability, for example, only for q > 0 ; the stability properties for q < 0 will then be known. Below, we consider only the case where the rotation rate increases with depth, i.e., q > 0 . Transformation (18) also shows that stability properties depend on the sign of the azimuthal wave number m . This dependence usually implies that unstable modes possess a finite helicity (Rüdiger et al. 2012). The absolute helicity in the linear problem is indefinite, but the relative helicity can be defined † . The angular brackets here denote the azimuthal averaging: For axisymmetric modes ( m = 0 ), this corresponds to averaging over the oscillation phase φ (the linear solutions are determined to within phase factor e i φ ). The overline in (19) and below denotes averaging over a spherical surface: For barotropic fluids, the total (volume-integrated) kinetic helicity is an integral of motion. For baroclinic fluids, this is not the case. As we will see, the unstable modes of baroclinic instability are indeed helical.", "pages": [ 7, 8 ] }, { "title": "Two Modes of Stable Oscillations", "content": "The solutions for special limiting cases are helpful in discussing the results to follow. In this Section, we consider uniform rotation ( q = 0 ) in the absence of dissipation ( χ = ν = 0 ). In the limiting case of a 'very stable' stratification, ˆ λ /greatermuch 1 , or N /greatermuch Ω kr , two modes of stable oscillations can be revealed: (1) For one of them, the frequency is low, ˆ ω /lessmuch ˆ λ . It then follows from Eq. (14) that S = 0 and Eq.(11) gives V = 0 . The flow possesses no poloidal component and the spectrum of purely toroidal oscillations can be found from Eq. (15): These are the r -modes of global oscillations also known as Rossby waves. (2) There is another solution for which the frequency is not low, ˆ ω ∼ ˆ λ . In this case, Eq. (15) in the highest order in ˆ λ gives W = 0 . The flow does not contain a toroidal part. We write Eqs. (14) and (11), also in the highest order in ˆ λ , as ˆ ω ( ˆ LV ) = -ˆ λ 2 ( ˆ LS ) and ˆ ωS = ˆ LV , respectively. Poloidal oscillations with the following spectrum are found: As can be seen from the expression for the frequency rotation does not affect these high-frequency oscillations. These are the internal gravity waves or g -modes. As we will see, in a differentially rotating fluid with baroclinic stratification, both modes of global oscillations acquire positive growth rates, i.e., become unstable.", "pages": [ 9 ] }, { "title": "Stability Borders and Growth Rates", "content": "The lines separating the regions of stability and instability for disturbances with different equatorial and axial symmetries are shown in Fig. 2. The instability appears at a small rotation inhomogeneity. In the upper part of the solar radiation zone, N/ Ω ≈ 400 . Even /negationslash a weakly inhomogeneous rotation q ∼ 10 -4 is unstable. In this way the instability under consideration differs from the barotropic instabilities that appear at a relatively large rotation inhomogeneity. Another important difference is that baroclinic instability exists both for axisymmetric disturbances and for various azimuthal wave numbers m = 0 . However, the greater the number |m|, the larger rotation inhomogeneity is required for the onset of instability. This trend is confirmed by our calculations for | m |≤ 10 . The disturbances that are global in horizontal dimensions are most unstable. The lines for modes S-1 and S-2 in Fig. 2 have kinks. This implies that different line segments correspond to disturbances of different nature. Unstable disturbances close to the r - and g -modes of global oscillations are revealed. This can be seen from the Table, where the characteristics of unstable disturbances are given. The kinetic energy of the disturbances is the sum of the energies of their poloidal and toroidal components (Chandrasekhar 1961): The Table lists the closest frequencies of the poloidal g -modes (23) for unstable poloidal disturbances ( u 2 p /u 2 t > 1 ) and the closest frequencies of the toroidal r -modes (22) for toroidal disturbances ( u 2 p /u 2 t < 1 ). The frequencies of the unstable disturbances and the corresponding oscillation modes differ little; there is a correspondence to the largest-scale oscillations. For example, the frequency of the unstable poloidal disturbance A3 is 13.6. The lowest value of l in expansion (16) for the poloidal potential of this disturbance is l = 4 . For l = 4 we find a frequency of 13.4 close to that of the unstable disturbance from (23). The cases where there is no correspondence to the largest-scale oscillation mode are marked with an asterisk in the Table. For example, the expansion of the toroidal potential for the toroidal mode A-3 with a frequency of 0.199 begins from l = 3 , but the r -mode (22) with the next l = 5 has the closest frequency ˆ ω r = 0 . 2 (the summation in (16) for disturbances with a certain equatorial symmetry is over either even or odd l ). The correspondence of the frequencies and the poloidal or toroidal character of growing disturbances to the r - and g -modes of global oscillations allows us to interpret the instability as the loss of stability against the excitation of these global oscillations. The question of what determines the transport of angular momentum in differentially rotating stars, the instability or the g -modes (Spruit 1987; Charbonnel & Talon 2005), may find an unexpected answer: the instability excites the g -modes. The Table also gives the correlation of the entropy and radial velocity perturbations, to which the power supplied by buoyancy forces is proportional. This correlation is positive for all unstable modes. Calculations show that this correlation can be negative only for damped disturbances. The energy of the growing disturbances increases due to the work of buoyancy forces, as it should be for baroclinic instability (Fig. 1). The Table gives the disturbance growth rate normalized to the rotation period, i.e., the disturbances grow by a factor of e ˆ γ in one stellar rotation. The growth rates are small. The star makes about 10 000 rotations in a disturbance e-folding time. Even for slowly rotating stars, however, this time ( ∼ 1000 yr) is short compared to evolutionary time scales. In Fig. 3, the growth rate of disturbances is plotted against their dimensionless wavelength ˆ λ (12). The equatorial symmetry does not determine the properties of unstable modes uniquely. For example, there is a discrete spectrum of modes S1. Figure 3 shows the highest growth rates. The kinks in the lines for modes with negative m correspond to a change in the type of the most rapidly growing disturbance. The highest growth rates belong to the r -modes for relatively small ˆ λ and to the g -modes for large ˆ λ . In Fig. 4, the highest growth rates are plotted against the rotation inhomogeneity parameter Q (12). For relatively large Q , these dependencies are nearly linear, ˆ γ ∼ Q .", "pages": [ 9, 10, 11, 12 ] }, { "title": "Dependence on Thermal Conductivity", "content": "The dependence on thermal conductivity is of interest in connection with the possible influence of chemical composition inhomogeneity. Such inhomogeneity is important for stability. The increase in mean molecular weight µ with depth makes the stratification 'more stable'. This can be taken into account by replacing the frequency N (4) with its effective value N ∗ , (Kippenhahn & Weigert 1990). This, however, is not the only effect of the compositional gradient. The diffusivity for chemical inhomogeneities in stellar radiation zones is much smaller than the thermal diffusivity. Therefore, the inhomogeneity of µ reduces the dissipation rate of density inhomogeneities in unstable disturbances. Here, we do not account for composition inhomogeneity, but the character of its influence can be seen by analyzing dependence of the instability on thermal conductivity. Figure 5 shows the growth rates of the unstable g -mode A0 for three values of the dimensionless thermal diffusivity /epsilon1 χ (13). Similar results are also obtained for other unstable modes. An increase in /epsilon1 χ suppresses the instability. It is generally believed that conduction of heat amplifies the instabilities in stellar radiation zones. Radial displacements produce the temperature and density disturbances and are, therefore, opposed by buoyancy. The dissipation of temperature inhomogeneities reduces the stabilizing buoyancy effect, thereby amplifying the instabilities. Figure 5 shows that the opposite is true of the baroclinic instability. This instability is peculiar in that it emerges precisely due to special features of the radiation zone stratification and is produced by buoyancy forces (Fig. 1). Therefore, an increase in thermal conductivity suppresses this instability. Assertions in the literature that the compositional gradient in stellar radiation zones switches off baroclinic instability seem questionable.", "pages": [ 12, 13, 14 ] }, { "title": "Helicity and the Possibility of Dynamo", "content": "Figure 6 shows the distributions of the relative helicity (19) for three g -modes of the instability under consideration. Positive and negative helicities dominate in the northern and southern hemispheres, respectively. In the dynamo theory, the helicity is known to be important for magnetic field generation. The origin of magnetic fields in stellar radiation zones presents a problem. Solar-type stars at early evolutionary stages are fully convective for more than a million years, which is approximately a factor of 10 4 longer than the turbulent diffusion time. A hydromagnetic dynamo can operate in such fully convective stars (Dudorov et al. 1989). Subsequently, a radiative core emerges and grows in the central part of the star. During its growth, it can capture the magnetic field from the surrounding convective envelope. However, this field is weak (<1 G), because the convective dynamo field is oscillating and the frequency of its oscillations is much higher than the growth rate of the radiation zone (Kitchatinov et al. 2001). The helicity of the eigenmodes of baroclinic instability (Fig. 6) points to another possibility - the dynamo action in a differentially rotating unstable radiation zone. The radiation zones of solar-type stars are deep beneath the surface and are inaccessible to direct observations. Higher-mass stars have outer radiative envelopes. Differential rotation can be present in such stars as they approach the main sequence due to radially nonuniform contraction. Recently, Alecian et al. (2013) detected rapid (in several years) changes of the global magnetic field on one of such Herbig Ae/Be stars with an extended outer radiation zone. They interpreted these changes as a manifestation of a deep dynamo in the newly-born convective core. However, an alternative explanation is also possible - the dynamo action due to baroclinic instability in a differentially rotating radiative envelope.", "pages": [ 15, 16 ] }, { "title": "CONCLUDING REMARKS", "content": "Linear analysis does not permit determination of the final state to which instability growth will lead. However, one might expect fully developed turbulence in view of the great variety of baroclinic instability modes. Turbulence in the radiation zone, irrespective of its source, is highly anisotropic with a predominance of horizontal flows, u 2 r /u 2 ∼ Ω 2 / ( τ 2 N 4 ) /lessmuch 1 , where τ is the eddy turnover time (Kitchatinov & Brandenburg 2012). Such turbulence efficiently transports angular momentum, removing rotation inhomogeneity. Note that the transport of angular momentum by anisotropic turbulence is not reduced to the action of eddy viscosity (Lebedinskii 1941). There are nondissipative angular momentum flows; as a result, the smoothing of rotation inhomogeneities in stellar radiation zones is much faster than the diffusion of chemical species. Since the threshold value of differential rotation for the onset of baroclinic instability is very low (Fig. 2), this instability can lead to an essentially uniform rotation of the radiation zone, which is revealed by helioseismology. Baroclinic instability can also have a bearing on the origin of magnetic fields in stellar radiation zones. The convective instability in rotating stars is known to be capable of generating magnetic fields. The helicity of convective flows plays the most important role in this process (see, e.g., Vainshtein et al. 1980). The growing global modes of baroclinic instability also possess helicity and may be capable of generating magnetic fields. Baroclinic instability has a bearing not only on stars. Already Tassoul & Tassoul (1983) pointed to this instability as a possible cause of turbulence in accretion disks. Subsequently, Klahr & Bodenhaimer (2003) analyzed this possibility. The so-called stratorotational instability of a Couette flow (Shalybkov & Rüdiger 2005) is also most likely of the baroclinic type. Figure 2 shows that the threshold rotation inhomogeneity for the onset of instability decreases with increasing radial scale of the g -modes. Therefore, a stability analysis for disturbances that are global not only horizontally but also radially can be a perspective for further study of baroclinic instability. Acknowledgements. This work was supported by the Russian Foundation for Basic Research (project no. 12-02-92691_Ind) and the Ministry of Education and Science of the Russian Federation (contract 8407 and State contract 14.518.11.7047).", "pages": [ 16, 17 ] }, { "title": "REFERENCES", "content": "Acheson, D. J. 1978, Phyl. Trans. Roy. Soc. London A289 , 459 Alecian, E., Neiner, C., Mathis, S. et al. 2013, A&A 549 , L8 Barnes, S. A. 2003, ApJ 586 , 464 Chandrasekhar, S. 1961, Hydrodynamic and Hydromagnetic Stability, Oxford, Clarendon Press, p.622 Charbonneau, P., Dikpati, M., & Gilman, P. A. 1999, ApJ 526 , 523 Charbonnel, C., & Talon, S. 2005, Science 309 , 2189 Denissenkov, P. A., Pinsonneault, M., Terndrup, D. M., & Newsham, G. 2010, ApJ 716 , 1269 Dicke, R. H. 1970, ARA&A 8 , 297 Dudorov, A. E., Krivodubskii, V. N., Ruzmaikina, T. V., & Ruzmaikin, A. A. 1989, Astron. Rep. 66 , 809 Gilman, P. A., Dikpati, M., & Miesch, M. S. 2007, ApJS 170 , 203 Goldreich, P, & Schubert, G. 1967, ApJ 150 , 571 Hartmann, L.W., & Noyes, R.W. 1987, ARA&A 25 , 271 Kippenhahn, R., & Weigert, A. 1990, Stellar Structure and Evolution, Berlin, Springer Kitchatinov, L. L. 2008, Astron. Rep. 85 , 279 Kitchatinov, L. L. 2010, Astron. Rep. 87 , 3 Kitchatinov, L. L., & Brandenburg, A. 2012, Astron. Nachr. 333 , 230 Kitchatinov, L. L., & Rüdiger, G. 2008, A&A 478 , 1 Kitchatinov, L. L., Jardine, M., & Collier Cameron, A. 2001, A&A 374 , 250 Klahr, H., & Bodenhaimer, P. 2003, ApJ 582 , 869 Korycansky, D. G. 1991, ApJ 381 , 515 Lebedinskii, A. I. 1941, Astron. Zh. 18 , 10 Pinsonneault, M. 1997, ARA&A 35 , 557 Rüdiger, G., Kitchatinov, L. L., & Elsther, D. 2012, MNRAS 425 , 2267 Shalybkov, D., & Rüdiger, G. 2005, A&A 438 , 411 Shibahashi, H. 1980 PASJ 32 , 341 Shou, J., Antia, H. M., Basu, S. et al. 1998, ApJ 505 , 390 Skumanich, A. 1972, ApJ 171 , 565 Spruit, H. C. 1987, The Internal Solar Angular Velocity , Ed. B.R.Durney, S.Sofia, Dordrecht: D. Reidel Publ., p.185 Spruit, H. C., & Knobloch, E. 1984, A&A 132 , 89 Tassoul, J.-L. 1982, Theory of Rotating Stars , Princeton, Princeton Univ. Press Tassoul, M., & Tassoul, J.-L. 1983, ApJ 271 , 315 Turbulent Dynamo in Vainshyein, S. I., Zeldovich, Ya. B., & Ruzmaikin, A. A. 1980, Astrophysics , Moscow, Nauka (in Russian)", "pages": [ 18, 19 ] } ]

[ { "title": "EVOLUTION AND PULSATION PERIOD CHANGE IN THE LARGE MAGELLANIC CLOUD CEPHEIDS", "content": "Yu. A. Fadeyev ∗ Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya ul. 48, Moscow, 109017 Russia Received June 3, 2013 Abstract -Theoretical estimates of the pulsation period change rates in LMC Cepheids are obtained from consistent calculation of stellar evolution and nonlinear stellar pulsation for stars with initial chemical composition X = 0 . 7, Z = 0 . 008, initial masses 5 M /circledot ≤ M ZAMS ≤ 9 M /circledot and pulsation periods ranged from 2.2 to 29 day. The Cepheid hydrodynamical models correspond to the evolutionary stage of thermonuclear core helium burning. During evolution across the instability strip in the HR diagram the pulsation period Π of Cepheids is the quadratic function of the evolution time for the both fundamental mode and first overtone. Cepheids with initial masses M ZAMS ≥ 7 M /circledot pulsate in the fundamental mode and the period change rate ˙ Π varies nearly by a factor of two for both crossings of the instability strip. In the period - period change rate diagram the values of Π and ˙ Π concentrate within the strips, their slope and halfwidth depending on both the direction of the movement in the HR-diagram and the pulsation mode. For oscillations in the fundamental mode the half-widths of the strip are δ log ˙ Π = 0 . 35 and δ log ˙ Π = 0 . 2 for the first and the secon crossings of the instability strip, respectively. Results of computations are compared with observations of nearly 700 LMC Cepheids. Within existing observational uncertainties of ˙ Π the theoretical dependences of the period change rate on the pulsation period are in a good agreement with observations. Keywords: stars: variable and peculiar.", "pages": [ 1 ] }, { "title": "introduction", "content": "Periods of light variations in many δ Cep pulsating type variables (Cepheids) are known with eight significant digits (Samus et al. 2012). So high accuracy of determination of the period Π is due excellent repetition of pulsation motions and also is owing to the fact that photographic observations of Cepheids are carried out since the end of the XIX century, so that photometric measurements of some stars of this type cover as many as several thousands oscillation cycles. In such a case the long-term observations allow us to significantly correct the value of the period with the O -C diagram. At the same time as early as in the thirties of the XX century the O -C diagrams of some Cepheids were found to have the quadratic term indicating the secular period change (Kukarkin and Florja 1932). Interest in such a property grew after works by Hofmeister et al. (1964) and Iben (1966) where the evolutionary state of Cepheids was determined and long-term period changes were thought to be due to evolutionary changes of the stellar structure during thermonuclear core helium burning (Fernie 1979; Mahmoud and Szabados 1980; Szabados 1983; Deasy and Wayman 1985). In recent years a great deal of observational data on long-term pulsation period changes in the Large Magellanic Cloud (LMC) Cepheids was obtained in ASAS, MACHO and OGLE projects. Pietrukowicz (2001) considered data on 378 LMC Cepheids and concluded that all studied variables with periods longer 8 days show period changes. Later Poleski (2008) carried out an analysis of 655 LMC Cepheids and found that 18% of fundamental mode and 41% of first overtone pulsators have evolutionary period changes. The estimation of the period change rate ˙ Π from observations is of great interest since it provides with the direct test of the stellar evolution theory. Unfortunately, theoretical studies of pulsation period changes in Cepheids based on consistent solution of the equations of stellar evolution and stellar pulsation have not been done yet. Pietrukowicz (2001) compared his observational data with evolutionary and pulsation models studied by Alibert et al. (1999) and Bono et al. (2000). However the pulsation period change rates ˙ Π were not evaluated in these theoretical works, so that Pietrukowicz (2001) used rough estimates from presented tabular data. Moreover, Alibert et al. (1999) in their linear analysis of pulsational instability did not take into account effects of convection. Such a simplification might be responsible for a large disagreement between theoretical models and observational estimates of ˙ Π (Pietrukowicz 2001). Poleski (2008) compared his observational data with stellar evolution theory using the approach by Turner et al. (2006) which is also based on strong simplifications. The goal of the present work is to obtain theoretical estimates of the pulsation period change rate ˙ Π as a function of the age of the Cepheid using the consistent calculations of stellar evolution and nonlinear stellar pulsation. Initial relative mass abundances of hydrogen and elements heavier than helium correspond to the LMC chemical composition: X = 0 . 7, Z = 0 . 008. In hydrodynamical calculations of nonlinear stellar pulsations we take into account effects of turbulent convection, so that the hydrodynamical models occupy the whole interval of effective temperatures bounded in the Hertzsprung-Russel (HR) diagram by the blue and red edges of the instability strip. Methods of stellar evolution calculation and basic equations of radiation hydrodynamics and turbulent convection used for calculation of nonlinear stellar pulsation are given in our previous paper (Fadeyev 2013).", "pages": [ 1, 2 ] }, { "title": "results of computations", "content": "Solution of the equations of hydrodynamics for nonlinear stellar oscillations as a function of time t was done with initial conditions taken in the form of stellar models of evolutionary sequences of stars with initial masses 5 M /circledot ≤ M ZAMS ≤ 9 M /circledot . Evolutionary tracks in the HR diagram of stars under consideration are shown in Fig. 1 where in dotted lines are shown parts of the track corresponding to the instability against radial oscillations. In the starting and in the ending track points the rate of the thermonuclear energy generation rate ε n , c and the rate of the gravitational energy production ε g , c in the stellar center are nearly the same: ε n , c ≈ ε g , c . Therefore the tracks displayed in Fig. 1 represent the evolutionary stage when the only source of energy generation in the stellar center is thermonuclear helium burning. For each evolutionary track the bounds of pulsational instability in the HR diagram were determined from hydrodynamical computations where as in our previous work (Fadeyev 2013) the kinetic energy of pulsation motions E K was calculated. The part of the evolutionary track with pulsational instability was determined from condition η > 0, where η = Π -1 d ln E Kmax /dt is the growth rate of the kinetic energy, E Kmax is the maximum value of the kinetic energy reached during one pulsational cycle. The pulsation period Π was evaluated from the discrete Fourier transform of the kinetic energy E K . It should be noted that the interval of time t within of which we integrate equations of hydrodynamics is comparable with the thermal scale of outer layers of the Cepheid and is much shorter in comparison with the nuclear evolution time scale. Fot example, in the Cepheid with initial mass M ZAMS = 7 M /circledot the evolution time between the red and blue edges of the instability strip is ∼ 10 5 years, whereas hydrodynamic computations of the instability growth with subsequent limit cycle attainment are done on the time interval of ∼ 10 years. In Fig. 2 we give the plots of the instability growth rate η versus effective temperature averaged over the pulsational cycle 〈 T eff 〉 for four Cepheid evolutionary sequences with initial masses form 5 M /circledot to 8 M /circledot . The evolutionary track crosses the instability strip twice in the HR diagram and therefore each evolutionary sequence is represented by two plots where the first one correponds to the movement across the HR diagram with increasing effective temperature (dotted lines) and the second plot corresponds to the movement in the opposite direction (dash-dotted lines). In Fig. 2 we use the averaged over the cycle effective temperature 〈 T eff 〉 as independent variable because in the hydrodynamical model the average radius of the photosphere 〈 r ph 〉 is smaller than the radius of the photosphere of the hydrostatically equilibrium model r ph , 0 . For hydrodynamical models of Cepheids calculated in the present study the ratio of the photosphere radii ranges within 0 . 975 ≤ 〈 r ph 〉 /r ph , 0 < 1 and edges of the instability strip shift to the blue in the HR diagram by 30 K < ∆ T eff < 70 K. The presence of two maxima in plots of η for M ZAMS ≤ 6 M /circledot is due to the fact that near the red edge of the instability strip radial pulsations are excited in the fundamental mode, whereas at higher effective temperatures pulsations are excited in the first overtone. Transition between oscillation modes takes place within the effective temperature range 6000 K < 〈 T eff 〉 < 6100 K. In Cepheids with initial mass M ZAMS = 7 M /circledot evolving blueward across the instability strip radial oscillations are due to instability of the fundamental mode and transition to the first overtone takes place just near the blue edge at 〈 T eff 〉 ≈ 6200 K. During the second crossing of the instability strip radial oscillations exist in the form of the fundamental mode. Pulsations of Cepheids with initial mass M ZAMS ≥ 8 M /circledot are always due to instability of the fundamental mode and the pulsation period Π gradually changes while the star moves in the HR diagram from one edge of the instability strip to another. The change of the pulsation period of the Cepheid with initial mass M ZAMS = 8 M /circledot is illustrated in Fig. 3 where for the sake of convenience we set the evolution time t ev to zero when the star crosses the edge of the instability strip and begins to oscillate. The plot with gradual decrease of the pulsation period corresponds to the first crossing of the instability strip and the plot with gradually increasing period corresponds to the second crossing. Hydrodynamical models with positive and negative growth rates η are shown in filled circles and opened circles, respectively. As is seen from shown plots the pulsation period Π is fitted by an algebraic polynomial Π( t ev ) = a 0 + a 1 t ev + a 2 t 2 ev for both evolutionary sequences with a good accuracy (i.e. with relative r.m.s. error less than one per cent). Polynomial approximation is shown in Fig. 3 by dotted and dash-dotted lines for the first crossing and the second crossing of the instability strip, respectively. Expression of the pulsation period Π as a quadratic polynomial of t ev was found to be a good approximation for all Cepheid models considered in the present study. The only exception is a discontinuity of the period due to transition from one pulsation mode to other. This is illustrated in Fig. 4 by the plots of the pulsation period for the Cepheid with initial mass M ZAMS = 6 M /circledot . However within the interval of the continuous change of Π the quadratic polynomial remains a quite good approximation. The quadratic dependence of the pulsation period Π on the evolutionary time t ev implies the linear change of ˙ Π which decreases during the first crossing of the instability strip and increases during the next crossing. In Cepheids pulsating in the fundamental mode within the whole instability strip the period change rate ˙ Π varies roughly by a factor of two. Typical values of ˙ Π can be found in the table where for the evolutionary sequences with initial masses 5 M /circledot ≤ M ZAMS ≤ 9 M /circledot we give the main properties of Cepheids at the points where the evolutionary track crosses the edges of the instability strip. Each evolutionary sequence is represented by four lines where the first pair of lines corresponds to the first crossing of the instability strip and the second pair of lines corresponds to the second crossing. In the second column of the table we give the evolution time ∆ t ev spent by the Cepheid within instability strip. In following columns we give main parameters of the Cepheid at the edge of the instability strip (i.e. for η = 0) which were obtained by linear interpolation of model parameters of adjacent hydrodynamical models with opposite signs of the growth rate of kinetic energy. The model parameters are as follows: the stellar mass M which is less than the initial mass M ZAMS due to effects of the stellar wind during the preceding evolution; the averaged over the cycle absolute bolometric luminosity L and effective temperature 〈 T eff 〉 ; the pulsation period Π, the dimensionless pulsation period change rate ˙ Π; the order of the pulsation mode k ( k = 0 for the fundamental mode and k = 1 for the first overtone).", "pages": [ 3, 4, 5 ] }, { "title": "comparison with observations", "content": "The period of light variations is the only quantity which can be determined from observations of the pulsating variable star with sufficiently high precision. Therefore for comparison of the results of theoretical computations with observational data we will consider the period change rate ˙ Π as a function of the pulsation period Π. In Fig. 5 the plots of the dimensionless period change rate ˙ Π are shown as a function of the pulsation period Π for the Cepheid models crossing blueward the instability strip. The plots are separated into two groups with Cepheids pulsating in the fundamental mode (6 . 9 day ≤ Π ≤ 28 day) and those pulsating in the first overtone (2 . 2 day ≤ Π ≤ 4 . 5 day). The plots locate along the dashed lines which are approximately given by following relations where the period Π is expressed in days, whereas k = 0 and k = 1 correspond to the fundamental mode and to the first overtone, respectively. Due to the finite width of the instability strip the plots of evolutionary sequences shown in Fig. 5 by solid lines are confined within the bands with half-width δ log ˙ Π = 0 . 035 for k = 0 and δ log ˙ Π ≈ 0 . 1 for k = 1. The diagram period - period change rate for Cepheids of the second crossing of the instability strip is shown in Fig. 6. Unfortunately, reliable estimates of the period change rate for Cepheid models with initial mass M ZAMS = 5 M /circledot evolving redward in the HR diagram were not obtained, so that in Fig. 6 the mean dependence of the period change rate is shown only for the fundamental mode: The half-width of the band in the period - period change rate diagram is δ log ˙ Π = 0 . 2. To compare results of our theoretical computations with observations we used observational estimates of the period Π and the period change rate ˙ Π from works by Pietrukowicz (2001) and Poleski (2008). Electronic tables 1 - 3 supplementing the paper by Pietrukowicz (2001) give data on 369 LMC Cepheids, whereas the period change rates were evaluated using the Harvard photographic observations obtained in time interval from 1910 to 1950. Observational data on LMC Cepheids obtained from the OGLE survey were received from the author (Poleski 2008). The diagram period - period change rate for Cepheids crossing the instability strip blueward with negative ˙ Π is presented in Fig. 7a and the same diagram for Cepheids evolving redward with positive ˙ Πis shown in Fig. 7b. Results of observations obtained by Pietrukowicz (2001) and Poleski (2008) are shown by filled circles and open circles, respectively. Theoretical dependences obtained in the present study for Cepheid evolutionary sequences with initial masses from 5 to 9 M /circledot are shown in solid lines. In general one can conclude that the theory of stellar evolution agrees with observations of Cepheids. At the same time one should note a disagreement between observational results by Pietrukowicz (2001) and those by Poleski (2008). A possible cause of such a difference seems to be a shorter time interval used by Poleski (2008) for evaluation of the Cepheid period changes.", "pages": [ 5, 6 ] }, { "title": "conclusion", "content": "Results of our calculations allow us to conclude that the oscillation period Π of the Cepheid is the quadratic function of the evolution time t ev and when the star crosses the instability strip the quantity ˙ Π changes by a factor of two. Earlier Deasy and Wayman (1985) noted the occurence of non-constant period change in LMC Cepheids. An interesting result of our calculations is that the dependence of the period change rate ˙ Π on the pulsation period Π for the fundamental mode differs from that for the first overtone. Unfortunately, at present observational confirmation of this feature seems to be impossible because of insufficiently high accuracy of observational evaluation of ˙ Π for the first overtone Cepheids. Significant scatter of points in Fig. 7 at short periods (Π < 7 day) is mainly due to the power law decrease of ˙ Π with decreasing pulsation period. Indeed, as is seen in Fig. 5 the period change rates in first overtone Cepheids are three orders of magnitude smaller in comparison with those in long period (Π ≈ 30 day) Cepheids. Therefore, to reduce the error of the observational estimate of ˙ Π in first overtone pulsators to the value comparable with that in long period Cepheids the time interval of the O -C diagram should be expanded by two orders of magnitude. Thus, for comparison of the stellar evolution calculations with observations of most interest are fundamental mode Cepheids. Here among the important questions we should emphasize the role of chemical composition and convective overshooting in the period - period change rate diagram. It should also be noted that the crossing time of the Cepheid instability strip by core helium burning stars with initial masses M ZAMS > 9 M /circledot becomes comparable with that during the gravitational contraction of the helium core before the stage of the red supergiant. Thefore the most massive long period Cepheids with gravitationally contracting core may play perceptible role in the period - period change rate diagram. The author thanks Radek Poleski who kindly placed the Cepheid OGLE data at his disposal. The study was supported by the Basic Research Program of the Russian Academy of Sciences 'Nonstationary phenomena in the Universe'.", "pages": [ 6, 7 ] } ]

[ { "title": "V.V. Bobylev", "content": "Pulkovo Astronomical Observatory, St. Petersburg, Russia Sobolev Astronomical Institute, St. Petersburg State University, Russia Abstract -The space velocities of 200 long-period ( P > 5 days) classical Cepheids with known proper motions and line-of-sight velocities whose distances were estimated from the period-luminosity relation have been analyzed. The linear Ogorodnikov-Milne model has been applied, with the Galactic rotation having been excluded from the observed velocities in advance. Two significant gradients have been found in the Cepheid velocities, ∂W/∂Y = -2 . 1 ± 0 . 7 km s -1 kpc -1 and ∂V/∂Z = 27 ± 10 km s -1 kpc -1 . In such a case, the angular velocity of solid-body rotation around the Galactic X axis directed to the Galactic center is -15 ± 5 km s -1 kpc -1 .", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "As analysis of the large-scale structure of neutral hydrogen showed, a warp of the gas disk is observed in the Galaxy (Westerhout 1957). The results of studying this structure using the currently available data on the HI and HII distributions are presented in Kalberla and Dedes (2008) and Cersosimo et al. (2009), respectively. The warp is seen in the distribution of stars and dust (Drimmel and Spergel 2001), pulsars (Yusifov 2004), OB stars from the Hipparcos catalogue (Miyamoto and Zhu 1998), and in the distribution of 2MASS red-giant-clump stars (Momany et al. 2006). The system of Cepheids also exhibits a similar feature (Fernie 1968; Berdnikov 1987; Bobylev 2013). Of great interest are the attempts to find a relationship between the kinematics of stars and the disk warp (Miyamoto et al. 1993; Miyamoto and Zhu 1998; Drimmel et al. 2000; Bobylev 2010). In particular, based on the proper motions of O-B5 stars, Miyamoto and Zhu (1998) found a positive rotation of this system of stars around the Galactic x axis with an angular velocity of about +4 km s -1 kpc -1 . In contrast, based on the proper motions of about 80 000 red-giant-clump stars, Bobylev (2010) found an opposite rotation of this system of stars around the x axis with an angular velocity of about -4 km s -1 kpc -1 . The stellar proper motions alone do not allow complete information to be obtained. In this respect, although the Cepheids are not all that many, they are a unique tool for studying the three-dimensional kinematics of the Galaxy: the distances, proper motions, and line-of-sight velocities are known for them. A number of models were proposed to explain the Galactic warp: (1) the interaction between the disk and a nonspherical dark matter halo (Sparke and Casertano 1988); (2) the gravitational influence from the Galaxy's nearest satellites (Bailin 2003); (3) the interaction of the disk with the flow near the Galaxy formed by high-velocity hydrogen clouds that resulted from mass exchange between the Galaxy and the Magellanic Clouds (Olano 2004); (4) the intergalactic flow (L'opez-Corredoira et al. 2002); and (5) the interaction with the intergalactic magnetic field (Battaner et al. 1990). Note that the term 'warp' implies some nonlinear dependence. However, we attempt to find a relationship between the kinematics of stars and the warp of the hydrogen layer in the form of a simple linear approach. For this purpose, we search, for example, for the rotation of the symmetry plane of the system of stars around some axis. Since the symmetry plane of the Cepheid system is inclined to the Galactic plane at an angle of ≈ -2 · in a direction of ≈ 270 · (Bobylev 2013), the most suitable manifestation of the relationship is the rotation of the system around the Galactic x axis. The goal of this study is to reveal the relationship between the Cepheid velocities and the warp of the stellar-gaseous Galactic disk. For this purpose, we use a sample of long-period classical Cepheids with measured proper motions and line-of-sight velocities and estimate their distances from the period-luminosity relation. We apply the linear Ogorodnikov-Milne model for our analysis and exclude the Galactic rotation from the observed velocities in advance, focusing our attention on the motion in the XZ and Y Z planes.", "pages": [ 1, 2 ] }, { "title": "DATA", "content": "We use Cepheids of the Galaxy's flat component classified as DCEP, DCEPS, CEP(B), CEP in the GCVS (Kazarovets et al. 2009) as well as CEPS used by other authors. To determine the distance based on from the period-luminosity relation, we used the calibration from Fouqu'e et al. (2007): 〈 M V 〉 = -1 . 275 -2 . 678 log P, where the period P is in days. Given 〈 M V 〉 , taking the period-averaged apparent magnitudes 〈 V 〉 and extinction A V = 3 . 23 E ( 〈 B 〉 - 〈 V 〉 ) mainly from Acharova et al. (2012) and, for several stars, from Feast and Whitelock (1997), we determine the distance r from the relation For a number of Cepheids (without extinction data), we used the distances from the catalog by Berdnikov et al. (2000) determined from infrared photometry. Data from Mishurov et al. (1997) and Gontcharov (2006) as well as from the SIMBAD and DDO databases served as the main sources of line-of-sight velocities for Cepheids. As a rule, the proper motions were taken from the UCAC4 catalog (Zacharias et al. 2013) and, in several cases, from TRC (Hog et al. 2000). Proceeding from the goals of our study, we concluded that it would be better not to use several stars located above the Galactic plane by more than 2 kpc and deep in the inner Galaxy. Thus, we used the constraints satisfied by 205 Cepheids. When calculating the velocity errors, we assumed the distance error to be 10%. In particular, the constraint for σ V in (2) is the random error in the total space velocity of a star. The constraint on the pulsation period P was chosen from the following considerations. Our analysis of the distribution of classical Cepheids (Bobylev 2013) shows that the oldest Cepheids with periods P < 5 d have a significantly different orientation than younger Cepheids. Bobylev et al. (2008) found the parameters of the Galactic rotation curve containing six terms of the Taylor expansion of the angular velocity of Galactic rotation Ω 0 for the Galactocentric distance of the Sun R 0 = 7 . 5 kpc. Data on hydrogen clouds at tangential points, on massive star-forming regions, and on the velocities of young open star clusters were used for this purpose. The more up-to-date value of R 0 is 8 kpc (Foster and Cooper 2010). Therefore, the parameters of the Galactic rotation curve were redetermined using the same sample but for R 0 = 8 kpc: Based on a sample of Cepheids, Bobylev and Bajkova (2012) found Ω 0 = -27 . 5 ± 0 . 5 km s -1 kpc -1 , Ω ' 0 = 4 . 12 ± 0 . 10 km s -1 kpc -2 and Ω '' 0 = -0 . 85 ± 0 . 07 km s -1 kpc -3 , which are in good agreement with the corresponding values (3). At the same time, the parameters (3) allow the Galactic rotation curve to be constructed in a wider range of Galactocentric distances R. This rotation curve is shown in Fig. 1. The parameters (3) were used to analyze the peculiar velocity V pec in (2). The constraint on the magnitude of V pec is an indirect constraint on the radius of the sample, which is r ≈ 6 kpc is our case.", "pages": [ 2, 3 ] }, { "title": "THE MODEL", "content": "We use a rectangular Galactic coordinate system with its axes directed from the observer toward the Galactic center (the x axis or axis 1), in the direction of Galactic rotation (the y axis or axis 2), and toward the North Galactic Pole (the z axis or axis 3). We apply the linear Ogorodnikov-Milne model (Ogorodnikov 1965), where the observed velocity V ( r ) of a star with a heliocentric radius vector r is described, to terms of the first order of smallness r/R 0 /lessmuch 1 , by the vector equation Here, V /circledot ( X /circledot , Y /circledot , Z /circledot ) is the Sun's peculiar velocity relative to the stars under consideration, V ' is the star's residual velocity, M is the displacement matrix (tensor) whose components are the partial derivatives of the velocity u ( u 1 , u 2 , u 3 ) with respect to the distance r ( r 1 , r 2 , r 3 ) , where u = V ( R ) -V ( R 0 ), R and R 0 are the Galactocentric distances of the star and the Sun, respectively. Then, taken at R = R 0 . All nine elements of the matrix M can be determined using three components of the observed velocities - the line-of-sight velocities V r and stellar proper motions µ l cos b, µ b : V r = -X /circledot cos b cos l -Y /circledot cos b sin l -Z /circledot sin b + + r [cos 2 b cos 2 lM 11 +cos 2 b cos l sin lM 12 +cos b sin b cos lM 13 + +cos 2 b sin l cos lM 21 +cos 2 b sin 2 lM 22 +cos b sin b sin lM 23 + +sin b cos b cos lM 31 +cos b sin b sin lM 32 +sin 2 bM 33 ] , 4 . 74 rµ l cos b = X /circledot sin l -Y /circledot cos l + + r [ -cos b cos l sin lM 11 -cos b sin 2 lM 12 -sin b sin lM 13 + +cos b cos 2 lM 21 +cos b sin l cos lM 22 +sin b cos lM 23 ] , 4 . 74 rµ b = X /circledot cos l sin b + Y /circledot sin l sin b -Z /circledot cos b + + r [ -sin b cos b cos 2 lM 11 -sin b cos b sin l cos lM 12 -sin 2 b cos lM 13 --sin b cos b sin l cos lM 21 -sin b cos b sin 2 lM 22 -sin 2 b sin lM 23 + +cos 2 b cos lM 31 +cos 2 b sin lM 32 +sin b cos bM 33 ] . (6) It is useful to divide the matrix M into its symmetric, M + (local deformation tensor), and antisymmetric, M -(rotation tensor), parts: where the subscript 0 means that the derivatives are taken at R = R 0 . The quantities M -32 , M -13 and M -21 are the components of the solid-body rotation vector of a small solar neighborhood around the x, y, z axes, respectively. In accordance with our chosen rectangular coordinate system, the positive rotations are those from axis 1 to axis 2 (Ω z ), from Note. The velocities X /circledot , Y /circledot , and Z /circledot are in km s -1 ; the remaining parameters are in km s -1 kpc -1 . axis 2 to axis 3 (Ω x ), and from axis 3 to axis 1 (Ω y ): The quantity M -21 is equivalent to the Oort constant B. Each of the quantities M + 12 , M + 13 and M + 23 describes the deformation in the corresponding plane; in particular, M + 12 is equivalent to the Oort constant A. The diagonal elements of the local deformation tensor M + 11 , M + 22 and M + 33 describe the general local compression or expansion of the entire stellar system (divergence). The set of conditional equations (6) includes twelve sought-for unknowns to be determined by the least-squares method.", "pages": [ 4, 5 ] }, { "title": "RESULTS AND DISCUSSION", "content": "The table gives the parameters of the Ogorognikov-Milne model found by simultaneously solving the set of equations (6) using a sample of 200 Cepheids. Two solutions are presented. The point is that the ICRS/Hipparcos (1997) system, whose extension is the UCAC4 catalog we use, has a small residual rotation relative to the initial frame of reference. The equatorial components of this vector are ( ω x , ω y , ω z ) = ( -0 . 11 , 0 . 24 , -0 . 52) ± (0 . 14 , 0 . 10 , 0 . 16) mas yr -1 (Bobylev 2010). Therefore, the table gives the parameters calculated for two cases: when the Cepheid proper motions were not corrected and when they were derived from the stellar proper motions after applying the correction ω z = -0 . 52 mas yr -1 . There are no significant differences between the two solutions. However, it can be noted that with the corrected proper motions, the parameter M 13 decreased to zero and the parameter M 32 slightly increased, which is important to us. Therefore, below we will use the results from the last column of the table. The XY plane. Since M 12 = -Ω 0 , the value of M 12 = -2 . 9 ± 0 . 7 km s -1 kpc -1 found shows that the Cepheid velocities were slightly overcorrected (we should have used Ω 0 ≈ -26 km s -1 kpc -1 precisely for this sample). This is of no serious importance for the goals of our study, because this is just a linear shift, while the nonlinear character of the Galactic rotation curve was taken into account well. The remaining parameters describing the kinematics in the XY plane, M 11 , M 21 and M 22 , are close to zero. The XZ plane. As can be seen from the table, none of the coefficients M 11 , M 13 , M 31 and M 33 , describing the kinematics in this plane differs significantly from zero. Figure 2 displays the corresponding distributions of stars. The YZ plane. Figure 3 shows the distributions of stars; the solid lines indicate two dependences plotted according to the data from the table: M 23 = ∂V/∂Z = 32 . 4 ± 11 . 9 km s -1 kpc -1 and M 32 = ∂W/∂Y = -2 . 1 ± 0 . 7 km s -1 kpc -1 . We refined the coefficient M 23 = 26 . 8 ± 10 . 2 km s -1 kpc -1 using a graphical method. For this purpose, we calculated the dependence V = f ( Z ) from the data of the corresponding graph in Fig. 3 with the constraint | Z | > 0 . 040 kpc (136 stars were used). Let us now consider the displacement tensor M W that we associate with the influence of the disk warp on the motion of the Cepheid system: According to the data from the table, both of its diagonal elements can be set equal to zero. This means that there are no motions like expansion-compression in this plane. Then, the deformation tensor (7) takes the form and the rotation tensor (7) is Based on (12), we may conclude that the angular velocity of solid-body rotation of the Cepheid system around the X axis is Ω W = M -32 = -15 ± 5 km s -1 kpc -1 . This is the minimum (but more reliable) estimate. If the deformations ( M + 23 ) are assumed to be also related to the effect under consideration, then the maximum angular velocity of rotation can be estimated as Ω W = M -32 -M + 23 = -27 ± 10 km s -1 kpc -1 . It is important that the direction of the rotation found (minus sign) is in agreement with the result of our analysis of the proper motions for red-giant clump stars (Bobylev 2010), where we used photometric distance estimates with errors e π /π ≈ 30%. The sign of the angular velocity Ω W depends on the sign of M 32 (Eqs. (7)-(8)), which was determined from Cepheids rather reliably owing to the wide range of coordinates ∆ Y ≈ 10 kpc. Since the range of coordinates ∆ Z ≈ 1 . 2 kpc is small when determining M 23 , the influence of random fluctuations in Cepheid velocities can be significant. The value of Ω W = -15 ± 5 km s -1 kpc -1 derived from Cepheids exceeds Ω W ≈ -4 ± 0 . 5 km s -1 kpc -1 obtained from red-giant-clump stars by Bobylev (2010) by a factor of 4. Such a difference may be related to the sample ages: the mean age of our sample of Cepheids is 77 Myr, while the mean age of the red-giant-clump stars is approximately 1 Gyr. However, this question requires a further study based on larger volumes of more accurate data.", "pages": [ 5, 6, 7, 8 ] }, { "title": "CONCLUSIONS", "content": "We considered the space velocities of about 200 long-period (with periods of more than 5 days) classical Cepheids with known proper motions and line-of-sight velocities whose distances were estimated from the period-luminosity relation. We applied the linear Ogorodnikov-Milne model to analyze their kinematics. The Galactic rotation that we found based on a more complex model was excluded from the observed velocities in advance. Two significant gradients were detected in the Cepheid velocities: ∂W/∂Y = -2 . 1 ± 0 . 7 km s -1 kpc -1 and ∂V/∂Z = 27 ± 10 km s -1 kpc -1 . This leads us to conclude that the angular velocity of solid-body rotation around the Galactic x axis is Ω W = -15 ± 5 km s -1 kpc -1 , which we associate with a manifestation of the warp of the stellar-gaseous Galactic disk. Indeed, the relationship between the spatial distribution of Cepheids and the warp of the stellar-gaseous Galactic disk may be considered to have been firmly established (Fernie 1968; Berdnikov 1987; Bobylev 2013). The results of our study show that the kinematic relationship of Cepheids to this phenomenon is also highly likely. The method considered here can be useful for a future analysis of large volumes of data, for example, from the GAIA space experiment or on masers with their trigonometric parallaxes measured by VLBI.", "pages": [ 8 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are grateful to the referee for helpful remarks that contributed to a improvement of the paper. This work was supported by the 'Nonstationary Phenomena in Objects of the Universe' Program of the Presidium of the Russian Academy of Sciences and the 'Multiwavelength Astrophysical Research' grant no. NSh-16245.2012.2 from the President of the Russian Federation.", "pages": [ 8, 9 ] } ]

[ { "title": "ABSTRACT", "content": "Noname manuscript No. (will be inserted by the editor)", "pages": [ 1 ] }, { "title": "Astrophysical black holes as natural laboratories for fundamental physics and strong-field gravity", "content": "Emanuele Berti Received: date / Accepted: date Abstract Astrophysical tests of general relativity belong to two categories: 1) 'internal', i.e. consistency tests within the theory (for example, tests that astrophysical black holes are indeed described by the Kerr solution and its perturbations), or 2) 'external', i.e. tests of the many proposed extensions of the theory. I review some ways in which astrophysical black holes can be used as natural laboratories for both 'internal' and 'external' tests of general relativity. The examples provided here (ringdown tests of the black hole 'no-hair' theorem, bosonic superradiant instabilities in rotating black holes and gravitational-wave tests of massive scalar-tensor theories) are shamelessly biased towards recent research by myself and my collaborators. Hopefully this colloquial introduction aimed mainly at astrophysicists will convince skeptics (if there are any) that space-based detectors will be crucial to study fundamental physics through gravitational-wave observations. Keywords General Relativity · Black Holes · Gravitational Radiation", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The foundations of Einstein's general relativity (GR) are very well tested in the regime of weak gravitational fields, small spacetime curvature and small velocities [1]. It is generally believed, on both theoretical and observational grounds (the most notable observational motivation being the dark energy problem), that Einstein's theory will require some modification or extension at high energies and strong gravitational fields, and these modifications generally require the introduction of additional degrees of freedom in the theory [2]. California Institute of Technology, Pasadena, CA 91109, USA E-mail: [email protected] Because GR is compatible with all observational tests in weak-gravity conditions, a major goal of present and future experiments is to probe astrophysical systems where gravity is, in some sense, strong. The strength of gravity can be measured either in terms of the gravitational field ϕ ∼ M/r , where M is the mass and r the size of the system in question 1 , or in terms of the curvature [3]. A quantitative measure of curvature are tidal forces, related to the components R r 0 r 0 ∼ M/r 3 of the Riemann tensor associated to the spacetime metric g ab [4]. The field strength is related to typical velocities of the system by the virial theorem ( v ∼ ϕ 1 / 2 ∼ √ M/r ) so it is essentially equivalent to the post-Newtonian small velocity parameter v (or v/c in 'standard' units). One could argue that 'strong curvature' is in some ways more fundamental than 'strong field', because Einstein's equations relate the stress-energy content of the spacetime to its curvature (so that 'curvature is energy') and because the curvature (not the field strength) enters the Lagrangian density in the action principle defining the theory: cf. e.g. Eq. (1) below. It is perhaps underappreciated that in astrophysical systems one can 'probe strong gravity' by observations of weak gravitational fields, and vice versa, observations in the strong-field regime may not be able to tell the difference between GR and its alternatives or extensions. The possibility to probe strong-field effects using weak-field binary dynamics is nicely illustrated by the 'spontaneous scalarization' phenomenon discovered by Damour and Esposito-Far'ese [5]. The idea is that the coupling of the scalar with matter can allow some scalar-tensor theories to pass all weak-field tests, while at the same time introducing macroscopically (and observationally) significant modifications in the structure of neutron stars (NSs). If spontaneous scalarization occurs 2 , the masses of the two stars in a binary can in principle be very different from their GR values. Therefore the dynamics of NS binaries will be significantly modified even when the binary members are sufficiently far apart that v ∼ √ M/r /lessmuch 1. For this reason, 'weak-field' observations of binary pulsars can strongly constrain a strong-field phenomenon such as spontaneous scalarization [12]. On the other hand, measurements of gas or particle dynamics in strongfield regions around the 'extremely relativistic' Kerr black hole (BH) spacetime are not necessarily smoking guns of hypothetical modifications to general relativity. The reason is that classic theorems in Brans-Dicke theory [15,16, 17], recently extended to generic scalar-tensor theories and f ( R ) theories [18, 19], show that solutions of the field equations in vacuum always include the Kerr metric as a special case. The main reason is that many generalizations of GR admit the vacuum equations of GR itself as a special case. This conclusion may be violated e.g. in the presence of time-varying boundary conditions, that could produce 'BH hair growth' on cosmological timescales [20] and dynamical horizons [21]. The Kerr solution is so ubiquitous that probes of the Kerr metric alone will not tell us whether the correct theory of gravity is indeed GR. However, the dynamics of BHs (as manifested in their behavior when they merge or are perturbed by external agents [22]) will be very different in GR and in alternative theories. In this sense, gravitational radiation (which bears the imprint of the dynamics of the gravitational field) has the potential to tell GR from its alternatives or extensions. To wrap up this introduction: our best bet to probe strong-field dynamics are certainly BHs and NSs, astronomical objects for which both ϕ ∼ M/r and the curvature ∼ M/r 3 are large. However: 1) there is the definite possibility that weak-field observations may probe strong gravity, as illustrated e.g. by the spontaneous scalarization phenomenon; and 2) measurements of the metric around BH spacetimes will not be sufficient to probe GR, but dynamical measurements of binary inspiral and merger dynamics will be sensitive to the dynamics of the theory.", "pages": [ 1, 2, 3 ] }, { "title": "2 Finding contenders to general relativity", "content": "Let us focus for the moment on 'external' tests, i.e. test of GR versus alternative theories of gravity. What extensions of GR can be considered serious contenders? A 'serious' contender (in this author's opinion) should at the very least be well defined in a mathematical sense, e.g. by having a well posed initial-value problem. From a phenomenological point of view, the theory must also be simple enough to make physical predictions that can be validated by experiments (it is perhaps a sad reflection on the current state of theoretical physics that one should make such a requirement explicit!). An elegant and comprehensive overview of theories that have been studied in the context of space-based gravitational-wave (GW) astronomy is presented in [23]. Here I focus on a special subclass of extensions of GR whose implications in the context of Solar-System tests, stellar structure and GW astronomy have been explored in some detail. I will give a 'minimal' discussion of these theories, with the main goal of justifying the choice of massive scalar-tensor theories as a particularly simple and interesting phenomenological playground. Among the several proposed extensions of GR (see e.g. [2] for an excellent review), theories that can be summarized via the Lagrangian density have rather well understood observational implications for cosmology, Solar System experiments, the structure of compact stars and gravitational radiation from binary systems. In the Lagrangian given above φ is a scalar-field degree of freedom (not to be confused with the gravitational field strength ϕ introduced earlier); R abcd is the Riemann tensor, R ab the Ricci tensor and R the Ricci scalar corresponding to the metric g ab ; Ψ denotes additional matter fields. The functions f i ( φ ) ( i = 0 , 1 , 2), M ( φ ) and A ( φ ) are in principle arbitrary, but they are not all independent. For example, field redefinitions allow us to set either f 0 ( φ ) = 1 or A ( φ ) = 1, which corresponds to working in the so-called 'Einstein' or 'Jordan' frames, respectively. This Lagrangian encompasses models in which gravity is coupled to a single scalar field φ in all possible ways, including all linearly independent quadratic curvature corrections to GR. Scalar-tensor gravity with generic coupling, sometimes called BergmannWagoner theory [24,25], corresponds to setting f 1 ( φ ) = f 2 ( φ ) = 0 in Eq. (1). This is one of the oldest and best-studied modifications of GR. If we further specialize to the case where A ( φ ) = 1, f 0 ( φ ) = φ , /pi1 ( φ ) = ω BD /φ and M ( φ ) = 0 we recover the 'standard' Brans-Dicke theory of gravity in the Jordan frame [26]; the Einstein frame corresponds to setting f 0 ( φ ) = 1 instead. In a Taylor expansion of M ( φ ), the term quadratic in φ introduces a nonzero mass for the scalar (see e.g. [27]). GR is recovered in the limit ω BD →∞ . Initially motivated by attempts to incorporate Mach's principle into GR, scalar-tensor theories have remained popular both because of their relative simplicity, and because scalar fields are the simplest prototype of the additional degrees of freedom predicted by most unification attempts [28]. BergmannWagoner theories are less well studied than one might expect, given their long history 3 . These theories can be seen as the low-energy limit of several proposed attempts to unify gravity with the other interactions or, more pragmatically, as mathematically consistent alternatives to GR that can be used to understand which features of the theory are well-tested, and which features need to be tested in more detail [30]. Most importantly, they meet all of the basic requirements of 'serious' contenders to GR, as defined above. They are well-posed and amenable to numerical evolutions [31], and in fact numerical evolutions of binary mergers in scalar-tensor theories have already been performed for both BH-BH [32] and NS-NS [33] binaries. At present, the most stringent bound on the coupling parameter of standard Brans-Dicke theory ( ω BD > 40 , 000) comes from Cassini measurements of the Shapiro time delay [1], but binary pulsar data are rapidly becoming competitive with the Cassini bound: observations of binary systems containing at least one pulsar, such as the pulsar-white dwarf binary PSR J1738+0333, already provide very stringent bounds on Bergmann-Wagoner theories [12]. The third line of the Lagrangian (1) describes theories quadratic in the curvature. The requirement that the field equations should be of second order means that corrections quadratic in the curvature must appear in the GaussBonnet (GB) combination R 2 GB = R 2 -4 R ab R ab + R abcd R abcd . We also allow for a dynamical Chern-Simons correction proportional to the wedge product R abcd ∗ R abcd [34]. Following [35], we will call these models 'extended scalartensor theories'. These theories have been extensively investigated from a phenomenological point of view: the literature includes studies of Solar-system tests [36,37], BH solutions and dynamics [38,39,40,41,42], NS structure [43,44, 35] and binary dynamics [45,46,47]. While the interest of this class of theories is undeniable, and recent work has highlighted very interesting phenomenological consequences for the dynamics of compact objects, it is presently unclear whether they admit a well defined initial value problem and whether they are amenable to numerical evolutions. In analytical treatments these theories are generally regarded as 'effective' rather than fundamental (see e.g. [45] for a discussion), and treated in a small-coupling approximation that simplifies the field equations and ensures that the field equations are of second order. The Lagrangian (1) is more generic than it may seem. For example, it describes - at least at the formal level - theories that replace the Ricci scalar R by a generic function f ( R ) in the Einstein-Hilbert action, because these theories can always be cast as (rather anomalous) scalar-tensor theories via appropriate variable redefinitions [18,48]. Unfortunately the mapping between f ( R ) theories and scalar-tensor theories is in general multivalued, and one should be very careful when considering the scalar-tensor 'equivalent' of an f ( R ) theory (see e.g. [49]). Recently popular theories that are not encompassed by the Lagrangian above include e.g. Einstein-aether theory [50], Hoˇrava gravity [51], Bekenstein's TeVeS [52], massive gravity theories [53] and 'Eddington inspired gravity' [54], which is equivalent to GR in vacuum, but differs from it in the coupling with matter. An overview of these theories is clearly beyond the scope of this paper. From now on I will focus on the surprisingly overlooked fact that theories of the Bergmann-Wagoner type, which are among the simplest options to modify GR, allow us to introduce very interesting dynamics by simply giving a nonzero mass to the scalar field. Scalar fields predicted in unification attempts are generally massive, so this 'requirement' is in fact very natural. I will now argue that massive scalar fields give rise to extremely interesting phenomena in BH physics (Section 3) and binary dynamics (Section 4).", "pages": [ 3, 4, 5 ] }, { "title": "3 Black hole dynamics and superradiance", "content": "With the caveat that measurements based on the Kerr metric alone do not necessarily differentiate between GR and alternative theories of gravity, BHs are ideal astrophysical laboratories for strong field gravity. Recent results in numerical relativity (see e.g. [55,56]) confirmed that the dynamics of BHs can be approximated surprisingly well using linear perturbation theory (see Chandrasekhar's classic monograph [57] for a review). In perturbation theory, the behavior of test fields of any spin (e.g. s = 0 , 1 , 2 for scalar, electromagnetic and gravitational fields) can be described in terms of an effective potential [57,58]. For massless scalar perturbations of a Kerr BH, the potential is such that: 1) it goes to zero at the BH horizon, which (introducing an appropriate radial 'tortoise coordinate' r ∗ [57]) corresponds to r ∗ →-∞ ; 2) it has a local maximum located (roughly) at the light ring; 3) it tends to zero as r ∗ →∞ . A nonzero scalar mass does not qualitatively alter features 1) and 2), but it creates a nonzero potential barrier such that V → m 2 (where m is the mass of the field in natural units G = c = /planckover2pi1 = 1) at infinity. Because of the nonzero potential barrier, the potential can now accommodate quasibound states in the potential well located between the light-ring maximum and the potential barrier at infinity (cf. e.g. Fig. 7 of [59]). These states are quasibound because the system is dissipative. In fact, under appropriate conditions the system can actually be unstable. The stable or unstable nature of BH perturbations is detemined by the shape of the potential and by a well-known feature of rotating BHs: the possibility of superradiant amplification of perturbation modes. I will begin by discussing stable perturbations in Section 3.1, and then I will turn to superradiantly unstable configurations in Section 3.2.", "pages": [ 5, 6 ] }, { "title": "3.1 Stable dynamics in GR: quasinormal modes", "content": "Massless (scalar, electromagnetic or gravitational) perturbations of a Kerr BH have a 'natural' set of boundary conditions: we must impose that waves can only be ingoing at the horizon (which is a one-way membrane) and outgoing at infinity, where the observer is located. Imposing these boundary conditions gives rise to an eigenvalue problem with complex eigenfrequencies, that correspond to the so-called BH quasinormal modes [58]. The nonzero imaginary part of the modes is due to damping (radiation leaves the system both at the horizon and at infinity), and its inverse corresponds to the damping time of the perturbation. By analogy with damped oscillations of a ringing bell, the gravitational radiation emitted in these modes is often called 'ringdown'. The direct detection of ringdown frequencies from perturbed BHs will provide stringent internal tests that astrophysical BHs are indeed described by the Kerr solution. The possibility to carry out such a test depends on the signalto-noise ratio (SNR) of the observed GWs: typically, SNRs larger than ∼ 30 should be sufficient to test the Kerr nature of the remnant [60]. While these tests may be possible using Earth-based detectors, they will probably require observations of relatively massive BH mergers with total mass ∼ 10 2 M /circledot . A detection of such high-mass mergers would be a great discovery in and by itself, given the dubious observational evidence for intermediate-mass BHs [61]. On the other hand, the existence of massive BHs with M /greaterorsimilar 10 5 M /circledot is well established, and space-based detectors such as (e)LISA [62,63,64] have a formidable potential for observing the mergers of the lightest supermassive BHs with large SNR throughout the Universe (see e.g. Fig. 16 of [63]). Any such observation would yield stringent 'internal' strong-field tests of GR. Furthermore, ringdown observations can be used to provide extremely precise measurements of the remnant spins [60]. Since the statistical distribution of BH spins encodes information on the past history of assembly and growth of the massive BH population in the Universe [65], spin measurements can be used to discrim- e between astrophysical models that make different assumptions on the birth and growth mechanism of massive BHs [66,67]. The potential of a space-based mission like (e)LISA to perform 'internal' tests of GR and constrain the merger history of massive BHs using ringdown observations is illustrated in Fig. 1. There we consider hypothetical (e)LISA detections of ringdown waves (computed using analytic prescriptions from [60]) within two different models for supermassive BH formation. Both models assume a hierarchical evolution starting from heavy BH seeds, but they differ in their prescription for the accretion mode, which is either coherent (leading on average to large spins) or chaotic (leading on average to small spins): see the LISA Parameter Estimation Taskforce study [65] for more details. The histograms show the distribution of SNR and spin measurement accuracy during the two-year nominal lifetime of the eLISA mission [63,64]. Independently of the accretion mode, both models predict that 1) more than ten events would have SNR larger than 30, and 2) a few tens of events would allow ringdownbased measurements of the remnant spin to an accuracy better than ∼ 10%. Space-based detectors with six links may identify electromagnetic counterparts to some of these merger events and determine their distance [68,63,64]. While extremely promising, this simple assessment of the potential of ringdown waves to test GR should still be viewed as somewhat pessimistic, because a statistical ensemble of events can bring significantly improvements over indi- dual observations: see e.g. [69] for a discussion of this point in the context of graviton-mass bounds with (e)LISA observations of inspiralling BH binaries 4 . 3.2 Unstable dynamics in the presence of massive bosons: superradiant instabilities As anticipated at the beginning of this section, the existence of a local minimum in the potential for massive scalar perturbations allows for the existence of quasibound states. Detweiler [72] computed analytically the frequencies of these quasibound states, finding that they can induce an instability in Kerr BHs. The physical origin of the instability is BH superradiance, as first pointed out by Press and Teukolsky [73] (see also [74,75,76]): scalar waves incident on a rotating BH with frequency 0 < ω < mΩ H (where Ω H is the angular frequency of the horizon) extract rotational energy from the BH and are reflected to infinity with an amplitude which is larger than the incident amplitude. The barrier at infinity acts as a reflecting mirror, so the wave is reflected and amplified again. The extraction of rotational energy and the amplification of the wave at each subsequent reflection trigger what Press and Teukolsky called the 'black-hole bomb' instability. Scalar fields. For scalar fields, results by Detweiler and others [72,73,77,78, 74,79,80] show that the strenght of the instability is regulated by the dimensionless parameter Mµ (in units G = c = 1), where M is the BH mass and m = µ /planckover2pi1 is the field mass, and it is strongest when the BH is maximally spinning and Mµ ∼ 1 (cf. [79]). For a solar mass BH and a field of mass m ∼ 1 eV the parameter Mµ ∼ 10 10 /greatermuch 1, and the instability is exponentially suppressed [78]. Therefore in many cases of astrophysical interest the instability timescale must be larger than the age of the Universe. Strong, astrophysically relevant superradiant instabilities with Mµ ∼ 1 can occur either for light primordial BHs which may have been produced in the early Universe, or for ultralight exotic particles found in some extensions of the standard model. An example is the 'string axiverse' scenario [81,59], according to which massive scalar fields with 10 -33 eV < m < 10 -18 eV could play a key role in cosmological models. Superradiant instabilities may allow us to probe the existence of such ultralight bosonic fields by producing gaps in the BH Regge plane [81, 59] (i.e. the mass/spin plane), by modifying the inspiral dynamics of compact binaries [82,83,27] or by inducing a 'bosenova', i.e. collapse of the axion cloud (see e.g. [84,85,86]). O Vector fields. It has long been believed that the 'BH bomb' instability should operate for all bosonic field perturbations in the Kerr spacetime, and in particular for massive spin-one (Proca) bosons 5 [79]. A proof of this conjecture was lacking until recently because of technical difficulties in separating the perturbation equations for massive spin-one (Proca) fields in the Kerr background. Pani et al. recently circumvented the problem using a slow-rotation expansion pushed to second order in rotation [91,92]. The Proca superradiant instability turns out to be stronger than the massive scalar field instability. Furthermore the Proca mass range where the instability would be active is very interesting from an experimental point of view: indeed, as shown in [91], astrophysical BH spin measurements are already setting the most stringent upper bound on the mass of spin-one fields. This can be seen in Fig. 2, which shows exclusion regions in the 'BH Regge plane' (cf. Fig. 3 of [59]) obtained by setting the instability timescale equal to the (Salpeter) accretion timescale τ Salpeter = 4 . 5 × 10 7 yr. The idea here is that a conservative bound on the critical mass of the Proca field corresponds to the case where the instability spins BHs down faster than accretion could possibly spin them up. Instability windows are shown for four different masses of the Proca field ( m v = 10 -18 eV, 10 -19 eV, 10 -20 eV and 2 × 10 -21 eV) and for two different classes of unstable Proca modes: 'axial' modes (bottom panel) and 'polar' modes with polarization index S = -1, which provides the strongest instability (top and middle panels). All regions above the instability window are ruled out 6 . The plot shows that essentially any spin measurement for supermassive BHs with 10 6 M /circledot /lessorsimilar M /lessorsimilar 10 9 M /circledot would exclude a wide range of vector field masses [91, 92]. Massive vector instabilities do not - strictly speaking - provide 'external' tests of GR, but rather tests of perturbative dynamics within GR; quite interestingly, they provide constraints on possible mechanisms to generate massive 'hidden' U(1) vector fields, which are predicted by various extensions of the Standard Model [93,94,95,96]. The results discussed in this section are quite remarkable, because they show that astrophysical measurements of nonzero spins for supermassive BHs can already place the strongest constraints on the mass of hypothetical vector bosons (for comparison, the Particle Data Group quotes an upper limit m< 10 -18 eV on the mass of the photon [97]).", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4 Present and future tests of massive scalar-tensor theories", "content": "So far I discussed 'internal' tests of GR from future GW observations of stable BH dynamics (ringdown waves). I also summarized how superradiant instabilities can be used to place bounds on the masses of scalar and vector fields, which emerge quite naturally in extensions of the Standard Model [81,59,93, 94,95,96]. In this Section I address a slightly different but related question, namely: what constraints on the mass and coupling of scalar fields are imposed by Solar System observations? Shall we be able to constrain these models better (or prove that scalar fields are indeed needed for a correct description of gravity) using future GW observations?", "pages": [ 10 ] }, { "title": "4.1 Solar System bounds", "content": "In [27] we investigated observational bounds on massive scalar-tensor theories of the Brans-Dicke type. In addition to deriving the orbital period derivative due to gravitational radiation, we also revisited the calculations of the Shapiro time delay and of the Nordtvedt effect in these theories (cf. [1] for a detailed and updated treatment of these tests). The comparison of our results for the orbital period derivative, Shapiro time delay and Nordtvedt parameter against recent observational data allows us to put constraints on the parameters of the theory: the scalar mass m s and the Brans-Dicke coupling parameter ω BD . These bounds are summarized in Figure 3. We find that the most stringent bounds come from the observations of the Shapiro time delay in the Solar System provided by the Cassini mission (which had already been studied in [98]). From the Cassini observations we obtain ω BD > 40 , 000 for m s < 2 . 5 × 10 -20 eV, while observations of the Nordtvedt effect using the Lunar Laser Ranging (LLR) experiment yield a slightly weaker bound of ω BD > 1 , 000 for m s < 2 . 5 × 10 -20 eV. Possibly our most interesting result concerns observations of the orbital period derivative of the circular white-dwarf neutron-star binary system PSR J1012+5307, which yield ω BD > 1 , 250 for m s < 10 -20 eV. The limiting factor here is our ability to obtain precise measurements of the masses of the component stars as well as of the orbital period derivative, once kinematic corrections have been accounted for. However, there is considerably more promise in the eccentric binary PSR J1141-6545, a system for which remarkably precise measurements of the orbital period derivative, the component star masses and the periastron shift are available. The calculation in [27] was limited to circular binaries, and we are currently working to generalize our treatment to eccentric binaries in order to carry out a more meaningful and precise comparison with observations of PSR J1141-6545.", "pages": [ 10, 11 ] }, { "title": "4.2 Gravitational-wave tests", "content": "Binary pulsar observations can test certain aspects of strong-field modifications to GR, such as the 'spontaneous scalarization' phenomenon in scalar-tensor theories [6], and interesting tests are also possible with current astronomical observations [3]. However a real breakthrough is expected to occur in the near future with the direct detection of GWs from the merger of compact binaries composed of BHs and/or NSs. One of the most exciting prospects of the future network of GW detectors (Advanced LIGO/Virgo [99], LIGO-India [100] and KAGRA [101] in the near future; third-generation Earth-based interferometers like the Einstein Telescope [102] and a space-based, LISA-like mission [62,63, 64] in the long term) is precisely their potential to test GR in strong-field, highvelocity regimes inaccessible to Solar System and binary pulsar experiments. Second-generation interferometers such as Advanced LIGO should detect a large number of compact binary coalescence events [103,104]. Unfortunately from the point of view of testing GR, most binary mergers detected by Advanced LIGO/Virgo are expected to have low signal-to-noise ratios (a possible exception being the observation of intermediate-mass BH mergers [105], that would be a great discovery in and by itself). Third-generation detectors such as the Einstein Telescope will perform significantly better in terms of parameter estimation and tests of alternative theories [106,107]. Here I will argue (using the example of massive scalar-tensor theories) that an (e)LISA-like mission will be an ideal instrument to test GR [63,64] by providing two examples: (1) bounds on massive scalar-tensor theories using (e)LISA observations of intermediate mass-ratio inspirals, and (2) the possibility to observe an exotic phenomenon related once again to superradiance, i.e., floating orbits. Bounds on massive scalar-tensor theories from intermediate massratio inspirals. In general, the gravitational radiation from a binary in massive scalar-tensor theories depends on both the scalar field mass m s and the coupling constant ω BD [108,27]. If the field is massless, corrections to the GW phasing are proportional to 1 /ω BD , and therefore comparisons of the phasing in GR and in scalar-tensor theories yield bounds on ω BD [109,108]. By computing the GW phase in the stationary-phase approximation, one finds that the scalar mass always contributes to the phase in the combination m 2 s /ω BD , so that GW observations of nonspinning, quasicircular inspirals can only set upper limits on m s / √ ω BD [110]. For large SNR ρ , the constraint is inversely proportional to ρ . The order of magnitude of the achievable bounds is essentially set by the lowest frequency accessible to the GW detector, and it can be understood by noting that the scalar mass and GW frequency are related (on dimensional grounds) by m s (eV) = 6 . 6 × 10 -16 f (Hz), or equivalently f (Hz) = 1 . 5 × 10 15 m s (eV). For eLISA, the lower cutoff frequency (imposed by acceleration noise) f cut ∼ 10 -5 Hz corresponds to a scalar of mass m s /similarequal 6 . 6 × 10 -21 eV. For Earth-based detectors the typical seismic cutoff frequency is f cut ∼ 10 Hz, corresponding to m s ∼ 6 . 6 × 10 -15 eV. This simple argument shows that space-based detectors can set ∼ 10 6 stronger bounds on the scalar mass than Earth-based detectors. An explicit calculation shows that the best bounds are obtained from (e)LISA observations of the intermediate mass-ratio inspiral of a neutron star into a BH of mass M BH /lessorsimilar 10 3 M /circledot , and that they would be of the order In summary, GW observations will provide two constraints: a lower limit on ω BD (corresponding to horizontal lines in Fig. 3) and an upper limit on m s / √ ω BD (corresponding to the straight diagonal lines in Fig. 3). Therefore GW observations would exclude the complement of a trapezoidal region on the top left of Fig. 3. Straight (dashed) lines show the bounds from eLISA observations of NS-BH binaries with SNR ρ = 10 when the BH has mass M BH = 300 M /circledot ( M BH = 3 × 10 4 M /circledot , respectively). The plot shows that GW observations with ρ = 10 become competitive with binary pulsar bounds when m s /greaterorsimilar 10 -19 eV, and competitive with Cassini bounds when m s /greaterorsimilar 10 -18 eV, with the exact 'transition point' depending on the SNR of the observation (for a GW observation with SNR ρ = 100 the 'straight line' bounds in Fig. 3 would be ten times higher). Therefore in this particular theory a single highSNR observation (or the statistical combination of several observations, see e.g. [69]) may yield better bounds on the scalar coupling than weak-gravity observations in the Solar System when m s /greaterorsimilar 10 -18 eV. Floating orbits. It is generally expected that small bodies orbiting around a BH will lose energy in gravitational waves, slowly inspiralling into the BH. In [82] we showed that the coupling of a massive scalar field to matter leads to a surprising effect: because of superradiance, orbiting objects can hover into 'floating orbits' for which the net gravitational energy loss at infinity is entirely provided by the BH's rotational energy. The idea is that a compact object around a rotating BH can excite superradiant modes to appreciable amplitudes when the frequency of the orbit matches the frequency of the unstable quasibound state. This follows from energy balance: if the orbital energy of the particle is E p , and the total (gravitational plus scalar) energy flux is ˙ E T = ˙ E g + ˙ E s , then Usually ˙ E g + ˙ E s > 0, and therefore the orbit shrinks with time. However it is possible that, due to superradiance, ˙ E g + ˙ E s = 0. In this case ˙ E p = 0, and the orbiting body can 'float' rather than spiralling in [111,73]. The system is essentially a 'BH laser', where the orbiting compact object is producing stimulated emission of gravitational radiation: because the massive scalar field acts as a mirror, negative scalar radiation ( ˙ E s < 0) is dumped into the horizon, while gravitational radiation can be detected at infinity. Orbiting bodies remain floating until they extract sufficient angular momentum from the BH, or until perturbations or nonlinear effects disrupt the orbit. For slowly rotating and nonrotating BHs floating orbits are unlikely to exist, but resonances at orbital frequencies corresponding to quasibound states of the scalar field can speed up the inspiral, so that the orbiting body 'sinks'. A detector like (e)LISA could easily observe these effects [82,83], that would be spectacular smoking guns of deviations from general relativity.", "pages": [ 12, 13, 14 ] }, { "title": "5 Conclusions", "content": "The three examples discussed in this paper (ringdown tests of the BH nohair theorem, bosonic superradiant instabilities in rotating BHs and GW tests of massive scalar-tensor theories) illustrate that astrophysical BHs, either in isolation or in compact binaries, can be spectacular nature-given laboratories for fundamental physics. We can already use astrophysical observations to do fundamental physics (e.g. by setting bounds on the masses of scalar and vector fields using supermassive BH spin measurements), but the real goldmine for the future of 'fundamental astrophysics' will be GW observations. In order to fully realize the promise of GWs as probes of strong-field gravity we will need several detections with large SNR. Second- and third-generation Earth-based interferometers will certainly deliver interesting science, but a full realization of strong-field tests and fundamental physics with GW observations may have to wait for space-based GW detectors. We'd better make sure they happen in our lifetime. Acknowledgements The research reviewed in this paper was supported by NSF CAREER Grant No. PHY-1055103. I thank my collaborators on various aspects of the work described in this paper: Justin Alsing, Vitor Cardoso, Sayan Chakrabarti, Jonathan Gair, Leonardo Gualtieri, Michael Horbatsch, Akihiro Ishibashi, Paolo Pani, Alberto Sesana, Ulrich Sperhake, Marta Volonteri, Clifford Will and Helmut Zaglauer. Special thanks go to Paolo Pani for comments on an early draft and to Alberto Sesana for preparing Fig. 1, as well as excellent mojitos.", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "Noname manuscript No. (will be inserted by the editor)", "pages": [ 1 ] }, { "title": "The co-evolution of galaxies and supermassive black holes in the near Universe", "content": "Thaisa Storchi-Bergmann Received: date / Accepted: date Abstract Afundamental role is attributed to supermassive black holes (SMBH), and the feedback they generate, in the evolution of galaxies. But theoretical models trying to reproduce the M SMBH -σ relation (between the SMBH mass and stellar velocity dispersion of the galaxy bulge) make broad assumptions about the physical processes involved. These assumptions are needed due to the scarcity of observational constraints on the relevant physical processes which occur when the SMBH is being fed via mass accretion in Active Galactic Nuclei (AGN). In search for these constraints, our group AGN Integral Field Spectroscopy (AGNIFS) - has been mapping the gas kinematics as well as the stellar population properties of the inner few hundred parsecs of a sample of nearby AGN hosts. In this contribution, I report results obtained so far which show gas inflows along nuclear spirals and compact disks in the inner tens to hundreds of pc in nearby AGN hosts which seem to be the sources of fuel to the AGN. As the inflow rates are much larger than the AGN accretion rate, the excess gas must be depleted via formation of new stars in the bulge. Indeed, in many cases, we find ∼ 100pc circumnuclear rings of recent star formation (ages ∼ 10-500Myr) that can be interpreted as a signature of co-evolution of the host galaxy and its AGN. I also report the mapping of outflows in ionized gas, which are ubiquitous in Seyfert galaxies, and discuss mass outflow rates and powers. Keywords galaxies: active · galaxies: nuclei · supermassive black holes · mass accretion rate", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The correlation between the mass of the Supermassive Black Hole (hereafter SMBH) present in the nuclei of most galaxies - the so-called M SMBH -σ relation - has been interpreted as indicating a coupling between the growth of SMBHand that of their host galaxies [28]. As pointed out by Tiziana di Matteo in this conference (see also [6,7]), the bulk of galaxy and SMBH growth must have taken place in the first 1-2 Gyr of the Universe. The observation of gas rich and star-forming galaxies, as well as the nuclear activity in Quasars at redshifts z ≥ 6 support this conclusion and have motivated the cosmological simulations which show that interactions can send gas towards the galaxy centers triggering episodes of star formation as well as luminous nuclear activity. Feeding and feedback processes which occur in Active Galactic Nuclei (AGN) are now a paradigm of galaxy evolution models in constraining the co-evolution of galaxies and SMBHs, but their implementation have been simplistic [29,30,31] because the physical processes involved are not well constrained by observations. This is due to the fact that they occur within the inner few hundred parsecs, which cannot be spatially resolved at z ≥ 2 where the co-evolution of galaxies and SMBH largely occurs. It is nearby galaxies that offer the only opportunity to test in detail the prescriptions used in models of galaxy and BH co-evolution. In this contribution, I discuss results of recent studies of the gas kinematics within the inner few hundred parsecs of nearby active galaxies performed by my research group - called AGNIFS (AGN Integral Field Spectroscopy) , at spatial resolution of a few to tens of parsecs, which do resolve the gas kinematics in the nuclear region and can be used to constrain the processes of AGN feeding and feedback. In a few cases, we have been also able to map the stellar population in the nuclear region, in search of recent episodes of star formation which could trace the growth of the galaxy bulge.", "pages": [ 2 ] }, { "title": "2 Observations", "content": "We have used integral field spectroscopy (IFS) at the Gemini telescopes, the final product of which are 'datacubes', which have two spatial dimensions allowing the extraction of images at a desired wavelength range - and one spectral dimension - allowing the extraction of spectral information of each spatial element. In the optical, we have used the Integral Field Unit of the Gemini MultiObject Spectrograph (IFU-GMOS), which has a field-of-view of 3 . '' 5 × 5 '' in one-slit mode or 5 '' × 7 '' in two-slit mode at a sampling of 0 . '' 2 and angular resolution (dictated by the seeing) of 0 . '' 6, on average. The resolving power is R ≈ 2500. In the near-infrared (near-IR) we have used the Near-Infrared Integral Field Spectrograph (NIFS) together with the adaptative optics module ALTAIR (ALTtitude conjugate Adaptive optics for the InfraRed), which delivers an angular resolution of ∼ 0 . '' 1. The field-of-view is 3 '' × 3 '' at a sampling of 0 . '' 04 × 0 . '' 1 and the spectral resolution is R ≈ 5300 at the Z, J, H and K bands.", "pages": [ 2, 3 ] }, { "title": "3 Inflows", "content": "Nuclear spirals - on scales of hundred parsecs - are frequently observed around AGN in images obtained with the Hubble Space Telescope (HST) [11]. [10] have shown that these spirals may be the channels through which matter is being transferred to the nucleus to feed the AGN. This interpretation is supported by models [9] and by results such as those from [21]. In the latter paper, we have built 'structure maps' using images obtained with the HST Wide-Field and Planetary Camera 2 (WFPC2) through the F606W filter of a sample of AGN and a control sample of non-active galaxies. The structure maps revealed dusty nuclear spirals in all early-type AGN hosts, but in only ∼ 25% of the non-AGN, indicating that these spirals are strongly linked to the nuclear activity and thus should map the matter in its way to feed the SMBH at the nucleus. But in order to test this hypothesis, based only on morphology, it was necessary to map the gas kinematics in these spirals, what can be best done with IFUs.", "pages": [ 3 ] }, { "title": "3.1 IFU observations in the optical", "content": "We show in Fig. 1 recent observations and measurements of the stellar and gas kinematics of the inner 0.7 kpc × 1kpc of the LINER/Seyfert1 galaxy NGC 7213, from [20]. The observations, obtained with the GMOS IFU, cover the wavelength range 5700 ˚ A-6900 ˚ A. Although the galaxy appears to be close to face on, the stellar kinematics, obtained from the absorption lines, shows a rotation pattern with an amplitude of ≈ 50kms -1 , with the line of nodes oriented approximately along NorthSouth (white line in Fig. 1). The gas kinematics - obtained over the whole field-of-view in the [N II] λ 6584 ˚ A emission line - is nevertheless completely distinct, showing a much larger amplitude (velocities up to 200 km s -1 ), and a 'distorted' rotation pattern with the largest velocity gradient running at a large angle to the line of nodes of the stellar kinematics. The 'distortions' in the gas velocity field are clearly correlated with the nuclear spirals seen in the structure map (shown in the rightmost panel of Fig. 1). Considering that the near side of the galaxy is the West (to the top and right of the line of nodes in the first panel of Fig. 1), and the far side is the East, and assuming that the emitting gas is in the plane of the galaxy, the redshifts observed to the West and the blueshifts observed to the East can be interpreted as due to gas inflows towards the central region of the galaxy. In order to obtain an estimate for the mass inflow rate, we have integrated the mass flux through concentric 'shallow' cylinders around the nucleus, obtaining a mass inflow rate at a distance of ≈ 100pc from the nucleus of 0.2 M /circledot yr -1 . Similar observations and velocity fields were obtained for three other LINER galaxies. In M 81 (LINER/Seyfert 1) we [19] have also observed rotation in the stellar velocity field within the inner 100 pc radius, but a totally distinct kinematics for the gas. The ionized gas kinematics show inflows along the galaxy minor axis that seem to be correlated with a nuclear spiral seen in a structure map. The estimated ionized gas mass inflow rate in M 81 is smaller than that in NGC 7213, being of the order of the AGN accretion rate. Signatures of inflows along nuclear spirals at similar mass inflow rates as that obtained for M 81 were also seen in the LINER/Seyfert 1 galaxy NGC 1097 [8] and in the LINER galaxy NGC6951 [24]. We note that these inflows were obtained only from observations of the ionized gas, and thus can be considered a lower limit for the total mass gas inflow rate, which is probably dominated by cold molecular gas with possible contribution of neutral gas as well.", "pages": [ 3, 4 ] }, { "title": "3.2 IFU observations in the near-IR", "content": "In [14] we have obtained the stellar and molecular gas kinematics of the inner 100pc of the nearby Seyfert 1 galaxy NGC 4051 at a spatial resolution of 5pc, via a datacube obtained with NIFS in the near-IR K band. The stellar kinematics shows again rotation with the lines of nodes running approximately North-South. As the spectral resolution of NIFS is almost three times higher than that of the GMOS IFU, we were able to obtain the gas kinematics using channel maps along the H 2 λ 2 . 122 µ memission line. We found mostly blueshifts in a spiral arm to the East, - the far side of the galaxy, and mostly redshifts to the West - the near side of the galaxy, which can be interpreted as inflows towards the galaxy center if we assume again that the gas is in the plane of the galaxy. The H 2 emission in the K band maps the 'warm' molecular gas (temperature T ∼ 2000K) which is probably only the 'skin' of a much larger cold molecular gas reservoir which emits in the millimetric wavelength range (and should thus be observable with the Atacame Large Millimetric Array - ALMA , for example). In fact, the warm H 2 mass inflow rate within the inner 100 pc of NGC4051 is of the order of only 10 -5 M /circledot yr -1 . But previous observations of both the warm and cold molecular gas in a sample of AGN host galaxies show typical ratios cold/warm H 2 masses ranging between 10 5 and 10 7 . Applying this ratio to NGC 4051 leads to a total gas mass inflow rate of ≥ 1M /circledot yr -1 . Similar inflows in warm H 2 gas were observed (using again NIFS) in the inner 350pc of the Seyfert 2 galaxy Mrk 1066 at 35 pc spatial resolution [15] along spiral arms which seem to feed a compact rotating disk with a 70 pc radius. The mass of warm H 2 gas is estimated as 3300 M /circledot , that, corrected for the 10 5 -7 factor to account for the cold component, would imply a reservoir of at least 10 8 M /circledot of cold molecular gas. The mass inflow rate, when corrected for this same ratio results 0.6 M /circledot yr -1 . Other authors have also reported inflows in near-IR observations of the H 2 kinematics, such as [5] in the nuclear spirals of NGC 1097 and [13] between the nuclear ring and the AGN in NGC 1068.", "pages": [ 4, 5 ] }, { "title": "4 Outflows", "content": "Our IFS observations of the inner few hundred parsecs of nearby active galaxies have also revealed outflows. In the near-IR, while the H 2 gas kinematics is dominated by circular rotation or inflows in the plane of the galaxy, the ionized gas emission, and in particular in the emission lines [Fe II] λ 1 . 644 µ m and [Fe II] λ 1 . 257 µ m usually traces outflows extending to high galactic latitudes. While the inflow velocities are of the order of 50 km s -1 , and the rotation velocities reach at most 200 km s -1 , the outflow velocities reach up to 1000kms -1 . One example is the case of Mrk 1066. The NIFS observations of the inner 350pc (radius) show a distinct flux distributions and kinematics for the ionized gas when compared to that of the H 2 . The H 2 emitting gas is distributed all over the inner disk of the galaxy, showing circular rotation in the galaxy plane with velocities smaller than 200 km s -1 . The [Fe II] λ 1.644 µ m emitting gas, on the other hand, is collimated along a nuclear radio jet [15] and reaches outflow velocities of up to 500 km s -1 . From the velocity field and geometry of the outflow we estimate a mass outflow rate in ionized gas of ≈ 0.5 M /circledot yr -1 , a value which is of the same order as that of the mass inflow rate in H 2 . NIFS observations of the inner 560pc × 200pc of the Seyfert 1.5 galaxy NGC4151 at ≈ 7pc spatial resolution [25,26] also show that the H 2 and [Fe II] flux distributions and kinematics are very distinct, presenting the same behavior to that observed in Mrk 1066: the H 2 shows a very small velocity amplitude ( ≤ 100kms -1 ) and its kinematics is dominated by rotation, while the velocities observed for [Fe II] reach 600 km s -1 and the kinematics is dominated by a conical outflow. The H 2 gas is concentrated within the inner 50 pc around the nucleus and is strongest along the direction of the galactic bar, consis- with an origin in an inflow along the bar. From the many [Fe II] and H 2 emission lines it was possible to obtain the electronic gas temperatures: T(H 2 ) ≈ 2000K, while T([Fe II]) ≈ 15000K, confirming that they correspond to distinct gas components. From the observed velocities and inferred geometry, we were able to estimate the mass flow rate along the conical outflow: ≈ 2 M /circledot yr -1 . We were also able to obtain the kinetic power of the outflow which is only ≈ 0.3% of the bolometric luminosity of the AGN. In Fig.2 we show channel maps of the inner 200 pc radius of the 'prototypical' Seyfert 2 galaxy NGC 1068 in the [Fe II] λ 1 . 64 µ m(in red) and H 2 λ 2 . 122 µ m (in green) emission lines, at a spatial resolution of 7 pc [3]. These channel maps show the flux distributions at the velocities indicated in each panel. The total flux distribution in [Fe II] emission shows an hourglass structure, while in the channel maps of Fig.2 it shows an ' α -shaped' structure in the blueshifted channels and a 'fan-shaped' structure in the redshifted channels. We attribute the blueshifted emission to the front part of the gas outflow modeled by [4], while the redshifted emission is attributed to the back part of the outflow. We note that the fan-shaped structure is very similar to that observed in Planetary Nebulae (e.g. NGC 6302), suggesting a similar mechanism for the origin of the outflow. Using the inferred geometry and velocity field, we have calculate a mass outflow rate of 6 M /circledot yr -1 . Fig. 2 also shows that the H 2 flux distribution is once more completely distinct from that of [Fe II], presenting a ring-like (radius ≈ 100pc) morphology. The H 2 kinematics shows again much smaller velocities than those observed in the [Fe II] emission and in common with other active galaxies shows also rotation. Nevertheless, something that was not seen in the other galaxies is the presence of expansion in the ring in the plane of the galaxy: Fig.2 shows that the H 2 emission is blueshifted in the near side of the galaxy and redshifted in the far side. If the gas is in the plane of the galaxy, this implies expansion of the ring. Additional results of IFS observations of outflows in active galaxies can be found in [2,16,18].", "pages": [ 5, 6, 7 ] }, { "title": "5 Stellar Population", "content": "In at least three studies with NIFS in the near-IR we have been able not only to map the flux distribution and kinematics but also the stellar population via spectral synthesis. Our study of the stellar population in Mrk 1066 [15] was the first resolved two-dimensional stellar population study of an AGN host in the near-IR. The spectral synthesis of the region within 350 pc from the nucleus revealed a 300 pc circumnuclear ring of 500 Myr old stellar population which is correlated with a ring of low stellar velocity dispersions. This result has been interpreted as due to capture of gas to the nuclear region with enough gas mass to trigger the formation of new stars 500 Myr ago. The low velocity dispersion indicates that these stars still keep the 'cold' kinematics of the gas from which they were formed. In [17], a similar study revealed almost the same result for another Seyfert 2 galaxy, Mrk 1157. More recently, in [27], we have found a smaller (100 pc radius) ring of star formation, with a younger age, of ≈ 30Myr, in NGC1068. This ring seems to be correlated with the molecular (H 2 ) ring described above, and shown in Fig. 3 (in green). In [1], in a study of the gas and stellar kinematics of the inner kiloparsec of six nearby Seyfert galaxies using the GMOS IFU, we have also found that four of them showed circumnuclear rings of low velocity dispersion, supporting the presence of young to intermediate age stars with 'cold' kinematics.", "pages": [ 7 ] }, { "title": "6 Summary and Conclusions", "content": "Our observations of the gas kinematics within the inner few hundred parsecs of nearby active galaxies have revealed inflows towards the center along nuclear spirals, with inflow velocities in the range 50-100km s -1 and mass flow rates ranging from 0.01 to 1 M /circledot yr -1 in ionized gas via observations in the optical. In the near-IR, we have also found inflow along nuclear spirals in warm molecular gas emission (T ∼ 2000K), although at much smaller inflow rates. We have concluded that the warm molecular gas is only the hot 'skin' of the bulk inflow that should be dominated by cold molecular gas (observable with ALMA). Using a typical conversion factor between the mass of warm to cold molecular gas (derived from an active galactic sample for which both millimetric and near-IR observations are available), we find values close to 1 M /circledot yr -1 . Our previous study [21] showed a clear dichotomy between the nuclear region of early-type AGN hosts - which always show excess of dust - and that of non-AGN, supporting the hypothesis that the nuclear spirals and filaments are a necessary condition for the presence of nuclear activity. Nevertheless, the mass accretion rate necessary to feed the AGN ( ≈ 10 -3 M /circledot yr -1 ) is typically much smaller than the above mass inflow rate. For example, at 1 M /circledot yr -1 , in 10 7 -8 yr (the expected duration of an activity cycle), at least 10 7 M /circledot of gas will be accumulated in the inner few hundred parsecs. This number is supported by our recent study [12] in which we have used Spitzer photometry of the sample of [21] to obtain dust masses. The average value for the AGN hosts is 10 5 . 5 M /circledot , and for a typical ratio of ≈ 100 between the gas and dust masses, the AGN hosts should thus have, on average, 10 7 . 5 M /circledot in gas in the inner few hundred parsecs, in agreement with our estimate above on the basis of the mass inflow rate. The accumulation of at least 10 7 M /circledot in the nuclear region will probably lead to the formation of new stars in the galaxy bulge. Signatures of this recent star formation have been indeed seen, in the form of rings at 100 pc scales with significant contribution from stars of ages in the range 30 Myr ≤ age ≤ 700Myr. These results suggest that we are witnessing the co-evolution of the SMBH and their host galaxies in the near Universe: while the SMBH at the center grows at typical rates of 10 -3 M /circledot yr -1 , the bulge growths at typical rates of 0.1-1M /circledot yr -1 . A similar evolution scenario has been previously proposed by [22]. Finally, the coupling between the SMBH and galaxy evolution depends also on the AGN feedback. Our IFS observations usually reveal ionized gas outflows in the nuclear region, mainly in Seyfert hosts. These outflows are oriented at random angles to the galaxy plane, and reach velocities in the range 200800kms -1 . The total mass of ionized gas in these outflows are of the order of 10 6 -7 M /circledot , and the mass outflow rates are in the range 0.5-10M /circledot yr -1 , which is similar to the range of the mass inflow rates, although only in a couple of galaxies we observe both inflows and outflows (e.g. in Mrk 1066). The fact that the mass outflow rates are about 1000 times the AGN accretion rate supports the idea that the observed outflows are due to mass loading of an AGN outflow (which should be at most equal to the AGN accretion rate) as it moves through the circumnuclear interstellar medium of the host galaxy.", "pages": [ 8, 9 ] } ]

[ { "title": "Interacting generalized ghost dark energy in a non-flat universe", "content": "Esmaeil Ebrahimi 1 , 3 ∗ , Ahmad Sheykhi 2 , 3 † and Hamzeh Alavirad 4 ‡ 2 Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran 3 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O. Box 55134-441, Maragha, Iran We investigate the generalized Quantum Chromodynamics (QCD) ghost model of dark energy in the framework of Einstein gravity. First, we study the non-interacting generalized ghost dark energy in a flat Friedmann-Robertson-Walker (FRW) background. We obtain the equation of state parameter, w D = p/ρ , the deceleration parameter, and the evolution equation of the generalized ghost dark energy. We find that, in this case, w D cannot cross the phantom line ( w D > -1) and eventually the universe approaches a de-Sitter phase of expansion ( w D → -1). Then, we extend the study to the interacting ghost dark energy in both a flat and non-flat FRW universe. We find that the equation of state parameter of the interacting generalized ghost dark energy can cross the phantom line ( w D < -1) provided the parameters of the model are chosen suitably. Finally, we constrain the model parameters by using the Markov Chain Monte Carlo (MCMC) method and a combined dataset of SNIa, CMB, BAO and X-ray gas mass fraction. Keywords : ghost; dark energy; acceleration; observational constraints.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The cosmological data from type Ia Supernova, Large Scale Structure(LSS) and Cosmic Microwave Background (CMB) indicate that our universe is currently accelerating [1]. To explain such an acceleration in the framework of standard cosmology, one is required to introduce a new type of energy with a negative pressure usually called 'dark energy' (DE) in the literature. A great variety of DE scenarios have been proposed to explain the acceleration of the universe's expansion. One can refer to [2, 3] for a review of DE models. On the other hand, many people believe in a modification of gravity, seeking an explanation for the late time acceleration. According to this idea the acceleration will be a part of the universe's expansion and does not need to invoke any kind of DE component. As examples of this approach one can look at Refs. [4-8]. It is important to note that the detection of gravitational waves should be the ultimate test for general relativity or alternatively the definitive endorsement for extended theories [9]. In most scenarios for DE, people usually need to consider a new degree of freedom or a new parameter, in order to explain the acceleration of the cosmic expansion (see e.g. [10] and references therein). However, it would be nice to resolve the DE puzzle without presenting any new degree of freedom or any new parameter in the theory. One of the successful and beautiful theories of modern physics is QCD which describes the strong interaction in nature. However, resolution of one of its mysteries, the U(1) problem, has remained somewhat unsatisfying. Veneziano ghost field explained the U(1) problem in QCD [11]. Vacuum energy of the ghost field can be used to explain the time-varying cosmological constant in a spacetime with nontrivial topology, since the ghost field has no contribution to the vacuum energy in the Minkowskian spacetime [12]. The energy density of the vacuum ghost field is proportional to Λ 3 QCD H , where Λ QCD is the QCD mass scale and H is the Hubble parameter [13]. It is well-known that the cosmological constant model of DE suffers the coincidence and the fine tuning problems. However, with correct choice of Λ QCD , the ghost dark energy (GDE) model does not encounter the fine tuning problem anymore [12, 13]. Phenomenological implications of the GDE model were discussed in [14]. In [15] GDE in a non-flat universe in the presence of interaction between DE and dark matter was explored. The instability of the GDE model against perturbations was studied in [16]. It was argued that the perfect fluid for GDE is classically unstable against perturbations. Other features of the GDE model have been investigated in Refs. [17-24]. In all the above references ([14-24]) the GDE was assumed to have the energy density of the form ρ D = αH , while, in general, the vacuum energy of the Veneziano ghost field in QCD is of the form H + O ( H 2 ) [25]. This indicates that in the previous works on the GDE model, only the leading term H has been considered. Motivated by the argument given in [26], one may expect that the subleading term H 2 in the GDE model might play a crucial role in the early evolution of the universe, acting as the early DE. It was shown [27] that taking the second term into account can give better agreement with observational data compared to the usual GDE. Hereafter we call this model the generalized ghost dark energy (GGDE) and our main task in this paper is to investigate the main properties of this model. In this model the energy density is written in the form ρ D = αH + βH 2 , where β is a constant. In addition to the DE component, there is also another unknown component of energy in our universe called 'dark matter' (DM). Since the nature of these two dark components are still a mystery and they seem to have different gravitational behaviour, people usually consider them separately and take their evolution independent of each other. However, there exist observational evidence of signatures of interaction between the two dark components [28, 29]. On the other hand, based on the cosmological principle the universe has three distinct geometries, namely open, flat and closed geometry corresponding to k = -1 , 0 , +1, respectively. For a long time it was a general belief that the universe has a flat ( k = 0) geometry, mainly based on the inflation theory [30]. With the development of observational techniques people found deviations from the flat geometry [31]. For example, CMB experiments [32], supernova measurements [33], and WMAP data [34] indicate that our universe has positive curvature. All the above reasons indicate that although people believe in a flat geometry for the universe, astronomical observations leave enough room for considering a nonflat geometry. Also about the interaction between DM and DE there are several signals from nature which guides us to let the models explain such behaviour. Based on these motivations we would like here to extend the studies on GGDE, to a non-flat FRW spacetime in the presence of an interaction term. Our work differs from [15, 19] in that we consider the GGDE model while in [15] and [19], the original GDE model in Einstein and Brans-Dicke theory were studied, respectively. To check the viability of our model, we also perform the cosmological constraints on the interacting GGDE in a non-flat universe by using the Marko Chain Monte Carlo (MCMC) method. We use the following observational datasets: Cosmic Microwave Background Radiation (CMB) data from WMAP7 [35], 557 Union2 dataset of type Ia supernova [36], baryon acoustic oscillation (BAO) data from SDSS DR7 [37], and the cluster X-ray gas mass fraction data from the Chandra X-ray observations [38]. To put the constraints, we modify the public available CosmoMC [39]. The outline of this paper is as follows. In section III, we study the cosmological implications of the GGDE scenario in the absence of interaction between DE and DM. In section III, we consider interacting GGDE in a flat geometry. In section IV, we generalize the study to the universe with spacial curvature in the presence of interaction between DM and DE. In section V, cosmological constraints on the parameters of the model are performed by using the Marko Chain Monte Carlo (MCMC) method. We summarize our results in section VI.", "pages": [ 1, 2 ] }, { "title": "II. GGDE MODEL IN A FLAT UNIVERSE", "content": "Consider a flat homogeneous and isotropic FRW universe, the corresponding Friedmann equation is where ρ m and ρ D are, the energy densities of pressureless DM and DE, respectively. The generalized ghost energy density may be written as [27] where α is a constant of order Λ 3 QCD and Λ QCD is QCD mass scale, and β is also a constant. In the original GDE ( β = 0) with Λ QCD ∼ 100 MeV and H ∼ 10 -33 eV , Λ 3 QCD H gives the right order of magnitude ∼ (3 × 10 -3 eV) 4 for the observed DE density [13]. In the GGDE, β is a free parameter and can be adjusted for better agreement with observations. As usual we introduce the fractional energy density parameters as where ρ cr = 3 H 2 / (8 πG ). Thus, we can rewrite the first Friedmann equation as Through this section we consider GGDE in the absence of the interaction term, thus DE and DM evolves independent of each other and hence they satisfy the following conservation equations If we take the derivative of relations (1) and (2) with respect to the cosmic time, we arrive at where u = ρ m /ρ D . Combining relations (7) and (8) with continuity equation (6), we get Solving the above equation for w D and noticing that u = Ω m / Ω D , and we obtain where ξ = 8 πGβ 3 . It is clear that this relation reduces to its respective one in the GDE when ξ = 0 [15]. In Fig. 1a we have plotted the evolution of w D versus Ω D . It is easy to see that at the late time where Ω D → 1, we have w D →-1, which implies that the GGDE model mimics a cosmological constant behaviour. One should notice that this behaviour is the same as for the original GDE model. This is expected since the subleading term H 2 in the late time can be ignored due to the smallness of H and the difference between these two models appears only at the early epoches of the universe. From figure (1a) we see that w D of the GGDE model cannot cross the phantom divide and the universe has a de Sitter phase at the late time. It is important to note that the universe is filled with two dark components namely, DM and GGDE. Thus to discuss the acceleration of the universe we should define the effective EoS parameter, w eff , as where ρ t and p t are, respectively, the total energy density and the total pressure of the universe. As usual, we have assumed the DM is in the form of pressureless fluid ( p m =0). Using relation (4) for the spatially flat universe, one can find Let us now turn to the deceleration parameter which is defined as where a is the scale factor. Using Eq. (7) and definition Ω D in (3) we obtain Replacing this relation into (14), and using (11) we find One can easily check that the deceleration parameter in GDE is retrieved for ξ = 0 [15]. We can also take a look at the early and the late time behaviour of the deceleration parameter. At the early stage of the universe where Ω D → 0, the deceleration parameter becomes which indicates that for ξ < 2 the universe is at the deceleration phase at early times while for ξ > 2, the universe could experience an acceleration phase, the former is consistent with the definition ξ = 8 πGβ 3 . On the other side, we find that at the late time where the DE dominates (Ω D → 1), independent of the value of the ξ , we have q = -1. We have plotted the behaviour of q in Fig. 1b. Besides, taking Ω D 0 = 0 . 72 and adjusting ξ = 0 . 01 we obtain q 0 ≈ -0 . 34, in agreement with observations [40]. Choosing the same set of parameters leads to w D 0 ≈ -0 . 78 and w eff0 ≈ -0 . 56. One should note that as we already mentioned about w D , the squared term in the GGDE density has a negative contribution in the role of the DE in the universe. We mean by negative contribution that arises by taking the squared term into account, the evolution of the universe will be slowed. For example, the universe will enter the acceleration phase later than the original GDE. This behaviour is clearly seen in both parts of Fig. 1. At the end of this section we present the evolution equation of the DE density parameter Ω D . To this goal we take the time derivative of Eq. (3), after using relation ˙ Ω D = H d Ω D d ln a as well as Eq. (14) we reach Using Eq. (11) we get Once again for the limiting case ξ = 0, the above relation reduces to its respective evolution equation for the original GDE presented in [15].", "pages": [ 2, 3, 4 ] }, { "title": "III. INTERACTING GGDE IN A FLAT UNIVERSE", "content": "In the previous section, the evolution of the DE and DM components were discussed separately. Here we would like to extend the study to the interacting case, seeking new features of GGDE. In the first look investigating interacting models of DE are valuable from two perspective. The first is the theoretical one, which states that we have no reason against interaction between DE and DM components. For example, in the unified models of field theory DM and DE can be explained by a single scalar field, thus they will be allowed to interact minimally. Besides, one can get rid of the coincidence problem by taking into account the interaction term between DM and DE. One can refer to[41-45] for detailed discussion. The other feature which motivates us to consider interacting models of DE and DM comes from observations which indicate the interaction between two dark components of our universe [28]. Thus, there exist enough motivations to consider the GGDE in the presence of an interaction term. To this end, we start with the energy balance equations for DE and DM, namely ˙ ρ D +3 Hρ D (1 + w D ) = - Q, (21) where Q > 0 represents the interaction term which allows the transition of energy from DE to DM. The form of Q is a matter of choice and can be taken as [15] with b 2 being a coupling constant. Inserting Eqs. (8) and (22) in Eq. (21) and taking into account u = Ω m Ω D , we find At first look one can find that setting b = 0, w D reduces to the respective relation in the absence of interaction obtained in Eq. (11). When ξ = 0 the result recovers those in [15] for original GDE. The first interesting point about the EoS parameter of the GGDE is that in the interacting case independent of the interaction parameter, b , for 0 < ξ < 1, w D can cross the phantom line in the future where Ω D → 1. At the present time, by choosing ξ = 0 . 03, b = 0 . 15 and Ω D 0 = 0 . 72, we find that w D 0 = -0 . 82 and w eff0 = -0 . 59 which the latter favored by observations. One can easily check that for a same coupling constant these values for the original GDE are w D 0 = -0 . 83 and w eff0 = -0 . 60 which clearly show that the square term in the energy density of the GGDE slow down the evolution of the universe compared to the original GDE model. For a better insight we have plotted w D against Ω D in Fig. 2a. This value for coupling constant, b , in the figure is consistent with recent observations [46]. It is worth mentioning that at the late time where Ω D → 1 the effective EoS parameter approaches less than -1, i.e. w eff < -1, which reminds a super acceleration for the universe in the future. Next we take a look at the deceleration parameter in the presence of an interaction term. Substituting (15) in (14) and using (23) yields Once again it is clear that setting b = 0, the respective relation in the previous section is retrieved. When ξ = 0 the result of [15] is recovered. For the set of parameters ( ξ = 0 . 03 , b = 0 . 15 , Ω D 0 = 0 . 72), we find that according to the GGDE the universe enters the acceleration phase at Ω = 0 . 48 while this transition happens earlier for the GDE model. This point is clear from Fig.2b. The present value of the deceleration parameter for the interacting GGDE model is q 0 = -0 . 38 which is consistent with observations [40]. Finally, we would like to obtain the evolution equation of DE in the presence of interaction. First we take the time derivative of (3) and obtain Using relation (21) as well as (15), it is a matter of calculation to show In the limiting case ξ = 0 the equation of motion of interacting GDE is recovered [15].", "pages": [ 4, 5 ] }, { "title": "IV. INTERACTING GGDE IN A NON-FLAT UNIVERSE", "content": "The flatness problem in standard cosmology was resolved by considering an inflation phase in the evolution history of the universe. Following this theory it became a general belief that our universe is spatially flat. However, later it was shown that exact flatness is not a necessary consequence of inflation if the number of e-foldings is not very large [47]. So it is still possible that there exists a contribution to the Friedmann equation from the spatial curvature, though much smaller than other energy components according to observations. Thus, theoretically the possibility of a curved FRW background is not rejected. In addition, recent observations support the possibility of a non-flat universe and detect a small deviation from k = 0 [48-51]. Furthermore, the parameter Ω k represents the contribution to the total energy density from the spatial curvature and it is constrained as -0 . 0175 < Ω k < 0 . 0085 with 95% confidence level by current observations [52]. Our aim in this section is to study the dynamic evolution of the GGDE in a universe with spatial curvature. The first Friedmann equation in a non-flat universe is written as where k is the curvature parameter with k = -1 , 0 , 1 corresponding to open, flat, and closed universes, respectively. Taking the energy density parameters (3) into account and defining the energy density parameter for the curvature term as Ω k = k/ ( a 2 H 2 ), the Friedmann equation can be rewritten in the following form Using the above equation the energy density ratio becomes The second Friedmann equation reads while the time derivative of GGDE density is Inserting Eq. (30) into (31) and combining the resulting relation with the conservation equation for DE component (21), after using (22) and (29), we find the EoS parameter of interacting GGDE in non-flat universe From the second Friedmann equation, (30), one can easily obtain and therefore the deceleration parameter in a non-flat background is obtained as Substituting Eqs. (29) and (32) in (34) we obtain In a non-flat FRW universe, the equation of motion of interacting GGDE is obtained following the method of the previous section. The result is In the limiting case Ω k = 0, the results of this section restore their respective equations in a flat FRW universe derived in the previous sections, while for ξ = 0 the respective relations in [15] are retrieved. The evolutions of w D and q against Ω D for a non-flat interacting GGDE and GDE models are plotted in Fig.3. Let us explore different features of GGDE in non-flat universe by a numerical study. First of all we study the EoS parameter of the GGDE in the future where Ω D → 1. In this case, taking ξ = 0 . 1, b = 0 . 15 and Ω k = 0 . 01 leads to w D = -1 . 05 which indicates that the GGDE is capable to cross the phantom line in the future. The present stage of the universe can be achieved by the same set of parameters but Ω D = 0 . 72. In such a case we see that w D 0 = -0 . 78 while the effective EoS parameter becomes w eff0 = -0 . 6 which is consistent with observations. The deceleration parameter of the model can also be obtained which is in agreement with observational evidences. For example, for the above choice of parameters one finds q 0 = -0 . 34 [40]. Transition from deceleration to the acceleration phase, in the interacting non-flat case, take place at Ω D = 0 . 52.", "pages": [ 5, 6, 7 ] }, { "title": "V. COSMOLOGICAL CONSTRAINTS", "content": "In order to constrain our model parameters space and check its viability, we apply the Marcov Chain Monte Carlo (MCMC) method. Observational constraints on the original GDE with and without bulk viscosity, was already performed [24]. Our work differs from [24] in that we consider the GGDE with energy density ρ D = αH + βH 2 , while the authors of [24] studied the original GDE with energy density ρ D = αH . Besides, we have extended here the study to the universe with any spacial curvature. To make a fitting on the cosmological parameters the public available CosmoMC package [39] has been modified.", "pages": [ 7 ] }, { "title": "A. Method", "content": "We want to get the best value of the parameters with 1 σ error at least. Thus, following [24], we employ the maximum likelihood method where the total likelihood function L = e -χ 2 / 2 is the product of the separate likelihood functions Here SNIa stands for type Ia supernova, BAO for baryon acoustic oscillation and gas stands for X -ray gas mass fraction data. The best fitting values of parameters are obtained by minimizing χ 2 tot . In the next subsection, every dataset will be discussed separately. We employ the following datasets. CMB data from WMAP7 [35], 557 Union2 dataset of type Ia supernova [36], baryon acoustic oscillation (BAO) data from SDSS DR7 [37], and the cluster X-ray gas mass fraction data from the Chandra X-ray observations datasets [38].", "pages": [ 7 ] }, { "title": "1. Cosmic Microwave Background", "content": "For the CMB data, we use the WMAP7 dataset [35]. The shift parameter R, which parametrize the changes in the amplitude of the acoustic peaks is given by [53] where z ∗ is the redshift of decoupling. In addition, the acoustic scale l A , which characterizes the changes of the peaks of CMB via the angular diameter distance out to the decoupling is defined as well in [53] by The comoving distance r ( z ) is defined and the comoving sound horizon at the recombination r s ( z ∗ ) is written and the sound speed c s ( a ) is defined by where the seven-year WMAP observations gives Ω γ 0 = 2 . 469 × 10 -5 h -2 [35]. The redshift z ∗ is obtained by using the fitting function proposed by Hu and Sugiyama [54] where Then one can define χ 2 CMB as χ 2 CMB = X T C -1 CMB X , with [24, 35] where C -1 CMB is the inverse covariant matrix.", "pages": [ 7, 8 ] }, { "title": "2. Type Ia Supernovae Data", "content": "We shall use the SNIa Union2 dataset [36] which includes 577 SNIa. The Hubble parameter H ( z ) determines the history of the universe. However, H ( z ) is specified by the underlying theory of gravity. To test this model, we can use the observational data for some predictable cosmological parameter such as luminosity distance d L . One may note that the Hubble parameter H ( z ; α 1 , ..., α n ) can describe the universe, where parameters ( α 1 , ...α n ) are predicted by the cosmological model. For such a cosmological model we can define the theoretical 'Hubble-constant free' luminosity distance as where E ≡ H H 0 , z is the redshift parameter, and Then one can write the theoretical modulus distance where µ 0 = 5log 10 ( cH -1 0 /Mpc ) + 25. On the other hand, the observational modulus distance of the SNIa, µ obs ( z i ), at redshift z i is given by where m and M are apparent and absolute magnitudes of SNIa respectively. Then the parameters of the theoretical model, α i s, can be determined by a likelihood analysis by defining χ 2 SNIa ( α i , M ' ) in Eq. (37) as where the nuisance parameter, M ' = µ 0 + M , can be marginalized over as", "pages": [ 8, 9 ] }, { "title": "3. Baryon Acoustic Oscillation", "content": "The baryon acoustic oscillations data from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) [37] is used here for constraining the model parameters. The data constrains d z ≡ r s ( z d ) /D V ( z ), where r s ( z d ) is the comoving sound horizon at the drag epoch (where baryons were released from photons) and D V is given by [55] The drag redshift is given by the fitting formula [56] where Then we can obtain χ 2 BAO by χ 2 BAO = Y T C -1 BAO Y , where and its covariance matrix is given by [37] These results are similar to those obtained in [24] for original GDE in flat universe. The ratio of the X-ray gas mass to the total mass of a cluster is defined as X-ray gas mass fraction [38]. The ΛCDM model proposed [38] The elements in Eq. (56) are defined as follows: D Λ CDM A ( z ) and D A ( z ) are the proper angular diameter distance in the ΛCDM and the interested model respectively. Angular correction factor A is caused by the change in angle for the our interested model θ 2500 in comparison with θ Λ CDM 2500 , where η = 0 . 214 ± 0 . 022 [38] is the slope of the f gas ( r/r 2500 ) data within the radius r 2500 . The proper angular diameter distance is given by The bias factor b ( z ) in Eq. (56) contains information about the uncertainties in the cluster depletion factor b ( z ) = b 0 (1+ α b z ), the parameter γ accounts for departures from the hydrostatic equilibrium. The function s ( z ) = s 0 (1+ α s z ) denotes the uncertainties of the baryonic mass fraction in stars with a Gaussian prior for s 0 , with s 0 = (0 . 16 ± 0 . 05) h 0 . 5 70 [38]. The factor K describes the combined effects of the residual uncertainties, such as the instrumental calibration, and a Gaussian prior for the 'calibration' factor is considered by K = 1 . 0 ± 0 . 1 [38]. Then, χ 2 gas is defined as [38] with the statistical uncertainties σ f gas ( z i ).", "pages": [ 9, 10 ] }, { "title": "B. Results", "content": "Finally, the maximum likelihood method is applied for the interacting GGDE in a non-flat universe by using the CosmoMc code [39]. Figure. 5 shows 2-D contours with 1 σ and 2 σ confidence levels where 1-D distribution of the model parameters are shown as well. Best fit parameter values are shown in Table. I with 1 σ and 2 σ confidence levels. From Table I we can see that the best fit results are given as: Ω 0 DE = 0 . 7145 +0 . 0427+0 . 0484 -0 . 0264 -0 . 0452 , Ω 0 m = 0 . 2854 +0 . 0264+0 . 0452 -0 . 0427 -0 . 0467 , Ω 0 k = 0 . 0285 +0 . 0014 -0 . 0274 . In addition for the model parameters the best fit values are obtained as: ξ = 0 . 2300 +0 . 4769 -0 . 0129 , b = 0 . 0592 +0 . 1407 -0 . 0492 . The age of the universe in this model is given by 13 . 7385 +0 . 3302+0 . 3796 -0 . 2907 -0 . 3313 Gyr. We have also plotted the evolution of ω D , Ω D and q against the scale factor a for the interacting GGDE in a nonflat universe by using the best fit values of the model parameters.", "pages": [ 10 ] }, { "title": "VI. SUMMARY AND DISCUSSION", "content": "In order to resolve the DE puzzle, people usually prefer to handle the problem by using existing degree's of freedom. GDE is a prototype of these models which discusses the acceleration of the universe and originates from vacuum energy of the Veneziano ghost field in QCD. This model can address the fine tuning problem [15]. An extended version of this model called GGDE was recently proposed by Cai et. al., [27], seeking a better agreement with observations. In this paper we explored some features of GGDE in both flat and non-flat FRW universe in the presence of an interaction term between the two dark components of the universe. In section II, we discussed the GGDE in a flat FRW background. We found that the EoS parameter approaches -1 which is the same as the cosmological constant. The next section was devoted to the interacting GGDE in a flat geometry. An interesting feature which we found was the capability of crossing the phantom line in this case. This behaviour is also seen in the last section for interacting GGDE in a universe with spatial curvature. Then, we applied the Markov Chain Monte Carlo method together with the latest observational data to constrain the model parameters. The results are presented in Table I and Fig. 5. The main result found through this paper is that in the GGDE model, there is a delay in different epoches of the cosmic evolution in comparison with original GDE model. This result was also pointed out in [27] due to the negative contribution of the square term in the energy density of GGDE.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "We are grateful to the referees for constructive comments which helped us to improve the paper significantly. A. Sheykhi thanks from the Research Council of Shiraz University. This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM) Iran. b DM 2 Ω Ω D. N. Spergel, Astrophys. J. Suppl. 148 (2003) 175; M. Tegmark, et al., Phys. Rev. D 69 (2004) 103501;", "pages": [ 11, 12, 13 ] } ]

[ { "title": "VLADIMIR CHERNOV AND STEFAN NEMIROVSKI", "content": "Abstract. It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R 4 . Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold N and R and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to N × R . Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on (3 + 1)-dimensional spacetimes. Introduction. One form of the strong cosmic censorship hypothesis proposed by Penrose asserts that 'physically relevant' spacetimes should be globally hyperbolic (see [16]). The purpose of this note is to point out that global hyperbolicity imposes strong restrictions on the differential topology of the spacetime. The starting point of all our considerations will be the smooth splitting theorem for globally hyperbolic spacetimes established by Bernal and S'anchez [2, 3]. All manifolds will be assumed Hausdorff and paracompact, since Hausdorff spacetimes are necessarily paracompact by [6, pp. 1743-1744]. The first result is valid in all dimensions but seems to be particularly interesting for (3 + 1)-dimensional spacetimes. In that case, the argument makes essential use of the three-dimensional Poincar'e conjecture proved by Perelman [17, 18, 19]. Theorem A. Let ( X,g ) be a globally hyperbolic ( n + 1) -dimensional spacetime. Suppose that X is contractible. Then X is diffeomorphic to the standard R n +1 . For every n ≥ 3, there exist uncountably many contractible smooth n -manifolds that are not homeomorphic to R n (see [12], [7] and [5]). In dimension four, in addition to that there are uncountably many smooth four-manifolds that are homeomorphic but not diffeomorphic to R 4 (the so-called exotic R 4 's, see [8] and [22]). The theorem shows that none of those carry globally hyperbolic Lorentz metrics. The topological argument used to prove Theorem A in the (3+1)-dimensional case was first applied in the context of Lorentz geometry by Newman and Clarke [15]. They showed that a globally hyperbolic spacetime which is diffeomorphic to R 4 can have any contractible 3-manifold as its Cauchy surface, see Remark 2.3. Global hyperbolicity singles out 'standard' smooth structures on another large class of (3 + 1)-dimensional spacetimes as well. The following result is based on Perelman's geometrization theorem for 3-manifolds and the work of Turaev [24]. Theorem B. Let ( X,g ) be a globally hyperbolic (3 + 1) -dimensional spacetime. Suppose that X is homeomorphic to the product of a closed oriented 3 -manifold N and R . Then X is diffeomorphic to N × R , where N and R have their unique smooth structures. In fact, we do not know an example of a topological 4-manifold admitting two non-diffeomorphic smooth structures each of which is the smooth structure of a globally hyperbolic spacetime. However, such manifolds exist in higher dimensions (for instance, S 7 × R ). To show that 4-dimensional examples do not exist, one would need to prove Theorem B for a 3-manifold N that may be non-compact or non-orientable. For two points x, y ∈ X , we write x ≤ y if either x = y or there exists a futurepointing curve connecting x to y . A spacetime is called causal if ≤ defines a partial order on it, that is, if there are no closed non-trivial future-pointing curves. A spacetime ( X,g ) is globally hyperbolic if it is causal and the 'causal segments' I x,y = { z ∈ X | x ≤ z ≤ y } are compact for all x, y ∈ X . (This definition is equivalent to the classical one [10, § 6.6] by [4, Theorem 3.2].) A Cauchy surface in a spacetime is a subset such that every endless future-pointing curve meets it exactly once. It is a classical fact [10, pp. 211-212] that a spacetime is globally hyperbolic if and only if it contains a Cauchy surface. It has long been conjectured (and sometimes tacitly assumed) that Cauchy surfaces can be chosen to be smooth and spacelike and that a globally hyperbolic spacetime must be diffeomorphic to the product of its Cauchy surface with R ; this was finally proved by Bernal and S'anchez in 2003. Theorem 1.1 (Bernal-S'anchez [2, 3]) . For a globally hyperbolic ( n +1) -dimensional spacetime ( X,g ) , there exist an n -dimensional smooth manifold M and a diffeomorphism h : M × R → X such that Note that it follows by projecting along the timelike t -direction that all smooth spacelike Cauchy surfaces in ( X,g ) are diffeomorphic to the same manifold M . Proposition 2.1 (McMillan [11, 13], Stallings [21]) . Suppose that M is a contractible smooth n -manifold. Then M × R is diffeomorphic to R n +1 . Proof. The proof splits into three cases according to the dimension of M . The result is obvious because the only contractible manifolds of dimension ≤ 2 are R and R 2 . We outline McMillan's argument [11, 13] trying to give precise references for each step. For an introduction to the relevant topological methods, the reader may consult the book by Rourke and Sanderson [20]. McMillan [11, Theorem 1] proved that if the three-dimensional Poincar'e conjecture holds true , then M can be exhausted by compact subsets PL-homeomorphic to handlebodies with handles of index one. It follows by an engulfing argument [11, Proof of Theorem 2, p. 513] that M × R is the union of compact subsets B n ⊂ M × R such that B n ⊂ Int B n +1 and each B n is PL-homeomorphic to the 4-ball. McMillan and Zeeman observed [13, Lemma 4] that this implies that M × R is PL-homeomorphic to R 4 . However, if a smooth manifold is PL-homeomorphic to R n , then it is diffeomorphic to R n by a result of Munkres [14, Corollary 6.6]. Since the Poincar'e conjecture is now known to be true because of Perelman's work [17, 18, 19], the result follows. 3. dim M ≥ 4. This is a special case of a result of Stallings [21, Corollary 5.3]. /square Remark 2.2 (The rˆole of the Poincar'e conjecture) . The three dimensional Poincar'e conjecture enters the preceding argument in the case n = 3 through the proof of [11, Theorem 1]. It is used there in the form of the following statement: A nullhomotopic embedded 2 -sphere in a three-manifold bounds a 3 -ball. The assertion that such a sphere bounds a homotopy ball is classical and 'elementary' (see e. g. [9, Proposition 3.10]); the Poincar'e conjecture ensures that the only homotopy 3-ball is the usual one. Remark 2.3 (Standard spacetimes vs non-standard Cauchy surfaces) . Following Newman and Clarke [15], let us show that although the underlying manifolds of contractible globally hyperbolic spacetimes are standard, their Cauchy surfaces can be completely arbitrary: For every contractible smooth n -manifold M , there exists a globally hyperbolic spacetime of the form ( R n +1 , g ) with Cauchy surface diffeomorphic to M . Indeed, take any complete Riemann metric ¯ g on M , then ( M × R , ¯ g ⊕-dt 2 ) is a globally hyperbolic spacetime. By Proposition 2.1 the manifold M × R is diffeomorphic to R n +1 . Note first that and hence M is closed and orientable. Turaev [24, Theorem 1.4, p. 293] proved that two orientable closed geometric 3manifolds are homeomorphic if and only if they are topologically h -cobordant. In [24, p. 291] geometric 3-manifolds were defined as connected sums of Seifert fibred, hyperbolic, and Haken manifolds. It is now known by the work of Perelman [17, 18] that a non-Haken (and hence atoroidal) irreducible orientable closed 3-manifold is either Seifert fibred (which includes all spherical 3-manifolds [23, p. 248]) or hyperbolic, see e. g. [1, Theorem 1.1.6]. Thus, all closed orientable 3-manifolds are geometric in the sense of [24]. It remains to construct a topological h -cobordism between N and M . Let ψ : M × R → N × R be a homeomorphism. Since ψ ( M ×{ 0 } ) is compact, it is contained in N × ( -∞ , T ) for some T /greatermuch 0. Reversing the R -factor in M × R if necessary, we may assume that N ×{ T } ⊂ ψ ( M × (0 , + ∞ )). Set This is a compact topological manifold with boundary that defines a topological cobordism between M top = ψ ( M ×{ 0 } ) and N = N ×{ T } . By the definition of an h -cobordism, we have to check now that the inclusions of the boundary components into W are homotopy equivalences. Let r M : M × R /dblarrowheadright M × [0 , + ∞ ) and r N : N × R /dblarrowheadright N × ( -∞ , T ] be the obvious strong deformation retractions. Then is a strong deformation retraction. Hence, the inclusion W ↪ → N × R is a homotopy equivalence. Since the inclusions N ×{ T } ↪ → N × R and ψ ( M ×{ 0 } ) ↪ → N × R are also homotopy equivalences, it follows that W is a topological h -cobordism indeed, which completes the proof of Theorem B.", "pages": [ 1, 2, 3, 4 ] }, { "title": "References", "content": "Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, NH 03755, USA E-mail address : [email protected] Steklov Mathematical Institute, 119991 Moscow, Russia; Mathematisches Institut, Ruhr-Universitat Bochum, 44780 Bochum, Germany E-mail address : [email protected]", "pages": [ 5 ] } ]

[ { "title": "C. M. WARNICK", "content": "Abstract. We consider the massive wave equation on asymptotically AdS spaces. We show that the timelike I behaves like a finite timelike boundary, on which one may impose the equivalent of Dirichlet, Neumann or Robin conditions for a range of (negative) mass parameter which includes the conformally coupled case. We demonstrate well posedness for the associated initial-boundary value problems at the H 1 level of regularity. We also prove that higher regularity may be obtained, together with an asymptotic expansion for the field near I . The proofs rely on energy methods, tailored to the modified energy introduced by Breitenlohner and Freedman. We do not assume the spacetime is stationary, nor that the wave equation separates. ALBERTA THY 3-12", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Among the solutions of Einstein's general theory of relativity, the maximally symmetric spacetimes hold a privileged position. Owing to the high level of symmetry, they serve as plausible 'ground states' for the gravitational field, so there is great interest in spacetimes which approach a maximally symmetric spacetime in some asymptotic region. Such a spacetime would represent an 'isolated gravitating system'. Historically, asymptotically flat spacetimes have been the most studied, however, recently there has been great interest in the asymptotically anti-de Sitter (AdS) spacetimes motivated by the putative AdS/CFT correspondence [1]. In the study of classical General Relativity, there have also been some very interesting recent results regarding the question of black hole stability [2, 3, 4] for asymptotically AdS black holes. The asymptotically AdS spacetimes approach (the covering space of) the spacetime of constant sectional curvature -3 l 2 , which we shall refer to as the AdS spacetime. In so called 'global coordinates', the metric takes the form In contrast to the Minkowski spacetime, AdS has a timelike conformal boundary, I . Accordingly one expects that in order to have a well posed time evolution for the equations of physics in this background it is necessary to specify some boundary condition on I . In pioneering work [5], Breitenlohner and Freedman considered the massive wave equation on a fixed anti-de Sitter background of constant sectional curvature -3 l 2 . They were able to solve the wave equation by separation of variables, making use of the SO (2 , 3) symmetry of AdS. The second order ordinary differential equation governing the radial part of the wave equation has a regular singular point at infinity, hence the field has an expansion near infinity: where λ ± = 3 2 ± √ 9 4 + λ . When the mass parameter is in the range -9 4 < λ < 0, both branches decay towards infinity. For a well posed problem it is necessary to place some constraints on the functions ψ ± . The usual choice would be to insist that ψ -= 0, which is analogous to imposing a Dirichlet condition at I . This corresponds to requiring a solution of finite energy (we shall elaborate on this point later). Breitenlohner and Freedman showed that for -9 4 < λ < -5 4 the wave equation can also be solved on the exact anti-de Sitter space under the assumption that ψ + = 0, analogous to a Neumann condition. 1 Breitenlohner and Freedman also introduced a modified or renormalised energy which is finite for both branches of the solution. As in the case of a finite domain, the Dirichlet and Neumann boundary conditions are not the only possible choices. We summarise some other possible boundary conditions in Table 1 below (the list is not exhaustive). The homogeneous conditions (1), (3), and (5) were considered by Ishibashi and Wald [6, 7], who showed that they give rise to a well defined unitary evolution for the scalar wave, Maxwell and gravitational perturbation equations in the exact anti-de Sitter spacetime. This work has been extended to the Dirac equation in the work of Bachelot [8]. These papers use methods based on self-adjoint extensions of the elliptic part of the wave operator. They make crucial use of properties of the exact AdS space (in particular staticity and separability) which are not shared by the general class of asymptotically AdS spaces. For Dirichlet boundary conditions (1), (2), the work of Holzegel [9] (using energy space methods) and Vasy [10] (using microlocal analysis) provides well posedness results for a more general class of asymptotically AdS spaces for the range λ > -9 4 , but does not treat the other possible boundary conditions. The aim of this work is to treat all of the boundary conditions (1)-(6), without making any assumptions regarding the staticity or separability of the metric. The natural approach is that of energy estimates, however, we are forced to confront the problem that for boundary conditions (2)-(6) we cannot expect the standard energy to be finite. To deal with this, we use the renormalised energy of [5]. We show that the natural Hilbert spaces associated with this energy are generalisations of the usual Sobolev spaces where 'twisted' derivatives of the form are supposed to exist in a distributional sense and belong to an appropriate L 2 space, for some appropriately chosen ρ , related to the distance to the boundary. To simplify the analysis at the expense of losing the geometrical structure, we map the problem to a more general problem in a finite region of R N which is very closely analogous to that of a finite initial boundary value problem (IBVP). We introduce weak formulations at the H 1 level for all of the boundary conditions, and are able to show that these weak problems admit a unique solution. The method used to show existence is to approximate the problem by a suitable hyperbolic IBVP in a finite cylinder, and let the cylinder approach I . We then use energy estimates to extract a weakly convergent subsequence. in this way, we can show well posedness results for (1)-(6). We are also able to recover higher regularity for the solution, if more assumptions are made on the data. We further provide an asymptotic expansion for the solutions near infinity. The paper will be structured as follows. We first define the asymptotically AdS spaces we consider in § 2. We then introduce the modified energy in § 3 and use it in § 4 to motivate weak formulations of the Dirichlet and Neumann problems, which we then show to be well posed. In § 5 we show that under stronger assumptions on data improved regularity can be obtained, together with the asymptotic behaviour of the solution. Finally, in § 6 we discuss briefly the inhomogeneous and Robin boundary conditions and remark on the connection to methods involving self-adjoint extensions. We assume throughout a degree of familiarity with the theory of the finite IBVP, as developed for example in [11, 12]. Acknowledgements. I would like to thank Gustav Holzegel for introducing me to this problem, and for helpful comments. I would also like to thank Mihalis Dafermos, Julian Sonner, Pau Figueras as well as the anonymous referees for comments. I would like to acknowledge funding from PIMS and NSERC. The early stages of this project were supported by Queens' College, Cambridge.", "pages": [ 1, 2, 3 ] }, { "title": "2. Asymptotically AdS spaces", "content": "Definition 1. Let X be a n + 1 dimensional manifold with boundary 2 ∂X , and g be a smooth Lorentzian metric on ˚ X . We say that a connected component I of ∂X is an asymptotically anti-de Sitter end of ( ˚ X,g ) with radius l if: We say that r is the asymptotic radial coordinate and I is the conformal infinity of this where g αβ dx α dx β is a Lorentzian metric on I . iii) r -2 g extends as a smooth metric on a neighbourhood of I . end. We make here a few remarks about these assumptions", "pages": [ 4 ] }, { "title": "3. The modified energy", "content": "We will now consider for a moment the case of the exact AdS spacetime. The usual energy one associates to solutions of the massive wave equation is given by where K µ is the timelike Killing vector and the energy-momentum tensor is given by and satisfies ∇ µ T µν = 0 when φ is a solution of (1.1). If φ has Dirichlet decay then one expects, by power counting, that E [Σ t ] will be finite. However, if we have Neumann decay then the integral in (3.1) fails to converge near infinity. In order to deal with this problem, Breitenlohner and Freedman modified the energy momentum tensor (3.2) to give a new tensor The modification satisfies and so for the exact AdS spacetime, ˜ T µν will also give rise to a formally conserved energy. It transpires that the new energy ˜ E [Σ t ] differs from the original energy E [Σ t ] by a surface term. This surface term vanishes when φ decays like the Dirichlet branch, but diverges when φ decays like the Neumann branch. By choosing κ appropriately, it is possible to construct an energy which is positive, finite and conserved for both Dirichlet and Neumann decay conditions. Our proofs will make use of the modified energy associated to the timelike vector ∂ t which, however, will only be approximately conserved since we no longer assume an exactly stationary spacetime. Rather than using the definition (3.3), it is in practice more straightforward to work directly from the PDE. In order to see how this works, we will briefly discuss a toy model which captures the salient features of the problem.", "pages": [ 4, 5 ] }, { "title": "3.1. A toy model. Consider the wave equation", "content": "where 0 < α < 1, subject to initial conditions Considering the behaviour near x = 0, we hope to impose as boundary conditions either At x = 1 we will require that u = 0. Suppose we have a suitably smooth solution to this equation. We can multiply (3.4) by xu t and, after integrating by parts, deduce the conservation law for the standard energy: For Dirichlet boundary conditions at x = 0 the right hand side vanishes, however, for Neumann boundary conditions it is infinite. In order to introduce the modified energy, it is convenient to re-write the equation in the following form which gives (3.4) upon expanding using Leibniz rule. We can multiply (3.7) by xu t and integrate by parts to deduce Now, for Dirichlet conditions at x = 0 we again find that the right hand side vanishes, however we now find that it also vanishes for the Neumann behaviour at x = 0. The two energies differ by a surface term: which vanishes for Dirichlet conditions at x = 0 and is infinite for the Neumann conditions. In this sense we can view ˜ E as a 'renormalized' energy, since we have formally subtracted an infinite boundary term from the infinite energy to get a finite result. Thus even though the standard energy is infinite for the Neumann behaviour, we can modify it to get a conserved, finite, positive energy. We see now the justification for using the terms 'Dirichlet' and 'Neumann' to describe the boundary conditions. The Dirichlet condition requires u → 0 as x → 0, while the Neumann condition requires x -α ∂ x ( x α u ) → 0 as x → 0. This discussion also suggests that it will be fruitful to re-formulate the equation in terms of 'twisted' derivatives of the form x -α ∂ x ( x α · ). We shall do so in the next section and this will lead us to the appropriate setting in which to discuss the well posedness of (1.1).", "pages": [ 5, 6 ] }, { "title": "4. Well Posedness of the Weak Formulation", "content": "4.1. Defining the problem. Motivated by the discussion of the previous section, we can now define the framework in which we shall work. We assume that U ⊂ R N is a bounded subset of R N with compact C ∞ boundary ∂U . This means that in the neighbourhood of any point P ∈ ∂U , there exists an open neighbourhood W P ⊂ U of P and a smooth bijection Φ P : W P → R N + ∩ B ( 0 , δ P ), where R N + = { ( x, x a ) ∈ R N : x ≥ 0 } and B ( x , r ) is the open Euclidean ball centred at x with radius r . We're also going to assume that there exists a smooth function ρ : U → R + , which vanishes only on ∂U and such that there exists ˜ /epsilon1 so that if d ( x, ∂U ) < ˜ /epsilon1 , we have ρ ( x ) = d ( x, ∂U ) and if d ( x, ∂U ) > ˜ /epsilon1 , ρ ( x ) > ˜ /epsilon1 . We will set n i = ∂ i ρ , which extends the unit normal of ∂U into the interior of U . We may assume that the neighbourhoods W P are such that ρ · Φ -1 P ( x, x a ) = x and δ P = ˜ /epsilon1 . We denote by U T the timelike cylinder (0 , T ) × U and by ∂U T the boundary (0 , T ) × ∂U . We define our twisted derivatives in a similar vein to above. For a differentiable function, we set Throughout, we will assume 0 < α < 1. We can see that this restriction is necessary from the toy model, since if α is outside this range, only the Dirichlet behaviour is compatible with finite energy even after renormalisation. We may now define the equation in which we are interested: where 4 subject to the initial conditions We assume all coefficients a ij , b i , c are in C ∞ ( U T ), however this is certainly stronger than necessary 5 . We will assume throughout that a ij is a symmetric matrix such that the uniform ellipticity condition holds: for any ξ i ∈ R N , where θ is uniform in both time and space coordinates, and furthermore that n i a ij is independent of t , on the boundary ∂U . For the time being, there are two possible boundary conditions in which we shall be interested. We will consider both Dirichlet: and Neumann: boundary conditions. To justify considering this equation, we have the following Lemma Lemma 4.1.1. Suppose I is an asymptotically AdS end of ( ˚ X n +1 , g ) with radius l , and let P ∈ I . Then there exists a smooth Lorentzian metric , ˜ g , on the solid cylinder U T = [ -T, T ] × B (0 , 1) ⊂ R n +1 together with a neighbourhood of P which embeds isometrically into ( U T , ˜ g ) , with I mapped to (a portion of) the boundary of the cylinder. Furthermore, setting φ = p r n/ 2 u for some p ∈ C ∞ ( U T ) depending only on g , the wave equation may be cast in the form (4.1) for some ρ , a ij , b i , c satisfying the assumptions above, with ρ = r -1 + O ( r -3 ) and 6 α = √ n 2 4 + λ . Proof. Define s = r -1 , so that I = { s = 0 } and ˆ g = s 2 g is a smooth metric on X , with I a totally geodesic submanifold. Pick a spacelike surface Σ 0 , containing P such that, Σ 0 is orthogonal to I and has normal n Σ 0 with respect to ˆ g . We can push forward n Σ 0 using the geodesic flow of ˆ g on TX to give a smooth unit vector field T , with associated diffeomorphism ψ t , in a neighbourhood of P . Now pick coordinates x i = ( ρ, x a ) on Σ 0 near P , such that ρ = s | Σ 0 , and extend them off Σ 0 by requiring Tρ = Tx a = 0. Near P we may take as coordinate functions ( t, ρ, x a ), and in these coordinates, the metric coefficients satisfy as ρ → 0, for some h ab independent of ρ . We can assume that the coordinate neighbourhood, V we constructed is in fact contained in a coordinate neighbourhood of the boundary of U T , and extend the metric ˆ g smoothly to a metric on the whole of U T , and define ˜ g = ρ -2 ˆ g . This agrees with g in V . Now note that √ g = ρ -( n +1) ( √ h + ρ 2 and it may then be verified that (4.6) may be cast in the form (4.1), with α = √ n 2 4 + λ , a ij = ρ -2 g ij and b i , c similarly given by functions constructed from the metric and its derivatives which are smooth up to ρ = 0. /square Making use of the finite speed of propagation for solutions of hyperbolic equations, any well posedness results for the problem (4.1) may be extended to regions of ˚ X , assuming some global causality conditions. In particular, well posedness of (4.1) with appropriate boundary conditions implies well posedness in the region D + [Σ ∪ ( I + (Σ) ∩ I )] for any spacelike hypersurface Σ, with initial data specified on Σ. where dv is the Lebesgue measure. This is clearly a Hilbert space with inner product Now, we note that for smooth functions φ, ψ of compact support we may integrate by parts to find This allows us to define a weak version of ∂ α . We say that v i = ∂ α i u is the weak α -twisted derivative of u if for all φ ∈ C ∞ c ( V ). We say that u ∈ H 1 ( V ) if ∂ α u exists in a weak sense and ∂ α i u ∈ L 2 ( V ). We can define a norm and inner product on H 1 ( V ) as follows: Next we define H 1 0 ( V ) to be the completion of C ∞ c ( V ) with respect to the norm ||·|| H 1 ( V ) . We shall often take V = U . On any subset compactly contained in U , these spaces are simply equivalent to the standard Sobolev spaces. We note at this stage that ∂ 1 -α i is the formal adjoint of ∂ α i with respect to the L 2 inner product. Thus the second order operator ∂ 1 -α i ( a ij ∂ α j · ) appearing in (4.1) is formally self-adjoint. When we come to consider higher regularity, we shall need the Sobolev space associated to the adjoint derivative operator. In particular u ∈ ˜ H 1 ( V ) if ∂ 1 -α i u exist in a weak sense and ∂ 1 -α i u ∈ L 2 ( V ), with the obvious inner product and norm. Again we define ˜ H 1 0 ( V ) to be the completion of C ∞ c ( V ) with respect to the norm ||·|| ˜ H 1 ( V ) . Let us state some properties of functions in these spaces. Lemma 4.2.1. (i) Functions of the form u = ρ -α v , with v ∈ C ∞ ( U ) are dense in H 1 ( U ) . where u 0 ∈ L 2 ( ∂U ) , with u 0 = 0 iff u ∈ H 1 0 ( U ) . Furthermore, for any δ > 0 , there exists a C δ such that Similar results hold for ˜ H , but with α replaced by 1 -α . Part ( i ) follows from a result of Kufner [13], and parts ( ii )-( iii ) may be derived by showing that the inequalities hold on suitable dense subsets. From this we see that if u ∈ H 1 ( U ), then u may 'blow up like ρ -α near ∂U ', whereas if u ∈ H 1 0 ( U ) then u is 'bounded near ∂U ' in some appropriate sense. These spaces thus capture, to a certain degree, the boundary behaviour we hope for in our solutions. A consequence of the proof of ( ii ) is that H 1 0 ( U ) = ˜ H 1 0 ( U ). In fact, we can prove a sharper result about the range of the trace operator, together with an extension result: Lemma 4.2.2. The operator T · ρ α , where T is the trace operator, maps H 1 ( U ) into H α ( ∂U ) , and the map is surjective. Furthermore there exists a bounded right inverse so that corresponding to any u 0 ∈ H α ( ∂U ) , there exists a u ∈ H 1 ( U ) with ρ α u | ∂U = u 0 in the trace sense, with the estimate where C is independent of u 0 . This follows from the fact that ρ α u belongs to a weighted Sobolev space, to which one may apply the results of [14]. We will also require the spaces H 1 0 ( U ) ∗ and H 1 ( U ) ∗ , the dual spaces of H 1 0 ( U ) and H 1 ( U ) respectively. If f ∈ X ∗ , u ∈ X we denote the pairing by and define It will be convenient, for notational compactness, to define the following spaces and norms. The Neumann data space H 1 data, N ( V ) consists of triples ( u 0 , u 1 , f ) with u 0 ∈ H 1 ( V ), u 1 ∈ L 2 ( V ) and f ∈ L 2 ([0 , T ]; L 2 ( V )), whereas for the Dirichlet data space H 1 data, D ( V ) we additionally require u 0 ∈ H 1 0 . For both spaces, we define We take H 1 sol., D ( V ) to consist of u ∈ L ∞ ([0 , T ]; H 1 0 ( V )) with and H 1 sol., N ( V ) to consist of u with 4.3. The Weak Formulations. In order to motivate the definition of the weak solutions, let us suppose that we have a solution to which is sufficiently smooth for the following operations to make sense. We can multiply the equation by a smooth function v , integrate over U and integrate by parts to establish The surface term will vanish either if u satisfies the Neumann boundary conditions, or else if v satisfies the Dirichlet conditions. We define the following bilinear form on H 1 ( V ) If B has no subscript, we assume the range to be U . Now we may define the weak Dirichlet and Neumann problems: Definition 2 (Weak Dirichlet IBVP) . Suppose ( u 0 , u 1 , f ) ∈ H 1 data, D ( U ) . We say that u ∈ H 1 sol., D ( U ) is a weak solution of the Dirichlet IBVP if Definition 3 (Weak Neumann IBVP) . Suppose ( u 0 , u 1 , f ) ∈ H 1 data, N ( U ) . We say that u ∈ H 1 sol., N ( U ) is a weak solution of the Neumann IBVP if We note that by the calculation above, a strong solution obeying the Dirichlet (resp. Neumann) condition on the boundary is necessarily a weak Dirichlet (resp. Neumann) solution. The converse of course need not be true, however if we have enough regularity to integrate by parts then taking an arbitrary v ∈ H 1 0 ( U ) we conclude that (4.13) holds almost everywhere in U in both the Dirichlet and Neumann case. Noting in the Neumann case that the trace of ρ α v is arbitrary on the boundary we can, with care, deduce that n i a ij ∂ α j u ∈ ˜ H 1 0 ( U ). We will see this in more detail later when we consider the asymptotics of the solutions. Theorem 4.1 (Uniqueness of weak solutions) . Suppose u is a weak solution of either the Dirichlet IBVP or of the Neumann IBVP. Then u is unique. Proof. The proof of uniqueness for the weak solutions proceeds almost identically to the proof of uniqueness of weak solutions to a finite IBVP. Without loss of generality, one may assume trivial data. In both cases one may take as test function and then integrate the weak equation over 0 ≤ t ≤ s . Standard manipulations making use of the uniform hyperbolicity condition (4.3) then show u = 0. For example, one may take the proof of Evans [11, p. 385] and replace the standard spatial derivatives with twisted derivatives. /square Theorem 4.2 (Existence of weak solutions) . (i) Given ( u 0 , u 1 , f ) ∈ H 1 data, D ( U ) , there exists a weak solution to the Dirichlet IBVP corresponding to this data. (ii) Given data ( u 0 , u 1 , f ) ∈ H 1 data, N ( U ) , there exists a weak solution to the Neumann IBVP corresponding to this data. In both cases, we have the following estimate where C depends on T, U, α and the coefficients of the equation. † stands for D or N as appropriate. It is convenient to divide the proof of Theorem 4.2 into several Lemmas. We start by picking a sequence 0 < a k < /epsilon1 which decreases monotonically to zero, and define the sets V k = { x : ρ ( x ) > a k } . The broad strategy is to solve the finite IBVP on each V k , where the equation becomes strictly hyperbolic and classical theory applies, and then find a way of passing to the limit ' k →∞ '. Lemma 4.4.1. (i) Data ( u 0 , u 1 , f ) such that for all k > k 0 the problem has a solution which is C ∞ ([0 , T ] × V k ) form a dense linear subspace of H 1 data, D ( U ) (ii) Data ( u 0 , u 1 , f ) such that for all k > k 0 the problem has a solution which is C ∞ ([0 , T ] × V k ) form a dense linear subspace of H 1 data, N ( U ) Proof. For ( i ), we may take u 0 , u 1 ∈ C ∞ c ( U ) and f ∈ C ∞ c ([0 , T ] × U ). For large enough k the data are supported inside V k , and data of this form a dense linear subspace of H 1 data, D ( U ). For (ii), we need the fact that smooth functions u for which n i a ij ∂ α j u = 0 outside a compact set are dense in H 1 ( U ). To see this, we first note that for any u ∈ H 1 ( U ) we may take u /epsilon1 = ρ -α v /epsilon1 , where v /epsilon1 ∈ C ∞ ( U ) and We define v /epsilon1 0 in a collar neighbourhood of the boundary [0 , δ ) × ∂U to satisfy Here n i is the unit normal of ρ = const. , which defines a smooth vector field provided δ is sufficiently small. Take χ ( ρ ) to be a smooth function, equal to 1 for ρ < δ/ 2 and vanishing for ρ > 3 δ/ 4. Now, u /epsilon1 -ρ -α v /epsilon1 0 χ ( ρ ) = ˜ u /epsilon1 ∈ H 1 0 ( U ), so there exists w ∈ C ∞ c ( U ) such that || ˜ u /epsilon1 -w /epsilon1 || H 1 ( U ) < /epsilon1 . Consider the function y /epsilon1 = ρ -α v /epsilon1 0 + w /epsilon1 . This satisfies and We can suppose then that ρ α u 0 ∈ C ∞ ( U ) with n i a ij ∂ α j u 0 = 0 near ∂U , and take u 1 ∈ C ∞ c ( U ). Finally we can take a smooth f which is a sum of one component in C ∞ c ( U ) and another of arbitrarily small L 2 ([0 , T ]; L 2 ( U )) norm which ensures the higher order compatibility conditions vanish to all orders on t = 0. For large enough k this data will launch a smooth solution and such data are dense in H 1 data, N ( U ). /square Lemma 4.4.2. Suppose u is a smooth solution of with either u k = 0 or n i a ij ∂ α j u k = 0 on ∂V k . Then u k satisfies the estimate ∣ ∣ ∣ ∣ where C is uniform in k . † stands for D or N as appropriate. Proof. We drop the superscript on the solutions u k for convenience. Multiplying by u t and integrating by parts, using the boundary condition to neglect the boundary term, one has We also note that Taking (4.20) and adding it to γ times (4.21), we arrive at the equality Note that we have a bound ∣ ∣ which together with the uniform hyperbolicity condition: implies that there exist γ, M , independent of k such that for each t holds for all smooth u . To see this, recall from (4.14) that Applying the uniform hyperbolicity estimate to the first term on the right hand side and the Cauchy-Schwarz inequality to the second term, we have that for any δ > 0 where Taking δ = θ/ 2 and γ > C δ , we conclude that (4.23) holds with M = 1 + 2 /θ . We can now estimate from (4.22), (4.23) and making use of the fact that we have bounds on the coefficients which are uniform in k : with C independent of k . Using Gronwall's lemma, together with a further application of (4.23) we arrive at (4.19). /square Lemma 4.4.3 (Weak compactness) . (i) Suppose u k ∈ H 1 sol . D ( V k ) , with Then there exists u ∈ H 1 sol . D ( U ) with || u || H 1 sol . D ( U ) ≤ C and a subsequence u k l such that for any v ∈ H 1 0 ( V m ) , taking l large enough that k l > m we have for almost every t : (ii) Suppose u k ∈ H 1 sol . N ( V k ) , with Then there exists u ∈ H 1 sol . N ( U ) with || u || H 1 sol . N ( U ) ≤ C and a subsequence u k l such that for any v ∈ H 1 ( U ) , we have for almost every t : Proof. We demonstrate first the proof for u k ∈ H 1 ( V k ), || u || H 1 ( V k ) ≤ C , i.e. the first part of (ii). We define u k ∈ L 2 ( U ) to agree with u k on V k and to vanish on U \\ V k . Similarly, we define ∂ α i u k ∈ L 2 ( U ) to agree with ∂ α i u k on V k and to vanish on U \\ V k . Weak compactness of L 2 ( U ) gives a weakly convergent subsequence ( u k l , ∂ α i u k l ) ⇀ ( u, v i ). It remains to show that v i = ∂ α i u in the weak sense. To show this, multiply ∂ α i u k l by φ ∈ C ∞ c ( U ) and integrate over U . For l large enough that supp φ ⊂ V k l , we have ∫ U φ∂ α i u k l ρdx = -∫ U ∂ 1 -α i φu k l ρdx , so by taking weak limits we're done. Similar considerations may be applied to the other results in the Lemma, after applying Riesz representation theorem to u . /square Remark : This Lemma can be extended to apply to higher spatial derivatives of u , in an essentially unchanged fashion. Now we can combine the results above to show that there exists a solution to the weak problems. Proof of Theorem 4.2. (i) Suppose we have data ( u 0 , u 1 , f ) such that for all k > k 0 the problem has a solution, u k which is C ∞ ([0 , T ] × V k ). By Lemma 4.4.2, we have the estimate ∣ ∣ ∣ ∣ And we also know that for k > m and for any v ∈ H 1 0 ( V m ) we have Applying Lemma 4.4.3 we conclude the existence of u satisfying and for any v ∈ H 1 0 ( V m ): Noting that functions v ∈ H 1 0 ( V m ) are dense in H 1 0 ( U ), we conclude that u satisfies the first condition to be a weak solution of the Dirichlet IBVP. We must now check that the weak solution we have constructed satisfies the initial conditions. For this, choose any function v ∈ C 2 (0 , T ; C ∞ c ( U )), with v ( T ) = ˙ v ( T ) = 0. Integrating (4.30) in time, we have after twice integrating by parts similarly, we have from (4.28) Setting k = k l , passing to the limit we have: Since v (0) , ˙ v (0) are arbitrary, we conclude that u (0) = u 0 , ˙ u (0) = u 1 and we're done. Finally, we make use of Lemma 4.4.1 together with the uniqueness result Theorem 4.1 and a standard argument based on continuity, using (4.29), to show that our result holds for any ( u 0 , u 1 , f ) ∈ H 1 data, D ( U ). /square", "pages": [ 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ] }, { "title": "5. Higher Regularity and Asymptotics", "content": "We now wish to show that if more assumptions are made on the data, the weak solution can be shown to have improved regularity. In order to do this, we will require some elliptic estimates, enabling us to control some appropriate H 2 norm of u in terms of L u . Definition 4. We say that a function u ∈ H 1 ( U ) belongs to H 2 ( U ) , provided for all A,B , and we define the norm:", "pages": [ 16, 17 ] }, { "title": "Remarks :", "content": "Definition 5. Suppose f ∈ H 1 0 ( U ) ∗ (resp. H 1 ( U ) ∗ ). We say that u ∈ H 1 0 ( U ) (resp. H 1 ( U ) ) is a weak solution of the Dirichlet (resp. Neumann) problem with u = 0 (resp. n i a ij ∂ α j u = 0 ) on ∂U , if for all v ∈ H 1 0 ( U ) (resp. H 1 ( U ) ). Theorem 5.1 (Elliptic Estimates) . Suppose u is a weak solution of either the Dirichlet or Neumann problem (5.1) and suppose that in fact f ∈ L 2 ( U ) . Then u ∈ H 2 ( U ) with the estimate Furthermore, in the Dirichlet case T i ∂ α i u ∈ H 1 0 ( U ) and in the Neumann case N i ∂ α i ∈ ˜ H 1 0 ( U ) . We split the result into several Lemmas Lemma 5.2.1. There exist constants C 1 , C 2 and µ 0 ≥ 0 such that Proof. This is a standard manipulation, making use of the uniform ellipticity of a ij . /square Lemma 5.2.2. There exists µ 0 ∈ R such that for all µ > µ 0 , f ∈ L 2 ( U ) the equation with either Dirichlet or Neumann boundary conditions, has a unique weak solution satisfying Proof. Because of the estimates in the previous lemma, we may apply the Lax-Milgram theorem to B U [ u, v ] + µ ( u, v ) L 2 ( U ) thought of as a bilinear form on either H 1 0 ( U ) or H 1 ( U ) for Dirichlet, Neumann conditions respectively. /square Lemma 5.2.3. Suppose f ∈ C ∞ c ( U ) , and k is sufficiently large that supp f ⊂ V k . Then for µ > µ 0 has a unique solution in C ∞ ( V k ) . Furthermore, this solution obeys the estimate with C uniform in k . Proof. We can apply Lemmas 5.2.1, 5.2.2 on V k to deduce the existence of a unique weak solution to (5.3) with the appropriate boundary conditions, satisfying where C is independent of k . On V k , the operator L is uniformly elliptic in the standard sense, so classical elliptic estimates imply that u is smooth. We have that where ∣ ∣ ∣ ∣ ∣ ∣ ˜ f ∣ ∣ ∣ ∣ ∣ ∣ L 2 ( V k ) ≤ C || f || L 2 ( U ) . Focusing on a coordinate patch, we can work assuming V k = { ( x, x a ) : x > /epsilon1 k , x 2 + x a x a < ˜ /epsilon1 } , ρ = x and assume that ζ is a smooth cut-off function on { ( x, x a ) : x ≥ 0 , x 2 + x a x a < ˜ /epsilon1 } which vanishes on the curved part of the boundary. We note that ∂ α a = ∂ a and that ∂ α i ∂ a = ∂ a ∂ α i . Now consider the following integral, where the index A ∈ { 2 , 3 , . . . , N } is a fixed index, with no summation over it. ∣ ∣ ∣ ∣ By choosing C large enough, we may take δ to be arbitrarily small. Now, since ζ vanishes on the curved part of ∂V and either u or n i a ij ∂ α j u vanishes on the flat part, we may integrate by parts twice to find ∣ ∣ ∣ ∣ where in the last line we have used the uniform ellipticity of a ij . The constant C here depends on the functions a ij , ζ , which are uniformly bounded in k . Now taking (5.4), (5.5), (5.6) together, and choosing δ sufficiently small, we have that ∣ ∣ ∣ ∣ with C uniform in k , so we have estimated the tangential derivatives. Returning now to the equation, we can write Multiplying by ζ 2 , we readily estimate ∣ ∣ ∣ ∣ Combining these estimates with a partition of unity subordinate to a set of coordinate patches covering the boundary and an interior estimate which follows from standard elliptic theory, we're done. /square Proof of Theorem 5.1. First we note that if u is a weak solution of (5.1) with either Dirichlet or Neumann boundary conditions, then u is the unique weak solution of with ˜ f = f + µu for sufficiently large µ . Suppose ˜ f ∈ C ∞ c ( U ). Then we can solve the finite problems on V k , with the estimate ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ with C uniform in k . We deduce the existence of a subsequence which tends weakly to u in H 2 , as in the remark after Lemma 4.4.3, so we find that ∣ ∣ ∣ ∣ now, we may relax the condition that ˜ f ∈ C ∞ c ( U ) since such functions are dense in L 2 ( U ). Finally, replacing ˜ f , we deduce Finally, note that for Dirichlet conditions we have u k ∈ H 1 0 ( V k ) and for Neumann we have n i a ij ∂ α j u k ∈ H 1 0 ( V k ), so that in the limit u ∈ H 1 0 ( U ) or n i a ij ∂ α j u ∈ H 1 0 ( U ) respectively. /square We would like to also prove elliptic estimates at a higher level of regularity than H 2 . Unfortunately, the behaviour of the solutions near the boundary doesn't lend itself to a description in terms of a global Sobolev space. Accordingly then, we first consider interior regularity. Theorem 5.2. Suppose u is a weak solution of either the Dirichlet or Neumann problem (5.1) and suppose that in fact f ∈ L 2 ( U ) ∩ H m loc. ( U ) . Then u ∈ H 2 ( U ) ∩ H m +2 loc. ( U ) Proof. This follows from standard elliptic estimates and the fact that L is uniformly elliptic on any V ⊂⊂ U . /square To say more about the behaviour near the boundary, we shall once again need to distinguish directions tangent and normal to the boundary. It's convenient to introduce the space H m T ( U ), consisting of all functions u such that for any l ≤ m smooth vector fields T ( i ) tangent to the boundary ∂U . To capture the behaviour normal to the boundary, we work with an asymptotic expansion. Theorem 5.3. Suppose u is a weak solution of either the Dirichlet or Neumann problem (5.1) where f ∈ H m T ( U ) ∩ H m loc. ( U ) , where m ≥ 0 . Suppose further that if m ≥ 1 near ∂U we have the following expansion for f : and || ρ -a O ( ρ a ) || L 2 ( ∂U ) is bounded as ρ → 0 . Then u ∈ H 2 ( U ) ∩ H m +2 T ( U ) ∩ H m +2 loc. ( U ) has the following expansion for m ≥ 0 : Furthermore if u satisfies the Dirichlet conditions, u -0 = u -1 = 0 , while if u satisfies the Neumann conditions u + 1 = 0 . Proof. The proof is by induction. To establish the m = 0 case, we apply Lemma 4.2.1 and Theorem 5.1 to deduce that in a coordinate patch near the boundary with c ± ∈ L 2 ( ∂U ). We thus have that ∂ α x u = x -α [˜ c -+ O ( x α )] + x α -1 [˜ c + + O ( x 1 -α ) ] , and integrating this gives (5.11) for m = 0, with u + 1 , u -0 , u -1 ∈ L 2 ( ∂U ). Finally we note that the second identity of (5.12) implies u -0 ∈ H 1 ( ∂U ). In order to get the induction step, we first commute with a vector field tangent to the boundary, which establishes all but the highest order in ρ of (5.11) by the induction assumption. To get the highest order terms, we re-arrange the equation L u = f to give an equation for ∂ 1 -α x ∂ α x u , making use of the induction assumptions and integrate twice. Taking care of the boundary conditions imposed shows that for Dirichlet conditions, we have u -0 = u -1 = 0, while for Neumann u + 1 = 0. /square Taking a little more care about the origin of terms in the series, we can easily show Corollary 5.4. (i) If u , f satisfy the conditions for Theorem 5.3 with Dirichlet boundary conditions and furthermore f -i = 0 for 0 ≤ i ≤ m -1 , then u -i = 0 for 0 ≤ i ≤ m +1 . 5.3. Higher regularity. We define the higher regularity data spaces inductively as follows. We say ( u 0 , u 1 , f ) ∈ H 2 data, D if u 0 ∈ H 2 ( U ) , u 1 ∈ H 1 0 ( U ) , f ∈ H 1 ([0 , T ]; L 2 ( U )) with the product norm. In the Neumann case, ( u 0 , u 1 , f ) ∈ H 2 data, N if u 0 ∈ H 2 ( U ) , a ij ∂ α j u 0 ∈ H 1 0 ( U ) , u 1 ∈ H 1 ( U ) , f ∈ H 1 ([0 , T ]; L 2 ( U )). Next, we define Here L ( i ) is the second order operator given by differentiating the coefficients of L i times with respect to t . For m > 2, we say ( u 0 , u 1 , f ) ∈ H m data, † if ( u 0 , u 1 , f ) ∈ H m -1 data, † , f ( i ) ∈ L 2 ([0 , T ]; H m -i -1 loc. ( U )) for 0 ≤ i ≤ m -1 and ( g m -1 , g m , f ( m -1) ) ∈ H 1 data, † . Here as usual † stands for D or N as appropriate. We define the norms ∣ ∣ ∣ ∣ these spaces are chosen so that the relevant 'compatibility conditions' hold. We may show that if ( u 0 , u 1 , f ) ∈ H k data, † , then u 0 ∈ H k loc. ( U ), u 1 ∈ H k -1 loc. ( U ). Theorem 5.5. (i) Suppose u is a weak solution of the Dirichlet IBVP corresponding to data ( u 0 , u 1 , f ) ∈ H 1 data, D ( U ) . Suppose in addition, ( u 0 , u 1 , f ) ∈ H 2 data, D ( U ) then with the estimate (ii) Suppose u is a weak solution of the Neumann IBVP corresponding to data ( u 0 , u 1 , f ) ∈ H 1 data, N ( U ) . Suppose in addition, ( u 0 , u 1 , f ) ∈ H 2 data, N ( U ) then with the estimate Proof. First note that without loss of generality, we may take u 0 = 0, so that data which give rise to a smooth solution to the restricted problem on V k for k sufficiently large are again dense. We return to the approximating sequence u k we established in proving the existence of a weak solution. Commuting the equation with ∂ t and making use of the elliptic estimates on V k established in the previous section it is straightforward to derive bounds for ∣ ∣ ∣ ∣ ˙ u k ∣ ∣ ∣ ∣ L ∞ ([0 ,T ]; H 1 ( V k )) , ∣ ∣ ∣ ∣ u k ∣ ∣ ∣ ∣ L ∞ ([0 ,T ]; L 2 ( V k )) and the relevant norm of ... u which are uniform in k . Passing to a weak limit and applying Theorem 5.1 to deduce u ∈ L ∞ ([0 , T ]; H 2 ( U )), we're done. /square Commuting further with ∂ t , it can be shown that the following theorem holds: Theorem 5.6 (Higher Regularity) . (i) Assume ( u 0 , u 1 , f ) ∈ H m data,D and suppose also u is the weak solution of the Dirichlet IBVP problem with this data. Then in fact ∣ ∣ ∣ ∣ where C is a constant which depends on T and α and the coefficients of the equation. Furthermore (ii) Assume ( u 0 , u 1 , f ) ∈ H m data,N and suppose also u is the weak solution of the Neumann IBVP problem with this data. Then in fact ∣ ∣ ∣ ∣ where C is a constant which depends on T and α and the coefficients of the equation. Furthermore Note that we do not directly control higher spatial derivatives of u in global Sobolev norms, although we do have control of powers of the L acting on u . We can however make use of (a very slight adaptation of) Theorem 5.3 to give the following asymptotic expansion in the situation where f has the appropriate behaviour near the boundary: Theorem 5.7. Suppose u satisfies the conditions of Theorem 5.6 ( i ) or ( ii ) . Suppose that f ( i ) ∈ L 2 ([0 , T ]; H m -i -1 T ( U ) ∩ H m -i -1 loc. ( U )) . Suppose further that if m ≥ 3 near ∂U we have the following expansion for f : f = ρ α -1 [ f + 0 + ρf + 1 + . . . + ρ m -3 f + m -3 + O ( ρ m -2 -α ) ) ] (5.20) + ρ -α [ f -0 + ρf -1 + . . . + ρ m -3 f -m -3 + O ( ρ m -3+ α ) ) ] where Then u has the following expansion for m ≥ 1 : Furthermore if u satisfies the Dirichlet conditions, u -0 = u -1 = 0 , while if u satisfies the Neumann conditions u + 1 = 0 . For a formal power series approach to determining the coefficients of these expansions, see [15].", "pages": [ 17, 18, 19, 20, 21, 22, 23 ] }, { "title": "6. Other boundary conditions", "content": "We have now established well posedness and a regularity theory for solutions of (4.1) subject to either Dirichlet or Neumann homogeneous boundary conditions. We will discuss briefly some of the other possibilities listed in the introduction, although we shall not go into quite so much detail. Definition 6 (Weak Inhomogeneous Dirichlet IBVP) . Suppose u 0 , u 1 , f are as above with the additional condition ρ α u 0 | ∂U T = g 0 (0) . We say that u ∈ L ∞ ([0 , T ]; H 1 ( U )) with ˙ u ∈ L ∞ ([0 , T ]; L 2 ( U )) , u ∈ L 2 ([0 , T ]; ( H 1 0 ( U )) ∗ ) is a weak solution of the inhomogeneous Dirichlet IBVP: provided iii) We have the boundary condition Note that in contrast to the homogeneous Dirichlet problem, the unrenormalized energy will be infinite for a solution of the inhomogeneous Dirichlet problem with g 0 = 0. /negationslash Definition 7 (Weak Inhomogeneous Neumann IBVP) . Suppose u 0 , u 1 , f are as above. We say that u ∈ L ∞ ([0 , T ]; H 1 ( U )) with ˙ u ∈ L ∞ ([0 , T ]; L 2 ( U )) , u ∈ L 2 ([0 , T ]; ( H 1 ( U )) ∗ ) is a weak solution of the inhomogeneous Neumann IBVP: provided If we assume sufficient regularity, it is possible to show that the weak solutions are equivalent to strong solutions. by a standard integration by parts. It is clear that if v ∈ L 2 ([0 , T ]; H 2 ( U )) ∩ H 1 ([0 , T ]; H 1 ( U )) ∩ H 2 ([0 , T ]; L 2 ( U )) = H 2 ( U T ) satisfies the condition for inhomogeneous Dirichlet, or for inhomogeneous Neumann . ∣ then we can apply our previous weak well posedness results to the functions We'd like to know what conditions are required on g 0 , g 1 in order that such a v exists. The following Lemma gives the results we require, and comes from adapting Lemma 4.2.2 to H 2 ( U ). Lemma 6.1.1. (i) Suppose v ∈ H 2 ( U T ) , then ρ α v | ∂U T and ρ 1 -α a ij ∂ α j v ∣ ∣ ∣ ∂U T exist in a trace sense, and we have ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (ii) Suppose v 0 ∈ H 1+ α ( ∂U T ) and v 1 ∈ H 1 -α ( ∂U T ) . Then there exists ˜ v ∈ H 2 ( U T ) such that where the restriction is understood in the trace sense. Furthermore we may choose ˜ v such that with C independent of v i . Taking this with our previous results, we conclude Theorem 6.1. (i) Given u 0 , u 1 , f as above, g 0 ∈ H 1+ α ( ∂U T ) with ρ α u 0 | ∂U T = g 0 | t =0 . , there exists a unique weak solution to the Dirichlet IBVP corresponding to this data, with the estimate We shall not go through the proof in detail, but it is clear that the higher regularity results of § 5 can be extended to the inhomogeneous case. where β is some suitable function on ∂U T . We assume β ∈ C ∞ ( ∂U T ) to be concrete, but this is not necessary. This can be achieved in the weak formulation in a similar fashion to the introduction of an inhomogeneity for the Neumann condition. Definition 8 (Weak Inhomogeneous Robin IBVP) . Suppose u 0 , u 1 , f are as in § 6.1. We say that u ∈ L ∞ ([0 , T ]; H 1 ( U )) with ˙ u ∈ L ∞ ([0 , T ]; L 2 ( U )) , u ∈ L 2 ([0 , T ]; ( H 1 ( U )) ∗ ) is a weak solution of the Robin IBVP: provided It is straightforward to show that the well posedness and regularity results of § 4 and § 5 can be extended, where we require the estimate (4.12) to deal with the surface terms which arise. Remark: In the case where b i = 0, c ≥ 0 and a ij , c independent of time, we can relate our result to the theory of essentially self-adjoint operators. A consequence of Stone's Theorem (see for example [16]) for self-adjoint operators states: Theorem 6.2. Let L : D → H be a densely defined positive symmetric operator. Suppose for every f, g ∈ D there exists a twice continuously differentiable solution (in D ) to Then L is essentially self-adjoint. In our case H = L 2 ( U ), L = L , however this operator is not essentially self-adjoint on C ∞ 0 ( U ). Thus, the choice of dense subspace D determines a self-adjoint extension of L . If we take this gives the self-adjoint extension corresponding to Dirichlet boundary conditions, whereas if we take we have the self-adjoint extension corresponding to Neumann boundary conditions. Finally, taking gives the self-adjoint extension corresponding to the Robin boundary conditions. It is straightforward to check that L is positive and symmetric in all three cases, provided for the Robin case we take β ≥ 0. As a consequence, we can apply the functional analytic machinery of essentially selfadjoint operators to L in these situations. In fact, we can do better than this, based on the close analogy with the finite case. It can be shown that L with homogeneous Dirichlet, Neumann or Robin boundary conditions has a countable set of eigenvalues with corresponding eigenfunctions, smooth in the interior of U , which form an orthonormal basis for L 2 ( U ). This result comes from first establishing that H 1 ( U ) is compactly embedded in L 2 ( U ), and using this fact to apply the Fredholm alternative to a suitably chosen compact operator.", "pages": [ 23, 24, 25, 26 ] } ]

[ { "title": "Valentina Baccetti, Prado Mart´ın-Moruno, and Matt Visser", "content": "School of Mathematics, Statistics, and Operations Research Victoria University of Wellington PO Box 600, Wellington 6140, New Zealand E-mail: [email protected], [email protected] and [email protected] Abstract. We discuss the subtle relationship between massive gravity and bimetric gravity, focusing particularly on the manner in which massive gravity may be viewed as a suitable limit of bimetric gravity. The limiting procedure is more delicate than currently appreciated. Specifically, this limiting procedure should not unnecessarily constrain the background metric, which must be externally specified by the theory of massive gravity itself. The fact that in bimetric theories one always has two sets of metric equations of motion continues to have an effect even in the massive gravity limit, leading to additional constraints besides the one set of equations of motion naively expected. Thus, since solutions of bimetric gravity in the limit of vanishing kinetic term are also solutions of massive gravity, but the contrary statement is not necessarily true, there is not complete continuity in the parameter space of the theory. In particular, we study the massive cosmological solutions which are continuous in the parameter space, showing that many interesting cosmologies belong to this class. PACS numbers: 04.50.Kd, 98.80.Jk, 95.36.+x Keywords : graviton mass, massive gravity, bimetric gravity, background geometry, foreground geometry, arXiv:1205.2158 [gr-qc].", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Massive gravity has recently undergone a significant surge of renewed interest. Since de Rham and Gabadadze [1], (see also de Rham, Gabadadze, and Tolley [2]), demonstrated that it is possible to develop an extension of the Fierz-Pauli mass term for linearized gravity [3], one that avoids the appearance of the Boulware-Deser ghost [4], at least up to fourth order in non-linearities, activity on this topic has become intense - with over 50 articles appearing in the last two years. Subsequently, both Hassan and Rosen [5], and Koyama, Niz, and Tasinato [6], have independently re-expressed and re-derived the theory of de Rham and Gabadadze - simplifying the treatment and shedding additional light on its characteristics. In fact, Hassan and Rosen took a significant step further by extending the theory to a general background metric [5]. Rapidly thereafter, several papers appeared studying the foundations of this theory [7, 8, 9, 10, 11, 12], and proving the absence of ghosts in the nonlinear theory [13, 14, 15, 16]. In particular, Hassan and Rosen showed that the introduction of a kinetic term for the background metric in the ghost-free massive gravity leads to a bimetric gravity theory which is also ghost-free [17]. It should be noted that in this case the background metric is not only an externally-specified kinematic quantity, but also has its own dynamics, acquiring the same physical status as the 'foreground' metric [18]. More recently, ghost-free multi-metric theories have also been considered [19, 20]. Any cautious and physically compelling approach to massive gravity should not only respect the beauty of standard general relativity, but also enhance it in some manner, by embedding standard general relativity in some wider parameter space. Physically, one might hopefully expect that the observational predictions of the extended theory should be continuous in these extra parameters, and that general relativity would be recovered by taking the limit for a zero graviton mass in massive gravity theories. However, this is not necessarily the case, since (as is well known) the predictions of massive gravity often qualitatively differ from those of general relativity even when the graviton mass vanishes. This effect is known as the van Dam-Veltman-Zakharov (vDVZ) discontinuity [21, 22]. The Vainshtein mechanism provides us (under some appropriate conditions) with a way to avoid (or rather ameliorate) such a discontinuity [23] (see also [24, 26, 25, 27] and [6]). Furthermore, as has been pointed out in reference [5], the vDVZ discontinuity can sometimes be arranged to be absent when one considers the background metric to be non-flat in massive gravity, which is certainly the more general situation. That is, it seems that the predictions of massive gravity might be arranged to be continuous in parameter space when suitably taking into account a curved background [28, 29] - this is one of several reasons for being interested in non-trivial backgrounds. But why, in the first place, should one modify general relativity in such a manner and even entertain the possibility of massive gravity? References [28] and [29] provide historical reviews of the motivations. On the other hand, over the last decade the theoretical revolution based on the inferred accelerated expansion of the universe has become a good reason for considering a wide variety of possible modifications of general relativity [30, 31, 33, 32, 34]. Consequently, when de Rham, Gabadadze, and Tolley found a family of ghost-free (flat-background) massive gravity theories [1, 2], their cosmological consequences were quickly analyzed [35, 36, 37, 38], even though the theory had not yet been formulated to be compatible with a non-flat background metric. Although these studies can be considered as a first attempt to understand the cosmological consequences of ghost-free massive gravity, their results are severely limited by the (with hindsight unnecessary, and perhaps even physically inappropriate) choice of a flat background metric. Indeed, assuming a flat background is not the most general situation, and it is even contra-indicated when considering black hole geometries [43] or cosmological scenarios [39]. Later on, independently and almost simultaneously, three groups have considered cosmological solutions in bimetric gravity [40, 41, 42], while also studying massive-gravity cosmologies more or less in passing. However, as we will show in this paper, considering solutions of massive gravity as a limit of bigravity theories can be much more subtle than expected. At this point one might still reasonably wonder whether bimetric gravity is itself well-behaved in parameter space. Although a background kinematic metric has provided us with a continuous limit of massive gravity with respect to general relativity, it could still be that the introduction of a kinetic term for the background metric, (with its associated dynamics), disrupts coherence with respect to massive gravity. In other words, it may well be that the physical predictions of bimetric gravity differ from those of massive gravity in the limit where this theory should be recovered. To settle this issue, in this paper we study the massive gravity limit of bimetric gravity in full generality. As we will show, the solutions of bimetric gravity in the limit where the kinetic term for the background metric vanishes will also be solutions of massive gravity compatible with a non-flat background metric, but not necessarily vice versa . This paper is organized as follows: In section 2 we (briefly) summarize some previous results of massive gravity and bimetric gravity. In section 3 we explore various limits by which bimetric gravity might be used to reproduce massive gravity. In particular we investigate how the vanishing kinetic term limit should be considered, see section 3.1, concluding that the limit procedure is not implying the need for a flat background metric in massive gravity. Consideration of this limit allows us to obtain the cosmological solutions which are continuous in the parameter space of massive gravity, see section 3.2. We study these solutions which can be classified as general solutions, continuous cosmological solutions of any ghost-free theory, and special case solutions. Although the second group are solutions only for a particular kind of massive gravity models, we show that they present some features of particular interest. In section 3.3 we briefly comment the consequences of retaining some effects of the background matter when taking this limit. Section 4 contains some discussion and conclusions, while some purely technical formulae and computations are relegated to Appendix A, Appendix B, and Appendix C.", "pages": [ 2, 3 ] }, { "title": "2. Massive gravity and bimetric gravity", "content": "As is already rather well-known [4] (see also [39]), in order to consider massive gravity one is forced to introduce a new rank-two tensor f µν whose kinematics is at this stage externally specified, and not governed by the theory itself. This new rank-two tensor f µν can best be interpreted as a 'background' metric, not necessarily flat, with linearized fluctuations h µν of the physical 'foreground' metric g µν = f µν + h µν satisfying a massive Klein-Gordon equation [4] (see also [39] and [44]). That is, the (full non-linear) mass term appears in the action as some interaction term which depends algebraically on the tensors f µν and g µν , through the quantity and which in the linearized limit reproduces a suitable quadratic mass term. On the one hand, there is a non-denumerable infinity of such interaction Lagrangians, on the other hand, almost all of them lead to the physically unacceptable Boulware-Deser ghost [4]. For example, a particular specification of the interaction Lagrangian given in reference [39] led to the cosmological models presented in references [45, 46, 47, 48], which assumes a flat background metric contraindicated for this situation, and reference [50], which additionally contains specific technical criticisms to [45, 46, 47, 48]. Different approaches include the Lorentz-violating massive gravity [49, 51, 52] and the 3-dimensional 'new massive gravity' [53, 54]. The novelty of the last two years is that the interaction Lagrangian has now been very tightly constrained and fixed to depend only on a finite number of parameters [5] by requiring the theory to be ghost-free. A quite unexpected technical result (from the Boulware-Deser point of view [4], see also [39]) is that the dependence of this ghost-free interaction term on the background and foreground metrics is through the square-root quantity On the other hand, bimetric gravity was first introduced by Isham, Salam and Strathdee [18] (see also [55, 56, 57]) to account for some features of strong interactions, and it was later rejuvenated by Damour and Kogan in order to address new physics scenarios [58] (see also [59, 60, 61]). It has also been recently proven to be ghost-free when considering the same interaction term as for massive gravity [17]. Let us now focus our attention on 4-dimensional (and Lorentz invariant) massive gravity. We can express the action generally as with S (m) describing the usual matter action, with matter fields coupled only to the foreground metric g µν , (and the measure √ -g ), to agree with the Einstein equivalence principle. The parameter m sets the scale for the graviton mass, and the interaction term L int ( g -1 f ) is a scalar chosen to be dimensionless. Regarding bimetric gravity, in addition to the kinetic term of the background metric, one must also consider the possibility of a background cosmological constant ¯ Λ, and 'background matter' ¯ S (m) coupling to f µν . This now leads to The effective Newton constant for the background spacetime/matter interaction is /epsilon1G/κ . The two parameters κ and /epsilon1 can in principle be adjusted independently. The interaction term of the ghost-free theories can without loss of generality be written as [5, 17] (see also the discussion of Appendix B) with and the polynomials e i (see Appendix A for a more formal definition and properties) are where [ K ] = tr( K µ ν ), and our definition of K , equation (6), agrees with that of references [6] and [40], but differs from that of reference [5]. One can now consider the foreground tetrad e µ A , and the background inverse tetrad w µ A defined by [40] (see also references [62, 36, 8, 19]) where no direct analogy between these expressions and the definition of the Stuckelberg fields should necessarily be deduced (the relation between quantities defined in the tangent space and in the tetrad basis cannot be thought of as in any way recovering any gauge freedom). This formalism allows one to write the term involving the square root as by requiring the consistency condition which leaves the equations of motion unchanged [40]. The equations of motion for massive gravity are obtained by varying the action (3) with respect to g µν . These are with T (m) µ ν denoting the usual stress-energy tensor of the matter fields, while is the dimensionless graviton-mass-induced contribution to the stress-energy. It can be noted that τ µ ν can be written as for any interaction Lagrangian which depends on the metrics through terms of the form [ K n ], with n an integer, (or equivalently [ γ n ]). In addition, the Einstein tensor of equation (11) needs to satisfy the (contracted) Bianchi identity. Taking into account the invariance of S (m) under diffeomorphisms, which implies ∇ µ T (m) µ ν = 0, that identity leads to a constraint on the graviton-mass-induced effective stress-energy: On the other hand, the bimetric gravity theory now has two sets of equations of motion, obtained by varying with respect to the two metrics. Thus, in addition to equation (11) one has Furthermore, the Bianchi-inspired constraint which follows from equation (15) is equivalent to that already obtained in equation (14), see [40, 41]. It should be emphasized that the equations of motion (11), (15) the definition of the effective energy-momentum tensor, equations (12), (13), (16) and the constraint (14), are all completely independent of the particular form of the interaction term, this being an automatic result of the fact that the interaction term depends only on the quantity g -1 f through terms of the form [ γ n ]. See for instance reference [39]. ( Warning: There is a significant typo in reference [39], amounting to accidentally dropping the scalar factor √ f/g in the effective stress-energy due to the graviton mass term - see reference [50] fortunately this does not quantitatively affect weak-field physics, nor does it qualitatively affect strong-field physics - though it will certainly significantly change many of the details.) If we now consider the specific family of ghost-free theories given by (5), we can explicitly express τ µ ν in terms of matrices as", "pages": [ 4, 5, 6 ] }, { "title": "3. Continuity of massive gravity with respect to bimetric gravity", "content": "An implication of the discussion in the previous section is that if one considers some particular solutions of a bimetric gravity theory, then those solutions will be more constrained than those corresponding to a massive gravity theory with the same interaction term. That is because when one considers f µν to be non-dynamical (being externally specified by the definition of the theory), the kinetic term for this metric is no longer present and one should not consider the variation of the action with respect to this metric. Following this spirit, von Strauss et al . have considered in reference [41] cosmological solutions for the ghost-free bimetric and massive gravity theories, given by the actions (4) and (3), respectively, and the interaction term (5). They have noted consistently that in massive gravity one loses the equations of motion given by equation (15). Nevertheless, in references [40, 42] the authors also study cosmological solutions of massive gravity but considering it as a particular limit of bigravity theory. They have obtained different and non-equivalent results to that presented in [41] because when considering massive gravity as a limit of bimetric gravity, one has not only the equations of motion of the foreground metric but also additional constraints. Thus, one may reasonably wonder whether the consideration of this limit implies that there is some kind of physical discontinuity in the parameter space of massive gravity. On the other hand, if one wants to consider massive gravity as a limit of bimetric gravity, then this limit should be carefully taken to avoid inconsistent results. In several references, presenting both pre-ghost-free and ghost-free analyses, the authors have concluded or implied that the background metric should be flat (Riemann-flat), an interpretation that can hide some problems of the theory, since a flat metric is an incompatible background for the cosmological scenarios that they have studied. The recent models for ghost-free massive gravity considered in references [40, 42] amount in the current language (and now including the possibility of a background cosmological constant) to taking the limit κ → ∞ while holding m fixed and setting /epsilon1 = 0. The background equations of motion then degenerate to G µν -¯ Λ f µν = 0, so that the background spacetime is some Einstein spacetime - for example, Schwarzschild/ Kerr/ de Sitter/ anti-de Sitter, or even Milne spacetime, not necessarily Minkowski spacetime. (Implicitly ignoring any background cosmological constant, as in [40, 42], leads to a Ricci-flat space for finite T µν .) However, following the philosophy of massive gravity, the background metric should not be constrained by the limiting procedure, but externally specified by the theory itself. Thus, if one wants to recover massive gravity as a limit of bimetric gravity, then it seems inconsistent to consider a limit which (unnecessarily) fixes the background metric.", "pages": [ 7 ] }, { "title": "3.1. The non-dynamical background limit", "content": "If we consider the action of bimetric gravity given by equation (4), it is easy to see that one can recover the action of massive gravity, equation (3), simply by simultaneously taking the limits κ → 0 and /epsilon1 → 0. Furthermore, the consideration of the limit κ → 0 will imply some constraints on the kind of interaction between both gravitational sectors in bimetric gravity, leading to one that must be also compatible with massive gravity to recover this theory, while not fixing the background metric as in references [40, 42], at least in principle. For consistency it can be checked that the same constraints coming from the consideration of this limit can be obtained by taking directly the variation of the action of massive gravity (3) with respect to the background metric f µν . Therefore, if one were interested in recovering the predictions of massive gravity in a certain limit of bigravity theory, then it would be natural to consider such a limit, which corresponds to the limit of vanishing kinetic term (that is the limit of a vanishing effective Planck mass for one of the metrics). Considering κ → 0 and /epsilon1 → 0 in the background equation of motion (15), and using the definition of T µν embodied in equation (16), the following perhaps unexpected constraint can be obtained: Moreover, taking into account equation (12), one can note that equations (18) imply that the effective energy-momentum tensor appearing in the equations of motion for the dynamical metric, equation (11), can be written as Thus, the constraint (14), ultimately coming from the contracted Bianchi identity, implies Therefore, from equations (19) and (20) we can conclude that the modification in the equations of motion (11) due to a putative non zero-mass for the graviton, considered as a limit of a theory with two dynamical metrics, is equivalent from the point of view of the physical foreground metric to simply introducing a cosmological constant, one which can at least in principle be either positive or negative. That is On the other hand, although in the next section we will focus our attention on theories of massive gravity with a spherically symmetric background metric, it should be pointed out that we have obtained no restriction about the curvature or dynamics of the background metric. In particular, there is no requirement for this metric to be Riemann flat (or Ricci-flat, or even an Einstein spacetime). It could be argued that one can choose to consider massive gravity with an Einstein background metric, and one can in fact do it. Nevertheless, that cannot be a requirement coming from seeing this theory as a particular limit of bimetric gravity, because in that case the adoption of that particular point of view would change the philosophy of the theory itself. Finally, it should be noted that up to now, we have not assumed any particular form for the interaction term (only that it depends on the metrics through [ γ n ]), neither have we assumed any symmetry for the metrics. Therefore, these results are completely general when studying the solutions for any massive gravity theory of that class as a limit of bimetric gravity. Thus, one can already conclude that there are fewer solutions that are solutions simultaneously of massive gravity and of bimetric gravity in the limit κ → 0 and /epsilon1 → 0, than of massive gravity alone. This is because the former solutions must be also solutions of massive gravity and in addition fulfill some extra constraints, namely equations (18), (19) and (20). Moreover, those solutions, if any, would be equivalent to simply considering a foreground cosmological constant.", "pages": [ 7, 8, 9 ] }, { "title": "3.2. Ghost-free cosmologies", "content": "We now focus our attention on the ghost-free case, which corresponds to the specific interaction term given by equation (5), and study solutions of massive gravity which are continuous in the parameter space (that is solutions of both massive gravity and bimetric gravity in the non-dynamical background limit). As we have concluded that those solutions mimic the effect of a cosmological constant in the foreground space, it is of particular interest to consider cosmological solutions. Thus, we assume a spherically symmetric situation, where both metrics fulfill this symmetry. Both for massive gravity and for bigravity theories these metrics can be written in general as [40] and where all the metric coefficients are functions of t and r . As is well known [56, 57, 60] (see also reference [40] for the ghost-free theory), in bimetric gravity the requirement of T (eff) 0 r = 0 (or, equivalently, τ 0 r = 0), which is implied by the consideration of a spherically symmetric scenario, leads in general to two classes of solutions: (i) those in which both metrics can be written in a diagonal way using the same coordinate patch, and (ii) those with metrics which are not commonly diagonal. Nevertheless, when one considers solutions of bimetric gravity with a vanishing kinetic term, such solutions must fulfill constraints (18) instead of the equations of motion of the background metric (15), leading G µ ν unspecified and, therefore, being compatible with massive gravity. This requirement leads to the conclusion that there is no Lorentzian-signature solution for equations (18) if C = 0 (see Appendix C for details) without conflicting with [56, 57, 60, 40] . So we can simplify the background metric and take /negationslash Considering metrics (22) and (24), it can be seen that there are two general solutions, and additionally two special-case solutions, to equations (18) and (20) for any ghost-free massive gravity theory whose interaction term can be described by (5). 3.2.1. First general solution. The first general solution we discuss is particularly interesting, since it relates the two metrics in a very simple way - a positionindependent rescaling. The metrics satisfy where D is a constant with D = 1 such that Explicitly For this solution the graviton mass produces a term in the modified Einstein equations (11), which can be described by the effective stress tensor which mimics the behavior of a positive cosmological constant if c 3 (1 -D ) > 3, and a negative one otherwise. Therefore, these solutions can describe a universe which is accelerating as if this acceleration would be originated by a cosmological constant. Moreover, this foreground universe would have exactly the same symmetry as the background metric - that is having a homogeneous and isotropic foreground universe is possible only if the background metric is also FLRW with the same sign of spatial curvature. We might have argued on general grounds for the existence of a solution of this type for a generic interaction term and in the absence of any symmetry of the metrics once taking into account equation (18). When calculating τ µ ν , notice that it is proportional to ∂L int /∂γ ν µ . Thus, one might guess the existence of some general (not necessarily unique) solution with γ µ ν = Dδ µ ν , where the value of D is fixed by the theory. In fact, in view of the specific form of L int , one might reasonably infer a generic polynomial constraint on D . (That is because τ ∝ ∂L int /∂γ ∼ ∑ i ∂e i /∂γ ∼ ∑ k p k ( γ ) γ k ; so if γ = D I is a solution to τ = 0, then one must have [ ∂L int /∂γ ] γ = D I = P ( D ) I = 0 for some polynomial P ( D ).) It must be pointed out that Blas, Deffayet, and Garriga already claimed for the existence of solutions of the kind considered in equation (25) in bigravity for an arbitrary interaction term between the metrics, where D would be determined by the particular interaction term [60]. The novelty in our study resides in the fact that we have concluded that those solutions are solutions of massive gravity which are continuous in the parameter space. Moreover, for the particular interaction Lagrangian considered in this paper, solution (25) is not the only solution implying an Einstein manifold g , as it was the case for the particular theory considered in reference [60] (see also reference [55] for a particular model), because any solution of bimetric gravity with τ µ ν ∝ δ µ ν would lead to a foreground metric of that kind. /negationslash 3.2.2. Second general solution. The second general solution corresponds to the metric where D and ¯ D are two separate constants satisfying and In this case, the extra term in the equations of motion (11) behaves as a positive contribution to the cosmological constant if c 3 ( D -1) > -1, leading to a foreground universe with accelerating expansion. This solution is similar to the previous one, with the only difference that the t -r sector of both metrics is conformally related through one constant whereas the angular sectors, in which we imposed the symmetry, are related through another constant. This fact would not lead to great differences, at least in principle. 3.2.3. Two special case solutions. There are two additional special-case solutions that hold only for a specific relation between the parameters c 3 and c 4 . In particular, we need to impose that c 4 = -3 / 4 c 2 3 . Both solutions lead to an effect in the modified Einstein equations (11) equivalent to a negative contribution to the cosmological constant. For the first special case solution, the background metric can be written as whereas for the second one Although these solutions are compatible only with a particular sub-class of models, they now allow us to have two possibly different cosmological metrics in the background and foreground. Regarding the first kind of solutions, where the foreground and background metric are given by equations (22) and (32), respectively, let assume that there are solutions with a foreground metric of the FLRW-kind. That is with t being the so-called FLRW conformal time coordinate (it is not the more usually occurring FLRW proper time coordinate) and the scale factor a ( t ) fulfilling the equation of motion (11), which, for this symmetry and this coordinate system, can be expressed as Thus, although the effect of the graviton mass is equivalent to a negative contribution to the cosmological constant, the solution is accelerating if Λ eff > 0, with Λ eff = Λ -m 2 ( D -1) 2 . The foreground spacetime (34) is a solution of massive gravity continuous in the parameter space, if we consider a theory with a background metric which can be written as (see Appendix C) As the function B ( r, t ) is not constrained by our analysis, it can take any form, not necessarily related with a ( t ). Thus, all the massive gravity theories defined with a background metric of the form given by equation (36) have FLRW solutions which are continuous in the parameter space. One interesting example would be to consider B 2 ( r, t ) = D 2 a 2 ( t ) / (1 -¯ kr 2 ), in such a way that the background metric now has different spatial curvature as the physical metric. One could also consider more exotic cosmologies compatible with metric (36). We can also reverse the logic of the problem and take a massive gravity theory defined with an isotropic and homogeneous background metric. That is which includes the particular case of massive gravity theories with a de Sitter background metric. In this case, requiring the fulfillment of the constraints (18) (and (20)), it can be seen that the foreground metric can be written as Note that the background scale factor a ( t ) does not now fulfill any Friedmann-like equation, since the background metric is not constrained but externally specified by the theory itself. We can obtain various different kinds of physical cosmologies, being described by the foreground metric, by changing the function N ( r, t ). As already mentioned, one particular solution belonging to this class would be that assuming that both spaces are FLRW, obtained by fixing N ( r, t ) = a 2 ( t ) / [ D 2 (1 -¯ kr 2 )]. Nevertheless, it is probably more interesting to consider the solutions implying an anisotropic expanding universe for a massive gravity theory defined by using a FLRW background metric, which could even have the maximal symmetry being a de Sitter space. That would be the case of any solution obtained by choosing N ( r, t ) = b 2 ( t ) / [ D 2 (1 -¯ kr 2 )], with b ( t ) = a ( t ). /negationslash On the other hand, those solutions in which the background metric takes the form given by metric (33) have similar characteristics as those with (32). The only difference is that in this case one can arbitrarily choose the function appearing in the temporal component of the metric. That is, for a theory with a FLRW background metric (37), one has which again can be fixed to be homogeneous and isotropic but it also allows the consideration of more exotic situations. Finally, it must be pointed out that the solutions presented here are, on one hand, more general that those already studied in the literature (for instance they can describe anisotropic cosmologies in one sector) and, on the other hand, more restrictive than other solutions, as they are only solutions for a particular model (with the parameters appearing in the Lagrangian fulfilling a particular relation).", "pages": [ 9, 10, 11, 12, 13 ] }, { "title": "3.3. Limit procedure without vanishing background matter", "content": "Let us briefly consider a slightly different limit which retains some effects of the background matter, that is κ → 0 with /epsilon1 nonzero and fixed. In this case the action becomes with dependence on the background metric f µν both in the obvious place L int ( f, g ) and in the action /epsilon1 ¯ S (m) for the background matter fields, which now have only an extremely indirect influence on the foreground physical sector. In this case the equation of motion for the background metric f µν becomes a purely algebraic one in terms of the background matter. One still has some equation constraining the background metric, and there will be some constraint intertwining the background and foreground metrics, and this is basically unavoidable in any reasonable limit of bimetric gravity that is based on tuning the parameters in the bigravity action to specific values. Nevertheless, it should be noted that one would not be able to directly detect the existence of any matter content in the background. Therefore, although the theory resulting when considering the limit κ → 0 (with /epsilon1 nonzero and fixed) has a very different motivation that massive gravity, it seems that we could not distinguish between them by measuring their physical 'foreground' consequences. In other words, we cannot think in any physical prediction which would be affected by considering that f µν is given by the theory, or constrained by some matter invisible to us.", "pages": [ 13 ] }, { "title": "4. Summary and Discussion", "content": "In this work we have explored the cosmological solutions for a massive gravity theory when viewed as a limit of taking a vanishing kinetic term in a bimetric theory, paying particular attention in the way this limit is taken. We have used a vierbein formulation based on that developed by Volkov in reference [40] (see also references [62, 36, 8, 19]), a formalism that proved to be very powerful in the treatment of this type of calculation. A first step is to realize that the solutions of massive gravity, taken as the limit of a bigravity theory, are in general more constrained than the general solutions of massive gravity. That is because in massive gravity there is only one set of equations, whereas if one considers this theory as a particular limit of bimetric gravity, then additional constraints must be taken into account. Thus, one cannot recover complete continuity in the physical predictions of the theory, since the solutions continuous in the parameter space of the theory are not the complete class of solutions. We have argued that massive gravity can be recovered from bimetric gravity by suitably taking the limit for κ → 0 in the action describing the latter. We have shown that this limit implies that the modification to the equation of motion introduced by a non-zero mass of the graviton is equivalent to introducing an extra contribution to the cosmological constant which, at this stage, can be positive or negative. An important fact to stress is that no restriction on curvature or dynamics of the background metric has been imposed. In particular, we have focused on ghost-free cosmologies with both metrics, foreground and background, being spherically symmetric. In the first place, we have shown that there are no non commonly diagonal solutions continuous in the parameter space. In the second place, we have found two kinds of solutions for any ghost-free theory. The first kind of solutions implies that the two metrics are proportional to each other, with the graviton mass interaction Lagrangian producing an extra contribution to the cosmological constant that can be either positive or negative, depending on some relation between the parameters of the theory. The net result is that this solution describes an accelerating or decelerating universe which has the same symmetry as the background metric. The second kind of solutions implies a more complicated relation between the two metrics, but we still obtain a positive cosmological constant for certain values of c 3 . Moreover, in the third place, we have also seen that two more kind of solutions can be obtained by considering a very specific relation between the parameters c 3 and c 4 , and that these solutions are equivalent to inducing an extra negative contribution to the cosmological constant in the modified Einstein theory. These solutions allow us to consider different cosmological metrics, for the background and foreground, related by some unconstrained arbitrary functions. In the first kind of solutions, when considering a spherically symmetric background metric and a FLRW physical metric, with a scale factor fulfilling the Einstein equations, we can obtain a different spatial curvature for the two metrics, by particularly tuning the function appearing in the radial component of the metric. If instead we consider the background metric to be FLRW, the (background) scale factor is now not forced to fulfill any Friedmann equations since this metric does not have a dynamics. A particular interesting case is to consider a massive gravity theory with an isotropic and homogeneous background metric which can lead to anisotropic expansion of the foreground space. On the other hand, in the second kind of solutions, we can obtain different cosmologies by changing the function in the temporal component of the metric. It must be emphasized that the solutions obtained in this paper are solutions of massive gravity continuous in the parameter space of this theory; that is, they are solutions simultaneously of both massive gravity and bimetric gravity in the limit of a vanishing kinetic term. Thus, these solutions are completely different from those of massive cosmologies presented in references [35, 36, 37, 38, 41], which are only solutions of massive gravity, and from those bimetric cosmologies studied in references [40, 41, 42], which are not solutions when the limit of vanishing kinetic term is taken. On the other hand, these solutions must also not be confused with the solutions obtained in references [42, 40] where a different limit of bimetric gravity is taken to recover massive gravity, namely κ →∞ . On the other hand, in view of the mentioned results, one could consider that the consideration of massive gravity as a limit of bimetric gravity leads to a theory which is equivalent to general relativity. Nevertheless, if this conclusion remain unchanged by the consideration of perturbative effects in the metric (note that the effective cosmological constant appears when considering the particular solutions and not directly in the action), then that would not be a problem of our treatment but of the theory of massive gravity itself. In that case one should conclude that: either (i) massive gravity is continuous on the parameter space only when it is equivalent to general relativity (with a particular value for the cosmological constant), or (ii) /epsilon1 cannot be zero and the dynamics of our universe is affected by some background invisible matter, or (iii) massive gravity cannot be seen as a particular limit of bimetric gravity. We have briefly considered option (ii) in this paper, but there is a rich phenomenology of models that might be explored in this case. It could be interesting to consider whether the kinds of models that should be considered could be constrained by some property of the hidden matter sector. For example, could it be reasonable to require that the classical energy conditions be satisfied in this sector? Should there be some relationship between the standard model of particle physics in both sectors, or can they be completely independent? Or, could this hidden matter be the 'mirror matter' speculated to restore complete symmetry in the fundamental interactions [63, 64]? In any case, in many ways it seems that bimetric theories and massive gravity should be kept in conceptually separate compartments. While there is no doubt that the relevant actions are very closely related, as we have seen treating massive gravity as a limit of bimetric gravity is fraught with considerable difficulty. In fact, whereas in massive gravity L int simply gives mass to the graviton, in bimetric (and multimetric) gravity this term is actually describing an interaction between two (or more) metrics. Therefore, bimetric gravity could be considered as a model of a bi-universe, in which two different physical worlds are coexisting, and interacting only through the gravitational sector (at a kinematical level the 'analogue spacetime' programme can be used to provide examples of multi-metric, though not multi-gravity, universes [65, 66, 68, 67]). Moreover, such an interpretation would lead to a new multiversal framework when considering multimetric gravitational theories [19], which would be very different to those previously considered in the literature, since it is not resulting from quantum gravity [69], inflationary theory [70], string theory [71], or general relativity [72], but it is a consequence of abandoning Einstein's theory. This type of multiverse is also rather different from the more exotic multiverse concept developed in references [73, 74].", "pages": [ 13, 14, 15 ] }, { "title": "Acknowledgments", "content": "PMMacknowledges financial support from the Spanish Ministry of Education through a FECYT grant, via the postdoctoral mobility contract EX2010-0854. VB acknowledges support by a Victoria University PhD scholarship. MV acknowledges support via the Marsden Fund and via a James Cook Fellowship, both administered by the Royal Society of New Zealand.", "pages": [ 15, 16 ] }, { "title": "Appendix A. Some identities", "content": "In this appendix we collect some purely algebraic results. For a general n × n matrix X the symmetric polynomials e i ( X ) are defined by Thus, the symmetric polynomials can be recovered iteratively from Newton's identity taking into account e 0 ( x ) = 1. Since we have which is a purely algebraic result which allows us to rewrite the interaction term in bimetric gravity in various useful ways [17] (see Appendix B for more details). Furthermore, since we have the 'shifting theorem'", "pages": [ 16 ] }, { "title": "Appendix B. The interaction term in bimetric gravity", "content": "In this paper we have used an expression for the graviton mass term, equation (5), which emphasized the fact that ghost-free massive gravity only includes three parameters more than general relativity. The consideration of such a L int in massive gravity leads to the same equations of motion as those that can be obtained using a formulation of the interaction term in terms of γ as used in reference [5]. In this appendix we will show that this is also the case in bimetric gravity when one cannot throw away the term depending only on the background metric. That is, both formulations are equivalent also in bimetric gravity, at least when considering both a foreground and background cosmological constant. We note that the ghost-free foreground-background interaction terms can in all generality be written as Here we have used the explicit algebraic symmetry between e i ( X ) and e n -i ( X -1 ) to exhibit an explicit interchange symmetry between foreground and background geometries. (See reference [17].) Furthermore, in view of the fact that γ = I -K , the shifting theorem yields Consider the original 'minimal' Lagrangian for generating a graviton mass [13]: Using the last of the shifting theorem equivalences from Appendix A, which relates e 4 ( γ ) with the polynomials in K , we see When written in this way the e 4 ( γ ) term corresponds in bimetric gravity to a background cosmological constant, and in massive gravity to an irrelevant constant. The e 2 ( K ), e 3 ( K ), and e 4 ( K ) terms are manifestly quadratic, cubic, and quartic in K . So insofar as one is only interested in giving the graviton a mass the quantity does just as good a job (in fact arguably a better job) than the original minimal mass term. This argument generalizes, we can use the shifting theorem to rewrite the general interaction term as Explicitly separating off the first two terms and using again the last of the shifting theorem equivalences we see So we have This eliminates (or rather redistributes) the e 1 ( K ) term. For current purposes it is now useful to split off the top and bottom terms (corresponding to foreground and background cosmological constants) and deal with them separately. We have explicitly absorbed them into the bigravity kinetic terms. After factoring out an explicit m 2 , this finally leaves us with the three-term interaction Lagrangian we have used in the paper, equation (5). This is Thus, we have shown explicitly that the interaction term (B.8) describes the same bimetric gravity theory as that considered in (B.1) when one takes into account a cosmological constant for each metric. On the other hand, there is still a hidden 'symmetry' between foreground and background. We can define an equivalent K f for the background metric with respect to the foreground metric via and equivalently Note that Therefore, as could have been suspected from equation (B.1), one can equivalently consider that the interaction term is giving mass either to the graviton related with f µν or to that of g µν . In fact, the ghost-free bimetric gravity is giving us 7 degrees of freedom (14 considering also the conjugate momenta) to distribute between both gravitons (see reference [5]), without any particular preference as to which. Here resides the great conceptual difference between massive gravity and bimetric gravity, since whereas in the first theory L int is a term whose purpose is giving mass to the graviton (the only graviton present in this theory), in bimetric gravity the non-existence of a preferred metric leads one to interpret L int merely as an interaction term. In fact, one could even think of some kind of democratic principle for bimetric theories, by interpreting the degrees of freedom to be distributed in such a way that we have two massless gravitons, and an interaction between the two metrics mediated by one vectorial and one scalar degree of freedom.", "pages": [ 16, 17, 18 ] }, { "title": "Appendix C. Spherically symmetric solutions", "content": "Taking into account the spherically symmetric metrics (22) and (23) in equation (17), the non-vanishing components of τ µ ν can be written as [40] and finally In particular this implies which greatly simplifies some calculations. We must find solutions for all these components being set equal to zero. Let us start by requiring that τ 0 r = 0. This can be obtained in either of two ways: /negationslash", "pages": [ 19 ] }, { "title": "Appendix C.1. Non-diagonal background metric", "content": "If we wish to consider a non-diagonal background metric, then we must require U ( r, t ) = D · R ( r, t ) in order to have τ 0 r = 0, where D is a constant such that /negationslash and, of course, D = 1. Substituting this into τ r r -τ θ θ we find that τ r r -τ θ θ vanishes when implying The components of τ µ ν then reduce to Thus, one must require A = -C 2 S/ ( N B ) to have a non-trivial situation. The interaction term now reads Replacing the relation A = -C 2 S/ ( N B ) in equation (24) for the background metric, one has which is non-Lorentzian. In fact this metric can be written as which manifestly has unphysical null signature (0 , -sign[ B 2 -C 2 ] , -1 , -1). Therefore there is no physical solution of this kind. Appendix C.2. Diagonal background metric We now set C = 0. In order to have τ r r -τ 0 0 = 0 we have two possibilities, either U ( r, t ) = D · R ( r, t ) with D = 1 and such that /negationslash as before ('case I'), or BS = AN ('case II'). Case I: Solutions for particular models. If we consider the first case, U = DR , then in order to have also τ r r -τ θ θ = 0, there are three options: thus, the theory should have to have solutions. The Lagrangian is L int = ( D -1) 2 , which fulfills the Bianchiinspired constraint. (iii) B = DN : Now which must vanish. Therefore, we have again c 3 and c 4 given by equation (C.15), and the same L int as in the previous case. Case II: Solutions for all the ghost-free models. We now consider A = BS/N . It can be seen that there are two cases in which τ r r -τ θ θ = 0. These are: These quantities vanish for U = D · R , with D such that which implies L int = (3 + c 3 ( D -1))( D -1) 2 . Thus, we should require D = 1 to have a (non-trivial) massive gravity theory. /negationslash", "pages": [ 19, 20, 21 ] }, { "title": "(ii) Consider", "content": "In this case we have which again vanishes if U = D · R , but now we must have D such that The mass term now reads It should be noted that there is yet one more formal solution for the equations τ µ ν = 0, but one which at the end of the day implies B = 0. Thus, for that solution the background metric is again non-Lorentzian and unphysical, this time with null signature (+1 , 0 , -1 , -1). This completes our consideration of the various explicit constraints on the background geometry arising from the equation τ = 0 in the case of spherical symmetry.", "pages": [ 21 ] } ]

[ { "title": "One vertex spin-foams with the Dipole Cosmology boundary", "content": "Marcin Kisielowski 1 , 2 , Jerzy Lewandowski 1 and Jacek Puchta 1 , 3 E-mail: [email protected] , [email protected] , [email protected] Abstract. We find all the spin-foams contributing in the first order of the vertex expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole Cosmology model. Our algorithm is general and provides spin-foams of arbitrarily given, fixed: boundary and, respectively, a number of internal vertices. We use the recently introduced Operator Spin-Network Diagrams framework. PACS numbers: 04.60.Pp, 04.60.Gw", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Spin-foams are quantum histories of states of the gravitational field according to the Spin-Foam Models of quantum gravity. In the usual formulation, a spin-foam is a two-complex, whose faces are colored with representations of a given group (depending on a model, for example SU(2)) and edges are colored with invariants of the tensor products [1, 2, 3, 4], or equivalently with operators if one uses the Operator Spin-Foam framework [5]. The spin-foams encode the data necessary to calculate the transition amplitude between states of Loop Quantum Gravity [4, 6, 7, 8, 9, 10] or more generally, the Rovelli boundary transition amplitude [4]. There are a few candidates for the spin-foam model of Quantum Gravity [11, 12, 13, 14, 15, 16, 17]. Important for the compatibility with Loop Quantum Gravity is to admit sufficiently general class of the 2-complexes, such that all the (closed, either abstract or embedded in a 3-manifold) graphs are obtained as their boundaries [18]. The optimal class of such 2-cell complexes was proposed in [20]. They are naturally provided in terms of the diagrammatic formalism introduced therein and called operator spin-network diagrams (OSN diagrams). A similar diagramatic framework for triangulations was introduced before in [19]. An additional advantage of our formalism, is that the OSN diagrams do not require neither 3d nor 4d imagination, they are easy to use and to classify possible spin-foams. We utilize and even improve these technical advantages in the current work. The generalized (to a non-simplicial 2-cell complex) EPRL vertex has been recently applied to introduce Dipole Cosmology, a quantum cosmological model which opens a new theory that can be called Spin-Foam Cosmology [21]. This application of spin foams in cosmology gave us the motivation to do the current research. We apply here the operator spin-network diagrams framework to find all spin-foams which contribute to the boundary amplitude of a fixed spinnetwork state in given order of the vertex expansion. The technical task one encounters when solving this problem is finding all the diagrams whose boundary is a given graph. We solve this problem in section 3 with the use of squid sets we introduce in section 2. The solution we present in that section is not limited to the Dipole Cosmology model only. Actually, it applies to the general spin-foam case. In section 4 we apply this general scheme to the model of Dipole Cosmology [21]. We find all the OSN-diagrams whose boundary is the boundary graph of Dipole Cosmology and which have exactly one interaction graph. Those diagrams contribute to the boundary amplitude in the first order of vertex expansion.", "pages": [ 1 ] }, { "title": "1.1. Definition of OSN-diagrams", "content": "In this subsection we recall the definition of OSN diagrams we introduced in [20]. In the analogy to spin-foams, which are colored 2-complexes, OSN-diagrams are colored graph diagrams. We first recall the definition of graph diagrams and then we recall the definition of coloring which turns a graph diagram into an OSN-diagram. One may think of graph diagrams as a way of building 2-complexes from building blocks which are (suitable) neighborhoods of vertices of the corresponding foam [1, 18]. Such a neighborhood is a 2-complexes obtained as an image of a homotopy of a graph. When glued together, they form a 2-complex. The way one glues them together is encoded in certain relations. Strictly speaking a graph diagram ( G , R ) consists of a set G of oriented, connected, closed graphs { Γ 1 , ..., Γ N } and a family R of relations defined as follows (see figure 1): glyph[negationslash] In order to be related, two nodes have to satisfy the consistency condition: the number of the incoming/outgoing links at each of them has to coincide with the number of the outgoing/incoming links at the other one (with possible closed links counted twice). Note that two graphs can be treated as one disconnected graph. Thus to reduce that ambiguity we assume that all the graphs defining the diagram are connected. An operator spin-network diagram ( G = { Γ 1 , ..., Γ N } , R , ρ, P, A ) is defined by using a compact group G and coloring a graph diagram ( G , R ) as follows (see figure 1): It is assumed that whenever two links glyph[lscript] and glyph[lscript] ' are mapped to each other by the link relation R nn ' link at some nodes n and n ' , then n ↦→ P n ∈ H n ⊗H ∗ n , (3) where H n is a Hilbert space defined at each node in the following way: where i / j labels the links incoming/outgoing at n . Whenever two nodes n and n ' are related by R node , then (from (2) and (4)) it follows that H n = H ∗ n ' and it is assumed about P that where n runs through the nodes of Γ I . Each graph Γ I itself defines a contractor, in the sense that there is a natural contraction defined by the graph Γ I and by the natural trace operation in ⊗ glyph[lscript] H glyph[lscript] ⊗H ∗ glyph[lscript] which contains ⊗ n H n , where n / glyph[lscript] ranges the set of nodes/links of Γ I . We denote this natural contractor by A Tr Γ I . However, that natural contraction is often preceded by some additional operations, like the EPRL embedding which gives rise to the EPRL operator spin-network diagrams.", "pages": [ 2, 3 ] }, { "title": "2. Characterization and construction of the diagrams", "content": "We will introduce now a useful characterization of the diagrams. The characterization will allow to control the structure and the complexity of the diagrams in a clear way. Finally it will lead to algorithms for construction of diagrams. In this section we will introduce the first, simpler algorithm. It will be improved to produce diagrams of a given boundary, in the next section.", "pages": [ 3 ] }, { "title": "2.1. Squids", "content": "An important element of our characterization of graph diagrams is an oriented squid set . Given a graph (oriented and closed), its oriented squid set is obtained by removing from each link of the graph a point of its entry and shaking the whole thing so that the disconnected parts of each link go apart (figure 2). Two different graphs may define a single squid set. Therefore, it makes sense to introduce and consider the notion of a squid set on its own. An oriented squid consist of 1 point called head , k + legs beginning at the head, and k -legs ending at the head. In other words it may be considered as the topological space obtained by glueing k + + k -oriented intervals with the head in the suitable way (figure 3a). An oriented squid set S is a disjoint finite union of squids (figure 3b). A graph Γ consists of a squid set S Γ and information about glueing the legs of the squids. Conversely, given a squid set S , a graph may be obtained by glueing the end of each outgoing leg with the beginning of arbitrarily chosen incoming leg of either the same or a different squid. However this procedure neither is unique, nor there is a guarantee it can be completed. Therefore, a single squid set can be the squid set of either more then one graph, or of an exactly one graph, or of no graph at all (figure 4). Agraph diagram (Γ , R ) can also be obtained from a squid set S Γ by endowing it with additional structure. The part of the structure has been already described above, it allows to reconstruct the graph Γ . The second element of the structure, the node relation R node , is given by indicating a set of pairs {{ λ 1 , λ ' 1 } , ..., { λ N , λ ' N }} of squids λ i , λ ' i ∈ S , whose heads are in the node relation R node . The last element of the graph diagram structure, a link relation, is defined for each pair { λ i , λ ' i } as a bijection between the incoming/outgoing legs of the squid λ i with the outgoing/incoming legs of the squid λ ' i (figure 5).", "pages": [ 3, 4, 5 ] }, { "title": "2.2. The algorithm", "content": "This characterization of an arbitrary graph diagram leads to the following algorithm for construction of all the graph diagrams: A graph diagram D resulting from ( i ) -( iv ) consist of a graph the disjoint union of connected graphs Γ I , I = 1 , .., N obtained by the glueing ( iv ) , a node relation given by ( ii ) and a link relation provided by ( iii ) . We denote it shortly by D = ( { Γ 1 , ..., Γ N } , R ) . It is turned into an operator spin-network diagram by coloring the links, the nodes and the connected components of Γ according to Section 1.1.", "pages": [ 5, 6 ] }, { "title": "2.3. Discussion", "content": "2.3.1. Colorings The colorings may be constrained by the geometry of the graph diagram and the relations. In order to control this constraint one may from the beginning fix the coloring of the boundary links (that is the legs of the squids whose heads are boundary nodes - the nodes which are unrelated by the node relation) by representations, and then allow only such glueings that agree with the coloring (i.e. two legs may be glued if and only if the are colored by the same representation). 2.3.2. Reorientation of the diagrams Graphs used in the definition of OSN diagram are oriented, and the orientation is relevant for the node and link relations. However, the operators evaluated from the OSN diagram are invariant with respect to consistent changes of the orientation accompanied with the dualization of the representation colors. Suppose ( { Γ ' 1 , ..., Γ ' N } , R ' , ρ ' , P ' , A ' ) is an OSN-diagram obtained from a given OSN-diagram ( { Γ 1 , ..., Γ N } , R , ρ, P, A ) by: flipping the orientation of some of the links glyph[lscript] 1 , ...glyph[lscript] k , leaving the same node relations, and leaving the same link relations, setting leaving for each unflipped link glyph[lscript] , and P ' = P as well as A ' = A . Notice, that the transformation of the orientations and the labelling ρ ↦→ ρ ' preserves the Hilbert spaces H n , where n ranges the set of the nodes of the diagram graph. That property makes the choice P ' = P and A ' = A possible. The OSN-diagram ( { Γ ' 1 , ..., Γ ' N } , R ' , ρ ' , P ' , A ' ) can be considered as reoriented OSNdiagram ( { Γ 1 , ..., Γ N } , R , ρ, P, A ) . The reorientation of any OSN-diagram does not change the Hilbert spaces and the resulting operator. Notice however, that given an OSN-diagram ( { Γ 1 , ..., Γ N } , R , ρ, P, A ) , we are not free to reorient any link we want, say exactly one arbitrarily selected link, to obtain a reoriented OSNdiagram ( { Γ ' 1 , ..., Γ ' N } , R ' ) . The possible changes of the orientation have to be consistent with the node-link relations R . For each of the flipped links glyph[lscript] i , a link which is in the link relation with glyph[lscript] i at one of its nodes has to be flipped too, and so on. In [20] we identified the chains of links related to each other with the equivalence classes of suitably defined face relation. We characterised the face relation classes in detail. It follows, that for each of the flipped links glyph[lscript] i , all the links, elements of its face relation equivalence class have to be flipped as well. This is the necessary and sufficient consistency condition for reorienting the links of a graph diagram in a way consistent with the node-link relations. 2.3.3. Advantages and a drawback The characterization and construction presented above allows to control the complexity of the diagrams by the following measures: the number of internal edges, the number of disconnected components of the graphs, the complexity of each disconnected component. There is one drawback though. From the point of view of the physical application, the boundary part of the diagram (consisting of the squids whose heads are not in the node relation with any other head - they form the boundary graph) describes either the initial and final state or, more generally, the surface state. The remaining part of the diagram consists of the pairs of the related nodes (internal edges) and describes the interaction. The two pieces of this information are entangled in the presented characterization. The reason is, that the diagrams, obtained with the algorithm above from a given squid set endowed with the node and link relations (the data ( i ) -( iii ) ) by implementing all the possible glueings (step ( iv ) ), in general have different boundary graphs (the graphs share a squid set, but have different graph structures). Therefore we do not control in that way the boundary of the diagram, that is the initial-final/boundary Hilbert space. We improve that characterization and the algorithm in the next section.", "pages": [ 6, 7 ] }, { "title": "3. Operator spin-network diagrams with fixed boundary", "content": "In this section we improve the construction and the algorithm introduced in the previous section. The improved algorithm provides all the operator spin-network diagrams which have a same, arbitrarily fixed boundary graph .", "pages": [ 7 ] }, { "title": "3.1. The idea and the trick", "content": "In [20] (see Sec. 6.2), for every graph Γ we introduced the graph diagram corresponding to the static spin-foam , that is the spin-foam describing the trivial evolution of the graph. Let us denote this diagram by D Γ and refer to it as the static graph diagram of Γ . The boundary of D Γ is the disjoint union Γ ∪ ¯ Γ where ¯ Γ is a graph obtained by switching the orientation of each of the links of Γ (figure 10). Given a coloring of the links of the graph Γ by representations, we endow the static graph diagram D Γ with the natural colorings (see [20], Sec. 6.2): ( i ) the coloring ρ of the links of the graphs in D Γ is the one induced by the coloring of the links of Γ ; ( ii ) the operator coloring P colors with the identity operators; ( iii ) finally, all the contractor coloring A assigns to each n -theta graph of D Γ the natural trace contractor A Tr . The result is the static OSN-diagram . The corresponding operator is the identity in the Hilbert space given by the coloring of the links of Γ . The technical definition of static diagram will be recalled in the next subsection. The diagrams we will construct with the improved algorithm introduced below, contain the boundary Γ together with its static diagram D Γ as a subdiagram. Given Γ , our construction will provide all the diagrams of this type, that is all the diagrams, modulo the static sub-diagram. In terms of the spin-foam formalism, we will construct all the spin-foams bounded by an arbitrarily fixed graph Γ . In each of the spin-foams, the neighborhood of Γ is homeomorphic to the cylinder Γ × [0 , 1] . The key trick behind our construction is the following observation: given a graph Γ of the squid set S Γ , and an arbitrary graph diagram D int called interaction diagram , whose boundary graph ∂ D int has the same squid set : we can combine the diagram D int with the static diagram D Γ into the new graph diagram D Γ # D int such that the graph Γ becomes its boundary. We achieve that by defining the graph of D Γ # D int to be the disjoint union of the graph of D Γ with the graph of D int and extending the node and link relations of the component diagrams such that each squid of the boundary ∂ D int is related to the corresponding squid of ¯ Γ (a part of the boundary of D Γ ) - see figure 11. The identification of the squid sets S Γ and ∂ D is defined modulo symmetries of S Γ (exchanging identical squids). In the consequence, the graphs ¯ Γ and ∂ D int may admit more than one way of relating their squid. In the algorithm below, the identification (a bijection) between the squid sets will be given, and the freedom will be in the glueing of the legs. Below we implement this idea in detail, beginning with recalling the exact definition of the static diagrams.", "pages": [ 8, 9 ] }, { "title": "3.2. Static diagrams", "content": "First, we recall the definition of the static diagram of a graph Γ . We use the squid set S Γ . For each squid λ ∈ S Γ we introduce a theta-like graph ˜ θ λ as follows (figure 12): The links of the graph ˜ θ λ are just the legs glyph[lscript] λ,i , ¯ glyph[lscript] λ,i , i = 1 , 2 , ... of the squids λ and ¯ λ respectively. The result of this procedure is a family of the graphs ˜ θ λ , one per each squid λ ∈ S Γ . The disjoint union ∐ λ ˜ θ λ is the graph of the static graph diagram D Γ . The node and the link relations in ∐ λ ˜ θ λ are induced by the the structure of the graph Γ - see figure 13. To define the node relation, notice that the set of the nodes of the graph ∐ λ ˜ θ λ consists of the nodes of two types: The link relation is defined as follows: Given a static graph diagram D Γ the natural coloring consists of: irreducible representations freely assigned to the links, the operators P n λ = id : H n λ → H n λ for every head n λ and every squid λ , and P n glyph[lscript] λ,i n glyph[lscript] ' λ ' ,i ' equal to the natural isomorphisms id : H n glyph[lscript] λ,i → H n glyph[lscript] ' λ ' ,i ' . Finally, the contractor assigned to each component graph ˜ θ λ is the natural A Tr ˜ θ .", "pages": [ 9, 10 ] }, { "title": "3.3. The improved algorithm", "content": "We present now our algorithm for the construction of the operator spin-network diagrams whose unoriented boundary is an arbitrarily fixed unoriented graph | Γ | . Notice, that it would be insufficient to fix one orientation of the boundary. A priori all the orientations have to be taken into account. On the other hand, in general, the algorithm will possibly give OSN-diagrams related by the reorientation (see the previous section). In specific cases, that redundancy should be reduced. The presented construction allows to control the level of complexity of resulting diagram by the level of complexity of the diagram D int . The complexity can be measured by the number of pairs of the nodes related by the node relation, that is the number of internal edges. The simplest case is zero internal edges, that is the squid set In that case all the graph used to define graph diagram D becomes the boundary graph ∂ D . int int A general example of the interaction graph diagram D int is given by a graph ˜ Γ int and a node relation R node int consisting of exactly N pairs of nodes. Increasing the number N we increase the complexity of the diagram D Γ # D int . In the previous subsection we have recalled the structure of colorings that turn graph diagrams into operator spin-network diagrams. In the case of the graph diagrams constructed in this subsection, without lack of generality, it is sufficient to consider colorings of the diagrams D Γ # D int provided by our algorithm, which reduced to the static graph diagram D Γ provides the static operator spin-network diagram defined above. A way to control the freedom in the colorings by representations is to fix a coloring of the links of the boundary graph Γ . In the next section we apply the algorithm to construct all the operator spin-network diagrams of the Rovelli-Vidotto dipole cosmology.", "pages": [ 10, 12 ] }, { "title": "4. Diagrams with boundary given by dipole cosmology graph", "content": "In this section we apply the algorithm presented above to a specific example of a boundary graph, namely the one given by the Dipole Cosmology model of Bianchi, Rovelli and Vidotto [21]. The Dipole Cosmology model is an attempt to test the behaviour of the spin-foam transition amplitudes in the limit of the homogeneous and isotropic boundary states. The boundary state of this model is supported on so called dipole graph (figure 19) which consists of two disjoint components, 4 -valent theta graphs representing the initial and, respectively, final geometry. The transition amplitude is calculated under several approximations. One of the approximations is the vertex expansion - i.e. at the first order one considers contribution of the spin-foams with four internal edges and one interaction vertex only.", "pages": [ 12, 13 ] }, { "title": "4.1. The improved algorithm in the Dipole Cosmology case", "content": "We apply now the algorithm of the previous section to construct all the operator spin-network diagrams whose boundary is fixed (modulo an orientation) to be the graph | Γ | which consists of two disjoint 4 -valent theta graphs | Γ 4 θ | (figure 19), and which correspond to 1-vertex spin-foams. In terms of the improved algorithm of the previous section this assumption means that the the interaction diagram D int consists of one graph Γ int whereas the node and link relations are trivial. That is its squid set equals the squid set of the boundary (the initial plus the final) graph. Specifically, for this Dipole Cosmology example and with the 1-vertex assumption the improved algorithm from section 3.3 reads: consists of four 4 -valent squids; two of them are oriented freely - their orientation determines the orientation of the remaining two squids and defines the orientation of Γ (figure 20b).", "pages": [ 13, 14 ] }, { "title": "4.2. All the possible interaction graphs", "content": "In this subsection we construct all the possible interaction graphs Γ int . We obtain each interaction graph Γ int by assigning an orientation to each link of an (unoriented) graph | Γ int | defined by the following two properties: We find below all possible graphs | Γ int | . We depicted the resulting graphs on figure 27. In order to obtain an interaction graph Γ int , we assign to each link of a graph | Γ int | an orientation consistent with the orientation of the boundary (and the squid set). Given a graph | Γ int | and (oriented) boundary graph Γ , such a choice of compatible orientation may be impossible. For example, take graph 1 from figure 27 as a graph | Γ int | . It is not possible to choose an orientation of links of this graph compatible with orientation of the boundary graph from figure 20a. This is because the boundary graph figure 20a has a node with three outgoing and one incoming link and such a structure of a node is not possible for graph 1 from figure 27 (since each link of this graph forms a loop, the number of incoming links and outgoing links needs to be equal at every node). Note, that there is a distinguished graph | Γ int | - the graph 20 from figure 27 used in [21]. This graph may be oriented in a way compatible with any boundary graph Γ . The natural question which arises is whether for every graph from figure 27 there is a boundary graph such that orientation of | Γ int | may be chosen to be compatible with this boundary graph. The answer is affirmative. It may be shown that orientation of links of each graph | Γ int | may be chosen to be compatible with a boundary graph oriented such that at every node a number of incoming links equals to a number of outgoing links. We now present in details the construction of unoriented graphs possessing exactly 4 nodes, all of which are four-valent, i.e. all possible graphs | Γ int | . It is well known that each unoriented graph may be encoded in adjacency matrix. It is a symmetric matrix A ∈ Sym( n ) with the number of columns/rows n equal to the number of the vertices of this graph. The entries A ij are equal to the numbers of links connecting node i with node j , with a specification that links forming closed loops (corresponding to diagonal entries) are counted twice. An example of such matrix and the corresponding graph is given on figure 22. However, given a graph there are many corresponding matrices, because for each permutation σ ∈ S n the matrices and A ij define the same graph. There is a natural bijective correspondence between graphs with n vertices and orbits, elements of Sym( n ) /S n . In our case graphs have four nodes. We are therefore interested in 4 × 4 matrices. The condition that each node is 4-valent corresponds to an assumption that the sum of numbers in each row/column is equal 4: The set of the possible interaction graphs G int is therefore characterised by the moduli space: First we introduce a parametrisation of the space of symmetric matrices satisfying (13) and then we find the moduli space using Wolfram's Mathematica 8.0. To define our parametrisation in a transparent way, we introduce a triple (K 4 , d, m ) (see figure 23): such that for all four nodes n 1 , n 2 , n 3 , n 4 of the graph K 4 the numbers d n 1 , d n 2 , d n 3 , d n 4 satisfy the generalized triangle inequalities: glyph[negationslash] such that glyph[negationslash] The condition that d n , n ∈ K (0) 4 satisfy the generalized triangle inequalities (16) ensures the existence of at least one labeling m ‡ To each triple (K 4 , d, m ) corresponds a (multi)graph Γ (K 4 ,d,m ) , defined in the following way (see also an example at figure 24): Alternatively one may read from (K 4 , d, m ) the corresponding adjacency matrix. Simply choose glyph[negationslash] some ordering of nodes (in our case it is a labelling of nodes with numbers { 1 , 2 , 3 , 4 } ) and let glyph[lscript] ij = glyph[lscript] ji , i = j be the link in K 4 connecting nodes n i and n j . Now glyph[negationslash] This correspondence is depicted on figure 26. Note that, having known the numbers d n , the total number of links going from nodes n 1 , n 2 to nodes n 3 , n 4 (we denote it by k ) and the number m glyph[lscript] 23 , we can reconstruct the remaining coloring of (K 4 , d, m ) . As a result those numbers give the parametrisation of adjacency matrix, we are using. Explicitly, we parametrize the solutions to equations (13) with The corresponding parametrisation is: We next find orbits of action of permutation group S 4 on the set of those solutions. To this end we used Mathematica 8.0. The resulting graphs are depicted on figure 27. Note that one could further restrict the number of matrices considered by requiring that the sequence ( d 1 , d 2 , d 3 , d 4 ) is monotonous and considering only orbits under action of S 4 /H , where H is the subgroup, which does not change the sequence ( d 1 , d 2 , d 3 , d 4 ) . This remark enables one to do the calculation without using computer. On the other hand, one could write a program which does not use the parametrisation we introduced - e.g. one could generate matrices with entries taking values in the set { 0 , 1 , 2 , 3 , 4 } (with even numbers on diagonal) and choose only those which satisfy equation (13) (a direct method). We have chosen the method we present here, because it gives better understanding of the structure of the graphs considered, it is less laborious than calculation by hand and the version we used is easier to implement than the direct method. It has the additional advantage that it is easily applicable to a more general case where the four nodes are not necessarily four-valent. When d 1 , d 2 , d 3 , d 4 are becoming larger, this method becomes considerably faster than the direct method.", "pages": [ 14, 15, 16, 17 ] }, { "title": "4.3. Possible graph diagrams and an interesting observation", "content": "As we explained in the previous subsection, there are exactly 20 interaction graphs. However, the number of the graph diagrams resulting from the procedure described above is different. In this subsection we discuss in more details the diversity of the resulting graph diagrams. Given an oriented interaction graph D int and a static diagram D Γ , there may be more than one graph diagrams D Γ # D int . The ambiguity is in the choice of the node relation and the link relations. another, different node relation obtained by switching the nodes n 1 and n 2 - see figure 28. Furthermore, some colorings of the boundary links may be incompatible with some interaction graphs - it may happen that the amplitude is zero for every coloring of a given interaction graph. In order to see how this limits the number of possible interaction graphs, consider the coloring of boundary graph depicted on figure 30a and OSN diagram on figure 30b (node relations and the corresponding coloring with operators is omitted for clarity). It is straightforward to see that the amplitude is non-zero only if the representations ρ 2 and ρ 3 and, respectively, the representations ρ 1 and ρ 5 are equal ( ρ 2 = ρ 3 , ρ 1 = ρ 5 ). Importantly, note that, because there are links forming closed loops in the interaction graph, the amplitude is non-zero only if among representations ρ 1 , ρ 2 , ρ 3 , ρ 4 or among representations ρ 5 , ρ 6 , ρ 7 , ρ 8 there is a pair of equal representations. In addition, since the interaction graph is connected the amplitude is non-zero only if there is a pair of equal representations ρ i = ρ j , such that i ∈ { 1 , 2 , 3 , 4 } , j ∈ { 5 , 6 , 7 , 8 } . Those two conditions do not depend on the choice of node and link relations but on the structure of interaction graph only. This example shows that there are colorings of boundary graph which are not compatible with the given interaction graph. A similar analysis may be performed for other interaction graphs. It leads to an interesting conclusion. There is a distinguished interaction graph, which is not limited by the coloring in the way described above - it is the graph 20 from figure 27 used in [21]. The corresponding amplitude is non-zero even if all eight links of the boundary graph are labeled with pairwise different representations. In this generic case, all other interaction graphs give identically zero amplitude. We expect therefore that for a generic boundary state (which is a linear combination of spin-network states of all possible spins), the graph diagram with this interaction graph gives major contribution. This conclusion needs however further justification.", "pages": [ 19, 20, 21 ] }, { "title": "5. Summary, conclusions and outlook", "content": "We presented a general algorithm for finding all spin-foams with given boundary graph in given order of vertex expansion. We applied this algorithm to the spin-foam cosmology model [21] and found all contributions in first order of vertex expansion which are compatible with generalization of EPRL vertex [18]. We expect that for a generic state the vertex used in [21] gives the main contribution to the transition amplitude. This scenario needs however more thorough calculations and we leave it for further research. The calculation we presented illustrates an application of OSN diagrams. The strength of this formalism lies in simplifying the classification of 2-complexes - listing those with given properties (such as order of vertex expansion or structure of boundary graph). It also gives precise definition of the class of 2-complexes one should consider.", "pages": [ 21, 22 ] }, { "title": "Acknowledgments", "content": "Marcin Kisielowski and Jacek Puchta acknowledges financial support from the project 'International PhD Studies in Fundamental Problems of Quantum Gravity and Quantum Field Theory' of Foundation for Polish Science, cofinanced from the programme IE OP 2007-2013 within European Regional Development Fund. The work was also partially supported by the grants N N202 104838, and 182/N-QGG/2008/0 (PMN) of Polish Ministerstwo Nauki i Szkolnictwa Wyższego. All the authors benefited from the travel grant of the ESF network Quantum Geometry and Quantum Gravity.", "pages": [ 22 ] } ]

[ { "title": "Shan Bai 1 , 2 and Niall ' O Murchadha 1", "content": "E-mail: [email protected] , [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "The existence of the initial value constraints means that specifying initial data for the Einstein equations is non-trivial. The standard method of constructing initial data in the asymptotically flat case is to choose an asymptotically flat 3-metric and a transverse-tracefree (TT) tensor on it. One can find a conformal transformation that maps these data into solutions of the constraints. In particular, the TT tensor becomes the extrinsic curvature of the 3-slice. We wish to understand how the physical solution changes as the free data is changed. In this paper we investigate an especially simple change: we multiply the TT tensor by a large constant. One might assume that this corresponds to pumping up the extrinsic curvature in the physical initial data. Unexpectedly, we show that, while the conformal factor monotonically increases, the physical extrinsic curvature decreases. The increase in the conformal factor however means that the physical volume increases in such a way that the ADM mass become unboundedly large. In turn, the blow-up of the mass combined with the control we have on the extrinsic curvature allows us to show that trapped surfaces, i.e., surfaces that are simultaneously future and past trapped, appear in the physical initial data. PACS numbers: 04.20.Cv", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "When we consider General Relativity as a dynamical system, or try to construct solutions to the Einstein equations numerically, we draw on our understanding of and experience with other theories of physics. As in the case of particle mechanics or electromagnetism, we can pose initial data, and use the field equations to propagate the system into the future (or even into the past if we wish). We need to give a Riemannian 3-metric g ij , and a symmetric tensor K ij that is the extrinsic curvature of the 3-geometry. All regular solutions to the equations can be obtained in this way. Just as in electromagnetism, the initial data must satisfy constraints. This means that we must first specify 'free' data, and then manipulate these data to map them into a solution of the constraints. What happens to the physical solution as one changes the free data? In this article we change the free data in a particularly simple way and investigate the consequences. Since electromagnetism is a massless spin-1 field theory, and gravity is a massless spin-2 field theory, it is not surprising that we can see parallels between them. Consider the Maxwell free data as a pair of 3-vectors, ( /vector A, /vector F ). /vector A is the magnetic vector potential, and /vector F is the vector field from which we extract the electric field. Thus, the magnetic and electric fields can be generated by /vector B = /vector ∇× /vector A and /vector E = /vector F -/vector ∇ V , where V is a scalar function chosen to satisfy ∇ 2 V = ∇ i F i . Electromagnetism as a theory can be expressed in terms of ( /vector A, /vector E ) rather than the standard ( /vector B, /vector E ). We can see that /vector A is analogous to the metric, since both are gauge dependent quantities, while /vector E is the analogue of the extrinsic curvature, because /vector E is essentially the time derivative of /vector A , and the extrinsic curvature is essentially the time derivative of the metric. Thus /vector F is the analogue of the free data from which we extract the extrinsic curvature. It is clear that if we multiply /vector F by any constant, then /vector E will then be multiplied by the same amount. As a result, the electromagnetic energy density, B 2 + E 2 , increases without limit, and then the total energy will also blow up. In this article we want to multiply the free data, which specifies the extrinsic curvature, by a large constant in order to see what happens. We will again show that the total gravitational energy increases without limit. Initial data for the Einstein equations consists of two parts: (1) a 3-dimensional spacelike slice with a Riemannian metric, ¯ g ij , and (2) a symmetric 3-tensor ¯ K ij . The spacelike slice is to be regarded as a slice through a 4-dimensional pseudo-Riemannian manifold that satisfies the Einstein equations, and ¯ K ij is the extrinsic curvature of the 3-surface embedded in the 4-manifold. This means that ¯ K ij is the Lie derivative of the 3-metric along the timelike normal to the slice. These initial data cannot be freely specified; rather, they satisfy 4 constraints, which, in vacuum, read where (3) ¯ R is the scalar curvature of the 3-metric, ¯ K = ¯ g ab ¯ K ab is the trace of the extrinsic curvature, and ¯ ∇ i is the covariant derivative with respect to ¯ g ij . Eq.(1) is called the Hamiltonian constraint, and Eq.(2) is called the momentum constraint. A comprehensive account of the constraints can be found in [1], especially in Chapter VII; the classic article is [2]. If ¯ K = 0, the momentum constraint shows that ¯ K ij is both divergence-free and trace-free. A symmetric 2-tensor is called a TT tensor if it is simultaneously divergence-free and trace-free. It turns out that TT tensors are conformally covariant in the sense that if F TT ij is TT with respect to a metric g ij , then φ -2 F TT ij is TT with respect to ¯ g ij = φ 4 g ij . The standard way of finding sets (¯ g ij , ¯ K ij ) that solve the constraints is the socalled conformal method which was initiated by Andr'e Lichnerowicz [3]. The idea is to specify a base-metric g ij and a TT tensor. The base-metric is conformally transformed via ¯ g ij = φ 4 g ij , choosing the conformal factor φ so as to satisfy simultaneously the Hamiltonian constraint and the momentum constraint. This article deals specifically with constructing initial data that are asymptotically flat. This means that we choose the base metric to be asymptotically flat, and seek a conformal factor, φ , such that φ → 1 at infinity to maintain asymptotic flatness. As discussed above, we will restrict our attention to 'maximal' slices, initial data for which K = 0. This is a non-trivial assumption: there exist spacetimes which contain no maximal slice, see, for example, [4]. On the other hand, there exist many spacetimes that do possess maximal slices. The maximal assumption is useful because it makes it easy to specify the 'free' data. In Section II, we will see what happens when we multiply the 'free' TT tensor by a large constant. This maintains TT-ness of the tensor, and we can always find a suitable conformal factor to solve the constraints. We will show that, despite the fact that we continue to demand φ → 1 at infinity, the conformal factor monotonically blows up as we increase the TT tensor. Further, we can show that the conformal factor behaves like φ ≈ 1+ E/ 2 r + . . . at infinity, where E is the ADM energy of the solution. In Section III we will show that E increases monotonically so that the energy becomes unboundedly large. In Section IV we show that trapped surfaces must appear in the physical data when the scaling is large. This paper is a companion to [5], in which we did the same analysis on a compact manifold. In [5] we found that the behaviour of the conformal factor was complicated; we got blow-up but not uniform blow-up. That paper therefore used a combination of numerical and analytic techniques to describe how the conformal factor changed. We assumed we would have similar difficulties in this calculation and did some numerical modeling. However, on viewing the numerics it was clear that the the conformal factor behaved in a much more regular fashion: we clearly had uniform blow-up. This inspired us to go back and look again at the analytical work. We were able to produce exact mathematical proofs of everything we wanted, and so we were able to eliminate the numerics. This, therefore, is a paper that has passed through a numerical phase and emerged finally as a purely analytic document.", "pages": [ 2, 3 ] }, { "title": "2. Scaling the Extrinsic Curvature", "content": "The Hamiltonian constraint is We assume that we are given a suitably asymptotically flat metric, g ij , and a suitably asymptotically flat TT tensor, K ij TT , with respect to the given metric. We map this set of initial data to an asymptotically flat solution of the Hamiltonian constraint by using ¯ g ij = φ 4 g ij , and ¯ K ij TT = φ -10 K ij TT , and finding an appropriate φ . Since the extrinsic curvature stays TT under this transformation, the momentum constraint is automatically satisfied. This reduces to solving the Lichnerowicz equation where A 2 = K TT ij K ij TT . We impose the boundary condition φ → 1 at infinity. If we have a maximal solution, i.e., ¯ K = 0, then the Hamiltonian constraint simplifies slightly to (3) ¯ R -¯ K ij ¯ K ij = 0, and an immediate consequence is (3) ¯ R ≥ 0. If we have a solution to the Lichnerowicz equation Eq.(4) we will conformally transform the base metric into one with non-negative scalar curvature. It turns out that there is an obstruction to this. Riemannian 3-manifolds split into three classes, called the Yamabe classes [7]. Only metrics in the positive Yamabe class can be conformally transformed into metrics with non-negative scalar curvature. These are the metrics that can be conformally mapped into a closed, without boundary, compact manifold with constant positive scalar curvature. All we need is that the AF metric be conformally flat at least to order 1 /r 2 . Obviously, if we want to construct a maximal solution to the Hamiltonian constraint via a conformal transformation, the base metric must be in the positive Yamabe class. One can show that Eq.(4) will have a unique positive solution if and only if the metric belongs to the positive Yamabe class [8]. We want to change the initial data and see what happens. In particular, we rescale the background TT tensor. We change K ij TT to α 4 K ij TT where α is a constant. The new K ij is still TT and we continue to be able to solve the Lichnerowicz equation. In this article we are interested in the behaviour of the conformal factor as α →∞ . This means that we write the Lichnerowicz equation as We are, of course, particularly interested in the properties of the physical initial data, (¯ g ij , ¯ K ij TT ) = ( φ 4 g ij , φ -10 α 4 K ij TT ), as α becomes large. Since we are in the positive Yamabe class, we can always make a preliminary conformal transformation to map the metric to one which satisfies R ≡ 0. This conformal transformation depends only on the base metric, i.e., it is independent of α . This reduces the equation we wish to analyse to with, of course, φ → 1. We know that this equation will have a regular positive solution for any finite α . What happens to φ as α becomes large? This equation would make no sense if φ remained regular and bounded as α →∞ , because we would have a finite term equalling a term that becomes unboundedly large. Therefore we expect that φ blows up as α becomes large. More precisely, we expect that φ blows up linearly with α . This is why we choose α 4 as the rescaling factor. We now introduce a normalized φ , ˆ φ = φ/α . This allows us to rewrite Eq.(6) as Obviously, ˆ φ depends on α . Let us consider a slightly different equation, This equation has a regular positive solution which is unique. This ψ , when used as a conformal factor, results in a compact without boundary manifold satisfying R = ψ -12 A 2 . We can do this in two stages. First: the Yamabe theorem [7] tells us that we can conformally map the asymptotically flat metric to a compact manifold of constant positive scalar curvature, R 0 . Second: following this transformation, Eq.(8) becomes This has a regular solution on the compact manifold. The solution of Eq.(8) is just the product of the compactifying conformal factor by the solution of Eq.(9). Note: this requires that ˆ A 2 be well behaved on the compact manifold. This is obviously satisfied if A 2 falls off like r -12 or faster on the asymptotically flat manifold. The key reason for introducing ψ is that we will show First : we can show that ˆ φ monotonically decreases as α increases. To see this, differentiate Eq.(7) by α to give It is clear that d ˆ φ/dα is negative at infinity. Let us assume that it is positive somewhere in the interior. If it is, then d ˆ φ/dα will have a positive maximum. At such a point, both of the terms in Eq.(11) are negative, and this cannot be the case. Therefore we have d ˆ φ/dα < 0. This means that ˆ φ monotonically decreases as α increases. Second : we can show that ˆ φ -1 /α monotonically increases as α increases. The equation that ˆ φ ' = ˆ φ -1 /α satisfies is Again, we differentiate this with respect to α and get Let us assume that d ˆ φ ' /dα is negative somewhere in the interior. Then there must exist a point where it is a negative minimum. At such a point all the three terms in Eq.(13) are positive which is an obvious contradiction. Therefore we have d ˆ φ ' /dα > 0. This means that ˆ φ ' = ˆ φ -1 /α monotonically increases as α increases. Furthermore, we can also show that ψ satisfies ˆ φ > ψ > ˆ φ ' . To see this, let us first subtract Eq.(8) from Eq.(7) to get Let us assume that ( ˆ φ -ψ ) goes negative in the interior. This means that there will be a negative minimum. At this point we have ∇ 2 ( ˆ φ -ψ ) ≥ 0 and, since ˆ φ < ψ we have ˆ φ -7 > ψ -7 . Therefore both terms in Eq.(14) are positive which cannot be the case. So we must have ˆ φ -ψ > 0 for all α . To show that ψ > ˆ φ ' , we need to subtract Eq.(12) from Eq.(8) to get Let us assume that we have a region with ψ < ˆ φ ' . In such a region ψ -7 > ˆ φ '-7 > ( ˆ φ ' +1 /α ) -7 . In this region ψ -ˆ φ ' will have a local minimum. At that point both terms in Eq.(15) will be positive, again a contradiction. This implies ψ > ˆ φ ' . Of course, we have ˆ φ -ˆ φ ' = 1 /α . This gap goes uniformly to zero as α →∞ and we have ˆ φ > ψ > ˆ φ ' . Therefore ˆ φ and ˆ φ ' uniformly approach ψ , one from above and one from below, as α →∞ .", "pages": [ 4, 5, 6 ] }, { "title": "3. Controlling the ADM mass", "content": "We know that the leading term of ψ is of the form C/ 2 r and we can easily show This means that C is positive and finite. Now ˆ φ will behave asymptotically like D/ 2 r + 1 /α with D depending on α . We will show that D → C as α →∞ . We have that We know that ˆ φ is bigger than ψ but less than ψ +1 /α . Thus Consequently which implies This can be immediately rewritten as Using the monotone convergence theorem we see that 1 16 π ∫ A 2 ( ψ + 1 /α ) -7 dv → C as α → ∞ . Therefore we get D → C . From this we conclude that the coefficient of the 1 / 2 r part of the 'physical' conformal factor, φ = α ˆ φ , behaves like αD ≈ αC . In other words, it blows up linearly with α . The energy of the physical metric is the sum of the monopole part of the 'background' which is finite and bounded plus the monopole part of the conformal factor which is positive and becomes large. Thus the total energy is positive and increases monotonically with α . This, by the way, is strong-field positive energy proof. The conformal factor which maps to a physical asymptotically flat data set is φ , the solution of Eq.(6). We use this to generate a solution of the constraints (¯ g ij , ¯ K ij TT ) = ( φ 4 g ij , φ -10 α 4 K ij TT ). From this we get ¯ K ij TT ¯ K TT ij = φ -12 α 8 K ij TT K TT ij . Since φ scales with α , this goes pointwise to zero like α -4 . The physics here is a bit tricky. We can think of ¯ K ij TT ¯ K TT ij as the analogue of E 2 in electromagnetism, and so can be thought of as the kinetic energy part of the gravitational wave energy density. This shrinks like α -4 . However, the physical volume increases as α 6 . This is why we see the total energy increasing despite the decrease of the energy density. Nevertheless, it is clear that the physical extrinsic curvature shrinks as α increases. Therefore the physical solution looks more and more like a moment of time symmetry data set. Each physical solution will have an ADM energy-momentum. Since we assume the extrinsic curvature falls off faster than 1 /r 2 , the ADM momentum vanishes, and the ADM energy becomes the ADM mass. This will be contained in the 1 /r part of the physical metric. This will be made up of two parts. One part will be the 'mass' contribution of the background metric, g ij , which is a fixed number. The other part is the coefficient of the 1 / 2 r term in the conformal factor, φ . This is αD ≈ αC . Therefore, for large α , the ADM mass diverges linearly with α . This is a 'strong-field' positive energy proof since we show that the energy is positive for large α without any restriction on the background energy. Further, we can multiply Eq.(19) by α to see that φ satisfies Therefore the conformal factor blows up uniformly and linearly with α .", "pages": [ 6, 7 ] }, { "title": "4. Appearance of trapped surfaces", "content": "Trapped surfaces are defined by negative outgoing null expansions. Consider a closed 2-surface in an initial data set, with outgoing unit spatial normal, n i . The outgoing null expansions are defined by Given a moment-of-time-symmetry slice in isotropic coordinates through the Schwarzschild spacetime, we know that the sphere defined by r = m/ 2 is the apparent horizon and all surfaces with r < m/ 2 are trapped. In particular, we can show that the mean curvature, k , of the surface defined by r = m/ 4 equals -2 /r = -8 /m . We have a sequence of physical metrics, labeled by α . On these metrics we can introduce quasiisotropic coordinates near infinity, and the surfaces labeled by r = m/ 4 will be included in these domains for large enough m . We work out the null expansion of these surfaces. This will have a negative leading term, -8 /m . Now we show that the corrections all fall off faster than 1 /r , i.e., faster than 1 /m . Thus, for large m , the leading negative term dominates and the surfaces are trapped. The key term in our analysis is k = ∇ i n i , the mean curvature of the 2-surface in the 3-space. We can ignore the extrinsic curvature term in this calculation because it falls off rapidly both with radius and with α . Therefore to find a trapped surface we need only find a 2-surface with negative mean curvature, k . This already shows that we are getting trapped surfaces in the language of Penrose. The surfaces will be simultaneously future and past trapped. To do this we need to understand the behaviour of φ at large radii. From Eq.(22) we know that we can write φ as where f is dominated by the dipole moment, so it falls off like 1 /r 2 , and grows proportional to α . This means f = O ( α/r 2 ). We also need control of the gradient of φ , i.e., the gradient of f . To do this, let us return to Eq.(7) and differentiate it with respect to, say, x to get where S is a source term which comes from differentiating A 2 and the metric. This is a nice linear elliptic equation which guarantees that d ˆ φ/dx is well behaved. Since φ ≈ α ˆ φ , we can deduce that df/dx = O ( α/r 3 ). Let us assume that the base metric is conformally flat outside some region of compact support and let us compute the mean curvature of a spherical surface r = r 0 , assuming that the physical metric is conformally flat, ¯ g ij = φ 4 δ ij . The flat space unit normal, q i = ( x/r, y/r, z/r ) is proportional to the normal in the physical space. We define | q | 2 = ¯ g ij q i q j . The unit normal to the surface is then n i = q i / | q | . We can now compute The mean curvature will be negative if We know that where C 3 is a constant independent of α . If we choose r = m/ 4 = αD/ 4, we get The third term will be less than 1 for α large and so the mean curvature goes negative. It is a straightforward exercise to show that if the base metric is not conformally flat, but is well behaved near infinity, the correction to k falls of quickly. In the same way, as we have indicated above, the correction due to the extrinsic curvature is also negligible. This article is closely related to those cited in [9]. The space of free data for the maximal constraints consists of smooth, Riemannian 3-metrics that have positive Yamabe constant together with finite TT tensors. We want to find out what happens as we approach the boundary of this space. In the articles cited in [9] we kept everything regular as we looked at a sequence of metrics along which the Yamabe constant went to zero. In this article we again keep everything regular but let the TT tensor diverge. This is a very different part of the boundary of the space of free data. Nevertheless, we find very similar behaviour. The conformal factor blows up, the ADM mass diverges to + ∞ and horizons appear. Unfortunately, this is not enough to construct a valid positive energy proof because one needs to investigate the other parts of the boundary, for example we need to consider sequences of metrics which either become 'rough' or cease to be Riemannian.", "pages": [ 7, 8, 9 ] }, { "title": "5. Appendix: Showing D → C", "content": "It is 'obvious' that as α → ∞ , 1 16 π ∫ A 2 ( ψ + 1 /α ) -7 dv → C , but we have difficulty proving it. Here we will prove something weaker, but this result is really all we need. Pick a radius r = r 0 such that where B ( r 0 ) is the ball inside r = r 0 . We know that both ψ and r 0 are independent of α . Since ψ is positive so it will have a minimum value inside B ( r 0 ). We pick α 0 so that 1 /α 0 equals this minimum value. Therefore, for all α > α 0 , we have that, for r < r 0 , ψ > 1 /α . Therefore for r < r 0 we have ψ +1 /α < 2 ψ . In turn we get This means that we can write Eq.(21) as We can adjust this proof so as to replace the 2 8 by a number which is as close to unity as we wish. Pick a β << 1. We move r 0 out until we get We also increase α 0 such that for all α > α 0 we have that, for r < r 0 , βψ > 1 /α . Therefore for r < r 0 we have ψ +1 /α < (1 + β ) ψ . Then we get This equation now holds for all α > α 0 . In other words, we can make D as close to C as we wish by picking a large α .", "pages": [ 9, 10 ] }, { "title": "Acknowledgments", "content": "We thank the anonymous referee for pointing out the use of the monotone convergence theorem to us. SB and N ' OMwere supported by Grant 07/RFP/PHYF148 from Science Foundation Ireland. SB is also supported by a grant from the state of Burgundy.", "pages": [ 10 ] } ]

[ { "title": "A finiteness bound for the EPRL/FK spin foam model", "content": "Aleksandar Mikovi'c ∗ Departamento de Matem'atica, Universidade Lus'ofona de Humanidades e Tecnologia, Av. do Campo Grande, 376, 1749-024, Lisboa, Portugal", "pages": [ 1 ] }, { "title": "Marko Vojinovi´c †", "content": "Grupo de F´ısica Matem´atica da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal (Dated: October 18, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We show that the EPRL/FK spin foam model of quantum gravity has an absolutely convergent partition function if the vertex amplitude is divided by an appropriate power p of the product of dimensions of the vertex spins. This power is independent of the spin foam 2-complex and we find that p > 2 insures the convergence of the state sum. Determining the convergence of the state sum for the values 0 ≤ p ≤ 2 requires the knowledge of the large-spin asymptotics of the vertex amplitude in the cases when some of the vertex spins are large and other are small. PACS numbers: 04.60.Pp", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Spin foam models are quantum gravity theories where the quantum geometry of spacetime is described by a colored two-complex where the colors are the spins, i.e. the irreducible SU (2) group representations and the corresponding intertwiners. By assigning appropriate weights for the simplexes of the 2-complex and by summing over the spins and the intertwiners, one obtains a state sum that can be interpreted as the transition amplitude for the boundary quantum geometries, which are described by spin networks [1]. A spin foam state sum can be considered as a path integral for general relativity. The most advanced spin foam model constructed so far is the EPRL/FK model, introduced in [2, 3]. The finiteness, as well as the semiclassical properties of a spin foam model, depend on the large-spin asymptotics of the vertex amplitude. This asymptotics was studied in [4-6] for the EPRL/FK case. The study of the finiteness of the model was started in [7], where only two simple spin foam amplitudes were studied (equivalent to loop Feynman diagrams with 2 and 5 vertices) in the Euclidean case. It was concluded that the degree of divergence of these two spin foam transition amplitudes depends on a choice of the normalization of the vertex amplitude. This normalization is a power of the product of the dimensions of the spins and the intertwiners which label the faces and the edges of the 4-simplex dual to a spin-foam vertex. One can exploit this freedom in the definition of the EPRL/FK vertex amplitude in order to achieve the finiteness of the model. Namely, an EPRL/FK vertex amplitude can be introduced such that it is the original one divided by a positive power p of the product of dimensions of the vertex spins ∆ v . This new amplitude will give the state sum with better convergence properties, and one can try to find a range of p for which the state sum is convergent. In this paper we will show that there are such values of p which are independent from the spin-foam 2-complex. Note that an equivalent approach was used in the case of the Barrett-Crane spin foam model, where the finiteness was achieved by introducing an appropriate edge amplitude [8, 9]. This is an equivalent approach to our approach because a state sum with a dual edge amplitude A 3 ( j ) = (dim j 1 ... dim j 4 ) q and a vertex amplitude A 4 ( j ) is the same as the state sum with ˜ A 3 ( j ) = 1 and ˜ A 4 ( j ) = (∆ v ) p ( j ) A 4 ( j ), where p is an appropriate power. Our paper is organized such that in section II we describe the EPRL/FK spin foam model and discuss the large-spin asymptotic properties of the vertex amplitude. In section III we show that the vertex amplitude divided by the product of the dimensions of the vertex spins is a bounded function of the spins. In section IV we introduce a rescaled EPRL/FK vertex amplitude, which is the original amplitude divided by the product of the dimensions of the vertex spins raised to a power p . We prove that the corresponding state sum is absolutely convergent for p > 2 by using the amplitude estimate from section III. In section V we discuss our results and present conclusions.", "pages": [ 2, 3 ] }, { "title": "II. THE VERTEX AMPLITUDE", "content": "The EPRL/FK spin foam model state sum is given by where T is a triangulation of the spacetime manifold, T ∗ is the dual simplicial complex, while e , f and v denote the edges, the faces and the vertices of T ∗ , respectively. The sum in (1) is over all possible assignements of SU (2) spins j f to the faces of T ∗ (triangles of T ) and over the corresponding intertwiner assignemets ι e to the edges of T ∗ (tetrahedrons of T ). A 2 is the face amplitude, and it can be fixed to be by using the consistent glueing reguirements for the transition amplitudes between threedimensional boundaries, see [12]. The vertex amplitude W can be written as where the 15 j symbol is for the unitary representations ( k, ρ ) of the SL (2 , C ) group, the universal covering group of the Lorentz group. The f ι e k e ρ e are the fusion coefficients, defined in detail in [2, 3, 10]. Instead of using the spin-intertwiner basis, one can rewrite (1) in the coherent state basis, introduced in [11]. In this basis, the state sum is given by The /vectorn ef is a unit three-dimensional vector associated to the triangle dual to a face f of the tetrahedron dual to an edge e which belongs to f (see [11] for details). For a geometric tetrahedron, the four vectors /vectorn can be identified with the unit normal vectors for the triangles. Note that the domain of integration for each such vector is a 2-sphere. The key property of W ( j, /vectorn ) amplitude, which was used to find the large-spin asymptotics, is that it can be written as an integral over the manifold SL (2 , C ) 4 × ( CP 1 ) 10 , see [6]. More precisely, where Ω is a slowly changing function and The functions w k are complex-valued, so that θ k = arg w k + 2 πm k , where m k are integers which have to be chosen such that log w k belong to the same branch of the logarithm. Since | w k | ≤ 1, it follows that ReS ≤ 0 and it can be shown that the large-spin asymptotics is given by for λ → + ∞ , where the sum is over the critical points x ∗ = ( g ∗ , z ∗ ) satisfying ∣ ∣ and H ( x ) is the Hessian for the function S ( x ). There are finitely many critical points, and it can be shown that the conditions (6) require that j k are proportional to the areas of triangles for a geometric 4-simplex, while /vectorn have to be the normal vectors for the triangles in a tetrahedron of a geometric 4-simplex and g ∗ have to be the corresponding holonomies. A geometric 4-simplex has a consistent assigment of the edge-lengths, and it can be shown that θ k ( /vectorn, x ∗ ) is proportional to the dehidral angle for a triangle in a geometric 4-simplex, so that corresponds to the Regge action for a 4-simplex. The Hessian H ( j, /vectorn, x ) is a 44 × 44 matrix, and since S is a linear function of j . Consequently is a homogeneous polinomial of degree 44 in j k variables. One also has that Re ( -H ) is a positive definite matrix.", "pages": [ 3, 4, 5 ] }, { "title": "III. A BOUND FOR THE VERTEX AMPLITUDE", "content": "We will now find a bound for the vertex amplitude by using the asymptotic formula (5) and its generalization for the case when some of the vertex spins are large and other are small. Since λS ( j, /vectorn, x ) = S ( λj, /vectorn, x ) and then the formula (5) can be rewritten as when j = ( j 1 , ..., j 10 ) → (+ ∞ , ..., + ∞ ) ≡ (+ ∞ ) 10 , because ∏ 10 k =1 dim j k scales as λ 10 for λ large. Therefore Note that and due to (8). The equations (9),(10) and (11) imply The equation (12) is equivalent to This implies that W is a bounded function in the region If we denote with D m the region where m < 10 spins are greater than δ and the rest are smaller or equal than δ , then Since the regions D m are not compact for m> 0, we do not know whether W is bounded in these regions. In order to determine this we need to know the asymptotics of W for the cases when some of the spins are large and other are small. This asymptotics can be obtained by using the same method as in the case when all the vertex spins are large. Let m be the number of large spins ( m ≥ 3 due to the triangle inequalities for the vertex spins) and let j ' = ( j 1 , ..., j m ). Then Therefore the asymptotic properties of W ( j ' , j '' , n ) will be determined by the critical points of S m ( j ' , n, x ). Consequently where r is the rank of the Hessian matrix H m for S m at a critical point x ∗ (1 ≤ r ≤ 44) and ˜ H m is the reduced Hessian matrix. ˜ H m is the restriction of the Hessian H m to the orthogonal complement of KerH m and ˜ H m has to be used if r < 44. The asymptotics (13) implies that the function W ( j ' , j '' , /vectorn ) will vanish for large j ' if r/ 2 -m > 0. If this was true for all m we could say that W ( j ) is a bounded function in R 10 + . However, calculating the values for r is not easy. Instead, we are going to estimate | W ( j ' , j '' , /vectorn ) | . Note that (13) is equivalent to for j ' → (+ ∞ ) m , since S m and ˜ H m are linear functions of the spins j ' and det( -˜ H m ) scales as λ r , while ∏ m k =1 dim j k scales as λ m when j ' → λj ' and λ is large. Hence for j ' → (+ ∞ ) m . From here it follows that for every m ≥ 3 since r ( m ) ≥ 1. Given that W = 0 in D 1 and D 2 , it follows that W ( j, /vectorn ) / ∏ 10 k =1 dim j k is a bounded function in R 10 + . Therefore exists C > 0 such that This bound can be rewritten as which is convenient for investigating the absolute convergence of the state sum.", "pages": [ 5, 6, 7 ] }, { "title": "IV. FINITENESS", "content": "We showed in the previous section that the vertex amplitude divided by the product of the dimensions of the vertex spins is a bounded function of spins. This result suggests to introduce a rescaled vertex amplitude W p as where p ≥ 0, in order to improve the convergence of the state sum. Given a triangulation T of a compact four-manifold M , we will consider the following state sum It is sufficient to consider T without a boundary, since if Z ( T ) is finite, then Z (Γ , T ) will be finite due to gluing properties, where Γ is the boundary spin network. The convergence of Z p will be determined by the large-spin asymptotics of the vertex amplitude W and the values of p . Since the asymptotics of W is not known completely, we will use the estimate (15) in order to find the values of p which make the state sum Z p convergent. Since and by using (15) we obtain where V is the total number of vertices in the triangulation T . At this point the integrand does not depend anymore on /vectorn ef , so the appropriate integration over 4 E 2-spheres can be performed. Here E is the total number of edges in σ , and it is multiplied by 4 since every edge is a boundary for exactly four faces. After the integration we obtain The sum over the spins in (19) can be rewritten as a product of single-spin sums. Let N f be the number of vertices bounding a given face f . Each vertex contributes with a factor (dim j f ) -p +1 , so the total contribution for each face f is (dim j f ) 1 -( p -1) N f . Thus we can rewrite (19) as The sum in (20) will be convergent if or for every N f . Since N f ≥ 2 for every face f , a sufficient condition for p is Therefore Z p is absolutely convergent for p > 2, which means that it is convergent for p > 2. As far as the convergence of Z p for p ≤ 2 cases is concerned, one has to calculate the ranks of the Hessians H m and use the following inequalities when possible. We expect that the inequalities (23) will hold for all m , since det( -˜ H m ) is a homogeneous polinomial of the spins of the degree r and Re ( -˜ H m ) is a positive definite matrix. Then for any j , where q = min { r/ 2 m | m = 3 , ..., 10 } . Since q > 0, the new bound (24) will be an improvment of the bound (15) and consequently Z p will be absolutely convergent for Given that r = 44 for m = 10, this implies that q ≥ 1 / 18 ( r = 1 and m = 9 case) and therefore p > 35 / 18.", "pages": [ 7, 8, 9 ] }, { "title": "V. CONCLUSIONS", "content": "We proved that the deformed partition function Z p for the EPRL-FK spin foam model is convergent for p > 2. We expect that the bound for p can be lowered below 2, since the inequalities (23) are likely to be true. In this way one can obtain that p > 35 / 18 without calculating the matrices H m . In order to find the exact value for q , the ranks r of the Hessians H m have to be calculated. If it turns out that q > 2, then the formula (25) will give that the p = 0 case is convergent. However, if it turns out that q ≤ 2, then the convergence of the p = 0 case has to be checked by some other method. If the p = 0 state sum is finite, our construction provides an infinite number of new models with better convergence properties. In any case, one has to decide which choices for p are physical. This can be done by analyzing the semiclassical limit of the corresponding EPRL/FK model. As shown in [13, 14], the parameter p appears in the first-order quantum correction to the classical Einstein-Hilbert term. It is therefore an experimental question to determine the value of p , provided that quantum gravity is described by an EPRL/FK spin foam model. Given that a p -deformed spin foam model is finite for p > 2 and any choice of the triangulation T , one can construct a quantum field theory whose Feynman diagrams are in one-to-one correspondence with the transition amplitudes for all triangulations T , see [15, 16]. Since all those amplitudes are finite by construction, the corresponding quantum field theory will be perturbatively finite. For such a theory, no regularization scheme is necessary and there is no necessity for a perturbative renormalization procedure. As the final remark, note that where T ' is a sub-complex of T obtained by removing one or more faces from T and Z ' is the state sum where the zero spins are absent. The state sums Z ' are considered more physical, because their spin foams correspond to simplicial complex geometries where all the triangles have a non-zero area. The relation (26) was used in [17] to define the sum over the spin foams, since if one chooses a very large σ , then (26) implies that Z ( σ ) is the result of a sum of the physical transition amplitudes for various spin foams. Since Z ( σ ) can be made finite for EPRL/FK model if one modifies the vertex amplitude as (16), one arrives at a concrete realization of the idea of summing over spin foams.", "pages": [ 9, 10 ] }, { "title": "Acknowledgments", "content": "We would like to thank John Barrett for discussions. AM was partially supported by the FCT grants PTDC/MAT/69635/2006 and PTDC/MAT/099880/2008 . MV was supported by grant SFRH/BPD/46376/2008 and partially by PTDC/MAT/099880/2008 .", "pages": [ 10 ] } ]

[ { "title": "Dynamical eigenfunctions and critical density in loop quantum cosmology", "content": "David A. Craig ∗ Perimeter Institute for Theoretical Physics Waterloo, Ontario, N2L 2Y5, Canada and Department of Physics, Le Moyne College Syracuse, New York, 13214, USA (Dated: September 20, 2021) We offer a new, physically transparent argument for the existence of the critical, universal maximum matter density in loop quantum cosmology for the case of a flat Friedmann-LemaˆıtreRobertson-Walker cosmology with scalar matter. The argument is based on the existence of a sharp exponential ultraviolet cutoff in momentum space on the eigenfunctions of the quantum cosmological dynamical evolution operator (the gravitational part of the Hamiltonian constraint), attributable to the fundamental discreteness of spatial volume in loop quantum cosmology. The existence of the cutoff is proved directly from recently found exact solutions for the eigenfunctions for this model. As a consequence, the operators corresponding to the momentum of the scalar field and the spatial volume approximately commute. The ultraviolet cutoff then implies that the scalar momentum, though not a bounded operator, is in effect bounded on subspaces of constant volume, leading to the upper bound on the expectation value of the matter density. The maximum matter density is universal (i.e. independent of the quantum state) because of the linear scaling of the cutoff with volume. These heuristic arguments are supplemented by a new proof in the volume representation of the existence of the maximum matter density. The techniques employed to demonstrate the existence of the cutoff also allow us to extract the large-volume limit of the exact eigenfunctions, confirming earlier numerical and analytical work showing that the eigenfunctions approach superpositions of the eigenfunctions of the Wheeler-DeWitt quantization of the same model. We argue that generic (not just semiclassical) quantum states approach symmetric superpositions of expanding and contracting universes. PACS numbers: 98.80.Qc,04.60.Pp,04.60.Ds,04.60.Kz", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Loop quantized cosmological models generically predict that the 'big bang' of classical general relativity is replaced by a quantum 'bounce' in the deep-Planckian regime, at which the density of matter is bounded by a maximum density, typically called the 'critical density' ρ crit . (See Refs. [1, 2] for recent reviews of loop quantum cosmology [LQC] and what is currently known about these bounds in various models, as well as references to the earlier literature.) In most models, the value of this critical density is inferred from numerical simulations of quasiclassical states. So far it has been possible in only a single model - the exactly solvable loop quantization, dubbed 'sLQC' [3], of a flat Friedmann-Lemaˆıtre-Robertson-Walker cosmology sourced by a massless, minimally coupled scalar field to demonstrate analytically the existence of ρ crit for generic quantum states. In this model, it was shown that ρ crit ≈ 0 . 41 ρ p , where ρ p is the Planck density. 1 As in other models, the bound for the density was first found numerically in Refs. [5, 6]. This value was then confirmed and given a clean analytic proof in Ref. [3]. In this paper we offer a new demonstration of the existence of a critical density in this model with the hope of enriching the understanding of existing results. The argument is rooted in a study of the behavior of the dynamical eigenfunctions of the model's evolution operator, the gravitational part of the Hamiltonian constraint, based on an explicit analytical solution for these eigenfunctions found recently in Refs. [7, 8]. We will show from this solution that the eigenfunctions exhibit an exponential cutoff in momentum space that is proportional to the spatial volume. This ultraviolet cutoff may be understood as a consequence of the fundamental discreteness of spatial volume exhibited by these models. As a consequence of the cutoff, the quantum operators corresponding to the scalar momentum and spatial volume approximately commute. The ultraviolet cutoff then implies that the scalar momentum - even though its spectrum is not bounded - is in effect bounded on subspaces of constant volume. The proportionality of the cutoff to the spatial volume then leads to the existence of a critical density that is universal in the sense that it is independent of the quantum state. It has long been understood in the loop quantum cosmology community that the behavior of the eigenfunctions of the gravitational Hamiltonian constraint operator is the key to understanding the physics of loop quantum models. In particular, the quantum 'repulsion' generated by quantum geometry at small volume - leading to the signature quantum bounce - was clearly recognized in the decay of the eigenfunctions at small volume in many examples [5, 6]. (See also e.g. Refs. [9-11], among many others.) It was also recognized numerically that the onset of this decay as a function of volume depended linearly on the constraint eigenvalue. (See e.g. Refs. [9, 10].) What is new in this work is the shift in focus to the behavior of the eigenfunctions as functions of the continuous variable k labeling the constraint/momentum eigenvalues. This allows certain insights that may not be as evident when they are considered as functions of the discrete volume variable ν . From the exact solutions for the eigenfunctions of sLQC, we are able to show a genuinely exponential cutoff in the eigenfunctions as functions of k that sets in at a value of k that is proportional to the spatial volume, thus confirming and grounding the numerical observations analytically. This, of course, is the same cutoff that manifests as the decay of the eigenfunctions at small volume, considered as functions of the volume - this is clearly evident in, for example, Fig. 3 - but seen from a complementary perspective that is in some ways cleaner because of the continuous nature of the variable k . From there, we go on to show how the linear scaling of the cutoff gives rise to the universal upper limit on the matter density. To our knowledge, this connection between the linear scaling of the cutoff on the eigenfunctions and the existence and value of the universal critical density has not previously been noted. Though appealingly intuitive, this argument is essentially heuristic, so we supplement it with a new proof in the volume representation of the existence of a ρ crit in this model. (The proof of Ref. [3] is in a different representation of the physical operators.) Thus we are able to offer a clear physical and mathematical account of the origin and value of the critical density, grounded analytically in the exact solutions for this model, that complements and confirms extensive numerical and analytic results extant in the literature. This perspective may be of some use in numerical and analytical investigations into the existence of a critical density in more complex models for which full analytical solutions are not available. We expand on this point in the discussion at the end, after we have developed the necessary details. As a by-product of the methods employed to reveal the ultraviolet cutoff on the dynamical eigenfunctions, the semiclassical (large volume) limit of the eigenfunctions is also obtained from the exact eigenfunctions. The result confirms the essence of the result obtained on the basis of analytical and numerical considerations in Refs. [5, 6], that the exact eigenfunctions approach a linear combination of the eigenfunctions for the Wheeler-DeWitt quantization of the same physical model. (See also Ref. [11], in which a careful analysis of the asymptotic limit of solutions to the gravitational constraint arrived at the same result as demonstrated here from the explicit solutions for the eigenfunctions.) The domain of applicability of this approximation is described. This result is then used to argue that, in the limit of large spatial volume, generic states in LQC - not just quasiclassical ones - become symmetric superpositions (in a precise sense to be specified) of expanding and contracting universes. The symmetry exhibited in numerical evolutions of semiclassical states - see e.g. Refs. [1, 3, 5, 6] - is therefore not an artifact of semiclassicality, but a generic property of all states in loop quantum cosmology. (Compare Refs. [11, 12] for analytic results bounding dispersions of states, showing they remain small on both sides of the bounce.) The plan of the paper is as follows. In Sec. II we summarize the loop quantization of a flat FLRW spacetime sourced by a massless scalar field. Sec. III studies the dynamical eigenfunctions e ( s ) k ( ν ) of the model in detail, exhibiting various explicit forms for the solutions, and works out the asymptotic behavior of the e ( s ) k ( ν ) in the limits | ν | glyph[greatermuch] λ | k | and λ | k | glyph[greatermuch] | ν | , where λ , defined in Eq. (2.7), is related to the LQC 'area gap'. (The ultraviolet cutoff on the e ( s ) k ( ν ) emerges from this analysis in Sec. III A 3.) In Sec. III B the cutoff is employed to place bounds on the matrix elements of the physical operators and argue that the scalar momentum is approximately diagonal in the volume representation. Section IV applies these results to show that generic states in sLQC are symmetric superpositions of expanding and contracting Wheeler-DeWitt universes at large volume. Finally, Sec. V offers an intuitive argument for the existence of a critical density in this model based on the UV cutoff for the eigenfunctions, as well as a new analytic proof in the volume representation. Section VI closes with some discussion.", "pages": [ 2, 3 ] }, { "title": "II. FLAT SCALAR FRW AND ITS LOOP QUANTIZATION", "content": "In this section we briefly describe the loop quantization of a flat ( k = 0) Friedmann-Robertson-Walker universe with a massless, minimally coupled scalar field as a matter source. The model is worked out in detail in Refs. [3, 5, 6] (see also Ref. [13]); see Ref. [7] for a summary with a useful perspective and Refs. [1, 2] for recent general reviews of results concerning loop quantizations of cosmological models.", "pages": [ 3 ] }, { "title": "A. Classical homogeneous and isotropic models", "content": "The starting point is a flat, fiducial metric ˚ q ab on a spatial manifold Σ in terms of which the physical 3-metric is given by q ab = a 2 ˚ q ab , where a is the scale factor. The full metric is given by where the normal n a = -N dt a to the fixed ( L t ˚ q ab = 0) spatial slices is given in terms of a global time t and lapse N ( t ), so that a = a ( t ). For the Hamiltonian formulation of the quantum theory spatial integrals over a finite volume are required. We may therefore either choose Σ to have topology T 3 with volume ˚ V with respect to ˚ q ab , or topology R 3 and choose a fixed fiducial cell V , also with volume ˚ V with respect to ˚ q ab . The choice plays no role in the sequel and we will proceed in the language of the latter choice. 2 The physical volume of V is therefore V = a 3 ˚ V . For a massless, minimally coupled scalar field, after the integration over the spatial cell V has been carried out the classical action is The classical Hamiltonian is thus where p a and p φ are the canonical momenta conjugate to the scale factor and scalar field. Solving Hamilton's equations yields the classical dynamical trajectories, for which p φ is a constant of the motion, and where V o and φ o are constants of integration. Regarding the value of the scalar field φ as an emergent internal physical 'clock', the classical trajectories correspond to disjoint expanding (+) and contracting ( -) branches. The expanding branch has a past singularity (the big bang) in the limit φ → -∞ , and the contracting branch a future singularity (big crunch) as φ → + ∞ . (See Fig. 1.) Note that all classical solutions of this model are singular in one of these limits. Finally, we observe that the matter density ρ on the spatial slices Σ at scalar field value φ is given in the classical theory by the ratio of the energy in the scalar field to the volume at that φ : Here ρ = T ab u a u b , where u a = ( d/dτ ) a and dτ = N dt .", "pages": [ 3, 4 ] }, { "title": "B. Loop quantization", "content": "In the quantum theory, following Ref. [3] we will discuss volume in terms of the variable ν , where γ is the Barbero-Immirzi parameter, l p = √ G glyph[planckover2pi1] is the Planck length (we take c = 1), and ε = ± 1 determines the orientation of the physical triad relative to the fiducial (co-)triad ˚ ω i a determining ˚ q ab (= ˚ ω i a ˚ ω j b δ ij ) - see Refs. [1, 3, 5, 6]. Thus -∞ < ν < + ∞ . Note that ν is dimensionful. For comparison to other work, note that ν = λ · v , where v is the dimensionless volume variable of Refs. [5, 6], and √ λ = ∆ · l p (2.7a) Here ∆ · l 2 p is the 'area gap' of loop quantum gravity. 3 Remarkably, when the physical model given by Eq. (2.2) is loop-quantized in these variables, the classical 'harmonic' gauge choice N ( t ) = a ( t ) 3 leads to an exactly solvable quantum theory, 4 referred to as 'sLQC' (for 'solvable LQC') [1, 3]. One finds that physical states Ψ( ν, φ ) may be chosen to be 'positive frequency' solutions to the quantum constraint, where the positive, self-adjoint 'evolution operator' Θ (the quantized gravitational constraint) is given in the ν -representation by a second-order difference operator, 5 Solutions to the full quantum constraint ( ˆ C = -[ ∂ 2 φ +Θ]) therefore decompose into disjoint sectors with support on the glyph[epsilon1] -lattices given by ν = 4 λn + glyph[epsilon1] , where glyph[epsilon1] ∈ [0 , 4 λ ) [5, 6]. In order not to exclude the classical singularity at ν = 0 from the start, we work exclusively on the lattice glyph[epsilon1] = 0, so that in this quantum cosmological model, the volume is discrete : Group averaging yields the physical inner product for some fiducial (but irrelevant) φ o . 6 According to Eq. (2.8), states at different values of the scalar field φ may be mapped onto one another by the unitary evolution It is natural therefore - though not essential [5] - to regard the scalar field φ as an emergent physical 'clock' or 'internal time' in which states evolve in this model. Eq. (2.12) shows that the inner product of Eq. (2.11) is independent of the choice of φ o , and is therefore preserved under evolution from one φ -'slice' to another. Finally, we note that in the absence of fermions, the action, dynamics, and other physics of the model are insensitive to the orientation of the physical triads [3, 5, 6, 15]. We may therefore restrict attention to the volume-symmetric sector of the theory in which Many further details concerning the quantization of this model and its observables may be found in Refs. [1, 3, 5, 6, 13].", "pages": [ 4, 5 ] }, { "title": "C. Observables", "content": "The basic variables in this representation are the scalar field φ and the volume ν . Employing φ as an internal time, the primary operators of interest are the volume, which acts as a multiplication operator, and the scalar momentum ˆ p φ , glyph[negationslash] (In this paper we will not have need of the (exponential of the) momentum b conjugate to ν [5, 6].) As in the classical theory, the scalar momentum ˆ p φ is a constant of the motion - it obviously commutes with the effective 'dynamics' given by √ Θ - and is therefore a Dirac observable. The volume ˆ ν is not, but the corresponding 'relational' observable ˆ ν | φ ∗ giving the volume at a fixed value φ ∗ of the internal time φ is. Defining the 'Heisenberg' operator ˆ ν | φ ∗ ( φ ) acting on states at φ is given by so that, for example, the physical volume ˆ V = 2 πγl 2 p | ˆ ν | of the cell V at φ ∗ is given by the operator It is straightforward to verify that ˆ p φ and ˆ V | φ ∗ ( φ ) commute with U ( φ ), and are therefore Dirac observables.", "pages": [ 5, 6 ] }, { "title": "III. EIGENFUNCTIONS OF THE EVOLUTION OPERATOR", "content": "General physical states Ψ( ν, φ ) may be readily expressed in terms of the eigenfunctions of the dynamical evolution operator Θ - the gravitational part of the Hamiltonian constraint - given by where and -∞ < k < ∞ is a dimensionless number labelling the 2-fold degenerate eigenvalues. Restricting to the symmetric lattice ν = 4 λn and physical states which satisfy Eq. (2.13), we usually choose to work with a symmetric basis of eigenfunctions e ( s ) k ( ν ) which satisfy e ( s ) k ( ν ) = e ( s ) k ( -ν ). In terms of these physical states may be expressed simply as For normalized states, ∑ ν =4 λn | Ψ( ν, φ ) | 2 = 1, Explicit analytic expressions for the eigenfunctions of Θ have recently been found [7, 8]. The symmetric eigenfunctions will eventually be expressed in terms of the primitive eigenfunctions [7] e 0 ( ν ) = δ 0 ,ν (3.5a) where A ( k ) is a normalization factor which for consistency will always be chosen to be The functions e k ( ν ) have support on both positive and negative ν and are not symmetric in ν . It is convenient to seek linear combinations e ± k ( ν ) of e k ( ν ) and e -k ( ν ) which have support only for ν ≷ 0. The correct combinations turn out to be [7] Clearly ∑ ν e ± k ( ν ) ∗ e ∓ k ' ( ν ) = 0. The choice of A ( k ) in Eq. (3.6) corresponds to the normalization where δ ( s ) ( k, k ' ) is the symmetric delta distribution (In contrast to Ref. [7], we choose to work with the full range of k , -∞ < k < ∞ . This leads to the second delta function appearing in Eq. (3.8) relative to Eq. (C13) of that reference. Since as we will see these functions are symmetric in k , the two approaches are of course equivalent, but do lead to some differences in choices of normalization.) The e ± k ( ν ) can be given explicitly as [7] recalling ν = 4 λn . Here I ( k, n ) = 0 for n < 0, and for n ≥ 0 is given by 7 where the second form follows from the first by simple manipulations of the definition of the hypergeometric function 2 F 1 ( a, b ; c ; z ) [16]. We will discuss the properties of I ( k, n ) in detail later. For now, note from Eq. (3.10) that and from Eq. (3.7) that Given the symmetry relation Eq. (3.12) it is clear that the symmetric eigenfunctions e ( s ) k ( ν ) are finally 8 The following symmetry properties may be verified: 9 The e ( s ) k ( ν ) with A ( k ) chosen as in Eq. (3.6) then satisfy the completeness relations ν where the symmetric Kronecker delta is defined analogously to Eq. (3.9). (The domains of these expressions are understood to be even functions of k and ν , respectively.) The symmetrized deltas arise because the functions e ( s ) k ( ν ) are symmetric in both ν and k . An expression for e ( s ) k ( ν ) we will find useful later is which follows from Eqs. (3.14a) and (3.7). As a useful aside, note it is easy to see from Eq. (3.5) that e ( s ) k ( ν ) ∗ = e ( s ) -k ( -ν ). Additionally, the change of variable b ' = -b + π/λ in Eq. (3.5) - remembering ν = 4 λn - reveals that Thus e k ( ν ) ∗ = e -k ( -ν ) = e k ( ν ), and both e k ( ν ) and e ( s ) k ( ν ) are therefore real. This completes the catalog of properties of the eigenfunctions we will require. We now describe the behavior of the functions e ( s ) k ( ν ) we seek to explain in the sequel. The results of the analysis will confirm and complement the understanding of earlier numerical and analytical work arrived at prior to the discovery of the exact solutions for this model. Plots of e ( s ) k ( ν ) are shown in Figs. 2, 3 and 4. Two behaviors are clearly evident in these plots. First, the dynamical eigenfunctions are exponentially damped as functions of k for | k | > 2 | n | = | ν/ 2 λ | (Figs. 2-3). This is the ultraviolet momentum space cutoff in the eigenfunctions described in the introduction. The cutoff can be understood as a consequence of the fundamental discreteness of volume in these quantum theories. Second, for | n | > | k | , the eigenfunctions e ( s ) k ( ν ) settle quickly into a decaying sinusoidal oscillation in n (Fig. 4). We will see that this oscillation corresponds to a specific symmetric superposition of the eigenfunctions for the Wheeler-DeWitt quantization of the same physical model. As a consequence, generic quantum states in this loop quantum cosmology will evolve to a symmetric superposition of an expanding and a collapsing Wheeler-DeWitt universe. glyph[negationslash]", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "A. Asymptotics", "content": "It is evident from Fig. 3 that, to an excellent approximation, the dynamical eigenfunctions e k ( ν as having support only in the wedge | k | glyph[lessorsimilar] 2 | n | in the ( k, n analysis of the exact solutions, Eq. (3.14), as well as the large volume limit and other features visible in Figs. 2-4. glyph[negationslash] ( s ) ) may be regarded ) plane. We now seek to explain this behavior based on an ( s ) √ ). Indeed, examwith no constant term, = 0 (cf. Eq. (3.5a)). ) (as in Fig. 2) shows that it always exhibits the maximum number of roots possible From Eqs. (3.14) and (3.11), e k ( ν ) is given by | n | / 2 | k · sinh( πk ) | times the polynomial I ( k, | n | ination of I ( k, | n | ), regarded as a polynomial in k , shows that it is an even polynomial in k whose terms alternate in sign. Thus we see immediately that e ( s ) k (0) = 0 and e ( s ) 0 ( ν ) = 0 for ν Examination of plots of I ( k, | n | (2 n -1) for such a polynomial; the alternating signs of the coefficients lead to the oscillations. Writing the product in Eq. (3.11a) as a 'falling factorial' [16], it is possible to arrive at an explicit expression for I ( k, | n | ) which is useful for some computations: 10 where the coefficients a ( | n | , j ) are given by Here the s ( p, q ) denote the (signed) Stirling numbers of the first kind. 11 It should be noted that the factor to the right of ( -1) j is always positive, leading to the alternating signs of these coefficients. For large | k | , I ( k, | n | ) is dominated by k 2 | n | , and therefore and the decay of the eigenfunctions is indeed exponential in k past the largest root of e ( s ) k ( ν ). In this limit and", "pages": [ 10, 11 ] }, { "title": "1. Steepest descents", "content": "To identify the value of k at which the decay of the symmetric eigenfunctions e ( s ) k ( ν ) sets in requires a bit more work. Recall from Eq. (3.17) that the e ( s ) k ( ν ) may be expressed in terms of the primitive eigenfunctions e k ( ν ) given by Eq. (3.5). The integral in this equation is of the form where This was the form from which the exact expression Eq. (3.10) was extracted in Ref. [7]. We however wish to evaluate this integral in the limits of large | k | and | ν | . While Eq. (3.22) is not quite of the same form for which the steepest descents approximation is normally discussed - a single large parameter multiplying an overall phase - the same arguments for the validity of the approximation apply. In regions where f ( b, k, ν ) is large, the integrand oscillates rapidly and contributions from neighboring values of b cancel one another. The dominant contributions to I ( k, ν ), therefore, come from regions close to the stationary points of f ( b, k, ν ) where f changes only slowly with b and the cancellations are not strong. This is the usual steepest-descents approximation, and in general one has [17] where the z i locate the stationary points f ' ( z i ) = 0 along the relevant contour. It is clear that when | k | or | ν | are large, f ( b, k, ν ) can become large, suppressing the value of I ( k, ν ), and so we seek the stationary points of f . First observe that f ( b, k, ν ) diverges at b = 0 and b = π/λ , so there is no contribution to I ( k, ν ) from the endpoints of the integration due to the rapid oscillation of the integrand there. Next, one finds and The stationary points therefore satisfy When solutions exist there are two roots b 1 and b 2 , given in the limit | ν | glyph[greatermuch] λ | k | by We now piece together these results to study the asymptotic limits of I ( k, ν ) and consequently e ( s ) k ( ν ). We will begin by considering the case in which k and ν are both positive, and return to the other possibilities shortly. When ν glyph[greatermuch] λk > 0, from Eq. (3.24) we find where to get to the last line we recall ν = 4 λn , so the final exponential factor in Eq. (3.31b) is unity. In the case where k and ν are both negative, the same results obtain, but now again leading to a cosine, but with k →| k | . Thus, from Eqs. (3.5), (3.6), and (3.31c), we find 12 where We have yet to consider the case where k and ν are opposite in sign. In this case note that since 0 ≤ b ≤ π/λ , there are no solutions to Eq. (3.27), and f ( b, k, ν ) has no stationary points in the domain of integration. Thus I ( k, ν ), and hence e k ( ν ), are strongly suppressed by the rapid oscillations of the integrand when k and ν are opposite in sign. We note from Eq. (3.17) that the symmetric eigenfunctions e ( s ) k ( ν ) are a linear combination of e k ( ν ) and e -k ( ν ). The functions e -k ( ν ) are, mutatis mutandis as above, strongly suppressed when ν and k have the same sign, and assume the limit Eq. (3.33) when k and ν are opposite in sign. The | ν | glyph[greatermuch] λ | k | limit of e ( s ) k ( ν ) will therefore pick up precisely one contribution of the form of Eq. (3.33) no matter the signs of k and ν . Observing that cosh( πk/ 2) / √ sinh( π | k | ) ≈ 1 / √ 2 for even very modest values of k glyph[greaterorsimilar] 1, we arrive finally at Fig. 5 shows that the exact eigenfunctions settle down to this asymptotic form very quickly. We will employ Eq. (3.35) to study in Sec. IV the large volume limit of flat scalar loop quantum universes. The asymptotic expression Eq. (3.35) was, in effect, arrived at on the basis of analytical and numerical considerations in Ref. [6], with a numerically motivated fit for the phase α ( k ). 13 In Ref. [11] an expression equivalent to Eq. (3.35) was derived from a careful analysis of the asymptotic limit of solutions to the constraint equation, including an expression for the phase α ( k ) equivalent to Eq. (3.34). (See Ref. [15] for a related analysis of this limit.) Here we have instead derived this asymptotic form from the exact eigenfunctions, explicitly confirming these prior analyses with the exact solutions for the model.", "pages": [ 11, 12 ] }, { "title": "3. Ultraviolet Cutoff", "content": "Figures 2-3 clearly exhibit the exponential ultraviolet cutoff in the eigenfunctions e ( s ) k ( ν ) for values of | k | > 2 | n | = | ν/ 2 λ | . We know already from Eq. (3.21) that an exponential decay will eventually set in. The only question is, at what value of k does that occur? We have, in fact, already seen the origin of this cutoff and its value. Eq. (3.27) shows that f ( b, k, | ν | ) has no stationary points when | 2 λk/ν | > 1. In other words, I ( k, ν ), hence e k ( ν ) and e ( s ) k ( ν ), are strongly suppressed unless This cutoff - in particular, its linear scaling with volume - may be understood physically as a consequence of the underlying discreteness of the quantum geometry. States with wave numbers | k | > 2 | n | (i.e. wavelengths shorter than the scale set by | λ/ν | ) are not supported. Alternately, it may be viewed as the manifestation in the eigenfunctions of the 'quantum repulsion' generated by quantum geometry at volumes smaller than the wave number.", "pages": [ 13 ] }, { "title": "4. Small volume limit", "content": "The same argument shows that the eigenstates will, equivalently, decay rapidly for small volume, when | n | glyph[lessorsimilar] | k | / 2, as is clear in Fig. 4. Eq. (3.21) tells us the decay in e ( s ) k ( ν ) as a function of k is exponential. The precise functional form of the decay as a function of n is less evident, but the figures show it is also quick. At this point a comment may be in order. It is tempting to study this question by regarding e ( s ) k ( ν ) as a function of a continuous variable ν . However, plotting the exact expressions for e ( s ) k ( ν ) for continuous values of ν on top of the values for ν = 4 λn should quickly disabuse one of the notion that there is a simple sense in which e ( s ) k ( ν ) is well approximated by its naive continuation to the continuum. In fact, as discussed in detail in Ref. [3], the convergence to the Wheeler-DeWitt theory in the continuum is not uniform, and must be extracted with some care in the limit the 'area gap' set by λ - fixed in loop quantum cosmology to the value of Eq. (2.7) - tends to 0. As noted in the Introduction, and as is clearly evident in Fig. 3, the ultraviolet cutoff in momentum space is the 'same' cutoff as the rapid decay at small volume as a function of volume that has long been known in loop quantum cosmology based on numerical solutions for the eigenfunctions [5, 6]. It was also known numerically that the onset of this decay was proportional to the eigenvalue ω k . (See e.g. Refs. [9, 10].) What is new in the present work, facilitated by the change in perspective to consideration of the behavior of the eigenfunctions as functions of the continuous variable k , is an analytic understanding of the linear cutoff grounded in a study of the model's exact solutions, its precise value, and its specific relation to the critical density.", "pages": [ 13, 14 ] }, { "title": "B. Representation of operators", "content": "As noted above in Eq. (2.13), we have restricted attention to the volume-symmetric sector of the theory. This is only possible because the physical operators preserve the symmetry of the quantum states. From Eq. (2.9), the matrix elements of Θ in the volume basis may be expressed as where ν = 4 λn and ν ' = 4 λn ' . Note these matrix elements satisfy the following properties: Owing to these relations, the operator Θ preserves the subspaces H ( s ) phys and H ( a ) phys of states that are even and odd in ν , so that P ( s ) Θ P ( a ) = 0, where P ( s ) and P ( a ) are the corresponding projections. (In other words, Θ commutes with the parity operator Π ν = P ( s ) -P ( a ) [5, 6].) On the symmetric subspace H ( s ) phys to which we have restricted ourselves, Θ | H ( s ) phys = P ( s ) Θ P ( s ) ≡ Θ ( s ) (and correspondingly p ( s ) φ = glyph[planckover2pi1] √ Θ ( s ) ) may be decomposed in terms of the symmetric basis of eigenstates | k ( s ) 〉 , where e ( s ) k ( ν ) ≡ 〈 ν | k ( s ) 〉 . The matrix elements of Θ ( s ) in the volume representation are related to those of Θ by 〈 ν | Θ | ν 〉 = 1 2 {〈 ν | Θ | ν 〉 + 〈 ν | Θ |- ν 〉} (3.40a) The actions of Θ and Θ ( s ) on H ( s ) phys are of course completely equivalent. Since ˆ p 2 φ = glyph[planckover2pi1] 2 Θ on H phys , these expressions give the matrix elements of ˆ p 2 φ on H ( s ) phys as well. √ The matrix elements of ˆ p φ = glyph[planckover2pi1] Θ are more complex. These are given in terms of derivatives of a generating function in Appendix C of Ref. [7]. Explicit expressions for the physical observables in another representation are also given in Ref. [3]. Here we note that on H ( s ) phys we may employ the e ( s ) k ( ν ) to calculate ˆ p ( s ) φ explicitly in the volume representation. Indeed, the polynomial solution Eq. (3.19) for I ( k, n ) makes it a straightforward matter to evaluate these matrix elements. The result is (with ν = 4 λn and ν ' = 4 λm ) where ζ ( z ) is the Riemann zeta-function. This expression gives the ( ν, ν ' ) matrix elements of ˆ p ( s ) φ / glyph[planckover2pi1] κ , or equivalently √ Θ ( s ) /κ . We observe from Eq. (3.37) that Θ is nearly diagonal in the volume representation, with only the n ' = n, n ± 1 elements not exactly zero. The same is therefore true of ˆ p 2 φ . Direct numerical evaluation of the expression Eq. (3.41) ( s ) ' ' ' reveals that ˆ p φ - and therefore √ Θ - are also nearly diagonal in the volume representation, with only the n ' = n, n ± 1 matrix elements significantly different from zero. (See Fig. 6.) In this case, however, the off-diagonal elements of ˆ p φ are merely very small, rather than precisely zero. The values of these matrix elements can be understood as a consequence of the ultraviolet cutoff, Eq. (3.36). Indeed, the exponential cutoff | k | glyph[lessorsimilar] | ν/ 2 λ | implies that the diagonal matrix elements are bounded, 14 (The 1 / 2 is a consequence of the symmetric normalization of the eigenfunctions, Eq. (3.16).) This bound on √ Θ ( s ) /κ may be compared with that set by the exact expression for Θ, Eq. (3.37). From Eq. (3.40), As it is always the case that 〈 ˆ A 2 〉 ≥ 〈 ˆ A 〉 2 , we see that a strict bound on the diagonal matrix elements of √ Θ ( s ) /κ is in agreement with the bound inferred from the UV cutoff. The off-diagonal elements may be bounded in a similar manner. For simplicity assume | ν | < | ν ' | . The exponential UV cutoff effectively restricts the range of integration to | k | glyph[lessorsimilar] | ν/ 2 λ | . The Cauchy-Schwarz inequality then gives Again, because the e ( s ) k ( ν ) are symmetrically normalized, the value of the first square root is essentially 1 / √ 2. As for the second, we note from Fig. 2 that the e ( s ) k ( ν ) execute approximately uniform amplitude oscillations, growing slowly with increasing k with a short lived increase before the exponential cutoff sets in at | k | = | ν/ 2 λ | . Therefore, for | ν | < | ν ' | we may estimate that at most showing that the cutoff alone implies that the off-diagonal terms are suppressed relative to the diagonal terms. Interference effects only reduce their values further; Fig. 6 shows that except for the n = n ' ± 1 elements - as with Θ itself - this suppression is dramatic. As a shorthand to express these bounds, we can say that ˆ p φ and | ˆ ν | - and therefore | ˆ ν | φ -approximately commute, in the sense that 〈 ν | ˆ p φ | ν ' 〉 is approximately diagonal. (See Fig. 6.) We will see in Sec. V that this helps explain why the matter density in these models remains bounded even though the spectrum of the scalar momentum ˆ p φ is not itself bounded.", "pages": [ 14, 15, 16 ] }, { "title": "IV. LARGE VOLUME LIMIT OF LOOP QUANTUM STATES", "content": "In Eq. (3.35) we have exhibited the large volume (more precisely, | ν | glyph[greatermuch] λ | k | ) limit of the basis e ( s ) k ( ν ) of symmetric states of flat scalar loop quantum cosmology. We extracted this limit from the exact solution for the model's eigenfunctions, essentially confirming prior numerical and analytical work. In this section we relate these states to the eigenstates in a Wheeler-DeWitt quantization of the same physical model. A complete, rigorous Hilbert space quantization of a flat Friedmann-Lemaˆıtre-Robertson-Walker cosmology sourced by a massless minimally coupled scalar field has been given in Refs. [5, 6] and compared to its loop quantization in detail in Ref. [3]. It is known rigorously that states in the Wheeler-DeWitt quantization are generically singular just as they are in the classical theory in the sense that all states assume arbitrarily small volume (equivalently, large density) at some point in their cosmic evolution in 'internal time' φ [3, 14]. The classical solutions are given in Eq. (2.4), corresponding to disjoint expanding and contracting branches which either begin or end in the classical singularity at V = 0. Solutions to the Wheeler-DeWitt quantum theory similarly divide into disjoint expanding and contracting branches, and as noted, are singular in the same way. The Wheeler-DeWitt version of the quantum constraint is 15 [6] Attention may again be restricted to symmetric (Eq. (2.13)), positive frequency solutions in the sense of Eq. (2.8). The symmetric eigenstates of Θ WdW ν satisfying Eqs. (3.1)-(3.2) are and are orthonormal (distributionally normalized to δ ( k, k ' )) in the inner product resulting from group averaging. Physical states may then be expressed as The orthogonal sectors of 'right-moving' (in a plot of φ vs. ν ) and 'left-moving' states clearly correspond to the expanding and contracting branches of the classical solutions, Eq. (2.4). A priori , note that ˜ Ψ WdW R ( k ) = ˜ Ψ WdW ( k ) ( k < 0) and ˜ Ψ WdW L ( k ) = ˜ Ψ WdW ( k ) ( k > 0) need not be in any way related in the Wheeler-DeWitt theory. We now show that generic states in the loop quantized theory decompose into symmetric superpositions of expanding and collapsing (right- and left-moving) Wheeler-DeWitt universes at large volume. This follows simply from Eq. (3.35), which may be written where z = √ 2 for the normalization of Eq. (4.2) appropriate to the range -∞ < ν < ∞ of Eq. (4.3), 16 and the √ λ is present because the e ( s ) k ( ν ) are dimensionless, whereas the e WdW k ( ν ) are not. The factor of z √ λ can be understood as arising from the difference between normalization of the Wheeler-DeWitt eigenfunctions on the continuous range -∞ < ν < ∞ vs. the normalization of loop quantum states on an infinite lattice with step-size 4 λ . (Compare Appendix B of Ref. [11].) The relationship expressed in Eq. (4.5) has long been known in loop quantum cosmology on the basis of both analytic and numerical arguments. See, for example, Eq. (5.3) of Ref. [6], as well as Eq. (3.1) of Ref. [11] - which also contains a careful analysis of the convergence properties of this limit - among many others. Here we have confirmed that the asymptotic behavior of the exact solutions agrees precisely with these earlier arguments. We know that the limit Eq. (4.5) is valid when | ν | glyph[greatermuch] 2 λ | k | , and more generally, that e ( s ) k ( ν ) has support only in the wedge | k | glyph[lessorsimilar] | ν | / 2 λ . Eq. (4.5) therefore holds inside this wedge of support but clearly breaks down near its boundary | k | = 2 | n | . From Eq. (3.3), quite generally It is noted in Ref. [11] that one must take care to draw conclusions concerning the asymptotic behavior of states in sLQC based on that of the eigenfunctions because the convergence of the sLQC basis e ( s ) k ( ν ) to that of the WheelerDeWitt theory is not uniform in k . Nonetheless, we argue that for a wide class of quantum states, there will be a well-defined region depending on the state in which this approximation will hold for that state. Specifically, replacement of e ( s ) k ( ν ) in the expression Eq. (4.6) with its asymptotic form Eq. (4.5) will be valid for values of the volume (significantly larger than that) for which the Fourier transform ˜ Ψ( k ) does not have significant support outside the wedge at that volume. Quantum states are normalized, so we know that ˜ Ψ( k ) is square-integrable. Because functions of compact support are dense in L 2 ( R ), there is a dense set of states for which, for every state Ψ( ν, φ ) in this set, there is some value of | k | whose value will in general depend on the state - call it k Ψ - outside of which ˜ Ψ( k ) has no support. Therefore, for a dense set of states in the quantum theory the replacement Eq. (4.5) will be a good approximation for | ν | glyph[greatermuch] 2 λ | k Ψ | . It is worth emphasizing, therefore, that the domain of applicability of the large-volume approximation is dependent upon the quantum state through the support of ˜ Ψ( k ). For states satisfying this condition and within that domain of applicability, we may write It will be seen shortly that the first term corresponds to a contracting universe, and the second, expanding. To begin, we note from Eqs. (3.2), (3.3), and (3.15b) that ˜ Ψ( k ) is even, ˜ Ψ( -k ) = ˜ Ψ( k ). Consider the first term alone. As Eq. (4.7) applies only for values of the volume for which ˜ Ψ( k ) has negligible support for | k | > | ν | / 2 λ , we may extend the range of k -integration to -∞ < k < ∞ . By separating the integral ∫ ∞ -∞ dk = ∫ 0 -∞ dk + ∫ ∞ 0 dk and making the change of variable k ' = -k in the first, one quickly finds As in the Wheeler-DeWitt case, Eq. (4.4), Ψ L ( ν, φ ) clearly corresponds to a contracting quantum universe, with equivalent Wheeler-DeWitt Fourier transform In an exactly similar way, the second term in Eq. (4.7) is which describes an expanding universe with equivalent Wheeler-DeWitt Fourier transform Therefore, for a dense set of quantum states and within the domain of applicability of the large volume approximation | ν | glyph[greatermuch] 2 λ | k Ψ | for the state Ψ( ν, φ ), we may always write Unlike the Wheeler-DeWitt case, however, Ψ L and Ψ R are not independent. In fact, owing to the symmetry of ˜ Ψ( k ) in the loop quantum case, the equivalent Wheeler-DeWitt Fourier transforms are essentially the same, having equal modulus | ˜ Ψ L ( -k ) | = | ˜ Ψ R ( k ) | and a fixed phase relation given by exp( iα ( | k | )) between them. (Note that expressions equivalent to Eqs. (4.9) and (4.11) may also be found in Ref. [11]. These relations are central to the 'scattering' picture of loop quantum cosmology developed in that reference.) Thus, we have shown from the exact solution that in the sense given by Eq. (4.12), at sufficiently large volume a dense set of states in flat scalar loop quantum cosmology may be written as symmetric superpositions of expanding and contracting universes. This is an essential feature of loop quantum cosmology, deeply connected with the fact that these cosmologies are non-singular [15] - all states 'bounce' with a finite maximum matter density. It was observed long ago from numerical solutions that semiclassical states 'bounce' symmetrically, including the dispersions of these states [5, 6]. Analytic bounds on the dispersions of all states in this model have been proved in Refs. [11, 12], in which further discussion of constraints on the sense in which such states are symmetric can also be found. (In this regard see also Refs. [13, 15].) Here we have demonstrated the symmetry of generic states (not just quasiclassical ones) at large volume, in the sense of Eq. (4.12), directly from the exact solutions.", "pages": [ 16, 17, 18 ] }, { "title": "V. CRITICAL DENSITY AND THE ULTRAVIOLET CUTOFF", "content": "A significant part of the interest in loop quantum cosmology has arisen from the fact that loop quantization seems to robustly and generically resolve cosmological singularities; see Ref. [1, 2] for recent overviews. This was noticed first in numerical results for semiclassical states [5, 6], subsequently observed in many other models (see e.g. Refs. A similar argument then shows where now | Ψ 〉 = ˆ V | φ | ω 〉 . For convenience, we will adopt this latter definition of the density in the sequel.", "pages": [ 18, 19 ] }, { "title": "A. Heuristic argument", "content": "We offer here a new perspective on the existence of a universal upper bound to the density by arguing that it can be seen as a consequence of the linear scaling of the ultraviolet cutoff in the e ( s ) k ( ν ) with volume, | k | glyph[lessorsimilar] | ν/ 2 λ | . We offer both a new proof of the existence of a critical density in this model in the volume representation, as well as an [9, 10]), and finally proved analytically for all quantum states in the model described in this paper in Ref. [3]. In that paper it is shown that the expectation value (and hence spectrum) of the matter density ˆ ρ | φ is bounded above by where ρ p is the Planck density and the value of the Barbero-Immirzi parameter γ ≈ 0 . 2375 inferred from black hole thermodynamics has been used [19]. We argue here that the existence of a universal upper bound to the density may be traced to the ultraviolet cutoff for values of | k | glyph[greaterorsimilar] | ν/ 2 λ | on the eigenfunctions e ( s ) k ( ν ). Classically, the matter density when the scalar field has value φ is given by the ratio of the energy in the scalar field to the volume, In Ref. [3] it is argued that a suitable definition for the corresponding quantum mechanical observable is where Even though the spectrum of this operator is not yet known, an upper bound on the spectrum of ˆ A | φ places a bound on the spectrum of ˆ A | 2 φ , thence on ˆ ρ | φ . Ref. [3] then observed that where | χ 〉 is defined through | Ψ 〉 = √ ˆ V | φ | χ 〉 . Thus, the expectation value of ˆ A | φ (in the state | Ψ 〉 ) may be expressed as the ratio of expectation values of the momentum and volume (in the state | χ 〉 ). Even though the spectrum of ˆ p φ is not bounded, they go on to show analytically that this ratio is nonetheless bounded above by √ 3 / 4 πγ 2 G/λ for all states in the domain of the physical observables, leading directly to the bound given by Eq. (5.1) on the density. Alternately, one might choose to define heuristic argument that has a clear and intuitive interpretation, making it a simple matter to calculate the value of the critical density simply from the slope of the scaling of the ultraviolet cutoff. In fact, heuristically speaking, using Eq. (5.7) we see that the UV cutoff | k | glyph[lessorsimilar] | ν/ 2 λ | implies that identical to the rigorous bound on the density - Eq. (5.1) - found in Ref. [3]. The linear scaling in the UV cutoff on the eigenfunctions thus, in this heuristic way, leads directly to the existence of the universal critical density. In particular, the slope of the scaling gives the value of the critical density correctly. This 'moral' argument is of course not rigorous since ˆ p φ and ˆ ν | φ do not commute. There is, however, an interesting reason the 'moral' argument works: as discussed in Sec. III B, the scalar momentum ˆ p φ and volume | ˆ ν | φ operators approximately commute, again as a consequence of the ultraviolet cutoff. Thus, also as a consequence of the ultraviolet cutoff, the operator ˆ p φ is, though its spectrum is not bounded, in effect bounded on subspaces of fixed volume. This leads immediately to the upper bound on the density, as in the 'moral' argument above. The intended meaning of these statements is the following. Because of the ultraviolet cutoff, Consider the subspace spanned by volume eigenstates with volume less than or equal to some V . While this subspace is not strictly invariant under the action of the operator ˆ p φ , it is approximately so because the off-diagonal matrix elements 〈 ν | ˆ p φ | ν ' 〉 are strongly suppressed. The norm of states restricted to this subspace is ‖ χ ‖ 2 V = ∑ | ν |≤V | χ ( ν ) | 2 . Then we have and we can see that ˆ p φ is in effect bounded on subspaces of volume less than a given value. Given Eq. (5.5) or (5.7), this helps explain why the heuristic argument above gives the correct value for the critical density.", "pages": [ 19, 20 ] }, { "title": "B. Proof in the volume representation", "content": "To complete this heuristic argument we offer a new, alternative proof of the existence of a critical density in this model in the volume representation, using the definition Eq. (5.6) for the density. (The original proof of Ref. [3] is in a different representation of the quantum states and operators and employs the definition Eq. (5.3) for the density, though their proof works for either definition.) The action of the gravitational constraint Θ in the volume representation, Eq. (2.9), may be written where is approximately the average of the values of ω on either side of the volume ν . In this notation, We wish to show this quantity is bounded above by ( glyph[planckover2pi1] κ/ 2 λ ) 2 〈 ˆ ν | 2 φ 〉 ω , in accord with the 'moral' argument of Eq. (5.8). To proceed, define where ν = 4 λn . Clearly ω '∗ ω ' = ω ∗ ω , so ω and ω ' have the same norm in the inner product of Eq. (2.11). However, Thus, Since Θ is a positive operator, 17 we find Eqs. (5.13) and (5.17) show that the absolute value of the sum of off-diagonal terms, ∑ ν ν 2 ω ∗ ω , is bounded above by the sum of the diagonal terms, 〈 ˆ ν | 2 φ 〉 ω . Thus, from Eq. (5.13) we see that as desired. With the definition Eq. (5.7) for the density, then, the heuristic 'moral' argument showing the relation between the slope of the scaling of the UV cutoff and the value of the critical density is supported by a direct calculation. A parallel demonstration in the volume representation using the definition Eq. (5.3) of the density would similarly show that the sum of the off-diagonal terms in 〈 ˆ p φ 〉 χ is bounded above by the sum of the diagonal terms. Though this can be plausibly argued on the basis of the observations in Sec. III B that the off-diagonal elements of ˆ p φ in the volume representation are bounded above by the diagonal elements, and strongly suppressed for elements connecting more than one step off the diagonal - and is of course known to be true because of the proof of Ref. [3] - a proof entirely in the volume representation at the same level of rigor as that possible for ˆ p 2 φ is more difficult because the matrix elements of ˆ p φ are so much more complicated. The 'moral' argument applies in either case.", "pages": [ 20, 21 ] }, { "title": "VI. DISCUSSION", "content": "Working from recent exact results for the eigenfunctions of the dynamical constraint operator in flat, scalar loop quantum cosmology, we have demonstrated the presence of a sharp momentum space cutoff in the eigenfunctions that sets in at wave numbers | k | = | ν/ 2 λ | that may be understood as an ultraviolet cutoff due to the discreteness of spatial volume in loop quantum gravity. Earlier numerical observations showing the onset of a rapid decay in the eigenfunctions at small volume at a volume proportional to the eigenvalue ω k are thus confirmed analytically in this model. We have argued that the existence of a maximum ('critical') value of the matter density ρ | φ = p 2 φ / 2 V | 2 φ that is universal in the sense that it is independent of the state can be viewed as a consequence of the ultraviolet cutoff since the minimum volume and maximum momentum scale in the same way. This bound holds for generic quantum states in the theory in the domain of the Dirac observables, not only states which are semiclassical at large volume. We have offered both an heuristic 'moral' argument based on the scaling of the UV cutoff, and a new direct proof in the volume representation. While the 'moral' argument for the critical density is not rigorous, it is physically and intuitively clear, and enables the value of the critical density to be calculated straightforwardly as in Eq. (5.8) once the slope of the scaling of the cutoff is known. Consistency with the bounds on the matrix elements of the physical operators set by the UV cutoff shows the overall coherence of these different points of view. It is our hope that this perspective on the origin of the critical density will have some use in the study of more complex models. In particular, while the dynamical eigenfunctions have been calculated analytically in this simple model, it is probably too much to hope that this will be accomplished in most other, more complicated, models. Rigorous proofs of the existence and value of a universal critical density may therefore be difficult to achieve in many models beyond sLQC. Nevertheless, in all models it should be possible to study solutions to the gravitational constraint numerically. With the recognition from Ref. [3] that the density is bounded by the ratio of the expectation value of the momentum to the volume, we have argued here that the existence of a universal critical density may be viewed as due to the linear scaling of the ultraviolet momentum space cutoff in the eigenfunctions e ( s ) k ( ν ) with volume. Therefore, in models in which analytical solutions are not available, numerical evidence for the existence of an ultraviolet cutoff in the eigenfunctions may nevertheless be employed to argue robustly for the existence of an upper bound to the matter density for generic quantum states in those models, and indeed, its precise value may be inferred from the slope of the cutoff scaling. The asymptotics enabling the demonstration of the ultraviolet cutoff in the eigenfunctions also enabled us to extract analytically the large volume limit of these eigenfunctions based on an analysis of the model's exact solutions. The result, consistent with considerable prior work in the field based on physical, analytical and numerical arguments, is that the eigenfunctions approach a particular linear combination of the eigenfunctions for the Wheeler-DeWitt quantization of the same physical model, with a precise determination of the phase, as well as some understanding of the domain of applicability of the approximation. In turn, this allowed us to show that generic quantum states in the theory approach symmetric linear combinations of 'expanding' and 'contracting' Wheeler-DeWitt universes at large volume, no matter how non-classical those states may be.", "pages": [ 22 ] }, { "title": "ACKNOWLEDGMENTS", "content": "D.C. would like to thank Parampreet Singh for the discussions which led to this work and for critical comments on an earlier version of the manuscript, and Marcus Appleby for helpful conversations. D.C. would also like to thank the Perimeter Institute, where much of this work was completed, for its hospitality. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.", "pages": [ 22 ] } ]

[ { "title": "New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory", "content": "N. Bodendorfer 1 , 2 ∗ , T. Thiemann 1 , 3 † , A. Thurn 1 ‡ 1 Inst. for Theoretical Physics III, FAU Erlangen - Nurnberg, Staudtstr. 7, 91058 Erlangen, Germany 2 Institute for Gravitation and the Cosmos & Physics Department, Penn State, University Park, PA 16802, U.S.A. 3 Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON N2L 2Y5, Canada October 29, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We quantise the new connection formulation of D +1 dimensional General Relativity developed in our companion papers by Loop Quantum Gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalise to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its 'diagonal' components acting at edges of spin network functions are easily solved, its 'off-diagonal' components acting at vertices are non trivial and require a more elaborate treatment.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In our companion papers [1, 2] we developed the classical framework for a new connection formulation of General Relativity that is applicable in all spacetime dimensions D +1 ≥ 3. In 3 + 1 dimensions, the current connection formulation is based on a triad and its corresponding spin connection. The miracle that happens in three spatial dimensions is that the defining representation of SO(3) is equivalent to its adjoint representation. Therefore, a connection and a triad carry the same number of degrees of freedom and can serve as a canonical pair on an extended phase space whose reduction by the SO(3) Gauß constraint leads back to the ADM phase space. In order that the connection is Poisson commuting, a further miracle has to happen, namely the spin connection is integrable, i.e. can be obtained from a functional by functional derivation. These two miracles are reserved for D = 3. The observation that enables a connection formulation in higher dimensions as well is that the mismatch between the number of degrees of freedom of the D -bein and its spin connection can be accounted for by a new constraint in addition to the Gauß constraint, which requires that the momentum conjugate to the connection comes from a D -bein. The details are a bit more complicated, we have to use SO( D +1) rather than SO( D ), the D -bein is a generalised D -bein and the spin connection is a generalised hybrid connection, but this is the rough idea. The final picture is therefore a SO( D +1) gauge theory subject to SO( D +1) Gauß constraint, simplicity constraint, spatial diffeomorphism constraint and Hamiltonian constraint. Apart from the different gauge group which however is compact and the additional simplicity constraint, the situation is precisely the same as for LQG and the quantisation of our connection formulation is therefore in complete analogy with LQG. We can therefore simply follow any standard text on LQG such as [3, 4] and follow all the quantisation steps. This way we arrive at the holonomy-flux algebra, its unique spatially diffeomorphism invariant state whose GNS data are the analogue for SO( D +1) of the Ashtekar-Isham-Lewandowski Hilbert space, the analogue of spin network functions, kinematical geometrical operators such as the volume operator which is pivotal for the quantisation of the Hamiltonian constraint, the SO( D + 1) Gauß constraint, the spatial diffeomorphism constraint, the Hamiltonian constraint and a corresponding Master constraint. The only structurally new ingredient is the simplicity constraint which constrains the type of allowed SO( D + 1) representations. When it acts at the interior point of edges, it requires that the corresponding SO( D +1) representation is simple. However, when it acts at a vertex, the constraint splits into several linearly independent ones which are not mutually commuting and do not close on themselves. The situation here is similar to the situation in spin foam models [5, 6, 7, 8, 9, 10] where similar constraints at the discretised level for SO(4) arise while ours are for SO( D +1) in the continuum. We propose to solve these anomalous components of the simplicity constraints as in [5, 6, 7, 8] by passing to a corresponding Master constraint and subtracting its spectral gap 1 . The manuscript is organised as follows: In section two we define the SO( D + 1) holonomy-flux algebra and the corresponding Hilbert space representation. In section three we implement the kinematical constraints, that is Gauß, simplicity and spatial diffeomorphism constraints. In section four we develop kinematical geometrical operators, specifically D -dimensional area and volume operators. Lower dimensional operators such as length operators etc. can be constructed similarly but are left for future publication. Finally, in section five we quantise the Hamiltonian constraint. The presentation will be brief since all the constructions literally parallel those of LQG. We therefore refer the interested reader to [4] for all the missing details.", "pages": [ 3, 4 ] }, { "title": "2 Kinematical Hilbert Space", "content": "The construction of the kinematical Hilbert has been performed in [11, 12, 13, 14, 15, 16] for four and higher space-time dimension and arbitrary compact gauge group. These results apply for the case considered here, since we are using the compact group SO( D +1) irrespective of the signature of the space-time metric. We therefore only cite the main results in this section and introduce notation needed later on. Since the Poisson brackets between A aIJ and π bKL are singular, we have to smear them with test functions. In order to obtain non-distributional Poisson brackets, smearing has to be done at least D -dimensional in total. A aIJ is a one-form, thus naturally smeared along a one-dimensional curve. From π aIJ , being a vector density of weight one, we can construct the so( D + 1) - valued pseudo ( D -1)-form ( ∗ π ) a 1 ...a D -1 := π aIJ glyph[epsilon1] aa 1 ...a D -1 τ IJ which is integrated over a ( D -1)-dimensional surface in a background-independent way. These considerations lead to the definitions of holonomies and fluxes, which yield a natural starting point for a background independent quantisation. In the following, we choose ( τ IJ ) K L = 1 2 ( δ K I δ JL -δ K J δ IL ) as a basis of the Lie algebra so( D +1).", "pages": [ 4 ] }, { "title": "2.1 Holonomies, Distributional Connections, Cylindrical Functions, Kinematical Hilbert Space and Spin-Network States", "content": "Denote by A the space of smooth connections over σ . We define the holonomy h c ( A ) ∈ SO( D +1) of the connection A ∈ A along a curve c : [0 , 1] → σ as the unique solution to the differential equation where c s ( t ) := c ( st ), s ∈ [0 , 1], A ( c ( s )) := A IJ a ( c ( s )) τ IJ ˙ c a ( s ). The solution is explicitly given by where P denotes the path ordering symbol which orders the smallest path parameter to the left. Like in 3 + 1 dimensional LQG, we will restrict ourselves to piecewise analytic and compactly supported curves. The holonomies coordinatise the classical configuration space. In quantum field theory it is generic that the measure underlying the scalar product of the theory is supported on a distributional extension of the classical configuration space. For gravity, this enlargement of the configuration space is done by generalising the idea of a holonomy. Since the equations hold, we see that an element A ∈ A is a homomorphism from the set of piecewise analytic paths with compact support P into the gauge group. We now introduce the set A := Hom( P , SO( D +1)) of all algebraic homomorphisms (without continuity assumptions) from P into the gauge group. This space A is called the space of distributional connections over σ and constitutes the quantum configuration space. The algebra of cylindrical functions Cyl( A ) on the space of distributional SO( D +1) connections is chosen as the algebra of kinematical observables. The former algebra can be written as the union of the set of functions of distributional connections defined on piecewise analytic graphs γ , Cyl( A ) = ∪ γ Cyl γ ( A ) / ∼ . Cyl γ ( A ) is defined as follows. A piecewise analytic graph γ ∈ σ consists of analytic edges e 1 ,..., e n , which meet at most at their endpoints, and vertices v 1 ,..., v m . We denote the edge and vertex set of γ by E ( γ ) ( | E ( γ ) | = n ) and V ( γ ) ( | V ( γ ) | = m ), respectively. A function f γ ∈ Cyl γ ( A ) is labelled by the graph γ and typically looks like f γ ( A ) = F γ ( h e 1 ( A ) , ..., h e | E | ( A ) ) , where F γ : SO( D +1) | E | → C . One and the same cylindrical function f ∈ Cyl( A ) can be represented on different graphs leading to cylindrically equivalent representations of that function. It is understood in the above union that such functions are identified. We will denote the pullback of a function f γ defined on γ on the bigger 2 graph γ ' glyph[follows] γ via the cylindrical projections by p ∗ γ ' γ . Then, the equivalence relation just mentioned can be made more explicit, f γ ∼ f ' γ ' iff p ∗ γ '' γ f γ = p ∗ γ '' γ ' f ' γ ' ∀ γ, γ ' ≺ γ '' . The pullback on the projective limit function space will be denoted by p ∗ γ . The functions cylindrical with respect to a graph that are N times differentiable with respect to the standard differentiable structure on SO( D +1) will be denoted by Cyl N γ ( A ) and Cyl N ( A ) := ∪ γ Cyl N γ ( A ) / ∼ . Since in the end we are interested only in gauge invariant quantities, after solving the Gauß constraint (classically oder quantum mechanically) we have to consider the algebra of cylindrical functions on the space of distributional connections modulo gauge transformations Cyl( A / G ). For representatives f γ of elements f of this space, the complex-valued function F γ on SO( D +1) | E | has to be such that f γ ( A ) is gauge invariant. We will slightly abuse notation and use the same notation for the new projectors p γ ' γ : A γ ' / G γ ' → A γ / G γ . There is a unique [17, 18] choice of a diffeomorphism invariant, faithful measure µ 0 on A / G which equips us with a kinematical, gauge invariant Hilbert space H 0 := L 2 ( A / G , dµ 0 ) appropriate for a representation in which A is diagonal. This measure is entirely characterised by its cylindrical projections defined by where µ H is the Haar probability measure on SO( D +1). An orthonormal basis on H 0 is given by spin-network states [19, 20, 21], which are defined as follows. Given a graph γ , label its edges e ∈ E ( γ ) with non-trivial irreducible representations π Λ e of SO( D +1), i.e. Λ e is the highest weight vector associated with e , and its vertices v ∈ V ( γ ) with intertwiners c v , i.e. matrices which contract all the matrices π Λ e ( h e ) for e incident at v in a gauge invariant way. A spin-network state is simply a C ∞ cylindrical function on A / G constructed on the above defined so-called spin-net, T γ, glyph[vector] Λ ,glyph[vector]c [ A ] := tr [ ⊗ | E | i =1 π Λ e i ( h e i ( A )) · ⊗ | V | j =1 c j ] , where glyph[vector] Λ = (Λ e ), glyph[vector]c = ( c v ) have indices corresponding to the edges and vertices of γ respectively.", "pages": [ 4, 5 ] }, { "title": "2.2 (Electric) Fluxes and Flux Vector Fields", "content": "Since π aIJ are Lie algebra-valued vector densities of weight one, ( ∗ π ) a 1 ...a D -1 := π aIJ glyph[epsilon1] aa 1 ...a D -1 τ IJ is a pseudo ( D -1)-form and is naturally integrated over a ( D -1)-dimensional face S . We therefore define the (electric) fluxes where n = n IJ τ IJ denotes a Lie algebra-valued scalar function of compact support. We again restrict to piecewise analytic surfaces S , to ensure finiteness of the number of isolated intersection points of S with a piecewise analytic path. In order to compute Poisson brackets, we have to suitably regularise the holonomies and fluxes to objects smeared in D spatial dimensions. A possible regularisation in any dimension is given in [4]. Removal of the regulator leads to the following action of the Hamiltonian vector fields Y n ( S ) corresponding to π n ( S ) on adapted representatives f γ S f γ S is an adapted representative of the cylindrical function f ∈ Cyl 1 ( A ) in the sense that all intersection points of S and γ S are beginning points b ( e ) of edges e ∈ E ( γ S ) (this can always be achieved by suitably splitting and inverting edges). In the above equation, glyph[epsilon1] ( e, S ) is a typeindicator function, which is +( -)1 if the beginning segment of the edge e lies above (below) the surface S and zero otherwise. R e IJ ( L e IJ ) is the right (left) invariant vector field on the copy of SO( D +1) labelled by e , The algebra of right (left) invariant vector fields is given by and analogously for L e IJ . We remark that, in order to calculate functional derivatives, we had to restrict f to A in the beginning. The end result (2.6), however, can be extended to all of A . Following the standard treatment, these vector fields are generalised from adapted to nonadapted graphs and shown to yield a cylindrically consistent family of vector fields, thus they define a vector field Y n ( S ) on A . The Y n ( S ) are called flux vector fields. On the Hilbert space defined in section 2.1, the elements of the classical holonomy-flux algebra become operators which act by where the right hand side is the action of the vector field Y n ( S ) on the cylindrical function ψ . The appearance of β is due to the fact that we defined the fluxes using π , whereas the momenta conjugate to the connection is given by ( β ) π = 1 β π . The momentum operators ˆ Y n ( S ), with dense domain Cyl 1 , can be shown to be essentially self-adjoint operators on H 0 analogously to the (3 + 1)-dimensional case [13].", "pages": [ 5, 6 ] }, { "title": "3.1 Gauß Constraint", "content": "Working with the gauge invariant Hilbert space from the beginning, the Gauß constraint is already solved. Yet we want to summarise its implementation on the gauge variant Hilbert space H = L 2 ( A , dµ ' 0 ) , since we want to compute quantum commutators of the constraint with the simplicity constraint in the next section. The implementation (as well as the solution) of the Gauß constraint can be copied from the (3 + 1)-dimensional case without modification. According to the RAQ programme, we choose the dense subspace Φ = Cyl ∞ ( A ) in the Hilbert space. Then, we are looking for an algebraic distribution L ∈ Φ ' such that the following equation holds for any v ∈ V ( γ ), any graph γ and f γ ∈ Cyl ∞ γ ( A ). The general solution for L is given by a linear combination of 〈 ψ, . 〉 , where ψ ∈ H 0 is gauge invariant. Thus, for an adapted graph γ ' (all edges outgoing from the vertex v in question), gauge invariance amounts to vanishing sum of all right invariant vector fields at a vertex,", "pages": [ 6, 7 ] }, { "title": "3.2.1 From Classical to Quantum", "content": "Classically, vanishing of the simplicity constraints S ab M ( x ) = 1 4 glyph[epsilon1] IJKLM π aIJ ( x ) π bKL ( x ) at all points x ∈ σ is completely equivalent to the vanishing of for all points x ∈ σ and all surfaces S x glyph[epsilon1] , S ' x glyph[epsilon1] ' ⊂ σ containing x and shrinking to x as glyph[epsilon1] , glyph[epsilon1] ' tend to zero. More precisely, we use faces of the form S x : ( -1 / 2 , 1 / 2) D -1 → σ ; ( u 1 , ..., u D -1 ) ↦→ S x ( u 1 , ..., u D -1 ) with semi-analytic but at least once differentiable functions S x ( u 1 , ..., u D -1 ) and S x (0 , ..., 0) = x , and define S x glyph[epsilon1] ( u 1 , ..., u D -1 ) := S x ( glyph[epsilon1]u 1 , ..., glyph[epsilon1]u D -1 ). We find that (2.5) becomes (with the choice n IJ = δ K [ I δ L J ] ) with n a ( S ) = glyph[epsilon1] aa 1 ...a D -1 ( ∂S xa 1 /∂u 1 )(0 , ..., 0) × ... × ( ∂S xa D -1 /∂u D -1 )(0 , ..., 0), from which the claim follows. Now, similar to the treatment of the area operator in section 4.1, we just plug in the known quantisation of the electric fluxes and hope to get a well-defined constraint operator in the end. Using the regularised action of the flux vector fields on cylindrical functions (2.6), we find for a representative f γ SS ' of f ∈ Cyl 2 ( A ) on a graph γ SS ' adapted to both S x and S ' x , The flux vector fields only act locally on the intersection points e ∩ S , e ∈ E ( γ SS ' ). Therefore, in the second line we used that for small surfaces S x glyph[epsilon1] , S ' x glyph[epsilon1] ' , the action of the constraint will be trivial expect for x (and of course only non-trivial if x is in the range of γ SS ' ), thus independent of glyph[epsilon1] . In the limit glyph[epsilon1], glyph[epsilon1] ' → 0 the expression in the last line of the above calculation clearly diverges except for ˆ ˜ Cf = 0, where the whole expression vanishes identically. Since the kernels of the constraint operators ˆ C and ˆ ˜ C coincide, we can work with the latter and propose the constraint (omitting the ∼ again) where R up ( ' ) IJ := ∑ e ∈ E ( γ SS ' ) ,b ( e )= x,glyph[epsilon1] ( e,S ( ' ))=1 R e IJ and similar for R down ( ' ) IJ . In the following, will drop the superscript x for the surfaces for simplicity. glyph[negationslash] The proof that the family ˆ C M γ ( S, S ' , x ) is consistent and defines a vector field ˆ C M ( S, S ' , x ) on A follows from the consistency of ˆ Y n ( S ). To see that the operator is essentially self-adjoint, let H 0 γ,glyph[vector]π be the finite-dimensional Hilbert subspace of H 0 given by the closed linear span of spin network functions over γ where all edges are labelled with the same irreducible representations given by glyph[vector]π , H 0 = ⊕ γ,glyph[vector]π H 0 γ,glyph[vector]π . Given any surfaces S , S ' we can restrict the sum over graphs to adapted ones since we have H 0 γ,glyph[vector]π ⊂ H 0 γ SS ' ,glyph[vector]π ' for the choice π ' e ' = π e with E ( γ SS ' ) glyph[owner] e ' ⊂ e ∈ E ( γ ). Since ˆ C M ( S, S ' , x ) preserves each H 0 γ,glyph[vector]π , its restriction is a symmetric operator on a finitedimensional Hilbert space, therefore self-adjoint. To see that it is symmetric, note that the right hand side of the first line of (3.6) consists of right-invariant vector fields which commute. This is obvious for the summands with vector fields acting on distinct edges e = e ' , and for e = e ' note that [ R e IJ , R e KL ] is antisymmetric in ( IJ ) ↔ ( KL ) and thus vanishes if contracted with glyph[epsilon1] IJKLM . Now it is straight forward to see that ˆ C M ( S, S ' , x ) itself is essentially self-adjoint. Note that we did not follow the standard route to quantise operators, which would be to adjust the density weight of the simplicity constraint to be +1 (in its current form it is +2) and quantise it using the methods in [22]. Rather, the quantisation displayed above parallels the quantisation of the (square of the) area operator in 3+1 dimensions and indeed we could have considered ∫ d D -1 u √ | n S a n S b S ab M | for arbitrary surfaces S and would have arrived at the above expression in the limit that S shrinks to a point without having to take away the regulator glyph[epsilon1] (the dependence on two rather than one surface can be achieved, to some extent, by an appeal to the polarisation identity). If we would have quantised it using the standard route then it would be necessary to have access to the volume operator. We will see in section 4.2 that for the derivation of the volume operator in certain dimensions in the form we propose, which is a generalisation of the 3 + 1 dimensional treatment, we need the above simplicity constraint operator to cancel some unwanted terms. Of course, there might be other proposals for volume operators which can be defined in any dimension without using the simplicity constraint. Still, the quantisation of the simplicity constraint presented here will (1) give contact to the simplicity constraints used in spin foam models and (2) enable us to solve the constraint in any dimension when acting on edges. Its action on the vertices, i.e. the requirements on the intertwiners, is more subtle and we propose to treat it using the Master constraint method. We will first present the action on edges and afterwards derive a suitable Master constraint. For following calculations, note that we always can adapt a graph to a finite number of surfaces. Furthermore, it is understood that all surfaces intersect γ ' in one point only (we may always shrink the surfaces until this is true).", "pages": [ 7, 8 ] }, { "title": "3.2.2 Edge Constraints and their Solution", "content": "The action of the quantum simplicity constraint at an interior point x of an analytic edge e = e 1 · ( e 2 ) -1 for both surfaces S , S ' not containing e (otherwise the action is trivial) is given by where the sign is + if the orientation of the two surface S , S ' with respect to e coincides and -otherwise. In the second and fourth step we used gauge invariance at the vertex v of an adapted graph, [ ∑ e ∈ E ( γ ); v = b ( e ) R e IJ ] f γ SS ' = 0, and in the third step we used that [ R e 1 , R e 2 ] = 0. This leads to the requirement on the generators of SO( D +1) for all edges The so-called simple representations of SO( D +1) satisfying this constraint were classified in [23]. Irreducible simple representations are given by homogeneous harmonic polynomials H ( D +1) N of degree N , in any dimension labelled by one positive integer N . In this sense, there is a similarity between the simple representations of SO( D + 1) and the representations of SO(3) (which all can be thought of as being simple). In particular, for D +1 = 4 we obtain the well-known simple representations of SO(4) used in spin foams labelled by j + = j = j -. The commutator with gauge transformations at an interior point x of an analytic edge e = e 1 · ( e 2 ) -1 ( e 1 , e 2 outgoing at x ) yields, analogously to the classical calculation, Two constraints acting at the same interior point x of an edge e = e 1 · ( e 2 ) -1 commute weakly. Using the gauge invariance of Cf if f is gauge invariant, we find which can be seen by the fact that the simplicity on an edge is quadratic in the rotation generator R e 1 on that edge, and we used the notation for a simplicity with a infinitesimal rotation acting on the multi-index M (cf. (3.9)). Here, we chose a graph γ adapted to all four surfaces S , S ' , S '' , S ''' . Note that classically, the Poisson bracket of two simplicity constraints vanishes strongly, whereas in the quantum theory this is only true in a weak sense. Still, the simplicity constraints acting on an edge are thus nonanomalous and can be solved by labelling all edges by simple representations of SO( D +1).", "pages": [ 9, 10 ] }, { "title": "3.2.3 Vertex Master Constraint", "content": "When acting on a node then, like the off-diagonal constraints in spin foam models, the simplicity constraints will not (weakly) commute anymore. Therefore, we are not allowed to introduce these constraints strongly and have the options of either trying to implement them weakly [5] or using a Master constraint. We will follow the latter route and give a proposal of how to construct a Master constraint of the simplicity constraints at the nodes. To reduce complexity, we try to find a both necessary and sufficient set of simple 'building blocks' of the simplicity constraint at the node and construct a Master constraint using these. Considering (3.6), an obviously sufficient set of building blocks at the vertex v is given by For necessity, we have to prove that we can choose surfaces in such a way that these building blocks follow. Note that it has already been shown in [24] that all right invariant vector fields R e for single edges e can be generated by the Y ( S ), but the construction involves commutators of the fluxes. Since the simplicity constraints acting on vertices are anomalous, we cannot use commutators in our argument. Instead, we will construct the right invariant vector fields R e by using linear combinations of fluxes only. To this end, we will prove the following lemma:", "pages": [ 10 ] }, { "title": "Lemma.", "content": "For each D -tuple of edges e 1 , . . . , e D there exists a choice of vectors glyph[vector]n 1 1 , . . . , glyph[vector]n 1 D , glyph[vector]n 2 1 , . . . , glyph[vector]n D D and a way to guide the limit ∆ 1 1 , ∆ 1 2 , . . . , ∆ D D → 0 such that vanishes if (a) if e 1 , . . . , e D do not all intersect p or Otherwise it tends to Here we have denoted by ˆ O e 1 ,...,e D ( p ) the trace(s) involved in the various terms of (5.8). We conclude that (5.8) reduces to lim ∆ D → 0 where v on the right-hand side is the intersection point of the D -tuple of edges and it is understood that we only sum over such D -tuples of edges which are incident at a common vertex and s ( e 1 , . . . , e D ) := sgn(det( ˙ e 1 (0) , . . . , ˙ e D (0))). Moreover, and is a right-invariant vector field in the τ IJ direction of SO( D +1), that is, R ( hg ) = R ( h ). We have also extended the values of the sign function to include 0, which takes care of the possibility that one has D -tuples of edges with linearly dependent tangents. glyph[negationslash] The final step is choosing ∆ 1 = . . . = ∆ D -1 and exponentiating the modulus by 1 / ( D -1). We replace the sum over all D -tuples incident at a common vertex ∑ e 1 ,...,e D by a sum over all vertices followed by a sum over all D -tuples incident at the same vertex ∑ v ∈ V ( γ ) ∑ e 1 ∩ ... ∩ e D = v . Now, for small enough ∆ and given p , at most one vertex contributes, that is, at most one of χ ∆ ( v, p ) = 0 because all vertices have finite separation. Then we can take the relevant χ ∆ ( p, v ) = χ ∆ ( p, v ) 2 out of the exponential and take the limit, which results in ˆ π ( p, ∆ 1 , . . . , ∆ D ) f", "pages": [ 18, 19 ] }, { "title": "2. For e ' ∈ E ( v ) , e ' = e : e ' ∩ S v,e = v , and for e ' / ∈ E ( v ), e ' ∩ S v,e = ∅ .", "content": "glyph[negationslash] is given in [24] and we summarise the result shortly. An analytic surface (edge) is completely determined by its germ [ S ] v ([ e ] v ) To ensure that s e ⊂ S v,e , we just need to choose a parametrisation of S such that S ( t, 0 , ..., 0) = e ( t ) which fixes the Taylor coefficients S ( m, 0 ,..., 0) (0 , ..., 0) = e ( m ) (0). For the finite number k = | E ( v ) | -1 of remaining edges at v , we can now use the freedom in choosing the other Taylor coefficients to assure that there are no (beginning segments of) other edges contained in S v,e [24]. In particular, only a finite number of Taylor coefficients is involved. glyph[negationslash] Now we state that the intersection properties of a finite number of transversal edges at v with any (sufficiently small) surface S are already fixed by a finite number of Taylor coefficients of S . We will discuss the case D = 3 for simplicity, higher dimensions are treated analogously. Locally around v we may always choose coordinates such that the surface is given by z = 0, S ( x, y ) = ( x, y, 0). The edge e contained in the surface is given by e ( t ) = ( x ( t ) , y ( t ) , 0) and for any transversal edge at v we find e ' ( t ) = ( x ' ( t ) , y ' ( t ) , z ' ( t )) where z ' ( t ) = t n -1 ( n -1)! z ' ( n -1) (0)+ O ( t n ), and n < ∞ since otherwise e ' would be contained in S . The sign of the lowest non-vanishing Taylor coefficient z ' ( n -1) (0) determines if the edge is 'up'- or 'down'-type locally. Set N = max e ' ∈ E ( v ) ,e ' = e ( n ), and obviously N < ∞ . Thus, we can e.g. by modifying S ( N, 0) (0 , 0) choose the surface ˜ S ( x, y ) = ( x, y, ± x N ), which locally has the same intersection properties with the edges e ' ∈ E ( v ) , e ' = e and certainly does not contain e anymore. glyph[negationslash] Coming back to the general case considered before, there always exists N < ∞ such that we can change S ( N, 0 ,..., 0) (0 , ..., 0) without modifying the intersection properties of any of the edges e ' ∈ E ( v ) , e ' = e , in particular the 'up'- or 'down'-type properties are unaffected. However, the edge e no longer is of the inside type, but becomes either 'up' or 'down' (depending on whether S ( N, 0 ,..., 0) (0 , ..., 0) is scaled up or down and on the orientation of S ). In general, new intersection points v ' ∈ E ( v ) ∩ S, v ' = v may occur when modifying the surface in the above described way, but we may always make S smaller to avoid them. glyph[negationslash] Now choose a pair of surfaces S , ˜ S for the edge e such that it is once 'up'- and once 'down'-type to obtain the desired result glyph[negationslash] Case (b): In the case that there is a partner ˜ e which is a analytic continuation of e through v , we cannot construct an analytic surface (without boundary) S v,e containing a beginning segment of e and not containing a segment of ˜ e . However, we can construct an analytic surface S v, { e, ˜ e } containing (beginning segments of) e , ˜ e and sharing the remaining properties with S v,e above. The method is the same as in case (a) [24]. Again, there always exists N < ∞ such that we can change S ( N, 0 ,..., 0) (0 , ..., 0) without modifying the intersection properties of any of the edges e ' ∈ E ( v ) , e ' = { e, ˜ e } , and such that both edges e , ˜ e become either 'up' or 'down'-type. Moreover, if we choose N even, then e , ˜ e will be of the same type with respect to the modified surface, while for N odd one edge will be 'up' and its partner will be 'down'. Calling the modified surface S for N even and ˜ S for N odd, we find with the same calculation (3.14) as in case (a) the desired result. This furnishes the proof of the above lemma 3 . Choosing the surfaces as described above, we find that the following linear combination proves the necessity of the building blocks. Using the fact that the edge representations are already simple, we can rewrite the building blocks as We proceed by showing that the building blocks are anomalous, starting with the case D = 3. We calculate for e = e ' = e '' = e glyph[negationslash] glyph[negationslash] glyph[negationslash] where we used the notation δ I 1 ...I n J 1 ...J n := n ! δ I 1 [ J 1 δ I 2 J 2 ...δ I n J n ] . To show that this expression can not be rewritten as a linear combination of the of building blocks (3.16), we antisymmetrise the indices [ ABIJ ], [ ABKC ] and [ IJKC ] and find in each case that the result is zero. Therefore, a simplicity building block can not be contained in any linear combination of terms of the type (3.17). For D > 3, we have Choosing M = E fixed, the anomaly is the same as above. A short remark concerning the terminology 'anomaly' is in order at this place. Normally, the term anomaly denotes that a certain classical structure, e.g. the constraint algebra, is not preserved at the quantum level, e.g. by factor ordering ambiguities. The non-commutativity of the simplicity constraints however can already be seen at the classical level when using holonomies and fluxes as basic variables. Thus, one could argue that it would be more precise to talk of a quantisation of second class constraints. On the other hand, since the holonomy-flux algebra is an integral part of the quantum theory and at the classical level it would be perfectly fine to use a non-singular smearing, we will nevertheless use the term anomaly to describe this phenomenon. Independently of the terminology chosen, we cannot quantise the simplicity constraints acting on vertices using the Dirac procedure since this will lead to the additional constraints (3.18) being imposed. The unique solution to these constraints has been worked out in [23] and is given by the Barrett-Crane intertwiner in four dimensions and a higher-dimensional analogue thereof. Several options are at our disposal at this point. Looking back at our companion paper [2], one could try to gauge unfix this second class system to obtain a first class system subject to only a subset of the vertex simplicity constraints. In this process, one would have to pick out a first class subset of the simplicity constraints which has a closing algebra with the remaining constraints. The construction of a possible choice of such a subset is discussed in our companion paper [25]. While the proposed subset is first class with respect to the other constraints, it suffers from the fact that the choice is based on a certain recoupling scheme and that a different choice of the recoupling scheme results in a different first class subset. This is not a problem for the theory itself, but it seems problematic when constructing a unitary map to SU(2) spin networks in four dimensions, as discussed in [25]. Another possibility is the construction of a Dirac bracket, which however would result in a non-commuting connection and the non-applicability of the LQG quantisation methods. The use of a weak implementation in the sense of Gupta and Bleuler is discussed in our companion paper [25]. While the results obtained in the context of the EPRL spin foam model can be also used in the canonical theory (up to certain subtleties discussed in [25]), they rely on specific properties of SO(4) which do not extend to higher dimensions. While equivalent at the classical level, the master constraint introduced in [26] allows to quantise also second class constraints by a strong operator equation. Due to the second class nature, one expects the master constraint operator to have an empty kernel or at least a kernel which is too small to describe the physical Hilbert space. Since we know that the BarrettCrane intertwiner is a solution to the strong imposition of all vertex simplicity constraints, we are in the second case. In order to find a larger kernel of the master constraint, one modifies it by adding terms to it which vanish in the classical limit, i.e. performs glyph[planckover2pi1] -corrections. The merits of this procedure are exemplified by the construction of the EPRL intertwiner [8] in four dimensions, which results from a master constraint for the linear simplicity constraint upon glyph[planckover2pi1] -corrections. A simplification arising in the treatment of the linear simplicity constraints, see e.g. [8] our companion paper [25], is that they act individually on every edge connected to the intertwiner. On the other hand, the quadratic constraints act on pairs of edges and the resulting algebraic structure of the master constraints is thus very different. Since we are not aware of a suitable solution for the quadratic vertex master simplicity constraint, we will contend ourselves by giving a definition of this constraint operator. The task remaining for solving the vertex simplicity master constraint operator is thus to find a proper glyph[planckover2pi1] -correction which results in a physical Hilbert space with the desired properties, e.g. that there exists a unitary map to SU(2) spin networks in four dimensions. A general simplicity Master constraint is given by glyph[negationslash] with a positive matrix c e '' e ''' ee ' MNOP IJKL , which we will choose diagonal for simplicity, c e '' e ''' ee ' MNOP IJKL = 1 4! c ee ' δ e '' ( e δ e ''' e ' ) δ MNOP IJKL . The diagonal elements c ee ' can be chosen symmetric because of the symmetry of the building blocks. We choose c ee ' = 1 ∀ e, e ' , e = e ' and c ee = 0 since the edge representations are already simple, leading to the final version of the Master constraint we propose, glyph[negationslash] Cylindrical consistency and essential self-adjointness follows analogously to the case of C ( S, S ' , x ) in section 3.2.1. For the case of SO(4), we can use the decomposition in self-dual and anti-selfdual generators to find that glyph[epsilon1] IJKL R e IJ R e ' KL = glyph[vector] J e + · glyph[vector] J e ' + -glyph[vector] J e -· glyph[vector] J e ' -, which implies This leads to the Master constraint glyph[negationslash] where + and -now label independent copies of SO(3). Thus, we can calculate the matrix elements of this constraint in a recoupling basis analogously to the standard LQG volume operator matrix elements [27]. As mentioned before, alternative routes to deal with the vertex simplicity constraints will be the subject of [25].", "pages": [ 11, 12, 13, 14 ] }, { "title": "3.3 Diffeomorphism Constraint", "content": "The diffeomorphism constraint can again be treated in exact agreement with the (3 + 1)dimensional case. To solve the diffeomorphism constraint, one proceeds as follows. Consider the set of smooth cylindrical functions Φ := Cyl ∞ ( A / G ) which can be shown to be dense in H 0 . By a distribution ψ ∈ Φ ' on Φ we simply mean a linear functional on Φ. The group average of a spin-network state T γ, glyph[vector] Λ ,glyph[vector]c is defined by the following well-defined distribution on Φ where [ γ ] denotes the orbit of γ under smooth diffeomorphisms of σ which preserve the analyticity of γ including an average over the graph symmetry group (see, e.g., [28] for technical details). Since we already solved the simplicity constraint on single edges, we can restrict attention to spin network states with edges labelled by simple SO( D +1) representations, Λ e = ( N e , 0 , ... ). The group average [ f ] of a general cylindrical function f is defined by demanding linearity of the averaging procedure, i.e. first decompose f into spin-network states and then average each of the spin-network states separately. An inner product for the diffeomorphism invariant Hilbert space can be constructed. We will not give details and refer the reader to [16, 28].", "pages": [ 14 ] }, { "title": "4.1 The D -1 Area Operator", "content": "The area operator was first considered in [29] and defined mathematically rigorously in the LQG representation in [30]. In [4], the results of [30] are generalised for arbitrary dimension D . Using the classical identity π aIJ π b IJ = 2 qq ab , we can basically copy the treatment found there. Let S be a surface and X : U 0 → S the associated embedding, where U 0 is an open submanifold of R D -1 . Then the area functional is given by Introduce U 0 = ∪ U ∈U U , a partition of U 0 by closed sets U with open interior, U being the collection of these sets. Then the area functional can be written as the limit as | U | → ∞ of the Riemann sum where S U = X ( U ) and π IJ ( S U ) is the electric flux with choice n IJ = δ I [ K δ J L ] , which has been quantised already. Let f ∈ Cyl 2 ( A ), choose a representative f γ and, using the known action of the quantised electric fluxes, obtain as in the (3 + 1)-dimensional case where γ S glyph[follows] γ is an adapted graph. The family of operators ̂ Ar γ [ S ] has dense domain Cyl 2 ( A ). Its independence of the adapted graph follows from that of the electric fluxes. Moreover, the properties of the area operator like cylindrical consistency, essential self-adjointness and discreteness of the spectrum can be shown analogously to [4]. The complete spectrum can be derived using the standard methods. We use where the ∆s are mutually commuting primitive Casimir operators of SO( D +1). Thus their spectrum is given by the Eigenvalues λ π > 0. We have to distinguish the cases D +1 = 2 n even, N glyph[owner] n ≥ 2 and D +1 = 2 n +1 odd, n ∈ N . In a representation of SO( D +1) with highest weight Λ = ( n 1 , ..., n n ), n i ∈ N 0 , we find for the eigenvalues of the Casimir 4 ∆ := -1 2 X IJ X IJ where we used the following notation such that f 1 ≥ f 2 ≥ ... ≥ f n . Note that the above formulas hold for general irreducible Spin( D +1) representations. Irreducible representations of SO( D +1) are found by the restriction that all f i be integers. Denoting by Π a collection of representatives of irreducible representations of SO( D +1), one for each equivalence class, we find for the area spectrum Note that the above formulas (4.5) significantly simplify if we restrict to simple representations, Λ 0 = ( N, 0 , 0 , ... ), We cannot use this simplified expression for the SO( D +1) Casimir operator in the general case (4.7), since in the decomposition of a tensor product of irreducible simple representations usually non-simple representations will appear 5 , but we can use it for a single edge. When acting on a single edge e = e 1 · ( e 2 ) -1 intersecting S transversally, we know that due to gauge invariance The action of the area operator on a single edge e , e ∩ S = ∅ is thus given by glyph[negationslash] where l ( D +1) p := D -1 √ glyph[planckover2pi1] G ( D +1) c 3 is the unique length in D +1 dimensions, and κ = 16 πG ( D +1) /c 3 in any dimension, where G ( D +1) denotes the gravitational constant. Note that for D = 3, we find the factor √ N ( N +2) in the area spectrum of an edge stemming from irreducible simple representations of SO(4). Replace the non-negative integer N labelling the weight by N = 2 j , j half integer, to find the factor 2 √ j ( j +1) of SO(4) spin foam models, which coincides with the usual spacing in (3 + 1)-dimensional LQG, In standard LQG, instead of the gauge group SO(3) one extends to the double cover Spin(3) ∼ = SU(2) and allows also for half integer representations. Note that in our case, we cannot allow for general Spin( D +1) representations at the edges, since the edge simplicity constraint is not satisfied in representations of Spin( D +1) which are not as well representations of SO( D +1), D ≥ 3 [23].", "pages": [ 14, 15, 16 ] }, { "title": "4.2 The Volume Operator", "content": "The derivation of the volume operator is analogous to the treatment in [4] and requires only a slight adjustment. The volume of a region R is classically measured by where √ q has to be expressed in terms of the canonical variables. The derivation is performed for β = 1, the general result is obtained by multiplying the resulting operator by β D/ ( D -1) .", "pages": [ 16 ] }, { "title": "4.2.1 D +1 Even", "content": "Let n = ( D -1) / 2. Let χ ∆ ( p, x ) be the characteristic function in the coordinate x of a hypercube with centre p spanned by the D vectors glyph[vector] ∆ i := ∆ i glyph[vector]n i , i = 1 , . . . , D , where glyph[vector]n i is a normal vector in the frame under consideration and which has coordinate volume vol = ∆ 1 . . . ∆ D det( glyph[vector]n 1 , . . . , glyph[vector]n D ) (we assume the vectors to be right-oriented). In other words, where < · , · > is the standard Euclidean inner product and Θ( y ) = 1 for y > 0 and zero otherwise. We will use lower indices (∆ 1 I , . . . , ∆ D I ) to label different hypercubes. It will turn out to be convenient to label the D edges appearing in the following formulae by e, e 1 , . . . , e n , e ' 1 , . . . , e ' n . We consider the smeared quantity Then it is easy to see that the classical identity holds. The canonical brackets give rise to the operator representation while the connection acts by multiplication. Let a graph γ be given. In order to simplify the notation, we subdivide each edge e with endpoints v, v ' which are vertices of γ into two segments s, s ' where e = s · ( s ' ) -1 and s has an orientation such that it is outgoing at v ' . This introduces new vertices s ∩ s ' which we will call pseudo-vertices because they are not points of non-semianalyticity of the graph. Let E ( γ ) be the set of these segments of γ but V ( γ ) the set of true (as opposed to pseudo) vertices of γ . Let us now evaluate the action of on a function f = p ∗ γ f γ cylindrical with respect to γ . We find ( e : [0 , 1] → σ, t → e ( t ) being a parametrisation of the edge e ) Here we have used (1) the fact that a cylindrical function is already determined by its values on A / G rather than A / G so that it makes sense to take the functional derivative, (2) the definition of the holonomy as the path-ordered exponential of ∫ e A with the smallest parameter value to the left, (3) A = dx a A aIJ τ IJ where τ IJ ∈ so( D + 1) and we have defined (4) tr( h T ∂/∂g ) = h AB ∂/∂ AB , A,B,C,... being SO( D +1) indices. The state that appears on the right-hand side of (4.19) is actually well-defined, in the sense of functions of connections, only when A is smooth for otherwise the integral over t does not exist, see [33] for details. However, as announced, we will be interested only in quantities constructed from operators of the form (4.19) and for which the limit of shrinking ∆ → 0 to a point has a meaning in the sense of H = L 2 ( A / G , dµ 0 ) and therefore will not be concerned with the actual range of the operator (4.19) for the moment. We now wish to evaluate the whole operator ˆ π ( p, ∆ 1 , . . . , ∆ D ) on f . It is clear that we obtain D types of terms, the first type comes from all three functional derivatives acting on f only, the second type comes from D -1 functional derivatives acting on f and the remaining one acting on the trace appearing in (4.19), and so forth. The first term (type) is explicitly given by χ ∆ 1 ( p, x 1 ) χ ∆ 2 (2 p, x 1 + x 2 ) . . . χ ∆ D ( Dp,x 1 + . . . + x D ) ˙ e a ( t ) ˙ e a 1 1 ( t 1 ) . . . ˙ e a n n ( t n ) ˙ e ' 1 b 1 ( t ' 1 ) . . . ˙ e ' n b n ( t ' n ) The other terms are vanishing due to either the same symmetry / anti-symmetry properties as in the usual treatment or the simplicity constraint in case the first derivative is involved. Given a D -tuple e 1 . . . e D of (not necessarily distinct) edges of γ , consider the functions This function has the interesting property that the Jacobian is given by which is precisely the form of the factor which enters the integral (5.8). We now consider the limit ∆ 1 , . . . , ∆ D → 0. The idea is that all quantities in (5.8) are meaningful in the sense of functions on smooth connections and thus limits of functions as ∆ → 0 are to be understood with respect to any Sobolev topology. The miracle is that the final function is again cylindrical and thus the operator that results in the limit has an extension to all of A / G .", "pages": [ 17, 18 ] }, { "title": "4.2.2 D +1 Odd", "content": "The case D + 1 uneven works analogously, except that the expression for det( q ) is changed a bit. With n = D/ 2, the result is", "pages": [ 20 ] }, { "title": "4.2.3 More Results and Open Questions", "content": "The derivations of cylindrical consistency, symmetry, positivity, self-adjointness and anomalyfreeness given in [4] generalise immediately to the higher dimensional volume operator. The question of uniqueness of the prefactor [34, 35] in front of the expression under the square root of the volume operator or the computation of the matrix elements [36, 36, 37, 38, 39] have not been addressed so far, however these are not necessary steps in order to use the volume operator for a consistent quantisation of the Hamiltonian constraint in what follows. We leave these open questions for future research.", "pages": [ 20 ] }, { "title": "5.1 Introductory Remarks", "content": "The implementation of the Hamiltonian constraint will follow along the lines of [4], see [40] for original literature and details. In our companion papers [1, 2], we derived the classical expression where a, b, c, . . . are spatial indices and I, J, K, . . . are so( D +1) indices. In order to have a well defined quantum version of this constraint, we have to express it in terms of holonomy and flux variables. As in the 3 + 1-dimensional case, the volume operator turns out to be a cornerstone of the quantisation. At first, we will introduce a graph adapted triangulation of σ in order to regularise the Hamiltonian constraint. Next, classical identities to express the Hamiltonian constraint in terms of holonomies and fluxes are derived. Since the complete expression for the Hamiltonian constraint will turn out to be rather laborious to write down, we will derive the regularisation piece by piece. Next, we show how to assemble the regularised pieces to the complete constraint and describe the quantisation. Finally, we construct a Hamiltonian Master constraint in order to avoid some of the usual difficulties associated with quantisation.", "pages": [ 20 ] }, { "title": "5.2 Triangulation", "content": "A natural choice for a triangulation turns out to be the following (we simplify the presentation drastically, the details can be found in [40]): given a graph γ one constructs a triangulation T ( γ, glyph[epsilon1] ) of σ adapted to γ which satisfies the following basic requirements. Requirement (a) prevents the action of the Hamiltonian constraint operator from being trivial. Requirement (b) guarantees that the regulated operator ˆ H glyph[epsilon1] ( N ) is densely defined for each glyph[epsilon1] . Requirements (c), (d) and (e) specify the triangulation in the neighbourhood of each vertex of γ and leave it unspecified outside of them. The reason why those D -simplices lying outside the neighbourhoods of the vertices described above are irrelevant will rest crucially on the choice of ordering with [ ˆ h -1 s , ˆ V ] on the rightmost: if f is a cylindrical function over γ and s has support outside the neighbourhood of any vertex of γ , then V ( γ ∪ s ) -V ( γ ) consists of planar at most four-valent vertices only so that [ ˆ h -1 s , ˆ V ] f = 0. We will define our operator on functions cylindrical over coloured graphs, that is, we define it on spin network functions. The domain for the operator that we will choose is a finite linear combination of spin-network functions, hence this defines the operator uniquely as a linear operator. Any operator automatically becomes consistent if one defines it on a basis, the consistency condition simply drops out. The volume operator will appear in every term of the regulated Hamiltonian constraint. We will choose a factor ordering such that the Hamiltonian constraint acts only on vertices. It is therefore sufficient to regularise the constraint at vertices. As in the usual treatment, we use the tangents to the edges at a vertex as tangent vectors spanning the tangent space of the spatial coordinates. To emphasise this, we will abuse the notation in the following way: Let e a (∆) denote the D edges incident at the vertex v of an analytic D -simplex ∆ ∈ T ( γ, glyph[epsilon1] ). The matrix consisting of the tangents of the edges e 1 (∆) , . . . , e D (∆) at v (in that sequence) has non-negative determinant, which induces an orientation of ∆. Furthermore, let α ab be the arc on the boundary of ∆ connecting the endpoints of e a (∆), e b (∆) such that the loop α ab (∆) = e a (∆) · a ab (∆) · e b (∆) -1 has positive orientation in the induced orientation of the boundary for a < b (modulo cyclic permutation) and negative in the remaining cases.", "pages": [ 21, 22 ] }, { "title": "5.3 Key Classical Identities", "content": "The following classical identities are key for the rest of the discussion.", "pages": [ 22 ] }, { "title": "5.3.1 D +1 ≥ 3 Arbitrary", "content": "We observe that where V ( x, glyph[epsilon1] ) := ∫ d D y χ glyph[epsilon1] ( x, y ) √ q is the volume of the region defined by χ glyph[epsilon1] ( x, y ) = 1 measured by q ab and χ glyph[epsilon1] ( x, y ) = ∏ D a =1 Θ( glyph[epsilon1]/ 2 -| x a -y a | ) is the characteristic function of a cube of coordinate volume glyph[epsilon1] D with centre x . Also, We can write the KKEE terms in the same way as in the usual 3 + 1-dimensional case, using Further, gives us access to all the needed terms.", "pages": [ 22 ] }, { "title": "5.3.2 D +1 Even", "content": "Let n = ( D -1) / 2. It is easy to see that The sign of the determinant of e I a where the internal space is the subspace perpendicular to n I is accessible through For the Euclidean part of the Hamiltonian constraint, we need Regarding quantisation, we have to choose a classical expression for π [ a | IK π b ] J K √ q ( x ). The above expression would be favourable by arguments of simplicity if it would not contain the additional factor of sgn(det( e I a ))( x ) which has to be accounted for. Therefore, we can equally well express the two factors of π aIJ separately and absorb the inverse square root into volume operators.", "pages": [ 22, 23 ] }, { "title": "5.3.3 D +1 Odd", "content": "Let n = ( D -2) / 2. With only minor modifications of the D +1 even case, we get with For the Euclidean part of the Hamiltonian constraint, we need and observe that the factor of sgn(det( e I a ))( x ) is canceled by another such factor coming from n I . The Euclidean part of the Hamiltonian constraint therefore has the same amount of complexity, measured by the 'number of involved operators', in even and odd dimensions.", "pages": [ 23 ] }, { "title": "5.4 General Scheme", "content": "The basic idea of the regularisation of the Hamiltonian constraint operator is to approximate the constraint operator on the graph adapted triangulation and then to take the limit of an infinitely refined triangulation. For this procedure to work, it is mandatory that the constraint operator has a density weight of +1. A typical term of the classical Hamiltonian constraint (or any other operator one wants to regulate) will, after using the above classical identities, consist of", "pages": [ 23 ] }, { "title": "· (covariant) derivatives.", "content": "Operators that are well defined on the kinematical Hilbert space are holonomies and the volume operator. We will show in the following that we can construct the Euclidean part of the Hamiltonian constraint operator, which gives us access to the remaining part of the constraint operator. As a start, it is therefore mandatory to write the Euclidean part of the Hamiltonian constraint in terms of holonomies and volume operators. We stress that we do not quantise the π aIJ as flux operators, which would also be possible. The reason is that the Hamiltonian constraint operator would not simplify significantly by using fluxes instead of derived flux operators. On the other hand, the appearance of fluxes only through volume operators can be seen as a certain simplification. Anyhow, different regularisations are possible and the discrimination between different regularisations has to be considered in the semiclassical limit. We begin with rewriting the integral. Given a D -tuple of edges ( e 1 , . . . , e D ) incident at v with outgoing orientation consider the D -simplex ∆ glyph[epsilon1] ( γ, e 1 , . . . , e D ) bounded by the D segments s glyph[epsilon1] γ,v,e 1 , . . . , s glyph[epsilon1] γ,v,e D incident at v and the D ( D -1) / 2 arcs a glyph[epsilon1] γ,v,e a ,e b , 1 ≤ a < b ≤ D . We now define the 'mirror images' glyph[negationslash] where p = p ' ∈ e 1 , . . . , e D and we have chosen some parametrisation of segments and arcs. Using the data (5.12) we build 2 D -1 more 'virtual' D -simplices bounded by these quantities so that we obtain altogether 2 D D -simplices that saturate v and triangulate a neighbourhood U glyph[epsilon1] γ,v,e 1 ,...,e D of v . Let U glyph[epsilon1] γ,v be the union of these neighbourhoods as we vary the ordered D -tuple of edges of γ incident at v . The U glyph[epsilon1] γ,v , v ∈ V ( γ ) were chosen to be mutually disjoint in point (d) above. Let now then we may write any classical integral (symbolically) as where in the last step we have noticed that classically the integral over U glyph[epsilon1] γ,v,e 1 ,...,e D converges to 2 D times the integral over ∆ glyph[epsilon1] γ,v,e 1 ,...,e D , ≈ means approximately and E ( v ) = ( n ( v ) D ) with n ( v ) being the valence of the vertex. Now when triangulating the regions of the integrals over ¯ U glyph[epsilon1] γ,v,e 1 ,...,e D and ¯ U glyph[epsilon1] γ in (5.14), regularisation and quantisation gives operators that vanish on f γ because the corresponding regions do not contain a non-planar vertex of γ . As a next step, we approximate the integral for some function g ( x ). Here we assumed the coordinate length of each segment s glyph[epsilon1] γ,v,e a to be glyph[epsilon1] . The general case of arbitrary coordinate length works analogously, since the factors of glyph[epsilon1] will be hidden in holonomies and derivatives contracted with an epsilon symbol which addresses each segment exactly once. The factor 1 /D ! accounts for the volume of a D -simplex. We now multiply the nominator and the denominator by glyph[epsilon1] D ( n -1) . Together with the factors √ q 1 -n ( v ) and the factor glyph[epsilon1] D from the integral, we get glyph[epsilon1] Dn /V ( v, glyph[epsilon1] ) n -1 . The volumes in the denominator are absorbed into the Poisson brackets by the standard technique. The factors of A aIJ are turned into holonomies ( h s a ) KL = δ KL + glyph[epsilon1] ˙ e a (0) A aIJ ( τ IJ ) KL + O ( glyph[epsilon1] 2 ) using the the same amount of factors of glyph[epsilon1] since we note that the zeroth order of the expansion of the holonomies vanishes when inserted into the Poisson brackets. We abbreviated s a = s glyph[epsilon1] γ,v,e a to simplify notation. The field strength tensors can be dealt with as follows. Let e, e ' be arbitrary paths which are images of the interval [0 , 1] under the corresponding embeddings, which we also denote by e, e ' such that v = e (0) = e ' (0). For any 0 < glyph[epsilon1] < 1 set e glyph[epsilon1] ( t ) := e ( glyph[epsilon1]t ) for t ∈ [0 , 1] and likewise for e ' . Then we expand h e glyph[epsilon1] ( A ) in powers of glyph[epsilon1] . Consider the loop α e glyph[epsilon1] ,e ' glyph[epsilon1] where in a coordinate neighbourhood Now expanding again in powers of glyph[epsilon1] we easily find h α eglyph[epsilon1] ,e ' glyph[epsilon1] = 1 D +1 + glyph[epsilon1] 2 F abIJ τ IJ ˙ e a (0) ˙ e ' b (0)+ O ( glyph[epsilon1] 3 ). Since the indices of the field strength tensors are contracted only with other antisymmetric index pairs, the zeroth order of the expansion vanishes as well as the orders beyond glyph[epsilon1] 2 in the limit glyph[epsilon1] → 0. The remaining factors of glyph[epsilon1] are absorbed into covariant derivatives using the approximation We note that partial derivatives can be dealt with in the same way. At this point, all factors of glyph[epsilon1] have been absorbed into holonomies and derivatives. It is key that the volume operators are ordered to the right in the quantum theory since then, the Hamiltonian constraint evaluated on a cylindrical function f γ will only act on the vertices of γ . The action at vertices however does not depend on the value of glyph[epsilon1] > 0 and we can take the limit glyph[epsilon1] → 0, thus removing the regulator. In order to quantise the Hamiltonian constraint, we have to replace the holonomies by multiplication operators, the volumes by volume operators, and the Poisson brackets by i/ glyph[planckover2pi1] times the commutator.", "pages": [ 24, 25 ] }, { "title": "5.5 Regularised Quantities", "content": "In order to construct a well defined Hamiltonian constraint operator, we have to express it in terms of operators well defined on the kinematical Hilbert space. Instead of writing down the explicit regularisation for the proposed Hamiltonian constraint, we want to provide a toolkit for a general class of operators. In the following, we will propose 'regulated' versions of the phase space variables, marked by an upper glyph[epsilon1] in front. The idea will be to replace all phase space variables in the classical Hamiltonian constraint by their corresponding regulated versions, do some additional minor modifications and directly arrive at the Hamiltonian constraint operator, without explicitly dealing with the triangulation and the correct powers of glyph[epsilon1] . Since the final constraint operator will only act on vertices of γ , it is sufficient to regularise the phase space variables at vertices v . In what follows, we use a graph adapted coordinate system, meaning that the spatial coordinates a, b, . . . = 1 , . . . , D enumerate the D edges incident at v of a D -simplex.", "pages": [ 25, 26 ] }, { "title": "5.5.1 D +1 ≥ 3 Arbitrary", "content": "We will express all the basic variables in terms of holonomies living on the edges of the adapted triangulation and volume operators acting on it. First, we notice that is gauge covariant and reduces to glyph[epsilon1] √ q x +1 π aIJ ( v ) in the limit glyph[epsilon1] → 0. The factor of glyph[epsilon1] is expected as the regulated quantity has a lower spatial index. In the end, when the complete constraint operator will be assembled, all factors of glyph[epsilon1] will cancel out. We restrict x > -1 because powers of the volume operator will be defined by the spectral theorem in the quantum theory. For the KKEE terms, we propose where the glyph[epsilon1] π aIJ will be defined below. Next, we regulate the gauge unfixing term DF -1 D with density weight 1. We will place zero density into F -1 and a density weight of 1 / 2 into each D . Accordingly, becomes with The D constraint contains a covariant derivative which we regularise as The full D constraint can thus be regularised as A different regularisation procedure for the DF -1 D part of the Hamiltonian constraint which is based on field strength tensors is outlined in appendix A. √ In general, a generic power of 1 / q needed to turn the individual terms with densities > 1 into densities of weight 1 can be constructed as with the usual x > -1. The field strength tensors are regularised as while we set", "pages": [ 26, 27 ] }, { "title": "5.5.2 D +1 Even", "content": "Let n = ( D -1) / 2. We 'regulate' and For the Euclidean part of the Hamiltonian constraint, we need As stressed before, the two possibilities to express the Euclidean part of the Hamiltonian constraint are equally complicated.", "pages": [ 27 ] }, { "title": "5.5.3 D +1 Odd", "content": "Let n = ( D -2) / 2. We 'regulate' and For the Euclidean part of the Hamiltonian constraint, we need", "pages": [ 28 ] }, { "title": "5.6 The Hamiltonian Constraint Operator", "content": "At this point, we are ready to assemble the Hamiltonian constraint operator. The general idea of the regularisation has been described in section 5.4. Here, we provide a toolkit in order to assemble the constraint operator. In order to understand the double sum over D -simplices appearing in the KKEE and the gauge unfixing term, consider the following argument given in a similar form in [22]: Since lim glyph[epsilon1] → 0 (1 /glyph[epsilon1] D ) χ glyph[epsilon1] ( x, y ) = δ D ( x, y ) we have lim glyph[epsilon1] → 0 (1 /glyph[epsilon1] D ) V ( x, glyph[epsilon1] ) = √ q ( x ). It is also easy to see that for each glyph[epsilon1] > 0 we have that δV/δπ aIJ ( x ) = δV ( x, glyph[epsilon1] ) /δπ aIJ ( x ). The terms under consideration are of the form where Z aIJ is a density of weight +1 and stands symbolically for the remaining terms, including a spatial glyph[epsilon1] -symbol with upper indices, one of which is a . We rewrite this expression as Triangulation leads to two sums over vertices and two sums over D -simplices containing the individual vertices. In the limit glyph[epsilon1] → 0 however the two sums over vertices collapse to a single sum over vertices due to the χ glyph[epsilon1] term and we have the desired result.", "pages": [ 28, 29 ] }, { "title": "5.7 Solution of the Hamiltonian Constraint", "content": "As in the 3 + 1-dimensional treatment, we realise that the only spin changing operation of the Hamiltonian constraint is performed by its Euclidean part. The construction of a set of rigorously defined solutions to the diffeomorphism and the Hamiltonian constraint described in [41] thus immediately generalises to our case.", "pages": [ 29 ] }, { "title": "5.8 Master Constraint", "content": "The implementation of the Master constraint works analogously to the 3 + 1-dimensional case described in [42]. The inverse square root is split up between the two Hamiltonian constraints and hidden by adjusting the power of the volume operators as before. The result of the derivation is the Master constraint operator with and l ( ˆ C † v T s 0 ([ s ]) ) being the evaluation of l on the Hamiltonian constraint operator with the additional 1 / 4 √ q hidden in the volume operator(s). The proof of the following theorem generalises with obvious modifications from the treatment in [4].", "pages": [ 29 ] }, { "title": "Theorem.", "content": "There is a unitary operator V such that V H θ diff is the direct integral Hilbert space where the measure class of µ and the Hilbert space H θ diff ( λ ) , in which V ˆ M V -1 acts by multiplication by λ , are uniquely determined. The physical Hilbert space is given by H θ phys = H θ diff (0) . We notice that we could define an extended Master Constraint that also involves the simplicity constraint.", "pages": [ 30 ] }, { "title": "5.9 Factor Ordering", "content": "In [34, 35], it has been shown that there is a unique factor ordering which results in a nonvanishing flux operator expressed through the volume operator and holonomies in the usual 3 + 1 dimensional LQG. The idea, translated to our case, is that the volume operator in the expression for glyph[epsilon1] π aIJ has to act on an at least D -valent non-planar vertex and the holonomies in the expression have to be ordered to the right for this to be ensured. Apart from ordering individual terms of the sums appearing differently (which would be highly unnatural), this leaves only one possible factor ordering. We remark that the proof of the equivalence of the 'normal' and 'derived' flux operator given in [34, 35] does not generalise trivially to our case since it is explicitly based on SU(2) as the internal gauge group. We leave this point open for further research. In order to ensure that the Hamiltonian constraint only acts on vertices, we order in all three terms either a commutator [ ˆ h -1 e , ˆ V ] or a double-commutator [ ˆ h -1 e , [ H E , ˆ V ] to the right. We leave the remaining details of the factor ordering open, as this paper only intends to show that a quantisation is possible in principle.", "pages": [ 30 ] }, { "title": "5.10 Outlook on Consistency Checks", "content": "At this point, one might ask if there are good indications whether the proposed theory is physically viable. In case of the usual formulation of LQG in terms of Ashtekar-Barbero variables, it was shown in [43] that a quantisation of Euclidean General Relativity in three dimensions with methods very similar to the ones used in LQG recovers the known solutions of threedimensional General Relativity familiar from other approaches. The reason why these theories match is that they both use the gauge groups SU(2) and that a suitable redefinition of the Lagrange multipliers of Euclidean three-dimensional General Relativity leads to a Hamiltonian constraint with the same algebraic structure as the Euclidean part of the constraint familiar from LQG. A similar check is conceivable for the presented theory in that we can describe Lorentzian three-dimensional General Relativity using SU(2) as a gauge group, which would result in a different Hamiltonian constraint. One could now check if the solution space of Lorentzian threedimensional General Relativity is reproduced when using SU(2) as a gauge group and thus mimicking the internal signature switch which is also done in this formulation. As for the simplicity constraint, we cannot use three-dimensional General Relativity as a testbed since the simplicity constraints only appear in four and higher dimensions. In this paper, two different regularisations for the gauge unfixing part of the Hamiltonian constraint were introduced, one in section 5.5 and one in appendix A. While the regularisation introduced in section 5.5 preserves the closure of the quantum constraint algebra, this is not obvious for the regularisation in appendix A since terms quadratic in the field strength appear. Another approach to consistency checks is to compare our formulation in four dimensions to the usual LQG formulation. In section 4.1, the area operator was shown to have the same spectrum as in standard LQG, which however does not come as a surprise regarding similar results from spin foam models. As for the volume operator, we do not know whether the spectrum matches the one of standard LQG. This is also tied to the fact that we are only interested in the spectrum on the solution space to the vertex simplicity constraint operators, for which we do not have a completely satisfactory proposal. We remark that a matching spectrum of the volume operator can be obtained by using a weak implementation of the linear vertex simplicity constraints [44]. However, as explained in our companion paper [25], this approach comes with its own problems in the canonical theory.", "pages": [ 30, 31 ] }, { "title": "6 Conclusion", "content": "In this paper we have demonstrated that by a straightforward adaption of the toolbox developed for LQG in 3 + 1 dimensions also the constraints of our new connection formulation of General Relativity in any dimension D +1 ≥ 3 can be quantised analogously and rigorously. The higher dimension does not require much more complexity than in 3 + 1 dimensions. We conclude that our new connection formulation has a consistent quantisation. The next task is to study matter coupling, in particular coupling to supersymmetric matter in interesting dimensions, where String theories and Supergravity theories are defined, and the quantisation thereof. This has to be done, as in 3 + 1 dimensions, in a background independent way, a task to which we turn in the next papers of this series [45, 46, 47]. In four dimensions, we now have the special situation that there are two formulations of LQG, one based on the usual Asthekar-Barbero variables, and one based on the variables proposed in this series of papers. From a direct comparison, one concludes that the new formulation is more complicated since the Hamiltonian constraint contains an additional term resulting from gauge unfixing. Two different regularisations for this term were introduced, the first one directly regularises the covariant derivatives in this term, the second one uses a Poisson bracket identity involving the Field strength and the whole expression is thus quadratic in the field strength. Both of these regularisations do no appear in the standard case and the Hamiltonian constraint operator is thus more complicated. On the other hand, since it is already hard to deal with the usual Hamiltonian constraint, we cannot conclude that our Hamiltonian constraint is significantly more complicated. The main problem remains the simplicity constraint for which a satisfactory implementation has to be found which is compatible with the action of the Hamiltonian constraint and allows for a unitary map to the Ashtekar-Lewandowski Hilbert space.", "pages": [ 31 ] }, { "title": "Acknowledgements", "content": "NB and AT thank Emanuele Alesci, Jonathan Engle, Alexander Stottmeister, and Antonia Zipfel for numerous discussions. NB and AT thank the Max Weber-Programm, the German National Merit Foundation, and the Leonardo-Kolleg of the FAU Erlangen-Nurnbeg for financial support. NB further acknowledges financial support by the Friedrich Naumann Foundation. The part of the research performed at the Perimeter Institute for Theoretical Physics was supported in part by funds from the Government of Canada through NSERC and from the Province of Ontario through MEDT. During final improvements of this work, NB was supported by the NSF grant PHY-1205388 and the Eberly research funds of The Pennsylvania State University.", "pages": [ 32 ] }, { "title": "A Alternative Regularisation of DF -1 D", "content": "It was suggested by Wieland [48] that one could simplify the D constraints by using the classical identity i.e. the torsion of the gravitational connection can be expressed using a Poisson bracket which will become a commutator in the quantum theory. Since the D constraints appear quadratically in ˜ H , this type of regularisation results in a more non-local operation of the Hamiltonian constraint. In order to apply the above identity, we recall from [2] that we can extend the covariant derivative in by a Christoffel symbol acting on spatial indices on the constraint surface. Therefore, and we calculate with In order to have direct access to ¯ K trace free aIJ , we can invert F ' as and write with We can also implement the above Poisson bracket identity without starting from the original D constraints but by trying to find an easier expression for ˜ H-H directly from D [ a π b ] IJ . It turns out that The obvious question at this point is which of the two expressions is suited better for a quantisation. Although a satisfactory answer might only be possible after studying the quantum dynamics, we see at the classical level that the second expression has a less complicated index structure due to the missing epsilon symbols. On the other hand, it contains correction terms proportional to ¯ K tr I , which are absent due to the epsilon symbols in the first expression. In the formulation studied in this paper, this does not affect the theory since ¯ K tr I ≈ 0 on the constraint surface [2]. In general, this won't be true any more when coupling fermions [45] or using the time normal n I as a independent field [46] in other papers of this series. Although introducing additional correction terms, an independent time normal would simplify the expression since the action of a multiplication operator corresponding to n I is simpler than the regularised version of n I n J ( π ).", "pages": [ 32, 33 ] } ]

[ { "title": "Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector", "content": "N. Bodendorfer 1 ∗ , T. Thiemann 1 , 2 † , A. Thurn 1 ‡ 1 Inst. for Theoretical Physics III, FAU Erlangen - Nurnberg, Staudtstr. 7, 91058 Erlangen, Germany and 2 Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON N2L 2Y5, Canada October 30, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "In our companion paper, we focussed on the quantisation of the Rarita-Schwinger sector of Supergravity theories in various dimensions by using an extension of Loop Quantum Gravity to all spacetime dimensions. In this paper, we extend this analysis by considering the quantisation of additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton and gravitino in various dimensions. As a generic example, we study concretely the quantisation of the 3 -index photon of minimal 11 d SUGRA, but our methods easily extend to more general p -form fields. Due to the presence of a Chern-Simons term for the 3 -index photon, which is due to local SUSY, the theory is self-interacting and its quantisation far from straightforward. Nevertheless, we show that a reduced phase space quantisation with respect to the 3 -index photon Gauß constraint is possible. Specifically, the Weyl algebra of observables, which deviates from the usual CCR Weyl algebras by an interesting twist contribution proportional to the level of the Chern-Simons theory, admits a background independent state of the Narnhofer-Thirring type.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In our companion papers [1, 2, 3, 4, 5], we studied the canonical formulation of General Relativity (gravitons) coupled to standard matter in terms of connection variables for a compact gauge group without second class constraints in order that Loop Quantum Gravity (LQG) quantisation methods, so far formulated only in three and four spacetime dimensions [6, 7], apply. The actual motivation for doing this comes from Supergravity and String theory [8, 9]: String theory is considered as a candidate for a UV completion of General Relativity, which in its present formulation requires extra dimensions and supersymmetry. Supergravity is considered as the low energy effective field theory limit of String theory. One may therefore call String theory a top - bottom approach. In this series of papers we take first steps towards a bottom - top approach in that we try to canonically quantise the Supergravity theories by LQG methods. While String theory in its present form needs a background dependent and perturbative quantum formulation, the LQG quantum formulation is by design background independent and non perturbative. On the other hand, quantum String theory is much richer above the low energy field theory limit, containing an infinite tower of higher excitation modes of the string, which come into play only when approaching the Planck scale and which are necessary in order to find a theory which is finite at least order by order in perturbation theory. The quantisation of Supergravity is therefore the ideal arena in which to compare these two complementary approaches to quantum gravity, which was not possible so far. At least at low energies, that is, in the semiclassical limit, the two theories should agree with each other, as otherwise they would quantise two different classical theories. Evidently, this opens the very exciting possibility of cross fertilisation between the two approaches, which we are going to address in future publications. The new field content of Supergravity theories as compared to standard matter Lagrangians are 1. Majorana (or Majorana-Weyl) spinor fields of spin 1 / 2 , 3 / 2 including the Rarita-Schwinger field (gravitino) and 2. additional bosonic fields that appear in order to obtain a complete supersymmetry multiplet in the dimension and the amount N of supersymmetry charges under consideration. The treatment of the Rarita-Schwinger sector and its embedding in the framework of [1, 2, 3, 4, 5] was accomplished in [10]. In this paper, we complete the quantisation of the extra matter content of many Supergravity theories by considering the quantisation of the additional bosonic fields, in particular, p -form fields. Specifically, for reasons of concreteness, we quantise the 3-index photon of 11 d Supergravity but it will transpire that the methods employed generalise to arbitrary p . What makes the quantisation possible is that the Gauß constraints of the 3-index photon form an Abelian ideal in the constraint algebra. If this ideal (or subalgebra) would be non - Abelian, then our methods would be insufficient and we most probably would have to use methods from higher gauge theory [11, 12, 13, 14, 15] such as p -groups, p -holonomies etc., a subject which at the moment is not yet sufficiently developed from the mathematical perspective (see [16] for the state of the art of the subject). Despite the Abelian character of this additional Gauß constraint, the quantisation of the theory is not straightforward and cannot be performed in complete analogy to the treatment of the Abelian Gauß constraint of standard 1-form matter [17]. This is due to a Chern-Simons term in the Supergravity action, whose presence is dictated by supersymmetry and which makes the theory in fact self-interacting, that is, the Hamiltonian is a fourth order polynomial in the 3-connection and its conjugate momentum just like in Yang-Mills theory. In particular, while one can define a holonomy flux algebra as for Abelian Maxwelltheory, the Ashtekar-Isham-Lewandowski representation [18, 19] is inadequate because the Abelian gauge group does not preserve the holonomy flux algebra. A solution to the problem lies in performing a reduced phase space quantisation in terms of a twisted holonomy flux algebra, which is in fact Gauß invariant. We were not able to find a background independent representation of the corresponding Heisenberg algebra, which also differs by a twist from the usual one, however, one succeeds when formulating the quantum theory in terms of the corresponding Weyl elements. The resulting Weyl algebra is not of standard form and to the best of our knowledge it has not been quantised before. We show that it admits a state of the Narnhofer-Thirring type [20] whence the Hilbert space representation follows by the GNS construction. The Hamiltonian (constraint) can be straightforwardly expressed in terms of the Weyl elements, in fact it is quadratic in terms of the classical observables, that is, the generators of the Heisenberg algebra. This paper's architecture is as follows: In section 2, we sketch the Hamiltonian analysis of the 3-index photon in a self-contained fashion for the benefit of the reader and in order to settle our notation. We also describe in detail why one cannot straightforwardly apply methods from LQG as mentioned above. In section 3, we display the reduced phase space quantisation solution in terms of the twisted holonomy flux algebra. Finally, in section 4, we summarise and conclude.", "pages": [ 2, 3 ] }, { "title": "2 Classical Hamiltonian Analysis of the 3-Index-Photon Action", "content": "The Hamiltonian analysis of the full 11 d SUGRA Lagrangian has been performed in [21]. We will review the analysis of the contribution of the 3-index-photon 3-form A µνρ = A [ µνρ ] to the 11 d SUGRA Lagrangian with Chern-Simons term. This part of the Lagrangian is given up to a numerical constant by Here, F = dA, F µ 1 ..µ 4 = ∂ [ µ 1 A µ 2 ..µ 4 ] is the curvature of the 3-index-photon and indices are moved with the spacetime metric g µν . Furthermore, J is a totally skew tensor current bilinear in the graviton field not containing derivatives, whose explicit form does not need to concern us here, except that it does not depend on any other fields. Finally, c, α are positive numerical constants whose value is fixed by the requirement of local supersymmetry [22]. The number c could be called the level of the Chern-Simons theory in analogy to d = D +1 = 3. We proceed to the 10 + 1 split of this Lagrangian in a coordinate system with coordinates t, x a ; a = 1 , .., 10 adapted to a foliation of the spacetime manifold. The result of a tedious calculation is given by where we used /epsilon1 a 1 ..a D = /epsilon1 ta 1 ..a D and defined j a 1 ..a 3 := J ta 1 ..a 3 . The potential terms V 1 , V 2 only depend on the spatial components of the curvature and do not contain time derivatives. Using we may perform the Legendre transform. The momentum conjugate to A reads We may solve (2.4) for F ta 1 a 2 a 3 where defines the inverse of G . Inverting (2.3) for ˙ A and using (2.4) and (2.2) we obtain for the Hamiltonian after a longer calculation where an integration by parts has been performed in order to isolate the Lagrange multiplier A ta 1 a 2 . Using the ADM frame metric components with q ab the induced metric on the spatial slices and lapse respectively shift functions N,N a we can easily decompose the piece of H independent of the 3-index Gauß constraint G a 1 a 2 C into the contributions N a H Ca + N H C to the spatial diffeomorphism constraint and Hamiltonian constraint, however, we will not need this at this point. We will drop the subscript C in what follows, since in this paper we are only interested in the p -form sector. We smear the Gauß constraint with a 2-form Λ, that is and study the gauge transformation behaviour of the canonical pair ( A abc , π abc ) with non-vanishing Poisson brackets We find These equations can be written more compactly in differential form language, in terms of which they are easier to memorise. Introducing the dual 7-pseudo-form 1 we may write (2.11) as Since the right hand side of (2.13) is closed, in fact exact, it would seem that the observables of the theory can be coordinatised by integrals of A and ∗ π respectively over closed 3-submanifolds or 7-submanifolds respectively. The G (Λ) generate an Abelian ideal in the constraint algebra since where H ( x ) is the integrand of H in (2.7) and since the only π or A dependent contributions to the Hamiltonian and spatial diffeomorphism constraints are contained in H ( x ). We see that due to the non vanishing Chern-Simons constant c , the transformation behaviour of ∗ π differs from the transformation behaviour with respect to the higher dimensional analog of the usual Maxwell type of Gauß law, which would be just the divergence term ∂ a 1 π a 1 ..a 3 . In particular, π abc itself is not gauge invariant. This 'twisted' Gauß constraint (2.7) can be written in the form which suggests to introduce a new momentum π ' . Unfortunately, this does not work because ∗ ( π ' -π ) = A ∧ F does not have a generating functional K with δK/δA = A ∧ F , since the only possible candidate K = ∫ A ∧ A ∧ F ≡ 0 identically vanishes in the dimensions considered here. Since this is not the case, the Poisson brackets of π ' with itself do not vanish and neither is π ' gauge invariant as we will see below, so that there is no advantage of working with π ' as compared to π . The presence of the twist term in the Gauß constraint leads to the following difficulty when trying to quantise the theory on the usual LQG type kinematical Hilbert space: Such a Hilbert space would roughly be generated by a holonomy flux algebra constructed from holonomies where e and S are oriented 3-dimensional and 7-dimensional submanifolds respectively, which we call 'edges' and surfaces in what follows. One could then study the GNS Hilbert space representation generated by the LQG type of positive linear functional where µ is an LQG type measure on a space of generalised connections A . One can define it abstractly by requiring that the charge network functions form an orthonormal basis in the corresponding H = L 2 ( A , µ ), see [7] for details. Here, a graph γ is a collection of edges which are disjoint up to intersections in 'vertices', which are oriented 2-manifolds. The possible intersection structure of these cobordisms should be tamed by requiring that all submanifolds are semi-analytic. Up to here everything is in full analogy with LQG. The problem is now to isolate the Gauß invariant subspace of the Hilbert space: While the connection transforms as in a theory with untwisted Gauß constraint, it appears that we can solve it by requiring that charges add up to zero at vertices. However, this does not work because while such a vector is annihilated by the divergence term in G ab , it is not by the second term ∝ A ∧ F . Even more disastrous, the term A ∧ F does not exist in this representation which is strongly discontinuous in the holonomies so that operators A,F do not exist. Finally, although π is not Gauß invariant, it leaves this would be gauge invariant subspace invariant, which reveals that this subspace is not the kernel of the twisted Gauß constraint. We therefore must be more sophisticated. Since the A dependent terms in G cannot be quantised on the kinematical Hilbert space, we must exponentiate it: Consider the Hamiltonian flow of G [Λ] which is a Poisson automorphism α Λ (canonical transformation) and one would like to secure that an implementation of the corresponding automorphism group α Λ · α Λ ' = α Λ+Λ ' by unitary operators U (Λ) exists. The U (Λ) would correspond to the desired exponentiation of the Gauß constraint. One way of securing this is by looking for an invariant state ω = ω · α Λ on the holonomy - flux algebra (see [23]) for the details for this construction). This would then open the possibility that the Gauß constraint can be solved by group averaging methods. The first problem is that the automorphisms do not preserve the holonomy flux algebra because there appears an F on the right hand side of (2.19) which should appear exponentiated in order that the algebra closes. This forces us to pass to exponentiated fluxes, that is, to the corresponding Weyl algebra defined by exponentials of π, A . This algebra is now preserved by the automorphisms, as one can see by an appeal to the Baker-Campbell-Hausdorff formula. However, we now see that the state (2.17) is not invariant, because for suitable choices of Λ. In the GNS Hilbert space we would like to have unitary operators U (Λ) such that for any element W in the Weyl algebra we have U (Λ) π ( W ) U (Λ) ∗ = π ( α Λ ( W )). Then (2.20) is compatible with unitarity only if the LQG vacuum Ω is not invariant under U (Λ). Now the operator U (Λ) should correspond to exp( iG [Λ]) and using a calculation similar to (2.14) and the BCH formula one shows that on the LQG vacuum Ω = 1 it reduces formally to U (Λ)Ω = exp( ic/ 2 ∫ Λ ∧ F ∧ F )Ω which is ill defined as it stands. We must therefore define U (Λ)Ω to be some state in the GNS Hilbert space which has a component orthogonal to the vacuum and such that the representation property U (Λ) U (Λ ' ) = U (Λ + Λ ' ) , U (Λ) ∗ = U ( -Λ) (possibly up to a projective twist) holds. We did not succeed to find a solution to this problem indicating that a unitary implementation of the Gauss constraint is impossible in the LQG representation and even it were possible, the strategy outlined in the next section is certainly more natural. We also remark that solving the constraint by group averaging methods becomes non trivial if not impossible in case of the non existence of U (Λ). Even if we could somehow construct the Gauß invariant Hilbert space, the observables A ( e ) , exp( iπ ( S )) with ∂e = ∂S = ∅ , which leave the physical Hilbert space invariant, are insufficient to approximate (for small e, S ) the π dependent terms appearing in the Hamiltonian (2.7), as one can check explicitly.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Reduced Phase Space Quantisation", "content": "In the previous section, we established that a quantisation in strict analogy to the procedure followed in LQG does not work. While a rigorous kinematical Hilbert space can be constructed, the Dirac operator constraint method of looking for the kernel of the Gauß constraint is problematic. As an alternative, a reduced phase space quantisation suggests itself. This has a chance to work due to the observation (2.14) which demonstrates that H ( x ) only depends on observables. Indeed, H ( x ) depends, except for G ab which is a trivial observable since it is constrained to vanish, only on the combination π + B + αj . Obviously j trivially Poisson commutes with G . Unpacking B from (2.5), we see that π + B is a linear combination (with only metric dependent coefficients) of F and which suggests that { G (Λ) , P abc ( x ) } = 0 because F is already invariant. This indeed can be verified using (2.13) Our classical observables therefore are coordinatised by the 4-form and 7-form F = dA and ∗ P = ∗ π + cA ∧ F respectively. Since F is exact, it is determined entirely by a 3-form modulo an exact form, which in turn is parametrised by a 2-form. This 2-form worth of gauge freedom matches the number of Gauß constraints which can be read as a condition on π . Thus, on the constraint surface, the number of degrees of freedom contained in F and P match. We compute the observable algebra. Let f be a 3-form and h a 6-form with dual ∗ h (a totally skew 4-times contravariant tensor pseudo density) and smear the observables with these Then, we find after a short computation Thus, the observable algebra closes but P is not conjugate to F . The form of the observable algebra (3.4) reveals the following: Typically, background independent representations tend to be discontinuous in at least one of the configuration or the momentum variable. For instance, in LQG electric fluxes exist in non exponentiated form, but connections do not. Let us assume that we find such a representation in which F [ h ] does not exist so that we have to consider instead its exponential (Weyl element). Then (3.4) tells us that in such a representation automatically also P [ f ] cannot be defined, because if it could, then its commutator would exist, which however is proportional to some F which is a contradiction. Hence, either both F, P exist or only both of their corresponding Weyl elements. We did not manage to find a representation in which the Weyl elements are strongly continuous operators in both f, h . However, we did find one in which they are discontinuous in both h, f . This representation was studied in the context of QED in [20] and was applied to an LQG type of quantisation of the closed bosonic string in [24]. Before we define it, we must first define the Weyl algebra generated by the Weyl elements (3.5). The ∗ -relations are obvious, However, the product relations are very interesting and non trivial, because they require the generalisation of the Baker-Campbell-Hausdorff formula [25, 26, 27, 28, 29, 30] to higher commutators [31]. Suppose that X,Y are operators on some Hilbert space such that the triple commutators [ X, [ X,Y ]] and [ Y, [ Y, X ]] commute with both X and Y . This formally applies to our case with X = F [ h ]+ P [ f ] , Y = F [ h ' ]+ P [ f ' ], which obey the canonical commutation relations (we set /planckover2pi1 = 1 for simplicity) From this follows for the triple commutators which thus are in the centre of the algebra. The BCH formula for the case of all triple commutators commuting with X,Y reads which can also be proved using elementary methods. From this it is easy to derive the also useful Zassenhaus formula [31] Putting all these together, we obtain the Weyl relations Hence also the Weyl relations get twisted as compared to the situation with c = 0. Notice that the first term in the phase is antisymmetric under the exchange ( h, f ) ↔ ( h ' , f ' ), while the second is symmetric. In order to obtain a representation of this ∗ -algebra A generated by the Weyl elements, it is sufficient to find a positive linear functional. We consider the Narnhofer-Thirring type of functional and show that it is positive definite on A . Let be a general element in A , where N ∈ N , c k ∈ C and the z k = ( h k , f k ) are arbitrary, where without loss of generality z k = z l for k = l . We have /negationslash /negationslash /negationslash /negationslash For k = l , we have z kl = α kl = 0 because f k , f l are 3-forms. For k = l , we must have either f k = f l or h k = h l or both. If f k = f l , then obviously z kl = 0. If f k = f l , then necessarily h k = h l and z kl = ( -h k + h l , 0) = 0. By definition (3.12) then /negationslash /negationslash /negationslash /negationslash /negationslash is positive definite. Thus, the left ideal I = { a ∈ A ; ω ( a ∗ a ) = 0 } = { 0 } is trivial and the Hilbert space representation is given by the GNS data [23]: The cyclic vector is Ω = 1 , the Hilbert space H is the Cauchy completion of A in the scalar product < a, b > := ω ( a ∗ b ) and the representation is simply π ( a ) b := ab on the common dense domain D = A . The representation is evidently strongly discontinuous in both h, f and while cyclic, it is not irreducible. Equivalently, ω is not a pure state [32, 33]. The question left open to answer is whether the algebra and the state ω are still well defined when restricting the smearing functions ( h, f ) to the form factors of 4-surfaces and 7-surfaces respectively. The bearing of this question is that in the Hamiltonian constraint the functions F and ∗ P appear in such a way, that in a discretisation of it, which results from replacing the integral by Riemann sums in the spirit of [34], these functions are naturally smeared over 4-surfaces and 7-surfaces respectively. They could thus be approximated by Weyl elements. To answer this question, let S 4 , S 7 be general 4 and 7 surfaces respectively. Consider the distributional forms ('form factors') Then Thus, the natural integrals of F, P over surfaces can be reexpressed in terms of distributional 6 forms and 4-forms respectively. It remains to check whether the exterior derivative and product combinations of these distributional forms appearing in the multiple Poisson brackets of (3.17) and in the Weyl relations remain meaningful. Three types of exterior derivative and product expressions appear. The first is, using formally Stokes theorem The integral is supported on ∂S 4 ∩ S 7 and we can decompose this set into components (submanifolds) which are 0,1,2,3-dimensional. The number of these components will be finite if the surfaces are semianalytic. We define the intersection number σ ( ∂S 4 , S 7 ) to be zero for the 1,2,3-dimensional components and by (3.18) for the isolated intersection points, which then takes the values ± 1. This can be justified by the same regularisation as in LQG for the holonomy flux algebra [7]. The second type of integral is given by F [ f S 7 ∧ f S ' 7 ]. The support of the integral will be on S S 7 ∩ S S ' 7 and in D = 10 dimensions this will decompose into components that are at least 4-dimensional. By the same regularisation as in [7], one can remove the higher dimensional components and thus keep only the 4-dimensional ones. In what follows, we thus assume that S 4 := S 7 ∩ S ' 7 is a single 4-dimensional component, otherwise the non vanishing contributions are over a sum of those. We have By assumption, we have embeddings with open subsets U, V of R 7 and an open subset W of R 4 respectively, whose coordinates will be denoted by u, v, w respectively. The condition X S 7 ( u ) = Y S ' 7 ( v ) = Z S 4 ( w ) is solved by solving u, v for w , which leads to u = u ( w ) , v = v ( w ). Since the integrals are reparametrisation invariant, in the neighbourhood of S 4 on both S 7 and S ' 7 therefore we may use adapted coordinates so that w I = u I = v I , I = 1 , .., 4 on S 4 and u I , v I , I = 5 , .., 7 denote the transversal coordinates, which take the value 0 on S 4 . In this parametrisation both U, V are of the form U = W × U ' , V = W × V ' for some 3-dimensional subsets U ' , V ' of R 3 . It follows Z ( w ) = X ( w, 0) = Y ( w, 0) in this parametrisation. The δ distribution is then supported on u I = v I , I = 1 , .., 4 and u I = v I = 0 , I = 5 , .., 7 and we have in the neighbourhood of S 4 We can now solve the δ distribution in (3.19) by performing the integral over u 5 , .., u 7 , v 1 , .., v 7 and find with the notation X a I = ∂X a S 7 ( u ) /∂u I and Y a I = ∂Y a S ' 7 ( v ) /∂v I etc. where the 10 d antisymmetric symbol is in terms of the coordinates u 5 , .., u 7 , v 1 , .., v 7 and in the last step we noticed that the range of I 1 ..I 4 is restricted to 1 .. 4. Also, we assumed that the sign function under the integral is constant and equal to ˜ σ ( S 7 , S ' 7 ) on S 4 (which defines this function), otherwise we must decompose S 4 further. Under this assumption, we conclude the form factor identity Finally, we consider the integral of the third type, which now combining (3.18) and (3.24) is easily calculated because ∂ ( S 7 ∩ S ' 7 ) ⊂ S 7 for which σ vanishes by definition. In order to make this restricted Weyl algebra close, we now have to decide whether the form factors should only be added with integer valued coefficients [17] or with real valued ones [35, 36, 37]. In the latter case we do not need to do anything and the restricted Weyl algebra already closes. In the former case we must replace the form factors f S 7 by 1 √ 3! 7! 3 c/ 2 f S 7 , such that in the simplest situation we have from which the general case can be easily deduced. We conclude that the restricted Weyl algebra is well defined in either case. Thus, wherever P or F appear in the Hamiltonian constraint, we follow the general regularisation procedure outlined in [34], which employs a combination of spatial diffeomorphism invariance and an infinite refinement limit of a Riemann sum approximation of the Hamiltonian constraint in terms of P [ S 7 ] and F [ S 4 ] = A [ ∂S 4 ], which we approximate for instance by sin( P [ S 7 ]) , sin( F [ S 4 ]) similar as in LQG. The details are obvious and are left to the interested reader.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4 Conclusions", "content": "/negationslash Supergravity theories typically need additional bosonic fields next to the graviton, in order to obtain a SUSY multiplet (representation) containing the gravitino. In this paper, we focussed on minimal 11 d SUGRA for reasons of concreteness (and its relevance for lower dimensional SUGRA theories), which contains the 3-index photon in the bosonic sector. However, our analysis is easily generalised to arbitrary p -form fields. Without the Chern-Simons term in the action (i.e. c = 0) the analysis would be straightforward and in complete analogy to the background independent treatment of Maxwell theory in D +1 = 4 dimensions [17]. In particular, the Hamiltonian constraint would be quadratic in the 3-form field and its conjugate momentum, which thus would reduce to a free field theory when switching off gravity. However, with the Chern-Simons term ( c = 0) the Hamiltonian constraint becomes in fact quartic in the connection and thus becomes self-interacting even when switching off gravity, just like in non Abelian Yang-Mills theories. It is therefore the more astonishing that we can quantise the resulting ∗ -algebra of observables (with respect to the 3-index-Gauß constraint) rigorously, even though the theory is self-interacting. In fact, in terms of the observables, the Hamiltonian constraint is a quadratic polynomial, however, the price to pay is that the observable algebra is non standard. Yet, the resulting Weyl algebra can be computed in closed form and we found at least one non trivial and background independent representation thereof, which nicely fits into the background independent quantisation of the gravitational degrees of freedom in the contribution to the Hamiltonian constraint depending on the 3-index-photon. There are many open questions arising from the present study. One of them concerns the reducibility of the GNS representation found, which involves a mixed state. It would be nice to have control over the superselection sectors of the theory and, in particular, to analyse whether the cyclic GNS vector is not already cyclic for the Abelian subalgebra generated by the W [ h, 0]. Next, it is worthwhile to study the question whether this algebra admits regular representations for both P and F , because then the GNS Hilbert space would admit a measure theoretic interpretation as an L 2 space. Finally, it is certainly necessary to work out the cobordism theory of relevance when restricting the Weyl algebra to distributional 4-form and 7-form factors as smearing functions which is only sketched in this paper. We plan to revisit these questions in future publications.", "pages": [ 10 ] }, { "title": "Acknowledgements", "content": "NB and AT thank Alexander Stottmeister, Derek Wise, and Antonia Zipfel for numerous discussions and the German National Merit Foundation for financial support. The part of the research performed at the Perimeter Institute for Theoretical Physics was supported in part by funds from the Government of Canada through NSERC and from the Province of Ontario through MEDT.", "pages": [ 10 ] } ]

[ { "title": "Casadio-Fabbri-Mazzacurati Black Strings and Braneworld-induced Quasars Luminosity Corrections", "content": "Rold˜ao da Rocha ∗ Centro de Matem'atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC 09210-170, Santo Andr'e, SP, Brazil.", "pages": [ 1 ] }, { "title": "A. Piloyan †", "content": "Institut fur Theoretische Physik, Philosophenweg 16 D-6912, Heidelberg, Germany Yerevan State Univ., Faculty of Physics, Alex Manoogian 1, Yerevan 0025, Armenia", "pages": [ 1 ] }, { "title": "A. M. Kuerten ‡", "content": "Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC 09210-170, Santo Andr'e, SP, Brazil.", "pages": [ 1 ] }, { "title": "C. H. Coimbra-Ara'ujo §", "content": "Campus Palotina, Universidade Federal do Paran'a, UFPR, 85950-000, Palotina, PR, Brazil. This paper aims to evince the corrections on the black string warped horizon in the braneworld paradigm, and their drastic physical consequences, as well as to provide subsequent applications in astrophysics. Our analysis concerning black holes on the brane departs from the Schwarzschild case, where the black string is unstable to large-scale perturbation. The cognizable measurability of the black string horizon corrections due to braneworld effects is investigated, as well as their applications in the variation of quasars luminosity. We delve into the case wherein two solutions of Einstein's equations proposed by Casadio, Fabbri, and Mazzacurati, regarding black hole metrics presenting a post-Newtonian parameter measured on the brane. In this scenario, it is possible to analyze purely the braneworld corrected variation in quasars luminosity, by an appropriate choice of the post-Newtonian parameter that precludes Hawking radiation on the brane: the variation in quasars luminosity is uniquely provided by pure braneworld effects, as the Hawking radiation on the brane is suppressed. PACS numbers: 04.50.Gh, 04.50.-h, 11.25.-w", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Black holes solutions of Einstein equations in general relativity are useful tools to investigate the space-time structure and underlying models for gravity and its quantum effects, as well as to study the astrophysics regarding supermassive objects, for instance. Extra-dimensional space-times are scenarios for extensions of the general relativity, providing solutions to the Einstein's field equations, as black holes metrics in higher dimensions, and some ensuing applications to cosmology in such a context. In addition, the recent effort to deal with the hierarchy problem, by inducing gravity to leak into extra dimensions [1], is explored in braneworld models. They are based on M-theory and string theory [2-4]. In particular, an useful approach to deal with the hierarchy is provided an effective 5D reduction of the Hoˇrava-Witten theory [3, 5, 6]. Impelled by a thorough development concerning gravity on 5D braneworld scenarios and some applications, in particular the black holes/black strings horizons variations induced by braneworld effects, [7-12], further aspects concerning corrections in the black string like objects and their warped horizons are here introduced. The CasadioFabbri-Mazzacurati metrics on the brane, namely the type I and type II black hole solutions [13, 14] are now analyzed and regarded as generating the bulk metric, inducing a black string like warped horizon. This procedure is well known for the Schwarzschild metric [9, 10, 12, 15]. The Casadio-Fabbri-Mazzacurati metrics depart from the Schwarzschild solution, possessing a post-Newtonian parameter. For some particular choice of this parameter, the black hole Hawking radiation on the brane is suppressed [13, 14]. The black holes Hawking radiation in braneworld scenarios was comprehensively investigated in, e. g., [16, 17]. This article is organized as follows: in Sec. II, after presenting the Einstein field equations in the brane, the deviation in Newton's 4D gravitational potential is revisited. For a static spherical metric on the brane, the propagating effect of 5D gravity is evinced from the Taylor expansion (along the extra dimension) of the metric. Such expansion is accomplished in powers of the normal coordinate - out of the brane - which provides the black string warped horizon profile. Such expansion can provide the bulk metric uniquely from the metric on the brane. In Sec. III, the type I and type II Casadio-Fabbri-Mazzacurati black string solutions and their respective warped horizons are obtained, analyzed and depicted. We analyze such solutions in the particular case where the associated post-Newtonian parameter makes the black hole Hawking radiation to be suppressed. Such analysis has paramount importance, since to measure pure effects to the corrections (by braneworld effects) for quasars luminosity is aimed. In Sec. IV, for an illustrative model for accretion in a supermassive black hole, the variation of luminosity in quasars is investigated more precisely for the two models provided by Casadio-Fabbri-Mazzacurati, and compared to the pure Schwarzschild black string. The correction effects on the black string warped horizon, induced and generated by braneworld models, preclude the Hawking radiation on the brane for the above mentioned suitable choice of the post-Newtonian parameter. All results are illustrated by graphics and figures, and the quasars luminosity provided by the Casadio-Fabbri-Mazzacurati black hole solution is compared with the Schwarzschild one.", "pages": [ 1, 2 ] }, { "title": "II. BLACK STRING BEHAVIOR ALONG THE EXTRA DIMENSION", "content": "Hereupon the notation in [15, 18, 19] is adopted, where { θ µ } , µ = 0 , 1 , 2 , 3 , typify a basis for the cotangent space T ∗ x M at a point x on a brane M embedded in a bulk. A frame θ A = dx A ( A = 0 , 1 , 2 , 3 , 4 ) in the bulk is represented in local coordinates. In the brane defined by y = 0, [hereon y denotes the associated Gaussian coordinate] dy = n A dx A is orthogonal to the brane. The metric ˚ g AB dx A dx B = g µν ( x α , y ) dx µ dx ν + dy 2 endows the bulk, which is related to the brane metric g µν by g µν =˚ g µν -n µ n ν . The bulk indexes A,B = 0 , . . . , 3, as ˚ g 44 = 1 and ˚ g µ 4 = 0 [15]. Hereupon the standard relations Λ 4 = κ 2 5 2 ( 1 6 κ 2 5 λ 2 + Λ ) and κ 2 4 = 1 6 λκ 4 5 are considered, where Λ 4 denotes the effective brane cosmological constant, and λ is the brane tension. The constant κ 5 = 8 πG 5 , where G 5 is the 5D Newton gravitational constant, denotes the 5D gravitational coupling, related to the 4D gravitational constant G by G 5 = Gglyph[lscript] Planck , where glyph[lscript] Planck = √ G glyph[planckover2pi1] /c 3 is the Planck length. The junction condition provides the extrinsic curvature tensor K µν = 1 2 £ n g µν by [15, 20] where T = T µ µ is the trace of the energy-momentum tensor. The 5D Weyl tensor is given by C µνσρ = (5) R µνσρ -2 3 (˚ g [ µσ (5) R ν ] ρ +˚ g [ νρ (5) R µ ] σ ) -1 6 (5) R (˚ g µ [ σ ˚ g νρ ] ) , where (5) R µνσρ denotes the components of the bulk Riemann tensor (as usual (5) R µν and (5) R are the associated Ricci tensor and the scalar curvature). The trace-free and symmetric components, respectively denoted by B µνα = g ρ µ g σ ν C ρσαβ n β and E µν = C µνσρ n σ n ρ , denote the magnetic and electric Weyl tensor components, respectively.", "pages": [ 2 ] }, { "title": "A. Brane field equations", "content": "The Einstein brane field equations can be expressed as The Weyl tensor electric term E µν carries an imprint of high-energy effects sourcing Kaluza-Klein (KK) modes. The gravitational potential V ( r ) = GM c 2 r , associated to the 4D classical gravity, is corrected by extra-dimensional effects [5, 15]: The parameter glyph[lscript] is associated with the bulk curvature radius and corresponds to the effective size of the extra dimension probed by a 5D graviton [5, 15, 21]. Indeed, the contribution of the massive KK modes sums to a correction of the 4D potential. At small scales r glyph[lessmuch] glyph[lscript] , one obtains the 5D features related to the potential V ( r ) ≈ GMglyph[lscript]/r 2 . For r glyph[greatermuch] glyph[lscript] the potential is provided by (2) reinforcing the gravitational field [5, 10, 15]. Considering vacuum on the brane, where T µν = 0 outside a black hole, the field equations G µν = -E µν -1 2 Λ 5 g µν and R = 0 = E µ µ hold for braneworlds with Z 2 -symmetry. The vacuum field equations in the brane are E µν = -R µν , where the bulk cosmological constant is comprised into the warp factor. The bulk can host for instance a plethora of non standard model fields, as dilatonic or moduli fields [22]. Although a Taylor expansion of the metric was used to probe properties of a black hole on the brane in, e. g., [15, 23], in order to enhance the range of our analysis throughout this paper a more complete approach to analyze braneworld corrections in the black string profile can be accomplished, based on [12]. A Taylor expansion of the metric along the extra dimension allows us to analyze the black string more deeply. The effective field equations are complemented by other ones, obtained from the 5D Einstein and Bianchi equations in Refs. [15, 18, 19]. Hereupon, since we are concerned with the Taylor expansion of the metric along the extra dimension up to the fourth order, besides the effective field equation £ n K µν = K µα K α ν -E µν -1 6 Λ 5 g µν , the effective equations are considered: These expressions are used to compute the terms in the Taylor expansion of the metric, along the extra dimension, providing the black string profile and further physical consequences as well. The effective field equations above were employed to construct a covariant analysis of the weak field [18]. Denoting K = K µ µ , the Taylor expansion is given by [12] [hereon we denote g µν ( x, 0) = g µν ]: Such an expansion was analyzed in [15, 24] only up to the second order, although it fizzled out to explain more reliably the black string horizon behavior along the extra dimension. In addition, this higher order expansion provides further physical features regarding variable tension braneworld scenarios, since the expansion terms beyond second order provide drastic modifications in the stability of black strings [12]. For an alternative method which does not take into account the Z 2 -symmetry, and some subsequent applications, see [25]. For a vacuum in the brane, Eq.(3) reads This expression is shown to be prominently relevant for our subsequent analysis. Hereon, the black hole horizon evolution along the extra dimension - the warped horizon [26] - shall be investigated, exploring the component g θθ ( x, y ) in (4). Indeed, let us consider any spherically symmetric metric associated to a black hole - in particular the Schwarzschild and the Casadio-Fabbri-Mazzacurati ones here investigated. Such metric has the radial coordinate given by √ g θθ ( x, 0) = r . The black hole solution, namely, the black string solution on the brane , is regarded when √ g θθ ( x, 0) = R , where R denotes the coordinate singularity, usually calculated by the component g -1 rr = 0 in the metric 1 . In the Schwarzschild metric R = R S = 2 GM c 2 r . The coordinate singularities for the Casadio-Fabbri-Mazzacurati metrics are going to be analyzed in what follows, in the black string context as well. Such singularities shall be shown to be also physical singularities (associated to the black holes and the black strings as well), by analyzing their respective four- and 5D Kretschmann scalars. In other words, in the analysis regarding the black string behavior along the extra dimension, we are concerned merely about the warped horizon behavior, which is provided uniquely by the value for the metric on the brane √ g θθ ( x, y ) | r = R S . More specifically, the black string horizon for the Schwarzschild metric - or warped horizon [26] - is defined when the radial coordinate r has the value r = R S = 2 GM c 2 , which is obtained when the coefficient ( 1 -2 GM c 2 r ) = g rr of the term dr 2 in the metric goes to infinity [16]. It corresponds to the black hole horizon on the brane. On the another hand, the (squared) general radial coordinate in spherical coordinates legitimately appears as the term g θθ dθ 2 = r 2 dθ 2 in the Schwarzschild metric. Our analysis of the term g θθ ( x, y ) (given by Eq.(3) for µ = θ = ν as the most general case, and provided by Eq.(4) for the Schwarzschild metric) holds for any value r . In particular, the term originally coined 'black string' corresponds to the Schwarzschild case [26], defined by the black hole horizon evolution along the extra dimension into the bulk. Hence, the black string regards solely the so called 'warped horizon', which is g θθ ( x, y ), for the particular case where r = R S is a coordinate singularity.", "pages": [ 2, 3, 4 ] }, { "title": "III. CASADIO-FABBRI-MAZZACURATI BRANEWORLD SOLUTIONS", "content": "The analysis of the gravitational field equations on the brane is not straightforward, due to the fact that the propagation of gravity into the bulk does not allow a complete presentation of the brane gravitational field equations as a closed form system [18]. The investigation concerning the gravitational collapse on the brane is therefore very complicated [27]. The solutions provided by Casadio, Fabbri, and Mazzacurati for the brane black holes metrics [13, 23, 28] take into account the post-Newtonian parameter β , measured on the brane. The case β = 1 generates forthwith an exact Schwarzschild solution on the brane, and elicits a black string prototype. Furthermore, it was observed in [13, 14] that β ≈ 1 holds in solar system scales measurements [24]. The parameter β is, furthermore, capable to indicate and to measure the difference between the inertial mass and the gravitational mass of a test body. This parameter also affects the perihelion shift and provides the Nordtvedt effect [24]. Moreover, measuring β gives information about the vacuum energy of the braneworld or, equivalently, the cosmological constant [13, 14, 29]. One of the main motivation regarding the Casadio-Fabbri-Mazzacurati setup is that black holes solutions of the Einstein equations on the brane must depart from the Schwarzschild solution. In particular, the Schwarzschild associated black string is unstable to large-scale perturbations [30, 31]: the associated Kretschmann scalar, regarding the 5D curvature, diverges on the Cauchy horizon [13, 15]. Indeed, it is important to emphasize that, as we shall see for the Casadio-Fabbri-Mazzacurati black string, it might be possible to find out points along the extra dimension for which the Kretschmann scalar (5) K = (5) R µνρσ (5) R µνρσ diverges, i. e., they are indeed naked singularities along the extra dimension. For instance, in order to identify singularities, for a Schwarzschild black string (5) K ∝ 1 /r 6 [31]. Hence there is a line singularity at r = 0 along the extra dimension, but not at the Schwarzschild horizon [15, 26]. Since the pure black string configuration is unstable [30], this structure is not physical ab initio. Anyway for y = 0 one reproduces the Kretschmann scalars standard 4D behavior. For the Schwarzschild solution, the singularity on the brane extends into the bulk and makes the AdS horizon singular. The Casadio-Fabbri-Mazzacurati black string solutions and their respective braneworld corrections are going to be presented and their stability analyzed as well. For the sake of completeness the next Section is briefly devoted to the braneworld corrections to the Schwarzschild solution [7, 15]. A static spherical metric on the brane is provided by g µν dx µ dx ν = -F ( r ) dt 2 + ( H ( r )) -1 dr 2 + r 2 d Ω 2 . The Schwarzschild metric is corresponds to F ( r ) = H ( r ) = 1 -2 GM c 2 r . Obtaining such functions remains an open problem in the black hole theory on the brane [13-15]. Considering the Weyl tensor projected electric component on the brane E θθ = 0 [15] yields [12] Note that obviously in the brane g θθ ( r, 0) = r 2 . Defining ψ ( r ) as the deviation from a Schwarzschild form for H ( r ) [14, 15, 32-36] as H ( r ) = 1 -2 GM c 2 r + ψ ( r ), for a large black hole with horizon scale r glyph[greatermuch] glyph[lscript] it follows from Eq.(2) that The formula above, together with Eq.(2), can be forthwith derived from the RS analysis concerning small gravitational fluctuations in terms of KK modes, where a curved background can support a bound state of the higher-dimensional graviton [5, 37]. Besides, the effect of the KK modes on the metric outside a specific matter distribution on the brane was incorporated by [37] in the form of the 1 /r 3 correction to the gravitational potential. Such corrections in the inverse-square law were experimentally shown in [38]. Now, given a general static spherically symmetric the Casadio-Fabbri-Mazzacurati 4D black hole solution was obtained in [13, 14]. The Schwarzschild 4D metric is obtained when N ( r ) = ( A ( r )) -1 and N ( r ) = 1 -2 GM c 2 r . Its unique extension into the bulk is a black string warped horizon, with the central singularity extending all along the extra dimension, and the bulk horizon singular [13, 14, 31]. If the Schwarzschild metric on the brane is demanded with a regular AdS horizon, there is no matter confinement on the brane: in this case matter percolates into the bulk [36]. The condition N ( r ) = ( A ( r )) -1 holds in the 4D case, although the most general solution is the Reissner-Nordstrom one [13, 14, 18], related to the case II analyzed in which follows. The case I below concerns the function N ( r ) = 1 -2 GM c 2 r like the Schwarzschild case, but this time A ( r ) to be calculated. Both cases are profoundly investigated, as well as their prominent applications.", "pages": [ 4, 5 ] }, { "title": "A. Casadio-Fabbri-Mazzacurati black string: Case I", "content": "This case was analyzed by Casadio, Fabbri, and Mazzacurati in [13, 14], regarding the 4D black hole solution (7). They obtained a solution of the Einstein's equations distinguished from the Schwarzschild one, provided by the metric coefficients to be considered in (7). The solution (8) depends on just one parameter and for M → 0 one recovers the Minkowski vacuum. The Casadio-Fabbri-Mazzacurati black string classical horizon, in the brane, is the solution of the algebraic equation ( A ( r )) -1 = 0. In order to extract phenomenological information of numerical calculations, we first consider in this Subsection the case where β = 5 / 4. Indeed, our aim is to analyze the pure braneworld corrected effects on the variation of luminosity in quasars, composed by a black hole which presents Hawking radiation in the brane equal to glyph[negationslash] zero [13, 14]. It makes feasible our analysis on the variation of quasars luminosity, purely due to braneworld effects. Hawking radiation in the context of black strings was investigated, e.g., in [16, 39]. Note that the metric above was also derived as a possible geometry outside a star on the brane [28]. The corresponding Hawking temperature is calculated in [13]. In comparison with Schwarzschild black holes, the black hole provided by this solution is either hotter or colder, depending upon the sign of ( β -1). The extension of these solutions into the bulk has prominent importance addressed in [13]. For the Schwarzschild case, the singularity on the brane extends into the bulk and makes the AdS horizon singular. Notwithstanding, according to the analysis illustrated by the graphics below, Eq.(3) asserts that the black string solutions might be regular for supermassive black holes. glyph[negationslash] Taking into account the metric in (7), the classical standard black hole radius is given by - supposing r = 3 GM 2 c 2 - two solutions of Schwarzschild type R S = 2 GM c 2 , for our choice of the parameter β = 5 / 4, providing zero Hawking black hole temperature. The assumption r = 3 GM/c 2 is quite natural: for this case the 4D Kretschmann scalar K ( I ) = R µνρσ R µνρσ diverges for r = 0 and r = 3 GM 2 c 2 (see the Appendix). Now, the Gauss equation is well known to relate the 5D and the 4D Riemann curvature tensor as By taking the junction conditions into account, where consequently K µν = -1 2 κ 2 5 ( T µν + 1 3 ( λ -T ) g µν ) , for the vacuum case here considered is follows that K µν = -1 6 κ 2 5 λg µν . By inserting it in the Gauss equation (9), it implies that the 5D Kretschmann scalar (5) K = (5) R µνρσ (5) R µνρσ for the Casadio-Fabbri-Mazzacurati type I black string also diverges for r = 0 and r = 3 GM 2 c 2 : the terms involving the extrinsic curvature in (9) above are not capable to cancel the divergence provided by the 4D Kretschmann scalar, in the computation for (5) K . Using the same procedure as [10, 15], one can use the metric coefficients (8) in Eq.(4) and calculate the black string warped horizon. As asserted, for instance in [13, 14], this analysis can be attempted either numerically or by Taylor expanding all 5D metric elements in powers of the extra coordinate. In the graphics below, we explicit the value for the black string warped horizon, provided by √ g θθ ( R S , y ), where R S is the Schwarzschild radius. Further, λ = Λ = 1 = κ 5 hereupon [ M glyph[circledot] denotes the sun mass]: /LParen1 /RParen1/Slash1 Fig. 1 evinces a very interesting profile for the black string horizon behavior along the extra dimension y in gaussian coordinates. It indicates a critical mass M (indeed our simulations provide M ∼ 73 M glyph[circledot] ) above which the associated black string warped horizon monotonically increases along the extra dimension. The black string is known to be placed in the bulk, in a tubular neighborhood along the axis of symmetry. A singularity associated to the black string is a fixed point y 0 (fixed) in the axis of symmetry along the extra dimension, such that the black string transversal slice has radius equal to zero. We show here that at the coordinate singularities r = 0 and r = 3 GM 2 c 2 there is a physical singularity for the black string at such values, irrespective of the value for y . In fact, the Kretschmann scalar K = (5) R µνρσ (5) R µνρσ diverges for such values (see the Appendix). Notwithstanding, the black string warped horizon √ g θθ ( R S , y ) does not equal to zero, as illustrated at the Fig. 1.", "pages": [ 5, 6 ] }, { "title": "B. Casadio-Fabbri-Mazzacurati black string: Case II", "content": "An alternative solution of (7) is obtained in [13, 14] where the metric coefficients are considered in (7). In order that the Hawking temperature be zero on the brane, the choice β = 3 / 2 is demanded [13]. The classical solution R for the black hole horizon is given by R = R S and R = 5 R S / 2, where R S denotes the Schwarzschild radius. It implies that the black string horizon now corrected by braneworld effects when (10) is substituted in (4), providing the graphic below. /LParen1 /RParen1/Slash1 The black string horizon profile along the extra dimension is qualitatively similar for all values of M depicted here: the warped horizon always increases monotonically. Furthermore, under a similar analysis accomplished this time for the case II Casadio-Fabbri-Mazzacurati, and by taking into account the Kretschmann scalar (A2) in the Appendix, we conclude that such expression diverges for r = 3 GM 2 c 2 , for r = 5 GM 2 c 2 and r = 2 GM c 2 . Contrary to the Schwarzschild metric, which presents the black hole horizon as a coordinate singularity - which can circumvented by, e. g., the KruskalSzekeres coordinates - and not as a physical singularity, the Kretschmann scalar for the Casadio-Fabbri-Mazzacurati case II metric indicates that each black hole horizon on the brane is a physical singularity, since it diverges for such values. Again, the terms involving the extrinsic curvature in (9) above are not able to cancel the divergence induced by the 4D Kretschmann scalar, when one calculates (5) K . Hence, the black string also diverges for such values. Fig. 2 indicates that the Casadio-Fabbri-Mazzacurati (case II) black string horizon always increases. Since the bulk has no fixed metric a priori, but it can be calculated from (3) taking into account the metric on the brane, we can calculate the bulk curvature using the metric coefficients in (3). Compact sources on the brane, such as stars and black holes, have been investigated extensively. However, their description has proven rather complicated and there is little hope to obtain analytic solutions. The present literature does in fact provide solutions on the brane [13, 23, 24, 28], perturbative results over the Randall-Sundrum background [37, 42], and numerical treatments [29]. In [43] the luminosity dissipation, the conditions for which a collapsing star generically evaporates and approaches the Hawking behavior as the (apparent) horizon is formed, are also analyzed.", "pages": [ 7 ] }, { "title": "IV. CORRECTIONS IN THE LUMINOSITY: BRANEWORLD EFFECTS", "content": "Once the black string behavior was previously analyzed along the extra dimension, we hereon aim to focus on the corrections now restricted to the phenomena on the brane. These corrections are shown here to induce dramatic consequences on the quasars luminosity variation, due to the braneworld model considered. Due to its prominent importance on the analysis hereupon, the effect of higher dimensions in the gravity sector might begin to make their presence felt as the black hole horizon is approached. The case of braneworld black holes horizon corrections is explored hereon. Quasars are astrophysical objects that can be found at large astronomical distances. Supermassive stars and the process of gravitational collapse are showed to be able to probe deviations from the 4D general relativity [10]. The observation of quasars in X -ray band can constrain the measure of the bulk curvature radius glyph[lscript] . Varied values for glyph[lscript] were used and tested, and no qualitative deviations have been detected. Table-top tests of Newton's law currently find no deviations down to the order glyph[lscript] glyph[lessorsimilar] 0.1 mm. A more accurate magnitude limit improvement on the AdS 5 curvature glyph[lscript] is provided in [15, 40] by analyzing the existence of stellar-mass black holes on long time scales and of black hole X -ray binaries. Furthermore, the failure of current experiments using torsion pendulums and mechanical oscillators to observe departures from Newtonian gravity at small scales have set the upper limit of glyph[lscript] in the region glyph[lscript] glyph[lessorsimilar] 0 . 2 mm [41]. Regarding a static black hole being accreted, in a straightforward model the accretion efficiency η is given by where R S denotes the black hole horizon, namely the black string horizon in the brane). The event horizon of the supermassive black hole is 10 15 times bigger than the bulk curvature parameter glyph[lscript] . This is not the case of mini black holes wherein the event horizon of magnitude orders smaller than glyph[lscript] . As proved in [10], the solution above for the black string horizon can be also found in terms of the curvature radius glyph[lscript] [9]. In the accretion rate model in [44], observational data for the luminosity L estimates a value for glyph[lscript] . The luminosity L , due to the accretion in a black hole composing a quasar, is a function of the bulk curvature radius parameter glyph[lscript] , and provided by where ˙ M denotes the mass accretion rate. For a typical black hole of 10 12 M glyph[circledot] in a supermassive quasar, the accretion rate is ˙ M ≈ 2 . 1 × 10 19 kg s -1 [10]. Supposing that the quasar radiates in the Eddington limit [44] L = L Edd ∼ 1 . 2 × 10 45 ( M 10 7 M glyph[circledot] ) erg s -1 , the luminosity is given by L ∼ 10 47 erg s -1 . From (11) and (12), the variation in the luminosity of a quasar composed by a supermassive black hole reads where R brane = √ g θθ ( r = R S , y ) denotes the black string corrected horizon. In the next Subsection the variation of the quasars luminosity for the two Casadio-Fabbri-Mazzacurati black holes are depicted and analyzed. Furthermore, the difference in the luminosity between the pure Schwarzschild and the case of the solutions (8) and (10) are computed and discussed in what follows.", "pages": [ 7, 8 ] }, { "title": "A. Corrections in the quasar luminosity for the both Casadio-Fabbri-Mazzacurati solutions", "content": "We want now to analyze how the corrections for the metric coefficients due to braneworld effects in [5, 9, 10, 15] can affect the luminosity emitted by quasars composed by black holes provided by the Casadio-Fabbri-Mazzacurati solutions (7), with coefficients (8, 10). The alteration in the black hole horizon definitely modifies the quasar luminosity. Its variation with respect to the pure Schwarzschild luminosity is provided by Eq.(13) and here depicted, for the Casadio-Fabbri-Mazzacurati type I metric: 0.2 0.20 Now the Casadio-Fabbri-Mazzacurati case II in Subsec. III B is analyzed, still adopting β = 3 / 2 in order to prevent Hawking radiation on the brane. It follows that similarly for the case I analyzed above, the corrected black string horizon on the brane (namely the black hole horizon). One can show that for the corrections on the black string horizon, as seen from the brane, ∆ L ∼ 10 30 ± 1 erg s -1 , for a typical supermassive black hole with M ≈ 10 9 M glyph[circledot] (case I and II respectively in the preceding Subsections III A and III B). The figures below regard respectively the cases I and II of Casadio-Fabbri-Mazzacurati black strings horizon on the brane, analyzed in Sec. III. In solar luminosity units L glyph[circledot] ∼ 3 . 9 × 10 33 erg s -1 , the variation of luminosity of a supermassive black hole quasar due to the correction of the horizon in a braneworld scenario is given by ∆ L ∼ 10 -3 L glyph[circledot] . Naturally, this small but cognizable correction in the horizon of supermassive black holes implies a consequent correction in the quasar luminosity, via accretion mechanism. This correction is clearly regarded in the luminosity integrated in all wavelength. The detection of these corrections works in particular selected wavelengths, since quasars emit radiation in the soft/hard X -ray band. We remark that the Schwarzschild braneworld corrected black string horizon on the brane was previously investigated in [10], and in that case ∆ L ∼ 10 -5 L glyph[circledot] . In addition, the variation of the quasar luminosity regarding the Casadio-Fabbri-Mazzacurati type II model, with respect to the pure Schwarzschild luminosity, is provided by Eq.(13) and here illustrated: y Figs. 3 and 4 evince the variation in the quasars luminosity in the Casadio-Fabbri-Mazzacurati types I and II metrics, respectively Eqs. (8, 10), with respect to the Schwarzschild black hole luminosity. The graphics reveal that the luminosity variation of the Casadio-Fabbri-Mazzacurati black holes, corrected by braneworld effects, is smaller compared to the Schwarzschild case. The exception is the curve in Figure 3 above the horizontal axis, which illustrates the general behavior of the Casadio-Fabbri-Mazzacurati type I black string warped horizon, associated to a black hole mass M glyph[lessorsimilar] 73 M glyph[circledot] . glyph[negationslash] Delving into the analysis concerning the figures above, the general different profile between the Casadio-FabbriMazzacurati black string warped horizon and the Schwarzschild horizon is expected. Their ratio(s) provides the Figures 3 and 4 by Eq.(13) and the profile is encrypted in the underlying structure of Eq. (4). Indeed, the warped horizon is provided by √ g θθ ( R S , y ) in (4) when µ = ν = θ . Besides, for the Schwarzschild metric the electric component of the Weyl tensor E θθ equals zero, what do not happen to the Casadio-Fabbri-Mazzacurati metrics. Indeed, taking into account the metric (7), in general E θθ = -1 + 1 A + r 2 A ( F ' F -A ' A ) = 0 for the coefficients in (8) and (10). As it is comprehensively discussed in [45-48], the black hole may recoil away from the brane by the emission of Hawking radiation into the bulk, but not on the brane. We would like to emphasize that only mini black holes in the Randall-Sundrum model are prevented to recoil away from the brane into the bulk [46]. Notwithstanding, here the Randall-Sundrum model is not required and all information about the bulk can be extracted from the Casadio-FabbriMazzacurati metrics on the brane - using (3) (and eventually further terms in | y | k , for any k positive, according to the required precision in the Taylor approximation). We considered terms up to y 4 , since irrespective of the black hole horizon radius, the effective distance y along the extra dimension equals the compactification radius [15], as well as the effective size of the extra dimension probed by a graviton. Besides, our procedure considers suitable values for the post-Newtonian parameter in order that the Hawking radiation on the brane is zero. The number of degrees of freedom of KK gravitons is much less than the number of standard model particles in the Hawking radiation in the bulk, and the black hole energy irradiated into the KK modes must be a small fraction of the total luminosity. Since the post-Newtonian parameter β was chosen to prevent Hawking radiation in the brane, standard model particles on the brane with high enough energy - larger than electroweak energy scale - are capable to overcome the confining mechanism [16]. In this case the bulk standard model fields should be included among the KK modes. Such mechanism responsible for the possible black hole recoil from the brane corroborates to the astrophysical phenomenology described in the figures above: the physical black hole radii R S = √ g θθ ( R S , 0) are now effectively dislocated into the bulk, and given by R brane = √ g θθ ( R S , y ) in Eq.(13). Further discussion and details on the general behavior encoded in the figures are presented in the next Section.", "pages": [ 8, 9, 10 ] }, { "title": "V. CONCLUDING REMARKS AND OUTLOOK", "content": "Any phenomenologically successful theory in which our Universe is viewed as a brane must reproduce the largescale predictions of general relativity on the brane. It implies that gravitational collapse of matter trapped on the brane provides the Casadio-Fabbri-Mazzacurati solutions on the brane: either a localized black hole or an extended black string solution, possessing a warped horizon. It is possible to intersect this solution with a vacuum domain wall and the induced metric is the ones presented in the analysis in Subsections III A and III B. In the case I, our analysis is restricted to the case where β = 5 / 4 since for this values there is a zero (Hawking) temperature black hole associated [13, 14]. Since we want to extract physical information on the braneworld effects on the variation of luminosity exclusively, we opted for this value for the parameter β , in such a way that the graphics, concerning this variation on the luminosity, take into account exclusively the braneworld effects, since Hawking radiation is shown to be suppressed with β = 5 / 4 for this metric. Analogously, for the case II the analysis is accomplished taking into account the value β = 3 / 2 in Eq.(10), as already discussed. For the Casadio-Fabbri-Mazzacurati (types I and II) black holes the variation of luminosity of a supermassive black hole quasar due to the correction of the horizon in a braneworld scenario is given by ∆ L ∼ 10 -3 L glyph[circledot] . On the other hand, the Schwarzschild braneworld corrected black string horizon on the brane was previously investigated in [10], and in that case ∆ L ∼ 10 -5 L glyph[circledot] . It shows that the Casadio-Fabbri-Mazzacurati black hole solutions, containing the post-Newtonian parameter, can probe two orders of magnitude more the variation in the quasars luminosity. Figs. 1 and 2 encode the brane effect-corrected black string horizon, respectively for the first and second black hole solution proposed by Casadio, Fabbri, and Mazzacurati, along the extra dimension y . The black string horizon behavior is obviously different for distinct values for the black hole mass. For the second case, the warped horizon is always an increasing function of the extra dimension. For the first one, instead, it holds only for a black hole with mass M glyph[greaterorsimilar] 73 M glyph[circledot] . Otherwise, the warped horizon is a decreasing function along the extra dimension. Once the corrections related to the black string horizon behavior along the extra dimension are obtained, we focused on how such corrections can also alter the black string warped horizon. Our results are exposed and conflated in Figs. 3 and 4 illustrating the variation of luminosity of quasars - supermassive black holes - when the Hawking radiation in the brane is precluded, and the pure braneworld effect can be analyzed. The corrections of the luminosity regarding the Schwarzschild black string, along the extra dimension, can be probed by a black hole by recoil effects from the brane. The variation of the quasar luminosity is considerable in this case. In [9] some properties of black holes were analyzed, in ADD [1] and Randall-Sundrum models. Mini black holes in ADD models have the first phase Hawking radiation mostly in the bulk and recoil effect to leave the brane. The analysis of the Casadio-Fabbri-Mazzacurati solutions in this paper sheds new light on mini black holes and their possible detection at LHC, since the preclusion of Hawking radiation can drastically modify the previous analysis about mini black holes in ADD and Randall-Sundrum braneworld models, as well as the mini black holes radiation in LHC measurements. The method here introduced can be immediately applied in such context.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "R. da Rocha is grateful to Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico (CNPq) 476580/2010-2, 303027/2012-6, and 304862/2009-6 for financial support, and to Prof. J. M. Hoff da Silva for fruitful and valuable discussions and suggestions as well. A. M. Kuerten thanks to CAPES for financial support. C. H. Coimbra-Ara'ujo thanks Funda¸c˜ao Arauc'aria and Itaipu Binacional for financial support.", "pages": [ 10 ] }, { "title": "Appendix A: Kretschmann scalars for Casadio-Fabbri-Mazzacurati metrics", "content": "The 5D Kretschmann scalars associated to the black strings here discussed can be obtained by the 4D ones via Eq.(9). Since we aim to investigate the particular case where β = 5 / 4 where there is no Hawking radiation, and pure braneworld effects can be probed, for this case the 4D Kretschmann scalar K ( I ) = R µνρσ R µνρσ -where the Riemann tensors used are the ones related to the Casadio-Fabbri-Mazzacurati case I - is given by which diverges at r = 3 GM 2 c 2 and r = 0. It agrees with the result in [13, 14] where K ( I ) ∝ (1 -3 GM 2 c 2 r ) -4 for values of r near to the respective singularity r = 3 GM 2 c 2 . Now, for the Casadio-Fabbri-Mazzacurati case II metric (10), the associated Kretschmann scalar, in the specific case here considered β = 3 / 2 the expression above is led to 35N4B (2005) 1129 [ arXiv:astro-ph/0509363 ]; C. H. Coimbra-Araujo, R. da Rocha, I. T. Pedron, Anti-de Sitter curvature radius constrained by quasars in brane-world scenarios, Int. J. Mod. Phys. D 14 (2005) 1883 [ arXiv:astro-ph/0505132 ]. [25] D. Jennings, I. R. Vernon, A. C. Davis, C. van de Bruck, Bulk black holes radiating in non-Z(2) brane-world spacetimes, JCAP 0504 (2005) 013. [ arXiv:hep-th/0412281 ]. [37] J. Garriga, T. Tanaka, Gravity in the Randall-Sundrum Brane World, Phys. Rev. Lett. 84 (2000) 2778 [ arXiv:hep-th/9911055 ]; H. Kudoh and T. Tanaka, Second order perturbations in the Randall-Sundrum infinite brane world model, Phys. Rev. D 64 (2001) 084022 [ arXiv:hep-th/0104049 ].", "pages": [ 10, 11, 12 ] } ]

[ { "title": "Felix M. Haehl", "content": "Institute for Theoretical Physics ETH Zurich CH-8093 Zurich Switzerland E-mail: [email protected] Abstract: We present a calculation of two-point correlation functions of the stress-energy tensor in the strongly-coupled, confining gauge theory which is holographically dual to the AdS soliton geometry. The fact that the AdS soliton smoothly caps off at a certain point along the holographic direction, ensures that these correlators are dominated by quasinormal mode contributions and thus show an exponential decay in position space. In order to study such a field theory on a curved spacetime, we foliate the six-dimensional AdS soliton with a Schwarzschild black hole. Via gauge/gravity duality, this new geometry describes a confining field theory with supersymmetry breaking boundary conditions on a non-dynamical Schwarzschild black hole background. We also calculate stress-energy correlators for this setting, thus demonstrating exponentially damped heat transport. This analysis is valid in the confined phase. We model a deconfinement transition by explicitly demonstrating a classical instability of Gregory-Laflamme-type of this bulk spacetime.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The AdS/CFT conjecture [1-3] has become a main tool to study strongly coupled gauge theories from the point of view of a dual description in terms of a suitable low energy limit of string theory. In particular, the dictionary of gauge/gravity duality provides a way to calculate correlation functions of field theory operators from the gravitational dynamics of the bulk. The stress-energy tensor of the field theory is induced by the asymptotic behavior of the bulk metric itself: The propagation of small perturbations of the quantum stress tensor can be studied by solving the problem of graviton propagation in the bulk spacetime. See [4] for the calculation of stress-energy tensor correlation functions in strongly coupled N = 4 supersymmetric Yang-Mills (SYM) theory, and [5, 6] for the generalization to thermal N = 4 SYM theory. The poles of these retarded Green's functions are usually easier to calculate because they are just the quasinormal modes in the language of bulk gravity [6, 7]. Besides understanding supersymmetric and conformal field theories (in particular N = 4 SYM), one would like to describe strongly coupled gauge theories that share more properties with (large N ) QCD. We will use an approach where we address the issue of finding bulk geometries which are appropriate to break supersymmetry and conformal invariance. Supersymmetry may be broken by considering a bulk which contains a Scherk-Schwarz compactified dimension and imposing antiperiodic boundary conditions on the fermions [8]. It has been proposed to consider the AdS soliton as a bulk which is only locally asymptotically AdS and has a compact dimension (an S 1 circle) [10, 13]. The introduction of a scale (the size of the compact dimension) also allows to break conformal invariance. The AdS soliton is constructed by a double analytic continuation of the planar AdS black hole such that a compactification takes place as one makes the original time coordinate Euclidean. This geometry is horizon-free and it may be visualized as the surface of one half of a cigar which caps off smoothly at its tip ( infrared floor ). The entropy density therefore vanishes in the classical limit and the dual field theory is in a confined phase. See [14] for a discussion along these lines, and [15] for an approach towards confinement. The AdS soliton is perturbatively stable and has the lowest energy in its class of spacetimes with the particular locally asymptotically AdS boundary conditions. One expects that the finite size of the geometry along the holographic direction induces a quantization of the graviton modes. These quasinormal frequencies appear as the poles of the retarded Green's function in the dual quantum theory, as one would expect on general grounds [6]; see also [17, 18]. We will explicitly calculate field theory correlators which describe momentum diffusion and we will see how the quantization of graviton modes leads to an exponential damping of such transport phenomena. After understanding some of these properties of the field theory dual to the AdS soliton, one can ask what happens if this theory lives on a non-trivial background spacetime. An interesting feature of the AdS soliton is that it can be foliated with arbitrary Ricci flat slices which take the role of the boundary geometry in holography. As one foliates the AdS soliton with a Schwarzschild black hole, one obtains a six-dimensional spacetime with two independent physical scales: the Schwarzschild radius r s and the size of the compact dimension L τ . This Schwarzschild-black string AdS soliton ( Schwarzschild soliton string for short) serves as a simple model of a bulk geometry that describes a non-supersymmetric, strongly coupled plasma around a (non-dynamical) black hole background. The geometric parameter r s corresponds to the plasma temperature T ∝ r s . We will address the question of how this field theory propagates thermal excitations near the black hole horizon towards infinity. Analogous considerations as in the case of a flat background lead us to expect an exponential damping. Besides our interest in stress tensor correlators on a black hole background, another closely related aspect of these field theories is their deconfinement transition as one lowers T [16]. Holographically, this phase transition can be understood in terms of a Hawking-Page transition between the Schwarzschild soliton string and the planar AdS black hole as one varies r s at fixed L τ . In order to determine the transition temperature, we will calculate at which point the Schwarzschild soliton string becomes unstable against small perturbations. This computation is closely related to the calculation of thermal Green's functions because it also involves solving the bulk graviton equations of motion. Qualitatively, one expects to find a Gregory-Laflamme-type instability [19] when the black hole horizon is small compared to the compact dimension. This paper is organized as follows. In section 2, we use the gauge/gravity duality to calculate a stress-energy tensor two-point function in the strongly coupled, confining gauge theory which is dual to the AdS soliton. In section 3, we introduce the Schwarzschild soliton string and confirm quantitatively that it is indeed unstable against small perturbations with tensor modes of a decomposition of the spacetime with respect to the base manifold B = Schw 4 (i.e. the Schwarzschild black hole). The instability problem reduces to a combination of the well-known Gregory-Laflamme instability and the propagation of a scalar in the AdS soliton. Once we have identified the stable phase of the Schwarzschild soliton string, section 4 will be concerned with the study of stress-energy correlation functions in the strongly coupled field theory on a Schwarzschild black hole background which is dual to the Schwarzschild soliton string in gravity. An important ingredient for this calculation will be the results from section 2. Due to the fact that the boundary metric is no longer translationally invariant, we will need to generalize the Fourier decomposition that could otherwise be used. We conclude with some remarks in section 5. In appendix A, details of the calculation of the AdS soliton quasinormal modes are outlined. We show in appendix B that vector and scalar perturbations do not destabilize the Schwarzschild soliton string.", "pages": [ 2, 3, 4 ] }, { "title": "2 Shear Diffusion in the AdS Soliton Dual", "content": "In this section we review some properties of the AdS soliton geometry and its dual confining field theory. We calculate the quasinormal modes (QNM) for a linearized perturbation of this geometry which propagates like a scalar field. The QNM contribution with longest wavelength dominates stress-energy correlators in the dual field theory. Besides being interesting for their own sake, results of this analysis will be needed for calculations in section 4. By viewing our quantum theory as a toy model for QCD, the poles of the Green's functions may be interpreted as glueball masses. Such holographic computations of glueball spectra in QCD 3 and QCD 4 have been carried out, see e.g. [9]. The authors of [10] even made use of the same AdS soliton geometry and calculated glueball masses with a WKB approach. We will make a comparison with their results. There have also been proposals for models which are more driven by phenomenology. In [11] the bulk is constructed as a stack of flat slices which are conformally rescaled in such a way as to reproduce known QCD phenomenology. This model relies on similar computations involving scalar field propagation and QNMs as the model that we will be studying. In this sense our calculation might be easily adopted to such more phenomenologically relevant scenarios. Models based on D-branes are also able to reproduce within certain bounds the phenomenology of QCD. In particular, the Sakai-Sugimoto ansatz models QCD holographically by investigating stacks of D-branes, see [12] and references therein.", "pages": [ 4 ] }, { "title": "2.1 The AdS Soliton", "content": "The AdS soliton has been described by Horowitz and Myers in [13]. To construct the AdS soliton, we start with the following AdS black hole solution to d -dimensional Einstein gravity with cosmological constant Λ < 0 : where glyph[lscript] = -( d -1)( d -2) / 2Λ is the AdS d radius, and i = 1 , . . . , d -2 . If we now perform a double analytic continuation of this metric, i.e. t → iτ and x d -2 → it , we obtain where η µν is the ( d -2) -dimensional Minkowski metric, and ( x µ ) = ( t, x 1 , . . . , x d -3 ) . Note that the coordinate τ has to be periodically identified in a Kaluza-Klein spirit in order to avoid a conical singularity at r = r 0 , i.e. τ ∼ τ +4 πglyph[lscript] 2 / ( d -1) r 0 . We anticipate already at this point the following nice feature of the geometry (2.2): The flat space metric η µν can be replaced by any Ricci flat manifold and (2.2) will still be a solution of Einstein gravity. Since this part of the metric corresponds to the non-compact boundary dimensions, we will be able to make a transition to boundary theories on a curved background (see section 3). Since the τ -circle closes smoothly at r = r 0 , the geometry just ends there and the entire spacetime (2.2) is horizon-free and everywhere smooth. At r = r 0 there is no singularity but an infrared floor where the spacetime ends in a cigar shaped geometry. The AdS soliton has a translational symmetry along the ( d -2) Minkowski coordinates, and a U (1) symmetry along the compact dimension. Due to the periodicity in τ the AdS soliton is only locally asymptotically AdS. What does the dual field theory look like? First of all, the compact dimension allows for supersymmetry breaking by means of imposing antiperiodic boundary conditions for the fermions along the τ -circle. This introduces a mass gap in the fermionic spectrum such that the massive excitations decouple in the low energy effective theory. The compact dimension also breaks conformal invariance which can be qualitatively understood by the fact that the decoupling of massive fermion modes changes the β -function (see also [15]). Since the AdS soliton geometry is horizon-free, the entropy of the AdS soliton vanishes to first order in N 2 (i.e. in the classical limit), as one would expect for a field theory in a confined phase. A first order confinement-deconfinement phase transition at a certain temperature T dec. > 0 is expected to happen in the field theory [14]. On the gravity side, this corresponds to a Hawking-Page transition between the AdS soliton and a Schwarzschild-AdS black hole [20].", "pages": [ 5 ] }, { "title": "2.2 Energy-Momentum Correlators: Analytic Approach", "content": "We now want to use gauge/gravity duality to calculate energy-momentum two-point functions in the boundary field theory of the AdS soliton. This analysis can be done in some analogy to [21, 22] since the boundary metric is flat and we can therefore use the same simplifications that are used to calculate correlators in thermal N = 4 SYM. In case of the AdS soliton we can consider fluctuations of φ ≡ h 2 1 that propagate in the x 3 -direction. This will eventually allow us to calculate the field theory correlator 〈 T 1 2 T 1 2 〉 holographically. By the same reasoning as in the case of thermal N = 4 SYM [21], the remaining O (2) symmetry of the background metric ensures that φ decouples and satisfies a massless scalar wave equation in the bulk metric. Rescaling the coordinates as z = r 0 /r and y = r 0 τ/glyph[lscript] 2 , the AdS soliton metric (2.2) becomes where α ≡ r 0 /glyph[lscript] 2 and ds 2 Mink. is the line element of ( d -2) -dimensional Minkowski space. From the massless scalar wave equation in this metric, ∂ a ( √ -gg ab ∂ b φ ) = 0 , we find where ˆ glyph[square] is the wave operator on ( d -2) -dimensional Minkowski space. From now on, we concentrate on d = 6 , although generalizations are straightforward. The case d = 6 describes the dual of a gauge theory in 3 + 1 Minkowski spacetime (times S 1 ) and is therefore particularly interesting. We make the additional assumption that the solution is independent of y , i.e. homogeneous along the compact circle. This assumption is well justified for the long wavelength limit that we are mainly interested in. We now make the Fourier ansatz such that φ (0) ( q ) is the Fourier transform of the boundary value φ ( t, x 3 , 0) , i.e. we demand the normalization φ q ( z = 0) = 1 . Eq. (2.4) thus yields the mode equation where ( q µ ) = ( ω, 0 , 0 , q ) in the zero frequency limit (i.e. ω = 0 ) such that m 2 = -q 2 is the boundary mass. 1 This equation has the following fundamental power series solutions near z = 0 : where the recursion relations for the coefficients a n and b n can be found in appendix A. It has been argued that in real time thermal AdS/CFT the incoming wave boundary condition at the horizon should be used to single out a unique solution [21]. However, in the case of the AdS soliton, the solution that we find near z = 1 is not of the form of an incoming or outgoing wave. This is related to the fact that the AdS soliton does not have a horizon and we have to impose another boundary condition. As pointed out by Witten [8], an important condition that should be imposed for any acceptable solution is the Neumann condition dφ q /dρ = 0 at the IR floor z = 1 . Here, ρ is the natural coordinate in which the 'tip' of the metric at z = 1 looks like the origin of polar coordinates 2 . The above condition then just expresses the fact that φ q is smooth at z = 1 . In order to impose this boundary condition, we look for a power series solution to Eq. (2.6) near the IR-floor. We find the Frobenius solution which looks as follows near z = 1 : where the c n are also recursively given in appendix A. There, it is also explained that the second independent solution near z = 1 cannot satisfy the above described boundary conditions due to a divergent term ∼ log(1 -z ) . Switching from ( z, y ) to the coordinates ( ρ, ϕ ) which look like usual two-dimensional polar coordinates with origin at z = 1 , one can easily verify that the solution (2.9) indeed satisfies the above mentioned Neumann condition. Knowing that (2.9) is a good solution near z = 1 and that any solution near z = 0 can be expressed as a linear combination of the solutions (2.7) and (2.8), we need to find out what the global solution is. We thus write the solution φ (1) q ( z ) satisfying the Neumann condition at z = 1 in the basis of the two fundamental solutions near the boundary: with connection coefficients A , B which might depend on q 2 /α 2 but not on z . Near the boundary, φ (0) q, 1 and φ (0) q, 2 have the forms (2.7, 2.8). In Eq. (2.10) we kept the normalization φ (1) q (1) = 1 . This could be changed arbitrarily, but as we will see, the Green's function will only depend on the ratio B / A , so it would not be affected by another normalization. We can now use the same prescription as in the case of thermal N = 4 SYM in order to to calculate the stress-energy correlator 〈 T 1 2 T 1 2 〉 [5, 23]. For this purpose we need to write down the action of our bulk theory. For approaches to embed the present theory in string theory, see e.g. [24-26]. We focus on a universal gauge/gravity duality and consider just the low energy gravitational sector which is described by the action with Λ = -10 /glyph[lscript] 2 and the six-dimensional gravitational constant κ 6 . By inserting the metric perturbation given by φ , the part of the (on-shell) action which is quadratic in the perturbation can be written as with the factor 4 π/ 5 coming from integrating out the compact dimension. The prescription for Lorentzian signature says that we get the retarded Green's function according to the following rule [5]: where we used the Dirichlet condition φ q ( z = ε → 0) = 1 for the purpose of finding the overall normalization. The poles of the retarded Green's function are given by the zeros of A . On the other hand, setting A = 0 in the matching condition (2.10) would correspond to imposing a vanishing Dirichlet condition at the boundary z = 0 , which defines just the QNM of the AdS soliton geometry. This conforms with the general arguments in [6]. If we want to calculate the correlation function in position space, the QNM become the essential ingredient because we can replace the Fourier integral by a sum over residues. Since we will be mainly interested in the zero frequency limit, we set ω = 0 , such that Instead of integrating q along the real line, we close the contour with a semicircle in the upper complex q -plane. Due to the Fourier exponential e iqx 3 , the arc doesn't contribute to the integral and we are left with a sum over QNM residues: The calculation of the QN frequencies and of the residues is decribed in the following paragraphs. As mentioned above, the simple poles of 1 / A are just the QN frequencies. Following the general methods in [6], for these particular values of q 2 /α 2 , the expansion (2.9) of φ (1) q is normalizable and can be matched smoothly with a linear combination of φ (0) q, 1 and φ (0) q, 2 over the entire interval z ∈ (0 , 1) ; see also [18] for an application of similar methods. The motivation for this is the observation that the underlying analytic solution is a power series that converges on the entire interval, independent of whether we expand around z = 0 or z = 1 . One can easily check numerically or by investigating the pole structure of the mode equation (2.6), that the radius of convergence of the power series of φ (1) q in Eq. (2.9) reaches z = 0 , such that the connection coefficient A can be found by evaluating the matching equation (2.10) at z = 0 with the involved functions φ (0) q, 1 , φ (0) q, 2 and φ (1) q being given by their power series expansions: The discrete set { q n | A ( q 2 n /α 2 ) = 0 } turns out to be purely imaginary. We determine these zeros numerically, using partial sums of the explicit expansion of A in Eq. (2.16). A WKB estimate for the same eigenvalues has been given in [10], where the set of m 2 n = -q 2 n has been associated with the glueball masses in the dual field theory. Their result (rewritten in terms of our conventions and parameters) is: The first QN frequencies are listed in table 1. Even beyond the shown accuracy, the values that we obtain from our matching method agree precisely with what we find by just using a finite differences algorithm to solve Eq. (2.6) numerically. We observe that the WKB results agree with these exact values to an accuracy which is in accordance with Eq. (2.17), becoming better for larger n . Note that the complex conjugates of all the q n are also zeros of A . However, we will not need them because we close the contour of the integral in Eq. (2.14) in the upper half plane. The functions φ q n ( z/z 0 ) for n = 3 , 5 are plotted in fig. 2.2. The expansion around z = 0 is shown on the interval [0 , 0 . 95] , and the expansion around z = 1 is shown on [0 , 1] . We cannot distinguish them in the plots since they match perfectly over the entire common interval when q ∈ { q n , q ∗ n } n . For all other values of q the boundary condition φ q (0) = 0 makes it impossible to match the two expansions. We can also express B as a function of A . Since all the power series expansions converge at z = 1 / 2 , we can evaluate Eq. (2.10) at this point 3 , and find We can now calculate the residues of B / A at the QNM poles. We take Eq. (2.16) and plot ( q -q n ) / A , which is a smooth function in the vicinity of q n , and determine the value at q n with high numerical precision. This result is multiplied with the value of B at the particular point, B ( q 2 n /α 2 ) . This latter value can easily be obtained from Eq. (2.18) with the second summand in the numerator, which is proportional to A , set to zero. This method works very well at least for the lower lying QN frequencies. The values of the first six residues are shown in the last line of table 1. Using these results, we can evaluate the expression in Eq. (2.15): This result confirms our expectation of an exponentially decaying correlation function in the long wavelength limit. Physically this means that shear diffusion to infinity is strongly supressed. The contribution of the lowest QNM dominates the sum over exponentially decaying terms. Although the values of the residues grow (see table 1), the exponentials make every higher QN frequency q n completely insignificant compared to the contribution of q n -1 .", "pages": [ 5, 6, 7, 8, 9, 10, 11 ] }, { "title": "3 Instability of the Schwarzschild Soliton String", "content": "We will now introduce a modification of the AdS soliton which contains a black hole in the boundary metric. Before we calculate Green's functions and holographic transport properties in this novel geometry, it will turn out to be useful to carry out a stability analysis in terms of linearized perturbations.", "pages": [ 11 ] }, { "title": "3.1 Generalizations of the AdS Soliton", "content": "It has been noted in [16] that the d -dimensional AdS soliton metric (2.2) can very easily be generalized. In fact, one can replace the Minkowski metric η µν in (2.2) by any Ricci flat metric g µν ( x µ ) and the resulting metric is still a solution of Einstein's vacuum equations: Again, τ ∼ τ +4 πglyph[lscript] 2 / ( d -1) r 0 needs to be periodically identified. If we choose for g µν the four-dimensional Schwarzschild metric, Eq. (3.1) describes the Schwarzschild soliton string. The geometry looks like Schw 4 × S 1 stretched out in a string along a AdS radial direction r that caps off smoothly at a finite value r = r 0 . The two relevant physical scales are the Schwarzschild radius r s of the black hole and the radius of the compact dimension, i.e. L τ = 2 glyph[lscript] 2 / 5 r 0 . In order to simplify calculations considerably, we perform the following transformations: where r is the original radial AdS coordinate, and ρ is the original radial coordinate in Schw 4 . This brings the Schwarzschild soliton string metric in the form where we defined the parameter α ≡ r 0 r s /glyph[lscript] 2 . The Schwarzschild soliton string is supposed to describe the bulk dual of a strongly coupled field theory in a Schwarzschild black hole background. It inherits the important property of the AdS soliton that it caps off smoothly at the IR floor deep in AdS. A more naive choice for a gravity dual of a field theory in a black hole background would be the AdS black string [28]. However, a serious problem would be that the AdS black string is nakedly singular at the end point along the string direction. Also, as discussed in [27], trying to cover this singularity by a horizon does not eliminate the presence of nakedly singular surfaces. The Schwarzschild soliton string clearly solves these problems: There is no naked singularity due to the special geometry inherited from the AdS soliton.", "pages": [ 11 ] }, { "title": "3.2 Reduction to Gregory-Laflamme Instability", "content": "We want to show that a decomposition of a perturbation of the Schwarzschild soliton string geometry in tensor, vector and scalar modes with respect to the base manifold B = Schw 4 produces an instability in the tensor sector. The decomposition of the metric (3.3) reads where µ, ν run over the indices of the four-dimensional Schwarzschild metric ˆ g µν in the coordinates of (3.3), g AB describes the two-dimensional orbit space which is parameterized by ( y, z ) , and a ( z ) ≡ glyph[lscript]α/z . Let us start with a transverse tracefree (TTF) tensor perturbation of the form where a, b = 0 , . . . , 5 run over all coordinates. For a solution of the Einstein equations of the form the linearized Einstein equations read where ∆ L is the Lichnerowicz Laplacian and ˆ ∆ L is the Lichnerowicz operator on the four-dimensional base manifold B . We choose a harmonic dependence on y . Furthermore, the z -dependence is the same for each component h ab . Writing h µν ( x µ , y, z ) = χ µν ( x µ ) e iνy H ( z ) /z 2 , we find that the z -dependence of this equation separates: where m 2 is the constant that comes from separating the variable z . For ν = 0 (no excitation in the compact dimension) the second of these equations is exactly the same as that of a scalar propagating in the AdS soliton, Eq. (2.6). We can therefore use the results that we derived in section 2.2: There is an infinite tower of discrete values of m 2 /α 2 > 0 for which Eq. (3.9) has a regular solution. The existence of an unstable mode thus depends on the existence of such a mode which solves Eq. (3.8). But Eq. (3.8) is just the equation that governs perturbations of the five-dimensional Schwarzschild black string, i.e. the well-known Gregory-Laflamme problem [19]. We assume that the time dependence of χ µν is of the form e Ω t . Using the same numerical methods as for the solution of Eq. (3.9), we can show that there exists a threshold mode with Ω = 0 which corresponds to the maximum boundary mass m 2 max ≈ 0 . 768 such that all 0 < m 2 < m 2 max. give modes which grow in time. This is in accordance with the results of [19]. However, we want to investigate Eq. (3.8) in some more detail since we will need the form of the analytic solution in section 4. We use the spherically symmetric threshold Gregory-Laflamme ansatz: where we introduced the coordinate x ≡ 1 / ¯ r with 0 < x ≤ 1 . Imposing the TTF gauge condition on this ansatz, the perturbation equations (3.8) reduce to the following set of equations which form an effectively one-dimensional problem: Solving the first of these equations (e.g. numerically with a finite differences algorithm and regular boundary conditions) indeed yields as the only non-negative eigenvalues of this system the zero mode m 2 = 0 and the Gregory-Laflamme threshold mode m 2 ≈ 0 . 768 .", "pages": [ 12, 13 ] }, { "title": "3.3 Relation Between Physical Scales", "content": "We can now draw conclusions about the critical values of the size of the compact dimension and of the Schwarzschild radius of the boundary black hole. To this end, we need to find the simultaneous eigenvalues m 2 of Eq. (3.8) and m 2 /α 2 of Eq. (3.9). Therefore, we need to combine the threshold eigenvalue m 2 max. ≈ 0 . 768 of the Lichnerowicz operator on the background metric with the discrete values for m 2 /α 2 ≡ -q 2 /α 2 from section 2.2. This yields This list continues and, in fact, it gives an infinite tower of discrete values for α which asymptotically approach 0 . See fig. 3.3 The largest value of α corresponds to the most unstable mode. Indeed, since we used the critical value m max. for which an instability can occur, the values of α for unstable linearized tensor modes cannot exceed the value α crit. ≈ 0 . 216 . Writing this result in terms of the physically relevant scales r s and the radius of the compact dimension L τ = 2 glyph[lscript] 2 / 5 r 0 , we conclude that the critical ratio for an instability to occur is It is the interplay between these two parameters which determines the stability of the Schwarzschild soliton string. If the horizon radius r s is small compared to the size of the compact circle (i.e. not bigger than allowed by the above equation), then the Schwarzschild soliton string is unstable. Note that we cannot straightforwardly reverse this reasoning without studying the vector and scalar sectors of perturbations. In appendix B we carry out the analyses of linearized perturbations in the vector and scalar sectors of the decomposition of the Schwarzschild soliton string with respect to B = Schw 4 . We demonstrate that neither vector nor scalar modes give rise to an instability. Holographically, we interpret the instability in terms of the confinement scale of the gauge theory. Since we associate the temperature of the field theory with the inverse period of the compact direction τ , we obtain a temperature for the deconfinement transition T dec. = (1 /L τ ) crit. for given r s . This confirms the conjecture of [16] in a quantitative way. Note that if we had not chosen ν = 0 to solve Eq. (3.9), the QN frequencies would have been all shifted to a bigger value such that the critical value of α for an instability to occur would have turned out to be smaller. This can easily be confirmed by solving Eq. (3.9) numerically for different values of ν . Therefore, in order to find the true critical (i.e. maximal) value of α it was safe to assume that the perturbation is not excited along the compact dimension.", "pages": [ 13, 14, 15 ] }, { "title": "4 Stress Tensor Correlators from the Schwarzschild Soliton String", "content": "We turn now to the task of calculating stress-energy two-point functions in the boundary field theory dual to the Schwarzschild soliton string. The analysis is complicated due to a lack of translational invariance of the boundary field theory metric. First of all, we need again to decide how we use the symmetries of the problem to decompose the perturbation. As a starting point, we take the Gregory-Laflamme mode and Eqs. (3.11-3.14) from the previous section. This corresponds to a TTF tensor mode with respect to the base manifold B = Schw 4 , and it will eventually allow us to calculate the correlators the first of which is particularly interesting: Since T tt is the energy density, the associated two-point function describes heat transport in the field theory. According to the dictionary of gauge/gravity duality, we need to find the solution to the equation of motion that separated into the equations (3.8) (or equivalently Eq. (3.11)) and (3.9). Then we need to calculate the on-shell action quadratic in the gravitational perturbation, and take appropriate functional derivatives. To our knowledge, an analytic solution to Eq. (3.11) does not exist. However, two observations suffice to calculate the desired Green's functions in a suitable limit. First, we write the equation in Sturm-Liouville form: Then, Sturm-Liouville theory ensures that whatever the exact solution to this equation is, in an appropriate normalization the solutions to different eigenvalues m 2 are orthonormal with respect to the inner product where the weight function w ( x ) is given by the factor multiplying the eigenvalue in Eq. (4.2): This is the first observation. The second observation concerns the asymptotic solution to Eq. (3.11). For the solution which is normalizable as x → 0 and regular near the horizon, we find the following asymptotics: We can recover the full tensor mode h µν ( x µ , y, z ) = χ µν ( x, θ ) e iνy H ( z ) /z 2 by integrating over the boundary masses. In particular, the time-independent and y -homogeneous 11 -component is given by where H ( m ) ( z ) is the function H ( z ) for a particular eigenvalue m 2 . Note that the correlators which we will calculate refer to the Schwarzschild soliton string in its stable phase. Therefore the integral (4.6) runs over m 2 > m 2 max. , where m max. is the threshold mode for the instability to occur. Eq. (4.6) is the analog of the spatial Fourier transform that we used in the translationally invariant case (section 2.2): The functions h ( m ) 2 take the role of the orthonormal set of base functions, and the c ( m ) correspond to the boundary values with respect to which we will take functional derivatives. We define them in analogy to what we did in the case of the AdS soliton, such that the normalization of the 'mode' function H ( m ) ( z ) is given by H ( m ) (0) = 1 . Let us again assume ν = 0 , i.e. no excitation in the compact dimension. Then the Eq. (3.9) for H ( m ) ( z ) is the same equation as for the mode φ q ( z ) in the case of the AdS soliton. Thus we conclude from our analysis in section 2.2 that the properly normalized solution is These information can be used to determine the Green's function that is induced by h 2 . In principle, we proceed as in the case of thermal N = 4 SYM and the AdS soliton (c.f. [21] and section 2.2, respectively). However, the details are more complicated due to a lack of translational invariance. Using the equations of motion, the part of the six-dimensional Einstein-Hilbert action which is quadratic in h 2 can again be written as a five-dimensional integral over the AdS boundary term. We start by identifying the relevant terms in the quadratic part of the action (2.11): where the dots stand for terms which do not contain the right number of derivatives in z , so they will be irrelevant in the eventual application of the gauge/gravity recipe. Furthermore, we have defined We proceed by using partial integration in order to remove all derivatives of h 2 and eventually make the x -integration trivial. The first step consits of integrating by parts the term ∼ ( h ' 2 ) 2 . This yields a new term which contains a second derivative, h '' 2 . If we express the full solution h 2 ( x ) as an integral over 'mode functions' h ( m ) 2 as in Eq. (4.6), the resulting integral reads where also the trivial integrals over θ , φ , y have been performed, and the time-integral has been deleted. The time integral can be omitted because we consider time-independent solutions, i.e. the final Green's function will not be localized in time. If we replace h ( m 2 ) 2 '' , using the equation of motion (3.11), there are only terms left which look like either ∼ h ( m 1 ) 2 h ( m 2 ) 2 or ∼ h ( m 1 ) 2 h ( m 2 ) 2 ' . Since the coefficient of the h ( m 1 ) 2 h ( m 2 ) 2 ' -term depends on m 1 and m 2 in the same way as in the definition (4.6), we can absorb the integrals over the boundary masses such that this term becomes ∼ h 2 h ' 2 = ∂ x [( h 2 ) 2 / 2] and another partial integration in x can be performed. This finally gives an expression that does not contain any derivatives of h ( m i ) 2 : The function w ( x ) is precisely the weight function from Eq. (4.4) for which Sturm-Liouville solutions h ( m ) 2 ( x ) are orthonormal. The integral over x thus yields a δ -function in the eigenvalue, δ ( m 2 1 -m 2 2 ) . We still have to integrate out the z -dependence, and find the boundary value of the action. First, we integrate by parts the term ∼ H ( m ) H ( m ) '' , such that it contributes with a negative sign to the part ∼ ( H ( m ) ' ) 2 . This produces two more terms, one of which is a boundary term that is precisely cancelled by the Gibbons-Hawking contribution to the Einstein-Hilbert action (2.11). The second unwanted term is ∝ H ( m ) H ( m ) ' and it is therefore not relevant for the purpose of applying the gauge/gravity recipe. The remaining term ∼ ( H ( m ) ' ) 2 can be integrated as in the translationally invariant case, and we find Following the gauge/gravity recipe, we take functional derivatives and obtain the final result for the Green's function in position space 4 : where we evaluated the integral over m 2 as follows: The poles of B / A lie on the integration contour. Therefore, the integral is given by its principal value which can be evaluated by deforming the contour such that it goes around the poles in small semicircles in the lower half plane. Closing the contour at infinity, we can apply the Cauchy residue theorem similar to what we did in the case of the AdS soliton. In order to determine the asymptotic behavior of the bi-tensor G R ab,cd ( x 1 , x 2 ) , we consider the limiting case that one point lies at the horizon, x 1 → 1 , and the point where the response is measured is far away, x 2 → 0 . For this purpose, we can use the approximate solutions (4.5). From the asymptotic form of h ( m ) 2 ( x 2 → 0) in Eq. (4.5), we see that the arc at infinity does not contribute to the Cauchy integral over m 2 in Eq. (4.13). We obtain for the retarded Green's function: with N = -192 iπ 3 glyph[lscript] 4 α 4 C 0 C 1 /κ 2 6 . We can clearly see the exponential decay which means that transport to infinity of fluctuations of radial momentum density is strongly supressed. Also the dominant role of the QNM contributions { m n } is clearly visible. We have not strictly proven the convergence of the sum in Eq. (4.14). But although the residues grow with n , one can easily check that for small x the exponential prefactors decay much faster for increasing n . We would like to find the same qualitative behavior for heat transport, i.e. for the Green's function of energy density correlations, G R 00 , 00 ( x 1 , x 2 ) . This can easily be achieved, using that h 0 is given by the action of a first order differential operator D x on h 2 : where the exact form of the operator D x can be read off from Eq. (3.12). We see immediately that this kind of transformation preserves the qualitative properties of the Green's function and in particular its exponential decay as x 2 → 0 + . For definiteness, we nevertheless give the result for the other Green's functions in the asymptotic limit: The exponential decay of correlators of T 0 0 shows that heat transport due to small perturbations in energy density near the horizon is exponentially supressed as one goes radially towards infinity.", "pages": [ 15, 16, 17, 18, 19 ] }, { "title": "5 Summary and Discussion", "content": "In order to get a step closer towards studying QCD-like theories via gauge/gravity duality, we started by investigating the six-dimensional AdS soliton which is completely smooth and horizon-free, but it has one compact dimension which allows to break supersymmetry and conformal invariance. In the context of gauge/gravity duality, the AdS soliton serves as a toy model to study a strongly coupled field theory with broken supersymmetry in a confined phase. On the other hand, we generalized the AdS soliton by observing that it can be foliated along the holographic direction with any Ricci flat metric, in particular with a Schwarzschild black hole, giving rise to the Schwarzschild soliton string. The Schwarzschild soliton string serves as a toy model to understand the field theory dual to the AdS soliton on a non-trivial background spacetime. By calculating the correlator 〈 T 1 2 ( x 1 ) T 1 2 ( x 2 ) 〉 via gauge/gravity duality, we have shown that momentum and energy diffusion to infinity is exponentially supressed in the quantum theory on the AdS soliton boundary. This behavior originates from the fact that the QN frequencies appear as the poles of the momentum space Green's function, such that the position space Green's functions are dominated by these QNM contributions. The quantization of the modes is due to the particular geometry, which caps off smoothly at the IR floor. We leave it to future studies to elaborate on other modes of perturbations of the AdS soliton. Qualitatively, one expects very similar results, although the analysis is more involved because the vector and scalar perturbations have more than just one non-zero component. In the case of the Schwarzschild soliton string we found similar behavior of stress-energy correlators. The transport of energy and momentum density from the near horizon region towards infinity is exponentially supressed in the boundary field theory. The physical reason is again the particular bulk geometry which leads to the dominance of QNM in the Green's functions. We found that due to the presence of another scale (the Schwarzschild radius r s of the black hole), the Schwarzschild soliton string shows an interesting classical behavior under small perturbations. The stability analysis of the tensor mode can be reduced to a combination of the classical Gregory-Laflamme instability and the propagation of a scalar field in the AdS soliton. Depending on the relation between the two physically relevant scales (i.e. the size L τ of the compact dimension, and r s ), this spacetime is unstable if the Schwarzschild radius is small compared to L τ . We found this instability at the level of spherically symmetric linearized tensor perturbations. We have also shown that the vector and scalar modes with respect to the base manifold Schw 4 do not develop instabilities. Via the holographic duality, this instability is interpreted as a deconfinement transition in the field theory. It is interesting to explore further holographic duals of field theory states on curved spacetimes. In [16], the possibility of black droplets and black funnels has been discussed. The black funnel is a solution with a single connected horizon that is dual to the HartleHawking state of a strongly coupled plasma around a black hole. Such solutions have recently been constructed numerically [31]. Black droplets, on the other hand, have been conjectured to describe the final state of the AdS black string instability [32] and are still to be constructed explicitly. Understanding the endpoint of the Schwarzschild soliton string instability in terms of such solutions might then allow for a complete survey of the deconfinement transition in a strongly coupled plasma in terms of different bulk geometries.", "pages": [ 19, 20 ] }, { "title": "Acknowledgments", "content": "It is a pleasure to thank my advisor Don Marolf for his support and guidance. I am also grateful to Jorge Santos for very useful discussions and substantial help with the numerics, and Mukund Rangamani for comments on a draft of this paper. I thank the University of California, Santa Barbara for their hospitality during the time when most of this work has been done. This research has been financially supported by funds from ETH Zurich and the University of California, by the US NSF grant PHY-0855415, and by the German Nationial Academic Foundation.", "pages": [ 20 ] }, { "title": "A Power Series Solution for Scalar Field in the AdS Soliton", "content": "In this appendix, we outline how a power series ansatz leads to the solutions (2.7, 2.8). Plugging the ansatz φ (0) q ( z ) = ∑ ∞ n =0 A n z n into Eq. (2.6), we find that the coefficients A 0 and A 5 are free, while all others are given by This yields the two Frobenius solutions (2.7) and (2.8) with exponents 0 and 5, respectively. The coefficients a n and b n are defined by the A n by setting either A 0 = 1 , A 5 = 0 or vice versa: On the other hand, we can also make an ansatz for a power series solution near the IR floor z = 1 , i.e. φ (1) q ( z ) = ∑ ∞ n =0 C n (1 -z ) n . This ansatz gives only one free coefficient C 0 and all others as proportional to it: This yields the solution (2.9) near z = 1 , where we defined again coefficients that are independent of the overall scaling C 0 : c n ≡ C n /C 0 . We find only one solution of this form near z = 1 because the second solution does not have the form of a simple power series. The indicial equation at z = 1 has zero as a double root, so the second independent solution near the IR-floor contains a term of the form ∼ log(1 -z ) φ (1) q ( z ) . Since this is divergent as z → 1 and does not satisfy the Neumann condition, we discard this solution.", "pages": [ 20, 21 ] }, { "title": "B Vector and Scalar Perturbations of the Schwarzschild Soliton String", "content": "We want to outline the analysis of linearized vector and scalar channel perturbations of the Schwarz-schild soliton string with respect to the base manifold B = Schw 4 . We will show that linearized, spherically symmetric vector and scalar perturbations do not give rise to an instability.", "pages": [ 21 ] }, { "title": "B.1 Vector Perturbations", "content": "Following the general methods for linearized gravitational perturbation theory as developed in [29, 30], we start with a vector harmonic on Schw 4 which we write as glyph[negationslash] where Greek indices and quantities with a hat refer to B = Schw 4 . Keeping the spherical symmetry, this vector harmonic does not have components on the sphere, and its remaining components are independent of φ . Also, the mode is assumed to be time-independent, which amounts to considering the threshold mode for which an instability could occur. From solving the condition ˆ ∇ µ V µ = 0 , one finds immediately that V 1 (¯ r, θ ) = ˜ v 1 ( θ )¯ r -2 (1 -1 / ¯ r ) -1 . We see that this solution is not regular at ¯ r = 1 . Indeed, also from looking at the defining equation for a harmonic vector on B , ( ˆ ∆ + k 2 V ) V µ = 0 , one can straightforwardly infer that for k 2 V = 0 , V 1 must vanish identically. The only remaining non-trivial condition from ( ˆ ∆+ k 2 V ) V µ = 0 reads which is just the eigenvalue equation for the Laplacian on S 2 . Considering one particular mode with 'angular momentum' l , we write V 0 (¯ r, θ ) = P l (cos θ ) v 0 (¯ r ) , and obtain The perturbation of the Schwarzschild metric which can be built from the harmonic vector V µ can be written as: with the tensor It turns out to be useful to formulate the problem in terms of the variables which are introduced in [29] for the gauge invariant master equation formalism. A gauge transformation adapted to the symmetry of the vector mode is described by a gauge vector ξ a = ( ξ µ , ξ A ) with ξ µ = aL ( x A ) V µ , ξ A = 0 . A gauge invariant variable is then given by where ∇ A is the covariant derivative with respect to the two-dimensional orbit space. Fixing the gauge with the variable H T is effectively set to zero and the perturbation is now TTF. According to Eq. (3.7), the linearized Einstein equations reduce for a TTF perturbation to ∆ L h ab = 0 . From this, one can immediately see that F y = 0 such that the Lichnerowicz equation reduces to a radial equation in ¯ r , and an equation for F z ( z ) . The radial equation is exactly the same as (B.3), and the equation for z reads This equation with regularity boundary conditions allows for discrete modes with k 2 V /α 2 < 0 , very similar to what we found for tensor modes. Since α 2 can by definition not be negative, the question is: Does Eq. (B.3) have regular solutions with negative eigenvalues k 2 V < 0 ? If so, then they would correspond to unstable vector modes. However, applying a finite differences algorithm, we find that independent of what value l ∈ { 0 , 1 , 2 , . . . } takes, the equation does not have any eigenvalues k 2 V < 0 that would correspond to an instability.", "pages": [ 21, 22 ] }, { "title": "B.2 Scalar Perturbations", "content": "The scalar harmonics S are defined by ( ˆ ∆+ k 2 S ) S = 0 . Assuming that our perturbation does not break the symmetry of the underlying 2-sphere, i.e. S is an eigenmode of the spherical Laplacian and can thus be written as S = P l (cos θ ) s (¯ r ) , the defining equation for S becomes We can again find the allowed eigenvalues k 2 S numerically. As it was the case for the vector modes, we only find eigenvalues k 2 S ≥ 0 . A comparison with vector modes thus already stronlgy suggests that there is no instability. However, for definiteness, we complete the perturbative analysis in the following paragraphs. We will show that all eigenvalues of the perturbation equations satisfiy k 2 S /α 2 ≤ 0 which is not compatible with the fact that Eq. (B.9) did not yield any negative eigenvalues k 2 S < 0 . Therefore the only consistent eigenvalue of the full problem will be k 2 S = 0 such that there is no threshold mode and no instability. As in [29], we construct the following vectors and tensors build out of the scalar harmonic S : where ˆ g is the four-dimensional Schwarzschild metric. From these building blocks we can construct a symmetry adapted scalar perturbation: We proceed as in the case of vector harmonics. First, we rewrite the degrees of freedom of the perturbation h ab in terms of the following (gauge invariant) variables: We fix the gauge by using the symmetry adapted gauge vector This yields the following simple expression for the scalar perturbation: We use this perturbation and study the resulting Einstein field equations (3.7). This yields a set of equations for F AB ( z ) and F ( z ) . After some algebraic operations, one finds that these equations can be decoupled: Solving these equations is slightly more complicated than in the vector and tensor sectors due to the fact that the eigenvalue k 2 S /α 2 appears quadratically in Eq. (B.18). We solve Eq. (B.18) directly by 'shooting' the eigenvalue such that the regularity boundary conditions are satisfied. We find that (just like in the case of vector and tensor perturbations) there are no positive eigenvalues k 2 S /α 2 > 0 . Instead, there are infinitely many discrete negative eigenvalues in addition to the eigenvalue 0 . Due to the lack of a compatible negative eigenvalue of Eq. (B.9), we conclude that at the level of linearized perturbations which have the previously assumed form in the scalar sector with respect to Schw 4 , there are no unstable modes.", "pages": [ 22, 23, 24 ] } ]

[ { "title": "On the space of generalized fluxes for loop quantum gravity 1", "content": "Bianca Dittrich, a,b Carlos Guedes, a Daniele Oriti a [email protected], [email protected], [email protected] Abstract: We show that the space of generalized fluxes - momentum space - for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections - position space - due to the non-abelianess of the gauge group SU(2). From the abelianization of SU(2), U(1) 3 , we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse-graining of the underlying graphs. We comment on the applications to loop quantum cosmology, in particular, how the characterization of the Bohr compactification of the real line as a projective limit opens the way for a similar analysis for LQC. Keywords: Loop quantum gravity, loop quantum cosmology, cylindrical consistency, inverse limit, direct limit", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In standard quantum mechanics in flat space, the standard Fourier transform relates the two (main) Hilbert space representations in terms of (wave-)functions of position and of momentum, defining a duality between them. 1 The availability of both is of course of practical utility, in that, depending on the system considered, each of them may be advantageous in bringing to the forefront different aspects of the system as well as for calculation purposes. Difficulties in defining a Fourier transform and a momentum space representation arise however as soon as the configuration space becomes non-trivial, in particular as soon as curvature is introduced, as in a gravitational context 2 . No such definition is available in the most general case, that is in the absence of symmetries. On the other hand, in the special case of group manifolds or homogeneous spaces, when there is a transitive action of a group of symmetries on the configuration space, harmonic analysis allows for a notion of Fourier transform in terms of irreducible representations of the relevant symmetry group. This includes the case of phase spaces given by the cotangent bundle of a Lie group, when momentum space is identified with the corresponding (dual of the) Lie algebra, as it happens in loop quantum gravity [3]. However, a different notion of group Fourier transform adapted to this group-theoretic setting has been proposed [4-9], and found several applications in quantum gravity models (see [10]). As it forms the basis of our analysis, we will introduce it in some detail in the following. Its roots can be traced back to the notions of quantum group Fourier transform [6] and deformation quantization, being a map to non-commutative functions on the Lie algebra endowed with a starproduct. The star-product reflects faithfully the choice of quantization map and ordering of the momentum space (Lie algebra) variables [11]. As a consequence of this last point, observables and states in the resulting dual representation (contrary to the representation obtained by harmonic analysis) maintain a direct resemblance to the classical quantities, simplifying their interpretation and analysis. Let us give a brief summary of the LQG framework. For more information about the intricacies of LQG refer to the original articles [12-15] or the comprehensive monograph [3]. Loop quantum gravity is formulated as a symplectic system, where the pair of conjugate variables is given by holonomies h e [ A ] of an su (2)-valued connection 1-form A (Ashtekar connection) smeared along 1-dimensional edges e , and densitized triads E smeared across 2-surfaces (electric fluxes). The smearing is crucial for quantization giving mathematical meaning to the distributional Poisson brackets, which among fundamental variables are where a, b, c, . . . are tangent space indices, and i, j, k, . . . refer to the su (2) Lie algebra. The same smearing leads to a definition of the classical phase space as well as of the space of quantum states based on graphs and associated dual surfaces. Since the smeared connection variables commute LQG is naturally defined in the connection representation. All holonomy operators can be diagonalized simultaneously and we thus have a functional calculus on a suitable space of generalized connections A . Avery important point is that despite the theory being defined on discrete graphs and associated surfaces, the set of graphs defines a directed and partially ordered set. Hence, refining any graph by a process called projective limit we are able to recover a notion of continuum limit. We will come back to this in the following, as we will attempt to define a similar continuum limit in terms of the conjugate variables. Even though the triad variables Poisson commute, that is no longer the case for the flux variables: For instance, for operators smeared by two different test fields f, g on the same 2-surface S we have if the Lie bracket [ f, g ] i = /epsilon1 i jk f j g k fails to vanish. Even smearing along two distinct surfaces, the commutator is again non-zero if the two surfaces intersect and the Lie bracket of the corresponding test fields is non-zero on the intersection. At first thought to be a quantization anomaly, in [16] it was shown to be a feature that can be traced back to the classical theory. Thus, a simple definition of a momentum representation in which functions of the fluxes would act as multiplicative operators is not available. In the simplest case, for a given fixed graph, fluxes across surfaces dual to a single edge act as invariant vector fields on the group, and have the symplectic structure of the su (2) Lie algebra. Therefore, after the smearing procedure, the phase space associated to a graph is a product over the edges of the graph of cotangent bundles T ∗ SU(2) /similarequal SU(2) × su (2) ∗ on the gauge group. Notwithstanding the fundamental non-commutativity of the fluxes, and taking advantage of the resulting Lie algebra structure and of the new notion of non-commutative Fourier transform mentioned above, in [17], a flux representation for loop quantum gravity was introduced. The work we are presenting here is an attempt to give a characterization of what the momentum space for LQG, defined through these new tools, should look like and how it can be constructed. Before presenting our results, let us motivate further the construction and use of such momentum/flux representation in (loop) quantum gravity (for an earlier attempt to define it, see [18]). First of all, any new representation of the states and observables of the theory will in principle allow for new calculation tools that could prove advantageous in some situations. Most importantly, however, is the fact that a flux representation makes the geometric content of the same states and observables (and, in the covariant formulation, of the quantum amplitudes for the fundamental transitions that are summed over) clearer, since the fluxes are nothing else than metric variables. For the same reason, one would expect a flux formulation to facilitate the calculation of geometric observables and the coarse graining of states and observables with respect to geometric constraints [19, 20] (we anticipate that the notion of coarse graining of geometric operators will be relevant also for our analysis of projective and inductive structures entering the construction of the space of generalized fluxes). The coupling of matter fields to quantum geometry is also most directly obtained in this representation [21]. Recently, new representations of the holonomy-flux algebra have been proposed for describing the physics of the theory around (condensate) vacua corresponding to diffeo-covariant, non-degenerate geometries [22], which are defined in terms of non-degenerate triad configurations, and could probably be developed further in a flux basis. As flux representations encode more directly the geometry of quantum states, such representations might be useful for the coarse graining of geometrical variables [23-25]. In particular, [26] presents a coarse graining in which the choice of representation and underlying vacuum is crucial. Finally, we recall that the flux representation has already found several applications in the related approaches of spin foam models and group field theories [10, 27-32], as well as in the analysis of simpler systems [9, 33]. This flux representation was found by defining a group Fourier transform together with a /star -product on its image, first introduced in [4-7] in the context of spin foam models. In this representation, flux operators act by /star -multiplication, and holonomies act as (exponentiated) translation operators. Using the projective limit construction of LQG, the group Fourier transform F γ was used to push-forward each level to its proper image, and in [17] the following diagram was shown to commute identifying the Hilbert space in the new triad representation as the completion of ( ∪ γ H /star,γ ) / ∼ . π and π /star are the canonical projections with respect to the equivalence relation ∼ which is inherited from the graph structure (see section 2). In the connection representation we know that the (kinematical) Hilbert space is given by ( ∪ γ H γ ) / ∼ /similarequal L 2 ( A , d µ 0 ), where A is the space of generalized connections and d µ 0 the Ashtekar-Lewandowski measure. Even if the Hilbert space in the triad representation can be defined by means of the projective limit, one would like to have a better characterization of the resulting space in terms of some functional calculus of generalized flux fields. Hence, the natural question is, can we write ( ∪ γ H /star,γ ) / ∼ /similarequal L 2 ( E , d µ /star, 0 ), for an appropriate space of generalized fluxes E and measure d µ /star, 0 ? This is precisely the issue we tackle in this paper. We will see here that there are several obstructions to such a construction. First of all, when translating the projective limit construction on the connection side over to the image of the Fourier transform the notion of cylindrical consistency is violated whenever the gauge group is non-abelian. Thus, it is not possible to define the relevant cylindrically consistent C ∗ -algebra. This result is crucial since the space of generalized fluxes would arise as the corresponding spectrum 3 of this algebra. We note that even if it was possible to define such a cylindrically consistent C ∗ -algebra, the characterization that we are looking for would require a generalization of the Gel'fand representation theorem to noncommutative C ∗ -algebras, as the multiplication in the algebra is a noncommutative /star -product. Nevertheless, we can still learn something about the space of generalized fluxes by considering the abelianization of SU(2), that is, U(1) 3 . In fact, it has been shown that the quantization of linearized gravity leads to the LQG framework with U(1) 3 as gauge group [34]. It is even enough to work with one single copy of U(1), since the case G = U(1) 3 is then simply obtained by a triple tensor product: not only the kinematical Hilbert space has this simple product structure, but also the respective gauge-invariant subspaces decompose the same way [35]. The outline of the paper is the following: in the next section we briefly review the projective limit structure of LQG together with the notion of cylindrical consistency. Section 3 is the bulk of the paper. In 3.1, we start by showing the non-cylindrical consistency of the /star -product for nonabelian gauge groups, and proceed to the U(1) case. In subsection 3.2 we show that the space of generalized fluxes for U(1)-LQG cannot be constructed as a projective limit, but in subsection 3.3 we show how it arises as an inductive limit. The space of functions is then determined by pull-back giving rise to a suitable proC ∗ -algebra. In the conclusion 4 we make some remarks on the analysis made here, and give an outlook on further work, in particular on the possibility of constructing a theory of loop quantum gravity tailored to the flux variables, and how the characterization of the Bohr compactification of the real line as a projective limit opens the way for a similar analysis for loop quantum cosmology.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 The notion of cylindrical consistency in a nutshell", "content": "After identifying the space of generalized connections, i.e. the set A = Hom( P , G ) of homomorphisms from the groupoid of paths P to the group G = SU(2), as the appropriate configuration space for loop quantum quantum gravity, the next step is to find the measure d µ 0 on this space to define the kinematical Hilbert space H 0 = L 2 ( A , d µ 0 ). This measure, called the Ashtekar-Lewandowski measure, which is gauge and diffeomorphism invariant, is built by realizing H 0 as an inductive limit (also called direct limit) of Hilbert spaces H γ = L 2 ( A γ , d µ γ ) associated to each graph γ (for the definition of and more details on projective/inductive limits, refer to the B). The inductive structure is inherited by pullback from the projective structure in A γ = Hom( γ, G ), where γ ⊂ P is the corresponding subgroupoid associated with γ . A γ is set-theoretically and topologically identified with G | γ | , with | γ | the number of edges in γ , and d µ γ defines the Haar measure on G | γ | . Then, the identification of A with the projective limit (inverse limit) of A γ gives where ∼ is an equivalence relation that determines the notion of cylindrical consistency (independence of representative) for the product between functions. We remark that this result relies heavily on the Gel'fand representation theorem which states that any commutative C ∗ -algebra A is isomorphic to the algebra of continuous functions that vanish at infinity over the spectrum of A , that is, A /similarequal C 0 (∆( A )). 4 The construction in (2.1) is done at the level of C ( A γ ) in the sense that the spectrum of the commutative C ∗ -algebra ∪ γ C ( A γ ) / ∼ (the algebra of cylindrical functions) coincides with the projective limit of the A γ 's. Compactness of A γ guarantees C ( A γ ) to be dense in L 2 ( A γ , d µ γ ). The existence of the measure is provided by the Riesz representation theorem (of linear functionals on function spaces) which basically states that linear functionals on spaces such as C ( A ) can be seen as integration against (Borel) measures. The linear functional on C ( A ) is then constructed by projective techniques through the linear functionals on C ( A γ ). As already remarked, the existence of the projective limit guarantees the existence of a continuum limit of the theory despite it being defined on discrete graphs, at least at a kinematical level. Since it will be important later on, let us describe in more detail the system of homomorphisms that give rise to the inductive/projective structure. Recall that the set of all embedded graphs in a (semi-) analytic manifold defines the index set over which the projective limit is taken. A graph γ = ( e 1 , . . . , e n ) is a finite set of analytic paths e i with 1 or 2-endpoint boundary (called edges), and we say that γ is smaller/coarser than a graph γ ' (thus, γ ' is bigger/finer than γ ), γ ≺ γ ' , when every edge in γ can be obtained from a sequence of edges in γ ' by composition and/or orientation reversal. Then ( A γ , ≺ ) defines a partially ordered and directed set, and we have, for γ ≺ γ ' , the natural (surjective) projections p γγ ' : A γ ' → A γ (restricting to A γ any morphism in A γ ' ). These projections go from a bigger graph to a smaller graph and they satisfy We thus have an inverse (or projective) system of objects and homomorphisms. These projections can be decomposed into three elementary ones associated to the three elementary moves from which one can obtain a larger graph from a smaller one compatible with operations on holonomies: (i) adding an edge, (ii) subdividing an edge, (iii) inverting an edge. See Figure 1. Then, The pullback of these defines the elementary injections for H γ which determines the basic elements in H 0 in the same equivalence class, i.e. [ f ] ∼ = { f, add · f, sub · f, inv · f, add · sub · f, add · inv · f, . . . } . Since p ∗ γγ ' : H γ →H γ ' go from a smaller graph to a bigger graph and satisfy we have a direct (or inductive) system of objects and homomorphisms. Let us check that the pointwise product in H 0 is indeed cylindrically consistent. Let f, f ' ∈ H 0 . By definition, we find graphs γ, γ ' and representatives f γ ∈ H γ , f ' γ ' ∈ H γ ' such that f = [ f γ ] ∼ , f ' = [ f ' γ ' ] ∼ . Embed γ, γ ' in the common larger graph γ '' , that is, γ, γ ' ≺ γ '' . Then f γ '' = p ∗ γγ '' f γ , f ' γ '' = p ∗ γ ' γ '' f ' γ ' , and p ∗ γγ '' f γ = p ∗ γ ' γ '' f γ ' , p ∗ γγ '' f ' γ = p ∗ γ ' γ '' f ' γ ' . Thus, i.e. f γ f ' γ ∼ f γ ' f ' γ ' , and the pointwise product does not depend on the representative chosen. In terms of add , sub , inv this amounts to for f, f ' ∈ H e . For a beautiful account on the structure of the space of generalized connections, refer to the article [36].", "pages": [ 5, 6, 7 ] }, { "title": "3 The space of generalized fluxes", "content": "This section constitutes the main part of the paper. The goal here is to define the analogue of the space of generalized connections A on the 'momentum side' of the LQG phase space, that is the flux variables. The resulting space will be called the space of generalized fluxes E . As said in the introduction, the natural approach for constructing such space fails for G = SU(2) (and any non-abelian group). The cylindrical consistency conditions used in defining the space of generalized connections are tailored to operations on holonomies ( h e 1 · e 2 = h e 1 h e 2 , h e -1 = h -1 e ) and the non-abelianess of the group makes the translation to similar conditions on fluxes ill-defined. To show explicitly this difficulty is our first result. We are then constrained to work with the abelianization of SU(2), U(1) 3 , or rather U(1). Pushing-forward under the Fourier transform the projective limit construction of the space of generalized connections would lead to a definition of the space of generalized fluxes also as a projective limit. However, we will show that there is also an obstacle to this construction hitting again on the fact that the fluxes are significantly different from the connections. Luckily, for G = U(1) the space of generalized connections is a true group opening in this way the possibility for a dual construction where the arrows are reversed. Thus, the projective limit is traded by an inductive limit and the previous problem disappears. The space of functions is finally defined by pull-back giving rise to a projective limit of C ∗ -algebras. Let us also note, at this point, that the U(1) case carries a further simplification, given by the fact that in this case the flux representation can be shown to be essentially equivalent to the charge network representation. We will clarify better in what sense this is true in section 3.3.", "pages": [ 7, 8 ] }, { "title": "3.1 The problems with SU(2)", "content": "Let us shortly summarize the commuting diagram from [17] For a single copy of SO(3) we define the noncommutative Fourier transform as the unitary map F from L 2 (SO(3) , d µ H ), equipped with Haar measure d µ H (recently generalized to SU(2) [37]), onto a space L 2 /star ( R 3 , d µ ) of functions on su (2) ∼ R 3 equipped with a noncommutative /star -product, and the standard Lebesgue measure: where d g is the normalized Haar measure on the group, and e g the appropriate plane-waves. The product is defined at the level of plane-waves as and extended by linearity to the image of F . As mentioned in the introduction, this non-commutative product is the result of a specific quantization map chosen for the Lie algebra part of the classical phase space [11]. The /star -product is crucial since it gives the natural algebra structure to the image of F , which is inherited from the convolution product in L 2 (SO(3)), that is, for f, f ' ∈ H e where the convolution product is as usual We say that the /star -product is dual to convolution. Extending to an arbitrary graph gives a family of unitary maps F γ : H γ → H /star,γ labelled by graphs γ , where H γ := L 2 ( A γ , d µ γ ) /similarequal L 2 (SO(3) | γ | , d µ γ ) and H /star,γ := L 2 /star ( R 3 ) ⊗| γ | . Thus, we have the unitary map and we want now to extend this to the full Hilbert space H 0 = ∪ γ H γ / ∼ . First, the family F γ gives a linear map ∪ γ H γ → ∪ γ H /star,γ . In order to project it onto a well-defined map on the equivalence classes, we introduce the equivalence relation on ∪ γ H /star,γ which is 'pushed-forward' by F γ : That is, we have the injections q /star,γγ ' : H /star,γ → H /star,γ ' for all γ ≺ γ ' defined dually by q /star,γγ ' F γ := F γ p ∗ γγ ' . Using the definition, it is easy to see that they satisfy i.e. we have an inductive system of objects and homomorphisms. Finally, completion is now given with respect to the inner product pushed-forward by ˜ F . That is, for any two elements u, v of the quotient with representatives u γ ∈ H /star,γ and v γ ' ∈ H /star,γ ' the inner product is given by choosing a graph γ '' with γ, γ ' ≺ γ '' and elements u γ '' ∼ u γ and v γ '' ∼ v γ ' in H /star,γ '' , and by setting Since F γ are unitary transformations, the r.h.s. does not depend on the representatives u γ , v γ ' nor on the graph γ '' . We thus have the complete definition of the full Hilbert space H /star, 0 = ( ∪ γ H /star,γ ) / ∼ as an inductive limit. In [17], the question of cylindrical consistency of the star product in H /star, 0 was not posed and, as a Hilbert space, H /star, 0 makes perfect sense. However, to give the desired intrinsic characterization for H /star, 0 analogous to H 0 , that is, to write H /star, 0 as L 2 ( E , d µ /star, 0 ) for some space of generalized fluxes E and measure d µ /star, 0 , we need to make sure that H /star, 0 is well-defined as a C ∗ -algebra, in particular, that the /star -product is cylindrically consistent. As we have seen in section 2 this amounts to the validity of Eqs. (2.3) for the /star -product, or by duality, for the convolution product. Then, for add we have and So, it works for add . sub gives and which matches (3.2) if and only if G is abelian. Lastly, inv and which again matches (3.3) if and only if G is abelian. Thus, the /star -product is not cylindrically consistent and consequently H /star, 0 is not a C ∗ -algebra. We emphasize that this result is independent of the specific Fourier transform used or of the specific form of the plane-waves, that is, any other quantization map chosen for the space of classical fluxes would have led to the same result. In order to have a well-defined algebra structure on the image of the Fourier transform we always need the multiplication to be dual to convolution, which, as we have just seen, is not cylindrically consistent unless the group G is abelian. Let us also stress that similar issues would arise whenever one tries to define a kinematical continuum limit in variables dual to the connection and associated to surfaces. In particular, they would appear even using representation variables resulting from the Peter-Weyl decomposition, as they do in recent attempts to define refinement limits for the 2-complexes in the spin foam context [38-43]. As already said, the cylindrical consistency conditions are tailored to operations on holonomies, hence it is not too surprising that fluxes should not satisfy the same 'gluing' conditions. Indeed, LQG kinematics treats connections and fluxes very asymmetrically. To understand better this asymmetry we will make the framework more symmetric by considering the abelianization of SU(2), U(1) 3 , where we can go further with the construction and still learn something about the space of generalized fluxes.", "pages": [ 8, 9, 10 ] }, { "title": "3.2 The space of generalized fluxes by group Fourier transform: the abelian case", "content": "Loop quantum gravity with U(1) as gauge group is simpler in many aspects. In particular, the U(1) group Fourier transform and the /star -product reduce to the usual Fourier transform on the circle and the pointwise product, respectively. To avoid detouring too much from the main ideas of the text, we relegate to the A an in-depth analysis of the U(1) group Fourier transform, where this is shown. Hence, F is the unitary map from L 2 (U(1)) /owner f onto /lscript 2 ( Z ), the space of square-summable sequences (which has C 0 ( Z ) as a dense subspace), and the product on the image of F is the usual pointwise product ( uv )( x ) = u ( x ) v ( x ) for u, v ∈ /lscript 2 ( Z ). However, bear in mind that the U(1) group Fourier transform is fully defined on R . That is, the conjugate variables to U(1) connections - the fluxes - are genuinely real numbers. It happens to be a feature of the U(1) group Fourier transform that it is sampled by its values on the integers 5 - cf. A. The extension to an arbitrary graph and the projection onto the equivalence classes works out as in the previous subsection; the main difference is the abelian ' /star -product' which now coincides with the pointwise product. It is still dual to convolution, but now that the group is abelian it is cylindrically consistent. We have the following result:", "pages": [ 10, 11 ] }, { "title": "Theorem 3.1. H /star, 0 is a non-unital commutative C ∗ -algebra.", "content": "Proof. Strictly speaking, we are now looking at the Hilbert spaces H /star,γ = /lscript 2 ( Z | γ | ) at the algebraic level C 0 ( Z | γ | ) (which form dense subspaces). Each of the spaces C 0 ( Z | γ | ) is a non-unital commutative C ∗ -algebra with respect to complex conjugation, sup-norm, and pointwise multiplication. Then, it just remains to check that the operations on the full algebra ∪ γ C 0 ( Z | γ | ) / ∼ , such as the product and the norm do not depend on the representative in each equivalence class, i.e. they are cylindrically consistent. For the ( /star -)product this amounts to The action of add , sub , and inv is given below, Eq. (3.4). Explicitly, for add , we have sub and inv can be shown similarly. The norm satisfies and is thus also well-defined. The definition of H /star, 0 as an inductive limit of abelian C ∗ -algebras sets us almost on the same footing as the standard kinematical Hilbert space for loop quantum gravity H 0 . The method used to determine the spectrum of the C ∗ -algebra relies heavily on the fact that if ( A α , p ∗ αβ , I ) is an inductive family of abelian C ∗ -algebras A α , where I is a partially ordered index set, the inductive limit A is a well-defined abelian C ∗ -algebra whose spectrum ∆( A ) is a locally compact Hausdorff space homeomorphic to the projective limit of the projective family (∆( A α ) , p αβ , I ). As for the space of cylindrical functions, the inductive system of homomorphisms splits into three elementary ones defined dually by q /star,γγ ' := F γ p ∗ γγ ' F -1 γ : which again determine the elements in ∪ γ H /star,γ / ∼ in the same equivalence class, i.e. [ u ] ∼ = { u, add /star · u, sub /star · u, inv /star · u, . . . } . Eqs. (3.4) define our inductive system of functions through the injections q /star,γγ ' . Recall that the usual procedure for LQG starts with the projections (2.2), and the injections at the level of functions are simply defined by pullback. Here we already have the system of injections (3.4) and, should they exist, we want to determine the system of projections p /star,γγ ' that give rise to these injections. That is, are the injections q /star,γγ ' 's the pullback of some projections p /star,γγ ' 's: q /star,γγ ' = p ∗ /star,γγ ' ? For the three elementary operations, we are looking for projections p /star, add , p /star, sub , and p /star, inv such that holds. Using the fact that u ∈ C 0 and thus vanish at infinity, we can naively define the projections as /negationslash /negationslash However, the above is rather formal and one runs into several technical problems in trying to justify the use of the infinity as an element of the target space of the projections. First of all, infinity does not belong to Z , so strictly speaking (3.5) does not define a map. One could consider the one-point compactification (or Alexandroff extension) of the integers but this amounts to change the algebra itself (as it now becomes unital) and functions will not vanish at infinity anymore. One could try to make the limiting procedure precise by using the very definition of u ∈ C 0 ( Z | γ | ), that is, since Z | γ | is locally compact, there exists a compact set K ⊆ Z | γ | such that | u ( x ) | < /epsilon1 for every /epsilon1 > 0 and for every x ∈ Z | γ | \\ K . But this means that the whole procedure would only allow an indirect characterization of the underlying space (of generalized fluxes) through the behaviour of the space of functions. As a result, any such definition would fail to provide the intrinsic characterization of E that we are looking for. Even though we do not have a proof that such projections p /star,γγ ' do not exist, it seems rather unnatural to force such a construction since the structure of connections and fluxes is significantly different. Indeed, one is the Fourier transform of the other and in the Fourier transform 'arrows' are naturally reversed. In particular projections are changed into inductions and vice versa, at least in this abelian case. Indeed, we will see in the next subsection how reversing the arrows in the categorical sense makes it possible to define the space of generalized fluxes as an inductive limit 6 .", "pages": [ 11, 12 ] }, { "title": "3.3 The space of generalized fluxes by duality", "content": "The framework of U(1)-LQG provides a different strategy for determining the space of generalized fluxes. The crucial point here is that for G = U(1) the space of generalized connections A is a true group, and the following theorem, giving the natural way of trading a projective system by an inductive system, is applicable. Theorem 3.2. Suppose A γ are abelian groups, and let A be the projective limit with projections p γ : A → A γ . Then, the dual group ̂ A equals the inductive limit of the dual groups ̂ A γ . Proof. Let p γ : A → A γ be the projections. Then, such that ˆ p γ ( χ γ )( g ) := χ γ ( p γ ( g )), g ∈ A , defines the morphisms in the dual system (direct). In particular, the inverse system of mappings where, for all γ ≺ γ ' ≺ γ '' , satisfy p γγ ' · p γ ' γ '' = p γγ '' , gives rise to the corresponding direct system of mappings where ˆ p γγ ' ( χ γ )( g γ ' ) := χ γ ( p γγ ' ( g γ ' )) = χ γ ( g γ ), for g γ ' ∈ A γ ' . Using the associativity for the inverse system, it is straightforward to show that that is, the mappings ˆ p γγ ' do indeed define an inductive system. We are now in position to determine the sought for dual construction, that is, the inductive system. Recall that A γ may be identified with U(1) | γ | and the Pontryagin dual is just ̂ A γ = Z | γ | through the identification χ x 1 ,...,x γ ( z 1 , . . . , z γ ) = z x 1 1 · · · z x γ γ , for z 1 , . . . , z γ ∈ U(1) and x 1 , . . . , x γ ∈ Z . Therefore, Eqs. (2.2) give whose actions on z 1 , z 2 ∈ U(1) are, respectively, Thus, the embeddings are simply and have a very nice flux interpretation which agrees with our intuition of how fluxes should behave under coarse-graining of the underlying graph: (i) adding an edge should not bring more information into the system, so the flux on the added edge is zero, (ii) subdividing an edge does not change anything and thus the flux through the subdivided edges is the same, (iii) inverting an edge just changes the direction of the flux, picking up a minus sign. See Figure 2. Using Theorem 3.2 we know that Eqs.(3.6) define an inductive system. Hence, we may define the space of generalized fluxes for U(1)-LQG as the inductive limit of Z | γ | 's which agrees with the Pontryagin dual E = ̂ A = ̂ Hom( P , U(1)) = Hom(Hom( P , U(1)) , C ). The important point here is not the explicit form, which is not very enlightening, but the fact that it can be defined consistently as an inductive limit and, above all, the gluing conditions (3.6) from which it arises. To finish this section we define the corresponding space of functions. The pullback of the embeddings (3.6) gives the following projections which give the consistency conditions to define the projective limit of the C ∗ -algebras C 0 ( X γ ) for X γ = Z | γ | . Notice that the partial order for X γ induces the same partial order for C 0 ( X γ ). An element ( u γ ) γ of the projective limit is an element of the product × γ C 0 ( Z | γ | ) subject to the conditions ˆ p ∗ γγ ' ( u γ ' ) = u γ for γ ≺ γ ' , and so a quite complicated object. A projective limit of C ∗ -algebras goes by the name proC ∗ -algebra (also known as LMC ∗ -algebra, locally C ∗ -algebra or σ -C ∗ -algebra). The Gel'fand duality theorem can be extended to commutative proC ∗ -algebras and from the perspective of non-commutative geometry, proC ∗ -algebras can be seen as non-commutative k -spaces [44]. Let us also remark that proC ∗ -algebras are in general not Hilbert spaces, although they might contain Hilbert subspaces. Therefore, they possess much more information than usual Hilbert spaces, as we detail in the next subsection. Let us, at this point, emphasize that in this U(1) case, the fluxes can be identified with the charge network basis, since in a sense their only relevant component is the modulus which corresponds to the charge. However, their modulus remains valued in the real numbers, as opposed to what we would have would the flux representation and charge network basis be exactly the same. What happens next, due to the sampling mentioned at the beginning of 3.2, is that the functions are fully specified by the evaluation on the integers and, therefore, in this simple U(1) case, working in the flux representation is fully equivalent to working with the, a priori different, charge network representation.", "pages": [ 12, 13, 14, 15 ] }, { "title": "3.4 More on projections and inductions", "content": "Let us give more detail on the system of projections and inductions defined above and on the Hilbert spaces we (would try to) construct from them. In particular, we clarify here in which sense the proC ∗ -algebra constructed above from the projective system is much bigger than the usual Hilbert space. Projective and inductive limit are in general related by duality. That is, a projective system of labels induces an inductive system of functions simply by pullback, and vice-versa. However, they can be both defined independently as is done in the B. Here, we deal with a system of projections π γγ ' and a system of inductions ι γ ' γ for one and the same space. First of all, note the following relation for any pair of graphs γ ≺ γ ' . It is straightforward to check that the inductions (3.4) and the projections (3.7) satisfy (3.8). Recall that π γγ ' : X γ ' → X γ , ι γ ' γ : X γ → X γ ' , for some collection of objects { X γ } γ ∈L and L a directed poset. We remark that the reverse equation ι γ ' γ · π γγ ' = id γ ' does not generically hold, since when we first project and then embed we are typically throwing some information away. With the inductions at hand one may define the inductive limit, while with the projections one may define the projective limit. Using (3.8) we will see how one can understand an element of the inductive limit in terms of an element of the projective limit, however, not vice-versa. Recall from B that elements of the projective limit X proj are nets (i.e. elements in the direct product over all graphs) subjected to a consistency condition While elements of the inductive limit consist of equivalence classes of elements of the disjoint union over all graphs where x γ ∼ x γ ' means that there exists γ '' such that ι γ '' γ ( x γ ) = ι γ '' γ ' ( x γ ' ) and γ, γ ' ≺ γ '' . Now, given an element y = [ y ] ∼ ∈ X ind we will construct an element x ∈ X proj , i.e. an assignment γ ↦→ x γ for all graphs γ , such that x γ ' = y γ ' for all γ ' for which a representative y γ ' in the equivalence class y exists. In this sense we can embed the inductive limit into the projective limit. Pick some element y γ in the equivalence class y . Then for any graph γ ' define the assignment γ ↦→ x γ as follows: choose γ '' such that γ, γ ' ≺ γ '' , then gives a consistent definition of an element of the projective limit, is independent of the choice y γ in y and moreover x γ agrees on all graphs on which a representative y γ of y exists. In this way, we can map the inductive limit into the projective limit, however not surjectively. The image of the inductive limit consists of elements x for which there exists a graph γ '' such that ι γ '' γ · π γγ ' ( x γ ' ) = x γ '' for all γ, γ ' with γ, γ ' ≺ γ '' . The existence of such a 'maximal graph' γ '' is however not guaranteed for generic elements of the projective limit. For this reason we cannot use the Hilbert space structure of the inductive limit to make the projective limit into a Hilbert space, confirming the fact that proC ∗ -algebras are much bigger objects than Hilbert spaces.", "pages": [ 15 ] }, { "title": "4 Conclusion and Outlook", "content": "The loop quantum gravity kinematics treats connections and fluxes very asymmetrically. Therefore, the projective limit construction of the space of generalized connections does not translate trivially over to the flux side. Let us summarize what we have done. Using the commuting diagram (1.1) from [17] we set out to give an intrinsic characterization of the space H /star, 0 = ( ∪ γ H /star,γ ) / ∼ in terms of some functional calculus of generalized flux fields. We have seen that the /star -product on the image of the Fourier transform is not cylindrically consistent unless the gauge group G is abelian, and consequently H /star, 0 is not a C ∗ -algebra. This result is important because it means we cannot make sense of the space of generalized fluxes as the spectrum of this (would be) algebra. In more physical terms, this result suggests that a definition of the continuum limit of the theory, even at the pure kinematical level, cannot rely on the cylindrical consistency conditions coming from operations on holonomies, if one wants it to imply also a continuum limit in the dual flux representation, and thus coming from a proper coarse-graining of fluxes. Rather, it seems to suggest that a new construction is needed. Nevertheless, we were still able to learn something about the space of generalized fluxes by considering the gauge group U(1). In this setting we found out that the space of generalized fluxes cannot be constructed as a projective limit, but arises naturally as an inductive limit. The cylindrical consistency conditions for the fluxes (3.6) turned out to have a very nice physical interpretation (see Figure 2): (i) adding an edge should not bring more information into the system, so the flux on the added edge is zero, (ii) subdividing an edge does not change anything and thus the flux through the subdivided edges is the same, (iii) inverting an edge just changes the direction of the flux, picking up a minus sign. Even though we determined the space of generalized fluxes for U(1)-LQG to be E = Hom(Hom( P , U(1)) , C ), one would like to have a better description of this space. Finally, the space of functions was defined by pull-back giving rise to a projective limit of C ∗ -algebras. We showed that this algebra is in general much bigger than the usual Hilbert space but, it might still be possible to improve its characterization by noting that the Gel'fand duality theorem can be extended to proC ∗ -algebras [44]. In light of our results, we conclude with an outlook on two issues worth pursuing further. LQG from scratch and coarse-graining of fluxes. Loop quantum gravity as it is formulated is entirely based on graphs. As we have seen the cylindrical consistency conditions do not translate easily to the flux side, specially because they are tailored to operations on holonomies. Therefore it seems misguided to force them on the flux variables. In the abelian case we learned that the fluxes compose according to (3.6), with the aforementioned natural geometric interpretation. However, this process does not correspond to a coarse-graining in the same way as the family of projections does for the holonomies: surfaces are added according to how their dual edges compose, i.e. the operation of 'adding' puts two surfaces 'parallel' to each other - but does not add them into a bigger surface. The question arises whether one can come up with a family of inductions that would rather represent these geometrically natural coarse-grainings. This direction of thoughts hits however on many difficulties encountered before: one, is the more complicated geometrical structure of surfaces as compared to edges; another, is that for a gauge covariant coarse-graining of fluxes one not only needs the fluxes but also the holonomies to parallel transport fluxes. To avoid these difficulties one could consider again Abelian groups in 2D space, where fluxes would be associated to (dual) edges. In this case flux and holonomy representation would be self-dual to each other (for finite Abelian groups), reflecting the well-known weak-strong coupling duality for 2D statistical (Ising like) models, see for instance [45]. Whereas the usual LQG vacuum based on projective maps for the holonomies leads to a vacuum underlying the strong coupling limit, projections representing the coarse graining of fluxes could lead to a vacuum underlying the weak coupling limit, see also [26]. This could be especially interesting for spin foam quantization, as it is based on BF theory, which is the weak coupling limit of lattice gauge theory. Loop quantum cosmology. Loop quantum cosmology is the quantization of symmetry reduced models of classical general relativity using the methods of loop quantum gravity [46-48]. The classical configuration space is the real line R , while the quantum configuration space is given by an extension of the real line to what is called the Bohr compactification of the real line R Bohr . This space can be given several independent descriptions. In particular, it can be understood as the Gel'fand spectrum of the algebra of almost periodic functions, which plays the role of the algebra of cylindrical functions for LQC. On the other hand, R Bohr can also be given a projective limit description [49]. Let us briefly recall this construction. For arbitrary n ∈ N consider the set of algebraically independent real numbers γ = { µ 1 , . . . , µ n } , that is Consider now the subgroups of R freely generated by the set γ This induces a partial order on the set of all γ 's: γ ≺ γ ' if G γ is a subgroup of G γ ' . The label set used in describing the projective structure of A consists of subgroupoids of P generated by finite collections of holonomically independent edges. Here, the label set is exactly the set of all γ 's: collections of real numbers on a discrete real line. The projective structure of R Bohr is now constructed by defining and the surjective projections Since G γ is freely generated by the set γ = { µ 1 , . . . , µ n } we can actually identify R γ with U(1) n . Finally, the family { R γ } γ forms a compact projective family, and its projective limit is homeomorphic to R Bohr . By definition the momentum space for LQC is R with discrete topology. Since each of the objects R γ is an abelian group, we are in the setting of Theorem 3.2. Thus, one may identify R with discrete topology with the inductive limit of the duals ̂ R γ = Z n . We easily note the similarity of LQC and U(1)-LQG from before, the subtlety being the index set over which the inductive/projective limit is taken. In [50] it was shown that the configuration space of LQC is not embeddable in the one of (SU(2)) LQG. The natural question to ask, then, in view of our results, is whether one can instead embed LQC into U(1)-LQG.", "pages": [ 16, 17 ] }, { "title": "Acknowledgments", "content": "We would like to thank Aristide Baratin for discussions at an earlier stage of this project, Matti Raasakka for general discussions and in particular for correcting a flaw on theorem A.4, and Johannes Tambornino for useful comments on an earlier draft of this article. CG is supported by the Portuguese Science Foundation ( Funda¸c˜ao para a Ciˆencia e a Tecnologia ) under research grant SFRH/BD/44329/2008, which he greatly acknowledges. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. DO acknowledges support from the A. von Humboldt Stiftung through a Sofja Kovalevskaja Prize.", "pages": [ 17, 18 ] }, { "title": "A The U(1) group Fourier transform and the /star -product", "content": "Let f ∈ C (U(1)), that is, f is a function of the form with -π < φ ≤ π ( -π and π obviously identified), and pointwise multiplication The following theorem allows one to move from C (U(1)) to L p (U(1)) which is much more structured. Theorem A.1 ([51]) . Let X be a locally compact metric space and let µ be a σ -finite regular Borel measure. Then the set C c ( X ) of continuous functions with compact support is dense in L p ( X, d µ ) , 1 ≤ p < ∞ . Since U(1) is compact, C (U(1)) = C c (U(1)) and we are done. L 2 are the only spaces of this class which are Hilbert spaces. Since we want to do quantum mechanics, we will stick to L 2 (U(1)). The inner product is Introduce the map F for any f in L 2 (U(1)) by where e φ ( x ) are the usual plane waves but for x ∈ R . If x ∈ Z we would have the usual Fourier transform. The Im F is a certain set of continuous functions on R , but certainly not all functions in C ( R ) are hit by F . The inverse transformation reads and converges pointwise. 7 Notice here already that only the values of F ( f ) on the integers are necessary to reconstruct back f . Now, instead of the usual pointwise multiplication, we equip Im F with a /star -product. It is defined at the level of plane waves as with D N ( φ ) the Dirichlet kernel, i.e., D N ( φ ) = 1 2 π ∑ N -N e iφx . Then, to prove that lim N →∞ S f N ( φ ) = f ( φ ), just use the fact that the Fourier coefficients of S f N ( φ ) -f ( φ ) tend to zero as N →∞ . and extended to Im F by linearity. Here [ φ + φ ' ] is the sum of two angles modulus 2 π such that -π < [ φ + φ ' ] ≤ π . Given u = F ( f ) and v = F ( h ) we have explicitly where for the last line we used that Then, /star is still commutative for U(1). In order to have a better characterization of Im F , we will give it a norm and an involution ∗ , u ∗ := ¯ u , i.e. complex conjugation. Theorem A.2. Im F with the star product (A.2) , norm (A.3) and complex conjugation as involution is a non-unital abelian C ∗ -algebra. Proof. First of all, we have to check that (A.3) is indeed a norm. We easily verify || αu || = | α | || u || , || u + v || ≤ || u || + || v || , for all u, v ∈ Im F , α ∈ C . To see positive definiteness, notice that the functions on Im F are already determined by the values of x ∈ Z , The C ∗ -identity holds, since for x ∈ Z Eq.(A.2) reduces to the usual pointwise product. Finally, we show that Im F is complete in this norm. Let u α ∈ Im F be a Cauchy sequence, that is, u α is of the form for some f α ∈ L 2 (U(1)). We see that u α is Cauchy iff f α is Cauchy. Since L 2 (U(1)) is complete, f α converges to some f ∈ L 2 (U(1)). This means that u α converges to some u of the form that is, u ∈ Im F . Therefore, Im F is complete as well. Using (A.2) it is easy to convince ourselves that we have no unit function on Im F . Furthermore, we have no constant functions at all. For instance, u ( x ) = 1 does not live on Im F since it would correspond to f ( φ ) = 2 πδ ( φ ), a distribution. Distributions do not belong to L 2 (U(1)), unless we extend the framework to rigged Hilbert spaces. On the other hand, (A.2) would give us Indeed, the /star -product is invariant under this transformation, that is, the /star -product does not see the difference between the l.h.s. and the r.h.s. of (A.4), as on the integers they coincide. Thus, Im F is a non-unital abelian C ∗ -algebra. Using the Gel'fand representation theorem we will now prove that Im F /similarequal C 0 ( Z ). Theorem A.3 (Gel'fand representation, [52]) . Let A be a (non-unital) commutative C ∗ -algebra. Then A is isomorphic to the algebra of continuous functions that vanish at infinity over the locally compact Hausdorff space ∆( A ) (the spectrum of A ), C 0 (∆( A )) . It remains to calculate the spectrum of Im F , that is, the set of all non-zero ∗ -homomorphisms χ : Im F → C . Theorem A.4. The spectrum of Im F is homeomorphic to Z , ∆(Im F ) /similarequal Z . Proof. The ∗ -homomorphisms are Clearly, χ x ( u /star v ) = ( u /star v )( x ) = u ( x ) v ( x ) = χ x ( u ) χ x ( v ), and χ x ( u ∗ ) = u ∗ ( x ) = u ( x ) = χ x ( u ). We have to show that is a homeomorphism (continuous bijection with continuous inverse). Define X ( x ) := χ x . Continuity of X : let ( x α ) be a net in Z converging to x , and let u ∈ Im F . First of all, notice that u ( x α ) → u ( x ) by continuity of the plane waves. Then, for all u ∈ Im F , hence X ( x α ) → X ( x ) in the Gel'fand topology. 8 Actually, any function on Z is continuous since the only topology available is the discrete one. /negationslash /negationslash Injectivity: suppose X ( x ) = X ( x ' ), then in particular [ X ( x )]( u ) = [ X ( x ' )]( u ) for all u ∈ Im F . We want to show that that x = x ' . With a bit of logic we can turn this statement into the much easier one: [ x = x ' ] implies [ u ( x ) = u ( x ' ) for some u ∈ Im F ]. Just pick u = (1 , 0 , . . . ). Thus, Im F separates the points of Z . Surjectivity: let χ ∈ Hom(Im F , C ) be given. We must construct x χ ∈ Z such that X ( x χ ) = χ . Since Z is a locally compact Hausdorff space, it is the spectrum of the abelian, non-unital C ∗ -algebra C 0 ( Z ), hence Z = Hom( C 0 ( Z ) , C ). It follows that there exists x χ ∈ Z such that χ ( u ) = u ( x χ ) for all u ∈ C 0 ( Z ). Continuity of X -1 : let ( χ α ) be a net in ∆(Im F ) converging to χ , so χ α ( u ) → χ ( u ) for any u ∈ Im F . Then X -1 ( χ α ) →X -1 ( χ ). Therefore, Im F /similarequal C 0 ( Z ), where Z is endowed with the discrete topology. We now have Im F = C 0 ( Z ). Once again we want to map C 0 ( Z ) to the much nicer (Hilbert) space L 2 ( Z ) = /lscript 2 ( Z ). There are two ways of doing this. The first starts by noticing that C c ( X ) = C 0 ( X ), that is the space of continuous functions with compact support is dense in the space of continuous functions that vanish at infinity. Thus, any function in C 0 ( X ) can be approximated by functions in C c ( X ). Using this fact and Theorem A.1 we translate C 0 ( Z ) naturally to /lscript 2 ( Z ) with the inner product The second method uses the GNS construction. However, this relies on the choice of a state which, of course, can be chosen appropriately to get /lscript 2 ( Z ). Let us then define a state for u ∈ C 0 ( Z ) as for all u, v ∈ C 0 ( Z ), and the GNS Hilbert space is just /lscript 2 ( Z ). Finally, this characterization of Im F upgrades F to an unitary transformation between L 2 (U(1)) and /lscript 2 ( Z ) Hence, F is just the usual Fourier transform on U(1).", "pages": [ 18, 19, 20, 21 ] }, { "title": "B Projective and inductive limits", "content": "Let L be a partially ordered and directed set, that is, we have a reflexive, antisymmetric, and transitive binary relation ≺ on the set L such that for any γ, γ ' ∈ L there exists a γ '' ∈ L satisfying γ, γ ' ≺ γ '' .", "pages": [ 21 ] }, { "title": "B.1 Inverse or projective limits", "content": "Let C be a category. An inverse system in C is a triple ( L , { X γ } , { p γγ ' } ), where L is a directed poset, { X γ } γ ∈L a collection of objects of C , and p γγ ' with γ ≺ γ ' morphisms (projections) p γγ ' : X γ ' → X γ satisfying An object X ∈ Ob( C ) is called an inverse or projective limit of the system ( L , { X γ } , { p γγ ' } ) and denoted lim ←-X γ , if there exist morphisms p γ : X → X γ for γ ∈ L such that The induced inner product is commutes; for γ ≺ γ ' , there exists an unique morphism m : Y → X such that the following diagram commutes. That is, if the inverse limit exists, it is unique up to C -isomorphism. Finally, we remark that inverse limits admit the following description:", "pages": [ 21, 22 ] }, { "title": "B.2 Direct or inductive limits", "content": "Let C be a category. A direct system in C is a triple ( L , { X γ } , { ι γ ' γ } ), where L is a directed poset, { X γ } γ ∈L a collection of objects of C , and ι γ ' γ with γ ≺ γ ' morphisms (injections) ι γ ' γ : X γ → X γ ' satisfying An object X ∈ Ob( C ) is called a direct or inductive limit of the system ( L , { X γ } , { ι γ ' γ } ) and denoted lim -→ X γ , if there exist morphisms ι γ : X γ → X for γ ∈ L such that commutes; for γ ≺ γ ' , there exists an unique morphism m : X → Y such that the following diagram commutes. That is, if the direct limit exists, it is unique up to C -isomorphism. We remark here that we may define the inductive limit differently. Let ∼ be the following relation on ∪ X γ : for x ∈ X γ and y ∈ X γ ' , then x ∼ y if there exists γ '' ∈ L such that ι γ '' γ ( x ) = ι γ '' γ ' ( y ) (identifying each X γ with its image in ∪ X γ ). Since L is a directed set, ∼ is an equivalence relation and one can show that", "pages": [ 22, 23 ] } ]

[ { "title": "D. M. Walsh ∗", "content": "Economics Department, Trinity College Dublin, Dublin 2, Ireland September 20, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the stability of solution branches for the Lichnerowicz-York equation at moment of time symmetry with constant unscaled energy density. We prove that the weakfield lower branch of solutions is stable whilst the upper branch of strong-field solutions is unstable. The existence of unstable solutions is interesting since a theorem by Sattinger proves that the sub-super solution monotone iteration method only gives stable solutions.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The loss of uniqueness is a salient feature of many nonlinear systems and is often accompanied by a loss or 'exchange' of stability as two solution branches meet. A number of recent papers have looked at the uniqueness of solutions of conformal formulations of the Einstein constraints. Recent numerical work by Holst and Kungurtsev in [9] has confirmed the assumption made in [19] that the linearisation of the (constant) energy density moment of time symmetry LichnerowiczYork equation has a one dimensional kernel and that a quadratic fold results giving two solutions for subcritical energy density ρ < ρ c , one solution at the critical value ρ = ρ c and none for energy density greater than a critical value ρ > ρ c . In [13], Maxwell studied the conformal formulations of the Einstein constraint equations without the usual constant mean curvature condition (CMC) so that the constraints do not decouple. One motivation for that study was to examine the uniqueness of solutions in the far-fromconstant mean curvature regime now that existence results for this case exist (see [8], [14]). He found interesting non-existence and non-uniqueness results showing that the constraints are ill-posed beyond the non-uniqueness that is introduced when one couples the lapse fixing equation to the four constraint equations, as in the extended conformal thin sandwich (XCTS) formulation (see [15], [2], [19]) and some constrained evolution schemes (see [16], [6] for a resolution of this scaling problem). The inherent ill-posedness of conformal formulations of the constraints found in [13] is worthy of further analysis. In this work we continue our bifurcation analysis of the Einstein constraints, begun in [19] with an analysis of the XCTS system, by studying a familiar problem and introducing the important related issue of the stability of the solutions obtained. The ultimate goal of this program is to better understand the mapping between general free initial data and solutions of the constraint equations in their various conformal formulations. See also the recent work of Holst and Meier [10] in this regard. In section 2 we study the solution branches of the constant density star found in [2], [19] and [9]. We prove that the lower branch of weak-field solutions is stable whereas the upper branch is unstable. (We also prove that the kernel of the linearisation is one-dimensional, which was shown numerically in [9]). In section 3 we follow [12] in applying LiapunovSchmidt methods, as in [19], to the stability analysis and find that the exchange of stability that occurs at the critical point of a fold is in fact generic under mild non-degeneracy conditions. The existence of unstable solutions to the constraint equations is interesting in itself because the most popular method for proving existence, the sub-super solution monotone convergence method, only yields stable solutions (a result proven by Sattinger in [18]).", "pages": [ 1, 2 ] }, { "title": "2 Stability of solutions", "content": "We shall be concerned with the Hamiltonian constraint, the Lichnerowicz-York equation, at moment of time symmetry, for a conformally-flat background metric and an unscaled constant energy density ρ : with φ > 0 . We work in spherical symmetry in this section so that ∇ 2 = d 2 dr 2 + 2 r d dr . For simplicity we assume that ρ is zero outside a ball of radius r = 1 with boundary conditions φ (1) = 1 , and dφ dr (0) = 0 as in [9]. We are interested in the stability of stationary solutions of the following parabolic problem: with ν ( t = 0 , x ) = φ 0 ( x ) . We are interested in whether, for initial data φ 0 ( x ) close to the stationary solution, the solution to this parabolic problem tends to the stationary solution as t →∞ . To motivate the definition of stability that follows we take where ˆ ν is the stationary solution and θ 1 is the eigenfunction corresponding to the principal eigenvalue µ 1 (the smallest eigenvalue) and glyph[epsilon1] is a small constant. Then substituting this into (2.2) gives to first order in glyph[epsilon1] . Definition We will say that a stationary solution to (2.2) is stable if the principal eigenvalue of the linearisation with θ k (1) = 0 and θ ' k (0) = 0 satisfies µ 1 ( ∇ 2 +10 πρφ 4 ) > 0 and otherwise that it is unstable. In the next Lemma we collect some facts about the eigenfunctions and eigenvalues of (2.3).", "pages": [ 2, 3 ] }, { "title": "Lemma 2.1", "content": "then f is proportional to θ 1 glyph[negationslash] where D is the set of smooth functions satisfying the boundary conditions η (1) = 0 , dη dr (0) = 0 Proof 1. It is easy to show that the eigenfunctions of (2.3) are orthogonal: and we have used Green's second identity with n the outward pointing normal to the surface element dS at r = 1 . We know that the eigenfunction corresponding to the first eigenvalue µ 1 is of definite sign so may be taken to be positive (see[7]). The orthogonality relation above then says that all higher eigenfunctions must have nodes. The proof that µ 1 is simple may be found in [7] for example. In [2] the authors worked in an unbounded domain which complicates the analysis of the spectrum of the linearisation. We work in a bounded domain corresponding to a ball of radius r = 1 with φ (1) = 1 . It is straightforward, using the ideas in [2], to generate a one parameter family of solutions with this boundary condition: where the boundary condition gives Note that the maximum value of ρ := ρ c occurs at α c = 1 and that the limit α →∞ corresponds to a lower branch of solutions and the limit α → 0 corresponds to an upper branch of solutions. It is easy to check that satisfies (2.3) with α = 1 in ρ and µ k = 0 . This eigenfunction has no nodes and therefore corresponds to the principle eigenvalue found numerically in [9]. So we have that µ 1 = 0 , we have found the principal eigenfunction at the critical value ρ c , α c = 1 . Since this eigenvalue is simple we have that the kernel of the linearisation at the critical energy density ρ c is one dimensional. In [2], [9] and above, global branches of solutions were found corresponding to a quadratic fold. For each ρ < ρ c we have an upper solution φ U and a lower solution φ L . (As in [2], a simple calculation reveals that there are apparent horizon's on the upper branch of solutions at the location r = α , and none on the lower branch). Before showing that the upper solutions φ U are unstable and the lower solutions φ L are stable, we motivate our analysis by the following observation. If we multiply the linearisation (2.3) by u := φ -1 and integrate twice by parts we get the following identity, having used (2.1): With θ 1 the first eigenfunction and therefore positive and u ≥ 0 since φ ≥ 1 we see that for weak-field solutions (4 u -1) is likely less than zero so that µ 1 > 0 and the solution is stable. But we can imagine a strong field solution whereby (4 u -1) > 0 so that µ 1 < 0 and we have an unstable solution.", "pages": [ 3, 4 ] }, { "title": "Theorem 2.2", "content": "Note that r ∈ [0 , 1] . On the lower branch of solutions 1 < α < ∞ , so that ρ ( α ) φ ( α ) 4 -ρ c φ 4 c < 0 , so Lemma 2.1.3 implies that so the lower branch of solutions is stable. Instead we utilise the fact that glyph[negationslash] glyph[negationslash] Using a trial function η T ∈ D then provides an upper bound for the principal eigenvalue corresponding to a particular choice of 0 < α < 1 , i.e. µ 1 ( α ) ≤ R ( α ) where R ( α ) is the Rayleigh quotient above. We shall choose and use MATLAB to perform the integration exactly. This yields a rather complicated expression given in Appendix A. A graph of R ( α ) is shown in Figure 1 and some explicit values for R ( α ) are given in Table 1 in Appendix A. We clearly see that for 0 < α < 1 , the upper branch of solutions, that µ 1 ( α upper ) ≤ R ( α upper ) < 0 so we have that the upper branch of solutions is unstable. QED", "pages": [ 5 ] }, { "title": "3 Stability of solutions via Liapunov-Schmidt reduction", "content": "Liapunov-Schmidt (LS) methods were applied to the Lichnerowicz-York equation with an unscaled source in [19] and more recently in [10]. Here we follow [12] and show that locally the exchange of stability witnessed in the previous section is in fact generic for quadratic fold-type branches. We again consider a nonlinear suitably smooth equation F ( x, ρ ) = 0 with x ∈ X , a Banach space and ρ ∈ R , ( F : X ∗ R → Z ) with X ⊂ Z and assume that zero is a simple eigenvalue _ of D x F ( x 0 , ρ 0 ) . We let θ denote the one-dimensional kernel and denote by θ ∗ the cokernel of D x F ( x 0 , ρ 0 ) normalised such that ∫ θθ ∗ = 1 (we assume the linearisation has a Fredholm index of zero). The LS construction allows us to construct a continuously differentiable curve of solutions, for small positive δ , { ( x ( s ) , ρ ( s )) | s ∈ ( -δ, δ ) , ( x (0) , ρ (0)) = ( x c , ρ c ) } , such that F ( x ( s ) , ρ ( s )) = 0 for all s ∈ ( -δ, δ ) . For the LS projection operators and splittings of the range and domain we follow the notation of [12] section 1.4-1.7, which is equivalent to [19]. In particular we have that where N denotes the kernel space and R the range of the linearisation. Note that if the following nondegeneracy conditions hold: glyph[negationslash] where a dot denotes differentiation with respect to s, and so the tangent vector to the solution curve ( x ( s ) , ρ ( s )) at ( x c , ρ c ) is ( θ, 0) (see corollary 1.4.2 in [12]) and we have a turning point or fold (sometimes called a 'saddle-node'). To analyse the stability of the 'branches' of the fold (upper and lower) we note the following result (Proposition 1.7.2 in [12]): then it is proven in [12] that", "pages": [ 5, 6 ] }, { "title": "Theorem 3.1", "content": "There exists a continuously differentiable curve of perturbed eigenvalues { µ ( s ) | s ∈ ( -δ, δ ) µ (0) = 0 } in R such that where { w ( s ) | s ∈ ( -δ, δ ) , w (0) = 0 } ⊂ R ∩ X is continuously differentiable (and the size of the interval ( -δ, δ ) is not necessarily the same as in the solution curve above, but possible shrunk). In this sense, µ ( s ) is the perturbation of the critical zero eigenvalue of D x F ( x c , ρ c ) . (Note that we used the opposite sign convection for the eigenvalues of the linearisation in section 2 where D x F ( x c , ρ c ) θ = -µθ ). It is then straightforward to show that (see 1.7.30 in [12]), glyph[negationslash] and Now µ (0) = 0 , ˙ µ (0) = 0 implies that µ ( s ) changes sign at s = 0 so that the stability of the solution curve changes at the turning point. glyph[negationslash] Applying this to the constant density problem of section 2 we see that for small s ∈ ( -δ, δ ) (and with α 2 and γ 2 two known constants obtained from (3.19) and (3.20)) which agrees with [19] and section 2 above, and which tells us that the lower branch of solutions s ∈ ( -δ, 0) is stable and the upper branch s ∈ (0 , δ ) is unstable, as expected. The solution branches for the conformal factor, lapse and shift vector found numerically in [15] for the XCTS formulation of the constraints are graphically similar to those studied above (LS methods were applied to this system in [19] under the assumption that the system developed a one dimensional kernel for sufficiently large initial data). It seems likely from this work that these branches also display an exchange of stability for a broad class of initial data such that the non-degeneracy conditions (3.16), (3.17) are satisfied. We conclude this section by mentioning a limitation on the application of the sub-super solution method that this work reveals. Solutions to elliptic equations obtained via the subsuper solution method are stable in the sense described in section 2 (see [17], [18]) so that the upper branch of solutions studied here are unattainable by this method. The general CMC Lichnerowicz-York equation with an unscaled source on an asymptotically Euclidean manifold was studied in [4] using the sub-super solution method, where r is proportional to the scalar curvature and a the traceless part of the conformally transformed extrinsic curvature squared (see section XII of that work). They found an open set of values of a and ρ such that existence could be proven-uniqueness was not proven. If, as seems likely from this work and [15], [2], [19],[9] and [10], that upper and lower branches of solutions exist for this equation for some combination of a and ρ then the sub-super solution and monotone iteration method will only converge to a stable solution. So if an upper branch of solutions is unstable, as above, then the sub-super solution method will not yield it. These comments are also relevant to the Einstein-Scalar field CMC Lichnerowicz-York equation as studied in [5] (see Theorem 8.8 in that work).", "pages": [ 7, 8 ] }, { "title": "4 Discussion", "content": "In this work we have concentrated on the non-standard case of an unscaled fluid with no momentum. (For a complete discussion of the role of conformal scaling of fluid sources see [3] or [4]). It is not of purely academic interest however because it serves as an excellent model of a poorly scaled system such as the XCTS system and also warns of the dangers of not scaling the extrinsic curvature as was common in numerical evolutions which started from moment of time symmetry initial data, see section 5 in [19]. In studying (2.1), we have been looking at the geometric problem of finding a conformal factor that maps from a scalar flat metric to one with scalar curvature equal to 16 πρ (for more details see the classic paper [11] where the authors allow a non-zero scalar curvature background metric). This is closely related to the Yamabe problem of finding a conformal factor that maps a given metric to one with constant scalar curvature. This has yielded many insights, particularly in relation to the existence theory of solutions to the Lichnerowicz-York equation, see [1] for an excellent review. It is clearly important to understand the strengths and limitations of solution methods. With this in mind, future work should determine the stability of non-unique solutions found in other works, for example in [15], [13]. The Lichnerowicz-York equation with an unscaled source also has a variational formulation since (2.1) is the Euler-Lagrange equation obtained by varying the following functional: It would be interesting to check which solutions (stable/lower or unstable/upper or both) application of the Mountain Pass theorem to this toy model yields.", "pages": [ 8 ] }, { "title": "Acknowledgements", "content": "I am grateful to Niall ' O Murchadha for helpful comments on an early draft of this work and an anonymous referee for suggestions that improved its presentation.", "pages": [ 8 ] }, { "title": "Appendix A", "content": "We used MATLAB to perform the following integration for the Rayleigh quotient for the trial function η T given by (2.13) Some specific values of the Rayleigh quotient are given in Table 1 below.", "pages": [ 9 ] } ]

[ { "title": "Andrei Khmelnitsky", "content": "Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universitat Munchen, 80333 Munich, Germany E-mail: [email protected] Abstract: We study the Gregory-Laflamme instability of a large uniform black brane wrapping a two-sphere compactification manifold. This paper continues the work [1], where the compactifications on p -torus were considered. The new features of the spherical case are the non-zero curvature of the compactification manifold and the absence of the rescaling symmetry due to a built-in stabilization mechanism. We calculate the order of the phase transition in dependence on the number d of extended dimensions using the Landau-Ginzburg approach. It is found that for d > 11 a uniform spherical black brane in microcanonical ensemble exhibits a smooth second order phase transition towards a stable branch of non-uniform black brane solutions. The critical number of extended dimensions, for which there is a change in the order of the phase transition, is different for microcanonical and canonical ensembles and does not coincide with the critical number of dimensions in the case of the flat toric compactifications. We briefly discuss the origin of this mismatch in the orders of phase transition for the different ensembles.", "pages": [ 1 ] }, { "title": "1 Introduction and summary", "content": "Uniform extended black branes in the presence of compact extra dimensions are unstable with respect to long wavelength perturbations if the black brane horizon size is substantially smaller than the size of the extra dimensions. This is known as the Gregory-Laflamme (GL) instability [2]. Therefore a large black brane should undergo a phase transition once its size becomes smaller than a certain critical size. By studying non-uniform perturbations on top of the critical black brane it is possible to find the order of this phase transition. It could be either a smooth second (or possibly higher) order transition when the black brane becomes slightly non-uniform in the compact dimension in a continuos way or a first order transition when below the critical size the black brane decays into some completely different solution. The final stage of the first order transition is in general unknown (see [3-5] for review). The first study of this kind was performed by Gubser for a five-dimensional black string on a single compact extra dimension in pure gravity in which case the transition is first order [6]. Recently this calculation was generalized by Sorkin to black strings in arbitrary number of extended dimensions. He found that the phase transition becomes second order in more than twelve extended dimensions [7]. It was also claimed that the critical number of dimensions, when there is a change in the phase transition order, depends on whether the phase transition happens at fixed black string mass (in microcanonical ensemble) or at fixed temperature (in canonical ensemble). In the latter case the transition becomes of the second order in more then eleven extended dimensions [8]. Later Kol and Sorkin considered the case with an arbitrary number d of extended dimensions and an arbitrary number p of the extra dimensions compactified on the torus T p [1]. In the special case when the sizes of all p circles are equal they found that the phase transition order depends only on the number d of extended dimensions and not on p . This result is explained by the fact that it is thermodynamically preferable for the GL instability to develop only along one of the circles on the torus. Therefore the toric black brane with p > 1 behaves effectively like the p = 1 black string. In this paper we determine the order of the phase transition for a black brane on a twosphere compactification manifold. We use the spontaneously compactified M d × S 2 solution of the Einstein-Maxwell theory in D = d +2 dimensions as the background geometry [9]. The presence of the Maxwell field is necessary in order to have a non-flat compactification manifold. Aside from having a non-gravitational matter field this case has two important features in comparison to the flat compactification set-ups studied earlier which affect the properties of the phase transition. First, the two-sphere is not a direct product of two flat compact dimensions, and thus it does not support a mode of instability analogous to the modes along a single circle on the torus. Instability on S 2 inevitably feels the presence of both compact dimensions and in this respect is more similar to the mode on the torus when the inhomogeneities along both circles are excited with equal amplitude (this mode is referred to as the 'diagonal' mode in [1]). The other important difference is that the size of the two-sphere is fixed by the parameters of the theory. Therefore there is no rescaling freedom which in case of the flat compactifications accounts for the fact that the size of the compact dimension can be set arbitrarily. In the terms of dimensional reduction, the radion field is stabilized and has a mass comparable to the inverse radius of the compact two-sphere. Because of the absence of an internal length scale for the flat compactification case, a set of thermodynamical quantities invariant under rescaling was introduced in [6] in order to study the phase transition. The results obtained in such a way correspond to the situation when the size of the compact extra dimension is held fixed and the radion is infinitely heavy. Since the critical black hole size is comparable to the radius of the twosphere, the radion mass in our set-up is naturally of the same order as the phase transition temperature. Therefore one can expect the presence of the dynamical radion to play a non-trivial role in the phase transition. The presence of these features suggests that the phase transition order for the spherical black brane could be different from the case of the flat toric compactification. In this paper we follow the method described in [1] to determine the order of the phase transition. The method employs the Landau-Ginzburg theory of phase transitions and is favoured in comparison to the original Gubser's computation since there is no need to compute any third order metric perturbations around the critical black brane. Considering the perturbations up to the second order is sufficient in order to compute the free energy and the entropy differences between the uniform and non-uniform black brane branches. The signs of these differences define the phase transition order in canonical and microcanonical ensembles respectively. As a result we find that the transition for the spherical black brane in microcanonical ensemble is of the second order when the number of extended dimensions exceeds eleven. This coincides with the critical dimension for the 'diagonal' mode on the two-torus found in [1]. In the canonical ensemble the difference of the free energy between the uniform and non-uniform black branes changes its sign when there are more than nine extended dimensions. It is lower than the corresponding critical dimension for the 'diagonal' mode on the two-torus. However, since the canonical ensemble itself is ill-defined for the black holes in asymptotically flat extended dimensions due to their negative specific heat, it does not make sense to associate this critical behaviour of the free energy with the change of the phase transition order. The relation between the order of phase transition in the canonical and microcanonical ensembles for a generic system is discussed in the appendix A. Therefore we indeed found that the phase transition order for the spherical black brane does not coincide with the case of the flat toric compactification. The rest of the paper is organized as follows. In section 2 we introduce the set-up and briefly describe how to apply the Landau-Ginzburg description of phase transitions to the black branes. We provide the details for the perturbative computation of inhomogeneous black brane solution in section 3. The results for the phase transition order in various number of extended dimension are presented and discussed in section 4.", "pages": [ 2, 3 ] }, { "title": "2 Set-up and Ginzburg-Landau method", "content": "One of the simplest non-flat compactifications with a stabilized size of the extra dimensions is the spontaneous compactification solution of the 6D Einstein-Maxwell theory described in a great detail in [9]. In this solution the Maxwell vector field has a magnetic monopole configuration on an external two-sphere. It is possible to fine-tune the six-dimensional cosmological constant with respect to the magnetic flux so that the four-dimensional cosmological constant is zero and the remaining four dimensions are flat. It is straightforward to generalize this solution to d extended dimensions. The fine-tuning condition between the cosmological constant, the vector field coupling, and the Newton's constant in D ≡ d +2 dimensions remains the same as in d = 4 case, i.e. The phase transition takes place for a uniform black brane which looks like a spherically symmetric Schwarzschild black hole in the d extended dimensions and completely wraps the external two-sphere. The Euclidean line element is given by where The first three terms in (2.2) correspond to a Schwarzschild-Tangherlini black hole in d -dimensional flat space, and the last two terms correspond to the two-sphere of the compact extra dimensions. The radius a of this sphere is determined by the cosmological constant scale: The mass and the inverse temperature of such a black brane are related to the horizon radius r 0 by the standard relations for a d -dimensional Schwarzschild black hole Here G d = G D / (4 πa 2 ) is the d -dimensional Newton's constant and Ω d -2 is the area of the ( d -2)-dimensional unit sphere. In the Landau-Ginzburg approach in order to study the order of phase transition in canonical ensemble it is sufficient to know the local behaviour of the free energy of the system around the critical point. A detailed description of this method in the context of black hole phase transitions is given in the reference [1] which we follow closely in our calculation. One first computes the local expansion of the free energy as a function of the order parameter and the temperature which in our case plays the role of the parameter that controls the onset of the transition. The role of the order parameter is played by the amplitude λ of the inhomogeneous perturbations in the metric and vector field. It shows the degree of the deviation of the black brane from the uniform solution and is also our perturbative expansion parameter. In order to compute other thermodynamic characteristics it is useful to know the free energy as the function of the inverse temperature. We expand the free energy up to the fourth order in λ around critical point: 1 where F 0 ( β ) = r d -3 0 is the free energy for the uniform unperturbed black brane 2 , β ∗ is the inverse critical temperature, and δβ ≡ β -β ∗ . The values of the coefficients A and C in the expansion (2.6) define the local thermodynamics completely. The phase transition occurs when the black brane becomes smaller than a certain critical size, which means that δβ becomes negative. This fixes the sign of A to be positive in order for the uniform phase λ = 0 to become an unstable extremum of the free energy for δβ < 0. In the case when C is positive there is a minimum of the free energy for δβ < 0 located at The presence of the non-trivial minimum in the vicinity of the uniform phase signals a smooth second order phase transition towards the slightly non-uniform phase with λ = λ ∗ . The difference in free energies between the non-uniform and uniform black branes is of the fourth order in perturbative expansion parameter λ and is given by The mass of the black brane can be obtained from the free energy using the relation The entropy S ( λ ; M ) can be computed by performing a Legendre transform of βF ( λ ; β ) with respect to β and is given by Here S 0 ( M ) = 4 π r d -2 0 is the uniform black brane entropy, M ∗ is the critical mass, and δM ≡ M -M ∗ . The entropy as a function of mass determines the behaviour of the system in microcanonical ensemble. In full analogy with the canonical ensemble case the order of the phase transition in the microcanonical ensemble is determined by the sign of the coefficient in front of the fourth order in λ term, which also gives the sign of the difference between the entropies of the non-uniform and uniform phases (cf. (2.8)): If the non-uniform branch has higher entropy than the uniform one, i.e. when σ 2 > 0, the phase transition in microcanonical ensemble is of the second order, and black brane settles in the stable non-uniform branch. The free energy of the black brane as a function of the metric and the vector field potential is given by the Euclidean action of the Einstein-Maxwell theory evaluated on the corresponding solution: Here R is the Ricci scalar, K is the extrinsic curvature on the boundary at infinity and F µν is the vector field strength. The surface term corresponding to a reference geometry with the extrinsic curvature K 0 has to be subtracted in order to make the resulting free energy finite. In our case the reference geometry is S 1 β × R d -1 × S 2 a with the Euclidean time period given by the inverse temperature β and a fixed radius a of the external two-sphere. The configuration space is spanned by the Euclidean solutions for g µν and V µ that asymptote to this reference geometry. The non-uniform solution can be found perturbatively by expanding the equations of motion and field deviations above the background in the powers of perturbative parameter λ . The parameter λ also plays the role of the order parameter in the free energy expansion (2.6). In order to compute the free energy up to the fourth order in λ it is sufficient to find the solution up to the second order. We collectively denote the metric and vector field deviations δg µν ≡ g µν -g (0) µν and δV µ ≡ V µ -V (0) µ by X ≡ { δg µν , δV µ } and expand them in powers of λ : The first order perturbation X (1) is nothing else but the static inhomogeneous GregoryLaflamme mode, and the second order perturbation X (2) corresponds to the back-reaction of the black brane on the GL mode. In practice one expands the free energy in powers of X and then plugs the solution for X (1) and X (2) . For determining the order of the phase transition one is interested in the O ( λ 4 ) term in the free energy. Using the equations of motion one arrives to the following expression for the quartic coefficient C [1]: Here the first term is the quartic in the field deviations X term of the free energy expansion, which is evaluated on the first order solution X (1) . The second term F 2 [ X (2) ] is the quadratic in X term evaluated on the second order perturbation X (2) .", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Perturbative solution", "content": "In this section we are looking for a slightly inhomogeneous black brane solution as a perturbation around the homogeneous black brane (2.2). The most general ansatz for Euclidean metric and vector field, which are static and spherically symmetric in d extended dimensions reads Here the functions { A,B,C,G,H,J,L } = X parametrize the metric and vector field deviations and depend only on the d -dimensional radial coordinate r and the angles on the external two-sphere θ and φ . One could, in principle, also include the components of the metric, which are odd under the inversion on the two-sphere: as well as the even under the inversion component of the vector field (the monopole background itself is odd): However it is always possible to choose a gauge in which these components are not excited and, therefore, we omit them. It is useful to expand the set of functions X in spherical harmonics on the external two-sphere. It is always possible to choose the direction of the inhomogeneous mode to contain only the harmonics with m = 0 that do not depend on φ . The spherical harmonics analysis also constrains considerably the possible θ -dependence: the first order mode X (1) contains only the first l = 1 harmonic, whereas the second order mode X (2) contains only the l = 0 and l = 2 harmonics. Moreover all the functions X can be subdivided according to their transformation properties under the coordinate transformations on the two-sphere into scalar, vector and tensor ones. The explicit expansion for the scalar quantities X s = { A,B,C,H } reads and the vector quantities X v = { G,L } are expanded as The tensor harmonic parametrized by J appears only at l = 2 and is given by Hence in this ansatz all the metric and vector field perturbations are parametrized by the set of functions x s i ≡ { a i , b i , c i , h i } , x v i ≡ { g i , l i } , and j 2 of a single variable r . The ansatz (3.1),(3.2) does not fix the coordinate transformation redundancy in the ( r, θ ) plane and two more gauge fixing conditions have to be specified. These conditions can be specified independently at each order of perturbation theory and for each spherical harmonic label l . We use this freedom later in order to simplify the resulting equations. The non-uniform black brane we are looking for has to satisfy the Euclidean equations of motion which in our case read After plugging in the ansatz (3.1),(3.2) and expanding up to the first order in λ one obtains a set of ordinary linear differential equations for the functions x s 1 ( r ) and x v 1 ( r ). After substituting also the background solution conditions (2.1) and (2.4) for the vector coupling and the cosmological constant, the remaining parameters in the resulting equations are the radius of the external sphere a and the black brane horizon size r 0 . For simplicity we set r 0 to be the unit of length. Then the only parameter in the equations is the dimensionless value of the radius a in r 0 units. A regular solution exists only for a particular value of a and defines the static Gregory-Laflamme mode on the black brane. In linear order we use the particular choice of the gauge fixing condition proposed in [1]: In this gauge all linear equations are reduced to a single second order equation for c 1 ( r ): Two conditions have to be imposed on the function c 1 in order to specify the solution. One of them corresponds to the choice of the normalization of c 1 . We fix it by imposing the condition c 1 (1) = 1 adopted in previous works on black string [1, 6-8]. The other condition arises if one requests the solution to be regular at the horizon r = 1 and is given by However for generic values of the parameter a this regular solution grows exponentially for large r . Therefore one is left with a one-parameter shooting problem in which the value of a is adjusted so that c 1 decays at large r . The equation (3.9) coincides with the first order equation (3.6) of [1] for the GL mode in the case of the black string and the black brane in the toric compactification. The critical size of the sphere a GL at which the GL instability sets in is related to the critical wavelength of the GL mode on the black string as: In the gauge (3.8) all other components of the metric can be explicitly expressed in terms of c 1 ( r ) as: At the second order the perturbations contain l = 0 and l = 2 modes which can be treated independently. The second order equations are inhomogeneous linear ODE's for x 0 , x 2 and j 2 that contain source terms quadratic in the first order perturbations found above. We start by considering the zero modes a 0 , b 0 , c 0 and h 0 . The function h 0 can be separated from the other zero modes and is defined by the equation The exact form of the source term S h 0 [ b 1 , c 1 , g 1 ] is given by (B.1). The regularity condition on the horizon fixes the value of the first derivative h ' 0 (1) in terms of the value of h 0 (1), while the latter is adjusted in order to find the solution that decays at large r . Instead of finding the solution for a 0 , b 0 and c 0 it is more efficient not to fix the gauge at all but to rewrite the equations in terms of the gauge-invariant combinations defined as: The free energy is obviously a gauge invariant quantity and can be expressed in terms of u and w . Thus there is no need to fix any particular gauge and determine a 0 , b 0 and c 0 . The equation for u is a first order differential equation with the source term given in (B.2). This equation can be straightforwardly integrated numerically. For the metric to match the flat reference geometry the constant of integration u (1) should be chosen so that u vanishes at infinity. The equation for w also happens to be of the first order: with the source given in (B.3). The equation can be integrated once the solution for u is known. The constant of integration w (1) can be fixed by considering the temperature of the perturbed black brane. More precisely the deviation of the temperature from the critical one can be expressed in terms of the metric perturbations as Thus different values of w (1) correspond to black branes with different temperatures. However, in order to obtain the free energy of the black brane at the critical point we consider only solutions with the critical temperature. The integration constant w (1) is then fixed by demanding the δβ to be zero. The freedom in the choice of w (1) is related to the fact that among the linear perturbations around any black hole there always exists a mode corresponding to the infinitesimal change of the size of the black hole. The equations for the l = 2 spherical harmonic components can also be separated in two groups. First we solve for the variables h 2 , j 2 , and l 2 , the equations for which are independent from the other variables. We adopt the same gauge condition l 2 = 0 as in the linear order. The system of equations for the h 2 and j 2 then takes the form: with the source terms given by equations (B.5) and (B.6). It is straightforward to bring this system to a solvable from by taking appropriate linear combinations of h 2 and j 2 . After that the equations for these linear combinations can be solved separately. In order to obtain solution one, as before, has to impose the regularity condition on the horizon and solve the one-parameter shooting problem by adjusting the values of the functions on the horizon in order to obtain the solution that decays at infinity. In analogy to the linear order one can fix the gauge in such a way that it would be possible to express the remaining variables a 2 , b 2 , and g 2 algebraically in terms of c 2 , which obeys a single second order ODE. The gauge fixing condition (3.8) is modified at the second order by the presence of a term quadratic in the first order variables: The equation for c 2 is similar to the equation (3.9) that defines the first order perturbation: with the source term given by (B.4). Thus finding the solution in this sector boils down to another one-parameter shooting problem for the value c 2 (1). After finding the solution for c 2 one can determine the remaining variables a 2 and g 2 as: The explicit expression for the source S g 2 can be found in (B.7).", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4 Results and discussion", "content": "Having found the solution for the metric and the vector field we can compute the coefficients in the free energy expansion (2.6). By substituting the solution with δβ = 0 in the Euclidean action (2.12) one obtains an expression for the free energy quartic in λ , from which the coefficient C can be determined. Alternatively one can use the expression (2.14), which is simpler to evaluate numerically. We found the numerical mismatch between the two different expressions for C to be less than a percent. By changing the parameters of numerical integration we found the change in the values of C to be at the same level, which thus can serve as the estimate of the numerical error. The values of C for a various number of extended dimensions d are listed in table 1. Note the change of sign of the quartic coefficient for d > 9. For a thermodynamically stable system it would mean that the phase transition for d > 9 is of the second order in canonical ensemble. It is instructive to compare the obtained behaviour of C for a black brane on a twosphere with the case of a flat compactification on the square two-torus T 2 considered by Kol and Sorkin in [1]. In the latter case there are two independent inhomogeneous modes corresponding to the two circles of T 2 . In order to study the free energy one can consider two limiting cases: when only a single mode along one of the two circles is excited, or when both of the modes are excited with equal amplitude, the so-called 'diagonal' mode [1]. The quartic coefficient C in all three cases is presented in dependence of the number of extended dimensions d in figure 1. We see that the single direction mode on a torus has lower free energy and thus thermodynamically favourable. Due to this fact the toric black branes during the phase transition effectively behave like black strings, with only the mode along a single circle being excited. In contrast, on the spherical black brane there is only one mode, and its dependence on the number of extended dimensions d is different from the modes on T 2 . The change of the sign of the coefficient C for the spherical black brane happens between d = 9 and 10. The behaviour of the black brane in microcanonical ensemble is determined by the sign of the coefficient σ 2 in the difference of the entropy between the non-uniform and uniform black branes (2.11). In order to find σ 2 the quadratic coefficient A should be determined. There are two independent methods how to compute A . First, one can take a first variation of the free energy (2.12) with respect to the temperature, which at the leading order in λ can be expressed using only the solution with δβ = 0. This method was applied in [1] and gives Alternatively one can use the solution of (3.19) with δβ = 0, i.e. keep the integration glyph[negationslash] constant for the zero harmonic w (1) initially unspecified. The temperature dependence of the free energy (2.12) is then obtained by using the relation (3.20) relating δβ to w (1). The coefficient A is given by where the asymptotic value w ( r →∞ ) is taken from the solution with δβ = 0. The resulting values of σ 2 in dependence on the number of extended dimensions d are given in the table 1 and presented in comparison to the case of the toric black brane in figure 2. For d > 11 the non-uniform black brane has larger entropy than the uniform one, and the phase transition in microcanonical becomes of the second order. We note that in microcanonical ensemble the critical number of extended dimensions for the spherical black brane coincides with the one for the diagonal inhomogeneous mode of the toric black brane. For d = 10 , 11 the signs of the quartic coefficients in the free energy and entropy, and consequently the predicted orders of phase transition in canonical and microcanonical ensembles, are different. The possibility of such a situation can be seen already from the equation (2.11) for the entropy difference, since the entropy difference can remain negative even if the coefficient C would turn to be positive. The details of this effect are discussed in appendix A.", "pages": [ 10, 11, 12 ] }, { "title": "Acknowledgments", "content": "The author is indebted to Sergey Sibiryakov, Gia Dvali, Dima Levkov and Valery Rubakov for fruitful discussions and L¯asma Alberte for careful reading of the draft. The research was supported by Alexander von Humboldt Foundation.", "pages": [ 12 ] }, { "title": "A The origin of the phase transition order mismatch in canonical and microcanonical ensembles", "content": "We consider a generic form of the free energy expansion in the vicinity of the critical temperature β ∗ where λ is the order parameter, and δβ ≡ β -β ∗ , cf. (2.6). The entropy is given by a Legendre transform of βF with respect to β and reads with δM ≡ M -M ∗ . The specific heat c u of the uniform black brane with λ = 0 kept fixed is given by By using this expression the entropy expansion can be rewritten in the following form: If the specific heat c u is negative, the coefficient ˜ C ≡ [ C + β ∗ A 2 2 c u ] , which determines the stability of the non-uniform branch in microcanonical ensemble, may become negative even for positive values of C . In the case of a positive C a non-trivial minimum of the free energy, located at appears for δβ < 0. Therefore, in canonical ensemble the system should settle in the non-uniform phase with λ = λ ∗ . The specific heat in this non-uniform phase is given by Comparison with (A.4) gives the following relationship between the quartic coefficients in the two ensembles and the specific heats in the different phases: This condition holds for any thermodynamic system and tells that the signs of the quartic coefficients in the free energy and the entropy expansions are different if and only if the specific heat has different sign in different phases. Hence, the mismatch between the phase transition orders in canonical and microcanonical ensembles can happen only if the specific heat changes its sign in the course of transition from the uniform to the non-uniform phase. Such a situation is indeed observed in some gravitational systems (c.f. [12] and references therein). If the system is thermodynamically stable before the phase transition, i.e c u > 0, then a situation is possible when the non-uniform branch is stable in the microcanonical ensemble ( ˜ C > 0) and unstable in the canonical ensemble ( C < 0). In such a case the system exhibits a second order phase transition in the microcanonical ensemble. In canonical ensemble the transition is of the first order and proceeds towards some third phase which is thermodynamically stable. This behaviour is attributed to the fact that in this case the system has negative specific heat in the non-uniform phase, as can be seen from (A.7). The opposite situation takes place for the spherical black brane in the cases d = 10 , 11. Specific heat flips its sign from negative on the uniform branch to positive on the nonuniform. Nevertheless it does not mean that the branch of the non-uniform branes is thermodynamically stable. Due to the change of sign of the specific heat the 'stable' non-uniform phase, which appears in canonical ensemble for δβ < 0, corresponds to the black branes with the mass larger than M ∗ . Thus the mass of the brane does not cross the GL critical value in the course of this transition, and the would be new phase is related to the local minimum of the entropy which appears for δM > 0. Thus it seems that positive specific heat of non-uniform branch in canonical ensemble is spurious, and the phase transition in canonical ensemble never proceeds towards the found non-uniform branch. Moreover in the case at hand the change of sign of the specific heat during the transition would mean that slightly non-uniform spherical black branes with flat asymptotics are thermodynamically stable in certain number of extended dimensions which does not seem to be the case.", "pages": [ 13, 14 ] }, { "title": "B The source terms for the second order perturbation equations", "content": "For the sake of completeness we present here the full expressions for the sources in backreaction equations. The source terms for the l = 0 mode equations (3.15), (3.18) and (3.19): The source term for the c 2 equation (3.24) reads The source terms for the rest of l = 2 mode equations (3.21), (3.22) and (3.26):", "pages": [ 15, 16 ] } ]

[ { "title": "Paulo H. F. Reimberg and L. Raul Abramo", "content": "Instituto de F'ısica, Universidade de S˜ao Paulo, CP 66318, 05314-970, S˜ao Paulo, Brazil E-mail: [email protected] Abstract. We present a formal derivation of the key equations governing gravitational lensing in arbitrary space-times, starting from the basic properties of Jacobi fields and their expressions in terms of the exponential map. A careful analysis of Jacobi fields and Jacobi classes near the origin of a light beam determines the nature of the singular behavior of the optical deformation matrix. We also show that potential problems that could arise from this singularity do not invalidate the conclusions of the original argument presented by Seitz, Schneider & Ehlers (1994). PACS numbers: 95.30Sf Submitted to: Class. Quantum Grav.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Gravitational lensing can be described, starting from very basic principles, in terms of the deviations of null geodesics with respect to a 'fiducial' ray in a light beam. The formal groundworks upon which this description is based were first spelled out by Seitz, Schneider and Ehlers (1994) [1], and their derivation has been widely used ever since [2, 3, 4, 5, 6]. The fundamental objects in this description are the separation vectors ξ , which determine how a beam of geodesics starting (or ending) at a given point deviates from the fiducial. The relevant components of these vectors naturally belong to a 2-dimensional space-like surface (the screen ) which is orthogonal to the direction of propagation of the null fiducial geodesic. And since, by construction, the beam is focused on the reference point, the separation vectors are such that ξ (0) = 0 . This description is time-symmetric, in the sense that the reference event can be regarded either as the original source of the beam (in which case the affine parameter is future-oriented), or as an observation event (in which case the affine parameter is past-oriented). The separation vectors are in fact the projection on the screen of Jacobi fields along the fiducial ray. A fundamental result in General Relativity is the fact that, to linear order in small perturbations around the fiducial geodesic, the Jacobi equation is linear . When projected on the screen, that equation leads to ξ '' ( λ ) = T ( λ ) ξ ( λ ) , where T is called the optical tidal matrix , and primes denote derivatives with respect to the affine parameter λ along the fiducial geodesic. The separation of a null geodesic from the fiducial ray at any given value of the affine parameter, ξ ( λ ), would then be given by the action of a linear map (the Jacobi map , [1]) on the velocity of separation of that geodesic at the reference point, ξ ' (0) =: θ . Ultimately, these two facts together allow us to frame gravitational lensing entirely in terms of the deviations θ on the screen at the reference point. We point out that previous demonstrations of these fundamental results have relied on a flawed argument which, if taken at face value, would imply that the projections of the Jacobi fields on the screen would vanish identically. E.g., the argument presented in [1] is the following: since the Jacobi equation is linear, the projection on the screen of the Jacobi fields, ξ ( λ ), can be related to the initial deviation ξ ' (0) = θ by a linear transformation, ξ ( λ ) = D ( λ ) θ . Substituting this expression back in the Jacobi equation, they are then able to derive the second-order linear differential equation which governs the evolution of the linear operator D ( λ ). They also claim that the linear operator D obeys the first-order differential equation D ' = S D , where S is called optical deformation matrix and is given in terms of optical observables for beams of null geodesics. This would imply, however, because the deviation can be written as ξ ' = D ' θ , that ξ ' = S ξ for all values of the affine parameter. The problem with this last identity is, of course, that a geodesic with ξ (0) = 0 would necessarily also have to satisfy ξ ' (0) = 0 , which would then imply that ξ = 0 for all values of the affine parameter. The only possible fix for this construction is for the optical deformation matrix to be singular at the reference point, in precisely the right way to introduce some geodesic deviation ξ ' at that point - but this then constitutes an incomplete argument. Here we will close the loophole in this argument. First, we will clarify the statement that the linearity of the Jacobi equation leads to the Jacobi map. Although our argument also relies on the linearity of the Jacobi equation, we present an explicit construction of the relation ξ ( λ ) = D ( λ ) θ where D ( λ ) is directly determined from the exponential map . Second, by employing the notion of Jacobi tensors we will show how to obtain a second-order linear differential equation for D which depends only on the Ricci and Weyl curvature tensors. In our demonstration we will use a few basic results from Lorentzian geometry, in particular the fact that basis vectors along null geodesics cannot be orthonormal, and decomposition of vectors and tensor in terms of these bases are not defined in the classical sense. We follow [7] in our construction of such a basis, over which separation vectors can be expressed. For this basis, we need to introduce the notions of the quotient by an equivalence relation. This then allows us to employ, in Section 4, Jacobi classes and Jacobi tensors [8], which turn out to be key to our argument. In Section 5 we construct the Jacobi map, and obtain the differential equation that is satisfied by the operator D ( λ ). After writing the basic equations that govern the lensing problem in terms of the exponential map, we reexamine the argument presented by [1]. Finally, we present some comments about conjugate points and critical behavior of beams of null geodesics. Our conventions are as follows. Space-time is assumed to be a Lorentzian manifold ( M,g ), and the tangent space to a point p ∈ M will be denoted by T p M . Our metric has signature -, + , + , +, and contractions with it are denoted as 〈 ., . 〉 , i.e., 〈 x, y 〉 = g αβ x α y β . Greek indices range from 0 to 3 and latin indices from 1 to 3, except the indices i and j , which can only take the values 1 or 2. The 2 × 2 identity matrix will be denoted by 1 . The Riemann and Ricci tensors are denoted by R ( . , . ) . and Ric( . , . ), respectively.", "pages": [ 1, 2, 3 ] }, { "title": "2. The exponential map and Jacobi fields", "content": "Let γ : [0 , 1] → M be a null geodesic, so, by definition, D dλ ( dγ dλ ) = 0 and 〈 dγ dλ , dγ dλ 〉 = 0. For given initial conditions γ (0) = p , and γ ' (0) = k ∈ T p M , the solution of the geodesic equation leads to a curve γ ( λ, p, k ). If the solution γ is defined, the affine parameter and the initial velocity can be scaled in such a way that γ (1 , p, k ) ∈ M . The point γ (1 , p, k ) ∈ M is called the image of k under the exponential map, for p fixed. In our notation, the exponential map exp p : T p M → M is defined for a given k ∈ T p M as the point obtained along the unique geodesic in M with γ (0 , p, k ) = p and γ ' (0 , p, k ) = k , after displacement by a length of 1 along this curve, i. e., exp p ( k ) := γ (1 , p, k ) [8]. An illustration of the action of the exponential map is shown in figure 1. Determining the explicit form of the exponential map is, in general, a very difficult task, and relies on the knowledge of all geodesics for a given space-time. Even without explicit formulas for the exponential map, however, much can be learned from its general properties. The geodesic homogeneity lemma, for example, establishes the aforementioned scaling relation between the affine parameter and the velocity, i. e., γ (1 , p, k ) = γ (1 /λ, p, λk ), and allows us to calculate the derivative of the exponential map at λ = 0: for any k ∈ T p M , The link between exponential map and variations of curves in the manifold is largely employed in the context of variational calculus [9], and can be established as follows: let k ( s ), s ∈ ( -ε, ε ) ⊂ R , be a curve in T p M such that k (0) = k, dk/ds (0) := w . Then f ( λ, s ) := γ (1 , p, λk ( s )) = exp p [ λk ( s )] defines a parametrized surface in M . For s = 0 we obtain the curve γ ( λ, p, k ), and other values of the parameter s generate variations of this original curve. A variational vector field J along γ constructed as J ( λ ) = ∂f ∂s ( λ, s = 0) is such that J (0) = 0 and J ' (0) := ∂ 2 f ∂λ∂s ( λ, s = 0) = w . Such a vector field is known as Jacobi field and satisfies the Jacobi equation [7]: Despite being broadly interpreted as geodesic deviation equation, the Jacobi equation has other solutions, depending on the initial conditions. For instance: if γ ( λ ) is a geodesic, then γ ' ( λ ) is a Jacobi field along γ , because γ ' ( λ ) is parallel-transported along itself, and because the Riemann curvature is anti-symmetric with respect to its two first arguments. Another example of a non-trivial Jacobi field along the curve γ ( λ ) is λγ ' ( λ ). Describing gravitational lensing, however, requires to take into account deformations of beams of null geodesics that begin at a given specific point in space-time and, therefore, conduce us to consider Jacobi fields that obey initial conditions J (0) = 0 and J ' (0) = w , where w is the initial velocity of separation between a given geodesic in the beam and the fiducial one. A fundamental property of these Jacobi fields, which lies at the heart of our discussion, is that the unique Jacobi field along γ with J (0) = 0 and J ' (0) = w is given by - see, e.g., Prop. 6 in chapter 8 of [10], or Prop 10.16 of [8]: To make more clear the notation employed in 3 and show that it defines, in fact, a Jacobi field, we can explicitly calculate: where we have used the chain rule to establish the third equality. Expressing the exponential map in terms of its definition, we can write λ ( d exp p ) λk w = ∂ ∂s γ (1 , p, λk ( s )) | s =0 . The differential of the exponential map ( d exp p ) λk is the linear map whose matrix form is given by the directional derivatives of exp p in the direction of the basis vectors of T λk ( T p M ), evaluated at λk . In general, for v ∈ T p M , the tangent space T v ( T p M ) is defined as: Therefore, the space T v ( T p M ) is isomorphic to T p M , so for our purposes the two can be identified. Consequently, ( d exp p ) λk will be understood as a linear map between the spaces T p M and T exp p ( λk ) M . In figure 2 we illustrate the action of the differential of the exponential map on vectors in the tangent space of a point p ∈ γ . To determine the behavior of the differential of the exponential map near the origin, we shall rewrite 1, inserting an intermediate equality (application of the chain rule) between the first and the last one: In other words, the differential of the exponential map calculated at the origin equals the identity matrix. This can be rephrased as the statement that the exponential map defines a diffeomorphism of a neighborhood of the origin of T p M into an open subset of M . This property also allows us to verify that J given in 3 does satisfy J ' (0) = w :", "pages": [ 3, 4, 5 ] }, { "title": "3. Construction of screen vectors", "content": "We now turn our attention to the Jacobi fields that can be expressed as in 3, and the way they can be decomposed in terms of a vectors basis along the null geodesic γ . Let's consider the vector basis along a null geodesic as introduced in Sec. 4.2 of [7]: since p = γ (0), we define a non-orthonormal basis { E 0 , E 1 , E 2 , E 3 } at T p M by requiring that E 0 = γ ' (0). By construction, 〈 E 0 , E 0 〉 = 0, and we take E 3 to be another null vector that obeys 〈 E 0 , E 3 〉 = -1. The remaining vectors E 1 , E 2 are then space-like and can be normalized. To extend this basis along γ we simply parallel-transport the basis vectors. With the help of this basis along γ , we can describe the separation of a bundle of geodesics that emerge from p with any velocity of separation. We will be primarily concerned with the deformation of light beams with respect to the fiducial ray γ , and will describe these other geodesics in terms of the basis defined in the tangent space of all points along the fiducial curve. Since the separation of geodesics is described by the Jacobi fields, and all Jacobi fields that satisfy J (0) = 0, J ' (0) = w have the form given in 3, then all the dependence on the separation between the geodesics is codified in J ' (0). If we decompose J ' (0) in the basis { E 0 , E 1 , E 2 , E 3 } , the component J ' 3 (0) is such that 〈 J ' 3 (0) , γ ' (0) 〉 /negationslash = 0. Hence, it follows from Gauss' Lemma that 〈 J 3 ( λ ) , γ ' ( λ ) 〉 /negationslash = 0 for all λ , where J 3 is the component of J in the direction E 3 . This component brings no information whatsoever about the problem we are interested in, so we shall only consider the vector space that results from the inverse image of 0 by the application 〈 . , γ ' ( λ ) 〉 . With a notational abuse, we shall call this space ˜ T p M , even if p was previously assumed to be fixed. There is, however, still one null Jacobi field remaining in ˜ T p M : as we have already remarked, γ ' ( λ ) is itself a Jacobi field along γ , but it does not describe any spatial separation between geodesics. In order to eliminate this component, consider two vectors W and V in ˜ T p M , and define W ∼ V if W -V ∈ span( γ ' ). It is easy to show that this is an equivalence relation in ˜ T p M . The set of equivalence classes in ˜ T p M by the relation ∼ inherit the structure of vector space from ˜ T p M . We shall denote ˜ T p M/ span( γ ' ) the set of equivalence classes of ˜ T p M by the relation ∼ . Given some V ∈ ˜ T p M , [ V ] ∈ ˜ T p M/ span( γ ' ) will denote the class corresponding to it. The same vector [ V ] corresponds to the family of vectors V + αγ ' ∈ ˜ T p M , α ∈ R , as illustrated in figure 3. The two-dimensional space ˜ T p M/ span( γ ' ) we have just constructed is called screen , and its elements are called screen vectors . The result of the construction above is a two-dimensional vector space associated to every point along γ , in terms of which separations of geodesics in the beam can be described. All vectors in this space are space-like, and correspond to the naive idea of projection to a space 'orthogonal' to γ ' ( λ ) - after taking into account that γ ' ( λ ) is orthogonal to itself.", "pages": [ 5, 6, 7 ] }, { "title": "4. Jacobi classes, Jacobi tensors and optical scalars", "content": "We have just constructed a vector space to which the objects we want to describe will be restricted. We should ask how the space-time geometry is expressed when restricted to this space. Firstly, we note that, given U, V ∈ ˜ T p M , then, since ˜ T p M has the structure of ˜ T p M/ span( γ ' ) ⊕ span( γ ' ), U = [ U ] + αk and V = [ V ] + βk for same α, β ∈ R . Therefore 〈 U, V 〉 = 〈 [ U ] + αk, [ V ] + βk 〉 = 〈 [ U ] , [ V ] 〉 . Similarly, /triangleinv γ ' [ U ] = [ /triangleinv γ ' U ]. Also [ R ( U, γ ' ) γ ' ] = [ R ([ U ] , γ ' ) γ ' + αR ( γ ' , γ ' ) γ ' ] = [ R ([ U ] , γ ' ) γ ' ] := [ R ]([ U ] , γ ' ) γ ' , due to the symmetries of Riemann tensor. We shall now consider the Jacobi fields restricted to the quotient space or, Jacobi classes , as these restrictions are usually called. In general, Jacobi classes are smooth vector fields in the quotient space satisfying the relation: It can be shown that, given a Jacobi class j ∈ ˜ T p M/ span( γ ' ), there exists a Jacobi field J in ˜ T p M such that [ J ] = j , and, conversely, if J is a Jacobi field in ˜ T p M , j = [ J ] is a Jacobi class in ˜ T p M/ span( γ ' ) [8]. We now introduce objects called Jacobi tensors [8], which are of fundamental importance in the discussion about optical properties of light beams, and in terms of which the optical scalars will be defined. Let A be a bilinear form A : ˜ T p M/ span( γ ' ) → ˜ T p M/ span( γ ' ). For any W ∈ ˜ T p M , [ R ] A ([ W ]) := [ R ]( A [ W ] , γ ' ) γ ' . We shall say that A is a Jacobi tensor if: subject to convenient initial conditions, and If the bilinear form A satisfies 7, then vector fields constructed by its action on vector basis E i will be Jacobi classes. The condition 8 eliminates trivial Jacobi classes from consideration: if j i = AE i are Jacobi classes along γ , since E i are parallel propagated, j ' i = A ' E i give their (covariant) derivatives. If at some point along γ the same linear combination of E i s belong to the kernel of both A and A ' , then the action of A on this linear combination will give rise to a trivial Jacobi class. To see that this is the case, we should recall that the dimension of the kernel of A is only non-null at conjugate points [8], and conclude, by taking a point λ ∗ conjugate to λ = 0 along γ for which j ' ( λ ∗ ) = [ γ ' ], that the only possible solution for 6 in this case does not represent separation of curves in a beam. The condition 8 has an important consequence: objects defined by A ' A -1 may be singular either because A is singular, or because A ' is singular, but never because both are singular at the same point. The optical scalars expansion , vorticity and shear of a geodesic beam are defined in terms of the Jacobi tensors, namely, through combinations of the object B =: A ' A -1 . Explicitly, we have [8]:", "pages": [ 7, 8 ] }, { "title": "(i) expansion", "content": "(ii) vorticity (iii) shear In terms of these objects the Raychaudhuri equation reads: Jacobi tensors are, then, key objects to understand and describe deformations of light beams. Next we will address the explicit construction of a Jacobi tensor that satisfies the physical restriction: since we are interested in gravitational lensing, all geodesics in the beam start at the same point and have different separation velocities with respect to a fiducial ray.", "pages": [ 9 ] }, { "title": "5. The Jacobi map", "content": "In what follows we will explicitly build the Jacobi map. The argument of Ref. [1] is that the linearity of the Jacobi equation implies that the Jacobi map should hold - although this is in fact true, we will see that it is far from trivial. By definition, in ˜ T p M/ span( γ ' ), where w = J ' (0) and i = 1 , 2. This initial velocity can be expressed as w = θ 1 E 1 (0) + θ 2 E 2 (0) + αE 0 and, since 〈 ( λd exp p ) λk E 0 , E i ( λ ) 〉 = 0, there is no loss of generality in taking w to be restricted to ˜ T p M/ span( γ ' ). This just means that we only have to determine the space-like components of the initial velocity of dispersion of geodesics in order to determine their future evolution. The tangential component of the geodesics may also deserve attention, for instance in cosmological contexts where redshifts or the Sachs-Wolfe [11] effect can take place, but in those cases the corrections can be treated separately. Taking the initial spread velocity restricted to ˜ T p M/ span( γ ' ), then 〈 [ λ ( d exp p ) λk . ] , [ . ] 〉 constitutes a bilinear form in ˜ T p M/ span( γ ' ), and therefore it admits a matrix representation: 13 expresses the separation of null geodesics in terms of the initial velocity of dispersion of the beam. If we denote the 2 × 2 matrix of 13 by D ( λ ), even if it depends on the point p ( λ = 0) and on the fiducial geodesic γ , we obtain ‡ : This is, in fact, what is usually called Jacobi map in the gravitational lensing literature [1]. Using the explicit form given in 13, it is easy to see, after 3, 1 and 5, that the matrix D satisfies the initial conditions D (0) = 0 and D ' (0) = 1 .", "pages": [ 9, 10 ] }, { "title": "6. A differential equation for D", "content": "We will now show that the matrix D which appears in the Jacobi map, 14, satisfies the differential equation 7. This can be verified with the help of the Jacobi equation 2. Let's define /epsilon1 = [ E 1 ] + i [ E 2 ], J = j 1 + ij 2 , and let ¯ /epsilon1 and ¯ J be their complex conjugates. Since for a given Jacobi field there is a unique Jacobi class associated with it, where R and F are given in terms of the Riemann curvature. Writing k = γ ' , it can be shown [12] that these objects are given by and where R µν and C αµβν are the components of the Ricci and Weyl tensor, respectively. Substituting 14 in 15 we obtain, indeed, 7. Explicitly, where T is the matrix form of the endomorphism [ R ]( . , γ ' ) γ ' acting on ˜ T p M/ span( γ ' ), and is usually called optical tidal matrix . In terms of R and F , T reads: As already remarked, 18 must be subjected to the initial conditions D (0) = 0 and D ' (0) = 1 .", "pages": [ 10 ] }, { "title": "7. The condition on the kernels", "content": "/negationslash In order to verify that D is in fact a Jacobi tensor, we must guarantee that the condition expressed by 8 is satisfied by the bilinear form constructed in 13, for all values of λ for which γ is defined. Since 8 holds for λ = 0 because of the initial conditions, we shall consider the case λ = 0. Keeping in mind that the vectors E i ( λ ) are parallel-transported along the fiducial ray, taking the derivative D ' ( λ ) leads to The Jacobi map for gravitational lensing: the role of the exponential map In order to determine if there are common elements in the kernels of D and D ' , we shall study if elements in the kernel of D may also be in the kernel of D ' . Given its explicit form of 13, and the fact that Jacobi fields expressed by 3 are linearly independent if the initial velocities are linearly independent, ker( D ( λ )) has non-trivial elements only at points which are conjugate points to λ = 0 along γ . At conjugate points, the first term in the right hand side of 20 is singular. The second term in the right hand side of 20, however, must not be singular for the same Jacobi class because, if it were, then the Jacobi class thus obtained would be the trivial one. We must, therefore, make the derivative term in the right-hand side of 20 to be singular because of the second Jacobi class, i.e., the second Jacobi class must reach an extremum exactly at the same value of the parameter where the first class vanishes. Although this is not sufficient to cause a violation of the condition expressed by 8, if we could have a sequence of conjugate points along the two Jacobi classes, as illustrated in figure 4 (i. e., between any two conjugate points of a class, there is a maximum of the other class), and if we were able to make all conjugate points accumulate in a neighborhood of some point in the interval, then there could be a way of taking this limit such that the condition 8 would fail in some point of that neighborhood. The Morse Index Theorem, however, states that the set of conjugate points along a null geodesic is a finite set [8]. Consequently, an accumulation of conjugate points cannot take place, and hence the limit depicted in figure 4 cannot exist. We conclude, then, that 8 holds for the bilinear form D , and therefore it determines a Jacobi tensor.", "pages": [ 10, 11 ] }, { "title": "8. The behavior near the origin and the sweep method", "content": "We now return to the argument presented in [1] to show that, despite the loophole in their argument leading to an equation equivalent to 18, this is of no consequence. Specifically, we will show that a matrix S such that D ' = SD and j ' = S j can be defined, but that it cannot be taken as a starting point for obtaining 18. Suppose that there exists a matrix S such that D ' ( λ ) = S ( λ ) D ( λ ), and that we restrict the domain of λ in such a way that there are no conjugate points to λ = 0 along γ . Then, from 18 it follows that S ' + S 2 = T . This equation is known as matrix Riccati equation , and corresponds to the usual association of Riccati's equation to second-order homogeneous differential equations. The initial conditions for this matrix Riccati equation should be carefully considered. If both D (0) = 0, D ' (0) = 1 and D ' ( λ ) = S ( λ ) D ( λ ) hold simultaneously, then S (0) must not be limited. Consequently, one cannot guarantee the existence of solutions for this first-order linear differential equation if λ is in the neighborhood of the origin ( λ = 0), and in that case one cannot assure a solution for the associated Riccati equation either. To avoid problems with the solution around λ = 0, we take λ > /epsilon1 > 0 and impose initial conditions D ( ε ) and D ' ( ε ), as illustrated in figure 5. The equation D ' = SD will then admit solutions for λ > ε , if S ( λ ) is bounded for λ > ε . If some further conditions D ( ¯ λ ) and D ' ( ¯ λ ), ¯ λ > ε , are also given, then one can solve for D ' = SD , subjected to these conditions, and check whether the solutions match those obtained by imposing initial conditions at λ = ε . If the two solutions match, the initial conditions given at ε and ¯ λ are said consistent, and the solution D will also be a solution of 18 in the same range of parameters. This method of obtaining solutions is known as 'sweep method' [13]. The sweep method would work well if geodesics in the beam did not cross each other - in other terms, if they formed a congruence. If the beam emerges from the same point, however, initial conditions at that point cannot be used for the 'forward sweep' step of the method, because one cannot guarantee the existence of solutions for the differential equation in this case. In spite of that, if one knows, by any other method, a solution for 18, then the behavior of this solution near λ = 0 can be investigated. We presented explicitly how D is given in terms of the exponential map in 13, and therefore we can investigate the properties of D ( ε ), ε ≈ 0. γ If one recalls that the entries of 13 are projections of Jacobi fields in base vectors, and that Jacobi fields in the neighborhood of the origin behave like ε + O ε 3 , we see that for ε close to zero, D ( ε ) ≈ ε 1 . Then, from 20 we obtain that D ' ( ε ) ≈ 1 . Hence, an equation like D ' = SD would only make sense if near λ = 0 the matrix S had a behavior like S ( ε ) ≈ (1 /ε ) 1 . We remark that, despite the fact that the equation D ' = SD may not be inconsistent with the initial conditions D (0) = 0 and D ' (0) = 1 , if S diverges in the prescribed way near the origin, the initial conditions for the Riccati equation cannot be provided at the origin, and the sweep forward cannot be employed. In other words, a general solution for 18 must be known in order to verify that D ' = SD is consistent near the origin but this last equation could never be used to derive or to generate solutions to 18 if the origin (or conjugate points) are in the domain of the solution.", "pages": [ 11, 12, 13 ] }, { "title": "9. Conjugate points and critical behavior", "content": "The set of conjugate points along the geodesic γ is not degenerate when the restriction to the quotient space is taken: it is in fact completely preserved, because the Jacobi field in the direction of γ ' (i.e., γ ' itself) does not vanish. All possible non-trivial Jacobi fields that satisfy J (0) = 0, and vanish at some other point γ ( λ 1 ), correspond to Jacobi classes satisfying [ J (0)] = [ γ ' (0)] and [ J ( λ 1 )] = [ γ ' ( λ 1 )]. The conjugate points along γ correspond to the critical points of the matrix D , and induce optical critical behavior which is codified by the diverging expansion given by 9. Because ˜ T p M/ span( γ ' ) is bi-dimensional, the maximal multiplicity of any conjugate point is two. After a conjugate point of multiplicity one, the source's image will appear inverted - something that does not occur after conjugate points of multiplicity two (or two conjugate points of multiplicity one). The presence of conjugate points along γ is much more significant in aspects other than classifying an image's orientation. [14] showed that conjugate points are necessary to form multiple images, and due to the connections between the existence of conjugate points along geodesics and the energy conditions [7], multiple images will always occur in a large class of space-times, as long as geodesics can be enough extended. As conjugate points are critical points of the energy functional, they also play an important role in the formulation of Morse Theory applied to light rays, in particular in the formulation of theorems on the possible number of images in lensing configurations [15].", "pages": [ 13, 14 ] }, { "title": "10. Discussion", "content": "The description of gravitational lensing presented here is central in many applications, especially in cosmology ([1, 2, 3, 4]), when 18 can be solved perturbatively to generate a lens map. Equations with the same general structure are also fundamental in the discussion of singularities in the Penrose limit [16] and can have other applications. Writing these equations in a rigorous way has allowed us to determine their explicit form in terms of the exponential map, beyond generating a clear interpretation for the Jacobi map and validate its centrality in the description of gravitational lensing. The limitations of the derivation presented in this work should also be clarified. First, the Jacobi equation itself is only approximative in first order to the problem of geodesic spread. Corrections may be included, such as non-linearities with respect to the velocity dispersion [17]. Second, despite these attempts to enlarge the limits of application of generalizations of the Jacobi equation, there are obstacles that may not be overpassed in this formulation, such as configurations in which cut points [8] exist, and a more careful geometrical analysis is required. One such example is lensing produced by a string, as introduced by [18], which induces a flat space-time that nevertheless generates multiple imaging. In full generality, gravitational lensing phenomena involve conjugate points and even cut points, and we can expect that an observer's past light cone will eventually reveal this complex geometry.", "pages": [ 14 ] }, { "title": "Acknowledgments", "content": "This work was supported by FAPESP.", "pages": [ 14 ] } ]

[ { "title": "How trapped surfaces jump in 2+1 dimensions", "content": "Emma Jakobsson ∗ Fysikum, Stockholms Universitet, S-106 91, Stockholm, Sweden", "pages": [ 1 ] }, { "title": "Abstract", "content": "When a lump of matter falls into a black hole it is expected that a marginally trapped tube when hit moves outwards everywhere, even in regions not yet in causal contact with the infalling matter. But to describe this phenomenon analytically in 3+1 dimensions is difficult since gravitational radiation is emitted. By considering a particle falling into a toy model of a black hole in 2+1 dimensions an exact description of this non-local behaviour of a marginally trapped tube is found.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "A black hole is defined by its event horizon; a boundary in spacetime, such that no event inside it can ever be seen from the outside. With this definition it is impossible to locate the event horizon without knowledge about the infinite future. Attempts to make alternative definitions of a black hole involve trapped surfaces that occur in the interior [1, 2, 3]. A trapped surface is a closed, spacelike surface such that both families of light rays orthogonal to it converge. The terminology of concepts closely related to these trapped surfaces might need to be made clear: A closed spacelike surface such that only one of the orthogonal families of light rays converges while the other has zero convergence, is referred to as a marginally trapped surface. If the surface is embedded in a hypersurface on which an outer direction is defined in a manner that would be intuitive in an asymptotically simple spacetime, and this surface is such that the outgoing family of light rays orthogonal to it converges, it is called outer trapped, regardless of the behaviour of the ingoing family of light rays. Marginally outer trapped surfaces are defined in a similar manner. While the event horizon is a globally defined property of spacetime - and therefore, as we will see, teleological in its nature - trapped surfaces are quasilocal, since their definition only involves the surfaces themselves and their infinitesimal surroundings. For this reason trapped surfaces are of importance to numerical relativists, since the occurence of such is the only practical way to identify a black hole in a simulated evolution of spacelike hypersurfaces. In such simulations the trapped surfaces sometimes make discontinuous 'jumps' outwards [4, 5]. This phenomenon is expected when matter is falling into the black hole [6]. A marginally trapped tube is a hypersurface foliated by marginally trapped surfaces. The marginally trapped tubes we will come across will be null and satisfy some other constraints that qualify them as isolated horizons [7]. It is desirable to find an exact description of how a marginally trapped tube is affected when hit by matter. This problem has also been studied in spherically symmetric cases [8, 9]. However, if a localized 'lump' of matter is falling into a black hole, it is much more difficult to find an analytical description since gravitational radiation is emitted. But it is expected that the jump in this case will be in some sense non-local; that the jump will take place also in regions not yet in causal contact with the infalling matter. There is no need to worry about causality violation; this effect is just a consequence of the quasilocal definition of a trapped surface. Light rays emitted from a region on a spacelike surface may converge, but whether the whole surface is closed - and thus trapped - or not depends on circumstances elsewhere. Because of the difficulties in 3+1 dimensions we instead tackle the problem in 2+1 dimensions [10] where there is no gravitational radiation. We consider a toy model of a black hole and let a point particle fall into it in order to find an exact description of how the marginally trapped tube jumps outwards in this non-local way.", "pages": [ 1, 2 ] }, { "title": "2 The black hole and trapped surfaces", "content": "The existence of a black hole in a 2+1-dimensional spacetime with constant negative curvature was first discovered by Bañados et al [11]. This is called a BTZ black hole. It is obtained by identifying points in anti-de Sitter space using an isometry [12]. embedded in a four dimensional spacetime with metric It has constant curvature which is negative. Each point can be represented by a matrix so that But this is a group element of SL (2 , R ) , consisting of all two by two matrices with real matrix elements and determinant one. Furthermore, any isometry can be described by letting the group act on itself. Isometries leaving the unit element fixed can be written where g 1 ∈ SL (2 , R ) . Transformations of the type (5) will have a line of fixed points and the nature of this line is determined by the trace of g 1 . If Tr g 1 < 2 it will be timelike, if Tr g 1 = 2 it will be lightlike and if Tr g 1 > 2 it will be spacelike. The embedding coordinates are convenient to use in calculations, but for visualization the intrinsic coordinates ( t, ρ, φ ) [13] are a better choice. They are given by The metric in these coordinates is With this choice of coordinates anti-de Sitter space is depicted as a cylinder. The timelike coordinate t runs along the cylinder, and the spatial slices of constant t are Poincaré disks. On the disk, ρ and φ are the radial and angular coordinates respectively and J is situated at the boundary ρ = 1 . To create a black hole we choose a group element The real constant µ will determine the mass of the black hole. Then we act with g BH on anti-de Sitter space through conjugation as in Eq. (5), and identify points that are transformed into each other. The region between the two surfaces Y = V tanh µ and Y = -V tanh µ can be taken to represent the resulting quotient space, as in Fig. 1. Due to the identification a spacelike slice now has the geometry of a cylinder, but space is still locally anti-de Sitter everywhere. Note that there are two asymptotic regions, as in the Schwarzschild solution in which one of the regions is considered unphysical. The fixed points of the transformation yielding the identification are located at the spacelike line Y = V = 0 . Starting from the slice t = -π/ 2 it is seen that the cylinders shrink in the periodical direction as t increases, until one dimension suddenly disappears at t = 0 , and all that is left is the line of fixed points. A geodesic ending at this singular line ends after only a finite parameter time, meaning that this spacetime is geodesically incomplete. The event horizon is the backward light cone of the last point on J , i.e. the point where the singular line meets J . There is one event horizon for each asymptotic region. In the embedding coordinates the event horizons are given as the quotient of each of the two surfaces X = ± U . The black hole spacetime is locally anti-de Sitter everywhere except at the singular line. On a spacelike surface, the only way to distinguish it from anti-de Sitter space is through the holonomy of the black hole: If a vector is parallel transported along a curve closed by the identification it will also be transformed by the group element effecting the identification. Finding trapped surfaces - or rather trapped curves, since we are in 2+1 dimensions - is easy. Consider the intersection of two light cones with vertices at the singularity. Light rays emanating orthogonally from such curves obviously converge. Moreover they coincide with flow lines of the identifying isometry and are therefore closed to smooth curves by the identification. Hence they are trapped. By letting one of the two vertices be on J , and varying the other, it is easily seen that the event horizon is a marginally trapped tube, that is a surface foliated by marginally trapped curves. Since trapped surfaces can not exist outside the event horizon according to the cosmic censorship hypothesis, the marginally trapped tube - that is the event horizon in this model - is also the boundary of the region containing trapped curves. In fact this is the complete picture: all marginally trapped curves lie on the event horizon. To see this, consider Raychaudhuri's equation [15] for the expansion θ of a congruence of lightlike geodesics in 2+1 dimensions. With k a being the tangent vector of a given geodesic we have If we impose Einstein's vacuum equation R ab = λg ab the second term vanishes since k 2 = 0 for a lightlike geodesic. We are left with which shows that a congruence of lightlike geodesics that have zero convergence at some point, must continue to have zero convergence. The conclusion is that a marginally trapped curve must lie on a null plane 1 , where a null plane is defined as a light cone with its vertex on J . It is not difficult to show that only the null plane containing a fixed point on J contains smooth and spacelike closed curves. As a side note, there is a theorem that says that a region of a spacelike hypersurface bounded by an outer trapped surface in one direction and by an outer untrapped surface in the other must contain a marginally outer trapped surface [16]. In this model the statement is almost obvious. Any smooth spacelike surface passing through the interior of the black hole will contain a smooth closed curve lying on the event horizon and thus being a marginally outer trapped curve. Since it lies on the event horizon it also separates the region containing trapped curves from the region not containing trapped curves on the surface.", "pages": [ 2, 3, 4, 5 ] }, { "title": "3 The infalling particle", "content": "Just like a black hole was obtained by identifying points, a point particle can be modelled using the same trick. Note that the matrix of Eq. (8) has a trace larger than two, and therefore has a spacelike line of fixed points. If we instead choose the group element with a being an arbitrary real constant, and identify points in anti-de Sitter space through conjugation, the line of fixed points will be lightlike since Tr g P = 2 . A fundamental region containing one representative of every point in the quotient space can be chosen by cutting away the wedge between the two identified surfaces Y = ± a ( X -V ) . The effect is that a surface of constant t now has the geometry of a cone, with the tip of the cone being a fixed point of the identification. This setup perfectly well describes a point particle [17, 18]. The particle is situated at the conical singularity, and it is a lightlike particle since its world line is lightlike. Let us consider a sequence of Poincaré disks. Before the time t = -π/ 2 there is no particle, just empty anti-de Sitter space. At t = -π/ 2 the particle comes in from infinity. Then it traverses the disk as t increases until it finally leaves at t = π/ 2 and we again are left with empty anti-de Sitter space. On the disk, space is locally anti-de Sitter everywhere except at the singularity, and the only way to notice the presence of the particle is to travel around it and reveal its holonomy. That the particle enters empty anti-de Sitter space from infinity is a property unique for lightlike particles in this construction. It is not crucial that the particle we use is lightlike, we might just as well consider a timelike particle. But the advantage of using a lightlike particle is that the starting point will be an undisturbed BTZ spacetime, instead of a white hole emitting massive particles. We are now ready to set up a model in which we let the particle fall into the black hole. The result is illustrated in Fig. 2. As the lightlike particle approaches the center of the disk it is seen how the identification surfaces of the particle eventually begin to intersect the identification surfaces of the black hole. These points of intersection are fixed points under the action of the combined holonomy g tot = g P g BH . Here the constants a and µ are chosen so that |Tr g tot | > 2 and consequently the transformation g → g tot gg -1 tot has a spacelike line of fixed points. This spacelike line is singular and appears at smaller t than the singularity of the original black hole. This means that the role of the original singularity is taken over by this new singular line. In turn this affects the location of the event horizon, shown as the dashed curves in Fig. 2. Also the mass of the black hole has been affected by the infalling particle. The change in mass is determined by the constant a . It turns out that the event horizon in this model has a kink before the particle crosses it. This kink nicely illustrates the teleological nature of the event horizon since it has acquired a kink not because of something that has happened to it in the past, but because of something that will happen to it in the future. Due to the kink the event horizon is not everywhere smooth, with the consequence that it is not completely foliated by marginally trapped curves. The question now is where the marginally trapped curves are in this model. We know that they are found on null planes and that a null plane is smooth only if it contains a fixed point on J . It is a crucial fact that the light cone on which the path of the particle lies splits the spacetime into two qualitatively different parts. In the outer region the holonomy is g tot . The event horizon is smooth and it contains the point on J that is a fixed point under the action of this holonomy. Therefore it is also foliated by marginally trapped curves. Moreover, the event horizon is the boundary of the region containing trapped curves since these can only appear in the interior of the black hole. In the inner region, on the other hand, the holonomy is g BH , and it is therefore isometric to a region of the BTZ spacetime. Restricted to this region, the situation is thus identical to that of a black hole with no infalling particle. All marginally trapped curves lie on the null plane that would have been the event horizon had the particle not been there. And, as we saw, this null plane is also the boundary of the region containing trapped curves. It is an isolated horizon in the terminology of ref. [7], as well as the event horizon in the outer region. But after it has been hit by the particle - in the outer region - it is no longer smooth. The marginally trapped tube thus consists of two parts: the two isolated horizons in the inner and the outer region respectively. All marginally trapped curves lie on the marginally trapped tube, and thus we have a complete knowledge of their whereabouts, independent of a given foliation of spacetime. When the particle hits the isolated horizon in the interior of the black hole, it is seemingly destroyed but then reappears on the event horizon in the outer region, it 'jumps'. This is clearly illustrated in the conformal diagram of Fig. 3. With this model in which the marginally trapped tube is discontinuous we have thus found a reasonable and exact illustration of how marginally trapped curves jump when hit by matter.", "pages": [ 5, 6, 7 ] }, { "title": "4 Conclusions", "content": "By considering a toy model of a black hole in 2+1 dimensions and letting a point particle fall into the black hole, we have seen how the marginally trapped tube splits into two parts. This exact description of the splitting illustrates the non-local jump described in the introduction. Similarly non-local jumps are expected in 3+1 dimensions, but most likely that case must be attacked numerically. As a concluding remark it is worth noting that since the world line of the particle is singular, the two parts of the marginally trapped tube can not be connected. To get around this problem one could consider a small tube of null dust instead of a point particle. It might be interesting to see what the marginally trapped tube would look like in this more complicated model; in particular if it would be smooth, and if so, if the smooth part joining the two isolated horizons would be timelike or spacelike.", "pages": [ 8 ] }, { "title": "Acknowledgements", "content": "I would like to thank Ingemar Bengtsson for bringing my attention to the problem and for his significant support, and two anonymous referees for helpful comments. I would also like to thank Sören Holst for accepting the similarities between Fig. 1 and his original.", "pages": [ 8 ] } ]

[ { "title": "AdS flowing black funnels: and heat transport in the dual CFT", "content": "Sebastian Fischetti 1 , Donald Marolf 1 , 2 , and Jorge E. Santos 1 1 Department of Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, U.S.A. 2 Department of Physics, University of Colorado, Boulder, CO 80309, U.S.A. September 9, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "/negationslash We construct stationary non-equilibrium black funnels locally asymptotic to global AdS 4 in vacuum Einstein-Hilbert gravity with negative cosmological constant. These are non-compactly-generated black holes in which a single connected bulk horizon extends to meet the conformal boundary. Thus the induced (conformal) boundary metric has smooth horizons as well. In our examples, the boundary spacetime contains a pair of black holes connected through the bulk by a tubular bulk horizon. Taking one boundary black hole to be hotter than the other (∆ T = 0) prohibits equilibrium. The result is a so-called flowing funnel, a stationary bulk black hole with a non-Killing horizon that may be said to transport heat toward the cooler boundary black hole. While generators of the bulk future horizon evolve toward zero expansion in the far future, they begin at finite affine parameter with infinite expansion on a singular past horizon characterized by power-law divergences with universal exponents. We explore both the horizon generators and the boundary stress tensor in detail. While most of our results are numerical, a semi-analytic fluid/gravity description can be obtained by passing to a one-parameter generalization of the above boundary conditions. The new parameter detunes the temperatures T bulk BH and T bndy BH of the bulk and boundary black holes, and we may then take α = T bndy BH T bulk BH and ∆ T small to control the accuracy of the fluid-gravity approximation. In the small α, ∆ T regime we find excellent agreement with our numerical solutions. For our cases the agreement also remains quite good even for α ∼ 0 . 8. In terms of a dual CFT, our α = 1 solutions describe heat transport via a large N version of Hawking radiation through a deconfined plasma that couples efficiently to both boundary black holes.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "We focus here on the classic problem of heat transport far from equilibrium, and away from the perturbative regime. If the system of interest is an appropriate strongly coupled large N conformal field theory (CFT), we may use gauge/gravity duality to exploit a perhapsmore-tractable description as a semi-classical bulk gravitational system. We will consider the classical limit in cases where the bulk description may be truncated to Λ < 0 EinsteinHilbert gravity. Our work complements perturbative computations of heat transport in this regime (e.g. [1]), as well as non-perturbative studies of thermalization (see e.g. [2, 3, 4] for recent examples and further references) and holographic shockwaves [5, 6]. Suppose in particular that we couple a CFT in d spacetime dimensions to heat sources or sinks of finite size and at finite locations. A convenient way to introduce such sources is to place the CFT on a background non-dynamical spacetime containing stationary black holes with surface gravity κ , which have temperatures κ/ 2 π due to the Hawking effect. As we review in section 2 below, this problem may also be generalized so that the field theory temperature at the black hole horizon differs from κ/ 2 π . But since no information can flow outward across the horizon, the choice of a black hole metric is nevertheless useful to decouple our CFT from the details of the heat sources and sinks. The problem of heat transport then becomes one of computing the expectation value of the stress tensor in the given background with the stated boundary conditions. Since the background spacetime is not dynamical, we can choose the metric at will. In particular, we can include as many black holes as we like at locations of our choosing, and we are free to assign their surface gravities as desired. Of course, since we consider CFTs, we may also conformally rescale the background metric to reinterpret our heat sources/sinks as being infinitely large and located at infinite distance; more will be said about this alternate interpretation in section 2 below. Gauge/gravity duality for large N field theories [7] has been used to study related settings in [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. In this context, the d -dimensional black hole spacetime on which the CFT lives becomes the conformal boundary of a ( D = d +1)-dimensional asymptotically locally anti-de Sitter (AlAdS) spacetime and we henceforth refer to our heat sources and sinks as boundary black holes. Though the above explorations in gauge/gravity duality involved certain tensions and subtleties, the picture that emerged in [22] (building on [19]) is one with two important phases for each boundary black hole, even when the CFT state is assumed to contain a deconfined plasma. See [30] for a condensed review. In the so-called 'funnel phase' a given boundary black hole is connected to distant regions of the boundary by a bulk horizon along which heat may be said to flow (say, if unequal temperatures are fixed at the two ends). But there is no such connection in the contrasting 'droplet phase.' Figure 1 depicts both phases for a simple case in which the boundary spacetime is asymptotically flat. In the CFT description, the funnel phase allows a given boundary black hole to exchange heat with distant regions much as in a free theory with a similar number of degrees of freedom. One may say that grey body factors are O (1) even at large N . But in the droplet phase there is no conduction of heat between a given black hole and the region far away at leading order in large N . In effect, all grey body factors associated with the black hole vanish at this order 1 , so that the black hole does not couple efficiently to the surrounding plasma. Additional phases are also possible that conduct heat between subsets of nearby black holes but not to infinity. Until recently, both funnel and droplet solutions were largely conjectural. Explicit examples were known only in rather contrived settings or in low enough dimensions that all properties were determined by conformal invariance. But numerical methods were used to construct more natural droplets in [28] and more natural funnels in [31]. An interesting detail is that the droplet solutions of [28] contain deformed planar black holes (see figure 1) with vanishing temperature. Constructing black droplet solutions that include finite-temperature (deformed) planar black holes remains an open technical challenge, though perturbative arguments give strong indications that they exist. The above funnel and droplet papers largely focussed on cases without heat flow; i.e., either on droplets (in which heat does not flow in the approximation that the bulk is classical) or on equilibrium funnels. The one exception was [30] which showed that, by changing conformal frames, rotating BTZ black holes [32, 33] in AdS 3 can be re-interpreted as describing heat transport in 1+1 CFTs. Here the standard left- and right-moving temperatures T L , T R of the BTZ black hole correspond directly to the temperatures of the left- and right-moving components of the CFT. Due to the strong constraints of conformal symmetry in low dimensions these components do not interact and the temperatures T L , T R must be constants /Bullet /Bullet /Bullet /Bullet if the heat flux is stationary. In addition, the flow of heat is necessarily isentropic (having no local generation of entropy). We refer to black funnels transporting heat as 'flowing funnels.' Since none of the above special properties should hold for d > 2 ( D > 3), higher dimensional flowing funnels should be quite different than those found in [30]. For example, a bulk horizon connecting two boundary black holes of different temperatures should (at least in some rough sense) be describable as having a temperature that varies along the horizon. But recall that there is no generally accepted definition of horizon temperature which allows this temperature to vary 2 . Indeed, the fact that any definition of temperature should vary implies that the horizon is not Killing, which is already a novel property for a stationary black hole 3 . This suggests that the horizon generators have positive expansion (though of course tending to zero in the far future), so that they caustic at finite affine parameter in the past. It is natural to expect this caustic to occur at a singular past horizon [22], and section 5 confirms this picture for our solutions. We focus below on what we call D = 4 global flowing funnels, by which we mean deformations of the global AdS 4 black string (also known as the Ba˜nados-Teitelboim-Zanelli (BTZ) black string; see e.g. [40] where the solution was obtained as a special case of the AdS C-metric). This reference solution may be constructed by starting with global AdS 4 written in coordinates for which slices of constant radial coordinate z are just AdS 3 . One then replaces each such AdS 3 slice with a BTZ metric [32, 33] having the correct z -dependence and which we chose to be nonrotating. The result is an AlAdS Einstein metric which may be written with H ( z ) = /lscript 3 cos( z//lscript 3 ) and f ( r ) = ( r 2 -r 2 0 ) //lscript 2 3 . The solution is sketched in figure 2. Here the horizon of the BTZ string is at r = r 0 , the parameter /lscript 4 is the AdS 4 length scale, and the AdS 3 length scale /lscript 3 of the BTZ foliations may be set to any desired value by rescaling z, r, r 0 . The two boundary black holes (at z//lscript 3 = ± π/ 2) have the same temperature, but we will seek deformations where these temperatures differ and heat flows between the two boundary black holes. The outline of the paper is as follows. Section 2 reviews how static black funnels (i.e., without flow) may be generalized by adding a parameter α = T bndy BH /T bulk BH which allows the temperature of bulk and boundary horizons to differ. In the small α limit, the analogous flowing funnels can be described in a derivative expansion; i.e., using the fluid/gravity correspondence of [34]. This correspondence is briefly reviewed and then applied to flowing funnels in section 3. Section 4 then explains how to formulate the construction of flowing funnels with any α in a manner where one can proceed numerically. The results of such numerics are presented in section 5 where they are compared to the fluid approximation of section 3. As expected, we find excellent agreement for small α , though for our cases the agreement remains good even for α close to 1. We close with some final discussion in section 6. Note: In the final stages of this work we learned of [41], which also addresses the construction of AdS black holes with non-Killing horizons and may have some overlap with our work. Their paper will appear simultaneously with ours on the arXiv.", "pages": [ 2, 3, 4, 5, 6 ] }, { "title": "2 Detuning the bulk and boundary black hole temperatures", "content": "/negationslash As mentioned in the introduction, even without heat flow the black funnel paradigm may be generalized by adding an extra parameter α = T bndy BH T bulk BH which allows us to detune to the temperatures of the bulk and boundary black holes. In terms of the dual field theory, taking α = 1 means that one considers a thermal ensemble at some temperature T Field Theory which differs from the natural temperature T bndy BH of the (say, static) boundary black hole spacetime on which the field theory lives. One may think of the resulting state as defined by a Euclidean path integral with period 1 /T Field Theory = 1 /T bndy BH and thus having a conical singularity at the horizon of the boundary black hole. What is interesting about this construction is that the gravitational dual can have a completely smooth Euclidean AlAdS bulk, with the conical singularity of the boundary geometry resulting only from a failure of the standard AlAdS boundary conditions at the singular boundary points [42, 43, 44]. Any smooth horizon then clearly has temperature T bulk BH = T Field Theory = T bndy BH as determined by the Euclidean period. /negationslash The prototypical detuned solution studied in [42, 43, 44] is just the general hyperbolic (sometimes referred to as 'topological') black hole of [45, 46, 47], whose metric in D = d +1 bulk spacetime dimensions may be written Here /lscript d +1 is the AdS length scale associated with the ( D = d +1)-dimensional cosmological constant, d Σ 2 d -1 = dξ 2 +sinh 2 ξd Ω 2 d -2 is the metric on the unit Euclidean hyperboloid, and r = r 0 is a smooth horizon of temperature Note that F ( r ) approaches r 2 //lscript 2 d +1 at large r . Making an obvious choice of boundary conformal frame, the boundary metric is just the hyperbolic cylinder H × R with ds 2 H × R = -dt 2 + /lscript 2 d +1 d Σ d -1 . But note that we may write where ρ//lscript d +1 = tanh ξ . Multiplying the right-hand side by (1 -ρ 2 //lscript 2 d +1 ) gives a metric on the static patch of the d -dimensional de Sitter space dS d with Hubble constant /lscript -1 d +1 . So by /negationslash changing conformal frames in this way we may regard the boundary of (2.1) as having de Sitter horizons with temperature 1 / 2 π/lscript d +1 . From the perspective of an observer in the static patch, the de Sitter horizon acts just like a black hole horizon with For general r 0 this temperature clearly differs from that of the bulk horizon. For the case where they agree, the hyperbolic black hole metric (2.1) is just pure AdS d +1 in appropriate hyperbolic coordinates. We recall that even for the tuned case α = 1 ref. [31] found the conformal frame (2.3) useful for constructing black funnel solutions numerically. Since the analysis of temperatures above involves only the horizons, it is clear that detuned bulk and boundary horizons should exist much more generally. Indeed, any static, spherically symmetric boundary metric with a pair of of smooth horizons at ρ = ± /lscript d +1 may be written in the form ˜ where ˜ F , ˜ G , and 1 / ˜ R 2 are smooth on some interval including ρ ∈ [ -/lscript d +1 , /lscript d +1 ], ˜ G has a second order zero at each of ρ = ± /lscript d +1 , and 1 / ˜ R 2 vanishes at ρ = ± /lscript d +1 . So after a conformal transformation (2.5) agrees with (2.3) to leading order in ρ for each term and in this sense may be said to approach H × R at large ρ . The ansatz (2.5) can equivalently be written where x 0 is some reference length scale and F and e ∓ 2 x/x 0 R 2 are smooth functions of e ∓ 2 x/x 0 at e ∓ 2 x/x 0 = 0. In particular, up to the conformal factor ds 2 H × R takes this form for x 0 = /lscript d +1 and R 2 = /lscript 2 d +1 sinh 2 ( x//lscript d +1 ). In terms of (2.6) the boundary black holes have temperatures It is therefore sensible to choose any r 0 and seek a smooth bulk solution in which each term approaches that of (2.1) to leading order in e -2 | x | /x 0 at large | x | ; see section 4 for a more complete analysis of these boundary conditions. Any static such solution will have a bulk horizon with temperature (2.2) and can again be interpreted as being dual to a field theory state of this temperature on a black hole background of temperature (2.4). In the next sections we will seek further generalizations with different values of r 0 (which we then call r ± ) at x = ±∞ . That is to say that for x → + ∞ the bulk solution will asymptote as above to (2.1) with r 0 = r + , while for x → -∞ it will analogously approach (2.1) with r 0 = r -. The bulk horizon may then be said to approach the temperatures T ± given by (2.2) with r 0 replaced r ± . We will also allow distinct temperatures T bndy BH ± for the x = ±∞ boundary black holes and introduce the parameters α ± = T bndy BH ± /T ± . In fact, we will always take α + = α -. Of course, we may also consider so-called ultrastatic conformal frames analogous to (2.3). Starting with (2.6) and multiplying by a conformal factor e 2 x/x 0 /F ( x ), one obtains the boundary metric for which ∂ t is a hypersurface-orthogonal Killing field of norm -1. In this frame, the boundary spacetime has two asymptotic regions, each asymptotic to H × R (say, with the same curvature scale /lscript d +1 ). Furthermore, in the CFT description each region contains an infinite reservoir of deconfined plasma. Such infinite reservoirs may act as heat baths, and indeed the boundary conditions imply that they are in thermal equilibrium at temperature T ± in the limits x →±∞ .", "pages": [ 6, 7, 8 ] }, { "title": "3 The fluid limit", "content": "While a general treatment of black funnels remains challenging, it is by now well known that the study of AdS black holes simplifies in the so-called hydrodynamic limit of the fluid/gravity correspondence [34] in which all other parameters vary slowly in comparison with the black hole temperature and the solution can be described using a derivative expansion. For any fixed boundary metric, taking the limit of large temperature (i.e., small α ± ) makes all metric derivatives small in this sense. We may thus expect a good hydrodynamic description if in addition we control temperature gradients by taking ∆ T = T + -T -small. The key point in the analysis of [34] is that, having chosen a boundary conformal frame with boundary metric g (0) ij , every AlAdS d +1 solution is associated with a d -dimensional boundary stress tensor T ij which is traceless and conserved on the boundary: where D i is the covariant derivative compatible with g (0) ij . Below, we use the boundary metric g (0) ij and its inverse to raise and lower indices i, j, k, l, . . . . As an example, consider the planar AdS-Schwarzschild black hole with r 0 = 4 π/lscript 2 d +1 T/d . Taking the boundary metric to be ds 2 bndy = -dt 2 + d x 2 d -1 , one finds which takes the form of an ideal fluid with velocity field u i ∂ i = ∂ t , transverse projector P ij = g ij + u i u j , and which of course satisfies (3.1). In (3.4), we have defined for convenience T ≡ 4 π/lscript d +1 T/d . By a simple Lorentz transformation we may obtain corresponding solutions with any constant (normalized) timelike d -velocity u i . The main result of [34] was to show that the temperature T and d -velocity u i may be promoted to slowly-varying functions of the boundary coordinates x , t (at which point we refer to them collectively as the hydrodynamic fields). Here the term 'slowly-varying' is defined with respect to the temperature as measured in the local rest frame selected by u i . In particular, under these conditions [34] showed that a smooth bulk solution may be constructed via a gradient expansion so long as u i is everywhere timelike and the associated boundary stress tensor does indeed satisfy (3.1). They further showed that at each order in this expansion the conditions (3.1) may be expressed as standard hydrodynamic equations for a (conformal) fluid with velocity field u i , which we take to satisfy u i u i = -1. This last step essentially just repeats the standard derivation of hydrodynamics from conservation laws. In particular, ref. [34] showed that the boundary stress tensor takes the form where Π ( n ) are dissipative terms that are n th order in derivatives of the hydrodynamic fields; for example, where is the shear viscosity, and θ = D i u i and are respectively the divergence and shear of the velocity field. In writing (3.6) there is a freedom to make certain field redefinitions which, following [34], we have removed by choosing the so-called Landau frame in which the Π ij ( n ) are taken to be purely transverse. Since by assumption derivatives of the hydrodynamic fields are parametrically small in some parameter /epsilon1 , Π ( n ) is of order /epsilon1 n . Below, we solve the fluid equations (3.1) at order n = 0 and n = 1 for the ultrastatic boundary metrics (2.8) and a purely radial velocity field (so that the only non-vanishing components are u t , u x ). We also assume the flow to be stationary, so that u i , T are independent of time. A new effect at first order is the appearance of dissipation, and thus the production of entropy. At zeroth order, the entropy current J i S takes the simple form ( J i S ) ideal = su i , where s ( x ) = T d -1 / 4 G ( d +1) is the entropy density. Using the equations of motion and thermodynamic relations, one can show [48] that At first order, the entropy current still takes the form ( J i S ) 1 = su i , but its divergence now becomes [48] showing that entropy is produced unless σ ij = 0.", "pages": [ 8, 9 ] }, { "title": "3.1 Ideal Fluid", "content": "We begin at order n = 0. We denote the associated fluid quantities T 0 , u i 0 and work in d = 3. Following [49], we project the fluid equations into components parallel and perpendicular to the velocity. These yield respectively or Thus in terms of integration constants that we have chosen to call T 2 ∞ / 2 a , T ∞ . Since u 2 = -1, it remains to solve a quadratic equation for T 0 , u i 0 . We of course obtain two solutions labeled by a choice of sign. The solution with finite and nonzero asymptotic temperatures T ± has Note that since R diverges at large x , at this order the asymptotic temperatures T ± at x →±∞ agree; i.e., ∆ T = T + -T -= O ( /epsilon1 ). We also find u x 0 → 0 at x = ±∞ .", "pages": [ 10 ] }, { "title": "3.2 First Order Corrections", "content": "To compute corrections to (3.14), (3.15), we choose to solve the fluid equations (3.1) iteratively. Introducing a bookkeeping parameter /epsilon1 to keep track of derivatives, we may write T = T m + O ( /epsilon1 m +1 ), u i = u i m + O ( /epsilon1 m +1 ) for each m . We compute T m , u i m by dropping terms with n > m in (3.5) and evaluating the remaining Π ij ( n ) on T m -n , u i m -n . Thus T m , u i m enter (3.1) only through T ij ideal and the equations to be solved are essentially just (3.11), (3.12) with additional source terms given by the Π ij ( n ) . The integration constants (as well as the sign choices that come from solving quadratic equations) may be fixed by requiring T m , u i m to approximate T m -1 , u i m -1 to the desired order as /epsilon1 → 0. To first order, one finds where we find for Noting that A ( x ) = O (1 /a ) 2 we then find and the square brackets [ · ] (0) indicate that the enclosed quantities are evaluated on the zeroth order solutions (3.14), (3.15). At this order, the asymptotic temperatures differ and are given by the (finite) expression so that It is useful to consider the further limit of small ∆ T , which greatly simplifies the above results. This is equivalent to taking a large. Since so that the non-zero components of the stress tensor are Note that the lowest order term in the energy flux T tx is linear in ∆ T ; this naturally leads to a notion of thermal conductivity. We first calculate the heat flux Φ as the energy flux integrated over a circle of constant x : We define the thermal conductivity as k := -d Φ /d ∆ T | ∆ T =0 so that We have also explored the analogous results at second order n = 2 in the hydrodynamic approximation. While the general expressions are unenlightening, each quantity above agrees with the n = 1 expression up to linear order in ∆ T for all T ∞ . In particular, the conductivity k is unchanged. Finally, the entropy current ( J i S ) 1 = su i for our solutions is which has divergence To lowest nonvanishing order in ∆ T , these become Note that the divergence of the current is of order ∆ T 2 as expected from (3.10). It turns out that (3.40) is unchanged when one passes to second order in the hydrodynamic expansion, though the entropy current J i S itself changes even at zeroth order in ∆ T . These expressions may of course be transformed to any other conformal frame. The ultrastatic frame (2.8) used above had the convenient feature that, at least at small velocity, the local fluid temperature (defined with respect to proper time in the fluid rest frame) coincides with the temperature defined with respect to the static Killing field ∂ t . In a more general conformal frame, these two temperatures do not coincide even at small velocity. Note that we will employ only time-independent conformal transformations below, so that ∂ t remains a Killing field in all frames. We will continue to refer to temperatures normalized (up to a boost to the fluid rest frame) with respect to ∂ t by T , while we denote the local fluid temperature (defined with respect to rest-frame proper time) as T loc . Thus T is unchanged by the conformal transformation while T loc is rescaled. For comparison with our later numerics, appendix A presents the results in the black hole frame for the explicit metric functions and in terms of the particular coordinates used in section 5 below. The resulting more explicit expressions are correspondingly more complicated than those above.", "pages": [ 10, 11, 12, 13 ] }, { "title": "4 How to flow a more general funnel", "content": "Our family of flowing funnels will be labeled by four parameters: the temperatures T bndy BH ± of the left- and right- boundary black holes and the temperatures T ± associated with the leftand right- ends of the bulk black hole. As discussed in section 2 these four temperatures are completely independent in principle, though in our simulations we will always set α + = α -which introduces one relation. The most generic ansatz compatible with our symmetry requirements depends on seven unknown functions: where A , B , F , S 1 , S 2 , χ 1 and χ 2 are all functions of y and w . In addition we have defined /negationslash The insertion of these factors will be justified later, when we will also see that β controls the temperature difference between the two boundary black holes, and y 0 is a parameter that controls the validity of the fluid approximation. Here y ranges over [0 , 1] and w ranges over [ -1 , 1], with y = 0 being the bulk horizon and y = 1 the conformal boundary. At least for y = 0 regions with w ∼ ± 1 are close (in the sense of a conformal diagram) to where either bulk horizon meets either the left or right boundary black hole (compare with figure 2). As we will explain below, the symbol ˜ t was used in (4.1) in order to save t for another coordinate associated with Fefferman-Graham gauge. However, ∂ ˜ t = ∂ t so we will refer to the time-translation as simply ∂ t .", "pages": [ 13, 14 ] }, { "title": "4.1 Boundary Conditions", "content": "At the conformal boundary ( y = 1) we impose the boundary conditions which ensures a boundary metric conformal to where ˆ ρ = ρ//lscript 4 . As in section 2, we refer to (4.4) as the boundary metric in the black hole conformal frame. In presenting our results in section 5 we will describe all boundary quantities, such as the stress energy tensor, with respect to this frame. The boundary metric ds 2 ∂ has horizons at ˆ ρ = ± 1 with Hawking temperatures We will extract the boundary stress tensor following the strategy of [31] and using the results of [50]. The only technical difference with respect to [31] involves the relation between the coordinates ( ˜ t, w, y, φ ) and Fefferman-Graham coordinates ( t, z, ρ, φ ). Due to the cross term χ 2 in Eq. (4.1) the map between ˜ t and t is not trivial, instead it is expressed as a powers series in z of the form: The left and right boundaries lie at w = ± 1. There we impose where for instance T 1 ( ρ ) = -Q ( ρ ) y 0 / (2(1 -ˆ ρ 2 )). which reduces Eq. (4.1) to Under the coordinate transformation: the line element (4.8) yields the large ξ limit of Eq. (2.1) with d = 3. The fact that our ansatz (4.1) reduces to a hyperbolic black hole at w = ± 1 displays the physical meaning of y 0 as an overall scale that controls the bulk horizon temperatures (and thus α ± ). Note that the line element (4.8) also defines T ± = T bndy BH ± M (0) / 2. If y 0 = 1, then T ± = T bndy BH ± , i.e. it represents the 'tuned' case α ± = 1. Thus the fluid approximation becomes more accurate as y 0 increases, or equivalently, as α ± decrease. We have imposed Dirichlet data at each of the above three edges of our computational domain. But it remains to specify boundary conditions at y = 0, the flowing funnel horizon. Here we demand that the line element (4.1) be smooth in ingoing Eddington-Finkelstein coordinates (which cover the future horizon). To understand the explicit form of this condition, we introduce local ingoing Eddington-Finkelstein coordinates ( v, ˜ w, ˜ y, φ ) through The terms omitted in the above ˜ y expansion can be chosen such that a line of constant ( v, ˜ w, φ ) is an ingoing null geodesic. Note that lines of constant v have d ˜ y/d ˜ t < 0, as required for ingoing coordinates. Furthermore, regularity of the line element (4.1) in the above coordinates requires /negationslash We will find χ 1 ( w, 0) to be finite and non-zero (at w = ± 1), so our original w is already an ingoing coordinate. It will thus be straightforward to read off results associated with the future horizon. The past horizon is more subtle. It is located at v →-∞ and can be reached along lines of constant ˜ w . Depending on the sign of χ 1 , this tends to drive w to either ± 1. Below, we consider T + > T -so that the hotter black hole is on the right. One might therefore expect w to decrease along the horizon generators so their coordinate velocity is toward the cooler black hole; i.e., one might expect χ 1 ( w, 0) > 0. But for the particular ansatz we have chosen our numerics turn out to give χ 1 < 0 (see section 5) so that the past horizon in fact lies at w = -1. This appears to be a coordinate artifact, though a full understanding is beyond the scope of this work. Below, we will solve the Einstein equations (with cosmological constant) in the form subject to the boundary conditions detailed above.", "pages": [ 14, 15 ] }, { "title": "4.2 The DeTurck Method", "content": "The diffeomorphism invariance of (4.12) means that these equations do not lead to a wellposed boundary value problem. While one could attempt to proceed by gauge-fixing, a clever trick known as the DeTurck method was introduced in [27] and in [28, 51] was shown to succeed (under rather general assumptions) when one seeks appropriate stationary equilibrium solutions of the vacuum Einstein equations, with or without a negative cosmological constant. Though our situation turns out to fall outside the bounds of the proof given in [28], we nevertheless employ this method successfully below. We begin with a brief review. The DeTurck method is based on the so called EinsteinDeTurck equation which differs from from Eq. (4.12) by the addition of -∇ ( a ˆ ξ b ) . Here ˆ ξ a = g cd [Γ a cd ( g ) -Γ a cd (¯ g )], Γ( g ) is the Levi-Civita connection associated with the metric g , and ¯ g is some specified nondynamical reference metric. Since ˆ ξ is defined by a difference between two connections, it transforms as a tensor. Hereafter ¯ g will be chosen to have the same asymptotics and horizon structure as g . In particular, it must satisfy the same Dirichlet boundary conditions as g . /negationslash Clearly any solution to E H ab = 0 with ˆ ξ = 0 also solves E ab = 0. But one may ask if (4.13) can have additional solutions that do not satisfy E ab = 0. Under a variety of circumstances one can show that solutions with ˆ ξ = 0, the so called Ricci solitons, cannot exist [28]. However, the assumptions used in [28] seem not to hold for our system of equations. In particular, after reduction along the symmetry directions t, φ our system turns out to have a mixed-elliptic hyperbolic nature. This is most easily seen from the fact that, while our system will be elliptic near infinity where ∂ t is timelike, we expect an ergoregion near the horizon where all linear combinations of ∂ t , ∂ φ are spacelike. So in this region reduction along ( t, φ ) gives a Lorentz-signature metric on the base space. This differs qualitatively from the case of Kerr, where ∂ t , ∂ φ span a timelike plane everywhere outside the horizon and reduction along ( t, φ ) gives a Euclidean-signature metric on the base space. See [51] for a more detailed discussion. The difference arises from the fact that the Kerr horizon 'flows' only along the Killing field ∂ φ while our horizon 'flows' in the w direction, which is not associated with any symmetry. Thus Ricci solitons may well exist in our case. But for any solution to (4.13) one may simply calculate ˆ ξ to see if it vanishes. For all of our flowing funnel solutions discussed below we find ˆ ξ = 0 to machine precision. It remains to specify our choice of reference metric ¯ g . We choose ¯ g to be given by the line element (4.1) with A = B = S 1 = S 2 = χ 2 = 1 and F = χ 1 = 0. This enforces all Dirichlet boundary conditions except those at the horizon, Eq. (4.11). To satisfy these remaining conditions we need only choose Q ( x ) as in Eq. (4.2).", "pages": [ 15, 16 ] }, { "title": "4.3 Numerical Method", "content": "We use a standard pseudospectral collocation approximation in w , y and solve the resulting non-linear algebraic equations using a damped Newton method with damping monitoring function | ˆ ξ t | . This ensures that Newton's method takes a path in the approximate solution space that decreases | ˆ ξ t | at each step. This method may also prove useful in solving more general mixed elliptic-hyperbolic systems. We represent the w and y dependence of all functions as a series in Chebyshev polynomials. As explained above, our integration domain lives on a rectangular grid, ( w, y ) ∈ [ -1 , 1] × [0 , 1]. To monitor the convergence of our method we have computed the total heat flux Φ (defined by the first equality in (3.32)) for several resolutions. We denote the number of grid points in w and y by N and compute ∆ N = | 1 -Φ N / Φ N +1 | for several values of N . The results for this procedure are illustrated in Fig. 3 for β = 0 . 1 and y 0 = 1. We find exponential convergence with N , as expected for pseudospectral collocation methods. Furthermore, in order to ensure that we are converging to an Einstein solution rather than a Ricci soliton we monitor all components of ˆ ξ . For all plots shown in this manuscript, each component of ˆ ξ a has absolute value smaller than 10 -10 .", "pages": [ 16, 17 ] }, { "title": "5 Results and comparisons", "content": "We now present the results of our numerical analysis and compare them with the first-order ( n = 1) hydrodynamic approximation. The plots below are labeled by a parameter T ∞ whose definition was inspired by the first-order hydrodynamic result (3.16). For small ∆ T we have T ∞ = ( T + + T -) / 2 + O (∆ T ) 2 . We note that all the numerical results we will present use units where /lscript 4 = 1 (so that ρ = ˆ ρ ) and 16 πG (4) = 1. We also take T + > T -so that the hotter black hole lies on the right. We begin with the norm | ∂ t | 2 of the time translation. Figure 4(a) shows a typical plot. To guide the eye we have also plotted a reference surface of constant | ∂ t | 2 = 0. The two surfaces intersect at the ergosurface, whose location we display separately in Fig. 4(b). Inside the ergoregion | ∂ t | 2 becomes positive, changing the character of Eq. (4.13) from elliptic to hyperbolic. This region is at the core of the difficulties in trying to prove that our numerical method ensures ˆ ξ = 0 on solutions of Eq. (4.13) with appropriate boundary conditions. Fig. 5 shows | ∂ t | 2 and, for comparison and later use, | ∂ φ | 2 as a function of w along the horizon. We remind the reader that ∂ t and ∂ φ are precisely orthogonal everywhere in our spacetime, so this describes the full induced metric h IJ (for I, J = t, φ ) in the 2-plane spanned by ∂ t , ∂ φ . Both norms are clearly positive everywhere on the horizon, though | ∂ t | 2 never becomes very large even with ∆ T/T ∞ = 0 . 2. This may help to explain why our numerical approach succeeded. Let us now discuss the behavior of the boundary stress tensor. For small α, ∆ T/T ∞ , this quantity may also be computed using the hydrodynamic approximation of section 3 and provides another good check of our numerics. Fig. 6 shows the components of the stress energy tensor as a function of the boundary coordinate ρ for several values of α at fixed β . The lines represent the first order hydrodynamic prediction and the symbols represent data extracted from our numerics. Large stress tensors correspond to larger values of α . We see that at least for small ∆ T/T ∞ the fluid gravity prediction works remarkably well even for for α ∼ 0 . 8. The agreement of all of these curves when α is small is a reassuring test of our numerics. However, at larger α qualitative differences from our hydrodynamic approximation begin to appear. For example, we note that while T t t is always negative (and thus the energy density is positive) in the hydrodynamic regime, for α /greaterorsimilar 1 our simulations show T t t becomes positive near the hotter black hole. From the standpoint of the dual CFT, the main physical result of our paper is displayed in Fig. 7. This plot shows how the total heat flux Φ varies for different values of ∆ T/T ∞ and α = α + = α -. We see that it increases in magnitude as ∆ T/T ∞ increases, and also as α decreases. This computation can be seen as a first principle calculation for the thermal conductivity of a strongly coupled plasma at large N beyond the hydrodynamic regime. Fig. 8 compares some α = const . cross-sections of Fig. 7 to the the results of first-order ( n = 1) hydrodynamics at linear order in ∆ T ; i.e., to (A.2)-(A.5). These show good agreement for small α and ∆ T , but deviate as expected at larger α . It remains to examine the horizon more closely. Our horizon is a three-dimensional null surface and, since ∂ t , ∂ φ are both spacelike and tangent to the horizon, any two null geodesics that generate the future horizon generators are related by some isometry. Thus all generators are equivalent, though it remains to understand the evolution of the spacetime along each generator. We compute the affine parameter, expansion, and shear along each generator using simple expressions in terms of the induced metric h IJ (for I, J = t, φ ) on the 2-plane spanned by ∂ t , ∂ φ . These expressions are given in appendix B. We study each of these quantities only on the surface y = 0. We begin with h IJ itself. Recall that w = ± 1 are the asymptotic regions of static hyperbolic black holes where the Killing field ∂ t becomes null at the horizon and | ∂ φ | 2 becomes large. These behaviors are clearly shown in figure 9(a). But these similarities between w = ± 1 are misleading and the actual behaviors at w = ± 1 are quite different. This may be seen from the plot of h = det h IJ = h tt h φφ = | ∂ t | 2 | ∂ φ | 2 in Fig. 9(a). This determinant vanishes at w = -1 but approaches a non-zero constant at w = +1. Note that h is monotonic along y = 0, as it must be along a smooth horizon. What is perhaps surprising is that h is an increasing function of w . This shows that w increases toward the future along the future horizon, so that the past horizon must lie at w = -1. In contrast, in the coordinates of e.g. [52], the coordinate velocity of the horizon generators would be in the direction of heat transport, and thus (since we take the cooler black hole to lie at w = -1) toward negative w . Standard coordinates for Kerr also behave like those of [52] and have the equivalent of our χ 1 being positive for positive angular /MedSolidDiamond velocity. In contrast, we find χ 1 to be negative at the horizon; see figure 9(b). Since χ 1 samples completely different metric components than h , we take this as a strong indication that our solutions are consistent despite the surprising location of the past horizon. Another strong indication of consistency is the above agreement between our boundary stress tensors and those predicted by the hydrodynamic approximation. Indeed, we have tested for various possible errors (such as inverting the sign of ∆ T ) in our code by examining the effect of h /LParen1 w /RParen1 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 /Minus 1.0 /Minus 0.5 0.0 0.5 1.0 w (a) various sign changes on Fig. 6 and found in each case that such changes would lead to notable discrepancies with hydrodynamics. In particular, we stress that our simulations give the physically correct sign for the heat flux T t ρ . The apparent proximity of the past horizon to the cooler black hole must thus be a coordinate artifact. We have confirmed this expectation by repeating our simulations in the coordinates defined by Eq. (4.10) and finding that the equivalent of χ 1 is positive for negative heat flux. For comparison, we mention that also note that a similarly surprising sign can be found in the 2+1 flowing funnels of [30]. In that case, writing the horizon generating Killing field in the Fefferman-Graham coordinates of [30] leads to a negative t component on part of the horizon, even though this component is everywhere positive at the AlAdS boundary. It would also be interesting to transform our current 3+1 solutions to the coordinates of [52] (say, for a solution deep within the hydrodynamic regime), though the additional numerics required places such an analysis is beyond the scope of this work. We may now proceed to investigate various quantities along the horizon. Perhaps the most important quantity is the affine parameter λ , which we show in Fig. 10(a) as function of w . Note that λ approaches a constant value at w = -1. This is to be expected, as we have already noted that w = -1 is the past horizon. Since the affine parameter is only defined up to affine transformations, this constant is arbitrary and we have set λ ( w = -1) = 0 for convenience. In contrast, the affine parameter diverges as we approach w = 1. Figure 10(b) shows the expansion θ as a function of λ . As expected on general grounds, θ is everywhere positive with dθ/dλ < 0 and θ asymptotes to zero at large λ . We see this as the most solid test of our numerics. Note that the sign of dθ/dλ < 0 is only guaranteed via Raychaudhuri's equation once the equations of motion are used. It is thus far from trivial that the sign comes out right. The expansion diverges at the past horizon ( λ = 0), indicating the presence of a caustic. Λ /LParen1 w /RParen1 8 6 4 2 0 /Minus 1.0 /Minus 0.5 0.0 0.5 w (a) 5 4 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 (c) In fact, it is easy to see that this caustic is a curvature singularity. To do so, note from 5(b) that | ∂ φ | diverges on the past horizon. But since Killing fields obey a second order differential equation governed by the Riemann tensor (see e.g. (C.3.6) of [53]) they can diverge at finite affine parameter only if R abcd diverges in all orthonormal frames. We now turn to the shear tensor ˆ σ IJ . From (B.4), (B.7) we see that since h IJ is diagonal, the same is true of ˆ σ IJ . Since ˆ σ IJ is also symmetric and traceless, it is completely characterized by the positive eigenvalue σ of ˆ σ I J , where the index was raised with the inverse h IJ of h IJ . Note that σ is a spacetime scalar. This eigenvalue is plotted as a function of λ in figure 10(c). As one might expect, it diverges at the caustic. What is interesting is that we find the same divergence structure for all α, ∆ T that we have studied. We quantify this behavior by fitting σ ( λ ) to a power law µλ η near λ = 0. We have extracted η for about 400 flowing funnels spanning the domain ( α, ∆ T/T ∞ ) ∈ (1 , 0 . 7) × (0 , 0 . 4). In all cases we find η = -0 . 82 ± 0 . 03, where this error is in fact the maximum deviation. We note that this number is remarkably close to -5 / 6. We can then use the Raychaudhuri equation (B.8) and the standard evolution equation for the shear (see (F.34) of [54]) to again show that R abcd diverges on the past horizon. In fact, for η = -5 / 6 one may show that some Weyl tensor component C abcd k b k d (where k a is an affinely parametrized tangent to a null generator of the horizon) must diverge like λ -11 / 6 . This fact merits an analytic explanation which we are unable to offer at this time.", "pages": [ 17, 18, 19, 20, 21, 22, 23 ] }, { "title": "6 Discussion", "content": "The above work constructs 'flowing funnel' stationary black hole solutions. Such solutions describe heat flow between reservoirs at unequal temperatures T ± . The particular solutions constructed are global AdS 4 flowing funnels which may be thought of as deformations of the BTZ black string (1.1). Thus each heat reservoir lies just outside a boundary black hole Σ /LParen1 Λ /RParen1 Λ of temperature α ± T ± . For the case α ± = 1, the CFT 3 duals of our bulk solutions describe heat transfer between two non-dynamical 3-dimensional black holes due to CFT 3 Hawking radiation . Our solutions display many properties expected on general grounds. There is a connected ergoregion near the horizon where ∂ t becomes spacelike. In fact, all Killing fields are spacelike at the future horizon H , so that H is not a Killing horizon. This is consistent with the rigidity theorems [37, 38, 39] since H is not compact. The expansion θ of the null generators is everywhere positive but decreases toward the future along each null generator. The generators extend to infinite affine parameter in the far future (where θ → 0) but reach a caustic ( θ →∞ ) at finite affine parameter toward the past. This caustic occurs on the past horizon, which is a curvature singularity characterized by a universal power law divergences for the shear σ ∼ λ -5 / 6 and for certain components of the Weyl tensor which grow like λ -11 / 6 in any orthonormal frame. These exponents were found numerically, but merit an analytic understanding. It remains an open question whether curvature scalars (e.g. the Kretschmann scalar R abcd R abcd ) might remain finite 4 . It would also be interesting to study the exponents governing the divergence of the expansion θ , the norm | ∂ φ | 2 and the inverse norm | ∂ t | -2 , though these have proved to be more difficult to extract from our numerics. Note that | ∂ φ | decreases with λ along the early part of the future horizon. But since θ > 0, the shrinkage of the φ circle with affine parameter λ is more than compensated by the growth in | ∂ t | . This positive expansion is associated with the expected generation of entropy due to the transport of heat from a hot source to a cold sink. In particular, it is the analogue at large α ± , ∆ T/T ∞ of the entropy generation term (3.10) seen in the hydrodynamic approximation. In our coordinate system, the horizon generators appear to flow toward the hotter black hole. While we have confirmed that this is a coordinate artifact, it would nevertheless be desirable to understand the effect in more detail. We studied the boundary stress tensor of such solutions both numerically and to first order in the hydrodynamic (fluid/gravity) approximation. In particular, we computed the total heat flux Φ for boundary metrics of the form (4.4) as a function of α, ∆ T/T ∞ ; see figure 7. It would clearly be of interest to study more general boundary metrics to understand which parts of this function are universal and which depend on detailed features of the boundary metric. The hydrodynamic approximation is a derivative expansion which, since we fix all other parameters in the boundary metric, is for us governed by the parameters α ± (which control the extent to which the bulk and boundary black hole temperatures are detuned) and ∆ T/ ( T + + T -) (which controls the temperature difference between the heat source and sink). As expected, we find excellent quantitative agreement when these parameters are small. Interestingly, we also find good qualitative agreement when these parameters are close to 1. This gives yet another confirmation of the robust nature of the fluid/gravity correspondence as seen previously in e.g. [3]. Of course, at large enough values of α ± , ∆ T/ ( T + + T -) we find both quantitative and qualitative disagreement. It would be interesting see to what extent agreement might be improved by incorporating higher order hydrodynamic corrections. A particularly notable feature at large α ( α + /greaterorsimilar 1 in our simulations) is that, while T t t is always negative in the hydrodynamic limit, it becomes positive close to the hotter boundary black hole black hole. It would be interesting to understand this feature analytically.", "pages": [ 23, 24, 25 ] }, { "title": "Acknowledgements", "content": "We thank Allen Adams, Pau Figueras, Veronika Hubeny, Mukund Rangamani, and Toby Wiseman for many stimulating discussions of flowing funnels and methods by which they might be constructed. We also thank Gary Horowitz for spotting an error in an early version of this manuscript. This work was supported in part by the National Science Foundation under Grant Nos PHY11-25915 and PHY08-55415, and by funds from the University of California. DM also thanks the University of Colorado, Boulder, for its hospitality during the completion of this work.", "pages": [ 25 ] }, { "title": "A Fluid results in the black hole frame", "content": "We may transform the hydrodynamic results of section 3.2 to the black hole frame associated with the metric (4.4) by implementing a boundary conformal transformation and an appropriate change of coordinates. Setting /lscript 4 = 1 the result is where ∆ T is again defined with respect to ∂ t , T loc is the local temperature with respect to proper time in the fluid rest frame, and setting H ( ρ ) := (1 -ρ 2 ) G ( ρ ) /y 0 we have defined", "pages": [ 25 ] }, { "title": "B The horizon-generating null congruence", "content": "We wish to study the expansion and the shear tensor associated with the null geodesic congruence that generates the future bulk horizon. Instead of solving the geodesic equation and taking derivatives of deviation vectors, we take advantage of the fact that our system is co-homogeneity 2 to compute these quantities directly (up to a position-dependent scale factor) from the induced metric h IJ on 2-dimensional surfaces tangent to the Killing fields ∂ t , ∂ φ . We then compute the affine parameter λ along these geodesics from the Raychaudhuri equation as explained below. Recall that we consider a future event horizon H of an AlAdS 4 spacetime with two commuting KVFs ∂ t and ∂ φ . The horizon is 3-dimensional, with a two-dimensional space of generators. So long as the horizon is not itself Killing, we see that any two generators are related by the actions of ∂ t and ∂ φ . Choose one horizon generator with affine parameter λ . We can extend λ to a scalar function on H by requiring it to be invariant under ∂ t , ∂ φ . In our case we can take λ = λ ( w ) since w is indeed invariant under both KVFs and is a good coordinate on H . Let k a be the tangent to our generator associated with affine parameter λ . Note that since ∂ t , ∂ φ are also tangent to the horizon we have k ⊥ ∂ t , ∂ φ . We also choose any /lscript a satisfying /lscript a k a = -1 and /lscript ⊥ ∂ t , ∂ φ . We then extend k, /lscript to vector fields defined across all of H by requiring them to be invariant under ∂ t , ∂ φ . We then define a 'deformation tensor' ˆ B ab associated with flow along the horizon generators by projecting B ab = ∇ b k a onto the space orthogonal to k, /lscript . See e.g. appendix F of [54]. Let us note that since ∂ t , ∂ φ commute they are surface-forming, and k is orthogonal to this surface. So k is hypersurface orthogonal and the twist ˆ ω ab = ˆ B [ ab ] vanishes. Thus ˆ B ab = ˆ B ba . We will use this symmetry below. Deviation vector fields for the horizon-generating null congruence are defined by the property that, when evaluated on a given horizon generator γ , they point to the same horizon generator γ ' for all λ . Let us consider a deviation vector field η orthogonal to both k and /lscript . Then (see e.g. appendix F of [54]) η satisfies Since translations along ∂ t , ∂ φ map one geodesic to another, both ∂ t and ∂ φ are deviation vectors. And both are orthogonal to k, /lscript . So we may choose η I = ∂ I for I = t, φ . Here the η I are two spacetime vectors, not the components of a single vector. Let us now consider the set of associated inner products In any coordinate system, η a t = ∂ t x a and η a φ = ∂ φ x a . So in particular in the coordinate system y, w, ˜ t, φ we have (since ∂ ˜ t = ∂ t ) i.e., this is just the induced metric on the 2-plane generated by ∂ t , ∂ φ in coordinates ( ˜ t, φ ) or, equivalently for this purpose, coordinates ( t, φ ). So it is easy to read off from our numerics. But note that h IJ was defined to be a set of scalars, so covariant derivatives of h IJ are just coordinate derivatives. The evolution of h IJ (with respect to λ , or equivalently with respect to w ) is governed by (B.1). Using (B.1) we compute where in the last step as for (B.3) above we have used ˆ B IJ to denote the ˜ t, φ components of ˆ B ac in the particular coordinate system y, w, ˜ t, φ (or equivalently the t, φ components). The deformation tensor ˆ B ab is by construction orthogonal to k, /lscript . Thus we may write ˆ B ab ∂ a ∂ b = ˆ B IJ ∂ I ∂ J . Furthermore, from (B.3) we have where K,L also range over φ, t . Thus we may safely use h IJ and its inverse h IJ to raise and lower indices I, J on ˆ B IJ . Now, the components ˆ B IJ are essentially exponentials of integrated versions of the expansion and shear. In particular, introducing the projector Q a b onto the subspace orthogonal to k, /lscript (i.e., onto the space spanned by ∂ t , ∂ φ ) we have and where D = 4 for AdS 4 . Of course, it remains to actually find the affine parameter λ used in the above definitions. We choose to calculate λ from Raychaudhuri's equation which, in the present context, may be written As usual, the symmetries of the Riemann tensor imply that Q ab R acbd k c k d = g ab R acbd k c k d , which is proportional to R cd k c k d . But R ab ∝ g ab by the equations of motion, so the final term in (B.8) vanishes. Since ˆ B ab is orthogonal to both k and /lscript we may then write In terms of a general coordinate w along the generators, (B.9) may be rearranged to yield where ' denotes the coordinate derivative d/dw and the last equality defines Z ( w ). This equation is then easily solved for λ in terms Z ( w ), which is relatively straightforward to extract from the numerics.", "pages": [ 26, 27, 28 ] } ]

[ { "title": "Keiju Murata", "content": "DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan March 8, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study linearized gravitational perturbations of extreme black hole solutions of the vacuum Einstein equation in any number of dimensions. We find that the equations governing such perturbations can be decoupled at the future event horizon. Using these equations, we show that transverse derivatives of certain gauge invariant quantities blow up at late time along the horizon if the black hole solution satisfies certain conditions. We find that these conditions are indeed satisfied by many extreme Myers-Perry solutions, including all such solutions in five dimensions.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Extreme black holes have theoretical importance in understanding of quantum theory of gravity. For example, Bekenstein-Hawking entropy of supersymmetric black holes was explained by counting BPS states in the view of string theory [1]. Furthermore, a duality called the Kerr/CFT correspondence between extreme black holes and a two-dimensional conformal field theory was proposed [2], and the entropy of the black holes was reproduced as the statistical entropy of the dual CFT. Recently, it was shown that extreme Reissner-Nordström and Kerr black holes are classically unstable against test scalar field perturbations [3-6]. Subsequently, the proof is extended to all other extreme black holes [7]. They showed that the second transverse derivative blows up at the horizons as ∂ 2 r φ ∼ v , where φ is the scalar field and we took the ingoing EddingtonFinkelstein coordinates, ( v, r ) . For extreme Kerr black holes, the similar instabilities were also found in gravitational and electromagnetic perturbations [7]. We arise following questions: Are all extreme black holes unstable against gravitational or electromagnetic perturbations? If not, what is the condition for the instability? In this paper, we address these questions by studying the perturbations of any extreme black holes. We use Geroch-Held-Penrose (GHP) formalism in higher dimensions developed in Refs. [8,9] to study the perturbations. So, in section.2, we give a brief review of the GHP formalism. We introduce gravitational, electromagnetic and scalar field perturbation equations based on the formalism. They are regarded as higher dimensional analogues of Teukolsky equations although they are not decoupled equations for gravitational and electromagnetic perturbations in general. In section.3, we introduce the most general expression for extreme black hole and express them in the view of the GHP formalism. In section.4, we study the scalar field perturbation. Although the scalar field perturbation on any extreme black holes has been already studied in Ref. [7], we revisit the problem using the GHP formalism. We find that all extreme black holes are unstable against scalar field perturbations as shown in Ref. [7]. In section.5, we study electromagnetic perturbations. We find that, near the horizon, electromagnetic perturbations satisfy decoupled equations. Using the decoupled equations, we show that the perturbations do not decay along the future event horizon if a certain operator on the horizon has a zero eigenvalue. In section.6, we study gravitational perturbations. By the similar way as electromagnetic perturbations, we can show the non-decay of the gravitational perturbations if a horizon operator has a zero eigenvalue. In addition to that, if the background geometry is algebraically special, the first or second transverse derivatives of the perturbation variables blow up along the horizon. The eigenvalues for the horizon operators have been calculated for some extreme black holes. In section.7, we see that there are zero eigenvalues in the horizon operators for all higher dimensional extreme black holes with zero cosmological constant as far as we calculated. The final section is devoted to discussions.", "pages": [ 1, 2 ] }, { "title": "2 Geroch-Held-Penrose formalism in higher dimensions", "content": "We study the perturbation of the general extreme black holes using the Geroch-HeldPenrose (GHP) formalism in higher dimensions developed in Refs. [8, 9]. In this section, we give a brief review of the GHP formalism. In the formalism, we use a null basis { e 0 , e 1 , e i } = { /lscript, n, m i } ( i = 2 , · · · , d -1) which satisfies We define the covariant derivatives of basis vectors as and The orthogonal relations (2.1) are invariant under spins, boosts and null rotations defined as follows. Spins are local SO ( d -2) rotations of the spacial basis { m i } : where X ij ∈ SO ( d -2) depends on the spacetime coordinate x µ . Boosts are local rescaling of the null basis: where λ is any real scalar function. Null rotations about /lscript and n are and where z i and z ' i are real functions of x µ . In the GHP formalism, we maintain the covariance with respective to spin and boost transformations. An object T i 1 ··· i s is a GHP scalar of spin s and boost weight b if it transforms by the spins and boosts as T i 1 ··· i s → X i 1 j 1 · · · X i s j s T j 1 ··· j s and T i 1 ··· i s → λ b T i 1 ··· i s . For example, the quantities ρ ij , τ i , κ i are GHP scalars with b = 1 , 0 , 2 , respectively. We also define priming operation: T i 1 ··· i s → T ' i 1 ··· i s , where T ' i 1 ··· i s is the object obtained by exchanging /lscript and n in the definition of T i 1 ··· i s . We define GHP scalars obtained from Weyl tensor C abcd as where Ω , Ψ , Φ , Ψ ' and Ω ' have boost weights b = 2 , 1 , 0 , -1 , -2 , respectively. The null vector /lscript is called multiple WAND (Weyl-aligned null direction) iff all boost weight +2 and +1 components of the Weyl tensor vanish. The spacetime admitting the multiple WAND is called algebraically special spacetime. We can also obtain GHP scalars from Maxwell field strength F ab are where ϕ , F and ϕ ' have boost weights b = 1 , 0 , -1 , respectively. The partial derivatives of GHP scalars, such as /lscript µ ∂ µ T i 1 ··· i s , n µ ∂ µ T i 1 ··· i s or m µ i ∂ µ T i 1 ··· i s , are not GHP scalars. It is convenient to define derivative operators which are covariant under spins and boosts as They are called GHP derivatives. We can check that þ T i 1 ··· i s , þ ' T i 1 ··· i s and ð i T j 1 ··· j s are all GHP scalars, with boost weight ( b +1 , b -1 , b ) and spins ( s, s, s +1) . The GHP scalars defined above are not independent because of Ricci equations, [ ∇ µ , ∇ ν ] V ρ = R µνρσ V σ , Bianchi equations, ∇ [ λ C µν | ρσ ] = 0 , and Maxwell equations, dF = d ∗ F = 0 . The relation for the GHP scalars in Einstein spacetimes R µν = Λ g µν are summarized in appendix.C. Since these equations are invariant under the spins and boosts, they are written by GHP scalars and their GHP derivatives. In the GHP formalism, the Klein-Gordon equation ( ∇ 2 -µ 2 ) φ = 0 is written as From appropriate linear combinations of equations in appendix.C, we can obtain useful equations for studying electromagnetic and gravitational perturbations [9]. They are written as and The right hand sides of these equations are very long so we wrote them schematically. In this paper, we do not need the detailed expressions of the right hand sides.", "pages": [ 2, 3, 4 ] }, { "title": "3 Extreme black holes in the GHP formalism", "content": "We consider general extreme black holes. The metric of the extreme black holes can be written as [10] where functions ( F, γ αβ , h α ) and L 2 are smooth function of { r, x a } and { x a } , respectively. The horizon of the spacetime is located at r = 0 . In the metric, there is a residual coordinate transformation, r → Γ( x ) r . We choose the free function Γ( x ) so that F ( r = 0 , x ) = 1 is satisfied. In this paper, we focus only on Einstein spacetimes satisfying R µν = Λ g µν . We assume that the background metric have n rotational symmetry generated by ∂/∂φ I ( I = 1 , 2 , · · · , n ) . Then, the metric can be written as We impose further assumption on metric functions as where k I are constants. These assumptions are true for a large class of extreme black holes [1115]. Under these assumptions, the near horizon geometry of the metric (3.2) takes 'standard' form: where we took the double scaling limit: r = /epsilon1R , v = V//epsilon1 and /epsilon1 → 0 . The induced metric on the horizon is written as as We take null basis { e 0 , e 1 , e i } = { /lscript, n, m i } in the general extreme black hole metric (3.1) where h i ≡ h α ˆ e i α and ˆ e i is an appropriate orthogonal basis for γ αβ . The null basis (3.6) is regular at the future horizon r = 0 . Using the basis, we can obtain GHP variables. The full expression of the GHP variables are summarized in appendix.A. Here, we focus on ρ ij and κ i since they will be important later. They are given as where ,i ≡ ˆ e µ i ∂ µ . We chose the residual gauge freedom so that F ( r = 0 , x ) = 1 is satisfied. Thus, we obtain F ,i = O ( r ) . Therefore, we have κ i = O ( r 3 ) . From the assumption (3.3), h α | r =0 is constant and we have h α ,j = O ( r ) . Thus, the first term in ρ ij is O ( r 2 ) . In the second term, there is a derivative operator, h k ∂ k = k I ∂ φ I + O ( r ) . Since the ∂/∂ φ I is a Killing vector, its operation to background variables vanishes and the second term is also O ( r 2 ) . The last term is trivially O ( r 2 ) . Therefore, we can conclude that ρ ij is second order in r . By the similar way, we obtain the near horizon expression of the GHP variables as Components of Weyl tensors with boost parameter +2 and +1 are given by Newman-Penrose equations (NP1) and (NP2) as Now, we consider Components of Weyl tensor with boost parameter 0 . In Ref. [8], it was shown that the induced Riemann tensor on a spacelike surface which is orthogonal to null vectors /lscript and n is written as Thus, we have where ˆ R ijkl is the induced Riemann tensor on the horizon (3.5). From the antisymmetric part of Eq.(NP4), we obtain The GHP derivatives are where ˆ ∇ is a covariant derivative with respect to the horizon induced metric (3.5).", "pages": [ 4, 5, 6 ] }, { "title": "4.1 Conserved quantity on the horizon", "content": "First, we consider the scalar field perturbation equation (2.15). The instability of the massless scalar field perturbation on any extreme black holes has been already shown in Ref. [7]. Here, we revisit the problem including the massive scalar field using the GHP formalism. Substituting near horizon expressions of GHP variables and derivatives (3.8-3.13) into the Klein-Gordon equation (2.15), we have the scalar field equation near the horizon as where the operator A 0 is defined as 1 where we decomposed { φ I } -dependence of φ by Fourier modes e im I φ I , that is, ∂ I φ = m I φ . We will do same decompositions for electromagnetic and gravitational perturbations. The derivation of the equation is given in appendix.B. Hereafter, we focus on axisymmetric perturbations m I = 0 . For axisymmetric perturbations, the operator A 0 is self-adjoint, ( Y 1 , A 0 Y 2 ) = ( A 0 Y 1 , Y 2 ) , with respect to an inner product where ∫ H = ∫ d d -2 x √ ˆ g and ˆ g is the determinant of the horizon induced metric defined in Eq.(3.5). We assume that operator A 0 has a zero eigenvalue. This is always true for massless case µ = 0 since we have A 0 Y = 0 when Y is a constant. For massive scalar field, A 0 can also have zero eigenvalues depending on the value of µ 2 and background geometries. We will discuss the existence of the zero eigenvalues in section.7.2. We denote the eigenfunction for the zero eigenvalue as Y . Operating ( Y, ∗ ) to both side of Eq.(4.1), we obtain /negationslash where we used ( Y, A 0 φ ) = ( A 0 Y, φ ) = 0 . Hence, I 0 is a conserved quantity along the horizon. Note that the GHP derivative þ ' contains only the radial derivative ∂ r . (See Eq.(3.13).) Thus, the integrand in the I 0 is written by the linear combination of ∂ r φ and φ . Therefore, we can conclude that if I 0 = 0 at an initial surface, ∂ r φ and φ do not both decay along the future horizon as v →∞ .", "pages": [ 6, 7 ] }, { "title": "4.2 Instability against scalar field perturbations", "content": "We assume that φ and its angular derivatives ∂/∂y A decay along the horizon. For extreme Reissner-Nordström and Kerr black holes, it was shown that φ decays along the horizon. So, this assumption seems likely also for any other extreme black holes. Then, at the late time, the conserved quantity I 0 approaches Now, we differentiate Eq.(2.15) by r . Near the horizon, the equation can be written as where, in the linear operator D 0 , there is no radial derivative ∂ r . Thus, we have D 0 φ → 0 as v →∞ . The derivation of above equation is in appendix.B. Operating ( Y, ∗ ) to both side of Eq.(4.6), we obtain Therefore, the quantity J 0 blows up linearly in time v as The integrand in the J 0 is written by the linear combination of ∂ 2 r φ , ∂ r φ and φ . Since we assumed that φ decays at the horizon, either ∂ 2 r φ or ∂ r φ blows up at the horizon. This implies the instability of the extreme black holes against scalar field perturbations.", "pages": [ 7 ] }, { "title": "5.1 Decoupled equation on the horizon", "content": "Secondly, we consider electromagnetic perturbations. We consider the Maxwell field as test field and, thus, it vanishes in the background. It follows that GHP scalars obtained from Maxwell field strength are all invariant under the infinitesimal coordinate transformations and basis transformations (2.4)-(2.7). The number of physical degrees of freedom of the Maxwell field in d -dimensions is d -2 . The number of components of ϕ i is also d -2 . Thus, we can expect that ϕ i has all physical degrees of freedom of electromagnetic perturbations and it would be nice if we can obtain decouple equations for ϕ i . The right hand side of Eq.(2.16) contains coupling terms between ϕ i and other components of the perturbation. So, ϕ i does not decouple in general. We can see that, however, all the terms in right hand side of Eq.(2.16) are multiplied by ρ , κ , Ω or Ψ . From Eqs.(3.8) and (3.9), they are at most O ( r 2 ) . Thus, the right hand side is O ( r 2 ) . Therefore, in Eq.(2.16) and its radial derivative, the right hand side is zero at the horizon.", "pages": [ 8 ] }, { "title": "5.2 Conserved quantity on the horizon", "content": "We consider left hand side of Eq.(2.16) neglecting the right hand side. Using near horizon expressions of GHP variables and derivatives (3.8-3.13), we obtain where we define the operator A 1 as The derivation of the equation is written in appendix.B. Hereafter, we focus on the axisymmetric perturbations m I = 0 . We define an inner product as For axisymmetric perturbations, we have ( Y 1 , A 1 Y 2 ) = ( A 1 Y 1 , Y 2 ) , that is, the operator A 1 is self-adjoint. We assume that the operator A 1 has zero eigenvalue. Although the existence of the zero eigenvalue is not obvious, we will see that many extreme black holes satisfy this assumption in section.7.2. We denote the eigenfunction by Y i . Operating ( Y, ∗ ) to Eq.(5.1), we have /negationslash where we used ( Y, A 1 ϕ ) = ( A 1 Y, ϕ ) = 0 . Therefore, I 1 is a conserved quantity along the horizon. Thus, If I 1 = 0 at an initial surface, ∂ r ϕ i and ϕ i do not both decay along the future horizon. We assume that ϕ i and its tangential derivatives along the horizon decay as v →∞ . Now, we differentiate Eq.(2.16) by r . Then, the right hand side becomes O ( r ) and still vanishes on the horizon. Thus, we have where, in the linear operator D 1 , there is no radial derivative ∂ r . Hence, we have D 1 ϕ i → 0 ( v →∞ ). The derivation of the equation is in appendix.B. Operating ( Y, ∗ ) to above equation and taking limit of v →∞ , we obtain /negationslash Therefore, the quantity dJ 1 /dv tends to be zero at the late time even if we consider initial data with I 1 = 0 . So, we cannot show the instability of extreme black holes against electromagnetic perturbations by the same way as scalar fields. We may be able to find instability in higher order derivatives ∂ n r ϕ i ( n > 2) . However, since the coupling terms are O ( r 2 ) in the equation, we cannot neglect these terms in the limit of r → 0 when we consider the higher order radial derivatives of Eq.(2.16). It seems to be difficult problem to show the instability taking into account the coupling terms.", "pages": [ 8, 9 ] }, { "title": "6.1 Gauge invariant variables on the horizon", "content": "Finally, we study gravitational perturbations. We consider perturbation of GHP variables as Ω ij → Ω ij + ˜ Ω ij , Ψ ijk → Ψ ijk + ˜ Ψ ijk , etc. Here, variables with tildes represent first order perturbations. Variables without tildes are background variables. The number of physical degrees of freedom of the gravitational perturbations is d ( d -3) / 2 . On the other hand, the number of components of ˜ Ω ij is also d ( d -3) / 2 . Thus, we can expect that ˜ Ω ij has all physical degrees of freedom of the gravitational perturbations. The perturbation variable ˜ Ω ij is transformed by gauge transformations as follows: Coordinate transformations ( x µ → x µ + ξ µ ( x ) ): Spins ( t ij ∈ so ( d -2) ): Boosts: Null rotations: Here, ξ µ , t ij , α and z k are infinitesimal functions depending on spacetime coordinates. In Eq.(3.9), we obtained Ω ij = O ( r 3 ) and Ψ ijk = O ( r 2 ) . Thus, under spin and boost transformations, we have ˜ Ω ij → ˜ Ω ij + O ( r 3 ) . On the other hand, under coordinate transformations and null rotations, we have ˜ Ω ij → ˜ Ω ij + O ( r 2 ) . Therefore, ˜ Ω ij | r =0 and ∂ r ˜ Ω ij | r =0 are gauge invariant, but ∂ 2 r ˜ Ω ij | r =0 is not gauge invariant in general. Thus, even if we could show that ∂ 2 r ˜ Ω ij | r =0 blows up along the horizon by the similar way as scalar field perturbations, we cannot determine if the instability is physical one or just a gauge mode. We will avoid this problem by assuming that the background geometry is algebraically special in section.6.4.", "pages": [ 9, 10 ] }, { "title": "6.2 Decoupled equations on the horizon", "content": "We consider the first order perturbation of Eq.(2.17). The right hand side contains coupling term between ˜ Ω ij and other perturbation variables. We can see that all terms in the right hand side are O ( r 2 ) . For example, we have ( τ ' ρ Ψ)˜ = ˜ τ ' ρ Ψ + τ ' ˜ ρ Ψ + τ ' ρ ˜ Ψ = O ( r 2 ) since ρ and Ψ are the second order in r . Thus, in Eq.(2.17) and its radial derivative, the right hand side is zero at the horizon. In the left hand side, there is another coupling term, 4 κ k þ ' (Ψ ( ij ) k +Ψ ( i δ j ) k ) which can be expanded as This coupling term is O ( r ) . (Recall that the GHP derivative þ ' contains the radial derivative ∂ r . Hence, we have þ ' Ψ ijk = O ( r ) .) Such a coupling term is harmless when we construct a conserved quantity at the horizon and we can show the non-decay of the gravitational perturbations by the same way as scalar and electromagnetic perturbations. However, when we prove that the perturbations blow up along the horizon, the coupling term is problematic since we differentiate Eq.(2.17) by r . In section.6.4, we will see that this problem can also be avoided by assuming that the background geometry is algebraically special.", "pages": [ 10 ] }, { "title": "6.3 Conserved quantity on the horizon", "content": "The coupling terms in Eq.(2.17) is O ( r ) and negligible near the horizon. Thus, near horizon, the equation becomes where the operator A 2 is defined as The derivation of the equation is written in appendix.B. For axisymmetric perturbations m I = 0 , the operator A 2 is self-adjoint with respect to an inner product Hereafter, we focus on axisymmetric perturbations. We assume that operator A 2 has a zero eigenvalue. In section.7.2, we will see that many extreme black holes satisfy this assumption. We denote the eigenfunction for the zero eigenvalue as Y ij . Operating ( Y, ∗ ) to both side of Eq.(6.6), we obtain where we used ( Y, A 2 ˜ Ω) = ( A 2 Y, ˜ Ω) = 0 . Therefore, I 2 is a conserved quantity along the horizon. Thus, If I 2 = 0 at an initial surface, ∂ r ˜ Ω ij and ˜ Ω ij do not both decay along the future horizon as v → ∞ . Recall that both of ∂ r ˜ Ω ij and ˜ Ω ij are gauge invariant at the horizon. /negationslash", "pages": [ 10, 11 ] }, { "title": "6.4 Null rotation to a multiple WAND", "content": "As explained in section.6.1 and 6.2, we require conditions Ψ ijk = O ( r 3 ) and Ω ij = O ( r 4 ) to show the instability of the gravitational perturbations. To satisfy these conditions, we assume that the background geometry is algebraically special. Then, there is a null rotation (2.7) which transforms the null vector /lscript to a multiple WAND, that is, Ψ ijk = Ω ij = 0 . In the near horizon geometry (3.4), we have already known the multiple WAND: /lscript NH = L -1 (2 ∂ V + r 2 ∂ R + 2 rk I ∂ I φ ) . We can expect that, in the near horizon limit: r = /epsilon1R , v = V//epsilon1 and /epsilon1 → 0 , the multiple WAND in the full geometry coincides with the /lscript NH modulo boost transformations. 2 The null vector /lscript defined in Eq.(3.6) satisfies this condition by itself. ( /lscript → /epsilon1/lscript NH in the near horizon limit.) Thus, z ' i in the null rotation (2.7) should be O ( r 2 ) . It follow that the near horizon expressions of GHP variables (3.8) are correct even after the this null rotation.", "pages": [ 11 ] }, { "title": "6.5 Instability against gravitational perturbations", "content": "We assume that ˜ Ω ij and its tangential derivatives along the horizon decay along the horizon. Then, at late time, the conserved quantity I 2 becomes Now, we differentiate Eq.(2.17) by r . Since we assumed that the background geometry is algebraically special, we can neglect the coupling term in the equation. Near the horizon, the equation can be written as where, in the linear operator D 2 , there is no radial derivative. Thus, we have D 2 ˜ Ω ij → 0 ( v →∞ ). The derivation of above equation is in appendix.B. Operating ( Y, ∗ ) to both side of Eq.(4.6), we obtain Therefore, the quantity J 2 blow up linearly in time v as Thus, either ∂ 2 r ˜ Ω ij or ∂ r ˜ Ω ij blows up along the horizon. This implies the instability of the extreme black holes against gravitational perturbations.", "pages": [ 11, 12 ] }, { "title": "7.1 Summary of our statement", "content": "Our statements obtained in this paper are as follows: If the operator A s has a zero eigenvalue for an axisymmetric perturbation and the horizon conserved quantity I s is nonzero, ∂ r ψ s and ψ s do not both decay along the future horizon as v → ∞ , where ψ 0 = φ , ψ 1 = ϕ i and ψ 2 = ˜ Ω ij . The explicit expressions of A s are given in Eqs.(4.2), (5.2) and (6.7). The horizon conserved quantities (4.4), (5.4) and (6.9) are written as where Y is the eigenfunction satisfying A s Y = 0 . In the proof of this statement, we used assumptions (3.3). Hereafter, we assume that ψ s and its tangential derivatives along the horizon decay as v →∞ . (Then, ∂ r ψ s cannot decay.) For scalar field perturbations, we can show that either ∂ 2 r φ or ∂ r φ blows up along the horizon. For gravitational perturbations, when the background geometry is algebraically special, either ∂ 2 r ˜ Ω ij or ∂ r ˜ Ω ij also blows up along the horizon. For electromagnetic perturbations, we could not find the instability in ∂ 2 r ϕ i or ∂ r ϕ i . (There may be instability in the higher order derivative by r .)", "pages": [ 12 ] }, { "title": "7.2 Eigenvalues of A s", "content": "The existence of a zero eigenvalue for the horizon operator A s is crucial for the proof of the instability. Surprisingly, in study of perturbations of near horizon geometries, the eigenvalues of A s have been calculated for some extreme black holes: 4-dimensional extreme Kerr black holes [19,20], all 5-dimensional black holes with two rotational symmetries for Λ = 0 [21] and Myers-Perry(-AdS) black holes with equal angular momenta [16,22]. In this subsection, we investigate the existence of zero eigenvalues of A s using their results. For massless scalar field, the existence of the zero eigenvalue is trivial since we have A 0 Y = 0 when Y is a constant. For massive case, there is no zero eigenvalue in general. In some cases, however, it has a zero eigenvalue depending on mass and background geometry. For example, in odd-dimensional Myers-Perry-AdS spacetimes with equal angular momenta, it was shown that the eigenvalue λ 0 is given by [16] where r + is the horizon radius and L 2 is a constant. The integer N relates to the spacetime dimension as d = 2 N + 3 . From this expression, when the scalar field mass is given by µ 2 = -4 κ ( κ + N ) /r 2 + , the operator A 0 has a zero mode. (If the horizon radius r + is sufficiently large, the µ 2 does not violate the Breitenlöhner-Freedman bound.) For electromagnetic and gravitational perturbations ( s = 1 , 2 ), the existence of a zero eigenvalue is not obvious. In 4-dimensional extreme Kerr geometry, the operator A s ( s = 1 , 2 ) does not have zero eigenvalue. However, if we consider perturbation equations for Ω ' ij instead of Ω ij , we can show the instability [7]. (In the 4-dimensional Kerr geometry, Ω ' ij satisfies a decoupled equation although it does not decouple in Myers-Perry spacetimes even if we consider the near horizon limit.) For all 5-dimensional black holes with two rotational symmetries, eigenvalues λ s are written as 3 where /lscript = 0 , 1 , 2 , · · · . We can find that λ 1 and λ 2 can be zero for /lscript = 0 and /lscript = 0 , 1 , respectively. Thus, in these spacetimes, the gravitational and electromagnetic perturbations do not decay in general. In particular, for 5-dimensional Myers-Perry black holes, we can show that gravitational perturbations (either ∂ 2 r ˜ Ω ij | r =0 or ∂ r ˜ Ω ij | r =0 ) blow up along the horizon since the spacetimes are known to be algebraically special [23]. The horizon induced metrics of Myers-Perry black holes with equal angular momenta can be viewed as Hopf fibration over CP N where N is the integer part of ( d -3) / 2 . Thus, the eigenfunction of A s can be decomposed into tensor, vector and scalar harmonics on the base space CP N . All eigenvalues for axisymmetric modes are given in Refs. [16, 22]. In scalar modes, we can always find zero eigenvalues. (Tensor and vector modes can also have zero eigenvalues depending on the spacetime dimension d .) Thus, Myers-Perry black holes with equal angular momenta in all dimensions are unstable against gravitational perturbations. (Electromagnetic perturbations do not decay at least.) Therefore, as far as we calculated, all vacuum extreme higher dimensional black holes with vanishing cosmological constant have zero eigenvalues in the horizon operator A s . It would be nice if we can show the existence of the zero eigenvalues for general black holes.", "pages": [ 12, 13 ] }, { "title": "8 Discussions", "content": "We studied perturbations in general extreme black hole spacetimes in all dimensions. We found a sufficient condition for instability which is summarized in section.7.1. Using the condition, we showed that 5-dimensional extreme Myers-Perry black holes are unstable against gravitational perturbations. For d ≥ 6 , we also found gravitational instability in extreme Myers-Perry black holes when they have equal angular momenta. In the study of perturbations of near horizon geometries [16,21,22], they considered dimensional reduction of perturbation equations and obtained effective equations of motion in AdS 2 . Then, they used a criterion m 2 < -1 / 4 to determine the instability of near horizon geometry, where m is the effective mass in the AdS 2 , since this implies violation of the BreitenlöhnerFreedman (BF) bound. The effective mass relates to eigenvalue of A s as m 2 = λ 0 , λ 1 , λ 2 +2 for scalar, electromagnetic and gravitational perturbations, respectively. We can see that the condition of the instability obtained in this paper ( λ s = 0 ) differs from theirs. This discrepancy comes from difference of type of instabilities. The violation of BF bound ( m 2 < -1 / 4 ) is considered as a condition for an exponential grow of the perturbations. This was explicitly shown for scalar field perturbations [16]. On the other hand, our condition λ s = 0 gives a power law grow of the perturbations, which is more modest than the exponential one. For ( d ≥ 6 )-dimensional extreme Myers-Perry black holes with equal angular momenta, it was shown that the BF bound in the near horizon geometries is violated for gravitational perturbations [16, 22]. In fact, in the case odd-dimensions, such an instability has been found in the full geometry near the extremality [24]. So, the power law instability found in this paper may not be important for these spacetimes. However, for 5-dimensional extreme Myers-Perry black holes with equal angular momenta, there is no violation of the BF bound in the near horizon geometries. In addition to that, from the study of perturbations of full geometries, strong evidence of stability for non-extreme black holes has been found in Ref. [25]. Thus, the power law instability found in this paper can be important for this spacetime. In this paper, we considered only vacuum black holes. Thus, our instability condition does not apply to gravitational and electromagnetic perturbations of Reissner-Nordström (RN) or Kerr-Newman black holes. Further work needs to be done to study their stability in extreme limit. They are solutions of N = 2 supergravity and extreme RN black holes are supersymmetric. For RN black holes, it is known that the perturbation equations are decoupled. Using the decoupled equations, we may be able to find conserved quantities on the horizons and show the instability [26]. For Kerr-Newman black holes, the decoupling of the perturbation equations has not been succeeded. Hence, even for the non-extreme case, their stability has not been studied. As we did in this paper, however, if the coupling terms in the perturbation equations are sufficiently small near the horizons, we can study the instability. It would be interesting to estimate the order of the coupling terms and study the instability of extreme Kerr-Newman black hole. It would make good progress in understanding of stability of Kerr-Newman black holes. One of the most interesting problems on the instability is its final state. We need to solve the time evolution of the instability taking into account the backreaction to specify the final state. For scalar field perturbations of RN black holes, we can find an instability for spherically symmetric modes [4]. Thus, the evolution equations of the instability are given by (1+1) -dimensional partial differential equations even if we consider the backreaction. Solving the PDEs and finding the final state would be another direction of the future research.", "pages": [ 13, 14 ] }, { "title": "Acknowledgments", "content": "We would like to thank to H. Reall for collaboration at an earlier stage and careful reading of the paper. KM is supported by JSPS Grant-in-Aid for Scientific Research No.24 · 2337. This work is partially supported by European Research Council grant no. ERC-2011-StG 279363- HiDGR.", "pages": [ 14, 15 ] }, { "title": "A GHP variables for extreme black holes", "content": "We take the null basis as in Eq.(3.6). The dual one-forms for these vectors are written as Using the Cartan equations de a + ω cab e c ∧ e b = 0 , we can calculate the spin connections ω abc . From the definition of L ab in Eq.(2.2), we have L ab = -ω b 0 a . Thus, GHP variables for the background spacetime are given as and where ˆ ω kij is defined by d ˆ e i + ˆ ω kij ˆ e k ∧ ˆ e j = 0 .", "pages": [ 15 ] }, { "title": "B Derivation of near horizon equations", "content": "Here, we derive near horizon equations (4.1), (4.6), (5.1), (5.5), (6.6) and (6.11). The equations for scalar, electromagnetic and gravitational perturbations are written in a unified form as where ψ 0 = φ , ψ 1 = ϕ i and ψ 2 = ˜ Ω ij . The angular operators B s are defined by In B s , there is no the radial derivative ∂ r . Up to second order in r , the GHP derivative þ can be written as where C is a operator in which the radial derivative ∂ r is not contained. From Eq.(3.13), we obtain [ þ ' , r ] = 1 / (2 L ) . Thus, we have where C ' = C + ρ ' ( ikm -b ) . We expand the operator B s as B s = B H s + r B 1 s + O ( r 2 ) . Then, Eq.(B.1) is written as where C '' = C ' + L B 1 s / 2 . Thus, from Eq.(B.7), we obtain where Eq.(B.8) expresses Eq.(4.1), (5.1) and (6.6). Differentiating Eq.(B.7) by r , we have where C ''' = C '' + 2( ikm -b )( þ ' -∂ r / (2 L )) . (Note that there is no radial derivative in the operator þ ' -∂ r / (2 L ) .) Setting m I = 0 in above equation, we obtain Eqs.(4.6), (5.5) and (6.11). The explicit expressions of A s can be obtained using near horizon expressions of GHP variables and derivatives (3.8), (3.11) and (3.13).", "pages": [ 15, 16 ] }, { "title": "C Useful GHP equations", "content": "We summarize the useful GHP equations for Einstein spacetime satisfying R µν = Λ g µ . These equations are firstly derived in Ref. [8].", "pages": [ 16 ] }, { "title": "C.1 Newman-Penrose equations", "content": "From the Ricci equations, [ ∇ µ , ∇ ν ] V ρ = R µνρσ V σ , we obtain following equations. Another four equations can be obtained by taking the prime ' of these four.", "pages": [ 17 ] }, { "title": "C.2 Bianchi equations", "content": "From Bianchi equations, ∇ [ λ C µν | ρσ ] = 0 , we obtain following equations. Boost weight +2:", "pages": [ 17 ] }, { "title": "Boost weight 0:", "content": "Another five equations are obtained by applying the prime operator to above equations.", "pages": [ 17 ] }, { "title": "C.3 Maxwell equations", "content": "From Maxwell equations, dF = d ∗ F = 0 , we obtain following equations. A further three equations can be obtained by priming above equations.", "pages": [ 18 ] }, { "title": "C.4 Commutators of derivatives", "content": "The commutation relations for GHP derivatives are given by The result for [ þ ' , ð i ] can be obtained from the prime operation of [ þ , ð i ] .", "pages": [ 18 ] } ]

[ { "title": "Variational approach to the time-dependent Schrodinger-Newton equations", "content": "Giovanni Manfredi and Paul-Antoine Hervieux Institut de Physique et Chimie des Mat´eriaux, CNRS and Universit´e de Strasbourg, BP 43, F-67034 Strasbourg, France", "pages": [ 1 ] }, { "title": "Fernando Haas", "content": "Departamento de F'ısica, Universidade Federal do Paran'a, 81531-990, Curitiba, Paran'a, Brazil (Dated: June 25, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "Using a variational approach based on a Lagrangian formulation and Gaussian trial functions, we derive a simple dynamical system that captures the main features of the time-dependent Schrodinger-Newton equations. With little analytical or numerical effort, the model furnishes information on the ground state density and energy eigenvalue, the linear frequencies, as well as the nonlinear long-time behaviour. Our results are in good agreement with those obtained through analytical estimates or numerical simulations of the full Schrodinger-Newton equations.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In recent years, there has been a renewal of interest in the set of nonlinear equations known as the Schrodinger-Newton (SN) equations. These consist of the ordinary Schrodinger equation where the gravitational potential V ( r , t ), in the Newtonian approximation, is obtained selfconsistently from Poisson's equation where m is the mass of the system and G is the gravitational constant. The source term in Poisson's equation is provided by a matter density ρ ( r , t ) = m | Ψ | 2 that is proportional to the probability density as given by the wavefunction Ψ( r , t ). The resulting equations are therefore nonlinear. SN-type equations have been proposed in various areas of physics and astrophysics. For instance, it has been suggested that gravitation, unlike other forces, may not be quantized at all [1]. In that case, the stress-energy tensor T µν in Einstein's equations should be replaced by its quantum-mechanical average 〈 T µν 〉 . The SN equations can thus be viewed as the nonrelativistic ( c →∞ ) and Newtonian ( G → 0) limit of the modified Einstein's equations G µν = (8 πG/c 4 ) 〈 T µν 〉 . More formally, Giulini and Großardt [2] recently showed that the SN equations can be derived in a WKB-like expansion in 1 /c from the Einstein-Klein-Gordon and Einstein-Dirac system. In another context, the SN equations have been proposed as a fundamental modification of the Schrodinger equation due to gravitational effects. Penrose [3, 4] and Diosi [5] postulated that gravity might be at the origin of the spontaneous collapse of the wavefunction and proposed the (stationary) SN equations as a possible candidate for an approximate description of such gravitationally-induced collapse. Finally, the SN equations have been used in an astrophysical context to study selfgravitating objects such as boson stars [6, 7] or to describe dark matter by means of a scalar field [8]. Whatever their present theoretical status and possible applications, the SN equations represent a minimal model in which nonrelativistic quantum mechanics is coupled self- consistently to Newtonian gravity. As such, they are worth investigating in some detail, both for their static and their dynamical properties. Many theoretical results on the SN equations were obtained in the past, using either analytical or numerical approaches [9]. For instance, the energy eigenvalues (all negative) have been determined numerically with good precision [10] and some analytical estimates exist on the lower bound for the ground state energy [11]. The linear stability properties of the ground state were also investigated [10]. In the time-dependent and fully nonlinear regime, virtually all results are numerical, with few exceptions whose validity is restricted to short time scales [12]. An unexpected result was published a few years ago by Salzman and Carlip [13]. In numerical simulations of spherically symmetric systems, these authors observed that, for masses above a certain critical value, the wavefunction 'collapsed' at the origin, at least within the accuracy of their simulations[22]. The most astonishing feature of these results was that the critical mass was far smaller than what could be expected from simple order-of-magnitude calculations. However, more recent calculations [12, 14] disagree with the results of Salzman and Carlip and set the critical mass at a value that is several order of magnitudes larger and consistent with analytical estimates. In this work, we revisit the SN equations using a Lagrangian variational method [15, 16]. With this approach, one can arrive at a single ordinary differential equation that describes the evolution of the width of the mass density. This method reproduces all the main results on the ground state and linear dynamics derived previously. In addition, this approach is not restricted to linear theory and can be used to investigate nonlinear oscillations or the long-time dynamics. Finally, the mathematical simplicity of the governing equation makes it easy to intuit at a glance the salient features of the solutions.", "pages": [ 2, 3 ] }, { "title": "A. Normalization", "content": "Let us first rewrite the SN equations (1)-(2) in dimensionless form, using the analog of atomic units for the gravitational interaction. Thus, lengths are measured in units of the gravitational 'Bohr radius' a G = ¯ h 2 / ( Gm 3 ), energy is measured in units of E G = m 5 G 2 / ¯ h 2 (the gravitational equivalent of the Hartree), and time in units of t G = ¯ h/E G . To ensure conservation of the wavefunction norm, one also needs to normalize Ψ to a -3 / 2 G . The dimensionless SN equations then read as with the normalization condition ∫ | Ψ | 2 d r = 1. Notice that Eqs. (3)-(4) are now free of all parameters. The evolution of the system is then entirely determined by its initial condition. For instance, if the initial condition is spherically symmetric and Gaussian (as will be the case in the rest of this paper), the only relevant dimensionless parameter is the width of the initial Gaussian measured in units of a G .", "pages": [ 3, 4 ] }, { "title": "B. Lagrangian approach", "content": "In this section, we will follow the derivation described in Refs. [15, 16] in the context of atomic or condensed matter physics, where the relevant interaction is Coulombian rather than gravitational. The SN equations (3)-(4) can be written in a hydrodynamical form by using the Madelung transformation Ψ = √ ρ exp( iS ), where √ ρ is the amplitude and S ( r , t ) is the phase of the wavefunction [17]. The hydrodynamical continuity and momentum equations read as: and the velocity is defined as the gradient of the phase, u = ∇ S . It can be shown that the above hydrodynamic equations (5)-(6), together with Poisson's equation (4), can be derived from the following Lagrangian density L [16]: So far, no approximation was made. The purpose is now to derive a set of evolution equations for a small number of macroscopic quantities that characterize the matter density profile. With this aim in mind, let us assume that the system is spherically symmetric and that the density profile is Gaussian: where r = | r | and R ( t ) is the time-dependent size of the density. For the above density profile, the exact solution of Poisson's equation (4) is where erf( x ) is the error function. In addition, the continuity equation (5) is exactly solved by the following velocity field: u = ( ˙ R/R ) r , which stems from the phase function S = ( ˙ R/ 2 R ) r 2 . The dot denotes derivation with respect to time. We can now compute the Lagrangian by plugging Eq. (8) and the above solutions for V and S into Eq. (7), and integrating over all space, i.e., L = -2 3 ∫ L d r , where the multiplicative factor was introduced for convenience of notation. The result is where C = 2 / (3 √ 2 π ) . The corresponding equations of motion are obtained from the Euler- Lagrange equations Equation (12) is equivalent to the Hamiltonian equation of motion of a pointlike particle evolving in the external potential U ( R ) = 1 / (2 R 2 ) -C/R (see Fig. 1). The first term is repulsive and represents kinetic energy due to velocity dispersion (uncertainty principle), whereas the second term is attractive and represents self-gravity. Note that this result was obtained from a rigorous development based on a Lagrangian variational principle. In particular, no assumptions of linear response were made in the derivation, so that Eq. (12) can be used to extract information on the nonlinear regime of the time-dependent SN equations. We also stress that the evolution obtained with the variational method is by construction unitary, since the trial Gaussian density [Eq. (8)] automatically satisfies: ∫ ρ d r = ∫ | Ψ | 2 d r = 1 for all times. which yield:", "pages": [ 4, 5 ] }, { "title": "III. RESULTS", "content": "The pseudo-potential U ( R ) is plotted in Fig. 1. It goes to infinity for R → 0 and goes to zero as R -1 for R →∞ . It crosses the horizontal axis at a point R 0 such that U ( R 0 ) = 0 and has a single minimum at R 1 , where U ' ( R 1 ) = 0. The values of these two points are easily determined and yield:", "pages": [ 6 ] }, { "title": "A. Ground state", "content": "The matter density profile in the ground state is given by Eq. (8), with R = R 1 , corresponding to the minimum of U . This profile is shown in Fig. 2 (dashed line), together with the ground state density obtained from a numerical solution of the stationary SN equations (solid line). The agreement is very good. Stationary solutions of the SN equations must satisfy the virial theorem, which states that the potential energy (in absolute value) is twice the kinetic energy. We can verify that this is the case using the wavefunction Ψ = √ ρ from Eq. (8) and the potential of Eq. (9). The kinetic energy is whereas the potential energy yields: It is readily checked that, when R = R 1 , | P | = 2 K = 1 / 3 π , so that the virial theorem is satisfied. The energy eigenvalue of the ground state (lowest energy state) has been computed numerically many times and an accepted value is E 0 = -0 . 163 [10, 18]. More accurate solutions may be obtained using the methods outlined in [19]. In our notation, E 0 = K +2 P = -3 K = -1 / 2 π ≈ -0 . 159, which is rather close to the numerical value (the error is less than 3%). Finally, we note that, when R = R 0 , one obtains P = -K so that the total energy is zero.", "pages": [ 6, 7 ] }, { "title": "B. Dynamics", "content": "One can linearize the equation of motion (12) around the equilibrium, by writing R = R 1 + δR , where δR ( t ) is a small perturbation. Substituting into Eq. (12) and taking the Fourier transform in the time variable (i.e., assuming δR ( t ) ∼ e i Ω t ) yields the following oscillation frequency: A perturbation analysis of the full time-dependent SN equations was performed by Harrison et al. [10]. The lowest oscillation frequency that these authors find (see Fig. 2 in Ref. [10]) is close to Ω Harr = 0 . 035, which is in very good agreement with Eq. (16) (the extra factor of 2 comes from the fact that Harrison et al. perturb the wavefunction instead of the density | Ψ | 2 ). In order to check this result, we solved the spherically symmetric time-dependent SN equations, using a second-order Crank-Nicolson method with centred differences for the spatial differentiation. The initial condition is the exact ground state computed numerically (solid curve in Fig. 2), to which a very small perturbation was added. The root mean square of the radius √ 〈 r 2 ( t ) 〉 is then computed using the standard quantum average. Its frequency spectrum is shown in Fig. 3 and displays a clear peak around Ω = 0 . 067, which is again very close to Eq. (16). For an initial condition that is slightly farther from the exact ground state, we expect the SN equations to display some nonlinear effects. This is apparent from Fig. 4, where the Gaussian profile given by Eq. (8) with R (0) = R 1 (dashed line in Fig. 2) was used as an initial condition. The time history of √ 〈 r 2 ( t ) 〉 clearly shows some nonlinear oscillations, although their frequency is still close to the linear estimate given by Eq. (16). It is also useful to monitor the evolution of the density ρ ( r , t ) in order to check that it stays sufficiently close to a Gaussian function, which is required for the validity of the /s32 variational approach. This is done in Fig. 5, where we plot the mass density obtained from numerical simulations of the full SN equations (solid lines) and compare it to a Gaussian density [Eq. (8)] with same width (dashed lines). The left panel refers to the same evolution as in Fig. 4 at t = 500. The right panel refers to a case where the density is initially localised near the origin, so that R (0) < R 1 . In this case, the system expands almost freely and at t = 200 (corresponding to the plot of Fig. 5) it has attained a considerable size. In both cases, the numerically-computed density is reasonably close to a Gaussian profile, thus strengthening our confidence in the present variational approach. We also stress that for similar problems involving the Coulomb interaction and rather strong nonlinearities (quartic confinement), the variational procedure appeared to work rather well [15]. Finally, we consider the long-time solutions of Eq. (12). From the shape of the pseudopotential U ( R ) (Fig. 1), it is clear that three different regimes are possible for an initial condition R (0) > 0, ˙ R (0) = 0: The three regimes described above are neatly reproduced in numerical simulations of Eq. (12), shown in Fig. 6. It is also worth to note that in cosmology these regimes correspond respectively to an open, closed, and Einstein-de Sitter universe (which expands as t 2 / 3 ). Of course, the full evolution of the wavefunction according to the SN equations can be much richer than this simple picture. For large masses, the wavepacket can break down into two parts, with some mass being ejected to infinity while the rest remains confined [12, 14]. This behaviour cannot be captured by our variational approach, which postulates that the density remains close to a Gaussian profile for all times. One could nevertheless extend the present model by considering more complicated trial functions involving more than one variational parameter. This would result in a set of coupled nonlinear differential equations that generalize Eq. (12).", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "IV. DISCUSSION", "content": "The main interest of the method outlined in this paper is that it relies on a rigorous development based on a variational principle, while at the same time yielding results that are simple and intuitive. The very shape of the pseudo-potential U ( R ) (Fig. 1) informs us on the type of motions that are to be expected. For instance, it is clear that wavepacket dispersion occurs if R < R 0 , whereas it is inhibited if R > R 0 . Further, if R > R 1 ≈ 3 . 76¯ h 2 / ( Gm 3 ) the wavepacket should start to contract right from the beginning of the evolution (for clarity, we restore dimensional units in this section). Now, we want to compare the above estimations with the numerical results of Giulini et al. [12], who considered a system of initial size R = 0 . 707 µ m ( a = 0 . 5 µ m in their notation). They observed a contracting wavepacket for masses greater than 7 × 10 9 amu (atomic mass units), which is rather close to the value m = 5 . 74 × 10 9 amu predicted by our formula Rm 3 = 3 . 76 ¯ h 2 /G . The results of other simulations [14] are also consistent with these findings. It is clear that the importance of self-gravitational effects in the SN equations depends on both the size R and the mass m of the object under consideration. Therefore, it is useful to plot a mass-radius diagram on a log-log scale (Fig. 7), where these two quantities appear explicitly. Gravitational effects should play a significant role for objects that fall in the region above the curve defined by Rm 3 = const . = 3 . 76 ¯ h 2 /G (solid line). Experiments aimed at detecting the role of gravity on quantum decoherence will probably involve studying the interference fringes of solid-state mesoscopic objects, which should be light enough for quantum coherence to be observable but also heavy enough for gravitational effects to play a measurable role. Interferometry experiments on small silica spheres [20] and gold clusters [21] are possible candidates for such studies. In order to fix ideas, let us focus on the case of gold or other metal clusters, for which the number density is typically n gold ≈ 5 × 10 28 m -3 . The dashed line on Fig. 7 represents the curve at constant density m [amu] /R 3 = n gold . The intersection of the dashed line with the the solid line Rm 3 = 3 . 76¯ h 2 /G yields the minimum mass and radius that gold clusters should possess for gravitational effects to play a significant role. This turns out to be of the order of a few microns in size and about 5 × 10 9 in atomic mass units. The same calculation performed for other metal clusters or the silica spheres mentioned above yields similar results. The experimental challenge will be to perform quantum interference experiments on such massive objects and to control other non-gravitational sources of decoherence. In practice, one may perform different experiments for increasing values of the cluster mass, thus moving from left to right on the dashed line in Fig. 7. When crossing the solid line, gravitational effects should be detected, perhaps as a reduction in the contrast of the interference fringes.", "pages": [ 11, 12, 13 ] } ]

[ { "title": "G. Papadopoulos", "content": "Department of Mathematics King's College London Strand London WC2R 2LS, UK", "pages": [ 1 ] }, { "title": "Abstract", "content": "We find under some mild assumptions that the most general potential of 1dimensional conformal systems with time independent couplings is expressed as V = V 0 + V 1 , where V 0 is a homogeneous function with respect to a homothetic motion in configuration space and V 1 is determined from an equation with source a homothetic potential. Such systems admit at most an SL (2 , R ) conformal symmetry which, depending on the couplings, is embedded in Diff( R ) in three different ways. In one case, SL (2 , R ) is also embedded in Diff( S 1 ). Examples of such models include those with potential V = αx 2 + βx -2 for arbitrary couplings α and β , the Calogero models with harmonic oscillator couplings and non-linear models with suitable metrics and potentials. In addition, we give the conditions on the couplings for a class of gauge theories to admit a SL (2 , R ) conformal symmetry. We present examples of such systems with general gauge groups and global symmetries that include the isometries of AdS 2 × S 3 and AdS 2 × S 3 × S 3 which arise as backgrounds in AdS 2 /CFT 1 .", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "It has been known for sometime that 1-dimensional models with potential V = βx -2 are conformally invariant [1, 2]. de Alfaro, Fubini and Furlan (DFF) explored the SL (2 , R ) conformal symmetry of this theory and noticed that the Hamiltonian operator does not have a ground state [2]. To overcome this problem, they suggested to choose the eigenstates of as a basis in the Hilbert space. O is not the Hamiltonian operator, but a linear combination of conserved charges associated with the SL (2 , R ) conformal symmetry of the theory. Choosing suitably the coupling constants α, β this operator exhibits a ground state and discrete energy spectrum. As a result the DFF formulation of the theory has been widely accepted in the literature. However, although a Hilbert space has been defined for the theory, the Hamiltonian operator is not diagonal in the chosen basis and so the energy levels of the theory cannot be identified. There have been many generalizations of the V = βx -2 model, see eg [3]-[8], including the construction of non-linear theories 1 [9, 10], which exhibit similar properties, see also reviews [11, 12] and references within. The DFF treatment of the theory and its generalizations have found widespread applications in the description of near horizon black hole dynamics [13, 14, 15, 16] and in the understanding of black hole moduli spaces [17, 18, 19, 20, 21]. Another application of conformal mechanics is in the context of AdS 2 /CFT 1 correspondence [22], and for further exploration see eg [23, 24] . It is expected that string theory or M-theory on a AdS 2 × X background is dual to a conformal theory on the boundary. After analytic continuation the Lorentzian boundary of AdS 2 , which is two copies of R , is mapped to a circle, see eg [23]. In the Euclidean regime, the associated dual theory should be a conformal theory defined on the circle. As we shall demonstrate, there are such conformal theories but they are based on different potentials 2 from V = -βx -2 . In this paper, we investigate the conformal properties of theories with Lagrangian where g is a metric on the configurations space, V is a potential and ˙ q is the time derivative of the position. The conditions required for such theories to be invariant under the conformal transformations (2.1) have been stated in (2.2). Assuming that the configuration space of these theories admits a homothetic vector field Z associated with a homothetic potential h , the conditions for conformal invariance (2.2) can be solved. The potential of the theory can be written as where V 0 is a homogeneous function with respect to the homothetic motion Z and V 1 obeys the inhomogeneous equation (2.13) which has source the homothetic potential h . The dimension of the conformal group of these models is at most three and one of the generators is time translations. This is because the parameter of the transformation obeys a third order equation (2.10). The maximal conformal group is SL (2 , R ) and it is embedded in Diff( R ) in three different ways generating the vector fields for some ω related to the couplings. /negationslash The first SL (2 , R ) embedding (i) in (1.4) is realized for the models with V 1 = 0. These class of models has a homogeneous potential V 0 and includes the DFF model, and its linear and non-linear generalizations [9, 10]. Furthermore, if V 1 = 0, the SL (2 , R ) conformal group is embedded in Diff( R ) generating the vector fields (ii) or (iii). These are Newton-Hooke transformations and the two cases are distinguished by the sign of the inhomogeneous term in the equation (2.13) which determines V 1 . The models with conformal transformation (ii) and (iii) are related by a naive analytic continuation, and the SL (2 , R ) group in the latter case can be embedded in Diff( S 1 ). The class of conformal models with conformal symmetry (ii) and (iii) in (1.4) includes those with potential [3] where V 0 = βx -2 and V 1 = αx 2 . For α < 0 the conformal group generates the vector fields (ii) in (1.4), while for α > 0 the conformal group generates the vector field (iii). There are also several multi-particle models which exhibit type (ii) and (iii) in (1.4) conformal symmetry. Such systems include the Calogero model with harmonic oscillator couplings of equal frequency [27], and the multi-particle linear models of [28] for which V 0 satisfies additional symmetries. We shall present some additional linear and non-linear systems with (ii) and (iii) conformal symmetries. Observe that the theories with α, β > 0 in (1.5) have a ground state and discrete energy spectrum, and so there is no need to choose another operator different from the Hamiltonian to give a basis in the Hilbert space of the theory. This also applies to several other models in this class. One result which follows from the general analysis of this paper is that the most general linear conformal model admits a potential (1.3), where V 0 is a homogeneous function of the positions q of degree -2 and V 1 = α | q | 2 . This rigidity result is based on the uniqueness of homothetic motions in flat space associated with a homothetic potential. The homothetic motion is the homogeneous scaling of all coordinates, q i → /lscriptq i . These models admit an SL (2 , R ) conformal symmetry generated by the vector fields (ii) and (iii) in (1.4) and depending on whether α < 0 or α > 0, respectively. More recently, conformal models in one dimension have been investigated which apart from scalar fields contain also vectors [25]. So far such theories have been based on gauging models with homogeneous potentials. We shall demonstrate that such models can be generalized to include potentials of the type (1.5). In particular, we derive the conditions (4.9) for gauged non-linear sigma models with Lagrangian (4.1) to admit a conformal symmetry, and determine the equations that restrict the potentials. We find that for a large class of such conformal theories the potential can be written as in (1.3), where both V 0 and V 1 must also be gauge invariant. In addition, we give some examples which include conformal models with a general gauged group and global symmetries. Some of these models exhibit the isometries of AdS 2 × S 3 and AdS 2 × S 3 × S 3 backgrounds as global symmetries. A class of these models is solvable, and the Hamiltonian has a ground state and discrete spectrum. A similar investigation of SL (2 , R ) symmetries in the context of matrix models has been done in [26] and the associated potentials have been identified. This paper is organized as follows. In section 2, we derive investigate the conditions for conformal invariance of non-linear 1-dimensional theories and derive the scalar potential (1.3). In section 3, we give several examples of such models. In section 4, we derive the conditions on the couplings gauged sigma models with a potential to admit conformal invariance, and give several examples. In section 5, we present our conclusions.", "pages": [ 2, 3, 4 ] }, { "title": "2.1 Lagrangian", "content": "Consider the Lagrangian (1.2) of a sigma model on a manifold M with metric g and with a potential V . This describes either the propagation of a non-relativistic particle in a curved manifold M or a multi-particle system with a non-trivial configuration space M . One can assign mass dimensions such that q is dimensionless [ q ] = 0 while [ t ] = -1. Thus [ L ] = 2 provided one takes the coupling V terms to have dimension 2. This is not the most general Lagrangian that one can consider as a coupling with dimension 1 has not been included. This will be done elsewhere [29].", "pages": [ 4 ] }, { "title": "2.2 Conformal transformations", "content": "All time re-parameterizations t ' = u ( t ) are conformal transformations of the Euclidean metric on R as ds 2 = ( dt ' ) 2 = ( ˙ u ) 2 dt 2 . Therefore, one can choose any of these transformations and demand that leave the action (1.2) invariant. Apart from time translations 3 , such transformations will not leave the action invariant unless there is a compensating additional transformation on the positions generated by a vector field X on M [9]. As a result, one considers the infinitesimal transformations [10] where /epsilon1 is a small parameter. The first term in the transformation of q is induced by the infinitesimal transformation δt = /epsilon1a ( t ), where a ( t ) is the vector field on R which generates the time re-parameterizations, while the second term containing X is the compensating transformation which may explicitly depend on t . The conditions for the invariance of the action (1.2), up to surface terms, under the transformations (2.1) are [10] where f = f ( t, q ) is the contribution from the surface term, and where ∂ t denotes differentiation of the explicit dependence of X and f on t , ie The conserved charges associated with the above symmetries are It can be easily shown that Q ( a, X ) is conserved subject to field equations.", "pages": [ 4, 5 ] }, { "title": "2.3 Solution of conformal conditions and new models", "content": "It is clear that the first condition in (2.2) implies that X generates a family of homothetic transformations on M which may depend on t . Since all Diff( R ) are conformal transformations, the system can be invariant under any subgroup of Diff( R ). So, one should consider at most as many homothetic motions in M as the dimension of the subgroup of conformal transformations. However, in most examples of interest M admits one homothetic motion generated by a vector field Z which does not depend explicitly on t where /lscript is a constant. Then, the first condition can be solved by setting Assuming that Z arises from a homothetic potential, ie where h = h ( q ), f can be chosen 4 The last equation in (2.2) can now be rewritten as /negationslash Since we are seeking to find potentials V which solve the above equations and do not depend explicitly on t , we have to take /negationslash where λ is a constant. Of course, if ˙ a = 0, there is no condition on V as the only symmetry of the action is time translations. Thus, we take ˙ a = 0 and as a result the equation which determines the potential is The general solution for the potential can be written as in (1.3), ie V = V 0 + V 1 , where V 0 is the most general solution of the homogenous equation and V 1 is a solution of Clearly, there are 3 cases to consider depending on whether λ = 0, or λ > 0 or λ < 0. In these three choices, the vector field a is determined from (2.10) as follows. For λ = 0, one has where a 0 , a 1 and a 2 are integration constants. For λ = ω 2 , one has and for λ = -ω 2 , one has where a 0 , b, c are integration constants. The new conformal models arise from the last two cases. Before we proceed to investigate individual models, let as examine the algebra of these transformations. A basis in the space of vector fields of the infinitesimal transformations (2.14), (2.15) and (2.16) is given in (i), (ii) and (iii) of (1.4), respectively, with | λ | = ω 2 . The group of transformations generated by (2.14), (2.15) and (2.16) is SL (2 , R ). However, SL (2 , R ) is embedded into Diff( R ) in three different ways 5 . The group of transformations generated by (2.16) is also embedded in the Diff( S 1 ) as the associated vector fields are periodic in t . The two cases (2.15) and (2.16) are related to each other by analytic continuation. Substituting the above expressions of X into the conserved charges and using the properties of the homothetic motion on M , one finds that These can be easily computed explicitly in the examples described below.", "pages": [ 5, 6 ] }, { "title": "3.1 Conformal particle in flat space", "content": "The most illuminating model is that of a single particle propagating on the real line. Here we shall show that (1.5), which has been found previously in [3], is the only potential consistent with conformal invariance. For this we shall take the Lagrangian and we shall determine V such that the action is conformally invariant. For this consider the homothetic vector field on the configuration space. For this choice of Z , /lscript = 1. The homothetic potential in this case is Then the equation (2.12) can be solved for V 0 to yield for some constant β , which the potential of the DFF model. However, we have seen that the potential V also receives a contribution from V 1 which is determined in (2.13). The latter equation can be solved as Thus the most general potential V = V 0 + V 1 of such conformal models is given in (1.5). The Hamiltonian of this class of conformal models is given in (1.1). As it has already been mentioned the associated Hamiltonian operator with α > 0 , β ≥ 0 has a ground state and discrete spectrum.", "pages": [ 7 ] }, { "title": "3.2 Conformal multi-particle systems", "content": "Consider next the linear model of N particles propagating in R and interacting with a potential V . The Lagrangian of such a system is To find the potentials V consistent with conformal invariance, consider the homothetic motion of R N configuration space. The homothetic potential in this case is As it has been mentioned in the introduction, Z in (3.7) is the unique homothetic motion in R N associated with a homothetic potential 6 up to an overall scale which does not affect the form of the potential. After solving the conditions (2.12) and (2.13), one finds that the potential V is and V 0 is a homogeneous function of degree -2 (3.9) is the most general potential of linear models. Of course, there are many choices for V 0 . A minimal choice for V 0 is V 0 = β | x | -2 . However, this is not unique. For example, one can also choose /negationslash The models with potentials V given in (3.9) and (3.11) are the Calogero models with harmonic couplings of equal frequency. Our results demonstrate that these models are conformally invariant. It is well-known that such models with α > 0 and β ≥ 0 have a vacuum state and discrete energy spectrum [27, 30]. Of course, there are many more potential functions V 0 which satisfy the homogeneity condition (3.10) above than those appearing in the Calogero models. The above models also include those presented in [28] where some additional symmetry assumptions were made on the form of V 0 potential. To summarize, we have shown that all the above models admit either an SL (2 , R ) conformal symmetry which is embedded in Diff( R ) as in (i), (ii) or (iii) of (1.4) depending on whether α = 0, α < 0 or α > 0, respectively. The associated conserved charges can be computed by a direct substitution in (2.17).", "pages": [ 7, 8 ] }, { "title": "3.3 Particles propagating on cones", "content": "So far, we have presented linear models as examples. For a non-linear example, consider particles propagating on a cone and interacting with a potential V . The Lagrangian of such a system is where γ is the metric of the cone section which does not depend on the radial coordinate r but it may depend on the rest of the coordinates x . The cone metric admits a homothetic motion generated by the vector field which homothetic potential where k is an arbitrary function of x . It is straightforward to show that the most general potential compatible with conformal symmetry is Again these models admit a SL (2 , R ) conformal symmetry generating the vector fields (i), (ii) or (iii) of (1.4) depending on whether α = 0, α < 0 or α > 0, respectively.", "pages": [ 8, 9 ] }, { "title": "4.1 Action", "content": "Motivated by applications in AdS/CFT , which typically requires dual theories with a gauge symmetry, and to enhance the class of 1-dimensional conformal systems, we shall also examine the conditions for a gauged sigma model to admit conformal invariance. For this, we assume that M admits a group of isometries G , generating the vector fields ξ , which leave V invariant. Gauging the isometries of (1.2), one finds the Lagrangian 7 where A is the gauge potential and f are the structure constants of G . We assign mass dimension to A as [ A ] = 1 so that L has mass dimension 2. The equations of motion of the theory are where Under certain conditions the gauge connection A can be eliminated from the equations of motion leading to a theory with dynamical variables just the q 's. In particular notice that the second equation of motion can be rewritten as where /lscript ab = g ij ξ i a ξ j b . If /lscript is invertible, then all A can be eliminated. However, we shall not elaborate on this here. Instead, we shall proceed to find the conditions such that the action (4.1) is invariant under some conformal symmetries.", "pages": [ 9, 10 ] }, { "title": "4.2 Conformal and gauge symmetries", "content": "The action (4.1) is invariant under the gauge transformations where η is the gauge infinitesimal parameter. Next as in the un-gauged case, one expects that the transformations on q and A , which induce the conformal symmetries of the action (4.1), to contain two parts. One part is associated with time re-parameterizations and an additional term which generates compensating transformations on the configuration space. As a result, we postulate the conformal transformations where the first term in the variation of q and the first two terms in the variation of A are the transformations induced on q and A from the infinitesimal re-parameterization of t , δt = /epsilon1a ( t ), and the rest are the compensating transformations. These transformation mix with the gauge transformations above. In particular, the coordinate transformation induced on A by a can be rewritten as a gauge transformation with parameter -aA a . Since the action is invariant under gauge transformations, this can be used to simplify the conformal transformations as For the same reason X and Z are not uniquely defined. In particular X and W are defined up to terms /lscript a ξ a and ∇ t /lscript a , respectively, where /lscript = /lscript ( t, q, A ). Assuming that X and W do not depend on time derivatives of q , a straightforward computation reveals that the conditions required for the invariance of the action, up to surface terms, are where f = f ( t, q ) is the contribution from the surface term. f is taken to be gauge invariant, ξ i a ∂ i f = 0. To find conformal models, one has to solve (4.9).", "pages": [ 10 ] }, { "title": "4.3 Solution of conformal conditions", "content": "Here, we shall not seek the most general solution to the conformal invariance conditions(4.9). Instead, we shall take In this case, the above conditions (4.9) reduce to those of (2.2) but with the additional assumption that f is gauge invariant. To find solutions, we proceed as in section 2.3. The potential is given as V = V 0 + V 1 , (1.3), with V 0 and V 1 determined by the equations (2.12) and (2.13), respectively. There is an additional restriction here that the homothetic potential h is gauge invariant, ξ i a ∂ i h = 0. As in the systems without gauge symmetry, there are three cases to consider depending on whether λ = 0, λ > 0 or λ < 0. In all cases the conformal group is SL (2 , R ) but it is embedded in three different ways into Diff( R ). The λ > 0 and λ < 0 models are related by analytic continuation.", "pages": [ 11 ] }, { "title": "4.4.1 Gauged nonlinear models on a cone", "content": "Examples of non-linear gauge theories exhibiting conformal symmetry are those that describe the propagation of particles on a cone. Assuming that the cone section metric γ admits a group of isometries generating the vector fields ξ , the Lagrangian of the theory can be written as where The homothetic vector field is again given by Z = 1 2 r∂ r and commutes with the Killing vector fields ξ a satisfying the assumption (4.10). The rest of the analysis proceed as in the cone example in section 3.3 for the un-gauged model yielding a potential where now β ( x ) and k ( x ) are gauge invariant functions of the cone section, ξ i a ∂ i β = ξ i a ∂ i k = 0. The simplest explicit example is to consider the flat cone R 2 and as the gauged symmetry the rotational symmetry. The potential of this model is given as in (4.24) with β and k constants.", "pages": [ 11 ] }, { "title": "4.4.2 Gauge theories", "content": "A large class of linear conformal models 8 can be constructed beginning from some gauge group G and some linear representation D of its Lie algebra g on a vector space V . Suppose that D leaves invariant a (constant) metric g on V . Then one can consider the Lagrangian where To determine V such that this theory is conformal, observe that the metric admits a homothetic motion generated by the vector field Moreover, this commutes with the Killing vector fields ie [ Z, ξ a ] = 0. As a consequence (4.10) is satisfied. Furthermore, the homothetic potential of Z is Using this, the potential V can be determined by solving (2.12) and (2.13) as and V 0 is a function of x of homogeneous degree -2, which is also invariant under G . The minimal choice is However such a choice is not unique for general gauge groups and representations D . A similar potential has been derived in the investigation of SL (2 , R ) invariant matrix models in [26]. Amongst these models, one can take as D = adj ⊗ I k , where adj is the adjoint representation of a group G and I is the trivial representation. In such a case, the Lagrangian can be written as where g ab is an invariant metric on the adjoint representation of G and κ a metric on the k-copies of the trivial representation. The potential in this case can be written as and V 0 is a function of x of homogeneous degree -2 which is also invariant under G . Now there are several options for V 0 . For example, V 0 can be any homogeneous function of degree -2 expressed in terms of the gauge invariant functions like and many others which can be constructed from all the invariant tensors of g under the action of the adjoint representation. One example is a gauged Calogero model for which the potential is given in (4.24) with /negationslash Further restrictions can be put on the form of the potential by requiring that the theory is invariant under the global symmetry × i O ( n i ) which leaves κ invariant. The above construction can also be done by replacing adj with another representation of the gauge group. This class of conformal theories has all the bosonic symmetries required for the CFT duals of backgrounds like AdS 2 × S 3 or AdS 2 × S 3 × S 3 . In particular, one can easily construct models with rigid symmetry SL (2 , R ) × SO (4), which is the isometry group of AdS 2 × S 3 , and any gauge symmetry including U ( N ), and similarly there are models which exhibit the isometries of AdS 2 × S 3 × S 3 backgrounds as symmetries. It is also worth remarking that the analytic continuation of a λ > 0 theory which exhibits SL (2 , R ) conformal symmetry is equivalent to taking λ to -λ and V 0 to -V 0 and leads to a model with SL (2 , R ) conformal invariance but now embedded in Diff( S 1 ) as expected in the context of AdS 2 /CFT 1 . /negationslash The quantum theory of the model with action (4.14) can be easily described in the case that V 0 = 0 and α > 0. The Hilbert space of these theories can be constructed starting from the Hilbert space of dim D harmonic oscillators. Then, gauge invariance requires that one has to consider only those states which are invariant under the gauge group. The Hamiltonian operator has a ground state and the spectrum is discrete. However the details of the construction depend on the choice of gauge group and representation D . If V 0 = 0, the quantum theory depends on the choice of V 0 . It is likely that some of the properties of the V 0 = 0 models can be maintained in the presence of a large class of V 0 potentials as it happens for the Calogero models with harmonic oscillator couplings.", "pages": [ 11, 12, 13 ] }, { "title": "5 Concluding remarks", "content": "We have demonstrated that the potential V of conformal mechanics models admitting a homothetic motion in configuration space can be expressed as a sum V = V 0 + V 1 , where V 0 is a homogeneous function of the homothetic motion and V 1 is determined from an equation which has as a source the homothetic potential. Depending on the couplings, the maximal conformal group SL (2 , R ) is embedded in Diff( R ) in three different ways. Furthermore, one of these can also be thought as an embedding of SL (2 , R ) in in Diff( S 1 ). This is significant from the point of view of AdS 2 /CFT 1 as the dual Euclidean theory must be defined on the boundary which is a circle. Examples of conformal 1-dimensional systems include models with potential V = αx 2 + βx -2 [3]. The SL (2 , R ) conformal symmetry of this model is embedded in Diff( R ) in three different ways depending on whether α = 0, α < 0 or α > 0, respectively. Moreover if α > 0, SL (2 , R ) can also be embedded in Diff( S 1 ). We have described all 1-dimensional linear conformal theories described by the Lagrangian (3.6). The potential of all such models is V = α | x | 2 + V 0 , where V 0 is a homogeneous of degree -2 function of the positions x . This rigidity result is based on the uniqueness of the homothetic motion in flat space associated with a homothetic potential and the analysis in section 2. Examples of such theories include the Calogero models with harmonic oscillator couplings of equal frequency as well as the models given in [28]. We have also presented examples of non-linear models. It is clear form the analysis of section 2 that if the configuration space of a system admits a single homothetic motion associated with a homothetic potential, then the vector field a ( t ) ∂ t which generates the time re-parameterizations obeys the third order equation (2.10). Because of this, the conformal group can be at most 3-dimensional. Therefore, if there are theories with larger conformal groups than SL (2 , R ), then necessarily must have additional fields, like vectors or spinors, and possibly must couple to gravity. As a consequence all linear models admit at most a SL (2 , R ) conformal symmetry. We have also investigated the conformal properties of 1-dimensional systems with scalar and vector fields based on the Lagrangian(4.1). We have derived the conditions for such systems to admit a conformal symmetry (4.9) and present several examples. The potential of a class of such theories is again the sum of a homogeneous function, under the action the homothetic motion, and a term that depends on the homothetic potential. Examples of such conformal models can exhibit general gauge groups and global symmetries. In particular, we have constructed models with arbitrary gauge group which have the isometries of AdS 2 × S 3 and AdS 2 × S 3 × S 3 backgrounds as global symmetries. Similar potentials have arisen in the investigation of matrix models with SL (2 , R ) invariance in [26]. Gravitational backgrounds that have applications in AdS 2 /CFT 1 typically preserve some of the spacetime supersymmetry and as a result the dual theories must be superconformal. The supersymmetric extension of some of the conformal models we have considered here has already been done, see eg [30] and [9, 10] for the supersymmetric extension of Calogero model with harmonic oscillator couplings and that of non-linear conformal theories with homogeneous potentials, respectively, see also [32] for matrix models. Conformal linear models with extended supersymmetry and homogenous potentials have been reviewed in [12], see also [33]. It is straightforward to construct superconformal models with potentials V = V 0 + V 1 specially those that exhibit a small number of supersymmetries. Such supersymmetric extensions can be based on the results of [31, 19] and they will be reported elsewhere.", "pages": [ 13, 14 ] }, { "title": "Acknowledgements", "content": "I would like to specially thank Anton Galajinsky and Jeong-Hyuck Park for their comments as they have led to significant improvements in the paper. I also like to thank Evgeny Ivanov and Roman Jackiw for correspondence. GP is partially supported by the STFC rolling grant ST/J002798/1.", "pages": [ 15 ] } ]

[ { "title": "An analogue of Hawking radiation in the quantum Hall effect", "content": "MICHAEL STONE University of Illinois, Department of Physics 1110 W. Green St. Urbana, IL 61801 USA E-mail: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "We use the identification of the edge mode of the filling fraction ν = 1 quantum Hall phase with a 1+1 dimensional chiral Dirac fermion to construct an analogue model for a chiral fermion in a space-time geometry possessing an event horizon. By solving the model in the lowest Landau level, we show that the event horizon emits particles and holes with a thermal spectrum. Each emitted quasiparticle is correlated with an opposite-energy partner on the other side of the event horizon. Once we trace out these 'unobservable' partners, we are left with a thermal density matrix. PACS numbers: 04.20.-q, 04.70.-s, 73.43.-f", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "There are several apparently different explanations for the origin of black hole radiation. In his original account [1] Hawking kept track of what one means by a 'particle' as a wavefunction propagates in the background geometry. A field theory derivation using the trace anomaly in the energy momentum tensor was given given by Christensen and Fulling [2], and more recently Robinson and Wilczek [3] and others [4, 5] have applied the twodimensional gravitational anomaly in the region near the horizon. Yet another route obtains the Hawking radiation from quantum tunnelling across the horizon [6]. (For a review of the tunnelling approach see [7].) Given these alternative derivations, it is reasonable to ask just what is required for an event horizon to emit thermal radiation. Is gravity really necessary? This question has led to the study of analogues of black holes and event horizons in other areas of wave propagation. The first such analogue was the acoustic black hole proposed by Unruh, who discovered that the wave equation for sound in a background fluid flow was equivalent to the wave equation for a scalar field in a curved space-time [8]. The subject has now developed extensively, with gravity and Hawking radiation analogues being proposed and constructed in quantum-fluids, optics, and solid-state devices. For review with an extensive list of references see [9]. The present paper proposes a conceptually simple, and possibly experimentally realizable, condensed matter model of quantum mode propagation in which an event horizon emits thermal radiation. The analogue space-time is flat, but consists of two causally disconnected halves. It is therefore a member of the general class of condensed-matter event horizons discussed by Volovik in [10]. Our model exploits the intepretation of the edge-modes of a filling fraction ν = 1 quantum Hall system as a massless chiral Dirac fermion whose local 'speed of light' is determined by the potential that confines the Hall fluid, and is therefore subject to external control. In the next section we describe the model in the language of first-quantized tunnelling. In the third section we adopt a second-quantized formalism so as to obtain a Bogoliubov transformation between two natural bases for the system. This allows us to display the physical 'vacuum' as a coherent superposition of particle-hole pairs that are entangled across the horizon. Just as the Minkowski pure-state vacuum is a thermal mixed state when seen by a Rindler co-ordinate observer [11, 12], our pure-state vacuum appears thermal when we trace out the 'unobservable' over-the-horizon member of each pair.", "pages": [ 2, 3 ] }, { "title": "II. LOWEST LANDAU LEVEL", "content": "We model our black hole as a two-dimensional electron gas (2DEG) in the ν = 1 quantum Hall phase. We arrange for the boundary of the 2DEG to lie along the y axis, with the region x < 0 occupied by the gas, and the region x > 0 empty. Now assume that we have engineered the 'confining' potential V to be of the form and have chosen the the direction of the perpendicular magnetic field B so that the classical guiding-centre drift velocity is The electrons move along equipotentials V ( x, y ) = E , which, for this potential, are rectangular hyperbolæ that have the x and y axes as asymptotes. In particular, the electrons at the edge of our 2DEG move vertically along the y axis at velocity This velocity is our analogue of the local speed of light. The edge modes in the regions y > 0 and y < 0 move in opposite directions, and so these two regions are causally disconnected. They are separated by an event horizon at y = 0. The price we pay for the event horizon is that the electrons in the occupied region with y < 0 (the interior of the black hole) are in a state of population inversion. In the absence of the magnetic field the electrons would rapidly fall into one of the the lower energy quadrants. Because of the strong field, however, and in the absence of inelastic or tunnelling processes, they are constrained to stay on their hyperbolic classical orbits. The inherent instability of the 'vacuum' in the black hole interior corresponds to the observation of Parikh and Wilczek [6] that a black hole must be thought of as highly excited quantum state. Except for the case E = 0, each of the classical equipotentials λxy = E consists of two disconnected branches and intitally all the particles lie on only one of these branches. The branches for small E approach each other near the origin. There is therefore a non-zero amplitude for a particle to tunnel from one branch to the other of the same energy. This tunnelling leads to electrons and holes being emitted from the event horizon. To calculate the tunneling amplitude, we will assume that the magnetic field is large enough that we can ignore all Landau levels except the lowest. The lowest Landau level (LLL) approximation is very natural as it is this situation that the excitations near the edge of a quantum Hall droplet can be identified with those of a 1+1 dimensional chiral fermion with Hamiltonian In this picture the 2DEG itself is the filled Dirac sea. We chose the symmetric gauge in which the LLL wave-functions are of the form where z = x + iy . All quantum information resides in the holomorphic factor ψ ( z ), and we will refer to this factor as the LLL 'wave-function.' We therefore regard the LLL Hilbert space as a Bargmann-Fock space of finite-norm holomorphic functions with inner product Bear in mind however that the LLL wavefunction should be multiplied by exp { -1 4 eB | z | 2 } before plotting probability densities or computing currents. The action of z on the LLL wavefunction is by simple multiplication, but multiplication by z takes us out of the space of holomorphic functions. The LLL operator corresponding to z becomes instead z † , where the adjoint is taken with respect to the Bargmann-Fock inner product. This identification makes For our potential the first-quantized eigenvalue problem therefore becomes Only the potential appears in the this equation as the LLL wavefunctions are annihilated by the electron kinetic energy operator. A rescaling gives us a standard form of Weber's equation (See [13] § 16.5, or [14] chapter 19.): with For simplicity we now set λ = eB = 1. We can always restore the general parameters by scaling the units of length and energy. If ϕ ( glyph[epsilon1], z ) is a solution of then so are ϕ ( glyph[epsilon1], -z ), ϕ ( -glyph[epsilon1], iz ) and ϕ ( -glyph[epsilon1], -iz ). At most two of these solutions can be linearly independent. A fundamental pair of independent solutions is Here 1 F 1 ( a, b, z ) is the confluent hypergeometric function. These functions are even and odd, respectively, under z ↔ -z . After multiplication by exp {-| z | 2 / 4 } the resulting wavefunctions are localized on the semiclassical orbits, which form the two disconnected branches of the rectangular hyperbola xy = glyph[epsilon1] (see figure 2). These solutions to the LLL potential have been studied in connection with Riemann hypothesis [15]. More useful to us is the solution of (13) given by the parabolic cylinder function Here D n ( z ) is Whittaker and Watson's notation for their parabolic cylinder function [13] , and U n ( z ) is the now more common notation used by Abramowitz and Stegun [14]. The essential properties of U n ( z ) are that it is an entire function, and that it decays rapidly as x → + ∞ for any real or complex n . The solution U -iglyph[epsilon1] ( z ) describes particles moving in from the left (the occupied region) in the lower left quadrant if glyph[epsilon1] > 0 and the upper left quadrant if glyph[epsilon1] < 0. They mostly remain in that quadrant, but there is some probability of tunnelling to the other branch of the hyperbola (see figure 3). If glyph[epsilon1] > 0 the result is that a tunnelled positive energy particle is emitted by the black hole, leaving a negative energy hole ( i.e. the absence of positive energy particle) inside the event horizon. If glyph[epsilon1] < 0 then a positive energy hole (the absence of a negative energy particle) is emitted by the black hole leaving a negative energy particle inside the event horizon. Along with the solution U -iglyph[epsilon1] ( z ) we have the solutions U -iglyph[epsilon1] ( -z ) and U iglyph[epsilon1] ( iz ) and U iglyph[epsilon1] ( -iz ). We will find use for all of these solutions, as they describe motion with different boundary conditions (see figures 4 and 5). To discover the relative amplitudes of the direct and tunnelled waves we can use the asymptotic expansion Near the y axis this reduces to The ratio of tunneled to direct amplitude is therefore exactly exp {-πglyph[epsilon1] } . We can conform this result by using the identity together with the fact that U -iglyph[epsilon1] ( z ) tends rapidly to zero in the right half-plane. Thus, if R is positive and The direct and tunneling amplitudes therefore have magnitude Note that | d ( glyph[epsilon1] ) | 2 + | t ( glyph[epsilon1] ) | 2 = 1 because Γ( z )Γ(1 -z ) = π cosec ( πz ) gives us The occupation probability of an outgoing particle or hole state with energy glyph[epsilon1] is therefore The chiral edge states emerging from the event horizon are therefore thermal with or once we restore parameters and units. Comparison of this with the usual Hawking radiation formula indicates that the 'surface gravity' κ of our analogue black hole is the edge-velocity acceleration .", "pages": [ 3, 4, 5, 6, 7, 8, 9 ] }, { "title": "III. SECOND QUANTIZATION, MODE EXPANSIONS, AND A BOGOLIUBOV TRANFORMATION", "content": "The space of LLL functions (5) does not contain the delta function. Its place is taken by a reproducing kernel If ψ ( x, y ) is of the form (5) then In particular { x 0 , y 0 | x 1 , y 1 } , considered as a function of ( x 1 , y 1 ), is of this form, so the kernel reproduces itself: When we expand out the second quantized LLL field operator in terms of the discrete set of normalized eigenmodes z n / √ 2 π 2 n n ! for the potential we find Here the operators ̂ a n obey and the usual canonical anticommutation relation { ̂ ψ † ( x ) , ̂ ψ ( x ' ) } = δ ( x -x ' ) for the field is replaced by If we retain only the holomorphic factors, then we have We can also expand in a continuous set of continuous set of eigenfunctions. For example, we can make use of energy E eigenfunctions for the potential V ( x, y ) = x . These are They have been normalized so that The holomorphic field operator is then with We easily confirm that (38) still satisfies (35). Similarly, we can expand the LLL field operator in terms of complete sets of parabolic cylinder functions. There are two distinct ways of doing this. Begin by defining The label 'in' designates that these wave-functions describe states that have a simple description prior to their opportunity for tunnelling (see figure 4) . These 'in' functions have been normalized so that and they obey the LLL completeness relation (Both normalization and completeness are easily established from the integral expression in the last line of (15).) Then we can set The 'in' vacuum is the appropriate many-body state for our initial conditions. It is characterized physically by the condition that no particle is approaching the 2DEG from the empty single-particle states to the right, and that all the single-particle states incoming from the left are occupied. It is characterized mathematically by the conditions The second set of functions is They are also orthogonal and obey the LLL completeness relation The labels 'ext' and 'int' indicate that the functions live mostly in the exterior ( y > 0) and interior ( y < 0) of the black hole. They decay rapidly in the other region (see figure 5). In terms of these new functions we have The 'out' operators ̂ a (out ,α ) glyph[epsilon1] and ( ̂ a (out ,α ) glyph[epsilon1] ) † create and annihilate particles that are simply described as excitations over the asymptotic na¨ıve vacuum in which every state in the region x < 0 is filled and every state in x > 0 is empty. For glyph[epsilon1] > 0 the operator ̂ a (out , ext) glyph[epsilon1] annihilates a positive energy particle in the asymptotic region y glyph[greatermuch] 0 outside the black hole. For glyph[epsilon1] < 0 it annihilates a particle in the 2DEG and so creates a positive energy hole in the same region. For the ̂ a (out , int) glyph[epsilon1] that act on states within the black hole the roles of hole creation and particle annihilation are reversed as the 2DEG consists of particles with positive energy. To stress the causally disconnected character of the interior and exterior regions, we will write 'out' vacuum as with and Comparing the two expressions for ̂ ψ ( z ) gives us the Bogoliubov transformation Similarly From the Bolgoluibov transformation and the mathematical characterization of | 0 , in 〉 we find that where N is a normalization factor. We have therefore exhibited the physical ground state as a sea of particle-hole pairs correlated between the interior and exterior regions. We now have the same formal situation as described in [12]. If we trace out the 'unobservable' interior of the black hole, we end up with density matrix is of the form where i labels the many-body state whose energy is glyph[epsilon1] ( i ). However, unlike the situation in the Unruh-Rindler vacuum [11, 12] our system contains genuine radiation rather that a thermal bath. This is because the chiral character of the particles means that they can only flow outwards.", "pages": [ 9, 10, 11, 12, 13, 14 ] }, { "title": "IV. DISCUSSION", "content": "The effective space-time metric in which the chiral edge-mode fermions move is The quantization of chiral fermions in such a background metric with general v edge ( y ) has been carried out in [16], although these authors did not consider the effect of an event horizon. In our case v edge = κy , κ = λ/eB , and a change to an exterior tortoise co-ordinate y ∗ = κ -1 ln( y ) in (60) leads to The new coordinates reveal that our space-time is flat, but the singularity at the horizon is not removed. It has been pushed to y ∗ = -∞ , and the interior of the black hole has become invisible. A superfluid system with this metric and event horizon was studied by Volovik in [10]. He uses a WKB analytic continuation method to compute the Bogoliubov coefficients, and finds the same Hawking temperature as our present calculation, but his non-chiral system has no actual radiation. The agreement in the temperature is perhaps not surprising. It must be obvious from looking at the classical trajectories of our particles that there is some connection between our 2DEG problem and that of Landau-Zener tunneling through an avoided level crossing. Indeed, although the physics is superficially different, the Landau-Zener time-dependent Schrodinger equation is solved using the same families of parabolic cylinder functions that we have used [17], and it is well known that an analytically continued form of the WKB approximation obtains the correct asymptotic Landau-Zener tunnelling probabilities [18]. The most remarkable property of the present model is that the emitted radiation is exactly thermal. There is no immediately obvious reason why the mathematical properties of the parabolic cylinder functions should lead to this result. In a real black hole the emitted radiation is modified by grey-body factors in dimensions greater than two, but that the hole can only be in equilibrium with radiation at T Hawking follows from the geometry of the Euclidean section of space-time being asymptotically periodic in imaginary time [19]. Does our space-time geometry tacitly force a Euclidean temporal periodicity? We can write so, up to a conformal factor κ -2 y -2 , the metric is that of Rindler space whose Euclidean section t ↦→ iτ has metric The absence of a conical singularity at y = 0 in the manifold described by (63) requires identifying κτ ∼ κτ +2 π and so implies a temperature T = glyph[planckover2pi1] κ/ 2 π -which is exactly what the tunneling calculation gives. However, given that it blows up at the point of interest, it seems unreasonable to ignore the conformal factor, making this argument at most suggestive.", "pages": [ 14, 15 ] }, { "title": "V. ACKNOWLEDGEMENTS", "content": "This project was supported by the National Science Foundation under grant DMR 0903291. I would like to thank Ted Jacobson for comments, and for drawing my attention to reference [10].", "pages": [ 15, 16 ] } ]

[ { "title": "Adrian P Gentle", "content": "Department of Mathematics, University of Southern Indiana, Evansville, IN 47712 E-mail: [email protected] Abstract. We revisit the Regge calculus model of the Kasner cosmology first considered by S. Lewis. One of the most highly symmetric applications of lattice gravity in the literature, Lewis' discrete model closely matched the degrees of freedom of the Kasner cosmology. As such, it was surprising that Lewis was unable to obtain the full set of Kasner-Einstein equations in the continuum limit. Indeed, an averaging procedure was required to ensure that the lattice equations were even consistent with the exact solution in this limit. We correct Lewis' calculations and show that the resulting Regge model converges quickly to the full set of Kasner-Einstein equations in the limit of very fine discretization. Numerical solutions to the discrete and continuoustime lattice equations are also considered. AMS classification scheme numbers: 83C27 PACS numbers: 04.20.-q, 04.25.Dm", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The discrete formulation of gravity proposed by T. Regge in 1961 [1] has been deployed in a wide variety of settings, from probing the foundations of gravity and the quantum realm [2, 3] to numerical studies of classical gravitating systems [2, 4]. Regge calculus continues to be used in new and diverse ways; recent examples include Ricci flow [5] and as an explanation of dark energy [6]. In this paper we re-examine the Regge calculus model of the vacuum Kasner cosmology first considered by Lewis [7], with the goal of gaining insight into the continuum limit of this discrete approach to gravity. In a general setting the structural differences between a continuous manifold and a discrete simplicial lattice lead to difficulties in directly comparing the Regge and Einstein equations or their solutions, with a single Regge equation per edge in the lattice compared with ten Einstein equations per event in spacetime. We expect many more simplicial equations than Einstein equations in a general simulation, and some form of averaging must be expected before the equations (or their solutions) can be compared. Lewis [7] studied both the Kasner and spatially flat Friedmann-LemaˆıtreRobertson-Walker (FLRW) cosmologies using a regular hypercubic lattice. We only consider the Kasner solution in this paper, where the high degree of symmetry, without the added complication of matter, allows explicit examination of the Regge equations in the continuum limit. By aligning the degrees of freedom of the lattice with the continuum metric components, Lewis was able to avoid the issue of averaging and make direct comparisons between the Regge equations and the Kasner-Einstein equations in the continuum limit. Unfortunately, Lewis was only able to recover one of the four Einstein equations in this limit, and even this was only possible after the equations were carefully averaged [7]. Without this averaging it is not clear that the equations obtained by Lewis actually represent the Kasner cosmology. We show that Lewis neglected a vital portion of the simplicial curvature arising from the two-dimensional spacelike lattice faces that lie on constant time hypersurfaces. The Kasner cosmology has zero intrinsic curvature on constant time hypersurfaces, so the lattice curvature concentrated on spacelike faces - measured on a plane with signature '-+' orthogonal to the face - make an important contribution to the total lattice curvature. We show that when these curvature terms are included, the discrete equations exactly reproduce the Kasner-Einstein equations in the limit of very fine triangulations without the need to average. In addition to reconsidering Lewis' analytic work on the spatially flat, anisotropic Kasner cosmology [7], we construct numerical solutions to the discrete and continuoustime lattice equations. This builds on the previous work of Collins and Williams [8] and Brewin [9] on highly symmetric, continuous time, closed FLRW cosmologies, and the (3+1)-dimensional numerical study of the Kasner cosmology by Gentle [10] with discrete time and coarse spatial resolution. We begin by briefly describing the continuum Kasner solution.", "pages": [ 1, 2 ] }, { "title": "2. The Kasner cosmology", "content": "The Kasner solution [11] is a vacuum, homogeneous, anisotropic cosmological solution of the Einstein equations with topology R × T 3 . Appropriate slicing creates flat spacelike hypersurfaces, while the global topology allows non-trivial vacuum solutions of the Einstein equations. The Kasner metric may be written in the form [12] where the functions f , g and h are determined by the vacuum Einstein equations Note that the equation G tt = 0 is a first integral of the remaining equations. The Kasner metric components are where the Kasner exponents p i are unknown constants. With this choice of metric functions the vacuum Kasner-Einstein equations reduce to two algebraic constraints, leaving a one parameter family of Kasner cosmologies. The Kasner solutions are the basis of the Mixmaster cosmologies [13], which may be regarded as a series of Kasner-like epochs undergoing an infinite series of 'bounces' from one set of Kasner exponents to the next. It is conjectured that these asymptotic velocity term dominated models embody the generic approach to singularity in crunch cosmologies, and it has been shown that the bounces represent a chaotic map on the Kasner exponents [14, 15].", "pages": [ 2, 3 ] }, { "title": "3. A homogeneous, anisotropic spacetime lattice", "content": "We follow Lewis and build a discrete approximation of the Kasner spacetime using a highly symmetric lattice of rectangular prisms. The regularity of the lattice implements homogeneity, while the rectangular prisms allow a degree of anisotropy. The complete four-geometry is constructed by extruding the initial three-geometry forward in time and filling the interior with four-dimensional prisms. Each flat T 3 hypersurface consists of a single rectangular prism with volume x i y i z i , where opposing faces are identified to give the global topology. This prism is subdivided into n 3 regular prisms with edge lengths u i , v i and w i , with and where the subscript i labels the Cauchy surface at time t i . The three-geometry is joined to a similar structure at time t = t i +1 = t i +∆ t i , where the prisms have edge lengths u i +1 , v i +1 and w i +1 . This structure is shown in figure 1. Time evolution of the initial surface maintains homogeneity, so within the worldtube of each prism there exists a local freely-falling inertial frame. In the coordinates of this frame the coordinates of vertex A in figure 1 can be written as and similarly, the coordinates for vertex A + (the counterpart of A on the next hypersurface) are The spacetime interval along the time-like edge joining A and A + is defined to be m 2 i < 0, and thus where the difference operator is defined as ∆ l i = l i +1 -l i . Note that the requirement that m 2 i < 0 implies a restriction on the choice of ∆ t i for a given value of the 'resolution' parameter n . Homogeneity guarantees that identical expressions hold for all timelike edges joining the spacelike hypersurfaces labeled t i and t i +1 . The discrete spacetime curvature in Regge calculus is manifest on the twodimensional faces on which three-dimensional blocks hinge [1], and is represented by the angle deficit (the difference from the flat space value) measured in the plane orthogonal to the face. There are two distinct classes of two-dimensional faces, or hinges, in the lattice: timelike areas formed by evolving a spacelike edge forward to the next hypersurface, and the rectangular faces of the prisms on t = constant slices. The timelike trapezoids formed when the spatial edge u i is carried forward in time from the hypersurface labeled t i to t i +1 , face ABA + B + in figure 1, has area which is real since m 2 j < 0. Likewise, the spacelike hinge ABCD shown in figure 1, consisting of the edges u i and v i , has area with the other spacelike and timelike hinges defined similarly. Turning now to the curvature about these faces, we note that there are four distinct three-dimensional prisms which hinge on the timelike face ABA + B + . Each of these is formed by dragging one of the two-dimensional faces containing the edge AB forward in time. This includes the prisms ABCDA + B + C + D + and ABEFA + B + E + F + . The remaining two prisms hinging on ABA + B + are not shown in figure 1. The homogeneity of the lattice ensures that the four hyper-dihedral angles that surround the hinge ABA + B + are the same. Denoting each of these angles as θ xt i , the angle defect (or deficit angle) about the timelike face ABA + B + is which measures the deviation of the total angle from the flat-space value of 2 π . The hyper-dihedral angle θ xt i is with analogous definitions for θ yt i and θ zt i , the hyper-dihedral angles about the remaining classes of spacelike hinge. To measure the deficit angles about the spacelike faces, consider the curvature /epsilon1 xy i about the hinge ABCD . Since this hinge is spacelike, the hyper-dihedral angles are boosts in the plane with signature -+ orthogonal to the hinge. The deficit angle for a spacelike hinge is [16] where the summation is over all boosts φ k which surround the hinge. The boost between the two three-dimensional prisms which hinge on ABCD , namely ABCDA + B + C + D + and ABCDEFGH , is √ and the hinge ABCD is surrounded by four such boosts, two identical boosts above and two below. Thus the final deficit angle measured about ABCD is Further details on calculating the deficit angles about a spacelike hinge are contained in a recent paper by Brewin [16]. We note that this type of hinge was not included in the calculations of Lewis [7], leading to errors in the resulting Regge equations. We return to this issue below.", "pages": [ 3, 4, 5 ] }, { "title": "4. The Regge calculus model", "content": "The vacuum Regge equations take the form [1] where the sum is over all triangles t with area A t that contain the edge L j , and /epsilon1 t the deficit angle about triangle t . Each edge in the lattice yields a single Regge equation, but the homogeneity and anisotropy of our model imply that there is one distinct equation for each of the four classes of edge in the lattice. The Regge equation associated with an individual timelike edge m 2 i involves six timelike lattice faces, and has the form which is the discrete counterpart of (1), the Einstein field equation G tt = 0. Similarly, the Regge equation which corresponds to the single spatial edge u i involves six lattice faces: two spacelike faces with area u i v i , two with area u i w i , plus the timelike faces formed by evolving u i both forwards and backwards in time. The vacuum Regge equation is with similar expressions for the equations arising from the spacelike edges v i and w i . Using the geometric information collected in section 3 we obtain a single Regge equation for each class of edge on the i th hypersurface, namely The structure of the Regge equations (7)-(10) is worth considering. Equation (7), associated with the timelike edge m 2 i , involves edges on, and between, two neighbouring hypersurfaces, whereas (8)-(10) involve information on and between three consecutive spatial hypersurfaces. Thus (7) is a first-order constraint, while (8)-(10) are second-order difference equations. This contrasts sharply with the equations derived by Lewis [7], who neglected the curvature associated with spacelike hinges, and was thus unable to derive the spatial equations (8)-(10). After correctly obtaining (7), Lewis found that he could only make sense of the truncated spatial equations by considering a careful average. This averaging resulted, once again, in the timelike equation (7). Without the spatial curvature terms /epsilon1 yz i , /epsilon1 xz i and /epsilon1 xy i Lewis was unable to build the second-order Regge equations (8)-(10). Before examining the continuum limit of the Regge model we consider solutions of the discrete equations. Initial data is constructed at t 0 = 1 to match the continuum Kasner solution as far as possible. Taking the exact initial data to be x (1) = y (1) = z (1) = 1 and ˙ x (1) = p 1 , ˙ y (1) = p 2 , and ˙ z (1) = p 3 , we mimic the properties of the exact solution in the lattice by setting u 0 = v 0 = v 0 = 1 and where α is an unknown parameter. This form is chosen to maintain the continuum condition ˙ x (1)+ ˙ y (1)+ ˙ z (1) = 1 to first order, which is physically equivalent to using the degrees of freedom in the initial data to generate linear expansion in the volume element. With these initial data the Regge constraint (7) is solved for the single parameter α . A typical solution of the discrete Regge equations is shown in figure 2 for the case p 1 = 0 . 75 and ∆ t = 0 . 01. The solution to the initial value problem in this case is α = 0 . 0214379, which represents a roughly 3% change in the initial rate of change of u i compared to the exact estimate. The evolution of the initial data is shown in figure 2a, while 2b shows the evolution of the fractional error in the Regge solutions compared with their exact counterparts. The fractional error in all edges remains in the 5% -10% range, and can be shown to shrink as the time step is reduced. The residual error in the first-order Regge constraint (7) is defined as √ √ √ and is a measure of the consistency amongst the Regge equations. Figure 2c shows R t as a function of time with ∆ t = 0 . 01, and clearly the residual R t remains small throughout the evolution. We repeated this process with different ∆ t to estimate the rate at which R t reduces as ∆ t tends to zero. Figure 2d shows second-order convergence in the mean value of R t over 1 < t < 100 as the timestep is reduced. We explore the issue of convergence in more detail in the following sections.", "pages": [ 5, 6, 7 ] }, { "title": "5. The continuous-time Regge model", "content": "Many of the early applications of Regge calculus to highly symmetric spacetimes considered the differential equations that arise in the limit of continuous time and discrete space [7, 8, 9]. In this section we derive the continuous time Regge equations and compare them with the results of Lewis [7] before considering numerical solutions. The continuous-time Regge model is developed from the discrete equations in section 4 in the limit of small ∆ t , with the assumption that spatial edges in the lattice approach continuous functions of time. For example, the spacelike edge u i is viewed as the value of a continuous function u ( t ) evaluated at t = t i . A power series expansion then relates the edge lengths on neighbouring surfaces, where all derivatives are evaluated at t = t i . Similar expressions hold for the edges u i -1 , v i +1 etc. The series expansion for the timelike edges m 2 i is obtained from (6). In the continuum limit the deficit angle about the spacelike area A xy i is and the deficit angle about the timelike face formed by the evolution of u i is with similar expressions for the remaining faces. Using these expansions, the Regge equations (7)-(10) are, to leading order, where /epsilon1 0 xt , /epsilon1 0 yt and /epsilon1 0 zt denote the zeroth order terms in the deficit angle expansions. The leading terms in (13)-(16) are the continuous-time Regge equations, a set of non-linear differential equations. Equation (13) is a first-order differential equation, while the remainder are second-order. The continuous-time Regge differential equations were solved numerically using Mathematica . Initial conditions were applied at t 0 = 1, and the initial data was chosen to ensure that the expansion of lattice volume elements is initially linear to match the exact solution in section 2. Once again introducing a parameter α , we set and use the first-order Regge initial value equation (13) to solve for α . This mimics the exact initial data, for which α = 0. The second-order, non-linear differential equations (14)-(16) are used to evolve the initial data forward in time. Figure 3 shows solutions to the continuous-time Regge equations with p 1 = 0 . 5. The solution to the initial value problem is α = 0 . 0204161, which represents a small deviation ( ≈ 4% change in the initial value of ˙ u ) from the exact initial data. As can be seen in the figure, the continuous-time Regge solutions are very similar to the exact Einstein solution, with the Regge edges deviating from the exact values by 5% -7%. In the next section we extend the limiting process to the spatial edges, and examine the difference between the exact and Regge equations more carefully.", "pages": [ 7, 8, 9 ] }, { "title": "6. The Regge equations in the limit of continuous space and time", "content": "In this section we explore the discrepancies between the discrete Regge model and the Kasner spacetime by examining the truncation error incurred when the Regge equations are viewed as approximations of the Kasner-Einstein equations (1)-(4). We consider the continuum limit of the Regge equations (7)-(10) as the spatial lattice is refined in both space and time. The continuous space and time limit of the Regge equations is obtained from the temporal series expansions (12) together with the link between lattice edges and the global length scales given by (5). Substituting these into the discrete Regge equations (7)-(10) and simultaneously increasing the number of prisms ( n → ∞ ) while reducing the timestep (∆ t → 0) we obtain a series expansion for the the discrete equations in the continuum limit. The refinement parameter n and timestep ∆ t are chosen so that m 2 j < 0. The series expansion of the temporal Regge equation (7) is which has truncation error that is first-order in the timestep ∆ t and second-order in the spatial discretization scale 1 /n . The spatial Regge equations (8)-(10) are to leading order in the continuum limit. The truncation error in these equations is second order in both the spatial and temporal discretization scales. The leading order terms in the continuous time and space Regge equations (17)(20) are identical to the Kasner-Einstein equations (1)-(4), so we expect solutions of the discrete lattice equations to approach the continuum solutions as length scales in the lattice are reduced. It is clear from the preceding equations that the truncation error for the Regge equations, when viewed as approximations to the Kasner-Einstein equations, are second order in the spatial discretization scale 1 /n . The truncation error is also second order in ∆ t for the spatial Regge equations (18)-(20). The truncation error in (17) implies that the Regge initial value equation (7) is only a first-order approximation to its continuum counterpart. This conflicts with the calculations in section 4, where the truncation error in the Regge constraint R t was found to converge to zero as the second power of ∆ t (see figure 2d). To understand this contradiction, we rewrite the coefficients of ∆ t in the expansion (17) as where the final equality follows from substitution of (18)-(20). Thus the truncation error in the Regge constraint (17) is formally of order ∆ t , but the coefficient of that term in the expansion is a linear combination of the spatial Kasner-Einstein equations. This should be zero to leading order for any solution of the spatial Regge equations. To clarify this argument, consider again the simulation in section 4. Once the initial data is set, the numerical evolution is achieved by the repeated solution of the spatial Regge equations (8)-(10). These are shown above to be second order accurate approximations of the Einstein equations in both space and time. We expect from (17) that the leading order error in the Regge constraint is first order in ∆ t . However, the coefficient of that error term is a linear combination of the Regge equations we are solving (to leading order), and thus the coefficient of ∆ t is itself zero to second order in ∆ t . Thus the effective leading order truncation in the Regge constraint equation (17) is of second order in both space and time. This is consistent with the numerical experiments in section 4.", "pages": [ 9, 10, 11 ] }, { "title": "7. Discussion", "content": "In the preceding sections we re-examined one of the most highly symmetric applications of Regge calculus to be found in the literature. The primary goal of this study was to examine the convergence properties of Regge calculus, and we have shown that for the discrete Kasner spacetime the equations of Regge calculus reduce identically to the corresponding Einstein equations in the continuum limit. The discrete lattice used by Lewis, outlined above, was specifically designed to guarantee a one-to-one correspondence between the degrees of freedom in the Regge lattice and the metric components in the continuum solution [7]. Despite this, Lewis needed to average the Regge evolution equations in order to obtain consistency in the continuum limit. The averaging process was chosen to obtain the first order Regge equation (7), but Lewis was still unable to derive the remaining spatial equations (8)(10). We showed in section 4 that once all lattice curvature elements are included in the calculations the Regge equations consist of one constraint and three evolution equations. In section 6 we showed that these lattice equations approach the full set of Kasner-Einstein equations in the continuum limit. It was also shown in section 6 that the discrete Regge equations are second order accurate approximations to the Einstein equations for the Kasner cosmology in the limit of very fine discretization. This convergence rate is in agreement with many previous numerical simulations, in particular the (3+1)-dimensional Regge calculus models of the Kasner cosmology that utilized general simplicial lattices [10, 19]. Unlike the current analysis, these simulations did not enforce the homogeneity and anisotropy of the Kasner model throughout the evolution, yet displayed second-order convergence to the continuum solution. These numerical simulations considered the convergence of solutions, rather than equations, and so are complementary to our analysis. In general applications of Regge calculus that utilize a simplicial lattice there will be many more Regge equations (one per lattice edge) than Einstein equations (10 per spacetime event). The direct comparison between individual Regge and continuum equations considered in this paper would not be possible, or even desirable. We expect that an appropriate average of the Regge equations would still correspond to the Einstein equations in the continuum limit [17, 18, 20], and several such averaging schemes have been suggested. Brewin considered a finite-element integration of the weak-field Einstein equations over a simplicial lattice (discussed in [18]), and suggested that the vertex-based equivalent of the vacuum Einstein equations are where ∆ x µ j is a vector oriented along edge L j in a coordinate system based at vertex v . The outer summation is over all triangles which meet at v , and the inner sum is over the edges on each triangle. These are essentially linear combinations of Regge equations, together with boundary terms [18]. Regardless of how one compares the continuum and discrete equations, it is ultimately the solutions that are of interest. The application of Regge calculus to the Kasner cosmology discussed in this paper demonstrates yet again that Regge calculus is a consistent second-order accurate discretization of general relativity, providing further support for the use of lattice gravity in numerical relativity and discrete quantum gravity.", "pages": [ 11, 12 ] } ]

[ { "title": "R. MILSON, L. WYLLEMAN", "content": "Abstract. We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of the Newman-Penrose formalism, and spinorial classification of the three-dimensional Ricci tensor.", "pages": [ 1 ] }, { "title": "1. Introduction and main result", "content": "We report on recent progress concerning the invariant classification problem for three-dimensional Lorentzian geometries. In a physical context, such geometries arise as exact solutions of three-dimensional theories of gravity, such as Topologically Massive Gravity (TMG), New Massive Gravity (NMG) and extensions of those. We refer to [9] and the introduction of [1] for reviews of the relevant literature. In [9] it was stressed that, when surveying the literature of exact solutions, it is often difficult to disentangle genuinely new solutions from those that are already known but written in different coordinate systems. To tackle this problem one needs a coordinate invariant local characterization of the geometry. A first step is to use the algebraic classification of the Ricci tensor, as was done in [9] to classify all TMG solutions known at that time. A complete answer to the problem (in any dimension in principle) is provided by the CartanKarlhede algorithm [8, 15]. The key quantities used here are so-called Cartan invariants , which are components of the Riemann tensor and a finite number of its covariant derivatives, relative to some maximally fixed vector frame associated to these tensors. Regarding three-dimensional Lorentzian geometries, we will show in the present paper that cases where one needs the theoretically maximal number of five derivatives for a complete classification do exist, but are limited to the metrics given in our main Theorem 1 below. This implies that any three-dimensional geometric theory of gravity whose field equations exclude the metrics of Theorem 1 requires at most four covariant derivatives of the Riemann tensor for a complete local invariant classification of its exact solutions. In the remainder of this introduction, we will outline the general mathematical context and background for the main theorem. Let ( M,g ) be a smooth, n -dimensional pseudo-Riemannian manifold, and let ( V, η ) be a real inner-product space having the same dimension and signature as ( M,g ). Henceforth, we use η ab to raise and lower frame indices, which we denote by a, b, c = 1 , . . . , n . Let O ( η ) be the group of automorphisms of η , and let o ( η ) be the corresponding Lie algebra of anti self-dual transformations. An η -orthogonal coframe is an inner-product isomorphism Let π : O ( η, M ) → M denote the principal O ( η )-bundle of all such. An η -orthogonal moving coframe is a local section of this bundle, or equivalently, a collection of 1forms ω a such that Set and let ˆ R ( p ) : O ( η, M ) →R p be the canonical, O ( η )-equivariant map defined by where the right hand side denotes the lift of the Riemann curvature tensor and its first p covariant derivatives to O ( η, M ). The following definitions are adapted from [20, Definitions 8.14 and 8.18]. Set r -1 = 0, and let r p denote the rank of ˆ R ( p ) , p = 0 , 1 , 2 , . . . . We say that ( M,g ) is fully regular if r p is constant for all p . Henceforth we assume that full regularity holds and let q = q M be the smallest integer such that r q -1 = r q . The integer q -1 is called the order of the metric [20, 26]. It can be shown[20, Theorem 12.11] that a fully regular metric of order q -1 is classified by ˆ R ( q ) , that is by q th-order differential invariants. The maximal order of a pseudo-Riemannian manifold, of fixed dimension and signature, is of particular interest. Cartan [8] established the upper bound Karlhede [15] improved Cartan's bound to where s 0 is the dimension of the automorphism group of the curvature tensor. The question of maximal order has received considerable attention in general relativity ( n = 4, Lorentzian signature) [10, 14, 23]. In that context, Karlhede's bound is q ≤ 7; recently, this bound was shown to be sharp [18]. The 4-dimensional metrics of maximal order describe a well-defined class of type N spacetimes with aligned null-radiation in an anti-deSitter background [21]. By contrast, Karlhede's bound in the generic Petrov type I case (for which s 0 = 0) is q ≤ 5, but at present we only have an example of a type I dust solution [29] with q = 3. In this paper, we investigate and classify 3-dimensional Lorentzian manifolds of maximal order. Our approach is grounded in Karlhede's refinement of the Cartan equivalence method [22], which is based on the notion of curvature normalization [15, 26]. A non-zero three-dimensional curvature tensor has vanishing Weyl part and is thus represented by its Ricci tensor, which may be regarded as a selfadjoint operator on the three-dimensional tangent space. Generically, the Ricci operator has a finite automorphism group (whence s 0 = 0). However, if two eigenvalues coincide or if the trace-free part of the operator is nilpotent, then s 0 = 1 is possible. Therefore, in the three-dimensional Lorentzian setting, Karlhede's bound is q ≤ 5 [25]. The question then becomes: Does there exist a 4th order, 3-dimensional Lorentzian metric, that is to say, a metric that is classified by 5th-order differential invariants? In 3-dimensional Lorentzian geometry, it is useful to make use of the real spinor representation of the Lorentz group. Such a spinor approach provides one with a natural null vector frame formalism. Moreover, the Petrov-Penrose classification of the curvature spinor (which, in three dimensions, is equivalent to the null alignment classification of the Ricci tensor) leads to a slight refinement of the usual Ricci-Segre classification. This is summarized in the appendices. Karlhede's result, which we formulate as Theorem 5 below, tells us that a metric which is classified by 5th order invariants, if one exists, is restricted to Petrov type D, type DZ (like type D, but the doubly aligned null directions are complex) and type N geometries. Below, we rule out the type DZ and N possibilities, and demonstrate that the q = 5 bound is realized for one very particular class of type D metrics. Theorem 1. The order of a curvature-regular, 3-dimensional Lorentzian manifold is bounded by This bound is sharp; every 4th order metric is locally isometric to /negationslash Here x, u, w are local coordinates. C, T are real constants such that C +2 T 2 = 0 , and F ( u ) is an arbitrary real function such that /negationslash /negationslash /negationslash Note 1: In the singular subcase of T = 0, the expression (1 -e 4 Tw ) / (2 T ) should be interpreted in the limit sense as being equal to -2 w . /negationslash Note 2: the expression δ C denotes 1 if C = 0 and 0 if C = 0. Note 3: curvature regularity is a strengthening of the full-regularity assumption that we impose in order to exclude 'type-changing' metrics (see Definition 2 below). The structure of this paper is as follows. In Section 2 we revise the relevant definitions and theorems regarding curvature normalization, leading to Karlhede's bound within his approach to the equivalence problem. The concepts of curvature homogeneity and pseudo-stabilization turn out to be the crucial ideas in the search for metrics of maximal order. In particular, the maximal order metrics shown in (4) enjoy the CH 1 (curvature homogeneous of order 1) property. The relevant definitions are given in Section 3. We isolate the structure equations for the maximal order metrics in Section 4. We then prove the main Theorem 1 by integrating these equations in Section 5. Relevant background material is put in four appendices: a three-dimensional analogue of the Newman-Penrose formalism, the transformation rules of connection and curvature variables under basic Lorentz transformations, the Petrov-Penrose classification of the three-dimensional Ricci tensor, and the structure equations obeyed by a CH 1 metric.", "pages": [ 1, 2, 3 ] }, { "title": "2. Curvature normalization and Karlhede's bound", "content": "A general approach towards finding metrics of maximal order was described in [10] and [19]. The approach is based on two key ideas: (i) curvature normalization, also known as the Karlhede algorithm [15], and (ii) curvature homogeneity [24]. Normalization of the curvature tensor and its covariant derivatives, also known as the Karlhede algorithm, splits the rank of the classifying map ˆ R ( p ) into horizontal and vertical subranks and thereby simplifies the equivalence problem. As was already mentioned, the rank r p is the maximal number of functionally independent component functions ( ˆ R abcd , . . . , ˆ R abcd ; e 1 ...e p ), where the latter are functions of both position and frame variables. In order to speak of horizontal rank, we need to assume that the above tensors can be normalized. The horizontal rank (see Definition 3 below) can then be defined as the the maximal number of functionally independent component functions of normalized curvature and its covariant derivatives. Definition 2. We say that a submanifold S ⊂ R p is a p th order normalizing cross-section for ( M,g ) provided: /negationslash denotes the curvature components relative to the coframe in question. If there exists a normalizing cross-section S ⊂ R p for every p = 0 , 1 , 2 , . . . we say that ( M,g ) is curvature regular . Suppose that curvature regularity holds. Normalizing R ( p ) reduces the structure group of the equivalence problem from O ( η ) to G p . Because of N2, the maximally normalized components ( R abcd , . . . , R abcd ; e 1 ...e p ) are locally defined functions on the base M . These differential invariants, commonly referred to as p th order Cartan invariants , suffice to invariantly classify ( M,g ) and to solve the metric equivalence problem [26, Chapter 9]. Definition 3. Suppose that ( M,g ) is curvature regular. We define relative to some choice of normalizing cross-section. We refer to s p as the p th order degree of frame freedom, and to t p as the p th order horizontal rank. Proposition 4. If ( M,g ) is curvature regular, then s p , t p do not vary with x ∈ M and are independent of the choice of normalizing cross-section. Furthermore, and Theorem 5 (Karlhede, Theorem 4.1 of [15], see also Section 9.2 of [26]) . Let ( M,g ) be a fully regular, curvature regular n -dimensional pseudo-Riemannian manifold with isometry group K . Let r p , t p , s p be as defined above, and let q be the smallest integer such that r q -1 = r q . Then, q is also the smallest integer such that s q -1 = s q and t q -1 = t q . Furthermore, we have that G q -1 ⊂ O ( η ) is isomorphic to the isotropy subgroups K x ⊂ O ( T x M ) , x ∈ M ; that and that n -t q is equal to the dimension of the K -orbits. In particular, (10) implies that By the regularity assumption, Applying the above inequality with p = q -1 and using (12) gives the Karlhede bound (3) as an immediate corollary.", "pages": [ 4, 5 ] }, { "title": "3. Curvature homogeneity and pseudo-stabilization", "content": "Suppose that ( M,g ) is fully regular and curvature regular. The curvaturehomogeneity condition admits several equivalent definitions [4, 12], but with the above assumptions, the following definition is the most convenient. Definition 6. A manifold ( M,g ) is curvature-homogeneous of order k , or CH k for short, if it is curvature regular and if the horizontal rank t k = 0. If t k = 0 and t k +1 > 0, we say that ( M,g ) is properly CH k . To put it another way, a properly curvature homogeneous manifold of order k has constant Cartan invariants of order ≤ k , with a non-constant invariant appearing at order k + 1. The main application of the curvature homogeneous concept was the following theorem [24]. Theorem 7 (Singer) . A manifold ( M,g ) is locally homogeneous if and only if it is CH k for all k = 0 , 1 , 2 , . . . . In other words, a locally homogeneous space is characterized by the property of having constant Cartan invariants. As such, Singer's theorem is an immediate corollary of Theorem 5. In this paper we are interested in curvature homogeneity for a different, but related reason. As was shown in [18], curvature homogeneity is also a key concept in the search for maximal order metrics. The relevant observation is that for a CH k geometry the rank r k is small because t k = 0, and this is exactly what is needed for maximal order. Let us explain further in the context of 3-dimensional Lorentzian metrics. Definition 8. We say that a curvature regular geometry has k th order pseudostabilization provided s k = s k -1 > s q . Our notion of pseudo-stabilization is different but conceptually related to the notion employed in [20, Theorem 5.37]. Notice that a k th order pseudo-stable geometry has t k > t k -1 by theorem 5. Proposition 9. A 4th order, 3-dimensional, Lorentz geometry, if one exists, is either properly CH 1 or is properly CH 0 with 1st order pseudo-stabilization. Proof. Table 1 reveals s 0 ≤ 1 for a non-homogeneous geometry (see Proposition 10 below). Hence, r 0 ≥ 2, and hence a 4th order geometry requires the following rank sequence: This can be achieved in essentially two ways: either by which describes a properly CH 1 geometry, or by by three possible sequences with s q = 0 and starting with which describes a properly CH 0 geometry with 1st order pseudo-stabilization. /square In the following section, we rule out the pseudo-stabilization and type DZ, N scenarios and show that a 4th order requires a type D, properly CH 1 geometry. We then explicitly write down the necessary structure equations and integrate them. The end result is Theorem 1.", "pages": [ 5, 6 ] }, { "title": "4. The equivalence problem", "content": "In this section we derive the necessary and sufficient conditions for a 4th order metric. Table 1 of the appendix shows that s 0 > 0 for curvature types O, N, DZ, and D. Type O can be ruled out by Schur's theorem. A proof can be found in [28, Cor. 2.2.5 and 2.2.7]. Proposition 10. If the curvature is type O at all points x ∈ M , then M is a locally homogeneous space, i.e. t p = 0 for all p . We are left with the following possibilities. Proposition 11. A 4th order metric, if one exists, requires curvature of type N, DZ, or D. According to Proposition 9, each of the above 3 cases further splits into two subcases, according to whether the geometry is properly CH 1 or properly CH 0 with 1st order pseudo-stabilization. We consider the above possibilities in turn. Five of the possibilities can be ruled out, and this leaves a unique configuration for a 4th order metric. Since in a CH 0 geometry the 0th order components R abcd are constant, the 1st order components R abcd ; e are quadratic expressions of certain spin coefficients. Therefore, in the analysis that follows it is more convenient to specify the Cartan invariants in terms of spin coefficients and their frame derivatives. This methodology for constructing invariants is related to the notion of essential torsion in the Cartan equivalence method. The relevant details and definitions are given in Appendix D. 4.1. Type N configurations. Taking the curvature canonical form of Table 1 for this case, and assuming the CH 0 property, we have where ˜ R is a real constant. The group G 0 preserving (15) is generated by null rotations (135) about /lscript and the reflections (163), (166). The type N 1st order torsion matrix (see Appendix D for the derivation) is Substituting (15) into the Bianchi equations (116)-(118) yields the relations We now consider the CH 1 and pseudo-stable cases in turn. Proposition 12. A type N, properly CH 1 geometry has order bounded by q -1 ≤ 3 . Proof. By assumption, after the first-order torsion is normalized, /epsilon1, τ, α, γ are constants. Hence, by (107) - (111) By (137)-(142), τ and α are invariant under any null rotation about /lscript , while γ transforms like /negationslash Since t 0 = t 1 = 0 and s 0 = 1 by assumption, s 1 = 1 would lead to q -1 = 0. Thus we assume s 1 = 0 henceforth. By (20) this entails 2 α + τ = 0, and hence by (19). We impose the normalizations which leaves G 1 as the discrete group generated by (163). Then equation (113) implies that From (21), (23) and s 1 = 0 it follows that the 2nd order invariants are generated by ν . If ν is constant then t 2 = 0 and q -1 = 1. Thus we assume henceforth that ν is non-constant, i.e. t 2 = 1. The remaining structure equations (114) and (115) reduce to /negationslash where ˜ τ, ˜ α are constants such that 2˜ α +˜ τ > 0. Suppose then that ∆ ν is functionally independent from ν , and hence that t 3 = 2 (else t 3 = 1 and q -1 = 2). By the (N2) curvature regularity assumption we have ∆ ν = 0 at each point and we fully fix the frame by normalizing ∆ ν > 0 . Now the 3rd order invariants are generated by ν, ∆ ν . Applying (91) and (92) to ν gives Hence, the 4th order invariants are generated by ν, ∆ ν and ∆ 2 ν . Applying (91) to ∆ ν gives and hence Therefore t 4 = 2, which implies that the order is q -1 = 3. /square Proposition 13. The order of a type N, CH 0 , pseudo-stable geometry is bounded by q -1 ≤ 3 . Proof. Referring to (16), the assumption s 1 = 1 implies that the remaining torsion scalars /epsilon1, τ, α and γ are invariant under arbitrary null rotations about /lscript and thus generate the 1st order Cartan invariants. By (139)-(142) this implies Hence the 1st order invariants R abcd ; e are generated by τ, γ . By the pseudostabilization assumption R abcd ; e is G 0 -invariant, and hence, using the notation of Appendix D, It follows that the 2nd order components are linear combinations of Dγ,Dτ,δτ, ∆ τ, δγ, ∆ γ and quadratic polynomials of τ, γ . Since the latter are null-rotation invariant, the 2nd order Cartan invariants are obtained by normalizing the former. From here equations (107)-(111) reduce to If t 1 > 1 then q -1 ≤ 3 automatically, so we may assume t 1 = 1. This implies d τ ∧ d γ = 0, and in particular Dγδτ = 0. Hence, which implies that τ = ˜ τ is a constant. This leaves γ as the only generator of the 1st order invariants. The transformation law (135) gives At this point, we must consider two cases. Case (a): suppose that ˜ τ = 0 Hence, ∆ γ is null-rotation invariant, and hence is a Cartan invariant. By (113) δγ = 0 . Since γ, ∆ γ generate the 2nd order invariants, we have s 2 = 1. Applying (91) and (92) to γ gives Hence, d γ ∧ d∆ γ = 0. This implies t 2 = 1 and thus q = 2. Hence, the corresponding geometries are not pseudo-stable. /negationslash Case (b): suppose that τ = 0. In view of (28), (164) and (167) we may fully fix the frame ( s 2 = 0) by imposing the normalizations By (113), It follows that t 2 = 1, and that the 1st and 2nd order invariants are generated by γ . Again, using the notation of Appendix D, By (27) (30) (31), the components ( ∇ 2 R ) abcde 1 e 2 ,f are generated by γ . Since the automorphism group of ∇ 2 R is trivial, equations (32) can be solved for Γ α f . It follows that λ, ν, π , together with γ , generate invariants of order 3 or less. The commutator relations (91) and (92) applied to γ give Hence, The remaining structure equations (114) and (115) reduce to Observe that dγ ∧ dν = 0 if and only if ∆ ν = 0; the corresponding rank sequence is ( s p ) = (1 , 1 , 0 , 0) , ( t p ) = (0 , 1 , 1 , 1) , and the order is q -1 = 2. Else we have t 3 = 2, the 4th order invariants being generated by γ, ν and ∆ ν . Applying (91) to ν gives D ∆ ν = 0 and thus Hence, the rank sequence is and the order is equal to q -1 = 3. /square 4.2. Type DZ configurations. For this case, combining the curvature canonical form of Table 1 and the CH 0 property t 0 = 0 gives /negationslash where ˜ Ψ 2 and ˜ R ∈ R are real constants. The group G 0 preserving (35) is generated by spins (151) and the reflections (163), (166), (169). The 1st order torsion is Substituting (35) into the Bianchi equations (116)-(118) yields Proposition 14. There does not exist a type DZ, CH 0 geometry with pseudostabilization. Proof. The assumption implies that t 0 = 0 , t 1 > 0 and that the 1st order torsion is spin-invariant. By (156), this entails Hence, α generates the 1st order invariants. However, (111) entails This implies that α is a constant, which contradicts the t 1 > 0 assumption. /square Proposition 15. A type DZ, properly CH 1 geometry does not exist. Proof. In addition to (35) we assume that t 1 = 0 , t 2 > 0 and that s 1 = 0. The t 1 = 0 assumption means that post-normalization, the torsion components (36) are constant, say Transformation law (156) now reads Since s 1 = 0 this cannot be zero and we may therefore impose the normalization Applying (40), (37) to equation (109) gives Then, (107)-(115) entail Hence, all 2nd order Cartan invariants are constant, a contradiction. /square", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4.3. Type D configurations.", "content": "Proposition 16. There does not exist a type D, properly CH 0 geometry with pseudo-stabilization. Proof. The curvature normalization from Table (1) and the CH 0 assumption give (42) Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = ˜ Ψ 2 , R = ˜ R, /negationslash where ˜ Ψ 2 = 0 and ˜ R are constants, and we also assume The 1st order torsion is where, by the Bianchi relations, By the boost transformation laws (121)-(126), in order to have s 1 = 1 we require κ = ν = 0 . Adding (110) to (112) yields which implies that τ is constant. Hence t 1 = 0, contradicting our assumption. /square Proposition 17. Up to O ( η ) conjugation, the unique type D, properly CH 1 configuration is where T and C are constants such that C +2 T 2 = 0 . /negationslash Proof. Suppose that the curvature is type D, and that As above we have (42) and (43). The curvature automorphism group G 0 is generated by the 1-dimensional group of boost transformations (119) and the discrete transformations (163), (166), (169). The corresponding transformation laws are shown in Appendix B. Since t 0 = t 1 = 0 this means that post-normalization, R, Ψ 2 , κ, σ, τ, π, λ, ν are all constant, where we put π = τ = T . As above, the Bianchi identities give (44). Equations (107), (108), (114) and (115) reduce to /negationslash If κ = ν ≡ 0, identically, then by (121) - (126) the first order torsion is boostinvariant, which violates the assumption s 1 = 0. Suppose then that κ = 0. By the (N2) maximality assumption in Definition 2, κ cannot change sign. Using (125) and (167), we impose the normalization /negationslash The 2nd order torsion is α, γ, /epsilon1 . If ν = 0 then, by (48) the 2nd order torsion vanishes, which violates the assumption t 2 > 0. Therefore, /negationslash identically. Hence, t 2 = 1. The case where κ ≡ 0 , ν = 0 does not not need to be analyzed, because it can be reduced to the present case by the Lorentz transformation (169), (170). Again by the (N2) assumption of maximal normalization, /epsilon1 must have definite sign. Using (163), (164) to impose the normalization /epsilon1 > 0 fully fixes the frame. Taking the second part of (45) as a definition for the constant C , equation (110) gives The rest of (107)-(115) are either satisfied identically, or reduce to (47). /square Above, we have derived a unique set of necessary conditions for a type D properly CH 1 geometry. In other words, if such a metric exists, then around every point there exists a unique null-orthogonal moving frame such that (45) - (47) hold. Such geometries feature 1st order invariants C, T , which must be constants, and a unique, up to functional dependence, non-constant 2nd order invariant /epsilon1 . This is the necessity question. Next, we consider sufficiency. The configuration equations (45) - (47) constitute a system of partial differential equations for type D, properly CH 1 metrics. We reformulate this system as the structure equations of a generalized Cartan realization problem [7, appendix] [11, Section 3] using Bryant's recent treatment [6] of the realization problem. To wit, (45) - (47) is equivalent to Proposition 18. Up to diffeomorphism, the general solution of (51) -(54) depends on one function of one variable. Proof. Writing a straightforward calculation shows that the differential ideal generated by (51)(54) is closed; i.e., d 2 = 0. The symbol tableau and its prolongation are Hence, the reduced characters are c 1 = 1 , c 2 = 0 , c 3 = 0, with The tableau is involutive of rank 1. The desired conclusion now follows by [6]. /square Proposition 19. Generically, the metric described by the preceding Proposition is classified by 5th order invariants. Proof. For generic solutions of (51)-(54), /epsilon1, P = D/epsilon1, P 1 = D 2 /epsilon1 are functionally independent. We already observed that /epsilon1 is a 2nd order invariant. Hence P, P 1 are a 3rd and a 4th order invariant, respectively. Generically, these will be functionally independent, and therefore, the rank sequence is as shown in (13). /square", "pages": [ 10, 11, 12 ] }, { "title": "5. Three-dimensional metrics of maximal order", "content": "In this section, we prove Theorem 1. Throughout, we assume full rank regularity and curvature regularity. By Propositions 12 - 17, all 4th order metrics are necessarily type D and properly CH 1 . By Propositions 18 and 19 such a geometry satisfies (51)-(54) and /negationslash /negationslash where We complete the proof of the main Theorem 1 by integrating (51)-(54) subject to the constraints (56). First assume T = 0. To integrate (51) we introduce an integrating factor: /negationslash Hence, for some functions u, x, w . Next, (52) gives with general solution Since ω 0 , ω 1 , ω 2 are linearly independent, u, w, x form a system of coordinates, and a is some, as yet undetermined, function of u, w, x . Solving (58) (59) (60) gives By (54), we have Hence, for some univariate function f ( u ). Taking the exterior derivative of (60) and using (53) gives gives /negationslash Therefore, for T = 0 the general solution of (51)-(54) is given by (58), (59), (60) and where f ( u ) , f 1 ( u ) are freely chosen functions. This solution form is invariant with respect to the following transformations: where φ ( U ) is an arbitrary strictly increasing function ( φ ' ( U ) > 0 everywhere). If T = 0 then one verifies that (51)-(54) is still equivalent to (58)-(63). Moreover, if (1 -e 4 Tw ) / (2 T ) is interpreted in the limit sense as being equal to -2 w , (66) remains valid. The form-preserving transformations are now where U 0 , W 0 are constants and φ ( U ) is an arbitrary function. /negationslash It follows by (68) and (71) that if C = 0, then one can normalize the above solution form by transforming f ( u ) → 0 identically. If C = 0 then T = 0 by assumption, and hence by (62) and (68) one can normalize the solution form by transforming f ( u ) → 2 T 2 . Evaluating 1 2 ( ω 1 ) 2 -2 ω 0 ω 2 gives the metric in (4). /negationslash Finally, a straightforward calculation relative to this metric form shows that the maximal order condition (56) is equivalent to (6). The above maximal order metrics are invariantly classified by the invariant scalars C, T and by the following Cartan invariants of orders 2 , 3 , 4 , 5, respectively: If C = 0, it is convenient to introduce the invariants /negationslash /negationslash Hence, the invariants J 1 and J 2 have order 4 and 5 respectively. If T = 0 then J = -C 2 is constant. In the generic case CT = 0, and in the light of (55) and analogous structure equations for dJ , the invariant J is non-constant and of order 3. Explicit calculations relative to the metric form (4) show that The latter two invariants have order 4 and 5, respectively. The metric is classified by the functional relationship between these invariants. Observe that the maximal order condition is B = 3 TA . /negationslash If C = 0 = T we define dimensionless invariants of order 3, 4 and 5: /negationslash Explicit calculations relative to (4) now give Hence, as above, the metric is classified by the functional relationship between a 4th and a 5th order invariant. The maximal order condition is V = 0. /negationslash", "pages": [ 12, 13, 14 ] }, { "title": "6. Conclusions and discussion", "content": "In this article we have demonstrated that 3-dimensional Lorentzian metrics may require 5th order differential invariants for their invariant classification. The class of maximal order metrics consists of a single, well-defined family of CH 1 solutions governed by a unified set of structure equations. This echoes a similar result in 4-dimensional Lorentzian geometry [18], although there the possibility of pseudostable geometries of maximal order was left open. Previously, 3-dimensional Lorentzian CH 1 metrics were studied in detail by Bueken and Djoric [5]. They already proved Proposition 15 and obtained the metrics covered by Propositions 12 and 17, albeit not in closed form but up to solving partial differential equations. The coordinate forms in [5] are therefore less convenient for invariant classification and the discussion of the order, whereas our work was more directly related to Cartan invariants. Even though our focus here was on type D metrics of maximal order, the type N, 3rd order CH 1 geometries from Proposition 12 also constitute an interesting class governed by a well-defined set of structure equations. A closed form for these metrics can be derived along the same lines as in the type D case outlined above, but we do not pursue this here. In [9] it was proved that the unique TMG solution of type D (dubbed type D s there, cf. table 1 of appendix C) is the homogeneous, biaxially spacelike-squashed AdS 3 metric family; this is the unique solution corresponding to the proof of Proposition 16. Type D NMG solutions with constant scalar curvature were fully classified in [2] and are also homogeneous. Hence, the metrics of Theorem 1 are not TMG nor NMG solutions. Therefore, our conclusion is that at most four covariant derivatives of the Riemann tensor are needed to invariant classify exact TMG and NMG solutions locally. In future work, we want to sharpen this result. Hereby, the technique we have followed in this paper to prove Propositions 12-17 not only provides a robust mechanism to invariantly characterize solutions, but also allows one to find new solutions, beyond the curvature homogeneity assumption. A first step, however, would be to classify all curvature homogeneous TMG and NMG solutions, in order to see whether the bound q -1 ≤ 3 for the TMG and NMG gravitational theories is sharp. Finally, the same argument given for Proposition 9 holds for Riemannian geometry as well. However, for Euclidean signature only the equivalent of type DZ curvature is possible and this suffices to rule out 4th order Riemannian metrics. However 3rd order, 3-dimensional order Riemannian metrics are possible. We will report on this fact elsewhere.", "pages": [ 14, 15 ] }, { "title": "Acknowledgments", "content": "RM was supported by an NSERC discovery grant. He thanks the Mathematical Institute of Utrecht University for its hospitality during a research visit. LW was supported by a BOF research grant of Ghent University, an FWO mobility grant No V4.356.10N to Utrecht University and an Yggdrasil mobility grant No 211109 to University of Stavanger while parts of this work were performed. He thanks the Department of Mathematics and Statistics of Dalhousie University for its hospitality during a research stay.", "pages": [ 15 ] }, { "title": "Appendix A. The three-dimensional formalism", "content": "Several three-dimensional NP-like formalisms, with different symbol choices, have been proposed in the context of exact solutions to topologically massive gravity [13, 3]. Our choice of symbols is close to [3], but differs slightly in the choice of normalization because we attempted to satisfy the following criteria: Let ( U, ε ) be a 2-dimensional symplectic, real vector space. The group of symplectic automorphisms is isomorphic to SL 2 R . The vector space V = S 2 U carries the natural structure of a Lorentzian inner product space with the inner product given by η = -ε 2 . Henceforth, we regard U as the space of spinors and V as the space of vectors. The group O ( η ) is isomorphic to SO(1 , 2); the group morphism SL 2 R → SO(1 , 2) gives the double cover of vectors by spinors. To facilitate frame calculations, we introduce a normalized spinor dyad o , ι : where the dyad indices A,B,... take values 0 or 1. Associated to this dyad, we define a null vector triad by where the triad indices a, b, c = 0 , 1 , 2 are doublets of symmetrized dyad indices: In this way, we have with all other components zero. Equivalently, with all other inner products equal to zero. Next, let ( M,g ) be a 3-dimensional Lorentzian manifold. A null triad at x ∈ M is an isomorphism ( V, η ) → ( T x M,g x ). A moving η -frame is a null triad at every x ∈ O for some open neighbourhood O ⊂ M . Equivalently, a null triad is a collection of vector fields /lscript , m , n that satisfy the relations with all other inner products zero. In other words, taking ( e 0 , e 1 , e 2 ) = ( /lscript , m , n ) gives In introducing symbols for the connection scalars, we wish to adapt the notation of the familiar four-dimensional NP formalism. To do so, it is convenient to regard the manifold M as a totally geodesic embedding (all geodesics in the submanifold are also geodesics of the surrounding manifold) φ : M ↪ → ˆ M in a 4dimensional Lorentzian manifold ( ˆ M, ˆ g ). This is equivalent to the condition that M be autoparallel, i.e., that the covariant derivative operator is closed with respect to vector fields that are tangent to M [16, Chapter 7, Sect. 8]. Recall that a null tetrad framing on ˆ M is a basis of vector fields ( ˆ m , ˆ m ∗ , ˆ n , ˆ /lscript ) such that with all other cross-products equal to zero. Here ˆ /lscript , ˆ n are real whereas ˆ m , ˆ m ∗ are complex conjugates. We relate the null tetrad on ˆ M to the null triad on M by setting Let ˆ ω i , i = 1 , 2 , 3 , 4 and ˆ Γ ij denote the dual coframe and the connection 1-form on ˆ M . Let denote the corresponding pullbacks to M . Henceforth, we use a tilde decoration to denote the pullback of objects from ˆ M to M . The pullback imposes the condition: The embedding of M into ˆ M induces an inclusion of the three-dimensional Lorentz group SO(1 , 2) into SO(1 , 3), the four-dimensional Lorentz group. The condition that M be autoparallel is equivalent to the condition that the pull-back of the connection 1-form take values in the subalgebra so (1 , 2). This imposes the following conditions on the pullback of the connection 1-form: Using the notation of [26, Section 7.2], the corresponding condition on the NP connection scalars is: Taking into account the difference in the ordering of the three-dimensional and the four-dimensional indices, we arrive at the following notation for the threedimensional connection 1-form and scalars: Writing we have by (76): where ˆ ψ is a scalar defined on ˆ M and ˜ ψ = φ ∗ ˆ ψ is its pullback to M . The threedimensional commutator relations can now be expressed as The above equations follow in a straightforward manner by applying symbol rules (85)-(87), (89) to the usual four-dimensional commutator relations, as shown for example in equations (7.6a)-(7.6c) of [26]. The three-dimensional curvature tensor R abcd decomposes into a curvature scalar and a trace-free part according to The image of the natural inclusion S 4 U ↪ → S 2 V is the 5-dimensional vector space of trace-free, symmetric tensors. Therefore, the trace-free part of a three-dimensional curvature tensor can be represented by means of a rank-4, symmetric curvature spinor: In this way, the definition of the curvature scalars Ψ 0 , Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 is formally identical to their four-dimensional counterparts; c.f. [26, Equation (3.76)]. We obtain the following representation of the trace-free part of the Ricci tensor and the curvature-two form: The three-dimensional curvature 2-form and curvature scalars are related to their four-dimensional counterparts as follows: with the right-hand sides of the above equations all real, as a consequence of equations (78) and (79). The three-dimensional version of the NP equations, or equivalently Cartan's second structure equations, take the form shown below. Using equations (85)-(87), (89), (101)-(106), it is straightforward to convert the four-dimensional NewmanPenrose equations into their three-dimensional counterparts. For example, the NP equations (7.21a) and (7.21b) of [26] read Note that all of the symbols in the above two equations should have hats, but we omit the decoration for the sake of simplicity. Taking the average of these two equations, pulling back and using (85)-(87), (89), (101)-(106) gives equation (107) below. The rest of the three-dimensional structure equations are obtained via the same reduction procedure. Likewise, the differential Bianchi equations are obtained by averaging the fourdimensional Bianchi equations and applying equations (85)-(87), (89), (101)-(106). They are:", "pages": [ 15, 16, 17, 18, 19, 20 ] }, { "title": "Appendix B. Lorentz Transformations", "content": "There are 3 different types of three-dimensional Lorentz transformations: boosts, spins, and null rotations. Each such transformation has a simple description as a transformation of spinor space, i.e., as an element of SL 2 R . Infinitesimally, boosts have non-zero, real eigenvalues, spins have imaginary eigenvalues, and null rotations have zero eigenvalues (in other words, an infinitesimal null rotation is a nilpotent transformation of spinor space). To facilitate calculations, we represent these transformations in a natural spinor dyad, and present their induced action on a suitable associated vector triad and on the corresponding connection and curvature scalars. Consistent with our philosophy of concordance between the three-dimensional and four-dimensional formalisms, all of the above equations are straightforward reductions of the four-dimensional transformation laws; c.f. [27, Appendix B]. A boost transformation corresponds to a real-diagonalizable element of SL 2 R . The corresponding spinor and vector actions are Boost transformations can also be realized as the 1-dimensional group of symmetries of the type D curvature spinor; c.f. line 6 of Table 1. The associated connection and curvature transformation laws are shown below. A null rotation corresponds to a unipotent, non-diagonalizable element of SL 2 R . The corresponding spinor and vector actions are Null rotations can also be realized as the 1-dimensional group of symmetries of the type N curvature spinor; c.f. line 9 of Table 1. The associated transformation laws for the connection and curvature scalars are shown below. A spin transformation corresponds to an element of SL 2 R with imaginary eigenvalues. As such, we have /lscript ( + n ) ' = LHS , Spin transformations can also be realized as the 1-parameter group of symmetries of the type DZ curvature spinor; cf line 7 of Table 1. The associated connection and curvature transformation laws are shown below. Finally, there are a number of discrete Lorentz transformations that lie outside the connected component of the identity in O ( η ). Given a null frame ( /lscript , m , n ) we define The transformation laws of the connection and curvature scalars under reflection of the vectors of the orthonormal triad ( t , m , x ) are also relevant for our purposes and are given below. Reflection of t ('time reversal'): (165) Reflection of m : (166) (167) Ψ 0 , Ψ 2 , Ψ 4 invariant , Ψ 1 , Ψ 3 change sign . m ↦→ -m : κ, τ, α, π, ν change sign , σ, /epsilon1, γ , λ invariant , Reflection of x : (170) κ ↔- ν, σ ↔- λ, τ ↔- π, /epsilon1 ↔- γ, α ' = - α, Appendix C. Petrov-Penrose classification of the three-dimensional Ricci tensor /negationslash Let /lscript , m , n a null vector triad for which Ψ 4 = 0. We introduce the threedimensional analogue of the Petrov-Penrose classification in terms of the root configurations of the real quartic We note that this classification forms a special case of the general null alignment classification for tensors in arbitrary dimensions [17], applied here to the threedimensional trace-free Ricci tensor S ab . Hence, in addition to the analogues of Petrov types I, II, D, III, N (where there are 4 real solutions z ) and type O, we /negationslash /negationslash have to account for the possibility that some or all of the roots of Ψ( z ) are complex. We will denote these additional root configurations as Petrov types IZ (two different real roots, two complex roots), IZZ (4 complex roots), IIZ (double real root, two complex roots), and DZ (the double roots are complex conjugate). but it is related to the form in the table by a Lorentz transformation. Analogously, a Lorentz-equivalent type D canonical form is Note that the Ricci-Petrov classification based on null alignment refines the RicciSegre type classification [13]. The distinction between Petrov types I and IZZ is the order of the timelike eigenvalue, relative to the spacelike eigenvalues. Regarding Segre type { 21 } , the spacelike or timelike character of the vector S ab l b , where the null vector /lscript lies in the 2-dimensional generalized eigenspace but is not an eigenvector, distinguishes between Petrov types II and IIZ. Also note that Petrov type O describes a constant curvature space.", "pages": [ 20, 21, 22, 23 ] }, { "title": "Appendix D. CH 1 structure equations", "content": "This appendix is devoted to an analysis of the algebraic data and the structure equations that underly curvature homogeneous geometries. In what follows a crucial, albeit technical, innovation allows us to simplify the form of higher order Cartan invariants by replacing them with certain connection scalars. The general theory is detailed in [19]. For the sake of concreteness we limit the discussion to the case of CH 1 geometries. We begin by recalling some preliminary notation and theory, and then turn to the description of CH 1 data and structure equations, which we call a CH 1 configuration. Let e a , a = 1 , . . . , n be a basis of V , and A α , α = 1 , . . . , n ( n -1) / 2 a basis of o ( η ). Let A a bα denote the matrix components of A α ; i.e., /negationslash Let C α βγ be the corresponding structure constants: Let ω a be an η -orthogonal coframe, Γ α , Ω α the corresponding connection 1-form and curvature 2-form, respectively. The latter are determined by the first and second structure equations: where Γ α a , R abcd are the connection and curvature components, respectively: The exterior derivative gives the algebraic and differential Bianchi relations: In Appendix A we introduced a convenient formalism that assigns specific symbols to the Γ α a , R abcd when n = 3. Our three-dimensional formalism is a suitable reduction of the well-known 4-dimensional Newman-Penrose (NP) formalism. In this reduced, 3-dimensional formalism, the first structure equations (173) correspond to the commutator relations (90)-(92); the second structure equations correspond to reduced NP equations (107)-(115). The component versions of the Bianchi relations are given by (116)-(118). The details of the formalism and of the reduction from 4 to 3 dimensions were given in Appendix A. Next, suppose that the CH 1 condition holds and let ω a be a curvature normalized η -orthogonal coframe. By the t 1 = 0 assumption, where the right hand sides denote arrays of constants. Let G 0 ⊂ O ( η ) be the automorphisms of ˜ R abcd and G 1 ⊂ G 0 the automorphisms of ˜ R abcde . Hence, (179) fixes the choice of coframe up to a G 1 gauge transformation. Set Let g 0 , g 1 denote the Lie algebra of G 0 , G 1 respectively. Introduce an adapted basis of g 1 ⊂ g 0 ⊂ g -1 consisting of where the A ξ are a basis of g 1 , the A λ are a basis of g 0 / g 1 and the A ρ are a basis of g -1 / g 0 . By the usual definition of the covariant derivative one has where for a rank k tensor T a 1 ...a k the notation denotes the infinitesimal action of a linear transformation on a covariant tensor. Since the R abcd are constant and since g 0 is the annihilator of ˜ R abcd , (180) becomes Equation (181) describes a linear system in Γ ρ e with maximal rank. Hence, Γ ρ e = ˜ Γ ρ e , where the latter are constants rationally dependent on ˜ R abcd , ˜ R abcde . Therefore, a CH 1 geometry is determined by constants ˜ R α ab = -˜ R α ba and constants ˜ Γ ρ a such that the following relations hold, relative to a normalized η -orthogonal coframe: Moreover, the Bianchi relations (177), (178) impose the following linear, respectively, bilinear constraints on the above constants: Similar to (181), the second order derivative of curvature is given by relative to a normalized coframe. Since the residual frame freedom is G 1 , the scalars Γ λ a obey an algebraic, G 1 -transformation law. The second structure equations (174) impose the following linear constraints on these scalars: where ρ 1 , ρ 2 = s 0 +1 , . . . , s -1 have the same range as ρ , and the following differential relations: where and where λ 1 , λ 2 = s 1 +1 , . . . s 0 have the same range as λ . We will refer to constants ˜ R α ab , ˜ Γ ρ a and scalars Γ λ a together with constraints (184)-(185) and (187)-(189) as a CH 1 configuration [19]. It is well known that the O ( η ) metric equivalence problem has trivial essential torsion [20, Section 12]. However, if we reduce the structure group to G 0 ⊂ O ( η ) by means of curvature normalization, we obtain the following reduced first structure equations: where are the G 0 -lifted 1-forms. The scalars ˆ Γ ρ a are well defined because the MaurerCartan term d XX -1 takes values in g 0 . Consequently, the scalars Γ ρ a have a G 0 -transformation law that does not depend on d X , and therefore constitute the essential torsion for the 1st iteration of the equivalence method. Thus, the scalars Γ ρ a can be interpreted as the essential torsion arising from the reduced G 0 -equivalence problem and the Γ λ a as essential torsion in the next iteration of the G 1 -equivalence problem. Therefore, we refer to the former as 1st order torsion, and to the latter as 2nd order torsion. By virtue of (181), normalizing the Γ ρ a is equivalent to normalizing R abcd ; e . The 1st order normalization reduces the structure group to G 1 ⊂ G 0 . If we suppose that the CH 1 property holds, then the resulting invariants are the constants ˜ Γ ρ a . The scalars Γ λ a are the essential torsion of the 2nd iteration of the equivalence method. By virtue of (181) (186), the 2nd order Cartan invariants are functions of the 0th order Cartan invariants R α ab and the 1st and 2nd order torsion scalars ˜ Γ ρ a , Γ λ a . Inversely, because of (184), ˜ R α ab is linearly dependent on ˜ R abcd , while (183) and (186) can be solved to give ˜ Γ ρ a as functions of ˜ R abcd , ˜ R abcde and Γ λ a as a function of ˜ R abcd , ˜ R abcde , R abcd ; ef .", "pages": [ 23, 24, 25, 26 ] }, { "title": "References", "content": "E-mail address : [email protected] E-mail address : [email protected] Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada Department of Mathematical Analysis, Ghent University, B-9000 Ghent, Belgium Mathematical Institute, Department of Mathematics, Utrecht University, 3584 CD Utrecht, The Netherlands Department of Mathematics and Natural Sciences, University of Stavanger, N-4036 Stavanger, Norway", "pages": [ 27 ] } ]

[ { "title": "ABSTRACT", "content": "To appear in Class. Quantum Grav.", "pages": [ 1 ] }, { "title": "L. Cosmai, G. Fanizza, M. Gasperini and L. Tedesco", "content": "Dipartimento di Fisica, Universit'a di Bari, Via G. Amendola 173, 70126 Bari, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy E-mail: [email protected], [email protected], [email protected], [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "The beginning of the cosmological phase bearing the direct kinematic imprints of supernovae dimming may significantly vary within different models of late-time cosmology, even if such models are able to fit present SNe data at a comparable level of statistical accuracy. This effect - useful in principle to discriminate among different physical interpretations of the luminosity-redshift relation - is illustrated here with a pedagogical example based on the LTB geometry. It is by now widely known that the observed luminosity-redshift distribution of type Ia supernovae [1] can be fitted even without dark energy, provided one introduces a sufficiently inhomogenous space-time geometry. A typical, very simple example of such a possibility is provided by matter-dominated cosmological models of the LemaˆıtreTolman-Bondi (LTB) type (see e.g. [2] for an incomplete list of papers on this subject), provided the observer is located near enough to the symmetry centre of the inhomogeneous - but isotropic - matter distribution [3, 4]. Such an example may be regarded as unnatural because of the amount of fine tuning required to localize the observer position [4], and also appears theoretically disfavoured by the possible presence of weak geometric singularities [5]. Nevertheless, the possible role of inhomogeneities in determining (or at least substantially contributing to) the large-scale dynamics should be - and, indeed, currently is [6] - seriously scrutinized and discussed, even in the presence of a dominant dark-energy cosmic component [7]. The general question that arises in such a context is how to distinguish different successful analyses of the SNe data based on different physical models and, in particular, on different (homogeneous versus inhomogeneous) large scale geometries. For inhomogeneous models of the LTB type various answers to this question are known, concerning the size of the voids described by the LTB geometry [8], the local expansion rate inside the voids [9], and the associated effect of redshift drift [10]. Other possibilities to test LTB models are provided by studies of scalar perturbations [11], of small-scale CMB effects [12], of the cosmic age parameter [13], and of BAO (baryon acoustic oscillations) data [14]. The main purpose of this note is to point out another possible difference between inhomogeneous and more conventional interpretations of the SNe data, not yet discussed in the literature; such a difference is based on the value of the redshift parameter z acc (to be defined below, see after Eq. (14)), marking the beginning of the regime directly characterized by the kinematic imprints of SNe dimming. The value of such parameter can be largely different even within models able to fit the presently observed luminosityredshift distributions at a comparable level of accuracy (see e.g. [15] for earlier studies on the beginning of the accelerated regime in the context of a homogeneous geometry). This suggests two possible experimental ways of discriminating among models of the luminosity-redshift relation. First, direct observations able to extend our present knowledge of the Hubble diagram up to values of z higher than those allowed by present SNe data: for instance, gamma-ray burst (as discussed in [16]), or even gravitational waves observations, through an analysis of the luminosity distance of the so-called 'standard sirens' [17]. Second, indirect observations which are sensitive to the timedependence of the so-called 'transfer function' [18], which controls the evolution of the primordial perturbation spectrum inside the horizon down to the present epoch, and which is crucially affected by the kinematics of the cosmological background (see e.g. [19]). The possible relevance of the parameter z acc will be illustrated in this paper by a simple exercise, in which the SNe data of the recent Union2 compilation [20] are fitted using a inhomogeneous, matter-dominated LTB model, and such a fit is compared with the standard one performed in the context of the flat concordance Λ CDM model. We stress that our aim is not to provide a realistic alternative to the successful concordance cosmology, but only to discuss how to distinguish, at least in principle, different fits of SNe data based on different geometric schemes. The proposed diagnostic may be added to other general methods aiming at discriminating the expansion history of competing models, like - in particular - dark-energy based diagnostics for homogeneous models [21]; a Friedmann equation diagnostic for homogeneous versus inhomogeneous models [22, 23]; and the already mentioned test of redshift drift [10, 23, 24]. The cosmological configuration we will consider refers to a late-time (in particular, post-reionization) Universe, characterized by a stochastic distribution of many overdense and underdense regions, of various possible sizes and shapes, possibly even incoherently superimposed among each other ∗ . Let us suppose that in such a context, and up to a given scale r V (to be specified below), the effective (averaged) large-scale geometry can be locally described by a model of the LTB type. Such a model is characterized in general by three arbitrary functions of the radial coordinate (see e.g. [26]). For the illustrative purpose of this paper, however, it will be enough to consider a simple example where the contribution of the spatial curvature is negligible and the gravitational sources are dominated by an isotropic cold dark matter (CDM) distribution (but the model could be easily generalized by the addition of an arbitrary cosmological constant). We will assume that the large-scale geometry around a given observer is described - in polar coordinates and in the synchronous gauge - by the following metric, where a prime denotes partial derivatives with respect to r and a dot with respect to t . In the limit A ( r, t ) = ra ( t ) one recovers the well known, spatially flat, FriedmanLemaˆıtre-Robertson-Walker (FLRW) metric. In general, the unknown function A ( r, t ) is to be determined by the Einstein equations, which in our case reduce to where H ( r, t ) = ˙ A/A and F ( r, t ) = ˙ A ' /A ' . The density profile of the CDM distribution around a central observer, ρ = ρ ( r, t ) satisfies the covariant conservation equation: while all the other Einstein equations are identically satisfied by the metric (1) (see e.g. [27]). The above cosmological equations can be integrated exactly, and in this paper we will adopt the particular exact solution normalized in such a way that A = r at t = 0. The arbitrary function H 0 ( r ) depends only on the radial coordinate, and the usual matter-dominated FLRW solution is exactly recovered in the limit H 0 = const. We will use, in particular, the parametrization already suggested in [27] for a similar LTB scenario (a brief discussion of other possible choices for the phenomenological profile H 0 ( r ) will be given in the final part of this paper). For the chosen profile the combination of parameters H + ∆ H ≡ H 0 (0) corresponds to the locally measured value of the Hubble constant, while the distance r V represents the typical distance scale above which inhomogeneity effects become rapidly negligible. To make contact with more general forms of the LTB metric appearing in the literature, and expressed in terms of three functions M ( r ), t B ( r ), E ( r ), (we are following the notations of [26]), it may be useful to report here the values of those functions for the model we are using. The effective gravitational mass with comoving radius r , for our solution, is given by M ( r ) = (1 / 2) r 3 H 2 0 ( r ). It can be easily checked that this function grow like r 3 for r glyph[lessmuch] r V and r glyph[greatermuch] r V , while, in the transition regime r ∼ r V , it is characterized by a fractal index D = 0 . 4, i.e. M ( r ) ∼ r 3 -D ∼ r 2 . 6 . The time scale t B - i.e., the local 'big-bang time' at which A ( r, t ) = 0 - in our case is given by t B ( r ) = -(2 / 3) H -1 0 ( r ). Finally, it is important to stress that the obtained solution is consistent with our assumption of vanishing spatial curvature, i.e. with the choice E ( r ) = 0. Perturbing the solution with the addition of scalar curvature (and assuming that E ( r ) ∼ r 2 as in the large-scale FLRW limit), we have checked indeed that the curvature contribution to the total energy density may have a variation which is at most of the order of 0 . 05% over length scales of order r V and time scales of order H -1 0 . Hence, if initially small but nonzero, it keeps small over the whole spatial and temporal range of interest for this paper. Let us now compute the luminosity distance d L of a source emitting light at a cosmic time t and a radial distance r from the origin. We will assume, for the moment, that the observer is also located at the origin (the consequences of a possible off-center position will be discussed later). The angular distance (or area distance) of the source, for the metric (1), is then given by d A = A ( r, t ), and the luminosity-distance, according to the so-called 'reciprocity law' [28], reduces to d L = (1 + z ) 2 A ( r, t ), where z is the redshift parameter evaluated along a null radial geodesic connecting the source to the origin. Calling u µ the static (time-like) geodesic vector field tangent to the worlines of source and observer, and k µ the null vector tangent to the null radial geodesic, we find in our metric u µ = dx µ /dτ = (1 , 0 , 0 , 0) and k µ = (( A ' ) -1 , -( A ' ) -2 , 0 , 0). Hence, for light emitted at time t , radial position r , and observed at the origin at t = t 0 , where A ' 0 ≡ A ' (0 , t 0 ) = const. For the phenomenological applications of this paper we need to express d L completely in terms of the redshift, namely we need to invert Eq. (6) to determine r ( z ) and t ( z ). We may consider, to this purpose, the differential variation of z with respect to the proper time interval dτ separating two different instants of light emission, at fixed observation coordinates: dz/dτ = u µ ∂ µ z = -(1+ z ) ˙ A ' /A ' . It follows that, along a null radial geodesic (where dt = -A ' dr ): For the model of Eq. (4), in particular, we obtain the differential equations Solving the above equations for t ( z ), r ( z ), and inserting the solutions into the explicit definition of d L , we are now in the position of comparing the predictions of our model with the observational data (as well as with the predictions of the standard ΛCDM scenario). Let us first recall that the Union2 compilation of the Supernova Cosmology Project [20] concerns redshift-magnitude measurements of 557 SNe of type Ia and provides, for each supernova, the observed distance modulus (with relative error) µ obs ( z i ) ± ∆ µ ( z i ), i = 1 , . . . , 557, for redshift values ranging from z 1 = 0 . 015 to z 557 = 1 . 4. The distance modulus µ ( z ) controls the difference between apparent and absolute magnitude, and is related to the luminosity distance d L ( z ) by: Here d L is given in units of Mpc, and the constant number 25 is determined by the conventional reference scale assumed for the absolute magnitude. The luminosity distance of Eq. (9), with H 0 ( r ) given by Eq. (5), is characterized in principle by three independent parameters, and can be applied to fit the experimental data by allowing free variations of H , ∆ H and r V . We have performed that exercise, and found that the resulting best fit provides for H 0 (0) ≡ H +∆ H a value very close to 70 Km s -1 Mpc -1 . We have thus chosen to concentrate the present discussion on a simpler, two-parameter fit of the data - which, in any case, is sufficiently accurate for the illustrative purpose of this paper - by imposing on our model the 'a priori' constraint H +∆ H = 70Kms -1 Mpc -1 . In this way we can eliminate, for instance, H , and we can fit the experimental points µ obs ( z i ) ± ∆ µ ( z i ) by performing a standard χ 2 analysis with The theoretical values µ ( z i , r V , ∆ H ) can be determined, for each value of z i , by numerically integrating the two equations (8), and computing the corresponding d L ( z i ) as a function of the two parameters r V , ∆ H . By minimizing the above χ 2 expression we have found the best fit values at a confidence level of 95%, and with a goodness of fit χ 2 / d . o . f . = 0 . 99. The minimization has been performed using the MINUIT package from CERNLIB [29]. The result of the fit is graphically illustrated by the red curve plotted in the left panel of Fig. 1, superimposed to the full set of Union2 data (reported with error bars). Consider now, for comparison, a fit of the same data performed in the context of a spatially flat FLRW geometry, with perfect fluid sources representing CDM and a cosmological constant Λ. Denoting with Ω m and Ω Λ the present critical fraction of dark matter and dark energy, we can express the luminosity distance in the usual integral form as (see e.g. [30]). Proceeding as in the previous case, we will reduce the number of parameters from 3 to 2 by imposing the same phenomenological constraint as before, which in this case amounts to the condition H 0 (Ω m +Ω Λ ) 1 / 2 = 70Kms -1 Mpc -1 . Using Eq. (13) to compute µ ( z i , Ω m , Ω Λ ), and minimizing the corresponding χ 2 expression, we obtain the best fit values Ω m = 0 . 27 ± 0 . 01, Ω Λ = 0 . 71 ± 0 . 03, at a confidence level of 95%, with χ 2 / d . o . f . = 0 . 98. The result of the fit is illustrated by the blue curve on the right panel of Fig. 1. The luminosity-redshift relations of the two models of Fig. 1 are in good agreement with the data, and in both cases the data points are fitted at a comparable level of statistical accuracy. However, we can disclose an important physical difference between the two fits if we subtract from the distance modulus of the two models the distance modulus µ Milne ( z ) of a linearly expanding (but globally flat) homogeneous Milne geometry (see e.g [31]), namely if we consider the quantity where H 0 is given in units of Mpc -1 . It is clear that positive or negative values of ∆ correspond to luminosity distances which are - at a given fixed z - respectively larger or smaller than the reference values of the Milne model. The case ∆ < 0 is typical of a decelerated Universe like that described by the standard cosmological scenario, where, at the same fixed z , the distances are smaller (or the received fluxes of radiation, i.e. the apparent magnitudes, are larger ) than predicted by a linearly expanding model. The case ∆ > 0, on the contrary, corresponds at the same z to larger distances (or smaller radiation fluxes) than predicted by linear expansion, and is only possible if the model undergoes a period of 'effective' accelerated expansion. In this last case, the transition across the value ∆ = 0 defines an epoch characterized by the parameter z acc such that ∆( z acc ) = 0 - marking the beginning of the cosmological phase directly imprinted by the kinematic effects of the acceleration. The plot of ∆( z ) is presented in Fig. 2 for three cases: the standard CDMdominated (always decelerated) model, and the two best-fit models of Fig. 1 (corresponding to our example of inhomogeneous geometry and to a typical example of homogeneous concordance cosmology). In the last two cases we have plotted the central values of the fit (solid curves), as well as the corresponding error bands ∗ at the 95% level of confidence (bounded by the dotted curves). We can see from Fig. 2 that ∆( z ) is always negative for the CDM model, as expected. For the other two models, instead, we have ∆( z ) > 0 in the redshift range z < z acc (because, as expected, a successful fit of the SNe data requires the presence of a phase describing - or mimicking - accelerated expansion). However, the values of z acc defined by the condition ∆( z acc ) = 0 are largely different in the two models. We find, in particular, and this difference falls outside the error bands illustrated in Fig. 2 (it is also much larger than the experimental uncertainty affecting present redshift measurements). This suggests that a precise (near-future?) determination of this parameter could provide a clear physical discrimination among different models implementing successful (and statistically equivalent) fits of SNe data. It should be mentioned, at this point, that in the computations of the error bands we have neglected the dispersion of data due to the possible presence of a cosmic background of stochastic perturbations: indeed, such a background may induce large errors at very small z , but in the range z ∼ 1 (typical of z acc ) the induced errors are typically lying in the few-percent range [7], hence are not expected to have a crucial impact on the results illustrated in Fig. 2. The same is expected to be true for the systematic errors - possibly slightly bigger than the previous ones, but in any case < ∼ 10% - induced on z LTB acc (but not on z ΛCDM acc ) by methods of SNe data reduction based on the assumption of standard homogeneous cosmology (and used in particular for the Union2 catalogue, see e.g. [32]). Finally, we should note that a value of z acc compatible with that of the inhomogeneous model considered here could be reproduced also in a homogeneous ΛCDM context, with realistic values of Ω m and Ω Λ , but only at the price of introducing a large enough negative spatial curvature, with Ω k ∼ 0 . 1 (for instance, a model with Ω m = 0 . 3, Ω Λ = 0 . 6, Ω k = 0 . 1 gives z ΛCDM acc = 1 . 087). In order to stress the importance of the parameter z acc let us now consider another possible form of the phenomenological profile H 0 ( r ) appearing in the LTB solution (4), for instance the profile ∗ We can then explicitly check that different models are characterized by largely different values of z acc even within the same class of inhomogenous geometries. By imposing, as before, the phenomenological constraint H 0 (0) = 70 Km s -1 Mpc -1 (in order to eliminate H ), we find that the new profile (16) provides indeed a satisfactory three-parameter fit of the Union2 data (see Fig. 3, left panel), with best fit values r 0 = 2500 ± 322 Mpc, ∆ r = 2387 ± 170 Mpc, ∆ H = 37 . 5 ± 2 . 8 Kms -1 Mpc -1 , at a confidence level of 95%, with χ 2 / d . o . f . = 1 . 31. However, the corresponding value of z acc for this model (called LTB 1 in Fig. 3) is significantly different from that of the previous LTB model, and, most important, the behaviour of ∆( z ) is exactly the opposite of the standard one, for the range of z of our interest (see Fig. 3, right panel). We have checked that the value of ∆( z ), for LTB 1 , turns back to the standard negative range only for z > ∼ 50. glyph[negationslash] Let us finally comment on the possibility that an off-center position of the observer embedded in a spherically symmetric LTB geometry may significantly affect the determination of z LTB acc , thus providing obstructions to a precise discrimination between LTB-based and a more conventional (homogeneous) fit of the SNe data. Indeed, if the observer is located at a distance r 0 = 0 from the center of a spherically symmetry geometry, the corresponding luminosity distance d L (referred to the position r 0 ) is no longer isotropic but acquires an angular dependence, and this in turn induces an angular dispersion of the value of z acc which depends on r 0 , and which obviously grows (in modulo) with the growth of r 0 . The luminosity distance of a source for off-center observers in a LTB geometry has been computed in [4] (see also [34]) as a function of z , of the distance r 0 from the centre, and of the polar observation angle γ (referred to r 0 ). We have applied the results of [4] to compute the directional variation of z acc , at fixed values of r 0 . We have considered, in particular, possible displacements from the centre in the range r 0 < ∼ 10 -2 r V , because - as discussed in [4] - higher values of r 0 would induce a dipole anisotropy too high to be compatible with present CMB observations. glyph[negationslash] The results of our exercise are illustrated in Fig. 4, where we have plotted the fractional variation ∆ z acc /z acc ≡ [ z acc ( r 0 , γ ) -z acc (0)] /z acc (0), for different values of r 0 up to 10 -2 r V , for the LTB model characterized by the parameter z LTB acc of Eq. (15). For the normalization of µ Milne we have consistently used H 0 ( r 0 ), but we have checked that using the fixed value H 0 = 70Kms -1 Mpc -1 simply rescales the zero of the difference ∆ z acc , without affecting the overall amplitude of the dispersion. As shown in Fig. 4, the angular variation of z acc induced by r 0 = 0 is bounded to be at most at the one-percent level, and has thus a negligible impact on the results of Fig. 2. In conclusion, we would like to stress again that the inhomogeneous model discussed in this paper should not be intended as a realistic alternative to the successful concordance cosmology, but only as a pedagogical example to learn how to distinguish different fits of SNe data based on different geometrical schemes. To this purpose we have shown, in particular, that in the model of this paper the Universe enters the regime directly affected the accelerated kinematics later than predicted by the ΛCDM scenario, i.e. z LTB acc < z ΛCDM acc . Hence, a precise determination of the transition epoch z acc (possibly through future extensions of the Hubble diagram to higher values of z , or through indirect studies of the transfer function of primordial perturbations [35]), could help us to physically discriminate among statistically equivalent fits.", "pages": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] }, { "title": "Acknowledgements", "content": "One of us (MG) is very grateful to Ido Ben-Dayan, Giovanni Marozzi, Fabien Nugier and Gabriele Veneziano for many useful discussions on the luminosity distance in the context of inhomogeneous cosmological models.", "pages": [ 10 ] } ]

[ { "title": "Smoothed Transitions in Higher Spin AdS Gravity", "content": "Shamik Banerjee, a Alejandra Castro, c Simeon Hellerman, d Eliot Hijano, b,e Arnaud Lepage-Jutier, b Alexander Maloney b and Stephen Shenker a", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider CFTs conjectured to be dual to higher spin theories of gravity in AdS 3 and AdS 4 . Two dimensional CFTs with W N symmetry are considered in the λ = 0 ( k →∞ ) limit where they are conjectured to be described by continuous orbifolds. The torus partition function is computed, using reasonable assumptions, and equals that of a free field theory. We find no phase transition at temperatures of order one; the usual Hawking-Page phase transition is removed by the highly degenerate light states associated with conical defect states in the bulk. Three dimensional Chern-Simons Matter CFTs with vector-like matter are considered on T 3 , where the dynamics is described by an effective theory for the eigenvalues of the holonomies. Likewise, we find no evidence for a Hawking-Page phase transition at large level k .", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Higher spin holography provides a simple and elegant framework to probe our understanding of quantum gravity. In four dimensions, the simplest version of this duality relates the singlet sector of the three dimensional O ( N ) model with Vasiliev's higher spin theory of gravity in AdS 4 [1, 2, 3]. The singlet constraint is implemented by coupling to a Chern-Simons gauge theory with large level [4]. In lower dimensions, minimal conformal field theories with W N symmetry are related to similar higher spin theories of gravity in AdS 3 [5, 6]. In both cases, the relative simplicity of the duality makes possible a variety of precise checks. The boundary theories are potentially exactly solvable due to the presence of an infinite number of conserved charges. Likewise, the gravity theories while complicated - appear simpler than full string theory in AdS. Perhaps the most notable feature of these dualities is the apparent paucity of bulk degrees of freedom. There are higher spin fields present but no conventional strings. But these higher spin theories contain more than higher spin fields. They contain so-called 'light states' [7, 8, 9, 10]. In the AdS 3 context these correspond to twisted states in a continuous orbifold description of the boundary theory [11] as we will review below. They have a bulk interpretation as 'conical defects' [16]. In the AdS 4 context, when the spatial boundary is T 2 , these states correspond to ChernSimons holonomies interacting with the vector-like matter [10]. In the gravitational dual theory, these are likely described by a topological closed string sector, while the Vasiliev degrees of freedom form an open string sector [17, 18, 19]. The goal of this paper is to understand the effect of these light states on the thermal behaviour of these theories. It is dramatic. We will argue that these states smooth out the Hawking-Page transition, at least in a limit. To study such thermal phenomena in the boundary field theory, we put the theory on Euclidean spaces with a compact time direction. In particular we will compute the partition function of W N minimal models on S 1 × S 1 = T 2 in the limit of small 't Hooft coupling λ . We will likewise study the partition function of three dimensional Chern-Simons theories with vector matter on T 2 × S 1 in the limit of large Chern-Simons level. The explicit computations rely on the conjecture that for λ → 0 the W N minimal model is described by a continuous orbifold [11]. 1 This gives a rather simple interpretation of the partition function as a gaussian path integral which can be computed exactly, albeit with some subtleties in the integration measure. In section 2 we will show that the partition function equals that of N -1 free bosons. To corroborate the conjecture and justify our assumptions we show that the low dimension part of the continuous spectrum of this free theory matches with the light states of the W N CFT as λ → 0. Our analysis shows explicitly that for the continuous orbifold theory the partition function (and its derivatives) is a smooth function of temperature. From the bulk gravity point of view this is a surprise. The AdS 3 theory has black hole solutions [20, 21, 22]. The entropy of the boundary W N theory agrees with the Bekenstein-Hawking entropy of these black holes at sufficiently high temperature [23, 24]. In AdS gravity, the formation of such black holes is typically associated with a Hawking-Page phase transition [25]. During this phase transition the theory jumps from a thermal gas of perturbative states in AdS (with entropy of order 1) to a black hole phase (with entropy of order N ). In the dual gauge theory, this is interpreted as a transition between a confined phase at low temperature and a deconfined phase at high temperature [26]. The W N theory, on the other hand, appears to be in a deconfined phase, with entropy of order N , at all temperatures. Unfortunately, it is difficult to extend this to the finite λ case. It is not obvious that the CFT spectrum is analytic with respect to λ , and therefore we cannot blindly extrapolate our results. Nevertheless, the effect of the light states is universal; even at finite λ the entropy at low temperatures is of order N . Thus there is no conventional Hawking-Page phase transition, even at finite 't Hooft coupling. This does not rule out the possibility of some other sort of phase transition at finite λ . In the 2+1 dimensional boundary dual to AdS 4 we find a similar story. On T 2 × S 1 we argue that the absence of a Vandermonde determinant in the measure for holonomies indicates the absence of a finite temperature phase transition, at least for large Chern-Simons level. Here the holonomies allow non-singlet matter states even at low temperatures, giving an intuition for the smoothing out of the Hawking Page transition. The bulk interpretation of these results requires a better understanding of both AdS 3 and AdS 4 higher spin gravity. There is no phase transition between an AdS ground state and a blackhole solution. Instead, there should be a continuous family of solutions that smoothly interpolate between these saddles. 2 This should be described by an appropriate moduli space of solutions in Vasiliev theory (and in AdS 4 its completion). The light states behave like a quantum mechanical system, rather than a local field theory. Our understanding of their geometrical and topological nature is incomplete. In AdS 3 , the light states are classically described by conical defects. In AdS 4 , on the other hand, the light states should be related to a topological closed string sector which couples to the open-string Vasiliev fields [17, 18, 19]. The organization of the paper is as follows. In section 2 we compute the partition function of the W N minimal models in the λ → 0 limit. The continuous part of the spectrum, due in part to light W N primaries, contributes logarithmically to the entropy of the system but dominates for finite temperature at small enough λ . In section 3 we interpret these results in the dual AdS 3 higher spin theory. In section 4 we discuss the analogous behavior for vector-like CFTs in three dimensions. We end in section 5 with a discussion of our results and their implications for the geometrical interpretation of Vasiliev's theory. In appendix A we compute the measure of flat connections used in section 2. In appendix B we compute the free energy and phase structure of a gas of W N descendants.", "pages": [ 2, 3 ] }, { "title": "2 Free theory analysis in (1+1) dimensions", "content": "In this section we will compute the partition function for 2D minimal models with W N symmetry and zero 't Hooft coupling. Our goal is to understand the phase structure of the CFT and its implications for the phase space and entropy of the dual gravitational higher spin theory. W N minimal models are coset WZW models of the form The central charge is The 't Hooft limit is defined by the large N and k limit, where the 't Hooft coupling is held fixed. In this limit the central charge becomes large and the theory is expected to be dual to a bulk higher spin theory [5]. When λ → 0, i.e. k → ∞ with N fixed so that the central charge approaches c = ( N -1), there is evidence that the theory is described by a continuous orbifold [11] (see also [12]). It is the SU ( N ) / Z N orbifold of N -1 bosons on the torus R N -1 /A N -1 , where A N -1 is the SU ( N ) root lattice. This was shown explicitly for N = 2 in [11]; here we will proceed under the assumption that the proposal is correct for any value of N . 3 We consider the partition function where is a complex linear combination of the angular potential θ and inverse temperature β . Since the CFT is rather simple, it is not difficult to write the partition function as a path integral. The partition function is where Z eff is the partition function of N -1 bosons on R N -1 /A N -1 with boundary conditions twisted by U and V in the space and (Euclidean) time directions, respectively. The integral is over commuting SU ( N ) holonomies U and V . As the holonomies commute we can take them to be simultaneously diagonalizable; we denote the eigenvalues e iψ j and e iχ j , j = 1 . . . N -1, and assemble them into vectors ψ and χ . To compute the measure in (2.6), we treat the holonomies as SU ( N ) gauge fields and take the gauge coupling to zero. In appendix A we compute the integration measure in detail using this approach. 4 The answer does not contain any Vandermonde determinants, so We have denoted the integration range by T , the maximal torus T ⊂ SU ( N ). One should in principle be careful about the integration range, as we must take into account both the Z N factor as well as that of the Weyl group S N on the eigenvalues. However, the integrand Z eff ( τ ) is invariant under the actions of Z N as well as the action of the Weyl group. Thus these will only contribute an overall τ -independent factor to the integral. We will ignore these and all other τ -independent factors, which can be absorbed into the path integral measure. We now compute the effective action Z eff = e S . Denote by X ∈ R N -1 /A N -1 the free bosons on the SU ( N ) torus. The classical solutions to the equations of motion are labelled by momentum and winding numbers n and w which take values in the lattice A N -1 and hence The classical action of such a solution is To include the effect of the SU ( N ) holonomies, recall that the gauge transformations in the maximal torus T ⊂ SU ( N ) generate translations in X . Effectively we are shifting the momentum and winding in (2.8) by Thus the classical action with twisted boundary conditions is so that Note that the one loop determinant is independent of the choice of classical solution (i.e. n and w ) because the path integral is gaussian. Moreover, up to a τ -independent constant, it is equal to the N th power of the determinant of a single free boson and is independent of ψ and χ . Here η ( τ ) is the Dedekind eta function. Combining (2.7), (2.13) and (2.14) with (2.6) we obtain Note that χ and w only appear in the combination χ +2 π w , and analogously for n and ψ . We can combine the integral over χ with the sum over w by shifting the integration variable χ → χ +2 π w so that In the first line T w denotes the shifted range of integration. In the second line we have used the sum over w to decompactify the range of χ integration. We can now perform the same manipulations for the n sum and the ψ integral to obtain The integral over ψ and χ is a finite constant independent of τ . Hence, the τ dependent part of the partition function is This equals the partition function of N -1 decompactified free bosons. This demonstrates that the spectrum of the continuous orbifold theory - and hence the thermodynamics - equals that of N -1 uncompactified bosons. However, the two theories certainly differ at the level of correlation functions. Nevertheless, this computation establishes that the theory has no phase transition in the large N limit. We should mention that the partition function is infinite due to the zero modes of the scalars. However we can systematically divide out this divergence (which is independent of the conformal structure τ ), and analyze the finite contribution given by (2.18). The normalization of equation (2.18) can be fixed by comparison with the spectrum of light states of the W N minimal model, as we will now see.", "pages": [ 4, 5, 6 ] }, { "title": "2.1 Spectrum revisited", "content": "We found that the partition function in the λ → 0 limit of the W N minimal equals that of N -1 uncompactified bosons. In particular the power of τ 2 in (2.18) reflects the continuum of operators e i k · X and contributes to the total entropy of the system logarithmically in T . More explicitly, in the low temperature and large N regime, we find where ' . . . ' correspond to subleading contributions in N and T ; the first term is the contribution from the ground state. The logarithmic term is subdominant at sufficiently low temperatures when T /lessmuch 1. If instead we are in the regime T /lessorsimilar 1, the continuum part of the spectrum is nonnegligible and the contribution to the entropy scales with N . 5 It is worth highlighting that the logarithmic correction to the free energy is the first indication of a departure from the Cardy regime for T ∼ O (1); this will be relevant when we discuss the gravitational interpretation of our results. In our derivation we have not made use of the W N coset construction of the theory. Thus it is unclear which features will persist at finite λ . Moreover, it is not clear that the spectrum is an analytic function of λ . This leaves room to speculate that the lack of a phase transition in the partition function (2.18) is an artifact of the free theory limit rather than a generic feature of the theory. However, in the following we will relate the continuum part of the spectrum to the corresponding W N primaries, and provide a simple extension of our results for λ = 0 to the finite case. 6 This will also provide an interpretation of the logarithmic growth (2.19) which is in accordance with the results reported in [10]. In general, states can be labelled (in the Drinfeld-Sokolov representation) by pairs of representations of SU ( N ), say (Λ + , Λ -) (see e.g. [6] for a review). As was discussed earlier by [7, 8, 9], there is a class of states with Λ + = Λ -= Λ whose weight is given by Here C 2 (Λ) is the quadratic Casimir of the SU ( N ) representation Λ. In terms of Young diagram data, we have where r i is the number of boxes in the i -th row of the Young diagram, and the r i are ordered. The quadratic Casimir is Note that the set of integers ( n i + B/N ) are distinct and there is no repetition among them. We will therefore take n 1 > n 2 > · · · . In the following, we will shift C 2 (Λ) by a constant such that the empty Young tableaux has weight zero. The distinctive feature of the states h (Λ , Λ) is that for finite B (i.e. finite number of boxes) the weight goes to zero as k go to infinity; no other primary states in the spectrum have this feature. The states h (Λ , Λ) are the so-called 'light states,' which form a continuum near the ground state. 7 The contribution of these light states to the partition function can be approximated as follows. Let ∆ = h (Λ , Λ) be the dimension of a given state. We want to consider ∆ /lessmuch 1 but fixed as k → ∞ . Equivalently, we want β fixed and large as k → ∞ . Thus we want to study C 2 /lessorsimilar k 2 , so that n i /lessorsimilar k . Next, we can treat (roughly) the n i as equal to r i with some offset as k /greatermuch N /greatermuch 1 . The offset B N acts like a center of mass coordinate for the r i , and the n i can be thought as relative coordinates. The n i add up to zero, but this is just one constraint on N variables, so we will ignore it. Integrating over the center of mass gives an extra volume factor which will be of order k which we ignore also. The range of n i (which can be positive or negative) is of order k and is large compared to the relevant scale determined by ∆, which is √ ∆ k . Still there is an ordering on the n i inherited from (2.22). The partition function becomes Define Then we can take a continuum approximation as k →∞ In the second line we removed the ordering by taking the unordered integral and dividing by ( N -1)!. From here it follows that at low temperatures ( β large) and large N we obtain We emphasize that the power law behavior will breakdown at some T /lessorsimilar 1. The integrals in (2.25) are well approximated by gaussians only for β large when compared to the volume spanned by the x i 's . This will be important when we compare our estimate in (2.26) to numerical data. The entropy attributed to the light states (excluding the ground state contribution) is then in the large N limit. Therefore, according to (2.25) and (2.27), the density of light states as a function of energy is well approximated by ρ (∆) ∼ ∆ ( N -3) / 2 for small values of ∆. (Here we have given the correct finite N expression.) This matches precisely the density of states of a N -1 uncompactified bosons, so provides a non-trivial check of the continuous orbifold result (2.18). We can check this analysis by directly computing the spectrum of light states for small N and large k . For N = 2 (Virasoro unitary minimal models) this can be done analytically. The primary states are labelled by pairs of integers ( r, s ) with (see e.g. [31]) The light states (2.20) are those with r = s . It is easy to check that as k →∞ the density of these states behaves like ρ (∆) ∼ ∆ -1 / 2 for small ∆. It is important to note, however, that the light states with r = s correctly reproduce this expected density of states only up to ∆ = 1 / 4. In order to reconstruct the continuous orbifold result ρ (∆) ∼ ∆ -1 / 2 for ∆ ≥ 1 / 4 one has to include 'non-light' states. For example, the states with r = s +1 have dimension ∆ ≥ 1 / 4 and also contribute to the continuous spectrum. In order to extend this discussion to N > 2 we numerically computed ρ (∆) for the light states. We evaluated C 2 (Λ) for all allowed representations for N = 3 , 4 , 5 and large values of k . In Figs. 1 and 2 we display our results. The density of light states perfectly matches the continuous orbifold result ρ (∆) ∼ ∆ ( N -3) / 2 for small values of ∆. Above a finite value of ∆ the light states fail to match the expected density of states. The critical value of the dimension - the N > 2 analogue of the ∆ = 1 / 4 point for the Virasoro minimal models - can be estimated numerically from Fig 1. /negationslash This is not a surprise. Just as in the N = 2 case, we expect that states with Λ + = Λ -will give important contributions to the spectrum above a certain critical value of the dimension. Without attempting a rigorous study of these non-light states, it is easy to see why they should give additional continuous contributions to the spectrum in the k → ∞ limit. In the language of highest weight representations the dimension of a state can be written as where ˆ ρ is the Weyl vector of su ( N ). Here p ≡ N + k and we have added an overall normalization constant so that h (0 , 0) = 0. We want to study the large p limit of this formula. Let us consider states with Λ 0 ≡ Λ + -Λ -fixed and finite as k →∞ . As in (2.24), we define which will be treated as a continuum variable in the large k limit. As we take k →∞ the range of Λ + is roughly bounded by k , hence we can view X as a vector whose components are continuous degrees of freedom of order one. The weight (2.29) becomes Thus a new continuum of states develops, now built on a ground state of positive dimension. Indeed, this is exactly the behaviour predicted by Fig. 1. It would be interesting to study in more detail the behaviour of the non-light states, and in particular to provide an analytic derivation of the critical dimension for the non-light states. To summarize, as k → ∞ , the light states of the W N minimal models are the source of the power law dependence in τ 2 for Z N at small temperatures. Their contribution to the entropy grows like N 2 log( T/λ 2 ). The divergent piece in the limit λ = 0 is the characteristic infinite entropy due to the zero modes of the free theory limit.", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "3 Implications for AdS 3 Higher Spin Gravity", "content": "The behavior of the entropy (2.27) is not typical of classical gravitational theories. In this section we will discuss the bulk interpretation of the logarithmic corrections to the entropy. This will have implications for the Hawking-Page phase transition for AdS 3 higher spin gravity. Most importantly, they indicate that the Bekenstein bound is violated in the bulk gravity theory. Before discussing the classical spectrum of the higher spin theory, we will review the basic mechanism of the Hawking-Page phase transition in AdS 3 [25]. We will discuss which conditions lead to the removal of this first order phase transition and discuss which features of the spectrum are responsible for this effect. This will provide some guidance for the comparison with the CFT partition function. For simplicity, we set to zero the angular potential θ and all other chemical potentials, and consider the free energy as a function of inverse temperature β . At any fixed temperature there are two relevant classical saddles: thermal AdS and the BTZ black hole. This statement is independent of the specific matter content of the theory, as the BTZ black hole is a quotient of AdS 3 and hence is guaranteed to be a solution of any gravitational theory in AdS [32, 33]. Further, the saddles are modular images of each other, with the thermal AdS saddle at inverse temperature β the modular transform of the black hole at 4 π 2 /β . If there is no exotic matter in the gravitational theory -e.g. scalar hair or higher spin fields- the classical limit of the gravitational path integral is dominated by the free energy of these two solutions where with /lscript the AdS radius and G Newton's constant. For the vector-like holography under consideration, we have c ∼ N so that the Planck length scales as G ∼ N -1 . Up to numerical factors, in the large N limit we have At β = 2 π there is a discontinuity in the first derivative of F . This is the Hawking-Page phase transition. We emphasize that in order for this phase transition to occur, the free energy must be dominated by the vacuum contribution for all β > 2 π , i.e. Note that (3.4) always holds at sufficiently low temperatures, e.g. if we scale temperature as β ∼ O ( N ). This follows from the existence of a gap in the spectrum above the ground state. However, the requirement that (3.4) holds up to temperatures of order one is a highly non-trivial constraint. We now reconsider this derivation for the higher spin duals to the 't Hooft limit of the W N minimal model. In the bulk we are studying a Vasiliev theory with gauge group hs [ λ ]. The theory contains an infinite tower of higher spin fields and one complex scalar with mass M 2 = -1 + λ in AdS units. We will not consider the scalar field in this discussion; the higher spin sector is then described by a pair of hs [ λ ] Chern-Simons theories. The local fields for the graviton (metric) and higher spin analogues can be constructed by the appropriate contractions of the Chern-Simons connections [34, 35, 36, 37, 38]. The classical phase space of the higher spin theory is defined as follows. It corresponds to the set of Chern-Simons connections that satisfy AdS fall-off conditions; this is analogous to the Brown-Henneaux boundary conditions for pure AdS 3 gravity [39]. The solutions also have to be smooth, i.e. the connection has to be globally well defined. In the Chern-Simons language this naturally translates to having trivial holonomy around contractible cycles; see [21, 40, 41] for a complete discussion of this condition. The authors in [16] found a family of novel solutions to SL ( N ) higher spin theories which satisfy this condition. 8 These solutions resemble conical defect geometries in a particular gauge. The description of these solutions as conical singularities is somewhat artificial since the solutions are - in every gauge-invariant sense - smooth, but we will continue to refer to these solutions as conical defects. The conical defects have an interesting feature: they are characterized by a set of N fractions m i such that ˆ m i ∈ Z and further with the constraint that m i = m j for i = j . Their energies are given by /negationslash /negationslash The solutions also carry higher spin charges; see [16] for further details. SL ( N ) Chern-Simons theory is a semiclassical theory with large central charge c and fixed N . This is not in the regime of validity of the W N minimal models since by construction in the CFT we have c ≤ N -1. Still, we can interpret the conical deficits solutions in the dual CFT as follows. If we compare the Young tableaux construction of the light states (2.20) with the spectrum of conical defects, we can identify m i = n i in (2.22). The couplings in front of (2.20) and (3.6) seem to disagree. This is an artifact which arises because the two expressions are written in different regimes of validity. Using (2.2), we can rewrite (2.20) as The authors of [42] showed that the large c and fixed N limit of the minimal models is mathematically well-defined and can be implemented as an analytic continuation of the couplings. 9 Implementing this here, it is clear that the large c and fixed N limit of (3.7) exactly gives (3.6). As shown in [16], the higher spin charges of the light states and conical defects match in this limit as well. To summarize, the bulk theory contains a large set of solutions - conical defects - which are in one-to-one correspondence with the light states of the boundary CFT. The computation of the path integral for higher spin gravity matches the computation of the free energy of the CFT, at least at low temperature. In particular, the conical defects with C 2 ( m ) finite should condense around thermal AdS in the hs [ λ ] theory, forming a continuum in the spectrum. Hence the path integral around thermal AdS will be corrected, just as in section 2.1. The free energy of the higher spin theory for small temperature is then given by (2.27) Although we can account in the bulk for the thermal behaviour of the light states, recall that for λ = 0 there are additional non-light states that contribute to the continuum spectrum, such as those of the form (2.31). It would be interesting to understand the gravity interpretation of these states. The behaviour (3.8) differs significantly from the free energy of a classical gravitational theory (3.3). It is clear that (3.8) does not obey the condition (3.4); there is no Hawking-Page phase transition. A consequence of this observation is that in higher spin gravity a given classical saddle never dominates the free energy at finite temperature. The condensation of light states smears out the phase transition. This indicates that it is impossible to attribute the thermodynamic behaviour, and hence the entropy, to any individual saddle; one can not isolate the contribution of classical solution to the free energy. Of course, if we scale T with λ and/or N then the thermal AdS or BTZ black hole will dominate the free energy, but not in the sharp sense defined by the Hawking-Page phase structure. A curiosity in this analysis is the apparent violation of the Bekenstein entropy bound. The basic intuition that motivates the holographic principle is that entropy in the bulk theory is proportional to area rather than volume. This is equivalent to a linear dependence of black hole entropy on T in three dimensional gravity. Our findings contradict this statement. The conglomeration of classical conical defects contributes a large amount of entropy, in particular a log T degeneracy per planck length G ∼ N -1 . If the behaviour is geometrical, the effective size of the gas of conical deficits is not linear with length or temperature: each individual conical deficit in the bulk will have some small entropy proportional to its size, however the ensemble of them gives rise to a continuous spectrum with logarithmic growth that dominates the semi-classical entropy. These conical defect solutions, and in particular their condensation into a continuous spectrum, appear to be special to higher spin theories. We are not aware of other Einstein-like theories of gravity which display a similar pattern. The thermodynamic properties of higher spin gravity do not resemble the universal features of general relativity and black hole physics. However, this logarithmic growth is typical of the spectrum of classical string world sheet solutions in AdS 3 [43]; long string states also form a continuous spectrum, mimicking some aspects of our discussion. The novelty of the higher spin theory is that the density of light states scales with N , making them abundant in the semiclassical limit. In contrast, the scalar representations of SL (2 , R ) -which represent these long string solutions- have a density of states that is independent of the coupling.", "pages": [ 11, 12, 13, 14 ] }, { "title": "4 Chern-Simons Matter analysis in (2+1)-dimensions", "content": "There are close similarities between the thermal behavior of the W N theory discussed above and that of SU ( N ) Chern-Simons theory coupled to fundamental scalar matter on a spatial torus T 2 . This system has light states [10] described by a global quantum mechanical system with holonomy degrees of freedom that at low energies is just given by N harmonic oscillators with /planckover2pi1 = 1 /k . Here k is the Chern-Simons level. The semiclassical partition function is the product of two integrals of the form (2.25), one for the N momenta and one for the N positions. These give a low temperature entropy where λ = N/k is the 't Hooft coupling. 10 This is analogous to (2.27) after noting that the gaps here are of order λ rather than λ 2 . To study the finite temperature partition function at large k we can first write the pure ChernSimons partition function in terms of holonomy eigenvalues and then include the effects of the scalar matter. On S 2 × S 1 the Vandermonde determinant causes eigenvalue repulsion while the scalar action favors all eigenvalues to be at the origin. These effects balance at a Gross-Witten-Wadia phase transition at a temperature of order √ N [29, 4]. On T 2 × S 1 the situation is quite different. There is no Vandermonde determinant for the eigenvalues in the pure Chern-Simons theory [44, 45] and so no balancing forces on the eigenvalues to cause a phase transition. So we conclude that at least in the limit of large k there is no phase transition in this system on a spatial T 2 . The eigenvalues of the 'thermal' circle are concentrated around the origin and so the system behaves much like one without a singlet constraint imposed. This absence of phase transition is analogous to the W N result discussed above. To extend this argument to large but finite k it will be necessary to study perturbative corrections to the measure on holonomies. We have not done this, but we do not expect a qualitative change. An intuitive reason for the difference between spatial S 2 and T 2 is the following. The Gauss Law constraint is, schematically: where F a 12 is the gauge field strength and j a 0 is the scalar charge. On S 2 the only solution at large k is j a 0 = 0, i.e. scalar singlet states. One has to go to temperatures T ∼ √ N to make this singlet constraint unimportant [29, 4]. On T 2 however the presence of almost flat connections allows states with F a 12 ∼ 1 /k . 11 These satisfy the Gauss Law constraint with nonzero j a 0 . So non-singlet scalar states are allowed. The presence of these states allows a smooth evolution to the high temperature NT 2 unconstrained scalar entropy.", "pages": [ 14, 15 ] }, { "title": "5 Discussion", "content": "The geometric interpretation of higher spin gravity remains puzzling. Among other issues, we must understand the black holes of the theory. In AdS 3 higher spin theories we have a sharp definition of the classical solitons, and thus explicit black hole solutions. In particular, it is also understood how to describe solutions carrying higher spin charges. A non-trivial test of vector-like dualities in AdS 3 /CFT 2 is the exact agreement between the thermodynamics of the higher spin black hole and the high temperature spectrum of the CFT [23, 24]. The black hole carries higher spin charge, thus additional sources are turned on in the CFT partition function. The high T density of states is determined by the modular properties of the partition function with arbitrary number of insertions of the zero mode of the higher spin charge. However, the theory remains enigmatic. In particular, other (apparently non-black hole) gravitational solutions are important. For instance, in AdS 3 higher spin gravity there are more smooth classical solitons - the conical defects- than is usually the case in gravitational theories. These solutions are crucial for the W N minimal model correspondence proposed by [5], as these conical solutions are in one-to-one correspondence with the light primary states of the dual theory [16, 42]. We have explored the effects of this light sector on the thermodynamics of the system. Our results reveal a novel feature of the bulk states, that they smooth out the Hawking-Page phase transition. It would be interesting to study the higher genus partition function of W N CFT, and investigate whether the higher genus version of the Hawking-Page transition is smoothed out as well. To understand the disappearance of the Hawking-Page transition in AdS 3 /CFT 2 , we evaluated the partition function at λ = 0 using the description of the WZW coset (2.1) as a continuous orbifold CFT. We emphasize that this is a conjecture; there is no proof that the W N theory reduces to this specific free theory at zero coupling. Our analysis in section 2.1 gives further evidence for this conjecture. We have matched the spectrum of the light states to the continuous spectrum of the free theory. The consequences of this smoothed transition are significant. The light states have an entropy of order log( T ), rather than T . This indicates that these states are quantum mechanical rather than field theoretic. They do not correspond to 'local' degrees of freedom. In the AdS 4 case, these states appear to be topological in nature. In the continuous orbifold theory, the light states effectively behave as the zero mode of the N -1 uncompactified bosons, which is the quantum mechanics of the continuum of operators e i k · X in the free field theory language. The presence of a continuous spectrum is novel but not disturbing. The shocking news is that there are a lot of them. In fact the entropy of these states dominates the entropy until a temperature where N log( T/λ 2 ) ∼ NT . This gives a scale at small λ . It will be important to understand the physical meaning of this scale. One possible approach would be to examine the bulk conical deficit solutions in the presence of a BTZ black hole. In a particular there has to be a dynamical mechanism that predicts the crossover in the bulk. Since the scale (5.1) is governed by the 't Hooft coupling, which also fixes the mass of the scalar in Vasiliev's theory, it might be necessary to include backreactions from the scalar field in the analysis. This field also adds local degrees of freedom - recall that hs [ λ ] Chern-Simons is topological - and hence might be a natural mechanism for the 'gravitational collapse' of a gas of conical deficits into a black hole. We emphasize that these last remarks are highly speculative, and to address this properly we must construct a representation of the conical solutions in the hs [ λ ] theory. The situation in AdS 4 /CFT 3 is somewhat similar. The partition function on T 2 × S 1 has a similar crossover and hence no phase transition. The entropy of the lights states is N log( T/λ ). The unconstrained scalar field (or conjectural black hole) entropy is NT 2 . So the crossover is at T 0 ∼ √ log(1 /λ ). In the case of Σ g × S 1 the scale is even higher. Here the entropy of the light states is N 2 log( T/λ ) so the crossover temperature is T 0 ∼ N log(1 /λ ). √ The bulk interpretation here is likely related to the observation that the light states found in [10] are described by a topological closed string sector of an open-closed string theory where the Vasiliev excitations are open string states [17, 18, 19].", "pages": [ 15, 16 ] }, { "title": "Acknowledgements", "content": "We are grateful to Chris Beasley, Tom Hartman, Shiraz Minwalla, Wei Song, Arkady Vainshtein, and Xi Yin for useful discussions and to Matthias Gaberdiel, Rajesh Gopakumar and Mukund Rangamani for valuable comments on an earlier version of this paper. We are especially grateful to Matthias Gaberdiel for discussions related to the continuous orbifold conjecture. In addition we thank the participants of the KITP program 'Bits, Branes and Black holes' and the ESI workshop on 'Higher Spin Gravity' for useful discussion. The research of SB and SS is supported by NSF grant 0756174 and the Stanford Institute for Theoretical Physics. The work of AC, ALJ and AM is supported by the National Science and Engineering Research Council of Canada. AC, AM and SS are supported in part by NSF under Grant No. PHY11-25915. EH acknowledges support from 'Fundaci'on Caja Madrid'. The work of S.H. was supported by the World Premier International Research Center Initiative, MEXT, Japan, and also by a Grant-in-Aid for Scientific Research (22740153) from the Japan Society for Promotion of Science (JSPS). The work of AC is also supported by the Fundamental Laws Initiative of the Center for the Fundamental Laws of Nature, Harvard University.", "pages": [ 16, 17 ] }, { "title": "A Measure factor", "content": "Our goal is to compute the integration measure of the path integral (2.6). A convenient way to cast the integral over the SU ( N ) holonomies U and V is by treating them as components of a SU ( N ) gauge field A µ and taking the gauge coupling to zero. The action for SU ( N ) gauge fields is as where g is the gauge coupling. The discussion follows [46]; the only minor difference is the inclusions of matter fields. We compactify the two dimensions on a torus of radii R µ . We first characterize the zero modes of our theory, i.e. the modes whose action vanishes. The zero modes for A µ consist of diagonal matrices with N eigenvalues ψ I µ , i.e. for { m ν } labelling the modes of the fields around the two cycles of the spacetime torus. Then the action to quadratic order is /negationslash where ∆ ψ µ IJ = ψ µ I -ψ µ J is the difference of eigenvalues. We now want to integrate out the gauge field KK modes ˆ A IJ µ, { m λ } . The naive result will be 1 / det M where M is a 2 by 2 matrix that can be written as The eigenvalues of this matrix are 0 and ∑ µ ( (∆ ψ µ ) IJ -2 πm µ R µ ) 2 . The existence of a vanishing eigenvalue is an indicator that we have forgotten about gauge symmetry. Convenient gauge fixing constraints are Up to a normalization, the Faddeev-Popov measure will come from the following determinant which evaluates to The gauge fixing constraints in terms of KK modes read /negationslash where r = 0. Exactly one component of the gauge field is eliminated by these constraints, which means that M is now a 1 by 1 matrix with eigenvalue ∏ I B Gas of Free Higher Spin Particles In this appendix we study the thermodynamics of a gas of higher spin particles in AdS 3 /CFT 2 . These theories possess both W N primary and W N descendant states; for simplicity we consider possible phase transitions due to descendant states. These are easiest to study in the case λ = 0, which corresponds to taking k → ∞ first before taking N → ∞ . In this case no null states are removed from the spectrum and the descendant states live in the W N versions of the Verma module. We rewrite the λ = 0 partition function (2.18) in terms of the W N characters with Z N = | q | -( N -1) / 12   1 τ ( N -1) / 2 2 ∣ ∣ ∣ ∣ ∣ N -1 ∏ n =1 (1 -q n ) n -N ∣ ∣ ∣ ∣ ∣ 2   χ N ¯ χ N , (B.1) χ N = N ∏ s =2 ∞ ∏ n = s (1 -q n ) = N -1 ∏ n =1 (1 -q n ) N -n P ( q ) N -1 , (B.2) and P ( q ) = q 1 / 24 η ( τ ) -1 . (B.3) The prefactor in (B.1) reflects the standard (cylinder) normalization where the ground state has dimension -c/ 24. The quantity in parenthesis encodes the density of states of W N primaries, and the free W N character is χ N . The partition function Z ( τ ) = | χ N ( τ ) | 2 , (B.4) describes a gas of free higher spin excitations (the analogues of boundary gravitons). Let us first verify that this gas of higher spin particles reproduces the expected Cardy behaviour. We will do this by studying the higher temperature ( β → 0) behaviour of χ N . The asymptotics of P ( q ) are easy to determine. At high temperature P ∼ exp ( π 2 6 β -1 + 1 2 log β -1 2 log 2 π -π 24 β + . . . ) . (B.5) It remains only to understand the asymptotics of the polynomial prefactor. We will define its logarithm to be g = log N -1 ∏ n =1 (1 -q ) N -n = N -1 ∑ n =1 ( N -n ) log(1 -q n ) = -∞ ∑ m =1 N -1 ∑ n =1 N -n m q nm = -∞ ∑ m =1 q Nm -Nq m + N -1 m ( q m -2 + q -m ) . (B.6) This can be approximated as β → 0, as g ∼ N ( N -1) 2 log β + g 0 -N ( N 2 -1) 12 β + . . . . (B.7) We see that at fixed N and large temperature ( β → 0) the W N character is dominated by P ( N ) log χ N ∼ π 2 ( N -1) 6 β -1 + . . . . (B.8) This agrees with the free energy (3.3) in the large N limit. We now wish to understand the robustness of this result in the large N limit. We know that for any fixed N the free energy will scale like β -2 at sufficiently high temperature, but we do not know how large β must be taken in order for this result to apply. It will turn out that F ∼ β -2 only when β vanishes more quickly than N -1 in the large N limit. In other words, Cardy's formula is only valid in the regime where the temperatures are large compared to N . Moreover we will not find a phase transition as a function of temperature. To see this, let us reconsider the formula for g above. We will scale the temperature with N as β = γN -p where γ is held fixed in the large N limit. It is straightforward to expand the polynomial appearing in g at large N . Let us first consider the case p < 1, where we obtain at large N g ∼ ∞ ∑ m =1 ( 1 m 3 β 2 -N m 2 β + . . . ) , (B.9) where the . . . denotes lower powers of β , which are subleading as β → 0. The sum over m gives 12 g ∼ ζ (3) β -2 -Nπ 2 6 β -1 + . . . . (B.10) Combining this with the asymptotic expression for P ( q ) we find that the β -1 terms cancel. We are left with F ∼ -ζ (3) β 3 . (B.11) Let us now consider the case where p > 1. One can again expand g at large N , and in this case it is easy to see that the leading terms at large N are all positive powers of β . The free energy is dominated by P ( q ), and we obtain the usual Cardy behaviour F ∼ -Nπ 2 3 β 2 . (B.12) In order to understand the transition between these two regimes, let us consider the case where p = 1, so that βN is held fixed in the large N limit. In this case g ∼ ∞ ∑ m =1 ( 1 -e -mNβ m 3 β 2 -N m 2 β ) = ζ (3) -Li 3 ( e -Nβ ) β 2 -Nπ 2 6 β -1 + . . . . (B.13) Now, as βN → 0 we have Li 3 ( e -Nβ ) ∼ ζ (3) -π 2 6 Nβ + . . . , (B.14) so that g approaches a constant as βN → 0. In this case the free energy has the Cardy behaviour. On the other hand, Li 3 ( e -βN ) vanishes exponentially when βN is large. This leads to the nonCardy, F ∼ 1 / ( Nβ 3 ) behaviour. At intermediate values of βN the function Li 3 is perfectly smooth. We conclude that when we take the large N limit with βN fixed, the free energy - regarded as a function of βN - smoothly interpolates between the two behaviours described above. It is worth noting that the behaviour described above does not constitute a phase transition in the usual sense, in that the free energy is a smooth function of β for each of the scaling behaviours described above. However, the free energy is non-analytic in the following sense. Instead of regard- ng the free energy F ( β, N ) as a function of β and N , we can let β = γN -p and regard the free energy as a function of N,γ and p . Then in the large N limit at fixed γ we have F ( N,γ,p ) →-{ ζ (3) γ -3 N 3 p p < 1 π 2 3 γ -2 N 2 p +1 p > 1 . (B.15) Thus, the free energy is not an analytic function of p , however it is important to keep in mind that the free energy at fixed p is an analytic function of γ , i.e. of the temperature. References [1] E. Sezgin and P. Sundell, 'Massless higher spins and holography,' Nucl. Phys. B 644 , 303 (2002) [Erratum-ibid. B 660 , 403 (2003)] [hep-th/0205131]. [2] I. R. Klebanov and A. M. Polyakov, 'AdS dual of the critical O(N) vector model,' Phys. Lett. B 550 , 213 (2002) [hep-th/0210114]. [3] For a recent review see: S. Giombi and X. Yin, 'The Higher Spin/Vector Model Duality,' arXiv:1208.4036 [hep-th]. [4] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin, 'Chern-Simons Theory with Vector Fermion Matter,' arXiv:1110.4386 [hep-th]. [5] M. R. Gaberdiel and R. Gopakumar, 'An AdS 3 Dual for Minimal Model CFTs,' Phys. Rev. D 83 , 066007 (2011) [arXiv:1011.2986 [hep-th]]. [6] For a recent review see: M. R. Gaberdiel and R. Gopakumar, 'Minimal Model Holography,' arXiv:1207.6697 [hep-th]. [7] M. R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, 'Partition Functions of Holographic Minimal Models,' JHEP 1108 , 077 (2011) [arXiv:1106.1897 [hep-th]]. [8] C. -M. Chang and X. Yin, 'Higher Spin Gravity with Matter in AdS 3 and Its CFT Dual,' arXiv:1106.2580 [hep-th]. [9] K. Papadodimas and S. Raju, 'Correlation Functions in Holographic Minimal Models,' Nucl. Phys. B 856 , 607 (2012) [arXiv:1108.3077 [hep-th]]. [10] S. Banerjee, S. Hellerman, J. Maltz and S. H. Shenker, 'Light States in Chern-Simons Theory Coupled to Fundamental Matter,' arXiv:1207.4195 [hep-th]. [11] M. R. Gaberdiel and P. Suchanek, 'Limits of Minimal Models and Continuous Orbifolds,' JHEP 1203 , 104 (2012) [arXiv:1112.1708 [hep-th]]. [12] S. Fredenhagen, 'Boundary conditions in Toda theories and minimal models,' JHEP 1102 , 052 (2011) [arXiv:1012.0485 [hep-th]]. [13] S. Fredenhagen, C. Restuccia and R. Sun, 'The limit of N=(2,2) superconformal minimal models,' arXiv:1204.0446 [hep-th]. [14] S. Fredenhagen and C. Restuccia, 'The geometry of the limit of N=2 minimal models,' arXiv:1208.6136 [hep-th]. [15] I. Runkel and G. M. T. Watts, 'A Nonrational CFT with c = 1 as a limit of minimal models,' JHEP 0109 , 006 (2001) [hep-th/0107118]. [16] A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, 'Conical Defects in Higher Spin Theories,' JHEP 1202 , 096 (2012) [arXiv:1111.3381 [hep-th]]. [17] O. Aharony, G. Gur-Ari and R. Yacoby, 'd=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories,' JHEP 1203 , 037 (2012) [arXiv:1110.4382 [hep-th]]. [18] C. -M. Chang, S. Minwalla, T. Sharma and X. Yin, 'ABJ Triality: from Higher Spin Fields to Strings,' arXiv:1207.4485 [hep-th]. [19] M. Aganagic, S. Hellerman, D. Jafferis and C. Vafa, In progress. [20] V. Didenko, A. Matveev, and M. Vasiliev, 'BTZ Black Hole as Solution of 3-D Higher Spin Gauge Theory,' Theor.Math.Phys. 153 (2007) 1487-1510, hep-th/0612161. [21] M. Gutperle and P. Kraus, 'Higher Spin Black Holes,' JHEP 1105 , 022 (2011) [arXiv:1103.4304 [hep-th]]. [22] For a recent review see: M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, 'Black holes in three dimensional higher spin gravity: A review,' [arXiv:1208.5182[hep-th]]. [23] P. Kraus and E. Perlmutter, 'Partition functions of higher spin black holes and their CFT duals,' JHEP 1111 , 061 (2011) [arXiv:1108.2567 [hep-th]]. [24] M. R. Gaberdiel, T. Hartman and K. Jin, 'Higher Spin Black Holes from CFT,' JHEP 1204 , 103 (2012) [arXiv:1203.0015 [hep-th]]. [25] S. W. Hawking and D. N. Page, 'Thermodynamics of Black Holes in anti-De Sitter Space,' Commun. Math. Phys. 87 , 577 (1983). [26] E. Witten, 'Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,' Adv. Theor. Math. Phys. 2 , 505 (1998) [hep-th/9803131]. [27] V. Didenko and M. Vasiliev, 'Static BPS black hole in 4d higher-spin gauge theory,' Phys.Lett. B682 (2009) 305-315, [arXiv:0906.3898 [hep-th]]. [28] C. Iazeolla and P. Sundell, 'Families of exact solutions to Vasiliev's 4D equations with spherical, cylindrical and biaxial symmetry,' JHEP 1112 , 084 (2011) [arXiv:1107.1217 [hep-th]]. [29] S. H. Shenker and X. Yin, 'Vector Models in the Singlet Sector at Finite Temperature,' arXiv:1109.3519 [hep-th]. [30] M. R. Gaberdiel, R. Gopakumar and M. Rangamani, Private communication. To appear . [31] P. Di Francesco, P. Mathieu, D. Senechal, 'Conformal field theory,' Springer (1997). [32] M. Banados, C. Teitelboim and J. Zanelli, 'The Black hole in three-dimensional space-time,' Phys. Rev. Lett. 69 , 1849 (1992) [hep-th/9204099]. [33] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, 'Geometry of the (2+1) black hole,' Phys. Rev. D 48 , 1506 (1993) [gr-qc/9302012]. [34] M. Henneaux and S. -J. Rey, 'Nonlinear W infinity as Asymptotic Symmetry of ThreeDimensional Higher Spin Anti-de Sitter Gravity,' JHEP 1012 (2010) 007 [arXiv:1008.4579 [hep-th]]. [35] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, 'Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields,' JHEP 1011 , 007 (2010) [arXiv:1008.4744 [hep-th]]. [36] M. R. Gaberdiel and T. Hartman, 'Symmetries of Holographic Minimal Models,' JHEP 1105 (2011) 031 [arXiv:1101.2910 [hep-th]]. [37] A. Campoleoni, S. Fredenhagen and S. Pfenninger, 'Asymptotic W-symmetries in threedimensional higher-spin gauge theories,' JHEP 1109 , 113 (2011) [arXiv:1107.0290 [hep-th]]. [38] M. Henneaux, G. Lucena Gomez, J. Park and S. -J. Rey, 'Super- W(infinity) Asymptotic Symmetry of Higher-Spin AdS 3 Supergravity,' JHEP 1206 (2012) 037 [arXiv:1203.5152 [hepth]]. [39] J. D. Brown and M. Henneaux, 'Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,' Commun. Math. Phys. 104 , 207 (1986). [40] M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, 'Spacetime Geometry in Higher Spin Gravity,' JHEP 1110 , 053 (2011) [arXiv:1106.4788 [hep-th]]. [41] A. Castro, E. Hijano, A. Lepage-Jutier and A. Maloney, 'Black Holes and Singularity Resolution in Higher Spin Gravity,' JHEP 1201 , 031 (2012) [arXiv:1110.4117 [hep-th]]. [42] M. R. Gaberdiel and R. Gopakumar, 'Triality in Minimal Model Holography,' JHEP 1207 , 127 (2012) [arXiv:1205.2472 [hep-th]]. [43] J. M. Maldacena and H. Ooguri, 'Strings in AdS(3) and SL(2,R) WZW model 1.: The Spectrum,' J. Math. Phys. 42 , 2929 (2001) [hep-th/0001053]. [44] M. Blau and G. Thompson, 'Derivation of the Verlinde formula from Chern-Simons theory and the G/G model,' Nucl. Phys. B 408 , 345 (1993) [hep-th/9305010]. [45] C. Beasley, Private communication. [46] O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, M. Van Raamsdonk and T. Wiseman, 'The Phase structure of low dimensional large N gauge theories on Tori,' JHEP 0601 , 140 (2006) [hep-th/0508077].", "pages": [ 17, 18 ] }, { "title": "B Gas of Free Higher Spin Particles", "content": "In this appendix we study the thermodynamics of a gas of higher spin particles in AdS 3 /CFT 2 . These theories possess both W N primary and W N descendant states; for simplicity we consider possible phase transitions due to descendant states. These are easiest to study in the case λ = 0, which corresponds to taking k → ∞ first before taking N → ∞ . In this case no null states are removed from the spectrum and the descendant states live in the W N versions of the Verma module. We rewrite the λ = 0 partition function (2.18) in terms of the W N characters with and The prefactor in (B.1) reflects the standard (cylinder) normalization where the ground state has dimension -c/ 24. The quantity in parenthesis encodes the density of states of W N primaries, and the free W N character is χ N . The partition function describes a gas of free higher spin excitations (the analogues of boundary gravitons). Let us first verify that this gas of higher spin particles reproduces the expected Cardy behaviour. We will do this by studying the higher temperature ( β → 0) behaviour of χ N . The asymptotics of P ( q ) are easy to determine. At high temperature It remains only to understand the asymptotics of the polynomial prefactor. We will define its logarithm to be This can be approximated as β → 0, as We see that at fixed N and large temperature ( β → 0) the W N character is dominated by P ( N ) This agrees with the free energy (3.3) in the large N limit. We now wish to understand the robustness of this result in the large N limit. We know that for any fixed N the free energy will scale like β -2 at sufficiently high temperature, but we do not know how large β must be taken in order for this result to apply. It will turn out that F ∼ β -2 only when β vanishes more quickly than N -1 in the large N limit. In other words, Cardy's formula is only valid in the regime where the temperatures are large compared to N . Moreover we will not find a phase transition as a function of temperature. To see this, let us reconsider the formula for g above. We will scale the temperature with N as β = γN -p where γ is held fixed in the large N limit. It is straightforward to expand the polynomial appearing in g at large N . Let us first consider the case p < 1, where we obtain at large N where the . . . denotes lower powers of β , which are subleading as β → 0. The sum over m gives 12 Combining this with the asymptotic expression for P ( q ) we find that the β -1 terms cancel. We are left with Let us now consider the case where p > 1. One can again expand g at large N , and in this case it is easy to see that the leading terms at large N are all positive powers of β . The free energy is dominated by P ( q ), and we obtain the usual Cardy behaviour In order to understand the transition between these two regimes, let us consider the case where p = 1, so that βN is held fixed in the large N limit. In this case Now, as βN → 0 we have so that g approaches a constant as βN → 0. In this case the free energy has the Cardy behaviour. On the other hand, Li 3 ( e -βN ) vanishes exponentially when βN is large. This leads to the nonCardy, F ∼ 1 / ( Nβ 3 ) behaviour. At intermediate values of βN the function Li 3 is perfectly smooth. We conclude that when we take the large N limit with βN fixed, the free energy - regarded as a function of βN - smoothly interpolates between the two behaviours described above. It is worth noting that the behaviour described above does not constitute a phase transition in the usual sense, in that the free energy is a smooth function of β for each of the scaling behaviours described above. However, the free energy is non-analytic in the following sense. Instead of regard- ng the free energy F ( β, N ) as a function of β and N , we can let β = γN -p and regard the free energy as a function of N,γ and p . Then in the large N limit at fixed γ we have Thus, the free energy is not an analytic function of p , however it is important to keep in mind that the free energy at fixed p is an analytic function of γ , i.e. of the temperature.", "pages": [ 18, 19, 20, 21 ] } ]

[ { "title": "Hiroyuki Funakoshi a and Kei Yamamoto a,b", "content": "a DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 9AL United Kingdom Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029, Blindern, N-0315 Oslo, Norway b E-mail: [email protected], [email protected] Abstract. We study the primordial bispectrum of curvature perturbation in the uniformdensity slicing generated by the interaction between the inflaton and isotropic background gauge fields. We derive the action up to cubic order in perturbation and take into account all the relevant effects in the leading order of slow-roll expansion. We first treat the quadratic vertices perturbatively and confirm the results of past studies, while identifying their regime of validity. We then extend the analysis to include the effect of the quadratic vertices to all orders by introducing exact linear mode functions, allowing us to make accurate predictions long after horizon crossing where the features of both the power spectrum and the bispectrum are drastically different. It is shown that the spectra become constant and scale-invariant in the limit of large e-folding. As a result, we are able to impose reliable constraints on the parameters of our theory using the recent observational data coming from Planck. Keywords: Non-Gaussianity, In-in formalism", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The prediction on the primordial density fluctuation from inflation offers an exciting opportunity to test the physics at high energy that is inaccessible for ground-based experiments. The advent of the Planck satellite, which is expected to improve the constraint on the threepoint and higher correlations of the density perturbation at recombination by a factor of 10 to 100, has prompted detailed theoretical investigations into the interaction of the inflaton [1]. So far, the efforts have been focused on the scalar self-interactions and interactions among multiple scalar fields. It is found that single-scalar models with a canonical kinetic term generically predict an undetectable level of non-Gaussian signals [2] while a scalar with the DBI action or multi-scalar dynamics, such as hybrid inflation or curvaton scenarios, can lead to significant higher-order correlations [3-5]. These models being based on the string theories in their origin of the inflaton, a detection of significant bispectrum or trispectrum can give us a clue for understanding the high-energy physics. In the context of unified theories of fundamental interactions, however, scalar fields cannot be the only ingredients of the universe. Most of the proposed theories such as superstring theories and M-theory rely on gauge symmetries, and gauge fields are indispensable to mediate interactions among the fields and in some cases to preserve supersymmetries. Even when they are absent in the fundamental Lagrangians, it is a generic prediction of dimensional reduction that a typical scalar field is coupled to some gauge fields [6, 7]. Earlier attempts to drive inflation with vector fields [8-11] turned out to be largely unsuccessful since one needs to abandon gauge symmetries, which results in the introduction of additional degrees of freedom and various instabilities [12-16]. More recently, interactions between the inflaton and gauge fields, motivated by those unified theories of interactions, have been taken into account in the context of preheating [17-22]. In addition to interesting phenomenologies including non-Gaussianity, primordial magnetic fields and gravitational waves [23-32], it was realized that the back reaction of gauge fields on the inflaton can effectively act as an extra friction term so that they slow down the rolling of the scalar field and help causing an accelerated expansion [33-39]. In fact, this back reaction can be so strong that it may generate a significant vacuum expectation value of the gauge fields and violate the isotropy of the universe. On the other hand, there has been a growing interest to maintain a small, but nonvanishing amplitude of classical gauge fields during inflation in order to explain the reported statistical anisotropy of cosmic microwave background radiation (CMBR) in WMAP 7-year [40-42]. By taking into account the aforementioned back reaction classically, the same types of scalar-gauge interactions arising from the high-energy particle theories have been found to enable acceleration of the cosmic expansion without requiring a sufficiently flat potential for the inflaton [43, 44]. This scenario turned out to be free of any classical instabilities or fine-tuning [45-50]. There have also been extensive studies on its potential imprints on CMBR and it was revealed that even a very small amplitude of background energy density of the gauge fields could result in a significant statistical anisotropy in the curvature fluctuation [51-55]. While it implies such an anisotropic vacuum expectation value of gauge fields must be severely constrained, a recent study suggests that their effect on primordial bispectum is as drastic as its linear counterpart and the resulting non-Gaussianity may still be observable. In another recent development, it has been shown that multiple vector degrees of freedom generically suppress the residual anisotropy of the background space-time through a dynamical attractor mechanism. In particular, when three or more gauge fields are coupled to a scalar field via a common gauge-kinetic function, the final state of the universe is completely isotropic regardless of initial conditions [56]. Such a circumstance may naturally be realized by non-Abelian gauge fields since the equal coupling is guaranteed by the symmetry [57]. There are other instances of isotropic inflation involving non-Abelian gauge fields which also exhibit similar attractor behaviours [58-63]. The linear perturbation of this isotropic inflation with background gauge fields has been studied, which has found that the primordial power spectrum is not strongly constrained by the current observations since the spectrum is perfectly isotropic and almost scale-invariant [64]. In this paper, we investigate the second-order perturbation of an isotropic universe containing three U (1) gauge fields and a scalar inflaton. We compute the bispectrum of curvature perturbation by deploying the in-in formalism and compare the results with the corresponding work in an anisotropic background [55], which is expected to be qualitatively similar. There are theoretical, phenomenological, and technical reasons for this particular model to be studied: Besides, it is worth emphasizing that the interactions under discussion frequently appear in supergravity theories that are low-energy effective theories of superstring theories and Mtheory. It is therefore of great interest to study observational consequences of these models as they are beyond the reach of any ground-based experiments. Our results reproduce the previous studies when the e-folding number is relatively small while extending the analysis so that it is applicable to the period long after the horizon exit. A full perturbative expansion of the Lagrangian up to cubic order is carried out and several interaction terms that are not suppressed by any of the slow-roll parameters are identified. We first treat all the interaction terms, both quadratic and cubic, as perturbation and compute the three-point function for the curvature ζ . The amplitude is solely controlled by the parameter I 2 that represents the ratio of background energy density of the gauge fields to the scalar kinetic energy density. We explicitly show that the leading contribution comes from the vertex involving a scalar field and two gauge fields, which confirms the claim of [55]. The three-point function scales as ∝ I 2 N 3 k where N k is the e-folding number after horizon exit for a mode with wavenumber k . The shape is local as has been shown in the previous studies. This results in a large f NL when the modification to the power spectrum is assumed to be small. However, we find that this conclusion is valid only if I 2 /lessmuch N -2 k , which is not satisfactory for this isotropic model since I 2 is not necessarily that small in contrast to the anisotropic cases where this is required to keep the background anisotropy within the range allowed by the observations. A reason for the limited applicability is the quadratic vertices that generate an infinite number of Feynman diagrams in the perturbative expansion even at tree level. In the second half of this paper, we take into account this fact by introducing exact linear mode functions. It turns out that one can solve the linear evolution equations analytically at superhorizon scales. By exploiting their general features, we shall prove that both power spectrum and bispectrum are convergent in the limit N k → ∞ , determining the late-time value of f NL . In order to obtain more quantitative estimates and handle the intermediate regime, we also solve the linear equations numerically from deep inside the horizon and use them in the integrand of the three-point correlators. We confirm the initial logarithmic behaviours in both power spectrum and bispectrum and their convergence at late times. It turns out that the time evolution of f NL (squeezed) displays some interesting features. It first peaks at N k ∼ 0 . 3 I -1 where the peak value scales as I -1 ; thus for certain small values of I , the latest Planck data appear to rule out the possibily of the observable modes in the CMBR arising from this intermediate phase. Then, f NL monotonically decreases and converges to a negative I -independent constant, -5 / 3. The paper is organized as follows. In the next section, after sketching the dynamics of the background evolution and introducing relevant parameters, we derive the perturbed Lagrangian up to cubic order. Section 3 gives the detailed procedure of computing the three-point function by perturbative expansion with respect to free de-Sitter mode functions. We calculate all the relevant contributions and identify the leading order term. Section 4 discusses the importance of the deviation from the de-Sitter mode functions on superhorizon scales. In the end we provide estimates for the late-time values for the power spectrum and bispectrum. In section 5, we numerically confirm these analytical results and make the prediction on non-Gaussianity more quantitative. Concluding remarks are given in section 6.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 Perturbative expansion up to cubic order", "content": "Our model contains a scalar field and several gauge fields minimally coupled to gravity: R is the Ricci scalar curvature and F a µν = ∂ µ A a ν -∂ ν A a µ ; a = 1 , 2 , 3 are three copies of U (1) gauge field strengths. These types of actions have been well studied in the context of magnetogenesis, preheating and anisotropic inflation. It has been realised that the coupling between the scalar field, which is identified to be the inflaton, and gauge fields enables an accelerated phase of expansion even with a relatively steep potential, as we will see later. We note that the energy density of the gauge fields stays constant in the first approximation in the inflating universe, violating the cosmic-no-hair conjecture. It has also been shown that the isotropic configuration of the gauge fields is a dynamical attractor of the system. Based on this result, we study the perturbation of this theory around the isotropic background with a non-vanishing triad of the gauge fields. We set 8 πG = 1 and follow the ADM formalism [65] and parametrize the metric as where The normalized extrinsic curvature of the constant time slice is given by and its intrinsic scalar curvature is Electric fields are defined to be", "pages": [ 5 ] }, { "title": "2.1 Gravity and the scalar field", "content": "The Einstein-Hilbert action in ADM formalism is given by We assume that the background is a flat Friedmann-Lemaˆıtre-Robertson-Walker space-time and write the perturbed metric components as When the problem concerns perturbation beyond linear order, one has to be careful in choosing the small quantities with respect to which the order of perturbation is determined. In the present case, we will solve the constraint equations so that φ and β i are expressed in terms of γ ij and the other matter variables. Thus, their order in perturbative expansion is subject to the equations to be solved and we should distinguish different orders as On the other hand, we avoid a similar expansion for γ ij since it will hardly appear in the following analysis as we are primarily working in the flat gauge, where the perturbation is set to be zero at each order by the choice of gauge. The only exception is the curvature perturbation on the uniform-density slice that will be expanded in terms of the dynamical variables in the flat gauge . As usual, the scalar-vector-tensor decomposition is made in order to decouple the linear-order equations. It is defined by In the uniform-density gauge, we denote the curvature perturbation -ζ = ψ and expand it as The 1 + 3 decomposition of the action for the scalar field is given by where primes denote derivatives with respect to the conformal time η . We split ϕ into the background and the perturbation: π will be treated as the dynamical variable in terms of which the perturbative expansion is defined so that we do not need to expand it further.", "pages": [ 6 ] }, { "title": "2.2 Gauge field perturbations", "content": "The Maxwell Lagrangian in the ADM formalism reads The perturbative expansion of the vector potentials yields where the background quantity A ( η ) behaves effectively as a second scalar field. The background values of A a 0 are taken to be zero by a gauge choice. As in the gravity sector, we should in principle distinguish the different orders of perturbation for the variables that are expanded in terms of the dynamical ones. However, after the adoption of flat slicing and U (1) gauge fixing, we are left only with the dynamical variables from this sector. Hence we suppress this distinction and the scalar-vector-tensor decomposition is carried out as follows:", "pages": [ 7 ] }, { "title": "2.3 Background dynamics and parameters", "content": "Before going into the perturbative analysis, we briefly review the background evolution of the system and identify the relevant parameters. The Maxwell's equation can be trivially integrated to give where c is an integration constant. As usual, we introduce the 'slow-roll' parameters which characterize the evolution of the scale factor a ( η ). The Raychaudhuri equation tells that the potential energy has to dominate over the scalar kinetic energy and the energy of gauge fields in order to have an accelerated expansion. This suggests the introduction of another parameter which controls the evolution of the inflaton. Combined with the Friedmann equation one derives which is the representative of the energy density for the gauge fields. Note that /epsilon1 ϕ ≤ /epsilon1 H where equality holds when the gauge fields vanish. Since this deviation from the single-scalar inflation plays a central role, we define the parameter which measures the ratio between the energy density of the gauge fields and the scalar kinetic energy. We note that I does not have to be small as far as the background dynamics and the power spectrum are concerned. Without loss of generality, we can assume ¯ ϕ ' > 0 and use ¯ ϕ ' = √ 2 /epsilon1 ϕ H . Now the equation of motion for ¯ ϕ gives where = ( /epsilon1 In principle, this quantity does not have to be small as long as η H /lessmuch 1, but we do assume that it is in order to control the perturbative expansion. Now by differentiating 2 2 c f H - /epsilon1 ϕ ) H a , one obtains and using the equation of motion for the scalar field yields √ The first expression tells that the slope of f ( ϕ ) must be steep in order to maintain the amplitude of gauge fields during inflation. The second implies that the gradient of potential is not necessarily small if /epsilon1 ϕ /lessmuch /epsilon1 H , or equivalently, if I /greatermuch 1. The reason is that the slow roll of the inflaton can be achieved by transferring the scalar kinetic energy to the gauge fields through the coupling f ( ϕ ). It later turns out that the perturbative approach breaks down when I > 1 anyway, so we assume that I < 1, where the usual intuition from single-scalar model works well. The higher order derivatives of V and f take complicated forms in general, but assuming the constancy of η H,ϕ and keeping only the leading-order terms in the small parameters, we obtain and Finally, we emphasize that this regime of accelerated expansion aided by gauge fields is a dynamical attractor for a wide range of potential and coupling. The readers are referred to ref. [56]. 2 2", "pages": [ 7, 8 ] }, { "title": "2.4 The cubic action for scalar perturbations", "content": "Since the curvature perturbation does not receive any contribution from vector or tensor modes at the linear order, we can eliminate them from the tree-level calculations of threepoint correlation functions arising from cubic interactions. Since their power spectra are known to be small, the higher order contribution is expected to be negligible. Hence, we focus on scalar perturbations hereafter. For the scalar perturbation, the gauge field variables are given by We use the U (1) gauge freedom to set θ ' + µ = 0. It follows that where δ indicates the perturbation of the following variables. For the metric, we adopt the flat slicing ψ = E = 0. Focusing on scalar modes, we can completely ignore γ ij . The gravitational action drastically simplifies up to cubic order to become The scalar part is the same as the standard: After some straightforward algebra, one obtains the gauge Lagrangian as where where the quadratic Lagrangian is given by It should be mentioned that we dropped the terms involving φ (2) and B (2) from the beginning since they multiply the background and linear-order constraint equations, which would be automatically satisfied in our formulation.", "pages": [ 9, 10 ] }, { "title": "2.5 Solving the linear constraints", "content": "Using the background equations and parameters, the quadratic Lagrangian can be rewritten as where we discarded the surface term and defined Varying B (1) determines φ (1) as Using this, variation of φ (1) leads to These relations will be substituted into the Lagrangian derived in the previous subsection and the curvature perturbation in the uniform-density gauge introduced in the following.", "pages": [ 10 ] }, { "title": "2.6 Curvature of the uniform-density surface", "content": "For the purpose of quantum field theory calculations in the multi-field dynamics, the most convenient gauge is the flat gauge where γ ij = h ij [3]. However, the observationally relevant quantity is the curvature perturbation in the comoving gauge R c that coincides with the curvature in the uniform-density gauge ζ beyond the horizon scale. The latter is more often picked up as done here since it possesses a desirable mathematical property. Hence, we need the transformation law between flat gauge and uniform-density gauge, which we cite from [66] as and where the right-hand sides are evaluated in the flat gauge. We defined the perturbative expansion of the energy density and a quadratic expression The background sound speed c 2 s in the present setting is with ¯ p being the background pressure. At the linear order, the energy density in the flat gauge is neatly written as The background energy density satisfies thus Therefore, the first-order curvature perturbation is given by The second-order part will be discussed later.", "pages": [ 11 ] }, { "title": "3 Analytical estimate of the bispectrum in the limit of small I", "content": "In this section, we apply the standard methods of the in-in formalism to the Lagrangian obtained in the previous section. In order to render the problem tractable, we keep only the leading-order contributions in the small parameters /epsilon1 H,ϕ , η H,ϕ . An interesting point is that even in the limit of de-Sitter space-time, the key parameter I does not necessarily vanish. Using equations (2.1) - (2.12) and substituting (2.15) and (2.16), the cubic Lagrangian (2.13) becomes We dropped τ for a reason that becomes clear soon. We assume the background is close to de-Sitter, which implies and discarded all the terms higher order in slow roll. One can rescale α to set f 0 = 1 without loss of generality. We further demand I < 1 since we would like to treat all but kinetic terms perturbatively. The factors of √ /epsilon1 ϕ -1 appearing in the cubic terms might look worrying for the validity of the perturbative approach. But when the action is written in terms of ζ , they are of the same order as the quadratic kinetic terms and the perturbative expansion should be marginally applicable. The following analysis is expected to be valid for /epsilon1 H /lessmuch I < 1. We are concerned with the three-point correlation function of the curvature perturbation which is obtained from (2.19) with (2.15) and (2.16), neglecting all the higher order terms in slow roll. The absence of τ at this linear order justifies its omission from the Lagrangian.", "pages": [ 12 ] }, { "title": "3.1 Notations", "content": "We take the free massless part of the action to be the background and treat all the other terms perturbatively. In the interaction picture, we set where the de-Sitter mode functions are defined to be The interaction Hamiltonian is given by where The term with higher spatial derivatives has been omitted. We often drop the subscript I . Note that it was claimed in [55] that H C I gives the leading contribution to the bispectrum. We shall explicitly confirm that it is the case as long as we remain within the regime of validity for the perturbative treatment of the quadratic vertices (i.e. H q I and the mass term for π ). We are going to compute the three-point correlation function in Fourier space defined by We often abbreviate it as 〈 ζ 3 k 〉 .", "pages": [ 12, 13 ] }, { "title": "3.2 The outline of the calculation", "content": "Introducing an auxiliary function the three-point function for ζ can be written as Despite the appearance of lower powers of I in the Lagrangian, the leading-order contribution to 〈 ζ 3 〉 turns out to be quadratic. 1 Then, the π 2 term in the interaction Hamiltonian is clearly irrelevant. However, we do have to keep H q I since it affects, for example 〈 ξ 3 〉 with H B I at this order. More specifically, 〈 ξ 3 〉 can be written as follows: In a similar way, 〈 ξ 2 α ' 〉 contains the quadratic term in the Hamiltonian given as Finally, 〈 ξα ' 2 〉 receives no contribution from the quadratic interaction and becomes Hence, we have to compute ten distinct integrations to fully work out 〈 ζ 3 〉 at the quadratic order in I 2 . We focus on the superhorizon limit of the spectrum, i.e. -k i η /lessmuch 0 for all i = 1 , 2 , 3.", "pages": [ 13, 14 ] }, { "title": "3.3 Summary of the results", "content": "We list the contributions from each of the ten integrations in the limit of -k i η → 0. The detailed calculations are presented in the appendix.", "pages": [ 14 ] }, { "title": "3-vertex contributions", "content": "Note that all of the remaining integrals can be carried out in the limit -k i η → 0, which result in a logarithm of -η for each integration. Therefore, 3-vertex contributions dominate over the others in superhorizon limit, as claimed in [55]. In addition, there is a contribution to bispectrum arising from second- and higher order perturbations of ζ in terms of the field variables π and α . It is evaluated for the second-order term in the appendix and shown to be of order I 2 , hence subdominant compared to the logarithms from the integrations listed above. In the end, our result is summarized as follows. At the order of I 2 , the tree-level amplitude of the three-point function in the super horizon limit becomes /negationslash where K is a reference momentum, say K = 1 3 ( k 1 + k 2 + k 3 ). While the ambiguity of K arising from the lower limits of the integrations leads to errors of order ln( k i /k j ) , i = j , for the wavelengths of interests, this should be of order 10. Since there are many other contributions of similar order which we have already ignored, it does not make sense to overly worry about this reference momentum. As we can see, the bispectrum is of local shape. In order to estimate the f NL in the squeezed limit, which is defined as we quote the result from [64] for the power spectrum Under the condition which implies we can replace /epsilon1 ϕ with /epsilon1 H , we obtain where N K is the number of efoldings experienced by the relevent modes after horizon crossing. This result qualitatively agrees with the one derived in [55] if the correction term in the denominator is ignored. However, there are a few unsatisfactory features in this result. The first is the limitation arising from our perturbative approach. Since we are sticking to perturbative expansion in terms of I , the formula (3.22) can be trusted only for since otherwise we would have to take into account the higher order terms from the Taylor expansion of the denominator. However, the condition (3.23) is much more strict than the generic one (3.21) considering that N K = -ln( -Kη ) for the modes relevant in CMBR are of order 50. Namely, the applicability of the analysis so far is limited to I 2 /lessorsimilar 10 -4 and we are unable to say anything about f NL for the range 10 -4 /lessorsimilar I 2 /lessorsimilar 1. Furthermore, the fact that 〈 ζ 3 k 〉 may grow indefinitely as long as inflation continues sounds unpleasant considering the classical stability of the quasi-de-Sitter background. It is distinct from the infrared divergence discussed in [55] which concerns the back reaction of the quantum fluctuations and loop corrections which is beyond the scope of the present article. The divergence is already there at the tree-level calculation. Motivated by this, in the next section, we shall give a more careful analysis on the superhorizon dynamics of the fluctuations.", "pages": [ 16, 17 ] }, { "title": "4 Non-perturbative treatment of the quadratic vertices", "content": "It is clear that the above approach based on the perturbative expansion in terms of I bares a limited applicability even if I /lessmuch 1. From the point of view of the classical stability of this inflationary regime shown in [56], the apparent indefinite growth of the correlation functions after horizon exit should halt sooner or later if all the relevant effects are taken into account. In the Lagrangian (3.1), we have regarded the quadratic interaction terms as perturbative corrections along with the cubic ones. In this way, the proper tree-level amplitude involves an infinite number of Feynman diagrams generated by those quadratic vertices. While we have avoided this issue by focusing on the leading-order contribution in I , one should expect a convergent result if the higher order corrections are treated appropriately. For this purpose, we investigate the linear perturbation more closely and show that both the power spectrum and the bispectrum become constant in the limit of η → 0 despite the appearance of logarithmic divergence ln( -kη ) in the perturbative analysis.", "pages": [ 17 ] }, { "title": "4.1 Linear evolution equations and their superhorizon solutions", "content": "The equations of motion at linear order are given by It turns out that one can write down analytic expressions for the solutions in the superhorizon limit. Ignoring the spatial gradients, the second immediately integrates to give where c 0 is an integration constant. We suppress its k -dependence since there should be no confusion as far as the linear theory is concerned. The same applies to the rest of the integration constants. Plugging this into the first equation, we derive whose general solution can be written as with two arbitrary constant c ± . The power exponents are given by The corresponding α is with the fourth integration constant c 1 . We used the relations", "pages": [ 17, 18 ] }, { "title": "4.2 Canonical mode functions", "content": "When the off-diagonal terms in the quadratic Lagrangian are taken into account, the introduction of mode functions is not so straightforward as with independent free fields. In this subsection, we look into the canonical formulation of the field theory. From the Lagrangian (3.1), we read off as the canonically normalised field variables. Their conjugate momenta are given by and we impose the canonical commutation relations with all the cross commutators being zero. To diagonalize the Hamiltonian, we introduce the creation and annihilation operators and expand the field operators in terms of the mode functions: Here, ( π a k , α a k ) , a = 1 , 2 are two independent solutions of equations (4.1) and (4.2). As an example, for the superhorizon solutions derived in the previous subsection, the mode functions become which are characterized by eight complex constants. In this way, we see that each field operator may excite two different particles. Conversely, for each particle species a , there are two associated mode functions ˆ u a k and ˆ v a k . This simply reflects the fact that the fields themselves do not define particles when there is a quadratic mixing term. This formulation is consistent as long as the mode functions satisfy the following conditions arising from the canonical commutators (from here on, the summation convention for indices a, b, · · · is assumed): It can be checked that they are preserved by the evolution equations (4.1) and (4.2) if they are satisfied at an initial time. Later, it proves to be useful to write down these conditions specifically for the superhorizon mode functions. Those six equations translate into algebraic conditions on the integration constants:", "pages": [ 18, 19 ] }, { "title": "4.3 Matching with the de-Sitter mode functions", "content": "From the solutions (4.3) and (4.4), it is almost obvious that the power spectrum should converge to a constant proportional to | c a 0 | 2 . In order to estimate its magnitude, however, one needs to determine the constants c a α , α = 0 , 1 , ± from appropriately initial conditions set deep inside the horizon. As a first approximation, we can match the de-Sitter mode functions: with the superhorizon counterparts (4.5) and (4.6) at the horizon crossing kη = -1. We evaluate them and their first derivatives and equate each other. This leads to the following eight equations: These easily solve as We can now estimate the final amplitude of the two-point function. Clearly, the dominant contribution at late time comes from c a 0 s. The mode functions asymptotically approach Using we derive Although the exact numerical factors should not be trusted due to the errors coming from the matching, the dependence on I 2 is generic (we will confirm this numerically). The fact that the final amplitude is divergent in the limit small I 2 might look worrying. But it is also the case in the standard single-scalar inflation where the power spectrum is formally infinite when the background is exactly de-Sitter. The same nature can be seen taking the limit of I → 0 and large e-folding number ( -kη → 0), rewriting the amplitude of the power spectrum as In the last approximation, we used (2.5) and This result is reasonable: the dependence of the power spectrum on the parameters is the same as the single-scalar inflation except that /epsilon1 ϕ is now replaced by /epsilon1 H -/epsilon1 ϕ . Recalling that it represents the energy density of the gauge fields in the unit of H 2 , the power spectrum is inversely proportional to the energy density of the background gauge fields instead of the background scalar kinetic energy. We emphasise that the only approximation we used to derive (4.10) was the matching with de-Sitter mode functions. Hence, we expect the expression is valid even for I /greaterorsimilar 1 in the superhorizon limit. Comparing the expression (4.10) with the perturbative result (3.20), one can estimate the time when the power spectrum settles down to a constant value after horizon exit. We simply equate these two in the limit of small I and infer that Beyond this point, 〈 ζ 2 k 〉 is conserved as it is in the usual adiabatic perturbation.", "pages": [ 19, 20, 21 ] }, { "title": "4.4 Estimating the superhorizon contribution to the late-time bispectrum", "content": "Under the general conditions (4.7) - (4.9) arising from the requirement of canonical commutation relations, one can show that the tree-level amplitude of the three-point function is convergent in the superhorizon limit too. First of all, let us introduce the mode functions for ζ by whose superhorizon limit becomes We restored the k -dependence of the coefficients. Now the tree-level amplitude does not involve any multiple integrals and we derive Note that Because of the conditions (4.9), we have Thus, we see that the lowest power of the integrand for the first term must come from The time dependence of its dominant contribution is given by both of which have the total power of -1 and contain a positive power of η , which implies the integration in the limit -η → 0 is convergent. The bispectrum generated by π 3 vertex long after horizon exit is therefore Similarly, using one can show that the second and third integrals give convergent results as and respectively. In the end, the late-time contribution to the three-point function becomes Assuming the final bispectrum is dominated by the superhorizon contribution, which appears to be the case in the evidence of the numerical study in the next section, one can now estimate the final value of f NL in the squeezed limit k 1 /lessmuch k 2 ∼ k 3 . Note that the dependence of c a 0 ( k ) on k derived by matching is rather generic. Then, in this limit, we have Combined with (4.10), the appropriately normalised f NL is computed as where the last limit was taken for I → 0 ⇔ /epsilon1 ϕ → /epsilon1 H . This beautiful result will be confirmed in the following section.", "pages": [ 21, 22, 23 ] }, { "title": "5 Numerical calculation of exact tree-level amplitude", "content": "Following from the previous section, here we treat the quadratic vertices non-pertubatively with the only difference being that now we calculate most of the contributions numerically. The aim is to negate the need for making any approximations and therefore to make our result more quantatively accurate. In the analytic results from section 4, we derived the qualitative features of power spectrum and bispectrum by assuming that at horizon crossing the mode functions are those of the free de-Sitter case and applying the superhorizon approximation ( kη = 0) as soon as the mode crosses the horizon ( kη = -1). We have been able to estimate the final amplitude of power spectrum, the time of transition from the perturbative regime discussed in section 3 to the one dictated by the superhorizon mode functions, and calculate the superhorizon contribution to the bispectrum in the limit kη → 0. However, we are yet to have a reliable estimate for the time evolution (or equivalently scale dependence) of the bispectrum. Here, we calculate the exact mode function, first setting the π and α fields in the Bunch-Davies vacuum deep inside the horizon, solve the coupled linear equations of motion numerically until the modes are far into the superhorizon regime. At this point we switch to using the superhorizon equations of motion and use the analytic solution - this is simply to avoid numerical instabilities encountered in this calculation. Now we only use the analytic superhorizon solution for -kη /lessmuch 1, so the error introduced by doing so is negligible. This leaves the factors of /epsilon1 H and /epsilon1 ϕ in the Lagrangian and the definition of the curvature perturbation; in the numerical calculation below, they will be set to 1. It can be easily seen that these two parameters can be reintroduced at the end as an overall multiplicative factor of /epsilon1 H //epsilon1 ϕ = 1 + I 2 for the value of f NL computed. The ζ mode functions, power spectrum and bispectrum will need to be multiplied by H (1 + I 2 ) -1 /epsilon1 -1 2 ϕ , H 2 (1 + I 2 ) -2 /epsilon1 -1 ϕ and H 4 (1 + I 2 ) -3 /epsilon1 -2 ϕ respectively, to restore the dependence on these constants. Ultimately, we will be interested in the value of f NL (squeezed) here. Factors of f 0 H present in the Lagrangian (3.1) will be absorbed into the definition of the α field, and the overall multiplicative factor of H -2 in front of the Lagrangian will not affect the value of f NL . We therefore set H = 1; for quantities such as the power spectrum or bispectrum, reintroducing H will be a matter of an overall multiplicative factor which will be included in the plots. When reintroducing H , I will need to be replaced with I /H .", "pages": [ 23 ] }, { "title": "5.1 Subhorizon linear evolution and initial conditions", "content": "Again, the linear equations of motion (4.1) and (4.2) are used, this time keeping the gradient terms. Since the evolution equations do not admit an analytic solution, we will find solutions numerically. For computational convenience, the canonical variables used in this section are π and α , and so their conjugate momenta are given by The initial conditions for the mode functions are given by the Bunch-Davies condition, expressed here in terms of the canonical variables for each k mode as: The point here is that these are the conditions required on the mode functions for the fields to be in the Bunch-Davies vacuum deep inside the horizon, and for the canonical commutation relations to hold. Given the definitions of the conjugate momenta, the initial conditions for solving the linear evolution equations will then be given by (5.1) and (5.2) along with in place of (5.3) for some -kη /greatermuch 1. For the results in this section, the initial conditions for the modefunctions were set at ( η ) init = -1000.", "pages": [ 23, 24, 25 ] }, { "title": "5.2 Numerical calculation of the ζ power spectrum", "content": "Now we calculate the power spectrum for the curvature perturbation. A similar analysis has been carried out in [52], so the results in this section are to recap these results, and to verify that these results are consistent with the perturbative expression for the curvature power spectrum [56]. Using the π and α mode functions we are able to define the ζ mode function as and hence The ζ mode functions are plotted in figures 1, 2, 3 and 4, while the numerical and analytic results for the power spectrum are shown in figures 5, 6 and 7. The perturbative solution for the time evolution of 〈 ζ k 2 〉 is shown to be useful only for small I (figure 5), while the analytic estimate for the final value of the power spectrum, derived in section 4, is valid for values of I up to around 0.2 (figure 7). The lack of quantitative agreement beyond I ∼ 0 . 2 is presumably due to the error arising from the matching since for larger values of I , the numerical calculations (figrues 1, 2, 3, 4) show a significant deviation from the de-Sitter mode functions around horizon crossing. The characteristic timescale for the time evolution for 〈 ζ k 2 〉 before it reaches constant is shown to be I -2 for I /lessorsimilar 0 . 1, in agreement with the analytical estimate (4.12) from the previous section (figure 6). This result has a significant implication on the validity of the perturbative treatment of quadratic vertices discussed at the end of section 3. The transition to constant regime occurs around which is much later than the time at which the correction term to the power spectrum becomes comparable to the leading-order term N k ∼ I -1 . In fact, the numerical evidence suggests that the perturbative formula (3.20) is valid right up to I 2 /lessorsimilar N -1 k , or for CMBR scale ( N k ∼ 50), I /lessorsimilar O (0 . 1). This observation plays a key role in imposing the observational constraint from Planck later.", "pages": [ 25, 26 ] }, { "title": "5.3 Numerical calculation of the ζ bispectrum", "content": "By solving the coupled linear evolution equations, we in effect include the contribution from the infinitely many tree-level Feynman diagrams coming from the quadratic H q term, and hence obtain a result correct to all orders in I (provided loop contributions are negligible). Therefore, the exact tree-level amplitude for the bispectrum, by standard application of Wick's theorem, is given by In particular, we now only have to compute 1-vertex terms. The evaluation of the integrand turns out to be a numerically unstable process sufficiently far outside the horizon, requiring a very precise cancellation of terms. It therefore becomes impractical to carry out the calculation with the numerically solved mode functions beyond a certain point. To overcome this difficulty, for -kη < 10 -5 we switch to using the analytic superhorizon solution discussed in the previous section. The only difference is in the matching of the analytic superhorizon solution; here we evaluate the numerical mode functions (and their time derivative) at -kη = 10 -5 and use these as the matching conditions for the analytic superhorizon solution. Then, for -kη < 10 -5 the time integrals in the above expression are computed analytically, therefore avoiding the problem of numerical instabilities. When performing the first stage of this computation (the numerical stage), we employ the technique recently developed in [67]. As we will see later, the bispectrum (or more precisely, the shape function) is peaked in the squeezed limit, and therefore to concentrate on the salient features we will restrict most of our analysis to the squeezed limit. We start by crosschecking our numerical calculations against the perturbative results from section 3 (figure 8), for small I (= 10 -3 ) for sometime after horizon exit where the perturbative treatment of the quadratic vertex H q is justified. This underwrites the overall consistency between the analytical and numerical methods. In figure 9, we confirm the convergence of the bispectrum generated by each cubic vertex. As one can see, while H C is the dominant contribution in the perturbative regime as it grows the fastest (ln( -kη )) 3 , it is overtaken by H B when the perturbative approximation breaks down. It also exhibits an approximate I -4 scaling of the final value of bispectrum, which is equivalent to the I independence of f NL that was inferred at the end of the previous section. The characteristic timescale for the transition to constant is again shown to be I -2 . In figure 11, we plot the intermediate time evolution of f NL , with the numerical calculation on the left panel and perturbative result on the right. For N k /lessmuch I -1 , the power spectrum is essentially constant and f NL grows as N 3 k . When N k /greaterorsimilar 0 . 1 I -1 , the power spectrum starts to be overtaken by the correction term and scale as N 2 k , which results in the peak around N k ∼ 0 . 3 I -1 . The maximum value appears to scale as I -1 , which means it may well be observable for a small value of I . Since these peaks occur on timescales ∝ I -1 , the time dependence (and therefore scale dependence) of f NL around this maximum can be understood by the perturbative results where analytical expressions are available. From the (ln( -kη )) 3 perturbative growth in the bispectrum, one may expect that the superhorizon contribution to the bispectrum dominates over the subhorizon contribution; in figure 10 we verify that this is indeed the case for most values of I . The bispectrum is evaluated at η = 0, and the kη < -1 (subhorizon) and kη > -1 (superhorizon) contributions to the time integral are plotted separately as a function of I . The two become comparable only as I reaches order unity, and for I /lessorsimilar 0 . 1 the subhorizon contribution is negligible. k 3 However, for later times /greaterorsimilar I -2 this is no longer the case, with f NL turning negative; this behaviour is shown in figure 12, where we plot the late-time evolution of f NL in the squeezed limit for different values of I . The final convergent value appears to be -5 / 3 independent of I , in agreement with the results of the previous section. It is also seen in figure 13 where the final value is presented as a function of I . We suspect that the cause of irregular behaviour for I /greaterorsimilar 0 . 1 is due to the significant contribution from the subhorizon evolution. We also note that given that there are two scalar degrees of freedom here, the single-field consistency relation [2] does not hold. We conclude this section by mentioning a few words about the shape of the bispectrum. As can be seen from figure 13, the bispectrum is peaked in the squeezed limit, which is expected given the fact that it is predominantly determined by the superhorizon evolution which tends to generate local bispectra. Provided that we wait until all relevant modes have become constant (as done for the plot), it is perfectly scale-invariant too. This can be understood by noting that since the background geometry is de-Sitter and the interaction terms are de-Sitter invariant, the correlation functions for the perturbations which have become constant are scale-invariant; in particular, for the modes which have settled down to the final value, both the power spectrum and the bispectrum are scale-invariant. By similar arguments, where we have plotted the time evolution of any quantity such as f NL , they can be used to read off the scale dependence at any given time.", "pages": [ 26, 27, 28 ] }, { "title": "6 Implications and concluding remarks", "content": "We have studied the perturbation of a model of inflation where a stable isotropic phase of inflation is realized by a scalar field coupled with a triplet of Abelian gauge fields. We derived the general action for scalar perturbation up to cubic order and identified all the relevant terms in the limit of vanishing slow-roll parameters. Using the standard method of in-in formalism, we first treated both the quadratic and cubic vertices perturbatively and computed the bispectrum at the leading order in the expansion parameter I . The resulting expression was consistent with the previous studies and f NL in the squeezed limit was shown to be proportional to I 2 N 3 k where N k is the e-folding number after the relevant modes exit the horizon. We then pointed out the limited applicability of this approach even for I /lessmuch 1 and rectified it by introducing the exact linear mode functions which take into account the effect of the infinite number of tree-level diagrams generated by the quadratic vertices. Solving the linear evolution equations analytically in the superhorizon limit, we proved that both the power spectrum and the bispectrum are convergent in the limit N k →∞ , with the late-time bispectrum being local in shape. In order to obtain a more quantitative estimate of the bispectrum and f NL , we carried out an extensive numerical analysis employing in part the recently developed technique [67]. We confirmed the analytical results and found a number of interesting features. In calculating the time evolution of f NL in the squeezed limit, we find that it peaks at some characteristic time after horizon crossing, with this peak value scaling as I -1 . After peaking, it settles down to the same value (independent of I for small I ) as was estimated analytically: f NL = -5 3 . We first emphasise that our analysis here is complete at tree-level. It indicates the overall consistency of this model in the classical regime; any fluctuations present at horizon crossing remain bounded and so do their correlation functions (at the very least at the 2-point and 3-point level). For a certain range of values of I , the model is ruled out by the latest observational constraint on f NL by Planck. However, for large values of I approaching 1, the length scales currently observable would come from the late-time stage where f NL is of order unity and hence within Planck bounds. Similarly, for small I , f NL grows sufficiently slowly after horizon crossing and will remain within current observational constraints. We now discuss the implications of our calculations. Recent Planck [68] data suggest f NL should be of order unity. The modes observable in our Universe typically experience around 50 e-folds after horizon crossing and we already argued that the perturbative expression (3.22) is valid as long as I /lessorsimilar 0 . 1. Thus excluding f NL > 10, we can constrain I to satisfy either I 2 /lessorsimilar 10 -7 or I 2 /greaterorsimilar 10 -3 . For I /greaterorsimilar 0 . 1, our numerical calculations suggest that f NL is in the constant regime for N k ∼ 50 and its value is of order unity (left panel in figure 13). One can expect that the same qualitative features will also apply to the anisotropic models where the background is permeated by one or two gauge fields with non-vanishing vacuum expectation values. The difference is that there the vector fields also contribute to the spatial anisotropy and there is a strict upper bound for I . It is going to be difficult to repeat our analysis for anisotropic models since the consistent perturbative expansion requires the inclusion of vector and tensor modes which are coupled to the scalars through the background anisotropy. For this purpose, it would be interesting to look into the relation between our results and the deltaN formalism [69]. In fact, the isotropic case can be regarded as a particular two-scalar model and the formalism should apply without any problem. Since the convergence of the power spectrum and bispectrum is based on the superhorizon evolution, the deltaN formalism will reproduce them in a more elegant manner. Since its mathematical basis resides in the equivalence of the superhorizon curvature perturbation to the background evolution of the FLRW universe [70], an appropriate extension to anisotropic backgrounds sounds plausible and can be a powerful tool to handle the complicated interactions among different modes. Another important theoretical issue is consistency of the quantum field theory in the existence of background gauge fields. The authors of [55] claimed that the infrared contribution of the one-loop diagrams can be interpreted as the rescaling of the background vacuum expectation value of the gauge fields so as to take into account the quantum mechanically created modes that froze in outside the horizon. Although we have not discussed this issue in the present article, it will be certainly an interesting direction of further research. Finally, it is in principle straightforward to extend our analysis to inflationary models with non-Abelian gauge fields, either the one based on gauge-kinetic coupling [47, 57] or Chern-Simons coupling [60]. Given the qualitative similarity between Abelian and nonAbelian models when all the vertices are treated perturbatively, it is natural to expect that a similar convergent result in the limit of large e-folding can be established, although it is mathematically far from obvious. From a phenomenological point of view, it would be important to clarify the difference among different scenarios so that one is able to observationally distinguish between them.", "pages": [ 29, 30, 31, 32 ] }, { "title": "Acknowledgments", "content": "We would like to thank Xingang Chen, Misao Sasaki and Jiro Soda for useful comments and Federico Urban for interesting discussions. KY is also greateful to the support and hospitality of the Institute of Theoretical Astrophysics in the University of Oslo where a part of this work was completed.", "pages": [ 32 ] }, { "title": "A Details of the perturbative calculation of bispectrum", "content": "Here, we give the details of the integrations and handling of second-order perturbations necessary for determining the leading-order bispectrum (3.18). First of all, let us introduce the following mode functions: It will be useful later to note that", "pages": [ 32 ] }, { "title": "A.1 1-vertex contributions", "content": "Let us start from the single integration (3.8). One can easily go to Fourier space and derive At a first glance, the integral looks divergent as η → 0 even if the factor of η 3 in (3.7) is taken into account. However, it is not the case since we have Therefore, an integration by parts gives The same type of cancelation of power holds for the remaining integrals. It is also helped by the fact that which is a manifestation of the constancy after horizon exit of the de-Sitter mode functions. Repeating another integration by parts, we are left with The third line is finite. The leading contribution is logarithmically divergent in η as The cross correlations (3.14) and (3.17) are in principle similar. We have In carrying out these integrals, it is useful to note that It will be later useful to derive the explicit forms of T k and H k . Straightforward integrations by parts lead to and and All the terms with negative powers of η cancel when taking the imaginary parts, due to the rapid decay of the imaginary part of the propagators beyond the Hubble horizon; The end results are again logarithmic dependences on η ;", "pages": [ 32, 33, 34 ] }, { "title": "A.2 2-vertex contributions", "content": "Let us start from the term (3.9). By definition, the connected tree-level contribution is given by Given we can write down its Fourier transform as Taking the commutators, it becomes Now our task is to carry out the integral Substituting the integrated expression for T k 1 ( η 1 ), the single integral arising from its second line gives at most ∝ ln( -η ) in the limit η → 0 since any power divergence disappears after taking the imaginary part as demonstrated for the 1-vertex cases. 2 We then only have to check if the remaining term gives rise to similar logarithmic contributions. Evaluating the double integral, we obtain All the integrations are at most of order ln( -η ) as η → 0 except for the first line whose leading term yields and behaves as (ln( -η )) 2 . The leading-order behaviors for the other contributions are essentially the same. For (3.10), the amplitude reads Defining we retain only the most divergent term to obtain For (3.15), and in Fourier space, it becomes Then, we derive As before, integration goes as Finally, (3.16) yields The leading-order contribution is", "pages": [ 34, 35, 36 ] }, { "title": "A.3 3-vertex contributions", "content": "We saw that the 1-vertex terms that involve only single time integrals resulted in ∝ ln( -η ) while the leading contributions from 2-vertex terms come from double integrals and proportional to (ln( -η )) 2 . Hence, one expects that 3-vertex contributions behave like (ln( -η )) 3 and dominate the tree-level amplitude at the order I 2 . This was also the result of [55]. We explicitly prove it and derive the coefficients in front. The principle of the calculations is the same as the previous sections although the algebra gets increasingly complicated. First of all, we write ×〈 3 2 〉〈 3 1 〉 〈 3 〉〈 2 〉〈 1 〉 and In the end, our integrand is ×/Ifractur ( g k 1 u k 1 ( η 1 ) ) /Ifractur ( g k 2 u k 2 ( η 2 ) ) /Ifractur ( g k 3 h k 1 ( η 1 ) h k 2 ( η 2 ) h k 1 ( η 3 ) h k 2 ( η 3 ) u k 3 ( η 3 ) ) +5 perms Anticipating the cancellation of terms with negative powers of η , we seek the expected (ln( -η )) 3 contribution. It can only come from where we integrated by parts for η 2 . We perform another integration by parts with η 1 as follows: Only the second term can give rise to the sought dependence on η . We derive For (3.12), we have ×〈 3 2 〉〈 2 1 〉 〈 3 〉〈 2 〉〈 1 〉 The integration results in ×/Ifractur ( g k 1 u k 1 ( η 1 ) ) /Ifractur ( g k 2 h k 1 ( η 1 ) u k 2 ( η 2 ) h k 1 ( η 2 ) ) /Ifractur ( g k 3 h k 3 ( η 2 ) u k 3 ( η 3 ) h k 3 ( η 3 ) ) +5 perms For (3.13), we have and and Similar to the other two, we obtain", "pages": [ 37, 38, 39 ] }, { "title": "A.4 Second order curvature perturbation and summary", "content": "So far, we have only discussed the linear part of the curvature perturbation since it is the only term that picks up contributions from cubic vertices at tree level. The second- and higher order terms in ζ also contribute to the bispectrum, however, through the combinations such as 〈 ζ (1) ζ (1) ζ (2) 〉 . Since it is impossible to examine at all orders if they give any contribution within the order I 2 , here we just look at the second-order term and check that they do not become dominant over the 3-vertex contributions derived in the previous subsection. First, we note that ignoring higher order corrections in /epsilon1 H,ϕ , η H,ϕ and the terms with spatial derivatives and using (2.17), we can rewrite Ξ ij as Comparing equation (3.3) with we see the second term is suppressed by a factor of /epsilon1 H . Throwing it away, equation (2.18) yields Expanding the energy-momentum tensor up to second order, we find The appearance of φ (2) forces us to look into the constraint equations at the second order. In fact, they are not too bad for the scalar perturbations in the flat gauge. The relevant equation is obtained from variation of N i in the ADM formalism and reads Expanding it to the second order, we find In the end, its contribution is subdominant. Keeping the leading-order terms in slow roll and discarding higher spatial derivatives, we obtain The contribution to the bispectrum is The first and the last terms in (A.5) are subdominant. The term quadratic in ζ (1) gives contributions such as which exist regardless of the dynamics and give | f NL | /lessorsimilar 1. The rest are the generic effects of the background gauge fields. Looking at (2.19), we see that the leading order contributions in I are quadratic, which involve terms such as and At the leading order, π and α ' are essentially just u k ( η ) and h k ( η ) /η 4 , therefore their contribution will be constant of | f NL | ∼ O ( I 2 ).", "pages": [ 39, 40 ] } ]

[ { "title": "STEFFEN AKSTEINER AND LARS ANDERSSON", "content": "Abstract. Maxwell test fields as well as solutions of linearized gravity on the Kerr exterior admit non-radiating modes, i.e. non-trivial time-independent solutions. These are closely related to conserved charges. In this paper we discuss the non-radiating modes for linearized gravity, which may be seen to correspond to the Poincare Lie-algebra. The 2-dimensional isometry group of Kerr corresponds to a 2-parameter family of gauge-invariant non-radiating modes representing infinitesimal perturbations of mass and azimuthal angular momentum. We calculate the linearized mass charge in terms of linearized Newman-Penrose scalars.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The black hole stability problem, i.e. the problem of proving dynamical stability for the Kerr family of black hole spacetimes, is one of the central open problems in General Relativity. The analysis of linear test fields on the exterior Kerr spacetime is an important step towards the full non-linear stability problem. For test fields of spin 0, i.e. solutions of the wave equation ∇ a ∇ a ψ = 0 , estimates proving boundedness and decay in time are known to hold. See [20, 14, 2, 42] for references and background. The field equations for linear test fields of spins 1 and 2 are the Maxwell and linearized gravity 1 equations, respectively. These equations imply wave equations for the Newman-Penrose Maxwell and linearized Weyl scalars. In particular, the Newman-Penrose scalars of spin weight zero satisfy (assuming a suitable gauge condition for the case of linearized gravity) analogs of the Regge-Wheeler equation. These wave equations take the form where for spin s = 1 , c 1 = 2 , ψ 1 = Ψ -1 /slash.left 3 2 φ 1 , while for spin s = 2 , c 2 = 8 , and ψ 2 = Ψ -2 /slash.left 3 2 ˙ Ψ 2 . Here ˙ Ψ 2 is the linearized Weyl scalar of spin weight zero. See [1] for details. As these scalars can be used as potentials for the Maxwell and linearized Weyl fields, one may apply the techniques developed in the previously mentioned papers to prove estimates also for the Maxwell and linearized gravity equations. This approach has been applied in the case of the Maxwell field on the Schwarzschild background in [7]. In contrast to the spin-0 case, the spin 1 and 2 field equations on the Kerr exterior admit non-trivial finite energy time-independent solutions. We shall refer to time-independent solutions as non-radiating modes. There is a close relation between gauge-invariant non-radiating modes and conserved charge integrals. For the Maxwell field, there is a two-parameter family of non-radiating, Coulomb type solutions which carry the two conserved electric and magnetic charges. In fact, a Maxwell field on the Kerr exterior will disperse exactly when it has vanishing charges. For linearized gravity, however, there are both non-radiating modes corresponding to gauge-invariant conserved charges, and 'pure gauge' non-radiating modes. Thus conditions ensuring that a solution of linearized gravity will disperse must be a combination of charge-vanishing and gauge conditions. From the discussion above, it is clear that in order to prove boundedness and decay for higher spin test fields on the Kerr exterior, it is a necessary step to eliminate the non-radiating modes. Due in part to this additional difficulty, decay estimates for the higher spin fields have been proved only for Maxwell test fields. See [7] for the Schwarzschild case and [3] for the Kerr case. In view of the just mentioned relation between non-radiating modes and charges, an essential step in doing so involves setting conserved charges to zero. In order to make effective use of such charge vanishing conditions, it is necessary to have simple expressions for the charge integrals in terms of the field strengths. The main result of this paper is to provide an expression for the conserved charge corresponding to the linearized mass, in terms of linearized curvature quantities on the Kerr background. We start by discussing the relation between charges and non-radiating modes for the case of the Maxwell field. Let the symmetric valence-2 spinor φ AB be the Maxwell spinor 2 , i.e. a solution of the massless spin-1 (source-free Maxwell) equation and let F ab = φ AB /epsilon1 A ' B ' be the corresponding complex self-dual two-form. The Maxwell equation takes the form d F = 0 and hence the charge integral depends only on the homology class of the surface S . Here real and imaginary parts correspond to electric and magnetic charges, respectively. The Kerr exterior, being diffeomorphic to R 4 with a solid cylinder removed, contains topologically non-trivial 2-spheres, and hence the Maxwell equation on the Kerr exterior admits solutions with non-vanishing charges. In view of the fact that the charges are conserved, it is natural that there is a time-independent solution which 'carries' the charge. In Boyer-Lindquist coordinates, this takes the explicit form where c is a complex number, and ι A , o A are principal spinors for Kerr. In order to prove boundedness and decay for the Maxwell field, it is necessary to make use of the above mentioned facts, see [3]. In particular, one eliminates the non-radiating modes by imposing the charge vanishing condition Written in terms of the Newman-Penrose scalars φ I , I = 0 , 1 , 2 , the charge vanishing condition (2) in the Carter tetrad [45] takes the form [3] where S 2 ( t, r ) is a sphere of constant t, r in the Boyer-Lindquist coordinates, V L = ∆ /slash.left( r 2 + a 2 ) 2 and d µ = sin θ d θ d ϕ . This yields a relation between the /lscript = 0 , m = 0 spherical harmonic of φ 1 and the /lscript = 1 , m = 0 spherical harmonics with spin weights 1 , -1 of φ 0 , φ 2 , respectively. Next, we consider the spin-2 case. Recall that the Kerr spacetime is a vacuum space of Petrov type D and hence, in addition to the Killing vector fields ∂ t , ∂ φ admits a 'hidden symmetry' manifested by the existence of the valence-2 Killing spinor κ AB = ψι ( A o B ) . Here the scalar ψ is determined up to a constant, which we fix by setting 3 Mψ -3 = -Ψ 2 on a Kerr background. In this situation, one may consider the spin-lowered version of the Weyl spinor, which is again a massless spin-1 field and hence the complex self-dual two-form satisfies the Maxwell equations d M= 0 . The charge for this field defined on any topologically non-trivial 2-sphere in the Kerr exterior is cf. [32] for a tensorial version (the calculation has been done much earlier in [34], but not in the context of Killing spinors and spin-lowering). Here M is the ADM mass [4] of the Kerr spacetime 4 . The relation between the mass and charge for the spin-lowered Weyl tensor M is natural in view of the fact that the divergence is proportional to ∂ t , see the discussion in [38, Chapter 6]. Note that the charge (4) is in general complex. The imaginary part corresponds to the NUT charge, which is the gravitational analog of a magnetic charge. Details are not discussed in this paper, see [39] for the construction of charge integrals in NUT spacetime. For linearized gravity on the Kerr background, the non-radiating modes include perturbations within the Kerr family, i.e. infinitesmal changes of mass and axial rotation speed. We denote the parameters for these deformations ˙ M, ˙ a . Since M,a are gauge-invariant quantities, it is not possible to eliminate these modes by imposing a gauge condition. A canonical analysis along the lines of [28], see below, yields conserved charges corresponding to the Killing fields ∂ t , ∂ φ , which in turn correspond to the gauge invariant deformations ˙ M, ˙ a mentioned above. The infinitesimal boosts, translations and (non-axial) rotations of the black hole yield further non-radiating modes which are, however, 'pure gauge' in the sense that they are generated by infinitesimal coordinate changes. If one imposes suitable regularity 5 conditions on the perturbations which exclude e.g. those which turn on the NUT charge, a 10-dimensional space of non-radiating modes remains. This is spanned by the 2-dimensional space of non-gauge modes which carry the ˙ M, ˙ a charges, together with the 'pure gauge' non-radiating modes, and corresponds in a natural way to the Lie algebra of the Poincare group. It can be seen from this discussion that a combination of charge vanishing conditions and gauge conditions allows one to eliminate all non-radiating solutions of linearized gravity. The constraint equations implied by the Maxwell and linearized gravity equations are underdetermined elliptic systems, and therefore admit solutions of compact support, see [16] and references therein. In particular, one may find solutions of the constraint equations with arbitrarily rapid fall-off at infinity. The corresponding solutions of the Maxwell equations have vanishing charges. For the case of linearized gravity, the charges corresponding to ˙ M, ˙ a vanish for solutions of the field equations with rapid fall-off at infinity. For such solutions, all non-radiating modes may therefore be eliminated by imposing suitable gauge conditions. The following discussion may easily be extended to the Einstein-Maxwell equations. Given an asymptotically flat vacuum spacetime ( N,g ab ) , a solution of the linearized Einstein equations ˙ g ab (satisfying suitable asymptotic conditions) and a Killing field ξ a ∂ a we have that the variation of the Hamiltonian current is an exact form, which yields the relation For the case of ξ = ∂ t , and considering solutions of the linearized Einstein equations on the Kerr background we have, following the discussion above, Here, P ξ ; ∞ is the Hamiltonian charge at infinity, generating the action of ξ , Q [ ξ ] is the Noether charge two-form for ξ , and Θ is the symplectic current three-form, defined with respect to the variation ˙ g ab . We use a ˙ to denote variations along ˙ g ab , thus ˙ P ξ ; ∞ and ˙ Q [ ξ ] denote the variation of the Hamiltonian and the Noether two-form, respectively. The integral on the right hand side of (5) is evaluated over an arbitrary sphere, which generates the second homology class. Working with the Carter tetrad, let Ψ i , i = 0 , /uni22EF , 4 be the Weyl scalars and let Z I , I = 0 , 1 , 2 denote the corresponding basis for the space of complex, self-dual two-forms, see section 2 for details. In this paper we shall show that the natural linearization of the spin-lowered Weyl tensor M is the two-form As will be demonstrated, see section 5 below, ˙ M is closed, and hence the integral defines a conserved charge. A charge vanishing condition for the linearized mass, analogous to the one discussed above for the charges of the Maxwell field, may be introduced by requiring that this integral vanishes. The coordinate form of this charge vanishing condition is which should be compared to the corresponding condition for the Maxwell case, cf. (3). Here, ˙ ̂ Ψ 2 and ˙ Ψ diff are suitable combinations of the linearized curvature scalars ˙ Ψ 1 , ˙ Ψ 2 , ˙ Ψ 3 and linearized tetrad. Let ˙ g ab be a solution of the linearized Einstein equation on the Kerr background, satisfying suitable asymptotic conditions, and let ˙ M be the corresponding perturbation of the ADM mass. Letting S = S 2 ( t, r ) and evaluating the limit of (6) as r → ∞ one finds, in view of the fact that (6) is conserved, the identity for any smooth 2-sphere S in the exterior of the Kerr black hole. Thus we have the relation for any surface S in the Kerr exterior. We remark that the left hand side of (8) can be evaluated in terms of the metric perturbation using the expressions for Q and Θ given in [28, section V]. On the other hand, the right hand side has been calculated in terms of linearized curvature. It would be of interest to have a direct derivation of the resulting identity. The canonical analysis following [28] which has been discussed above shows that in addition to the conserved charge corresponding to ˙ M , equation (5) with ξ = ∂ φ , the angular Killing field, gives a conserved charge integral for linearized angular momentum ˙ a . If ∂ φ is tangent to S , then the term ∂ φ · Θ does not contribute in (5). We remark that an expression for ˙ a for linearized gravity on the Schwarzschild background was given in [30, section 3]. A charge integral for ˙ a for linearized gravity on the Kerr background will be considered in a future paper. Remark 1.1. (1) There are many candidates for a quasi-local mass expression in the literature including, to mention just a few, those put forward by Penrose, Brown and York, and Wang and Yau. See the review of Szabados [41] for background and references. Although as discussed above, cf. equation (4) , for a spacetime of type D, there is a quasi-local mass charge, it must be emphasized that for a general spacetime on cannot expect the existence of a quasi-local mass which is conserved , i.e. independent of the 2-surface used in its definition. The same is true for linearized gravity on a general background. Thus the existence of a conserved charge integral for the linearized mass is a feature which is special to linearized gravity on a background with Killing symmetries. This paper is organized as follows. In section 2, we introduce bivector formalism. Conformal Killing Yano tensors and Killing spinors are discussed in section 3. Section 4 deals with conserved charges for spin-2 fields on Minkowski (§4.1 ) and type D spacetimes (§4.2). The main result, a charge integral in terms of linearized curvature, is derived in section 5, and finally, section 6 contains some concluding remarks.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "2. Preliminaries and notation", "content": "Let ( N,g ab ) be a 4 dimensional Lorentzian spacetime of signature + - --, admitting a spinor structure. Although most of the results can be generalized to the electrovac case with cosmological constant, we restrict in this paper to the vacuum case. In particular, we consider test Maxwell fields and linearized gravity on vacuum type D background spacetimes. Let o A , ι A be a spinor dyad, normalized so that o A ι A = 1 , and let be the corresponding null tetrad, satisfying l a n a = -m a ¯ m a = 1 , the other inner products being zero. The 2-spinor calculus provides a powerful tool for computations in 4-dimensional geometry. The GHP formalism deals with dyad (or equivalently tetrad) components of geometric objects and exploits the simplifications arising by taking into account the action of dyad rescalings and permutations. These formalisms are closely related to the less widely used bivector formalism [34, 6, 9, 27] in which the basic quantity is a basis for the 3-dimensional space of complex selfdual two-forms. A two-form Z is called self-dual, if ∗ Z = i Z and anti self-dual, if ∗ Z = -i Z . Given a spinor dyad, a natural choice 6 is where the notation 2 x [ a y b ] = x a y b -y a x b for anti symmetrization and 2 x ( a y b ) = x a y b + y a x b for symmetrization is used. We use capital latin indices I, J, K taking values in 0 , 1 , 2 for the elements in the bivector triad Z I . The metric g ab induces a triad metric G IJ and its inverse G IJ given by Here, · is the induced inner product on two-form, Z I · Z J = 1 2 Z I ab Z Jab . Triad indices are raised and lowered with this metric, More general we have", "pages": [ 5, 6 ] }, { "title": "Proposition 2.1.", "content": "with /epsilon1 JKL the totally antisymmetric symbol fixed by /epsilon1 012 = 1 . A real two-form F ab , e.g. the Maxwell field strength, has spinor representation It is equivalent to the symmetric 2-spinor φ AB = φ 2 o A o B -2 φ 1 o ( A ι B ) + φ 0 ι A ι B , where the six real degress of freedom of F ab are encoded in 3 complex scalars So the real two-form has bivector representation or in index notation φ I = F · Z I and F = φ I Z I + φ I Z I . The Weyl tensor is a symmetric 2-tensor over bivector space and has spinor representation where Ψ ABCD is a completely symmetric 4-spinor. The 10 degrees of freedom of the Weyl tensor are given by 5 complex scalars 7 Similarly we could have used the Weyl 2-bivector which relates to the real Weyl tensor via Because of different conventions and normalisations in the literature [34, 6, 9, 27], we rederive here the equations of structure in bivector formalism. Based on Cartan's equations of structure for tetrad one-forms 8 Bianchi identities and definitions of connection one-forms σ J and curvature two-forms Σ J in bivector formalism, we find Proposition 2.2. The bivector equations of structure are while the Bianchi identities read Here ∧ is the usual wedge product of one-forms σ J and two-forms Z J , Σ J . Proof. Expanding the bivectors Z J = 1 2 Z J ab e a ∧ e b , we find where proposition 2.1 has been used in the third step. For the second equation of structure, we plug (14) into (12), Since Z J · ¯ Z K = 0 and proposition 2.1, the selfdual part reads Changing index positions by using det G JK = 1 2 gives the 2nd equation of structure. For the first Bianchi identity, look at where the identity /epsilon1 IJK /epsilon1 INM = δ J N δ K M -δ J M δ K N has been used. Finally, the second Bianchi identity is Remark 2.3. Instead of using Cartan equations for the tetrad one could have used the bivector connection form For later use it is convenient to write the components of the equations of structure explicitely. The connection one-forms for example can be expressed in terms of NP spin coefficients, The middle component σ 1 a collects all unweighted coefficients and so can be used to define the GHP covariant derivative Θ a η = ( ∇ a -pσ 1 a -qσ 1 a ) η . To avoid clutter in the notation, we write Γ ∶ = σ 0 and σ 2 = -Γ ' , where ' is the GHP prime operation[21]. Derivatives of the spinor dyad can now be written in the compact form Θ a o A = -Γ a ι A and Θ a ι A = -Γ ' a o A , and the components of the first equations of structure, which we present here for convenience with the usual exterior derivative and with weighted exterior derivative d Θ = d -pσ 1 ∧-qσ 1 ∧ , read Note that the middle component can be simplified to d Z 1 = -h ∧ Z 1 with the oneform h = 2 ( ρ ' l + ρn -τ ' m -τ ¯ m ) . This fact and a relation between type D curvature Ψ 2 and h will be crucial in the derivation of the conservation law in section 5. a ( b ; c ) In vacuum, we have for the curvature two-forms Σ J = C JK Z K and the components of the second equations of structure read Σ 1 C 1 J Z d σ 1 - Γ ∧ Γ (20b) = Finally the Bianchi identities are 3. Conformal Killing Yano tensors and Killing spinors Conformal Killing Yano tensors of rank 2 are two-forms Y ab solving the conformal Killing Yano equation, Y = g bc ξ a - g a ( b ξ c ) , where ξ a = 1 3 Y a ; b . (22) It is well known, that the divergence ξ a is a Killing vector and in case it vanishes, Y ab is called Killing Yano tensor. The symmetrised product X c ( a Y b ) c = ∶ K ab of Killing Yano tensors X ab , Y ab is a Killing tensor, ∇ ( a K bc ) = 0 , which can be used to construct a constant of motion or a symmetry operator for e.g. the scalar wave equation, known as Carter's constant and Carter operator, respectively. By inserting Y ab = κ AB /epsilon1 A ' B ' + ¯ κ A ' B ' /epsilon1 AB into (22) one can show that κ AB and ¯ κ A ' B ' satisfy the Killing spinor equation and its complex conjugated version. For the spinor components κ AB = κ 2 o A o B -2 κ 1 o ( A ι B ) + κ 0 ι A ι B (or equivalently the self dual bivector components of Y ab , we find the following set of eight scalar equations by projecting (23) into a spinor dyad. Thus, we have three different sets of equations, (22), (23), (24,25), which are equivalent and we will use the most appropriate for the problem at hand. As spin-s fields are heavily restricted on curved backgrounds (Buchdahl constraint, see equation (5.8.2) in [37]), so are Killing spinors. Consider a Killing spinor κ A 1 ...A n = κ ( A 1 ...A n ) which satisfies the Killing spinor equation of valence n Contracting a second derivative ∇ B ' C and symmetrising gives For Killing spinors of valence 1 (satisfying the twistor equation) this yields 0 = Ψ ABCD κ D as can be found in [38], eq.(6.1.6). For 2-spinors we find b J ' = For non trivial κ , this restricts the spacetime to be of Petrov type D,N or O . For a given spacetime of type D in a principal frame (only Ψ 2 ≠ 0 ) (27) becomes with constants C 1 , C 2 and it follows κ 0 ≡ 0 ≡ κ 2 . The remaining component satisfies the simplified equations (32), which have only one non trivial complex solution, cf. [22] where explicit integration of the conformal Killing Yano equation was done.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4. Conserved Charges", "content": "4.1. Conserved charges for Minkowski spacetime. The Killing spinor equation or conformal Killing Yano equation on Minkowski space has been widely discussed in the literature [38],[32], [25] and the explicit solution in cartesian coordinates is well known, Here U AB , W A ' B ' are constant, symmetric spinors and V B A ' a constant complex vector which yield 2 · 6 + 8 = 20 independent real solutions. Each solution gives a charge when contracted into a spin-2 field, e.g. the linearized Weyl tensor, and integrated over a 2-sphere. In [38, p.99], 10 of these charges are related to a source for linearized gravity in the following sense. Given a divergence free, symmetric energy momentum tensor T ab , one has for each Killing field ξ b the divergence free current J a = T ab ξ b . Using linearized Einstein equations and the conformal Killing Yano equation (22), they showed Here Σ denotes a 3 dimensional hypersurface with boundary ∂ Σ and e abcd is the Levi-Civita tensor. The left hand side is the charge integral described above, while the right hand side gives the more familiar form of a conserved three-form corresponding to a linarized source and a Killing vector ξ a = 1 3 Y ab ; b . Note that it is the dual conformal Killing Yano tensor on the left hand side, which gives the charge associated to the isometry ξ a . In cartesian coordinates x a = ( t, x, y, z ) the Poincaré isometries read and the relation to the charges is listed in table 1. The angular momentum around the z -axis is found in the component L xy = ∂ φ . Explicit expressions for linearized sources generating these charges can be found in [29, eq.27]. The 10 remaining charges cannot be generated this way, since the corresponding conformal Killing Yano tensors have vanishing divergence (they are Killing Yano tensors). One of these charges corresponds to the NUT parameter 9 , and the remaining nine are three dual linear momenta and six ofam 10 . In the expression (28) for a general Killing spinor, they correspond to U and the imaginary part of V . For a metric perturbation, which one might interpret as a potential for the linarized curvature, these 10 additional charges vanish, see [38, §6.5]. To understand the charges as projections into l = 0 and l = 1 mode, we rederive the complete set of solutions in spherical coordinates using spin weighted spherical harmonics. A null tetrad for Minkowski spacetime in spherical coordinates ( t, r, θ, φ ) (symmetric Carter tetrad) is given by with non vanishing spin coefficients A general two-form can be expanded and it is a conformal Killing Yano tensor, if the components κ i satisfy (24,25). The subset (24) of the Killing spinor equation becomes so κ 0 = f 0 ( t -r ) 1 Y 1 m and κ 2 = f 1 ( t + r ) -1 Y 1 m with functions f i depending on advanced and retarded coordinates only. Finally (25) can be solved for κ 1 , which is only possible for particular functions f i . The result is given in table 2. Ω 1 is one complex solution, while Ω i m , i = 0 , 1 , 2 represent 3 complex solutions each, ( m = 0 , ± 1 ). We find the following correspondence to the solutions (28) in cartesian coordinates 4.2. Conserved charges for type D spacetimes. The vacuum field equations in the algebraically special case of Petrov type D have been integrated explicitly by Kinnersley [36]. An explicit type D line element solving the Einstein-Maxwell equations with cosmological constant is known, from which all type D line elements of this type can be derived by certain limiting procedures, see [40, §19.1.2], see also [15]. The family of type D spacetimes contains the Kerr and Schwarzschild solutions, but also solutions with more complicated topology and asymptotic behaviour, such as the NUT- or C-metrics, and solutions whose orbits of the isometry group are null. In the following, we again restrict to the vacuum case. A Newman-Penrose tetrad such that the two real null vectors l a , n a are aligned with the two repeated principal null directions of a Weyl tensor of Petrov type D is called a principal tetrad. In this case, and Ψ 2 ≠ 0 . Due to the integrability condition (27), we have κ 0 = 0 = κ 2 . Hence, the components (24,25) of the Killing spinor equation simplify to Comparison with the Bianchi identities The divergence ξ AA ' = ∇ A ' B κ AB is a Killing vector field, which is proportional to a real Killing vector field for all type D spacetimes except for Kinnersley class IIIB, cf. [11]. If ξ AA ' is real, the imaginary part of κ AB is a Killing-Yano tensor. Spacetimes satisfying the just mentioned condition are called generalized KerrNUT spacetimes [19]. The square of the Killing-Yano tensor is a symmetric Killing tensor K ab = Y ac Y c b and it follows, that η a = K ab ξ b is a Killing vector. On a Kerr background, ξ a and η a are linearly independent and span the space of isometries, see [26]. In the special case of a Schwarzschild background, η a vanishes, see also [12] for details. shows that κ 1 ∶ = ψ ∝ Ψ -1 /slash.left 3 2 is a solution, and in fact up to a constant κ AB = ψo ( A ι B ) is the only solution of the Killing spinor equation. For Kerr spacetime in Boyer-Lindquist coordinates we find and we set the factor of proportionality to 1, so that the solution reduces to Ω 1 as given in table 2, in the Minkowski limit M,a → 0 . We find ∇ b /parenleft.alt1 ψZ 1 ab /parenright.alt1 = 3 ( ∂ t ) a . The Killing spinor with components given by (34) is We have ψZ 1 ab = κ AB /epsilon1 A ' B ' and therefore and it follows Spin lowering the Weyl spinor using (35) gives the Maxwell field ψ ABCD κ CD , which has charges proportional to mass and dual mass , see also [33]. Letting M ( C,κ ) denote the corresponding closed complex two-form we have Evaluating the charge for the Kerr metric yields where M is the ADM mass while the dual mass is zero. The closed two-form (36) has been derived much earlier by Jordan, Ehlers and Sachs [34]. We will repeat the derivation here, since this formulation can be generalized to linearized gravity most easily. On a type D background, the curvature forms and the connection simplify to so the middle Bianchi identity (21b) becomes where h = 2 ( ρ ' l + ρn -τ ' m -τ ¯ m ) was used. As noted in [18], the Bianchi identities (33) can be rewritten as 2dΨ 2 = 3 h Ψ 2 and one obtains We finally end up with the Jordan-Ehlers-Sachs conservation law [34], Using ψ ∝ Ψ -1 /slash.left 3 2 , this is the same result as (36). See also [27], where the conservation law is generalised to spacetimes of Petrov type II. The result for type D backgrounds fit into the picture of Penrose potentials[23] and in the next section we will see that it generalizes to linear perturbations.", "pages": [ 10, 11, 12, 13 ] }, { "title": "5. Fackerell's conservation law", "content": "We can of course linearize the two-form (36), which would provide a charge for perturbations within the class of type D spacetimes. But more generally, Fackerell [17] derived a closed two-form for arbitrary linear perturbations around a type D background 11 . Starting from this conservation law, Fackerell and Crossmann derived field equations for perturbations of Kerr-Newmann spacetime. Let us give a shortened derivation in the vacuum case. When linearizing (with parameter /epsilon1 ) the general bivector equations around a type D background in principal tetrad, we have Proof. Since and we have Now expanding the Bianchi identitiy (21b), we find dΣ 1 = 1 2 h ∧ Σ 1 + O ( /epsilon1 2 ) which can be written Because ( τ ' m -ρ ' l ) ∧ Z 2 = 0 this could be added which yields the result. /square In the background, this gives the Jordan-Ehlers-Sachs conservation law (39). For linearized gravity, making use of 3 h Ψ 2 = 2dΨ 2 , we find the identity were the linearized version of d Z 1 = -h ∧ Z 1 is used in the last step. Note, that also holds, which looks similar to Maxwell equations with a source. We summarize the above discussion by the following Theorem 5.1. For linearized gravity on a vacuum type D background in principal tetrad exists a closed two-form which can be used to calculate the 'linearized mass'. The integral is conserved, gauge invariant and gives the linearized ADM mass. The gauge invariance follows already from its relation to the ADM mass, but the integrand itself has interesting behaviour under gauge transformations. Beside infinitesimal changes of coordinates (coordinate gauge), there are infinitesimal Lorentz transformations of the tetrad (tetrad gauge). To discuss the second one, we need some notation. Following [13], introduce 4 real functions N 1 , N 2 , L 1 , L 2 and 6 complex functions L 3 , N 3 , M i , i = 1 , .., 4 to relate the linearized tetrad to the background tetrad These are 16 d.o.f. at a point, 10 correspond to metric perturbations and 6 are infinitesimal Lorentz transformations (tetrad gauge). The linearized tetrad oneforms have the representation It follows Linearization of the tetrad representation of the metric yields and therefore tr g h = -2 ( L 1 + N 2 + M 3 + M 3 ) . One should also note, that the selfdual components of ˙ Z 1 in ˙ M cancel some of the additional terms, not coming from the linearized Weyl tensor, Using these facts, we show Proposition 5.2. On a spacetime of Petrov type D, the two-form ˙ M is tetrad gauge invariant and changes only with a term χ which is exact, χ = d f , under coordinate gauge transformations. Remark 5.3. In the work of Fayos et al. [18] , a gauge in which d /parenleft.alt1 ψ Ψ 2 ˙ Z 1 /parenright.alt1 = 0 was used. It is not clear from that work whether this gauge condition is compatible with a hyperbolic system of evolution equations for linearized gravity. Proof of Proposition 5.2. Let us first look at the coordinate gauge. Under infinitesimal coordinate transformations x a → x a + ξ a , a tensor field transforms with Lie derivative, T → T -L ξ T . For linearized gravity, we write this as ˙ T → ˙ T + δ ˙ T = ˙ T -L ξ T . Now look at the middle bivector component Z 1 and use Cartan's identity L ξ ω = d ( ξ /uni2A3C ω ) + ξ /uni2A3C d ω , which holds for arbitrary forms ω . It follows for coordinate gauge transformations in ˙ M , where ξ /uni2A3C h = 2 3 Ψ -1 2 ξ ( Ψ 2 ) and ξ /uni2A3C ( h ∧ Z 1 ) = ( ξ /uni2A3C h ) Z 1 -h ∧ ( ξ /uni2A3C Z 1 ) was used. The two-form (49) is exact and hence integrates to zero. A tetrad gauge transformation changes the tetrad (45) as follows, with a, b complex and A,ϑ real valued. It follows, that the tetrad gauge dependent terms in (48a,48c) cancel the ones in (47). The anti selfdual part in (47) is invariant, as follows from (50). This shows the tetrad gauge invariance of ˙ M and therefore gauge invariance of (44). /square Finally, to express the charge integral in a form similar to the Maxwell case (3), we need the θφ components of the bivectors, The charge integral becomes with V L = ∆ /slash.left( r 2 + a 2 ) 2 , d µ = sin θ d θ d ϕ and", "pages": [ 13, 14, 15, 16 ] }, { "title": "6. Conclusions", "content": "For each isometry of a given background, there is a conserved charge for the linearized gravitational field. Working in terms of linearized curvature, we derived a linearized mass charge (corresponding to the time translation isometry) for Petrov type D backgrounds, by using Penrose's idea of spin-lowering with a Killing spinor. A second Killing spinor, corresponding to the axial isometry of Kerr spacetime does not exist, (32). Hence spin lowering cannot be used directly to derive a linearized angular momentum charge, even tough a canonical analysis provides one in terms of the linarized metric. For a Schwarzschild background, gauge conditions are known, which eliminate the gauge dependent non-radiating modes [44, 30]. Understanding these conditions in a geometric way and generalizing them to a Kerr background needs further investigation.", "pages": [ 16 ] }, { "title": "Appendix A. Coordinate expressions", "content": "Using a Carter tetrad, the bivectors and connection one-forms in Boyer-Lindquist coordinates are Here, we used Acknowledgements. We would like to thank Jiří Bičák, Pieter Blue, Domenico Giulini, Jacek Jezierski, Lionel Mason and Jean-Philippe Nicolas for helpful discussions. One of the authors (S.A.) gratefully acknowledges the support of the Centre for Quantum Engineering and Space-Time Research (QUEST) and the Center of Applied Space Technology and Microgravity (ZARM) , and thanks the Albert Einstein Institute, Potsdam for hospitality during part of the work on this paper. L.A. thanks the Department of Mathematics of the University of Miami for hospitality during part of the work on this paper.", "pages": [ 17 ] }, { "title": "References", "content": "E-mail address : [email protected] QUEST, Leibniz University Hannover, Welfengarten 1, D-30167 Hannover, Germany ZARM, University of Bremen, Am Fallturm 1, D-28359 Bremen, Germany E-mail address : [email protected] Albert Einstein Institute, Am Mühlenberg 1, D-14476 Potsdam, Germany", "pages": [ 19 ] } ]

[ { "title": "Disformal invariance of Maxwell's field equations", "content": "E. Goulart ∗ , F. T. Falciano † Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: October 17, 2018) We show that Maxwell's electrodynamics in vacuum is invariant under active transformations of the metric. These metrics are related by disformal mappings induced by derivatives of the gauge vector A µ such that the gauge symmetry is preserved. Our results generalize the well known conformal invariance of electrodynamics and characterize a new type of internal symmetry of the theory. The group structure associated with these transformations is also investigated in details. PACS numbers: 02.40.Ky, 03.50.De, 03.50.Kk, 04.20.Cv", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In a recent communication [1], we have shown that there exists a new symmetry in the relativistic wave equation for a scalar field in arbitrary dimensions. This symmetry is related to redefinitions of the metric tensor which implement a map between non-equivalent manifolds. We have encountered this result as a natural consequence of our work in analogue models of gravity, where we showed that it is possible to geometrize the dynamics of a generic nonlinear scalar field [2]. However, we have later realized that these metric mappings were in fact disformal transformations. Thus, in [1] we were in fact showing that the relativistic Klein-Gordon equation is invariant under disformal transformations. Firstly introduced by Bekenstein in [3, 4], disformal transformations typically evoke the presence of an auxiliary scalar field ψ ( x ) which appears explicitly in the transformed geometry. Thus, a disformal transformation is characterized by the relation where A and B are real functions constructed with the field's invariants and B is chosen to have dimensions of M -4 so that ψ has dimensions of M . This is the most general symmetric covariant object that can be constructed with the background metric g µν , the scalar field ψ and its first derivatives ∂ψ . We will call g µν and ˆ g µν 'disformally related metrics' and the second term in the transformation as the 'disformal term' so as to contrast to the conformal transformations that can be seen as special cases of disformal transformations with B = 0. From their own side, conformally related geometries appear in a variety of important physical situations. From condensed matter systems to string theory, they provide a rich source of insights and are deeply ingrained in some modern field theory approaches (see, for instance, [5]). In the last decade, asymptotic and theoretical problems in quantum field theories have led to a renew of interest in conformal field theory and in the mathematical structure of the restricted conformal group SO(2,4). The SO(2,4) encompass both the Poincar'e as well as the de Sitter group and is a basic ingredient for the ADS/CFT correspondence [6]. Conformal maps and disformal transformations can be viewed as complementary concepts. While the former implements rescaling of the metric which preserve angles, the latter deforms the spacetime fabric in an anisotropic manner according to a preferred direction characterized by the gradient of the dynamical fields. Accordingly, relation (1) does not preserve the causal structure of the original geometry. Thus, in general, the null vectors of g µν are different from the null vectors of ˆ g µν , i.e. their cones of influence are distinct. This property has been used, for instance, to generate cosmological scenarios with varying speed of light (VSL) [7]-[9]. In this case, the velocity may depend not only on the functions A and B , but also on the character of the gradient ∂ α ψ (see [10] for a detailed discussion). Since Bekentein's initial proposal, many aspects of disformal relations have been investigated. They appear in bi-metric theories [11], TeVeS models [12], massive gravity [13], DBI-Galileons [14], cosmic acceleration schemes [15] and others. In cosmology, for instance, they are able to reproduce many features of scalar field dark energy models: cosmological constant, quintessence, k-essence and tachyon condensates. The most important property of (1) is that it provides a natural and simple way to implement modifications of usual gravity [16]. Typically, according to the disformal prescription, one replaces g µν → ˆ g µν in some sector of the lagrangian and hence generates an effective coupling between the scalar field and the energy momentum tensor of the other fields describing the matter content. A fairly simple example is provided by a cosmological constant living in a disformal metric which mimics the behavior of a Chaplygin gas. The main theme of these scenarios is that there exists a duplicity of geometries. While the gravitational geometry g µν satisfies Einstein's equations, it is the physical geometry ˆ g µν that controls the dynamics of the matter fields. Although the role of conformal symmetries has already been extensively explored, the invariance under disformal transformations is by far less understood. It is not clear, for instance, when a given set of equations of motion are invariant under a disformal mapping. In addition, it might happen that such an invariance could help us to gain new insights in field theory as has been the case for conformal transformations. /negationslash In its seminal paper [4], Bekenstein states that 'Maxwell's equations, the Weyl equation for spinors, gauge field equations, etc. will all be invariant under the transformation with B = 0, but will not be invariant under g αβ → ˆ g µν with B = 0'. It is certainly true that for an arbitrary function B , these equations are not disformal invariant. However, in the present paper we shall show that it is possible to define a large class of disformal transformations with respect to which these theories are in fact disformal invariant. More specifically, we shall show that Maxwell's equations in vacuum are invariant under certain disformal transformations. In dealing with electrodynamics, instead of using a scalar field such as in (1), the disformal transformations here introduced depend on the gauge vector A µ . Thus, given a metric g µν and the electromagnetic two-form F µν = ∂ µ A ν -∂ ν A µ that satisfies Maxwell's equations, we will be concerned with disformal relations of the form where I 1 and I 2 are the electromagnetic gauge invariant scalars. In a sense, our result generalizes the usual conformal invariance of electrodynamics and constitutes a complementary internal symmetry of the theory.", "pages": [ 1, 2 ] }, { "title": "II. DEVELOPMENT", "content": "In this short introductory section we shall define some relevant objects and fix our notation. Let us start with an electromagnetic field F µν propagating in a globally hyperbolic spacetime with metric g µν that has signature (+ - --). Throughout our development, we shall consider source free field, hence, Maxwell's equations in vacuum read where the semicolon means covariant derivative with respect to g µν . The second set of equations guarantee that the electromagnetic field is completely characterized by a gauge vector A µ , i.e. F µν = A [ µ ; ν ] . Using the electromagnetic field and its dual, one can only construct two invariants, namely where the dual bi-vector is defined as and η αβµν is the completely antisymmetric Levi-Civita permutation tensor. The electric and magnetic fields /vector E and /vector H are spatial three-dimensional vector fields that are orthogonal to the observer's worldline. We also recall that the energy-momentum tensor associated with electromagnetic fields satisfying Maxwell's equations is given by One of the new feature of a disformal transformation is that the new metric may explicitly depend on the dynamical fields themselves. Note however that while the metric is a symmetric tensor, the electromagnetic twoform is antisymmetric which means that it cannot appears linearly. Thus, we need a procedure to construct a symmetric object using only the metric g µν , the electro- magnetic field F µν and its dual ∗ F µν . Fortunately, due to algebraic relations between these objects, this construction is unique. Indeed, the electromagnetic field and its dual satisfy the relations Therefore, any symmetric tensor ∆ µν that depends only on these three fields has to be of the form where a and b are two real functions that can depend on the coordinates, the electromagnetic field and its dual. Under certain mild conditions, the quantity ∆ µν is invertible which allow us to use (8) as a disformal transformation induced by the electromagnetic tensor F µν , i.e. /negationslash Note that the algebraic structure of the above disformal term is much more involved than the scalar field case. In analogy to (1), one could have expected to define the disformal term proportional to ∂ α A β + ∂ β A α . Notwithstanding, the maintenance of the gauge symmetry requires the use of the electromagnetic two-form which unavoidable leads us to (9). If b is zero we recover the usual conformal transformation and the causal structure of the theory is preserved. But when b = 0 the disformal transformation do not preserve angles between vectors and the causal structure changes drastically. The vectors k µ satisfying ∆ µν k µ k ν = 0 are, in general, not null with respect to g µν (see [17] for a detailed discussion) and hence the characteristic surfaces in these two situations are not the same. Only if there exist null eigenvectors of the disformal term alone the null cones of the two metrics may coincide along some specific directions. /negationslash We shall assume that ∆ µν is always nonsingular, i.e. det(∆ µν ) = 0. Thus, there exist a new tensor ∆ -1 µν such that In general, the inverse of an object of the form g µν + h µν , with arbitrary h µν , is given as an infinite series. Notwithstanding, due to the algebraic properties encoded in the disformal term, its inverse has also a binomial form. Indeed, a direct calculation yields with the coefficients A and B given in terms of the invariants I 1 , I 2 and the previous quantities a , b where p ≡ b/a is the 'disformal ratio' and for future convenience we have defined the auxiliary quantity Having established the general form of the disformal transformation (8), in what follows we shall define the two up to now arbitrary functions a and b in such a way that the disformal transformation leaves Maxwell's equations invariant. In other words, within a large class of disformal transformations Maxwell's equations are disformal invariant.", "pages": [ 2, 3 ] }, { "title": "III. DISFORMAL INVARIANCE", "content": "As it is well known, Maxwell's equations (3) are invariant under conformal transformations that here are described by (11) with B = 0. Thus, any electromagnetic configuration that is a solution of Maxwell's equation defined in the g µν manifold is also a solution of the same system of equations but in the ∆ -1 µν manifold. /negationslash However, in general, this property will not hold if B = 0, i.e. the presence of the anisotropic stretching deforms the equations of motion in a non-trivial way. Actually, the new system of equations will depend explicitly on the choice of A and B . Nevertheless, there is a specific choice of the function B where the above mentioned property also holds, hence, by a suitable choice of B any solution of Maxwell's equation in the g µν manifold is also a solution in the ∆ -1 µν manifold even if B = 0. Our first step is to calculate the action of the 'delta tensors' on the electromagnetic bi-vector. The electromagnetic two-form F µν is defined independently of any metric but its contravariant version does depend on which metric we are using to raise or lower the indices. To distinguish the two situation we shall use a hat over the tensor to indicate that it has been defined in the ∆ -1 µν manifold, i.e. /negationslash It is worth noticing that this is a highly non-linear transformation inasmuch the ∆ µν tensor already has a nontrivial dependency on F µν . A straightforward calculation using (8) and the algebraic relations (6)-(7) shows that ˆ F µν may be written as a combination of the field and its dual as with the functions ψ and χ given strictly in terms of the field invariants and the pair ( a, p ), i.e. The dynamical set of Maxwell's equation can be written in a more suggestive form. As long as we are only considering Riemannian manifolds we can re-write the first group of equations of (3) as Thus, it becomes evident that if we construct the disformal transformations such that then Maxwell's equations in vacuum will automatically be invariant under these transformations. To calculate the determinant of ∆ -1 µν we can use the Cayley-Hamilton theorem, which shows that the determinant of any mixed tensor T can be expanded in terms of its traces as In our case, a direct calculation shows that The algebraic properties of the energy-momentum tensor have important informations about the propagation of the electromagnetic discontinuities. This local analysis can be done by studying its eigenvalue problem. It can be shown (see [18]) that the electromagnetic energymomentum tensor has only two eigenvalues given by ± κ where /negationslash A field configuration is called algebraically general if κ = 0 and null if κ = 0. In order to show that indeed there are disformal transformations that satisfy (16) we shall consider separately the algebraically general and the null cases.", "pages": [ 3 ] }, { "title": "· General Field κ = 0", "content": "/negationslash We first note that (14) and (16) immediately imply that the term proportional to the dual must vanish so we need to impose χ = 0. There exist, in principle, two possibilities for p that makes χ = 0, i.e. /negationslash Evidently, we can have I 1 = 0 but κ = 0 so if I 1 = 0 we are force to consider p = 0. However, p = 0 only reproduce the conformal invariance of electrodynamics. In this manner we shall consider only the second solution which is the relevant one for our purposes. We proceed by calculating explicitly the quantity √ -∆ ψ and imposing the condition p = 4 /I 1 . Using (17) and (12) we obtain that (16) will be satisfied if Incidentally, this condition is automatically satisfied. In other words, the quantity √ -∆ ψ is constant (does not depend on the field invariants) independently on the value of the function a . Thus, in the case of an algebraically general field, the admissible pair is ( a, 4 /I 1 ) with a arbitrary.", "pages": [ 4 ] }, { "title": "· Null field κ = 0", "content": "Since the invariants appear quadratically in κ we must have I 1 = I 2 = 0. Substituting these values in ψ and χ , we obtain In the case of a null field, relation (16) is always satisfied independently of the particular realization of the pair ( a, p ). Therefore, (16) is valid for arbitrary values of the functions a and p . Let us summarize our result. In terms of the gauge vector A µ ( x ), Maxwell's equations (3) in vacuum may be recast in the form If A µ ( x ) satisfies Maxwell's equations in a spacetime endowed with metric g µν it also satisfies the same set of dynamical equations but in a different spacetime endowed with metric ∆ -1 µν . The metric of the latter manifold is constructed according with the proper choice of ( a, p ) discussed above. Defining the covariant derivative' || 'such that we immediately obtain Thus, in the same way that a gauge transformation A µ → A µ + ∂ µ Λ characterizes different representations of the same physical situation, we may say that, from the formal point of view, it is impossible to distinguish between different spacetimes related by the disformal transformation given by (9). In other words, all spacetimes ∆ µν constructed with A µ and the pair of functions ( a, p ) are compatible with the same potential configuration as a solution. This is a symmetry of Maxwell's electromagnetic theory.", "pages": [ 4 ] }, { "title": "A. Metrical Properties of the Energy-Momentum tensor", "content": "/negationslash A remarkable property of this new symmetry is that there exist an intimate relationship between the disformal metric ∆ µν and the energy-momentum tensor defined in the original geometry g µν . Let us concern ourselves to the case κ = 0. In this case, the disformal transformation (8) is given by with the function a completely arbitrary. Making the redefinition 4 a = -I 1 Ω 2 (we choose the minus sign to keep our signature convention intact), where Ω is an arbitrary function, and using (5) we recast the form of the admissible delta tensors as 1 In other words, the disformal metric is conformally related to the energy-momentum tensor of the original field configuration. Thus, things that look like energy and momentum in the former manifold appear as spacetime distances in the 'new' manifold. This is an interesting symmetry property of the dynamical equations that somehow generalize the concept of conformal transformations. The disformally related line element is given by In general, the riemannian spacetimes generated by the disformal transformations are non-flat and depend explicitly on the particular solutions of the gauge potential A µ ( x ). Having defined the metrical structure of a given manifold, one can study its properties by constructing the /negationslash geometrical objects and the Debever's invariants associated to them. Considering for instance the curvature tensor that has second derivatives of the metric ∆ µν , it happens that it will also have higher derivatives of the vector potential, i.e. Its explicit expression is a very long and involved equation that should be analyzed for each particular solution. Besides, there seems to have no natural way to separate and classify the terms appearing in its decomposition. It is also worth noting that, in general, the metric ∆ µν does not have the same isometries as the original g µν . There is no reason for these two metrics to share the same set of killing vectors. Our analysis has focused in the invariance of the equation of motion (3) under metric transformations and, as it is well known, symmetries of the equation of motion do not imply symmetries in the action. However, it is straightforward to show that transformation (9) is also a symmetry of the action. Indeed, the action integral is invariant under the map In a Taylor expansion around a fiducial point x 0 , the function a ( x ) or p ( x ) are characterized by an infinite number of parameters. Consequently, this symmetry of the action is also characterized by an infinite number of parameters [19].", "pages": [ 4, 5 ] }, { "title": "B. A particular realization of the symmetry: static point charge", "content": "As a simple example let us analyze the case of a motionless point-like charge. In spherical coordinates x µ = ( t, r, θ, φ ), the spherically symmetric and static solution of (3) is of the form where E ( r ) = Q/r 2 . A direct calculation yields the contravariant components of the energy-momentum tensor For the sake of simplicity we choose Ω 2 = 1. Thus, A direct calculation shows that, indeed, the field satisfies the equation which again is an explicit example of our results. The line element is given by Note that the radial coordinate becomes a timelike coordinate while the time coordinate becomes spacelike. This is an intriguing property that shall be investigated in a future work. Also one can calculate the curvature tensor of (34) and confirm that the ∆ -1 µν = T -1 µν manifold is indeed curved. Notwithstanding, the conformal tensor is identically zero which means that the electric point charge generates a class of conformally flat spacetimes via the disformal mapping.", "pages": [ 5 ] }, { "title": "IV. GROUP STRUCTURE", "content": "Another interesting property of the symmetry transformation generated by (9) is that together with its set of differential manifolds they form a group for each and every solution A µ ( x ) of the dynamical equations. To simplify notation we can define a symmetric tensor φ µν that using the algebraic relations (6)-(7) satisfies In addition, we will included two indices ( a, p ) to specify the transformation. From now on we shall change a bit our notation and drop the hat used to designate the transformed objects. The indices ( a, p ) should suffice to appropriately identify the transformation so that we write Once more, we shall separate our analysis in the two cases of an algebraic general or null electromagnetic field. /negationslash", "pages": [ 5 ] }, { "title": "A. General Field κ = 0", "content": "For the case of an algebraic general field, eq. (19) shows us that the disformal ratio is fix, i.e. we have only one free function. Thus, the disformal transformation reads where we have defined the quantity ξ ≡ I 2 I 1 The electromagnetic two-form in the ∆ -1 ( a ) µν geometry can be related with its counterpart in the g µν through Therefore it is straightforward to obtain the relations We define the transformation T acting on the metric g µν such that with ∆ -1 ( a ) µν defined by (39). According to our previous discussion the transformation symbol relates two nonequivalent manifolds. Let us apply a second transformation T b associated with the function b ( x ) on the metric ∆ -1 ( a ) µν . We have Replacing all g µν by ∆ -1 ( a ) µν into (39) one immediately obtains A direct calculation using explicitly (39)-(43) gives us which is conformally related to the g µν metric. The transformation (44) is homogeneous of order 1, hence any metric conformally related to g µν satisfies Therefore, an arbitrary number of successive transformations will generate only two types of metric, namely The collection of all metrics M ( a ) µν and N ( a ) µν together with transformation (44) can be viewed as a representation of a group G . The composition law of G can be depicted as With this composition law, it is straightforward to show that they indeed form a group. The collection of all M ( a ) 's forms a subgroup H ⊂ G . Actually, they are an invariant subgroup of G since Thus, we can define an equivalence relation between the left coset defined in terms of the subgroup H . Due to relations (48)-(50) there is actually only two coset since In addition, the two cosets satisfy Thus, the quotient group is G / H = Z 2", "pages": [ 5, 6 ] }, { "title": "B. The null Field κ = 0", "content": "The algebraic null field, contrary to the general case, has an extra free function to define the disformal transformation. Relations (35)-(37) show that a contravariant metric defined as has an inverse given by /negationslash where the only condition over the functions a and b comes from (21) that requires a = 0. The electromagnetic twoform in the ∆ -1 ( a,b ) µν geometry can be related with its counterpart in the g µν through Therefore the relations (40)-(43) now modify to In the same way, we define the transformation T acting on the metric g µν such that Applying a second transformation T ( c,d ) on the metric ∆ -1 ( a,b ) µν , we have Therefore, the group properties are Note that this is a non-abelian group, i.e. /negationslash It is straightforward to show that the collection of all T ( a, 0) 's forms a subgroup. However, this is not an invariant subgroup since Notwithstanding, there is another subgroup O formed by the collection of all T (1 ,b ) 's that are in fact an invariant subgroup. Indeed, we have Thus, we can establish an equivalence relation between the left coset defined in terms of this subgroup. The left cosets are of two types /negationslash /negationslash Contrary to the general case κ = 0, here there is an infinity of different left cosets labeled by the function c above. Again, we can establish an equivalence relation between elements of the same left coset. These cosets inherit from the former group the following composition rule: Thus, the quotient group associated with disformal transformation in the the null case is infinite and characterized by all analytical 4-dimension real functions.", "pages": [ 6, 7 ] }, { "title": "V. CONCLUSION", "content": "There is no question that symmetries play a fundamental role in modern physics. Noether's theorem, for instance, associate invariance of the action of a given physical system with conservation law's for tensor currents such as the energy-momentum tensor. In general, these symmetries are associated with variation of the action with respect to the spacetime coordinates and/or fields redefinitions. In this paper, we have developed a different concept of symmetry that is closely related to the conformal symmetry. The point of departure is the definition of a dynamical system in an arbitrary spacetime g µν and the specification of the solutions of these equations of motion. Thereon, we have shown that the dynamics of the physical fields are invariant with respect to redefinitions of the metric tensors that maps different riemannian manifolds. More specifically, Maxwell's electrodynamics is invariant with respect to a large class of disformal metric transformations. The disformal transformations together with the different manifolds that they generate has a group structure that is abelian for the algebraic general and non-abelian for the algebraic null case. There are several interesting issues related to the physical meaning of these disformally related manifolds and their mathematical structure that we shall address in future works.", "pages": [ 7 ] }, { "title": "VI. ACKNOWLEDGEMENTS", "content": "E. Goulart would like to thank FAPERJ for financial support. 085011.", "pages": [ 7 ] } ]

[ { "title": "Dynamical evaporation of quantum horizons", "content": "Daniele Pranzetti 1 ∗ 1 Max Planck Institute for Gravitational Physics (AEI), Am M¨uhlenberg 1, D-14476 Golm, Germany. (Dated: June 4, 2022) We describe the black hole evaporation process driven by the dynamical evolution of the quantum gravitational degrees of freedom resident at the horizon, as identified by the loop quantum gravity kinematics. Using a parallel with the Brownian motion, we interpret the first law of quantum dynamical horizon in terms of a fluctuation-dissipation relation. In this way, the horizon evolution is described in terms of relaxation to an equilibrium state balanced by the excitation of Planck scale constituents of the horizon. This discrete quantum hair structure associated to the horizon geometry produces a deviation from thermality in the radiation spectrum. We investigate the final stage of the evaporation process and show how the dynamics leads to the formation of a massive remnant, which can eventually decay. Implications for the information paradox are discussed.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In the late sixties, early seventies a fascinating parallel between black holes physics and thermodynamics started to be delineated, which culminated in the derivation of four laws analog to those of a thermodynamical system [1, 2]. However, this analogy was not originally taken too seriously by many since a black hole has classically zero temperature. A simple dimensional analysis shows how, in order to talk about black hole temperature, one needs to refer to the quantum theory. This motivated Hawking's seminal study of a quantum scalar field on a Schwarzschild background, which led to the discovery of black holes evaporation [3]. However, as soon realized by Hawking himself, this result, together with the singularity theorems [4, 5], leads to a violation of unitary evolution [6]. All these elements represent a quite strong evidence of the need of a quantum treatment of the gravitational field in order to fully understand black holes physics. In this regard, the main points a quantum theory of gravity should address and explain are: the microscopic origin of the entropy degrees of freedom and the dynamics of the evaporation process. While the former has received a lot of attention in the last twenty years, with several proposals within different approaches, more or less successful in reproducing the semi-classical result of Bekenstein and Hawking, the latter has been object of much less investigation. In this letter, we want to concentrate on the description of the dynamical evaporation of quantum horizons, as recently introduced in the context of the Loop Quantum Gravity (LQG) approach [7], and analyze its implications for the information paradox. The main idea at the core of our analysis is the notion of quantum horizon as a gas of particles. Such a notion can be traced back to an old proposal by Bekenstein [8] and it has been exploited within the years by different authors (see, e.g., [7, 9, 10]). We believe that the main motivations in support of such an analogy come from the points of view championed, among others, by Sorkin [11], Smolin [12], Jacobson and Rovelli [13-15] on the nature of black hole entropy. In particular, these authors argue that the entropy in the first law is the logarythm of the number of states of the black hole that can affect the exterior and such degrees of freedom have to reside on the horizon; moreover, the finiteness of the entropy has to be related to the deep, discrete structure of space-time. In the LQG picture, the information on the horizon accounting for the entropy is stored in the spin network links piercing the horizon [16-19], which encode the quantum fluctuations of the geometry: the 'charge' associated to the topological defects on the boundary correspond to a quantum hair for the black hole. Here, we want to exploit further this picture of a quantum horizon as a discrete system whose constituents represent the 'atoms' of space. In analogy to Einstein's treatment of Brownian motion, we show that the first law of quantum dynamical horizons [20, 21], based on the loop quantization of the Hamiltonian constraint, can be interpreted as a fluctuation-dissipation theorem for the horizon degrees of freedom. In this way, the radiation spectrum produced by the action of the Hamiltonian operator near the horizon could play a role, analog to that of Einstein's relation for the atomic theory of matter, in proving the atomic structure of quantum space, as described by the kinematical Hilbert space of LQG. This is the first main result of the paper and will be presented in Section II. Here we also explore the relation between the microscopic action of the Hamiltonian operator and the emergent macroscopic description of the horizon dynamics. In Section III we investigate the implications of the existence of this quantum hair at the horizon (associated to the quantum geometry d.o.f.) and of possible non-local effects in the quantum gravitational regime of the collapse to describe the last stage of evaporation process within the LQG framework. By implementing locally the dynamics encoded in the Hamiltonian operator till the horizon reaches a Planck scale size, we show that the horizon area operator result to be bounded from below, with the minimum eigenvalue allowed by the dynamics being 8 πβglyph[lscript] 2 p √ 2. The analysis shows that conservative scenarios can be realized, leading to either the formation of a massive remnant or the dissolution of the horizon. This is the second main result obtained here and it provides support to a singularity-free evolution from the full theory and a possible solution to the information paradox. Conclusions are presented in Section IV.", "pages": [ 2, 3 ] }, { "title": "II. FLUCTUATION-DISSIPATION THEOREM", "content": "Fluctuation-dissipation theorems represent very powerful tools to study the fluctuations of systems described by statistical mechanics. They express the existence of a relation between the spontaneous fluctuations and the response to external fields of physical observables. Fluctuationdissipation theorems are based on Onsager's principle in the theory of dissipative processes, stating that a linear system behaves on average in the same way in a given configuration whether it reached that configuration by a spontaneous fluctuation or by an externally induced perturbation. Motivated by a suggestion of Candelas and Sciama [22] to understand Hawking radiation in these terms, the main result of this section is to show how, within the dynamical horizons framework [20, 21] and a local statistical mechanical perspective [23], a physical process version of the first law can be understood as a fluctuation-dissipation theorem and used to describe the evaporation process. Such an area balance law can be written in terms of the Hamiltonian constraint of gravity. Then, the LQG quantization prescription for the Hamiltonian operator implemented locally near the horizon is used to relate the microscopic description of the geometry fluctuations to the radiation spectrum and the dissipative nature of the evaporation process. In this way, black hole radiance is described from a completely new perspective, namely from a local quantum gravity point of view. At the same time, this new level of analysis provides a new, deeper statistical mechanical understanding of black holes as thermodynamical systems. In order to present these ideas in a clearer and more pedagogical way, we first review in section II A a specific example of fluctuation-dissipation theorem successfully applied to connect the macroscopic and microscopic levels of description of a physical system, namely the Brownian motion. The new result is presented in section II B. We conclude in section II C with some speculative observations relating a modified first law for IH recently proposed in [24], a generalized Clausius law taking into account non-equilibrium dissipative processes and a coarse grained description of the Hamiltonian operator action emerging from the fluctuation-dissipation theorem interpretation.", "pages": [ 3, 4 ] }, { "title": "A. Brownian motion", "content": "The first example of fluctuation-dissipation theorems was provided by Einstein's work on Brownian motion, which culminated with the famous 'Einstein relation' expressing the diffusion coefficient D of a Brownian particle in terms of its mobility µ , through the temperature T of the fluid and Boltzmann's constant k . The experimental verification of Einstein relation allowed the determination of the Avogadro number (a microscopic quantity) from accessible macroscopic quantities, thus providing conclusive evidence of the existence of atoms. In order to understand how (1) encodes a relation between fluctuations and the response to external perturbations, let us quickly go through its derivation. Einstein's analysis can be divided into two parts. The first part of his argument consisted of considering the collective motion of Brownian particles and showing that the particle density n ( x, t ) satisfies the diffusion equation from which the mean square displacement grows in proportion to time according to The second part of Einstein's theory involves a dynamic equilibrium established between opposing forces and is what the fluctuation-dissipation theorem arises from. Following Langevin's approach, one can write down a stochastic differential equation taking into account the effect of molecular collisions by means of an average force, given by the fluid friction, and of a random fluctuating term, namely where, assuming Stokes law for the frictional force f v = -mγv , mγ = 6 πaη , with a the particle radius and η the fluid viscosity. The component R ( t ) of the force resulting from the action of the molecules of the fluid on the Brownian particle is a random fluctuating force, independent of the particle motion. By means of Fourier analysis applied to the random force R ( t ) and the velocity v ( t ) of the Brownian particle, the stochastic differential equation (4) can be written as where v ( ω ) , R ( ω ) are the Fourier modes. By means of the Wiener-Khintchine theorem that associate the correlation function of a given process z ( t ) to its intensity spectrum via eq. (5) gives Now, let us assume for simplicity that the power spectrum I R of the random force R ( t ) is independent of frequency and demand the equipartition law m /uni27E8 v 2 /uni27E9 = kT to hold also for the colloidal particle, as expected if this is kept for a sufficiently long time in the fluid. Then it follows from (7) with (6) that the response function (mobility) of the system µ = ( mγ ) -1 can be associated to the correlation function of the stochastic process v ( t ) through the relation Hence, knowing the power spectrum I R ( ω ) , eq. (7) converts it into I u ( ω ) , allowing to solve the initial Langevin equation (4) to the same extent. As a last step, we write the mean square average of the displacement of the Brownian particle in a time interval ( 0 , t ) as /uni27E8 x ( t ) 2 /uni27E9 = ∫ t 0 dt 1 ∫ t 0 dt 2 /uni27E8 v ( t 1 ) v ( t 2 )/uni27E9 and transform this into Thus, combining the result (3) of the first part of the analysis with the fluctuation-dissipation theorem (8)-which links the macroscopic (validity of the Stokes law) and microscopic (assumption that the Brownian particle is in statistical equilibrium with the molecules in the liquid) levels of description-yields immediately the Einstein relation (1).", "pages": [ 4, 5, 6 ] }, { "title": "B. Horizon dissipative processes", "content": "A couple of years after Hawking's derivation of black holes radiance [3], Candelas and Sciama [22] applied the concepts behind Onsager's principle to study the thermodynamics of dissipative processes associated to black holes physics. The main idea of Candelas and Sciama was to derive Hawking radiation by means of a fluctuation-dissipation theorem relating the black hole area dissipation rate to the fluctuations of a quantum shear operator associated to gravitational degrees of freedom on the horizon. In order to do so, in analogy to the Brownian motion example, they studied the effect of a gravitational perturbation encoded in a non-vanishing shear σ by analyzing the spontaneous vacuum fluctuations of the shear itself. Starting from the Hawking-Hartle relation [25] for an horizon area increase in presence of a purely gravitational stationary perturbation at lowest order-expressing, for example, the rate of slowing down of a black hole by a non-axisymmetric gravitational field produced by distant masses-, Candelas and Sciama interpret σ as a quantum operator and write down the following fluctuation-dissipation relation where t is a suitably defined time variable on the horizon and κ the surface gravity. By properly choosing the vacuum state, they compute the r.h.s. of (10) and shows that the shear fluctuations have the stochastic properties of black-body radiation at temperature κ /slash.left 2 π . Therefore, as the perturbation is dissipated away and a stationary state is approached, the horizon would emit gravitational radiation matching Hawking's result. Such a simple and elegant derivation of black holes radiance shows the power and usefulness of flcutuation-dissipation theorems in studying non-equilibrium statistical mechanics. We now want to interpret our analysis in [7] of the radiation process along those lines and make the idea behind the implementation of the near-horizon dynamics introduced there more precise. By doing so, we will argue how, in analogy to Einstein's work on the Brownian motion, black holes evaporation could provide a proof for the atomic structure of quantum space. In [7], we applied the dynamical horizons formalism developed in [20, 21] to study the transition between two equilibrium configurations of the horizon 1 . More precisely, together with the local statistical mechanical framework introduced in [23, 24], a physical process version of the first law derived in [21] has been used to implement the bulk dynamics near the horizon, as described by the LQG approach, and evolve the horizon quantum geometry. Such a first law was derived from an area balance law relating the change in the area of the dynamical horizon to the flux of matter and gravitational energy. In the vacuum, for a non-rotating horizon the canonical form of Einstein equation gives [21] where ∆ H is the portion of dynamical horizon bounded by the 2-sphere leaves S 1 , S 2 of radius r 1 , r 2 , N r a lapse function, H is the scalar Hamiltonian constraint and σ is the shear of a null vector field glyph[lscript] a normal to the leaves S foliating the horizon H . From eq. (12) then one obtains the analog of (10), namely where A = 4 πr 2 , κ r is the surface gravity associated with the vector field ξ a r = N r glyph[lscript] a and the lapse has been chosen such that d 3 V = N -1 r drd 2 V . The dynamical version of the first law (12) relates the infinitesimal change of the horizon area in 'time' (played by the radial coordinate r along ∆ H ) to the flux of gravitational energy associated with ξ a r . One can then use the freedom to reparametrize the time variable r with an arbitrary function f ( r ) -encoding the freedom to rescale the vector field ξ r -to identify the l.h.s of (12) with the local notion of horizon energy introduced in [23]. Namely, by choosing the function f ( r ) as the proper distance ρ of a preferred family of static observers hovering closely outside the horizon 2 , one gets where E = κ ρ A /slash.left 8 πG and κ ρ = ( dρ /slash.left dr ) κ r = 1 /slash.left ρ + o ( ρ ) represents the local surface gravity measured by the stationary observer. Considering the local perspective we are assuming here, the proper distance observable emerges as a natural choice, corresponding to a Rindler form of the near-horizon metric. In fact, it is the only choice such that, once a stationary equilibrium configuration is reached again, the energy associated to the corresponding Killing vector field matches the physical notion of energy derived in [23]. Relation (13) can now be used in the quantum theory to study the spectrum of the evaporation process, in analogy to [22]. The dynamical phase can be studied as a perturbation between two equilibrium states represented by the IH configurations. Classically, isolated horizons are defined as null internal boundaries of space-time whose congruence of null generator vector fields has vanishing expansion (see [27] for a detailed definition) plus some energy condition. From this definition one can show that on each 2-sphere foliating the horizon the following boundary conditions hold where A is the Ashtekar-Barbero connection, Σ the 2-form dual of the densitized triad conjugate to A , a H the horizon area and β the Barbero-Immirzi parameter. At the quantum level then the kinematical Hilbert space can be split in a bulk and a surface part. The bulk space geometry is described by the polymer-like excitations of the gravitational field encoded in the spin networks states, which span the kinematical Hilbert space of LQG. Some edges of those states can now pierce through the horizon surface, providing local quantum d.o.f. accounting for the horizon entropy. More precisely, for a fixed graph γ in the bulk M with end points on the isolated horizon IH , denoted γ ∩ IH , the quantum operator associated with Σ in (24) is where [ ˆ J i ( p ) , ˆ J j ( p )] = glyph[epsilon1] ij k ˆ J k ( p ) at each p ∈ γ ∩ IH . let us denote /divides.alt0{ j p , m p } n 1 ; /uni22EF /uni27E9 the boundary state, where j p and m p are the spins and magnetic numbers labeling the n edges puncturing the horizon at points x p (other labels are left implicit). The horizon area operator ˆ a H is diagonal on this state namely The boundary theory then is quantized as a Chern-Simons theory in presence of particles. In fact, by defining k ≡ a H /slash.left( 4 πglyph[lscript] 2 p β ( 1 -β 2 )) , the quantum boundary condition (24) can be rewritten as where we have identified the ˆ J i ( p ) LQG operators with the with the Chern-Simons particles spin oprators. From the eom (17) we see that the curvature of the Chern-Simons connection vanishes everywhere on IH except at the position of the defects where we find conical singularities of strength proportional to the defects' momenta [18]. Moreover, there is an important global constraint that follows from (17) implying that the Chern-Simons boundary Hilbert space be isomorphic to the SU ( 2 ) singlet space between all the punctures, once the large horizon area limit is taken. In this way, one recovers the SU ( 2 ) intertwiner model of [18] (see FIG. 1 below). The IH boundary conditions imply that lapse must be zero at the horizon so that the Hamiltonian constraint is only imposed in the bulk. Now we want to 'unfrozen' the bulk dynamics near the horizon by interpolating two IH configuration with a dynamical horizon phase ∆ H , as studied in [21]. As shown above, the Hamiltonian constraint ∫ ∆ H N r Hd 3 V = 0 takes the form (13) in the gauge f = ρ , corresponding to a local stationary observer point of view. In the quantum theory then the (deparametrized) version of eq. (11) can formally be written as where ˆ p ρ = ∆ ˆ E , given by the (variation of the) area Hamiltonian, plays the role of the momentum conjugate to the time variable ρ and corresponds to the energy of the emitted quantum of radiation; ˆ H 0 = ˆ σ 2 /slash.left 8 πG is the Hamiltonian related to a shear operator driving the area variation, hence evolving the boundary states 3 . Notice that in this dynamical context only ˆ p ρ = ∆ ˆ E is a Dirac observable and the imposition of the Hamiltonian constraint (18), i.e. the implementation of dynamics of the theory, corresponds to a relation between partial observable ρ and ˆ E [29]. More precisely, going to the Heisenberg picture, the dynamics encoded by (18) can be used to study the radiation spectrum induced by the dissipation of the horizon energy by means of the matrix elements of the Hamiltonian constraint operator, since (the matrix elements of) the dissipation rate of the horizon energy observable can be written as where /divides.alt0 i /uni27E9 ≡ /divides.alt0{ j i p , m i p } n i 1 ; /uni22EF /uni27E9 , /divides.alt0 f /uni27E9 ≡ /divides.alt0{ j f p , m f p } n f 1 ; /uni22EF /uni27E9 are the initial and final sets of punctures data, defining the eigenstates of the area Hamiltonian before and after the action of ˆ H . Eq. (19) allows us to relate the horizon energy dissipation rate to the spectrum of the full Hamiltonian operator. Let us explain more in detail the meaning of this proposal for the implementation of the near-horizon dynamics. At the quantum level, the scalar constraint acts locally at the vertices of the spin network states and changes the spin associated to the edges attached to the given vertex, hence inducing fluctuations of the quantum geometry. Using the kinematical picture defined above, if we consider Thiemann's proposal [30] and concentrate only on the Euclidean part for simplicity, the action of the Hamiltonian constraint on a 3-valent node having two edges piercing the horizon can be graphically represented as ˆ where the holonomies entering the regularization of H are taken in the fundamental representation. The action of the Hamiltonian operator ˆ H near the horizon makes the initial horizon area operator eigenstate /divides.alt0 i /uni27E9 on the left of (20) (the other punctures forming the boundary Hilbert space and not affected by the action of ˆ H are not shown in the picture) jump to a different one /divides.alt0 f /uni27E9 on the right, generating in this way a change in the local notion of horizon energy and hence producing radiation 4 . The spectrum of such an emission process results in a discrete set of lines depending on the matrix elements of the Hamiltonian operator. The imposition of the scalar constraint (18) on a (space-like) portion of dynamical horizon connecting the two stationary configurations encodes a relation between the horizon energy dissipation and the metric fluctuations induced by a nearhorizon geometrical operator during the transition phase between the two consecutive equilibrium states. In the LQG description, the action of the full Hamiltonian operator on a vertex near the horizon (20) can be used to 'evolve' the isolated horizon, through a dynamical phase, from one quantum configuration to another. This is how relation (19) should be understood. We can now see the fluctuation-dissipation theorem interpretation of the first law of dynamical horizons associated to the relation (13). The Hamiltonian constraint encodes the same characteristics of the force driving the Brownian motion: it's action induce both a frictional and a fluctuating effect on the horizon quantum geometry, with the 'viscous' dissipation rate of the horizon area (energy) related to its matrix elements. The dissipative effect of the quantum fluctuation of geometry is related to the form of the area operator spectrum in LQG; in fact, due to the decreasing gap between higher eigenvalues, the overall balance of the geometry fluctuations results in a rate between energy emission and absorption bigger than one. In this way, the quantum theory provides a microscopic explanation for the dissipative nature of the evaporation process. In this picture then, an eventual observation of black holes radiance could provide a macroscopic window on the microscopic world, playing a role analog to that of Einstein's relation in proving the existence of atoms. In fact, from such an observation one could derive the number of punctures of a given spin defining the quantum horizon geometry by means of the spectrum intensity relation analog of the Fermi golden rule where transition probabilities are computed using (19). In the previous equation ¯ s j is the expectation value for the occupation number of punctures with a given spin j , /uni27E8 ˆ H /uni27E9 are the LQG Hamiltonian operator matrix elements corresponding to transition amplitudes involving two punctures with spins p and q piercing the horizon and jumping to a different energy level, with ∆ E pq being the energy emission in this single process (the plot of the relevant lines of the emission spectrum (21) can be found in [7], where we used the matrix elements of ˆ H computed in [31]). The formula (21) represents the spectrum of Hawking radiation encoding quantum gravity effects.", "pages": [ 6, 7, 8, 9, 10, 11 ] }, { "title": "C. Modified first law", "content": "The interpretation at the quantum level of the first law (12) as a fluctuation-dissipation theorem reveals interesting analogies with the derivation of the Einstein equation from a non-equilibrium thermodynamical treatment of the gravitational d.o.f. performed in [32, 33] and, at the same time, sheds light on it. Here it was shown that, in the case of non-vanishing shear at the horizon, the thermodynamical argument can still be run as long as non-equilibrium consideration are applied. More precisely, the shear contribution in the Raychaudhuri equation leads to an entropy balance relation in which purely geometrical d.o.f. are encoded in entropy production terms associated to gravitational energy fluxes. The presence of this internal entropy term leads to a generalized Clausius relation of the form [33] where the δN term is related to the exictation of internal/purely gravitational d.o.f. whose macroscopic effect is encoded in the horizon shear viscosity. The association of this internal entropy contribution to some sort of viscous work on the microscopic d.o.f. of the system goes along with the description of the Hamiltonian operator action emerging from the fluctuation-dissipation theorem interpretation provided above. In fact, in the quantum microscopic theory, this non-equilibrium entropy contribution is related to the area dissipation effect associated to the elimination of a puncture from the horizon, providing a natural relation with the modified first law for IH proposed in [24], where the extra term on the r.h.s. is added in order to take into account, in the quantum geometry description of the horizon, the presence of the quantum hair associated to the number of punctures, as mentioned in the Introduction (more details on this follow below). The modification (23) can be understood as a non-equilibrium, dissipative contribution introduced by the excitation of the quantum gravitational d.o.f.: it contains a quantity playing the role of a chemical potential conjugate to the number of punctures and can therefore be related to the evaporation process described in [7]. Therefore, the analogy between the thermodynamical approach in the derivation of Einstein equation in presence on non-equilibrium processes [32, 33] and the dynamical evaporation of quantum horizons described in [7] provides important insights into the proposal (23). In fact, it suggests an interpretation of this extra term as an internal entropy production term, i.e. an entropy associated to the dynamics of the quantum gravitational d.o.f. and encoded in the correlations between the radiation and the internal state of the horizon. More precisely, as described more in detail below, the action of ˆ H induces the disappearance of a puncture inside the horizon, representing a bit of information no more available to the external observer. In this way, the radiation process creates correlations between fluctuations of quantum geometry just outside and inside the horizon, with emission of quanta associated to the creation of new links in the hole bulk, hence providing an entanglement entropy contribution 5 . Such an interpretation could be made more precise by (and at the same time could provide important insights into) the relation between the Boltzmann [14, 16] and the von Neumann [35, 36] derivations of black hole entropy in LQG recently found in [37]. There the two pictures have been shown to be two equivalent and complementary descriptions of the horizon degrees of freedom. Furthemore, the analogy strengthens further the macroscopic, coarse grained interpretation of the Hamiltonian operator action on a quantum horizon as a non-equilibrium dissipative process via gravitational/microscopic d.o.f.. Such an hydrodynamics intuition also suggests a parallel with the membrane paradigm [38] and the stretched horizon picture [39], providing a precise characterizations of the notion of 'atoms' used in that context-the analysis that follows presents some analogies with the scenario depicted in [39], even though the framework we work in is quite different.", "pages": [ 11, 12, 13 ] }, { "title": "III. INFORMATION PARADOX", "content": "Right after his discovery of black holes radiance, Hawking himself realized [6] that such a phenomenon would lead to a breakdown of unitary evolution of black holes. The problem, usually referred to as the information paradox (see [40, 41] for some reviews), is two-fold. A first loss of information concerns the matter degrees of freedom that collapsed and formed the black hole. This is related to the fact that Hawking radiation is semiclassically in a mixed (thermal) state and, due to the 'no hair' theorems, carries no information at all about the collapsing body. As a consequence of this feature, a second loss regards the quanta of the field giving rise to the radiation. In Hawking's description, the outgoing modes of the field are correlated to the ingoing ones, even though one has no access to the latter: this is why the radiation state is thermal and it has an entanglement entropy associated to it. While this is a common picture in statistical mechanics-if you drop a box containing a gas in a trash can you have no access any more to the degrees of freedom inside the box, but the final state is still pure, as long as you take into account the whole system-, the problem with a black hole arises when this evaporates completely: the quanta in the radiation outside the hole are left in a state that is mixed, even though there is nothing to be mixed with anymore. In other words, the initial pure state has evolved into a final state that is mixed in a fundamental way. Difficulties with the viability of Hawking's drastic proposal [6] of a modification of the fundamental laws of quantum mechanics, in order to deal with this violation of unitary evolution, have been emphasized in [42] (see, however, also [43, 44]). Another path often advocated as a possible way out of the paradox is to take into account some sort of non-locality [45]. In particular, this seems an unavoidable choice in string theory, where Bekenstein-Hawking entropy is interpreted in the strong form, as a measure of the number of degrees of freedom inside a black hole [46] 6 . The idea that non-locality could solve the paradox in string theory led to the introduction of notions such as 'black hole complementarity' [39, 49, 50] and 'holographic principle' [51], which eventually culminated in the celebrated AdS/CFT correspondence [52]. However, there are difficulties in finding evidence for such non-locality effects in string theory [53] and no detailed analysis of the evaporation process, in either the CFT or the gravity theory, has explicitly been worked out, showing how locality breakdown can be reconciled with ordinary quantum field theory on a macroscopic scale. More recently, another picture within the string theory framework of how information can come out of black holes has been developed by modeling the hole in terms of so-called 'fuzzball' states [41]. In such a picture, a traditional horizon never forms; instead, different states of the string (creating different fuzzballs) spread over a horizon sized region. In this way, the matter making the hole is not confined at the singularity, but fills up the entire horizon interior and the radiation emerging from the fuzzball can send its information out [54]. However, also in this scenario reconciliation with local QFT in low curvature space-time regions is in a conjectural state. Moreover, extension of the fuzzball conjecture to the non-extremal case is a difficult task. In the LQG approach, the degrees of freedom responsible for the black hole entropy are located on the horizon (i.e., horizon quantum geometry fluctuations [14]). Therefore, instead of locality, the assumption in Hawking's argument which is abandoned is the existence of an information-free horizon. In this section we are going to exploit this quantum hair structure associate to the fundamental discrete quantum horizon geometry to describe subtle modifications of the Hawking radiation. Moreover, by implementing the evaporation dynamics described in the previous section till the Planck regime is reached, we show how the classical singularity is resolved leading to the formation of a long-lived remnant at the end of the evaporation process. This represents a prediction following from the imposition of the LQG dynamics and is the main new result of this section. We then explain how the combination of these two elements allows us to outline a picture for a possible solution to the information paradox in LQG. Furthermore, the role of non-local effects in the deep Planck regime due to non-local spin network links is considered, allowing for an extension of space-time beyond the classical singularity. In section III A we analyze more in detail the quantum hair notion emerging from the kinemat- ical structure of the boundary Hilbert space. This section doesn't contain original material, but investigate the analogy between the LQG description of the IH quantum gravitational d.o.f. and the discrete gauge symmetries of black hole horizon studied in previous literature. Such analogy supports the microscopic understanding of black hole thermodynamics emerging from our analysis. In section III B the massive remnant formation at the end of the evaporation process is derived using the dynamical evolution of IH quantum configurations described in section II B. The robustness of this scenario and its implications for the information paradox are discussed in section III C, where possible implementation of non-local effects is also considered; this last section is of a more speculative nature.", "pages": [ 13, 14, 15 ] }, { "title": "A. Quantum Hair", "content": "In the picture emerging from the LQG description of quantum black hole, the information resident on the horizon is encoded in a quantum hair at each puncture piercing the horizon. This notion of quantum hair, while of a very different nature, carries some similarities with the black hole quantum numbers associated with discrete Z N gauge charge analyzed in [55]. While compatible with (classical) 'no-hair' theorems, the quantum hair considered by these authors, which are not associated with massless gauge fields, have semi-classical effects on the local observables outside the horizon and on black hole thermodynamics, affecting for instance the hole Hawking temperature. In particular, [55] showed how, given two black holes with the same mass, the one with larger Z N charge is cooler. As a consequence, quantum hair can inhibit the emission of Hawking radiation and therefore stop the evaporation process. We will show in the next section how the presence of a quantum hair associated to the quantum gravitational d.o.f. allows for a precise realization of such a stabilization mechanism by implementing the near-horizon quantum dynamics all the way till the hole reaches a Planck scale size. Moreover, the kind of quantum hair of the black hole analyzed in [55] is argued to lead to non-perturbative corrections to the area law in [56], providing a further analogy with the LQG case. Before analyzing the last stage of the evaporation process, let us make more explicit the parallel between the Z N quantum hair considered in [55] and the one introduced in the LQG framework. In [55] the discrete Z N gauge symmetry arises in the Higgs phase of a U ( 1 ) gauge theory when a scalar with charge Ne condenses. The residual Z N subgroup which survives is related to the fact that the condensate cannot screen the electric field of a charge modulo N . Since the Higgs phase with unbroken Z N local symmetry supports a 'cosmic string', a vortex with magnetic flux 2 π /slash.left Ne trapped in its core, the charge modulo N on the black hole can be detected by means of the Aharonov-Bohm phase exp ( ı2 πQ /slash.left Ne ) generated when a charge Q is transported around the string. In this way, the Z N electric hair induces an infinite range interaction between string and charge which has non-perturbative (in /uni0335 h ) effects on the dynamical properties of the hole. In LQG, as we recalled above, the quantum geometry d.o.f. on the horizon are described by a topological gauge theory with local defects [16-18], namely by a Chern-Simons theory on a punctured two-sphere. As a result of the quantum implementation of (24), the punctures coming from the bulk and piercing the boundary represent the quantum excitations of the gravitational field on the horizon, as described by the LQG kinematics. If one adopts the point of view of [57], the quantum version of (24) can be taken as the starting point for a full definition of quantum horizon within the LQG framework. In fact, the analysis of [57] provides a rigorous mathematical basis to realize the original intuition of [58] relating horizons in LQG to TQFT. The emerging picture is that of a quantum horizon as a brane of a flat connection with local excitations of the electric quantum field ˆ Σ. Let us now see how the analog of the discrete gauge symmetry of [55] arises in this context, when one restricts to the U(1) model. In [57] it is shown that the horizon quantum state Ψ constructed as a solution of the quantum version of (24) is invariant under diffeomorphisms that keep the punctures fixed. However, this (gauge 7 ) symmetry is broken if one considers the exchange of two punctures by diffeomorphisms that leave the other punctures invariant. This is a fundamental characteristic of the horizon state, since the distinguishibility of the punctures is a crucial property to recover the linear area behavior of the entropy, and it's a typical example of how the presence of a boundary can break the local symmetry and turn gauge d.o.f. into physical ones [59]. Nevertheless, there is a residual Z k symmetry left at the punctures related to the fact that, by adding to the integers m p (labeling the U ( 1 ) irreducible representations) associated to each puncture a multiple of the Chern-Simons level k , the horizon quantum state will not change. Such a symmetry, which is a well known property of Chern-Simons theory with punctures, derives from the boundary condition (24). In fact, by means of the Stokes theorem one has where h γ is the holonomy around γ = ∂S ; in the quantum theory then, when γ goes around a puncture p from which the local Z k symmetry of the horizon spin network quantum state Ψ follows. Notice that when the horizon has the topology of a 2-sphere, i.e. in the single intertwiner model, the punctures have to satisfy also the global constraint as a consequence of (25) when γ goes around all the punctures on the horizon. It is now clear the analogy with the quantum hair considered in [55]. Each puncture p carries a Z k electric charge given by its representation integer m p ; the Wilson loop for parallel transport around a puncture (25) defines an element of the residual gauge group Z k , measuring the flux inside with basic unit Φ k = 2 π /slash.left k . Therefore, the detection of the quantum hair effects by such observables is the analog of the Aharonov-Bohm interaction. In [57] a Lebesgue measure for the integral on connections satisfying (25) is defined by fixing the U ( 1 ) angle integration variable at each puncture p . In this way, a definition of 'quantum isolated horizon' (QIH) within the full theory can be introduced. In fact, exploiting the analogy with the notion of quantum hair investigated in [55], the relation (27) can be interpreted as containing no reference to classical horizon elements: the integer k can be assumed as a parameter proper of the topological quantum theory associated to a local, discrete quantum gauge symmetry; this definition of QIH would push further the point of view [57] and correspond to a realization at the quantum level of the paradigmshift introduced in [60, 61]. Moreover, the extension of such a definition to the SU ( 2 ) case, which requires a more involved analysis due to the highly nontrivial action of the operator on the r.h.s. of (24), seems to match the model of [17] and be compatible with the proposals of [35, 62].", "pages": [ 15, 16, 17 ] }, { "title": "B. Singularity Resolution", "content": "We now want to exploit the existence of this horizon quantum hair and the associated description of the radiation emission of section II B to explore the dynamics of the final stage of the evaporation process and its implication for the information loss. In particular, we want to investigate the idea that quantum geometry fluctuations may allow unitary evolution of the black hole beyond the classical singularity. The relevance of such a scenario for the resolution of the information paradox has been emphasized, for instance, in [63, 64]. In section II B we saw that the quantization of the IH boundary condition (24) yields a ChernSimons theory on the 2-sphere horizon with punctures. In the large area limit then the boundary Hilbert space is isomorphic to a SU ( 2 ) intertwiner space [16-18]. As we saw above, the approach of [57], where the IH d.o.f. are represented by elements of the holonomy-flux algebra of LQG via a modified Ashtekar-Lewandowski measure on the horizon implementing (24), also converge to the single intertwiner model and shares the same topological features for the quantum boundary theory. Moreover, this picture is also compatible with the approaches [35, 62] attempting to provide a definition of quantum horizon without referring directly to the IH framework. Therefore, let us concentrate on the SU ( 2 ) intertwiner model for the IH quantum state and in particular on the more general approach of [57]. This can be constructed starting from an arbitrary spin network state on the canonical hypersurface Σ with support on a graph Γ. We denote B a connected bounded region of Γ formed by a finite set of vertices and edges that connect them. The horizon surface IH is defined as the set of n edges with only one vertex in B , which defines the boundary ∂B of the connected region and is endowed with boundary data represented by the spin label j 1 , . . . , j n of the boundary edges. To each puncture p we can associate an elementary patch with a microscopic area given by the spin j p : the collection of all these patches forms the quantum horizon brane. These colorings also determine the expectation values of observables defined on the horizon, in particular Wilson loops around the punctures are fully constrained by imposition of the boundary condition (24) via the modified Ashtekar-Lewandowski measure on IH , as mentioned before. Moreover, in accordance with the expectation of the topological nature of the boundary theory, holonomies on all contractible loops in IH do not represent local gauge invariant d.o.f. and the range of the spin labels is restricted by the Chern-Simons level playing the role of a cut-off in the quantum theory, namely j p ≤ k /slash.left 2 8 . The intertwiner structure can now be obtained in the following way. The black hole interior geometry is fully described by the (superposition of) spin network states with support on the arbitrary complicated interior graph Γ B . However, as argued above, for the entropy calculation the relevant information is only the one that can affect the external observer and be read off the horizon. Therefore, the details of the spin network inside the black hole do not matter and to an outside observer the horizon state looks like an intertwiner between the SU ( 2 ) representations V j p . In other words, in the entropy calculation one traces over the bulk d.o.f. by coarse graining the interior geometry to a spin network having support on a graph with a single vertex inside the horizon, with the boundary edges coming out of it. The coarse graining techniques have been developed in detail in [35, 66], where it has been shown that the coarse-grained intertwiner in the boundary Hilbert space depends in general on the the coarse graining data { g e ∈ B /slash.left T } , where T is a maximal tree in B 9 . The group elements { g e ∈ B /slash.left T } represent the holonomies around each (noncontractible) loop of the interior graph Γ B , once all the group elements on the edges belonging to the tree have been fixed to the identity, by means of gauge invariance. Henceforth, the coarse graining data encode the information about the non-trivial topology of the quantum geometry of the black hole interior B . Following [35], having set the maximal number of holonomies to the identity, the whole graph B can be effectively reduced to a single vertex with n open edges (forming the boundary) and L B loops (see FIG. 2). The number of non-trivial loops of Γ B is given by L B = E B -V B -1, where /divides.alt0 T /divides.alt0 = V B -1 is the number of edges of the maximal tree T in B and E B the total number of edges in B . The quantum state of geometry of B is then represented by the contraction of the single intertwnier I ∶ V ⊗ n ⊗ V ⊗ 2 L B → C with the holonomies g e ∈ B /slash.left T along the loops. Notice that the analog of the global constraint (26) in the SU ( 2 ) case corresponds to the condition that the sum of all the spin labels j p be an integer. Let us now analyze how the evaporation process evolves this state by implementing the nearhorizon dynamics as described in [7]. We showed in section II B how Thiemann's Hamiltonian operator acting on a node right outside the horizon with two edges departing from it and piercing the boundary surface creates a new link of spin 1 /slash.left 2 (assuming that we regularize ˆ H in the fundamental representation) between the punctures and changes the spin associated to them. A typical process is when ˆ H acts on two spin-1/2 punctures and then one jumps to spin-0 while the other to spin-1 or on a spin-1/2 and a spin-1 punctures and they jump respectively to spin-0 and spin-1/2; graphically we have where gauge invariance at the new trivalent node created inside the horizon has been used to form the petal. In this way, a boundary puncture disappears (the horizon shrinks) and a new loop of spin-1/2 is created in the interior of the hole every time. Of course, there will be also transitions in which the quantum geometry fluctuations correspond to an absorption process leading to an area increase. This for instance can happen when acting on a spin-1/2 and a spin-1 like in (29) and both punctures jump to an higher spin, namely 1 and 3/2. However, as pointed out in section II B, assuming the standard area operator spectrum in LQG, the average process has a dissipative effect; this follows from the property of a decreasing gap between higher eigenvalues which makes most of the transitions induced by ˆ H correspond to an emission process. For example, among the six possible fluctuations allowed by the action of the Hamiltonian constraint for the initial configurations in (28) and (29) (which for a large black hole represent the most likely ones) only one is associated to an absorption process, namely the one mentioned above. Obviously, the velocity of the evaporation process depends on the specific form of the Hamiltonian operator matrix elements and in general it is expected to be very slow for large black holes, consistently with the semi-classical picture. We can now iterate this dynamics till the horizon reaches a Planck scale size in order to investigate the last stage of the evaporation process. Notice that in [7] terms arising from the action of ˆ H where new punctures are created (corresponding to an absorption process with area increase) were discarded due to the breaking of diffeomorphisms symmetry on the horizon in those cases (for instance, such an action would create pathological states like those analyzed in [67], making the horizon area observable ill defined). However, as the deep quantum regime is reached and the curvature becomes large, the notion of classical manifold breaks down. Therefore, in its final stage the horizon is expected to be in a quantum fluctuating state with no significant quantum gravitational radiation emission anymore: boundary punctures are continuously being created and annihilated. Such a final interior state carries resemblance with the picture of long hornlike ge- metries connected to the external space by tiny holes [48], which classically would evolve to reach infinite length and zero width in finite proper time. We can nevertheless wonder if the dynamics allows, at least in principle, a complete evaporation of the horizon where the area shrinks to zero, regardless of the velocity with which such a complete evaporation would take place. This can be investigated by implementing the action depicted in (28), (29) as far as possible. Namely, assuming an initial distribution of only spin-1/2 and spin-1 punctures forming the boundary state, through a sequence of processes depicted in (28) and (29), the quantum horizon could eventually reach a Planck scale state formed by only two punctures of spin-1/2 (notice that a Planckian state corresponding to two punctures of spin-1/2 and spin-1 is not allowed by the global gauge constraint mentioned above); if we now try to implement the action of the Hamiltonian operator (28) further in order to eliminate these residual two punctures, we realize that this is not allowed. The reason for this impossibility relies on the form of the matrix elements of (the Euclidean part of) ˆ H derived in [31], where it is shown that the probability for two edges with the same spin j to both jump to j -1 /slash.left 2 is zero. Therefore, only one of the two punctures can disappear inside the hole while the other has to jump to the spin-1 level, graphically The (daisy-)state represented on the r.h.s. of (30) corresponds to the allowed configuration of the quantum black hole with minimal area, showing how the horizon area operator is bounded from below by the dynamics of the theory, with a minimum value corresponding to 8 πβglyph[lscript] 2 p √ 2. Such an asymptotic state corresponds to a vanishing temperature limit of the evaporation process. A similar departure from the semi-classical scenario is found, for instance, also in [68], for the behavior of the surface gravity, and in [69], by means of dynamical arguments based on the generalized uncertainty principle. Therefore, the analysis shows that the quantum horizon can never shrink completely, i.e. it never hits the ' r = 0' point (see FIG. 3 below). In this way, the spacelike singularity inside the black hole is removed due to the quantum dynamics of the theory, confirming the results of previous analyses [70-74] based on the application of mini-superspaces to Schwarzschild interior. Notice that it is the '6j ' part of the euclidean Hamiltonian operator which is responsible of the avoidance of the complete evaporation of the horizon, as shown in (30). Henceforth, this feature of the quantum gravitational description of the black hole evolution is not related just to Thiemann's regularization scheme and, in particular, it is expected to be recovered also within the implementation the near-horizon dynamics in the spin foam formalism. In fact, since spin foam amplitude is expected to provide a definition of the physical scalar product in LQG, the fundamental action of ˆ H on a near-horizon node could be described by a vertex amplitude whose boundary graph pierces the horizon and contains the initial and final states as subgraphs. It is then easy to see how the part of the amplitude responsible for the disappearance of boundary punctures has the combinatorics of a 6j . Moreover, the vanishing of the transition amplitude corresponding to the complete evaporation of the horizon holds for both orderings concerning the position of the volume operator inside the regularized expression for the Hamiltonian constraint, as considered in [31]; henceforth, the remnant formation scenario is free from this ambiguity. We want to conclude this section with a remark concerning the first law. The identification in [23, 24] of the horizon area with a notion of local energy suggests a scheme for a microscopic derivation of it, within the framework we have been delineating. In fact, this local thermodynamics perspective is telling us that every flux of matter through the horizon leaves an imprint on it. From the microscopic theory, it is natural to see how this can be realized, since matter d.o.f. are associated to spin network links and, hence, matter falling into the black hole can happen only via the creation of new links piercing the horizon. This does not mean that the energy remains distributed on the horizon d.o.f.; indeed, it will fall inside and contribute to the interior state by creating new blu petals. However, the horizon Hilbert space will be modified by this flux and the surface system is left in a new ensemble. If now, according to the LQG recipe, we associate the black hole entropy to the dimension of the boundary Hilbert space 10 , we see how the horizon can keep track of the entropy it gains as a bit of energy flows through and the first law follows naturally.", "pages": [ 17, 18, 19, 20, 21, 22, 23 ] }, { "title": "C. Black Hole Remnants", "content": "The last stage of the evaporation process described in the previous section shows how the horizon never shrinks down completely but stabilizes at a finite (microscopic) size. Such a final state corresponds to the formation of a massive remnant and leads to a non-singular quantum space-time according to the definition introduced in [64], that is the dynamics defines a reversible linear map between the Hilbert spaces associated to two complete non-intersecting spacelike hypersurfaces. As argued in [64], this property of the quantum dynamics is enough to restore unitarity once the d.o.f. inside the remnant are taken into account.", "pages": [ 23 ] }, { "title": "1. Where the information goes", "content": "We can now see how information is not lost. First of all, let us point out how the d.o.f. associated to the collapsed matter that formed the black hole are encoded in both the boundary punctures forming the horizon and the coarse graining data associated to the non-trivial topology (the blue petals in FIG. 2) of the graph in the interior of the hole. The latter correspond to the large number of possible interior states, reflecting the variety of histories of the gravitational collapse, and, in the weak interpretation of the Bekenstein-Hawking entropy [11-13], they are not assumed to contribute since otherwise the result would be much larger. The second form of information that is assumed to be lost in the semi-classical scenario consists of the correlations between the quanta of the matter field which reaches future null infinity via the Hawking process and their partners that fall inside the singularity. In our description of the evaporation process, the analog of these correlations are between the quanta of radiation that a stationary observer hovering close outside the horizon sees and the new petals that form inside the hole as the boundary punctures disappear in the emission process. However, the two types of information are strictly related since the boundary punctures data encode part of the collapsed matter degrees of freedom. A fundamental difference with the semi-classical description is that, for a local fiducial observer hovering at close distance from the horizon, the spectrum of the radiation is not thermal anymore, but formed by a discrete set of lines [7]. While a smooth thermal envelope can be expected to be recovered for an asymptotic observer, for which the large number of punctures could compensate the suppression of more transition lines associated to small transition amplitudes, such a discrete structure would still emerge after a certain point from the beginning of the evaporation and before reaching the deep Planck regime. This is when the information associated to these correlations, and hence to the matter d.o.f., start to leak out. More precisely, by measuring the energy levels and the intensity of the lines, by means of the spectrum formula (21), one can eventually recover the information about the microscopic structure of the quantum horizon encoded in the punctures data (i.e. how many punctures are in a given spinj level). Notice that the observation of such a spectrum would allow us also to solve ambiguities present in the quantization of the Hamiltonian constraint, like the irreducible representation to take the holonomies in and ordering ambiguities, as described in [7]. However, to support this picture of information leakage a more detailed analysis of the radiation spectrum beyond the one-vertex approximation used in [7] is required. The rest of the information is stored behind the horizon of the fluctuating quantum final state of the evaporation process. No bit of information goes lost in the singularity, as the semi-classical analysis would suggest, since there is no singular final state anymore due to quantum dynamical effects. However, let us point out how the deviation from thermality associated to the quantum hair structure on the horizon could be enough also for an asymptotic observer to recover the full information at infinity. In fact, the quantum isolated horizon temperature has been recently derived in [68] from a local microscopic analysis. The final formula presents a quantum correction associated to the Chern-Simons level, which defines an effective temperature reproducing exactly the deviation from thermality of the radiation spectrum found in [75]. In [76] it has been argued how such a modification encodes correlations among quanta of Hawking radiation for a total amount of information corresponding to the Bekenstein-Hawking formula.", "pages": [ 23, 24 ] }, { "title": "2. Non-local effects", "content": "A natural question then is whether this remnant state is permanent and the external observer will never have access to the information stored inside again (FIG. 3) or it can actually decay, with the internal d.o.f. coupling again to the exterior spin network state, and the horizon disappear. As we saw above, the blue petals in the interior of the black hole are associated to the nontrivial topology of the graph inside and encode part of the collapsed matter d.o.f. that formed the horizon (or keep falling inside after its formation) and do not leave a direct imprint on the surface state. Therefore, they are to be interpreted as bulk d.o.f. and, in its weak interpretation, they do not contribute to the Bekenstein-Hawking entropy. Nevertheless, they may play an important role in the black hole evolution, since, due to quantum fluctuations of geometry induced by the dynamics, it could be possible for them to tunnel out (see FIG. 4) and, given that the boundary conditions defining the quantum horizon (no matter their specific choice) may not be preserved by this dynamical evolution violating its causal structure, to destroy the horizon. In this way, the space-time can extend beyond the classical singularity, corresponding to a sector of the phase space where the triad has reversed orientation. All the information that was trapped within the apparent horizon could eventually get out to infinity by means of the time-reversed action of ˆ H driving the evaporation process. However, given the local nature of the Hamiltonian operator action and the very large volume of the bulk region (due to the presence of a big intertwiner), one would expect the amplitude for this tunnel effect to be considerably suppressed and such a scenario highly unlikely. At this point, an important observation concerns the notion of locality for the graphs underlying the 3-geometry quantum states. In [77] (see also references therein) it has been shown how the notion of microlocality , associated to the connectivity of the combinatorial structure of the graph, and that of macrolocality , associated to an emergent classical space-time metric, need not to coincide in the definition of semi-classical states 11 . The authors show that states which are semiclassical but nonetheless contain non-local links are common in the physical Hilbert space of LQG. Moreover, it is argued that, starting with a local state associated with a classical three metric and implementing a long series of local moves induced by the LQG dynamics, one can create non-local links, which are not suppressed by further implementation of dynamics. This suggests a concrete realization of non-local effects in the evaporation process we described, which does not violate the local quantum field theory description of physics at low curvatures. Namely, the black hole bulk state can start out without any non-local links, allowing a local low energy limit of the quantum gravity theory in space-time regions where the semi-classical approximation is supposed to be valid. However, in the long term, over time scales large compared to the Planck time (as the horizon shrinks down and the curvature becomes big), non-local connections are introduced in the interior state. In this way, it would be possible for the in-falling initial d.o.f. in the deep Planck regime to tunnel out, as depicted in FIG. 5. Henceforth, the presence of these non-local effects induced by the LQG dynamics, while compatible with the semi-classical analysis in its regime of validity, can provide a concrete example of the notion of non-locality invoked in [78]. This is surely a scenario that deserves further investigation and at this stage it has to be taken just as a proposal for a possible realization from the full theory of the paradigm described in [63, 79].", "pages": [ 24, 25, 26 ] }, { "title": "3. Objections against remnants", "content": "Problems with permanent or long lived remnants have been raised and discussed by several authors in the literature (see for instance [40, 43, 48, 80, 81]). Some of these objections haven been addressed also in [64]. Our description weakens further the criticisms to the remnants scenario, the main one being the infinite pair production problem. Let us address this issue more in detail. First of all, the dynamical nature of the horizon and bulk states described above, due to quantum geometry fluctuations allowing for the internal d.o.f. to couple with the external ones, precludes the application of effective field theory with minimal coupling to remnants, as argued in [48, 64]. Treating black hole remnants as pointlike particles is not allowed, once quantum gravity dynamics is taken into account, and the tensor product structure of the Hilbert space becomes fuzzy in this regime. Another important observation that would invalidate an effective QFT treatment of the remnant is related to the modified first law (23). In fact, as noted by L. Freidel, such a modification, necessary in order to take into account dissipative effects on the horizon (as discussed in Section II), could allow the remnant to have a mass well above the Planck mass. Moreover, despite the fact that the formation of a big intertwiner in the interior of the remnant state allows for a region with very large volume (hidden behind the small horizon), the amount of information to be stored inside the remnant, in order to restore unitarity of the black hole evolution, is not as large as usually expected. In fact, as argued above, the radiation process we described allows to recover part of the information about the initial collapsed matter d.o.f. and the radiation correlations already before the evaporation process stops. This means that the infinite degeneracy of the interior state advocated to affect the effective field theory description of the coupling of the remnants to soft quanta is not such a valid objection in our case. By the time the remnant forms, an external observer has already had access to part of its internal structure states and the amount of information left to recover in order for the Bekenstein-Hawking entropy (in its weak interpretation) to vanish is not as large to require an infinite degeneracy of remnant species.", "pages": [ 26, 27 ] }, { "title": "IV. CONCLUSIONS", "content": "If we adopt the point of view that the entropy d.o.f. reside at the horizon and we refer to a local description of black hole thermodynamical properties, a theory of quantum gravity can be successfully applied to shed light on the fundamental problems of black hole physics. We have seen how the interpretation of a quantum horizon as a gas of punctures ('atoms' of space) whose microscopic dynamics is described by the LQG formalism, provides a thorough statistical mechanical analysis of the system. In [22] it was shown how Hawking radiation can be recovered entirely in terms of the viscous nature of the horizon associated to purely gravitational d.o.f.; if we take this indication seriously, at the local microscopical level then one should be able to describe the evaporation process by focusing exclusively on the dynamics of the quantum geometry constituents. In particular, the horizon evolution, when described in terms of a fluctuation-dissipation relation applied to the quantum hair associated to the fundamental discrete structure, provides a description of the radiation emission in terms of relaxation to an equilibrium state balanced by the excitation of Planck scale d.o.f. at the horizon. In this way, the space-time dissipative effects, encoded in a modification of the first law (23), are related to the quantum geometry fluctuation induced by the Hamiltonian operator, providing a connection between macroscopic and microscopic levels of descriptions. This new description of the evaporation process is one of the two main results presented in this paper and it has important implications for the information paradox. Namely, a fiducial observer hovering very close to the horizon and capable to perform measurements with sufficiently fine time resolution could reconcile the effective 'viscosity' of the horizon with the unitarity of the process. In fact, such an observer would be able to access most of the information of the matter d.o.f. that fell inside the black hole and left an imprint on the horizon from the details of the radiation spectrum she observes on extremely short distance and time scales. An asymptotic observer, however, can only discuss average properties of the hole and have access to the information only after a long time from the beginning of the evaporation process. It is this broadening of the spectrum lines, which is expected to take place in the initial phase of the evaporation for large black holes, the coarse-graining procedure necessary to prove the second law. In this regime, the distant observer would not be able to read off the correlations between the emitted quanta and the in-falling d.o.f. from the radiation spectrum, with the entropy increasing in time due to this constant injection of entanglement. However, as the horizon shrinks down in size, the spectrum starts to reveal its discrete structure and the correlations eventually become detectable. This is the point when the black hole entropy curve flip over and start descending, in a scenario similar to the one envisaged by [82], where an estimation of the information contained in the Hawking radiation subsystem (forming a random pure state with a second subsystem represented by the black hole) shows that information could indeed gradually come out via correlations between early and late radiation parts. In any case, the information that could not be recovered from the radiation content is not lost. In fact, when taking into account the local quantum dynamics of the gravitational field, black hole evolution is not singular and unitarity can be recovered. More precisely, a quantum mechanical description in terms of Hilbert space structures of the black hole evolution is valid and possible at all the stages. After the deep Planck regime is reached and part of the information has leaked out through the spectrum of the quantum gravitational radiation, the collapsed matter forms a massive remnant, which does not radiate anymore and whose d.o.f. are described by a separate Hilbert space. This is the second main result of our analysis. In this high curvature regime, if no non-local effects take place, the interior state information, whilst not lost, won't be able to come out and be accessible again to an exterior observer. On the other hand, if non-local effects of the kind described by the notion of disordered locality [77] develop, the horizon could dissolve in this quantum gravitational phase, with the consequent vanishing of the trapped surface (see FIG. 5). In this case, all the d.o.f. on a complete Cauchy surface can eventually be described again in terms of a single Hilbert space. Inclusion of this non-local effects and their possible connection with the complementarity ideas introduced in the string theory literature deserve and necessitate of a more detailed investigation. While each of these solutions, proposed at different stages in the literature, faces serious problems when taken singularly, the combination of them, allowed and actually realized by the unitary dynamics of LQG, provides a valid alternative to more drastic departures from semi-classical physics. Surely, the analysis presented here is far from conclusive. Among other things, a more thorough investigation of the radiation spectrum derived in [7] has to be carried out; a precise canonical definition of the quantum horizon from the full theory needs to be sort out, possibly allowing to derive a notion of Unruh temperature in terms of the horizon geometrical d.o.f.; matter should be included in the picture and the gravitational imprinting referred to at the end of Section III B described in detail; the Lorentzian part of the Hamiltonian constraint should be added to the derivation of the radiation spectrum and its implications for the formation of a massive remnant analyzed in order to make this scenario more robust. Nevertheless, we believe that the picture presented here provides a coherent and appealing description of the statistical mechanics understanding of black holes thermodynamics. Let us conclude by pointing out that our analysis provides a concrete realization of the conjectured scenario [83] in which black hole evaporation could cease once the hole gets close to the Planck mass, allowing for the formation of Planck relics which could contribute to the dark matter (see also [84] for a more recent proposal along these lines). Furthermore, the insights gained in the context of black holes on the interplay between microscopic and macroscopic scales might turn out to be useful in application of emergent space-time scenarios to the investigation of the semi-classical continuum limit of the theory [85]. Acknowledgements. I would like to thank E. Bianchi, B. Dittrich, D. Oriti, A. Perez, L. Sindoni and L. Smolin for useful discussions and comments which helped to improve this manuscript. Comments from an anonymous referee have also contributed to make the presentation clearer. geometry from quantum information: Area renormalisation, coarse-graining and entanglement on spin networks,' gr-qc/0603008. [hep-th/0604072].", "pages": [ 27, 28, 29, 30, 32, 33 ] } ]

[ { "title": "L. Iorio", "content": "Ministero dell'Istruzione, dell'Universit'a e della Ricerca (M.I.U.R.)-Istruzione Fellow of the Royal Astronomical Society (F.R.A.S.) Viale Unit'a di Italia 68, 70125, Bari (BA), Italy June 14, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "The dynamics of non-spherical systems described by MOND theories arising from generalizations of the Poisson equation is affected by an extra MONDian quadrupolar potential φ Q even if they are isolated (no EFE effect) and if they are in deep Newtonian regime. In general MOND theories quickly approaching Newtonian dynamics for accelerations beyond A 0 , φ Q is proportional to a coefficient α ∼ 1, while in MOND models becoming Newtonian beyond κA 0 , κ /greatermuch 1 , it is enhanced by κ 2 . We analytically work out some orbital effects due to φ Q in the framework of QUMOND, and compare them with the latest observational determinations of Solar System's planetary dynamics, exoplanets, double lined spectroscopic binary stars and binary radio pulsars. The current admissible range for the anomalous perihelion precession of Saturn -0 . 5 milliarcseconds per century ≤ ∆ ˙ /pi1 ≤ 0 . 8 milliarcseconds per century yields | κ | ≤ 3 . 5 × 10 3 , while the radial velocity of α Cen AB allows to infer | κ | ≤ 6 . 2 × 10 4 (A) and | κ | ≤ 4 . 2 × 10 4 (B). PACS: 04.80.-y; 04.80.Cc; 95.1O.Ce; 95.10.Km; 97.80.-d", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The MOdified Newtonian Dynamics (MOND) (see [1] for a recent review) is a theoretical framework proposed by Milgrom [2-4] to modify the laws of the gravitational interaction in a suitably defined low acceleration regime to explain the observed anomalous kinematics of certain astrophysical systems such as various kinds of galaxies [5-7]. Indeed, their behaviour does not agree with the predictions made with the usual Newtonian inverse-square law applied to the electromagnetically detected baryonic matter whose quantity appears to be insufficient. In the case of the mass discrepancy occurring in clusters of galaxies [8], MOND actually faces difficulties in explaining it [9-11]. In almost all its relativistic formulations, MOND implies a single 1 acceleration scale [13] A 0 = (1 . 2 ± 0 . 27) × 10 -10 m s -2 below which the laws of gravitation would suffer notable modifications mimicking the effect of the additional non-baryonic Dark Matter which is usually invoked to explain the observed discrepancy within the standard theoretical framework. In this paper, we propose to constrain a recently predicted strong-field effect of MOND [14] by using various observables pertaining different astronomical scenarios. In the following we will briefly outline the main features of such a novel prediction of MOND which occurs even if the system under consideration is isolated and if its characteristic accelerations are quite larger than A 0 . Let us consider an isolated, strongly gravitating system S of total mass M tot and extension where G is the Newtonian constant of gravitation. Let us also assume that the mass distribution of S , characterized by a generally anisotropic matter density /rho1 ( r ), varies over timescales much larger than t M . = d M /c, where c is the speed of light in vacuum. According to formulations of MOND based on extensions of the Poisson equation such as the nonlinear Poisson model by Bekenstein and Milgrom [15] and 2 QUMOND[17], it turns out [14] that, if on the one hand, the MOND field equations of S coincide with the usual linear Poisson equation for r ≤ R depending on how the MOND interpolating function µ is close to unity, on the other hand, they differ from it for r ≥ d M . This is a crucial feature since it implies that the solution φ of the Poisson equation for r ≤ R is, in general, different from the usual Newtonian one φ N , thus affecting the internal dynamics of S even if it is in the strong gravity regime. It is as if a hollow 'phantom' matter distribution, characterized by a phantom matter density /rho1 ph ( r ), was present at r ≥ d M in such a way that, in the quasi-static limit previously defined, /rho1 ( r ) instantaneously controls /rho1 ph ( r ) by fixing its symmetry properties. If /rho1 = /rho1 ( r ), i.e. if S is spherically symmetric, then the phantom matter density is spherically symmetric as well. In this case, the internal dynamics of S would not be affected by the peculiar boundary conditions on the MOND field equations at r ≥ d M or, equivalently, by the phantom matter. Indeed, it would be arranged in a hollow spherical shell; the dynamics of S would be Newtonian to the extent that the MOND interpolating function µ matches the unity. On the contrary, if /rho1 = /rho1 ( r ), i.e. if S is not spherically symmetric, the same occurs to the phantom matter as well. Thus, it does have an influence on the internal dynamics of S which, to the lowest order, can be approximated by an additional quadrupolar potential 3 φ Q = φ -φ N . By assuming µ = 1 to the desired accuracy everywhere within S and by using QUMOND [17], Milgrom [14] obtained with x i F , i = 1 , 2 , 3 in eq. (2) are the components of the position vector r F of a generic point F with respect to the barycenter of S , while x ' j , j = 1 , 2 , 3 in eq. (3) determine the barycentric position of the system's mass elements. The coefficient α depends on the specific form of the interpolating function chosen. Milgrom [14], by considering also the case in which the strong field regime is obtained in terms of a second, dimensionless constant κ /greatermuch 1 when the Newtonian acceleration is as large as ∼ κA 0 , picked up an interpolating function yielding In general, there should be many other interpolating functions that could be used with κ /greatermuch 1; in this paper, we will focus on eq. (4). Finally, we remark that Milgrom [14] felt that theories with κ /greatermuch 1 cannot be considered as generic MOND results. As stressed by Milgrom [14], the quadrupolar MOND effect of eq. (2) has not to be confused with some other MONDian features occurring in the strong acceleration regime which were previously examined in literature. In particular, it is not the quadrupolar effect [18, 19] due to the External Field Effect (EFE) [2,15,20] arising when the system under consideration is immersed in an external background field; indeed, here the system is considered isolated. Even so, residual MONDian effects in the strong acceleration regime exist, in general, because of the remaining departure of µ ( q ) from 1 when q /greatermuch 1; their consequences on orbital motions of Solar System objects were treated in, e.g., [2,18,21,22]. Nonetheless, they are different from the presently studied effect, for which it was posed µ = 1 to the desired accuracy. Finally, Milgrom [14] showed that the impact of the zero-gravity points [23-25] existing in high acceleration regions on the dynamics of the mass sources themselves is negligible with respect to the effect considered here. The plan of the paper is as follows. In Section 2 we analytically work out some orbital effects caused by eq. (2) to an isolated two-body system in the case of eq. (4). In Section 3 our results are compared to latest observations on Solar System planetary motions, extrasolar planets, and spectroscopic binary stars. Section 4 is devoted to summarizing our findings.", "pages": [ 1, 2, 3, 4 ] }, { "title": "2 Calculation of some orbital effects", "content": "Let us consider a typical non-spherical system such as a localized binary made of two point masses M and m with M b . = M tot = M + m . In a barycentric frame, its mass density /rho1 ( r F ) at a generic point F can be posed where r m and r M are the barycentric position vectors of m and M , respectively. After having calculated φ Q ( r F ) for eq. (5), its gradient with respect to r F yields the extra-acceleration A F of an unit mass at a generic point F . The extra-accelerations A m and A M experienced by m and M can be obtained by calculating A F for r m = MM -1 b r and for r M = -mM -1 b r , respectively, where r . = r m -r M is the relative position vector directed from M to m . It turns out that the accelerations felt by m and M are The relative extra-acceleration is, thus, [14] where µ b . = mMM -1 b is the binary's reduced mass. For the following developments, it is useful to remark that, formally, eq. (8) can be derived from the effective potential", "pages": [ 4, 5 ] }, { "title": "2.1 The pericenter rate for a two-body MOND quadrupole", "content": "The longitude of pericenter /pi1 . = Ω + ω is a 'broken' angle since the longitude of the ascending node Ω lies in the reference { x, y } plane from the reference x direction to the line of the nodes 4 , while the argument of pericenter ω reckons the position of the point of closest approach in the orbital plane with respect to the line of the nodes. The angle /pi1 is usually adopted in Solar System studies to put constraints on putative modifications of standard Newtonian/Einsteinian dynamics [26]. Its Lagrange perturbation equation is [27] where U pert is a small correction to the Newtonian potential; a is the relative semimajor axis, e is the orbital eccentricity, and I is the inclination of the orbital plane to the reference { x, y } plane. The brackets 〈 . . . 〉 in eq. (10) denote the average over one full orbital period P b = 2 πn -1 b = 2 π √ a 3 G -1 M -1 b . By adopting eq. (9) as perturbing potential U pert in eq. (10), one gets It should be remarked that eq. (9) and, thus, eq. (11) are valid just for a two-body MOND quadrupole Q ij . As a cross-check of the validity of our result, we repeated the calculation of the long-term precession of /pi1 by using eq. (8) as perturbing acceleration and the Gauss equations for the variations of the elements: we re-obtained eq. (11).", "pages": [ 5 ] }, { "title": "2.2 The timing in binary radiopulsars", "content": "The basic observable in binary pulsar systems is the periodic change δτ p in the time of arrivals (TOAs) τ p of the pulsar p due to the fact that it is gravitationally bounded to a generally unseen companion c, thus describing an orbital motion around the common barycenter. In a binary hosting an emitting radiopulsar, the Keplerian expression of δτ p is obtained by taking the ratio of the component ρ p of the barycentric pulsar's orbit along the line of sight to the speed of light c . Thus, one has Since the line of sight is customarily assumed as reference z axis, in eq. (12) it is as it can be inferred from the standard expressions for the orientation of the Keplerian ellipse in space. In eq. (13), r p is the distance of the pulsar from the system's center of mass, I is the inclination of the orbit to the plane of the sky, assumed as reference { x, y } plane, and f is the true anomaly reckoning the instantaneous position of the pulsar with respect to the periastron position. By using where a p is the semimajor axis of the the pulsar's barycentric orbit and E is the eccentric anomaly, from eq. (12)-eq. (13) one straightforwardly gets [28, 29] In eq. (17), x p . = a p sin I/c is the projected semimajor axis of the pulsar's barycentric orbit and has dimensions of time; by posing m p . = M,m c . = m , it is a p . = a M = mM -1 b a, where a is the semimajor axis of the pulsar-companion relative orbit. In general, the shift per orbit ∆ Y of an observable Y with respect to its classical expression due to the action of a perturbing acceleration such as either eq. (6) or eq. (7) can be computed as where M is the mean anomaly and ψ collectively denotes the other Keplerian orbital elements. The rates ˙ M , ˙ ψ entering eq. (18) are due to the perturbation and are instantaneous. As such, they are obtained by computing the right-hand-sides of either the Lagrange equations or the Gauss equations onto the unperturbed Keplerian ellipse without averaging them over P b . The derivatives ∂Y/∂E,∂Y/∂ψ in eq. (18) are computed by using the unperturbed expression for Y . By using eq. (18) and the MOND time shift perturbation can be computed as It is important to notice that eq. (20) is proportional to P 2 b and to e . At a first sight, it may be weird to see in eq. (20) a dependence on the speed of light c in a non-relativistic theory such as QUMOND; actually, it is not so because of the definition of τ p in eq. (12).", "pages": [ 6, 7 ] }, { "title": "2.3 The radial velocity", "content": "The radial velocity V ρ lc [30] is a standard observable in spectroscopic studies of binaries [31]. Up to the radial velocity of the binary's center of mass V 0 , the Keplerian expression of the radial velocity of the component of the binary whose light curve (lc) is available can be obtained by taking the time derivative of the projection ρ lc of the barycentric orbit of the visible component onto the line of sight. Thus, from eq. (13), it can be posed By using the standard Keplerian expressions where r lc and a lc refer to the barycentric orbit of the visible partner, eq. (21) straightforwardly yields In eq. (24), K is the semi-amplitude of the radial velocity. In the case of extrasolar planetary systems, the light curve is usually available only for the hosting star; thus, a lc . = a M = mM -1 b a . In the case of spectroscopic binary stars, it may happen that the light curves of both the components (double lined spectroscopic binary stars) are available. As for ∆ δτ p , also the perturbation ∆ V ρ lc of the radial velocity due to a disturbing extra-acceleration can be calculated from eq. (18). In this case, it is computationally more convenient to replace E with f throughout eq. (18); as a consequence, must be used. The MOND perturbation of V ρ lc turns out to be It is important to note the proportionality of eq. (26) to P b and to e .", "pages": [ 7, 8 ] }, { "title": "3.1 Planets of the Solar System", "content": "As far as the Solar System is concerned, t M = 39 d; thus the quasi-staticity condition is fully satisfied by the gaseous giant planets for which it is P b /greaterorsimilar 4300 d. Among them, Saturn, whose orbital period is as large as P b = 10759 d, is the most suitable to effectively constrain α since its orbit is nowadays known with ≈ 20 m accuracy [26] in view of the multi-year record of accurate radiotechnical data from the Cassini spacecraft. Looking at its perihelion, any deviation of its secular precession from the rate predicted by the standard Newtonian/Einsteinian dynamics can nowadays be constrained down to submilliarcseconds per century (mas cty -1 ) level, as shown by Table 1. If the case α κ = κ 2 α κ =1 , with α κ =1 ∼ 1 is considered, the two-body expression of eq. (11) and Table 1 yield larger values for | κ | would yield an anomalous secular perihelion precession exceeding the allowed bounds in Table 1. Actually, our analysis is incomplete since it is limited to a two-body scenario. As remarked by Milgrom himself [14], also the contribution of the other planets, especially the more massive ones, should be taken into account in the mass density /rho1 of S in eq. (3). The resulting constraints on κ may, thus, be altered with respect to eq. (27). We will face this issue in a numerical way by integrating the barycentric equations of motion of the Sun, Jupiter, Saturn, Uranus and Neptune modified with the inclusion of the accelerations due to eq. (2). Moreover, eq. (3) will be calculated by taking into account the contributions of Jupiter, Uranus and Neptune as well. The result is depicted in Figure 1. It shows that the inclusion of the other major bodies of the Solar System in the MOND planetary quadrupole of eq. (3) actually enhances its effect on the perihelion of Saturn. Thus, more stringent constraints on κ can be inferred: which is two orders of magnitude better than eq. (27). Remaining in the Solar System, other authors obtained looser constraints on κ from a different class of MOND phenomena occurring in the strong-field regime, i.e. the boundaries of the MOND domains around the zero-gravity points. Bekenstein and Magueijo [23] found κ = 1 . 75 × 10 5 , while Magueijo and Mozaffari [25] inferred κ /greaterorsimilar 1 . 6 × 10 6 . In principle, it may be argued that such constraints might be optimistic. Indeed, MOND was not included in the dynamical force models which were fitted to the real observations used to produce the INPOP10a ephemerides; thus, the putative MOND signature may have been partly removed from the real residuals in the estimation of, say, the planetary initial conditions. As a consequence, it would be more correct to reprocess the same data record by explicitly modeling the MOND dynamics and determine some dedicated solve-for parameters. On the other hand, it should be considered that, even in such a case, nothing would assure that the resulting constraints on κ would necessarily be more trustable than ours. Indeed, it could always be argued that some other mismodelled/unmodeled dynamical feature, either of classical or of exotic nature, may somehow creep into the estimated MOND parameter(s). About the issue of the potential partial removal of an unmodelled signature from the real residuals 5 , it is difficult to believe that it may be a general feature valid in every circumstances for every force models. Otherwise, it would be difficult to realize how Le Verrier [33] could have positively measured the general relativistic perihelion precession of Mercury [34] by processing the observations with purely Newtonian models for both the planetary dynamics and for the propagation of light. Here we are not even engaged in measuring some effects; more modestly, we are looking just for upper bounds. As another example, let us consider the Pioneer anomaly [35, 36]. In that case, we concluded [37] that it could not be due to a gravitational anomalous acceleration directed /OverBar /OverBar/OverBar /OverBar/OverBar /OverBar towards the Sun by comparing the predicted planetary perihelion precessions caused by it with the limits of the anomalous planetary perihelion precessions obtained by some astronomers without explicitly modeling such a putative acceleration. Our conclusions were substantially confirmed later by dedicated analyses of independent teams of astronomers. Indeed, either ad-hoc modified dynamical planetary theories were fitted by them to data records of increasing length and quality with quite negative results for values of the anomalous radial acceleration as large as the Pioneer one [38-41], or they explicitly modeled and solved for a constant, radial acceleration getting admissible upper bounds [42] not weaker than those obtained by us [43]. On the other hand, Blanchet and Novak [19] inferred their constraints on the EFE-induced MONDian quadrupole effect [18] with the same approach followed by us in this paper in obtaining eq. (27): they confronted their analytically calculated perihelion precessions with the admissible ranges for the anomalous precessions obtained by some astronomers without modeling MOND. Finally, our results support the guess by Milgrom [14] that values of κ > 10 5 might be excluded. The outer planets (Uranus, Neptune, Pluto) are not yet suitable for such kind of analyses: indeed, their orbits are still poorly known because of a lack of extended records of radio-technical data. As far as their perihelia are concerned, their anomalous precessions are constrained to a 4 -5 arcseconds per century ( '' cty -1 ) level [44]. To be more quantitative, a preliminary two-body analysis is adequate for them. From eq. (11) for moderate eccentricities it turns out In addition to Saturn ( m = 5 . 7 × 10 26 kg, a = 9 . 5 au), let us consider Pluto ( m = 1 . 3 × 10 22 kg, a = 39 . 2 au); Pitjeva [44] yields ∆ /pi1 = 2 . 84 ± 4 . 51 '' cty -1 for its anomalous perihelion precession. Thus, the constraint on κ from Pluto would be 1460 times less tight than eq. (27) inferred from Saturn. Although the orbit determination of Pluto will be improved by the ongoing New Horizons mission [45] to its system, its perihelion precession should be constrained down to a totally unrealistic 0 . 001 mas cty -1 level in order to yield constraints competitive with eq. (27). An analogous calculation for Neptune ( m = 1 × 10 26 kg, a = 30 . 1 au, ∆ ˙ /pi1 = -4 . 44 ± 5 . 40 '' cty -1 [44]) yields κ Neptune ∼ 28 κ Saturn . It implies that the anomalous perihelion precession of Neptune should be improved down to a 0 . 1 mas cty -1 level. At present, no missions to the Neptunian system are scheduled. Nonetheless, the OSS (Outer Solar System) mission [46], aimed to test fundamental and planetary physics with Neptune, Triton and the Kuiper Belt, has been recently proposed; further studies are required to investigate the possibility that, as a potential by-product of OSS, the orbit determination of Neptune can reach the aforementioned demanding level of accuracy. The situation for Jupiter ( m = 1 . 898 × 10 27 kg, a = 5 . 2 au) is, in perspective, more promising. At present, its perihelion precession is modestly constrained at a -41 ± 42 mas cty -1 level [26]; thus it is currently not competitive with Saturn. A 0 . 1 mas cty -1 level would be required: such a goal may, perhaps, not be too unrealistic in view of the ongoing Juno mission [47], which should reach Jupiter in 2016 for a year-long scientific phase, and of the approved 6 JUICE mission [48], to be launched in 2022, whose expected lifetime in the Jovian system is 3.5 yr.", "pages": [ 8, 9, 10, 11, 12, 13 ] }, { "title": "3.2 Radial velocities in binaries", "content": "In general, according to eq. (26), the most potentially promising binaries are necessarily those orbiting slowly enough to fulfil the quasi-staticity condition. Moreover, they should move in highly elliptical, non-face-on orbits, and their masses should be comparable. Finally, the data records should cover at least one full orbital revolution.", "pages": [ 13 ] }, { "title": "3.2.1 Exoplanets", "content": "The wealth of exoplanets discovered so far allows, at least in principle, to select some of them for our purposes. Let us consider 55 Cnc d [49] which is a Jupiter-like planet ( m sin I = 3 . 835 m J ) orbiting a Sun-like star ( M = 0 . 94 M /circledot ; t M = 38 d) along a moderately elliptic orbit ( e = 0 . 025) with a period P b = 14 . 28 yr = 5218 d; the other relevant parameters are ω = 181 . 3 · , I = 53 · . It was discovered spectroscopically; the accuracy in measuring the amplitude K of its radial velocity is [49] By using eq. (26) for 55 Cnc d and eq. (31), it turns out which is 3 orders of magnitude weaker than the constraint of eq. (27) inferred from the perihelion precession of Saturn. It should be noticed that the use of eq. (26), which refers to the shift of the radial velocity over one full orbital revolution, is fully justified since Fischer et al. [49] analyzed 18 years of Doppler shift measurements of 55 Cnc. Other wide systems may yield better constraints, although not yet competitive with those from our Solar System. For example, HD 168443c [50] ( M = 0 . 995 M /circledot , m sin I = 17 . 193 m J , t M = 39 . 8 d, P b = 4 . 79 yr = 1749 . 83 d, e = 0 . 2113, a = 2 . 8373 au, ω = 64 . 87 · , σ K = 0 . 68 m s -1 ) yields by assuming I = 50 · . Also in this case the use of eq. (26) is justified since the spectroscopic Doppler measurements cover more than one orbital period. A similar result may occur for 47 Uma d [51] ( M = 1 . 03 M /circledot , m sin I = 1 . 6 m J , t M = 40 d, P b = 38 . 3 yr = 14002 d, e = 0 . 16, a = 11 . 6 au, ω = 110 · , σ K = 2 . 9 m s -1 ), but, in this case, the data used by Gregory et al. [51] span a period of just 21 . 6 years.", "pages": [ 13, 14 ] }, { "title": "3.2.2 Spectroscopic stellar binaries", "content": "Looking at double lined spectroscopic binary stars, an interesting candidate is the α Cen AB system [52]. It is constituted by two Sun-like main sequence stars A ( M = 1 . 105 M /circledot ) and B ( m = 0 . 934 M /circledot ) revolving along a wide ( a = 23 . 52 au) and eccentric ( e = 0 . 5179) orbit with P b = 79 . 91 yr = 29187 . 12 d /greatermuch t M = 56 . 35 d. The standard deviations of their radial velocities are [52] σ V (A) ρ = 7 . 6 m s -1 , σ V (B) ρ = 4 . 3 m s -1 . Thus, from eq. (26) we obtain the tight constraints Such bounds are one order of magnitude tighter than the two-body limit of eq. (27) inferred from the perihelion precession of Saturn, but, on the other hand, the multi-body constraint of eq. (28) from Saturn's perihelion is better than eq. (34)-eq. (35) by about one order of magnitude. Other aspects of MOND, different from the effect treated here, were investigated with Proxima Centauri 7 ( α Cen C) [54-56].", "pages": [ 14 ] }, { "title": "3.3 Pulsars", "content": "In order to fruitfully use eq. (20), the orbital period of the binary chosen should be larger than t M ≈ 46 . 7 d, obtained by using the standard value for the pulsar's mass M = 1 . 4 M /circledot ; this implies that wide orbits are required. Moreover, they should be rather eccentric as well, and the mass m of the companion should not be too small with respect to the pulsar's one. Finally, timing observations should cover at least one full orbital revolution. As a consequence, most of the currently known binaries hosting at least one radiopulsar are to be excluded because they are often tight systems with very short periods. A partial exception is represented by the Earth-like planets [57] C ( P b = 66 . 5 d, m = 0 . 0163 m J , a = 0 . 36 au, e = 0 . 0186, I = 53 · , ω = 250 . 4) and D ( P b = 98 . 2 d, m = 0 . 0164 m J , a = 0 . 46 au, e = 0 . 0252, I = 47 · , ω = 108 . 3) discovered in 1991 around the PSR 1257+12 pulsar ( M = 1 . 4 M /circledot ) [58]; the post-fit residuals for the TOAs was σ δτ p = 3 . 0 µ s [57]. Applying eq. (20) to D yields Such a constraints is far not competitive with those inferred from Saturn (Section 3.1) and α Cen AB (Section 3.2.2).", "pages": [ 15 ] }, { "title": "4 Summary and conclusions", "content": "We looked at the newly predicted quadrupolar MOND effect occurring in non-spherical, isolated and quasi-static ( P b /greatermuch t M = √ GM tot A -1 0 c -2 ) systems in deep Newtonian regime, and calculated some orbital effects for a localized binary system in the framework of the QUMOND theory. In particular, we worked out the secular precession of the pericenter, the radial velocity and timing shifts per revolution for a two-body system. Our results are exact in the sense that no simplifying assumptions about the orbital geometry were used. By using the latest orbital determinations of the planets of the Solar System, we inferred | κ | ≤ 2 . 5 × 10 5 from the supplementary precession of the perihelion of Saturn. Such a bound is based on an expression for the MOND quadrupole which takes into account only the contributions of the Sun and of Saturn itself. Actually, the contributions of the other giant planets of the Solar System do have a non-negligible impact. We evaluated it by numerically integrating the planetary equations of motion. As a result, we found a tighter constraint from Saturn: | κ | ≤ 3 . 5 × 10 3 . The double lined spectroscopic binary α Cen AB allowed to obtain | κ | ≤ 6 . 2 × 10 4 (A) , | κ | ≤ 4 . 2 × 10 4 (B) from our prediction for the shift in the radial velocity. The bounds that can be obtained by extrasolar planets, including also those orbiting pulsars, are not yet competitive. In general, the best candidates are binary systems made of comparable masses moving along accurately determined wide and highly eccentric orbits. Our constraints are to be intended as somewhat preliminary because, strictly speaking, they did not come from a targeted data processing in which the MOND dynamics was explicitly modeled in processing the real observations and a dedicated solve-for MOND parameter such as κ was determined along with the other ones. Nonetheless, they are useful as indicative of the potentiality offered by the systems considered, and may focus the attention just to them for more refined analyses.", "pages": [ 15, 16 ] }, { "title": "References", "content": "Symposium #261, American Astronomical Society , S. A. Klioner, P. K. Seidelmann, and M. H. Soffel, eds., vol. 261, pp. 155-158. May, 2009. Johnson, C. McCarthy, and H. Isaacson, 'Five Planets Orbiting 55 Cancri,' The Astrophysical Journal 675 no. 1, (Mar., 2008) 790-801, arXiv:0712.3917 . arXiv:1111.4485 [astro-ph.GA] .", "pages": [ 21, 22 ] } ]

[ { "title": "Gravitational field of a slowly rotating black hole with phantom global monopole", "content": "Songbai Chen ∗ , Jiliang Jing † Institute of Physics and Department of Physics, Hunan Normal University, Changsha, Hunan 410081, People's Republic of China Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, People's Republic of China", "pages": [ 1 ] }, { "title": "Abstract", "content": "We present a slowly rotating black hole with phantom global monopole by solving Einstein's field equation and find that presence of global monopole changes the structure of black hole. The metric coefficient g tφ contains hypergeometric function of the polar coordinate r , which is more complex than that in the usual slowly rotating black hole. The energy scale of symmetry breaking η affects the black hole horizon and a deficit solid angle. Especially, the solid angle is surplus rather than deficit for a black hole with the phantom global monopole. We also study the correction originating from the global monopole to the angular velocity of the horizon Ω H , the Kepler's third law, the innermost stable circular orbit and the radiative efficiency /epsilon1 in the thin accretion disk model. Our results also show that for the phantom black hole the radiative efficiency /epsilon1 is positive only for the case η ≤ η c . The threshold value η c increases with the rotation parameter a . PACS numbers: 04.70.Dy, 95.30.Sf, 97.60.Lf", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Phantom field is a special kind of dark energy model with the negative kinetic energy [1], which is applied extensively in cosmology to explain the accelerating expansion of the current Universe [2-7]. Comparing with other dark energy models, the phantom field is more interesting because that the presence of negative kinetic energy results in that the equation of state of the phantom field is less than -1 and then the null energy condition is violated. Although the phantom field owns such exotic properties, it is still not excluded by recent precise observational data [8], which encourages many people to focus on investigating phantom field from various aspects of physics. Phantom field also exhibits some peculiar properties in the black hole physics. E. Babichev [9] found that the mass of a black hole decreases when it absorbs the phantom dark energy. This means that the cosmic censorship conjecture is challenged severely by a fact that the charge of a Reissner-Nordstrom-like black hole absorbing the phantom energy will be larger than its mass. We studied the wave dynamics of the phantom scalar perturbation in the Schwarzschild black hole spacetime and that in the late-time evolution the phantom scalar perturbation grows with an exponential rate rather than decays as the usual scalar perturbations [10, 11]. Moreover, we also find that the phantom scalar emission will enhance the Hawking radiation of a black hole [12]. Furthermore, some black hole solutions describing gravity coupled to phantom scalar fields or phantom Maxwell fields have also been found in [13-20]. The thermodynamics and the possibility of phase transitions in these phantom black holes are studied in [16, 17]. The gravitational collapse of a charged scalar field [18] and the light paths [19] are investigated in such kind of spacetimes. Moreover, S. Bolokhov et al also study the regular electrically and magnetically charged black hole with a phantom scalar in [20]. These investigations could help us to get a deeper understand about dark energy and black hole physics. A global monopole is one of the topological defects which could be formed during phase transitions in the evolution of the early Universe. The metric describing a static black hole with a global monopole was obtained by Barriola and Vilenkin [21], which arises from the breaking of global SO(3) symmetry of a triplet scalar field in a Schwarzschild background. Due to the presence of the global monopole, the black hole owns different topological structure from that of the Schwarzschild black hole. The physical properties of the black hole with a global monopole have been studied extensively in recent years [22-25]. The main purpose of this paper is to study the gravitational field of phantom global monopole arising from a triplet scalar field with negative kinetic energy and to see how the energy scale of symmetry breaking η influences the structure of black hole, the Kepler's third law, the innermost stable circular orbit and the radiative efficiency /epsilon1 in the thin accretion disk model. Moreover, we will explore how it differs from that in the black hole with ordinary global monopole. The paper is organized as follows: in the following section we will construct a static and spherical symmetric solution of a phantom global monopole from a triplet scalar field with negative kinetic energy, and then study the effect of the parameter η on the black hole. In Sec.III, we obtain a slowly rotating black hole with phantom global monopole by solving Einstein's field equation and find that presence of global monopole make the metric coefficient of black hole more complex. In Sec.IV, we will focus on investigating the effects of the parameter η on the Kepler's third law, the innermost stable circular orbit and the radiative efficiency /epsilon1 in the thin accretion disk model. We end the paper with a summary.", "pages": [ 2, 3 ] }, { "title": "II. A STATIC AND SPHERICAL SYMMETRIC BLACK HOLE WITH PHANTOM GLOBAL MONOPOLE", "content": "Let us now first study a static and spherical symmetric black hole with phantom global monopole formed by spontaneous symmetry breaking of a triplet of phantom scalar fields with a global symmetry group O (3). The action giving rise to the phantom global monopole is where ψ i is a triplet of scalar field with i = 1 , 2 , 3, η is the energy scale of symmetry breaking and λ is a constant. The coupling constant ξ in the kinetic term takes the value ξ = 1 corresponds to the case of the ordinary global monopole originating from the scalar field with the positive kinetic energy [21]. As the coupling constant ξ = -1, the kinetic energy of the scalar field is negative and then the phantom global monopole is formed. Following in Ref.[21], we can take ansatz describing a monopole as where x i x i = r 2 . Equipping with the general static and spherical symmetric metric one can find that the field equations for ψ i can be reduced to a single equation for f ( r ) Moreover, the energy-momentum tensor for the spacetime with a global monopole can be expressed as Similarly, as in Ref.[21], one can take an approximation f ( r ) = 1 outside the core due to a fact that f ( r ) grows linearly when r < ( η √ λ ) -1 and tends exponentially to unity as soon as r > ( η √ λ ) -1 . With this approximation, we can obtain a solution of the Einstein equations where M is an integrate constant. Obviously, the radius of event horizon is r H = 2 M/ (1 -8 πξη 2 ). Here we must point that it is possible that the radius of event horizon r H is larger than the monopole's core δ ∼ ( η √ λ ) -1 if the coupling constant λ /greatermuch (1 -8 πξη 2 ) 2 4 M 2 η 2 . In this case, the metric (3) can describe the geometry near the horizon in the spacetime with global monopole. With the increase of the energy scale of symmetry breaking η , one can find that the radius of event horizon r H increases for a Schwarzschild black hole with the ordinary global monopole ( S + ), but decreases for a Schwarzschild black hole with the phantom global monopole ( S -). Thus, comparing with the system S + , one can find that the system S -possesses the higher Hawking temperature and the lower entropy. In the low energy limit, the luminosity of Hawking radiation of a spherical symmetric black hole can be approximated as L = 2 π 3 r 2 H 15 T 4 H ∝ (1 -8 πξη 2 ) 4 /r 2 H , which tells us that the energy scale of symmetry breaking η enhances Hawking radiation for the system S -, but it decreases Hawking radiation in the system S + . The presence of the phantom field enhances the Hawking emission of Kerr black hole are also found in [12]. Introducing the following transformations one can rewrite the metric (3) with the functions (8) as It is clear that there exists a deficit solid angle (1 -8 πη 2 ) for the system S + . However, for the system S -(i.e., ξ = -1), one can find that the solid angle becomes (1 + 8 πη 2 ), which is surplus rather than deficit. This implies that the topological properties of a spacetime with the phantom global monopole is different from that of with a ordinary global monopole.", "pages": [ 3, 4 ] }, { "title": "III. A SLOWLY ROTATING BLACK HOLE WITH PHANTOM GLOBAL MONOPOLE", "content": "In this section, we first obtain the metric for a slowly rotating black hole with phantom global monopole by solving Einstein's field equation. And then, we will study the properties of the black hole spacetime. From the static and spherical symmetric solution (3) with the metric function (8), we can assume the metric has a form for a slowly rotating black hole with phantom global monopole where a is a parameter associated with its angular momentum. Moreover, we assume that in the slowly rotating spacetime (11) the ansatz describing a monopole is It is easy to find that the field equations for ψ i can also be reduced to a single equation for f ( r ) which is similar to that in the static and spherical symmetric spacetime (3). It is not surprising since the triplet scalar field ψ i does not depend on the time coordinate t . Inserting the metric (11) and the triplet scalar (12) into the Einstein field equation, we find that the nonvanishing components of field equation can be expanded to first order in the angular momentum parameter a as Solving the Einstein equations (14)-(17) with the approximation f ( r ) = 1 outside the core, we can obtain the metric coefficient Separating F ( r, θ ) = h ( r )Θ( θ ), we can obtain the equation for the angular part In order to that the coefficient g tφ can be reduced to that in the slowly rotating black hole without the global monopole, here we set λ = -2 and find that Θ( θ ) = sin 2 ( θ ) in this case. And then the radial part of Eq.(18) becomes Substituting Eq. (19) into the above radial equation, we obtain (see in appendix) where b = 8 πξη 2 and 2 F 1 [ a 1 , b 1 , c 1 ; x ] is the hypergeometric function. As a usual slowly rotating black hole, the horizon of black hole (11) is given by the zeros of the function U ( r ) = ( g rr ) -1 , i.e., r H = 2 M 1 -b , which is the same as that in the static and spherical symmetric case (3). The mainly reason is that we here expand the metric only to first order in the angular momentum parameter a . From Eq. (22), it is obvious to see that due to the presence of the global monopole the form of the metric coefficient g tφ becomes more complicated in the slowly rotating black hole (11). The dependence of h ( r ) on the parameter η is shown in Fig. (1), which tells us that for the slowly rotating black hole with the ordinary global monopole ( SR + ) the function h ( r ) increases with the parameter η near the horizon, but decreases at the far field region with the larger value of r . For a slowly rotating black hole with the phantom global monopole ( SR -), the behavior of h ( r ) is just the opposite. In a word, the dependence of the function h ( r ) on the η near the horizon is different that in the far field region in these two global monopole cases. As the parameter b → 0, one can find that h ( r ) → 2 M r , which recovers that of a slowly rotating black hole without global monopole. When the rotation parameter a vanishes, one can get the previous solution of a static and spherical symmetric black hole with phantom global monopole (3). For a rotating black hole, one of the important quantities is the angular velocity of the horizon Ω H , which affects the region where the super-radiance occurs in the black hole background. In the spacetime of a slowly rotating black hole with global monopole, the angular velocity of the horizon Ω H can be expressed as which depends on the energy scale of symmetry breaking η . We plot the change of the angular velocity Ω H with the parameter η in Fig.(2). For the system SR + , we find that the angular velocity Ω H first decreases slowly and then increases rapidly with the increase of η . There exists a minimum for Ω H at where η = 0 . 1725 (i.e., b = 0 . 7477). This means that there exist a minimum region for the occurrence of the super-radiance in the black hole with the ordinary global monopole for fixed a . For the the system SR -, Ω H increases monotonically with the energy scale of symmetry breaking η . Making the same transformations (9) as in the Sec.II, one find that the area of the horizon becomes A H = br 2 H . It means that for the system SR -, the solid angle is surplus rather than deficit, which is similar to that in the static and spherical symmetric case (3). with the effective potential where the overhead dot stands for a derivative with respect to the affine parameter. The constants E and L z correspond to the conserved energy and the ( z -component of) orbital angular momentum of the particle, respectively. For simplicity, we set the orbits on the equatorial plane. With the restriction that θ = π/ 2, one can find that for the stable circular orbit in the equatorial plane, the effective potential V eff ( r ) must obey Solving above equations, one can obtain where Ω is the angular velocity of particle moving in the orbits. From Eq.(29), one can obtain Kepler's third law in the slowly-rotating black-hole spacetime with the global monopole", "pages": [ 5, 6, 7, 8 ] }, { "title": "IV. KEPLER'S THIRD LAW AND THE INNERMOST STABLE CIRCULAR ORBIT", "content": "In this section, we will focus on how the symmetry breaking scale η of global monopole affects the Kepler's third law and the innermost stable circular orbit (ISCO) in this slowly rotating black hole. In the stationary and axially symmetric spacetime, one can find that the timelike geodesics take the form where T is the orbital period and R is the radius of the circular orbit. The later terms in the right hand side is the correction by the a and the symmetry breaking scale η of global monopole. From Eq.(30), we find that for fixed R the η affects the orbital period T only in the case with nonzero rotation parameter a . In Fig. (3), we present the change of the corrected term ∆ T 2 = 4 π 2 T 2 R 3 M -1 with the parameter η in this spacetime. It is shown that the absolute value | ∆ T 2 | increases with the scale η for the system SR + , but decreases with η in the system SR -. Moreover, we also find that the presence of the global monopole make the orbital period T increase for a prograde particle (i.e., a > 0) and decrease for a retrograde one (i.e., a < 0). The innermost stable circular orbit (ISCO) of the particle around the black hole is given by the condition V eff,rr = 0. For a slow rotating black hole with the global monopole (11), we obtain Obviously, the ISCO radius r ISCO decreases with the rotation parameter a , which is similar to that in the usual Kerr black hole. In Fig.(4), we set M = 1 and plotted the variety of the ISCO radius r ISCO with the parameter η in the slowly rotating black hole spacetime with the global monopole. It is shown that the ISCO radius r ISCO increases with the scale η for the system SR + , but decreases with η in the system SR -. Moreover, we find that the ISCO radius in the system SR + is larger than that in the system SR -for fixed a . Let us now compute the effect of the symmetry breaking scale η on the radiative efficiency /epsilon1 in the thin accretion disk model, which is defined by This quantity corresponds to the maximum fraction of energy being radiated when a test particle accretes into a central black hole. For Schwarzschild and extremal Kerr black holes, /epsilon1 ∼ 0 . 06 and /epsilon1 ∼ 0 . 42, respectively. For a slowly rotating black hole with the global monopole, the radiative efficiency /epsilon1 can be expressed as It is clear that the radiative efficiency /epsilon1 increases with the rotation parameter a in these two global monopole black holes. The effect of the symmetry breaking scale η on the radiative efficiency /epsilon1 is shown in Fig. (5). It tells us that with the scale η the radiative efficiency /epsilon1 increases for the system SR + , but decreases for the system SR -. Moreover, we also note that for the phantom black hole the radiative efficiency /epsilon1 is positive only for the case η ≤ η c . This could be explained by a fact that for a phantom black hole with η ≥ η c , its capability of capturing particle could become so weak that the accreted matter around it is very dilute, which could lead to that the radiation can not be generated because of lacking of the enough stress and dynamic friction in the accretion disk model. The critical value η c can be approximated as It means that the critical value η c increases with the rotation parameter a , which is also shown in Fig.(6).", "pages": [ 8, 9, 10, 11 ] }, { "title": "V. SUMMARY", "content": "In this paper we present firstly a four-dimensional spherical symmetric black hole with phantom global monopole and find that the scale of symmetry breaking η affects the radius of the black hole horizon and a deficit solid angle. For the system SR -, the solid angle is surplus rather than deficit as in the systems SR + . Then, we obtain a slowly rotating black hole solution with global monopole by solving Einstein's field equation. We find that presence of global monopole makes the metric coefficient g tφ contain the hypergeometric function of the polar coordinate r , which is more complex than that in the usual slowly rotating black hole. We study the property of the angular velocity of the horizon Ω H , which is connected with the region where the superradiance occurs in the black hole background. With the increase of η , the angular velocity Ω H first decreases slowly and then increases rapidly for the system SR + , but increases monotonically with η for the system SR -. We also analyze the effects of the scale of symmetry breaking η on the Kepler's third law, the innermost stable circular orbit and the radiative efficiency /epsilon1 in the thin accretion disk model. Our results show that only in the case with nonzero rotation parameter a the orbital period T depends on the scale of symmetry breaking η for fixed orbital radius R . The presence of the global monopole make the orbital period T increase for a prograde particle (i.e., a > 0) and decrease for a retrograde one (i.e., a < 0). All of the absolute value of the corrected term to Kepler's Third Law (i.e., | ∆ T 2 | ), the ISCO radius r ISCO and the radiative efficiency /epsilon1 in the thin accretion disk model increase with the scale η for the system SR + , but decrease with η in the system SR -. Moreover, we also find that for the phantom black hole the radiative efficiency /epsilon1 is positive only for the case η ≤ η c . The threshold value η c increases with the rotation parameter a .", "pages": [ 11, 12 ] }, { "title": "VI. ACKNOWLEDGMENTS", "content": "This work was partially supported by the National Natural Science Foundation of China under Grant No.11275065, the NCET under Grant No.10-0165, the PCSIRT under Grant No. IRT0964, the Hunan Provincial Natural Science Foundation of China (11JJ7001) and the construct program of key disciplines in Hunan Province. J. Jing's work was partially supported by the National Natural Science Foundation of China under Grant Nos. 11175065, 10935013; 973 Program Grant No. 2010CB833004.", "pages": [ 12 ] }, { "title": "Appendix A: The form of h ( r )", "content": "Here we present the form of h ( r ) by solving the radial equation (21). Defining z = 2 M (1 -b ) r , the radial equation (21) can be expressed as Employing the transformation h ( z ) = z α (1 -z ) β H ( z ), we can write Eq. (A1) into the standard form of the hypergeometric equation with Because of the constraint from coefficient of H ( z ), the power coefficients α and β must satisfy the second-order algebraic equations Considering the asymptotical behavior h ( r ) at spatial infinite r →∞ , we choose α = 1 2 [ √ 9 -b 1 -b -1]. Then, the function h ( r ) has the form Singh P, Sami M and Dadhich N 2003 Phys. Rev. D 68 023522 (arXiv: hep-th/0305110) Saridakis E N, Gonzalez-Diaz P F and Siguenza C L 2009 Class. Quant. Grav. 26 165003 Cl'ement G, Fabris J C and Rodrigues M E 2009 Phys. Rev. D 79 Azreg-Ainou M, Cl'ement G, Fabris J C and Rodrigues M E 2011 Phys. Rev. D 83 124001 (arXiv:1102.4093).", "pages": [ 12, 13 ] } ]

[ { "title": "Para-Complex Geometry and Gravitational Instantons", "content": "J. B. Gutowski 1 and W. A. Sabra 2 1 Department of Mathematics, King's College London Strand, London WC2R 2LS, UK. E-mail: [email protected] 2 Centre for Advanced Mathematical Sciences and Physics Department, American University of Beirut, Lebanon E-mail: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "We give a complete classification of supersymmetric gravitational instantons in Euclidean N=2 supergravity coupled to vector multiplets. An interesting class of solutions is found which corresponds to the Euclidean analogue of stationary black hole solutions of N=2 supergravity theories.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Instantons are of particular importance in theoretical physics and mathematics. For example, instantons are an essential ingredient in the non-perturbative analysis of non-Abelian gauge theories and quantum mechanical systems [1]. Moreover, the existence of spin-1/2 zero-mode of the instanton is linked to the Atiyah-Singer index theorem [2], a fact reflecting the intimate relation of non-Abelian gauge theory to the field of fibre bundles and differential geometry. An important example of Yang-Mills instanton solutions are those given in [3]. In finding the instanton solutions of [3], the self-duality (or anti-self-duality) is imposed on the Yang-Mills field strength, this leads to the fact that the Bianchi identity implies the Yang-Mills field equations. This considerably simplifies finding solutions, as instead of solving second order differential equations, one solves the Bianchi identities containing only first derivatives of the vector potential. Gravitational instantons are in general defined as non-singular complete solutions to the Euclidean Einstein equations of motion. Notable early examples of gravitational instantons are the Eguchi-Hanson instantons [4]. which are the first examples of the family of the Gibbons-Hawking instanton solutions [5]. In finding gravitational instantons [4, 5] and in analogy with the Yang-Mills case, the spin connection one-form is assumed to be self-dual (or anti-self-dual), which leads to a self-dual curvature two-form. This property together with the cyclic identity ensures that Einstein's equations of motion are satisfied. The equations coming from the self-duality of the spin connection are simpler as they contain only first derivatives of the spacetime metric. In recent years, a good deal of work has been done on the classification of solutions preserving fractions of supersymmetry in supergravity theories in various dimensions. It is clear that the quest of finding solutions admitting some supersymmetry is easier as one in these cases is simply dealing with first order Killing spinors differential equations rather than Einstein's equations of motion. Following the results of [6], a systematic classification for all metrics admitting Killing spinors in D = 4 EinsteinMaxwell theory, was performed in [7]. The solutions with time-like Killing spinors turn out to be the IWP (Isreal-Wilson-Perj'es) solutions [8] whose static limit is given by the the Majumdar-Papapetrou solutions [9]. It was shown by Hartle and Hawking that all the non-static solutions suffered from naked singularities [10, 11]. Using the two-component spinor calculus [12], the instanton analogue of the IWP metric was constructed in [13]. These solutions were also recovered in the complete classification of instanton solutions admitting Killing spinors using spinorial geometry techniques [14]. Spinorial geometry, partly based on [15, 16, 17], was first used in [18] and has also been a very powerful tool in the classification of solutions in lower dimensions (see for example [19]) and in the classification of supersymmetric solutions of Euclidean N = 4 super Yang-Mills theory [20]. Sometime ago general stationary solutions of N = 2 supergravity action coupled to N = 2 matter multiplets were found in [21]. These can be thought of as generalizations of the IWP solutions of Einstein-Maxwell theory to include more gauge and scalar fields. The symplectic formulation of the underlying special geometry played an important role in the construction of these solutions. The stationary solutions found are generalization of the double-extreme and static black hole solutions found in [22]. It was also shown in [23] that the solutions of [21] are the unique half-supersymmetric solutions with time-like Killing vector. The N = 2 solutions are covariantly formulated in terms of the underlying special geometry. The solution is defined in terms of the symplectic sections satisfying the so-called stabilization equations. In the present work we extend the construction of [14] to N = 2 Euclidean supergravities with gauge and scalar fields. A class of these theories were recently derived in [24] as a reduction of the five-dimensional N = 2 supergravity theories coupled to vector multiplets [25] on a time-like circle. The paper is organized as follows. In the next section, we will collect some formulae and expressions of N = 2 supergravity which will be important for the following discussion. Section three contains a derivation of the gravitational instantons using spinorial geometry method. The solutions found are the Euclidean analogues of the stationary black hole solutions of [21]. Section four contains a summary and some future directions. We include an Appendix containing a linear system of equations obtained from the Killing spinor equations.", "pages": [ 2, 3 ] }, { "title": "2 Special Geometry", "content": "In this section we review some of the structure and equations of the original theory of special geometry when formulated in (1 , 3) signature. We then briefly discuss the modifications one introduces for the Euclidean (0 , 4) signature. For further details on the subject the reader is referred to [26]. The bosonic Lagrangian of the fourdimensional N = 2 supergravity theory coupled to vector multiplets can be written as The n complex scalar fields z A of N = 2 vector multiplets are coordinates of a special Kahler manifold. F I are n +1 two-forms representing the gauge field strength two-forms and we have used the notation F · F = F µν F µν . A special Kahler manifold is a Kahler-Hodge manifold with conditions on the curvature Here g A ¯ B = ∂ A ∂ ¯ B K is the Kahler metric, K is the Kahler potential and C ABC is a completely symmetric covariantly holomorphic tensor. A Kahler-Hodge manifold has a U (1) bundle whose first Chern class coincides with the Kahler class, thus locally the U (1) connection A can be written as A useful definition of a special Kahler manifold can be given by introducing a (2 n +2)dimensional symplectic bundle over the Kahler-Hodge manifold with the covariantly holomorphic sections These sections obey the symplectic constraint One also defines In general one can write where N IJ is a symmetric complex matrix. It can be demonstrated that the constraint (2.2) can be obtained from the integrability conditions on the following differential constraints The Kahler potential is introduced via the definition of the holomorphic sections Using (2.4) we obtain Here we list some equations coming from special geometry Recently, Euclidean versions of special geometry have been investigated in the context of Euclidean supergravity theories [24]. The Euclidean theories are found by replacing i with e in the corresponding Lorentzian versions of the theories, where e has the properties e 2 = 1 and ¯ e = -e . Thus one has, in the Euclidean theory, the para-complex fields and as a result all quantities expressed in terms of L I become para-complex. One can also introduce the so-called adapted coordinates which are defined as and also set It should be noted that the replacement of i by e , was first done in the context of finding D-instanton solutions in type IIB supergravity [27]. This replacement is effectively the replacement of the complex structure by a para-complex structure. Details on para-complex geometry, para-holomorphic bundles, para-Kahler manifolds and affine special para-Kahler manifolds can be found in [28]. The Killing spinor equations in the Euclidean N = 2 supergravity theory were recently obtained in [29] by reducing those of the five-dimensional theory given in [25]. The equations of special geometry for either signatures can be considered in a unified manner by introducing the symbol i /epsilon1 , ¯ ı /epsilon1 = -i /epsilon1 , where i 2 /epsilon1 = /epsilon1, with /epsilon1 = -1 for theories with (1 , 3) signature and /epsilon1 = +1 for theories with (0 , 4) signature. Note for the adapted coordinates one uses the definition given in (2.15). From the above equations one can derive some useful relations which will be needed in our analysis. Using (2.4) and (2.6), we write which implies the relation after using Moreover from (2.10) we have Using (2.20), we obtain from (2.3) Also using (2.4), (2.6) and (2.7), one obtains This implies that It will be convenient to rewrite a number of these conditions in terms of adapted co-ordinates, which will be used for the Euclidean calculation in the following section. In particular, the relationship between M I and L I given in (2.7) is equivalent to The condition (2.12) is equivalent to and (2.13) is equivalent to Also, (2.18) is equivalent to and (2.23) is equivalent to In addition, note that on contracting (2.11) with g A ¯ B , and using (2.12), one finds that and hence This implies that", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Gravitational Instantons", "content": "In this section we classify the gravitational instanton solutions by solving the Killing spinor equations for the Euclidean theory [29]: We remark that if ε is a Killing spinor satisfying (3.1) and (3.2), then so is C ∗ Γ 5 ε , which is moreover linearly independent of ε (over C ); here C ∗ is a charge conjugation operator whose construction is defined in terms of spinorial geometry techniques in [14]. It follows that the complex space of Killing spinors must be of even dimension, i.e. the supersymmetric solutions must preserve either 4 or 8 real supersymmetries. If a solution is maximally supersymmetric, then the gaugino Killing spinor equation (3.2) implies that the scalars z A are constant. It then follows that the gravitino Killing spinor equation reduces to the gravitino Killing spinor equation of the minimal theory. As the maximally supersymmetric solutions of the minimal theory have already been fully classified in [14], for the remainder of this paper we shall consider solutions preserving half of the supersymmetry. In order to proceed with the analysis, we define the spacetime basis e 1 , e 2 , e ¯ 1 , e ¯ 2 , with respect to which the spacetime metric is The space of Dirac spinors is taken to be the complexified space of forms on R 2 , with basis { 1 , e 1 , e 2 , e 12 = e 1 ∧ e 2 } ; a generic Dirac spinor ε is a complex linear combination of these basis elements. In this basis, the action of the Dirac matrices Γ m on the Dirac spinors is given by for m = 1 , 2. We also define which acts on spinors via With this representation of the Dirac matrices acting on spinors, the resulting linear system obtained from (3.1) and (3.2) is listed in Appendix A. There are three non-trivial orbits of Spin (4) = Sp (1) × Sp (1) acting on the space of Dirac spinors. In our notation, one can use SU (2) transformations to rotate a generic spinor /epsilon1 into the canonical form [14, 30] /negationslash /negationslash /negationslash /negationslash /negationslash where λ, σ ∈ R . The three orbits mentioned above correspond to the cases λ = 0, σ = 0; λ = 0, σ = 0 and λ = 0, σ = 0. The orbits corresponding to λ = 0, σ = 0 and λ = 0, σ = 0 are equivalent under the action of Pin (4). We shall treat these orbits separately. /negationslash", "pages": [ 7 ] }, { "title": "3.1 Solutions with λ = 0 and σ = 0", "content": "/negationslash /negationslash /negationslash /negationslash For solutions with λ = 0 and σ = 0, the analysis of the linear system (A.3) obtained from the gravitino equation produces the following geometric conditions as well as the following conditions involving the gauge field strengths The geometric constraints (3.8) imply the following implying that Thus we introduce three real local coordinates x, y and z, such that Furthermore, the vector defined by is a Killing vector, and so we introduce a local-coordinate τ such that and where φ = φ x dx + φ y dy + φ z dz is a 1-form. We remark that the conditions imposed on the geometry by the gravitino Killing spinor equations imply that So, in the co-ordinates τ, x, y, z , the metric is where λσ and φ are independent of τ , and φ satisfies where ˆ ∗ denotes the Hodge dual on R 3 equipped with metric dx 2 + dy 2 + dz 2 and volume form dx ∧ dy ∧ dz . Next consider we consider the linear system (A.4) derived from (3.2). First, one finds that This condition implies that and hence the τ -independence of λσ , together with (3.16), imply that both λ and σ are independent of τ . Next, using (2.25), together with (3.9), one obtains from which we obtain In terms of the local co-ordinates τ, x, y, z , the gauge field strengths are Thus the Bianchi identity implies that where ˆ ∇ 2 is the Laplacian on R 3 . The dual gauge field strength ˜ F I is given by where the volume form satisfies /epsilon1 1 ¯ 12 ¯ 2 = 1. Hence In terms of the local co-ordinates τ, x, y, z one finds Evaluating Re N IJ F J +Im N IJ ˜ F J and making use of (2.28) we obtain The gauge field equations are which implies that Therefore Bianchi identities and Maxwell's equations imply These are the Euclidean version of the stabilisation conditions. Also (3.31) implies From the stablisation conditions (3.31) and (2.21) we obtain Returning to (3.18), (3.33) implies that This system of equations for Euclidean instantons can be analysed in a similar way to the analysis performed for the corresponding black hole solutions in the Lorentzian theory [31, 32].", "pages": [ 8, 9, 10 ] }, { "title": "3.2 Solutions with σ = 0 , λ = 0 and λ = 0 , σ = 0", "content": "/negationslash /negationslash The analysis of the linear system in (A.3) for the case of σ = 0 , λ = 0 , implies that the spin connections satisfy /negationslash The conditions on the spin connection (3.35) imply that the (anti-self-dual) almost complex structures I 1 , I 2 , I 3 defined by and which satisfy the algebra of the imaginary unit quaternions, are covariantly constant with respect to the Levi-Civita connection, i.e. the manifold is hyper-Kahler. The remaining conditions from (A.3) are and It is also straightforward to show that (A.4) implies that In the non-minimal theory, (3.39) implies that F I are self-dual. To see this, consider first F I 1 ¯ 1 + F I 2 ¯ 2 ; the first condition in (3.39) implies that where and hence where we have made use of (2.31). So, for the case of the non-minimal theory, n ≥ 1, and this condition implies that h = 0, and hence Similar reasoning applied to the second condition in (3.39) also implies that F I 12 = 0. It follows that F I . I 1 = F I . I 2 = F I . I 3 = 0, so as the hypercomplex structures are anti-self-dual, the F I must be self-dual. Next, consider the Einstein field equations; as the manifold is hyper-Kahler there is no contribution from the curvature terms. Also, as the F I are self-dual, the contribution from the gauge field strengths also vanishes. So, on taking the trace of the Einstein equations, one obtains Assuming that the scalar manifold metric is positive definite, this implies that the scalars are constant, and (3.37) then implies that λ is constant as well. /negationslash The analysis of the case λ = 0, σ = 0 proceeds in exactly the same fashion. In particular, the conditions on the geometry, and on the gauge field strengths, are identical to those of the λ = 0, σ = 0 case, modulo the interchange L I + ↔ L I -and (space-time frame index) 1 ↔ ¯ 1 throughout. Hence, it follows that the manifold is again hyper-Kahler, though now with self-dual hyper-complex structures, and the gauge field strengths must (for the non-minimal theory) be anti-self-dual. The scalars are again constant, as is σ . /negationslash", "pages": [ 11, 12 ] }, { "title": "4 Summary", "content": "In this paper we have classified the instanton solutions admitting Killing spinors for Euclidean N = 2 supergravity theory coupled to vector multiplets. The halfsupersymmetric solutions are either hyper-Kahler with constant scalars and self-dual (or anti-self-dual) gauge field strengths, or they are Euclidean analogues of the black hole solutions found in [21]. The stationary black holes of [21] were shown to be the unique solutions with time-like Killing vector, and admitting half of supersymmetry, in the systematic analysis of [23]. They can also be obtained using spinorial geometry techniques. Employing the results of [33], the time-like Killing spinors can be written in the canonical form which is obtained by gauge fixing the generic spinor of the form where e 1 , e 2 are 1-forms on R 2 , and i = 1 , 2; e 12 = e 1 ∧ e 2 . λ , µ i and σ are complex functions. The analysis of the Killing spinor equations then gives the solutions where β is a complex t -independent function. The gauge field strengths and scalars are given by and where H I and H I are harmonic functions. The equations (4.5) are the so-called generalised stabilisation equations for the scalar fields. Recently, instanton solutions of Einstein-Maxwell theory with non-zero cosmological constant were considered in [34]. In the analysis of the particular case with anti-self dual Maxwell field, the field equations of supersymmetric solutions were shown to reduce to the Einstein-Weyl system in three dimensions [35] which is integrable by a twistor construction. Also, it was demonstrated that the Maxwell field anti-self-duality implies Weyl tensor anti-self-duality. Moreover, interesting relations were discovered between gravitational instantons and the SU ( ∞ ) Toda equation. Following on from this, the anti-self-duality condition on the Maxwell field was relaxed, and supersymmetric gravitational instanton solutions were classified using spinorial geometry techniques [36]. An important generalisation of our work is the construction of Euclidean gauged supergravity theories and the analysis of their gravitational instanton solutions and their relations to Toda theories and integrable models. Lifting the solutions of this paper to higher dimensions as well as the analysis of the instanton moduli spaces are also left for future investigation.", "pages": [ 12, 13 ] }, { "title": "Appendix A Linear Systems", "content": "The (non-vanishing) actions of the Dirac matrices on the spinors are given by and one also obtains, for a 2-form T , The linear system obtained from (3.1) is The linear system obtained from (3.2) is", "pages": [ 14, 15 ] }, { "title": "Acknowledgements", "content": "JG is supported by the STFC grant, ST/1004874/1.", "pages": [ 15 ] } ]

[ { "title": "M. A. Cuyubamba", "content": "Universidade Federal do ABC (UFABC), Rua Aboli¸c˜ao, CEP: 09210-180, Santo Andr´e, SP, Brazil E-mail: [email protected] PACS numbers: 11.10.-z,47.35.-i,04.30.Nk", "pages": [ 1 ] }, { "title": "Abstract.", "content": "We study a gas flow in the Laval nozzle, which is a convergent-divergent tube that has a sonic point in its throat. We show how to obtain the appropriate form of the tube, so that the acoustic perturbations of the gas flow in it satisfy any given wavelike equation. With the help of the proposed method we find the Laval nozzle, which is an acoustic analogue of the massive scalar field in the background of the Schwarzschild black hole. This gives us a possibility to observe in a laboratory the quasinormal ringing of the massive scalar field, which, for special set of the parameters, can have infinitely long-living oscillations in its spectrum.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Massive fields in the vicinity of black holes have been studied during the last two decades (see [1] for review). It was found that their behaviour is qualitatively different from the behaviour of the massless fields. The response of a black hole upon any perturbations at late times can be described by a characteristic spectrum of exponentially damped oscillations. The spectrum of a massive field, for some particular values of the parameters, has the oscillations with a very small decay rate in their characteristic spectra. These oscillations, that behave similarly to standing waves, were called quasiresonances [2, 3]. The asymptotical behaviour of massive fields is also different: one observes the oscillating tails that decay as inverse power of time, which is universal at the asymptotically late times [4]. Yet, since massive fields are short-ranged, we cannot expect the observation of their signal from black holes in near-future experiments. An attractive possibility for experimental study of the massive fields in the background of a black hole is a consideration of the acoustic analogue. This is a well-known Unruh analogue of a black hole [5], which is an inhomogeneous fluid system, where the perturbations (sound waves) can be described by a Klein-Gordon equation in the background of some effective curved metric [6]. The sound waves in a fluid can propagate from a subsonic region to a supersonic one, but they cannot go back. Therefore, sonic points in a fluid with a space-dependent velocity form a one-way surface for the sound waves, which is called 'acoustic horizon' by similarity with the event horizon of the black hole. The works of Unruh stimulated the study of various acoustic systems, such as: Within the analogue-gravity approach one considers hydrodynamical equations as field equations in some effective background which is not a solution to Einstein equations. Using this approach the one-dimensional perturbations in the Laval nozzle were studied in [25]. It was found that the perturbations of the gas flow in the Laval nozzle can be described by a wave-like equation with the effective potential, which depends on the form of the tube. The inverse problem for the correspondence of the form of the Laval nozzle to the Schwarzschild black holes has been solved in [26], where the form of the Laval was found in order to obtain acoustic analogue for the perturbations of massless fields. Here we describe a method, which allows us to find an appropriate form of the Laval nozzle for any given effective potential. We use this method to obtain the acoustic analogue of the massive scalar field in the background of the Schwarzschild black hole. This paper is organized in the following form: In the section II we give the basic equations for a one-dimensional flow in the Laval nozzle and its perturbations. In the section III we describe the numerical method, which allows us to find the appropriate nozzle form in order to mimic any given effective potential. In the section IV we apply the method to find the form of the Laval nozzle, which is an acoustic analogue of the massive scalar field in the Schwarzschild background and show the corresponding timedomain profiles. Finally, in the conclusion, we discuss the obtained results and open questions.", "pages": [ 1, 2 ] }, { "title": "2. Basic equations", "content": "A perfect fluid in the Laval nozzle can be described by the continuity equation and the Euler equation, that read, respectively, where ρ is the density of a gas, /vectorυ is the fluid velocity, p is the pressure, and A is the cross-section area of the nozzle. Following [25] we assume that the fluid is isentropic and the pressure depends only on the density (3) where γ is the heat capacity ( γ = 1 . 4 for the air). Assuming that the flux is irrotational ∇× /vectorυ = /vector 0, the velocity can be expressed as /vectorυ = ∇ Φ, where Φ = ∫ υdx is the velocity potential, which satisfies the Bernoulli equation We study linear perturbations of the flux, i.e. we consider the fluid density ρ and the velocity potential Φ as where ¯ ρ , ¯ Φ are the background dynamical quantities which satisfy (1) and (4), δρ and φ describe the perturbations, which are considered small so that we neglect the higherorder corrections. We introduce the function H ω ( x ), with where c s is the sound speed, We find that H ω satisfies the Schrodinger-type wave-like equation with respect to the new variable where κ = ω/c s 0 and c s 0 is the stagnation sound speed. The coordinate x ∗ is the tortoise coordinate for the analogue black hole: x ∗ = -∞ at the throat and x ∗ = ∞ corresponds to the spatial infinity ( x = ∞ ). Following [26], we measure A and ρ , respectively, in the units of cross-sectional area at the throat ( A th ) and the flux stagnation density ( ρ 0 ), and choose the arbitrary factor for the function g in such a way that Then the cross section area can be expressed as a function of g as We find also that Since the gas velocity is equal to the sound velocity at the acoustic horizon, we obtain", "pages": [ 2, 3, 4 ] }, { "title": "3. The nozzle form from a given effective potential", "content": "Linear perturbations of a spherically-symmetric black hole, after decoupling of the time and angular variables, can always be reduced to the following wave-like equation where the effective potential V = V ( r ∗ ) depends on the parameters of the field and the black hole and the tortoise coordinate is defined as where f ( r ) depends on the parameters of the black hole. In order to find the form of the Laval nozzle which is an acoustic analogue of the black hole perturbations we equate the tortoise coordinates and the effective potentials of the equations (10) and (17) From dx ∗ = dr ∗ and equations (15) and (12) we find the relation between the coordinate of the nozzle and the radial coordinate of the metric r : If g ( r ) is known, from the equations (14) and (20), one can find the function A ( x ), which describes the nozzle form. In order to find g ( r ) we make the substitution g ( r ) = h ( r ) 2 . Then the differential equation (19) reads Since the function f ( r ) vanishes at the event horizon r = r + , the linear equation (21) always has a regular singular point there. Using the Frobenius method we expand the general solution to the differential equation near the event horizon as where c 1 and c 2 are arbitrary constants, when λ 1 -λ 2 is an integer, and otherwise, λ 2 ≤ λ 1 are the roots of the indicial equation and depend on the given functions f ( r ) and V ( r ). In order to satisfy (16), one of the roots must be zero. f ' ( r + ) > 0 implies that the other root is negative. Hence, for λ 2 ≤ λ 1 = 0, h 2 ( r ) is always divergent at the horizon r = r + and we choose c 2 = 0. Therefore, from (16) we find that We expand (23) near the event horizon and find h ' ( r +), which completely fixes the initial value problem at r = r + . Then, we are able to solve numerically the equation (21) using the Runge-Kutta method for r > r + .", "pages": [ 4, 5 ] }, { "title": "4. Acoustic analogue for the massive scalar field", "content": "We consider the massive scalar field in the background of the Schwarzschild black hole, given by the line element where M is the mass of the black hole. Hereafter we measure all the quantities in units of the black hole horizon, i.e. r + = 2 M = 1. The scalar field Ψ satisfies the Klein-Gordon equation where ∇ µ is the covariant derivative, m is the field mass. The equation (25) in the background (24) reads After the separation of the angular and time variables we obtain the wave-like equation (17) with the effective potential In order to show the time-domain evolution of perturbations we use the discretization scheme proposed by Gundlach, Price, and Pullin [27]. We consider the time-dependent equation Rewriting (29) in terms of the light-cone coordinates du = dt -dr ∗ and dv = dt + dr ∗ , we find that where the point N , M , E and S are the points of one square in a grid with step h in the u -v plane, as follows: S = ( u, v ), W = ( u + h, v ), E = ( u, v + h ) and N = ( u + h, v + h ). With the initial data specified on two null-surfaces u = u 0 and v = v 0 we are able to find values of the function Ψ at each of the points of the grid. On the figures (1) and (2) we show the forms of the nozzle that are acoustic analogs of the massive field with particular values of the mass for which nearly infinitely longliving oscillations exist. These oscillations called quasiresonances one can observe on the corresponding time-domain profiles. We see that in the tube of a particular form the decay rate of sound waves is almost zero for some tone which is an analogue of the quasiresonance of the massive scalar field. Although we present here only the nozzles where the quasiresonances can be observed, the method described above can be used to construct an analogue for any finite mass of the in such a way that the sound waves in the nozzle will have the same behaviour as the massive scalar field in the background of the Schwarzschild black hole. From the figures (1) and (2) one can observe that the higher mass is, the quicker the nozzle cross-section grows, diverging at the end. However, as it was pointed out in [26], this does not lead to a problem with the presented model because of the freedom of the choice of the units of length. One can rescale the nozzle along the transversal axis in order to make the cross-section change as slowly as one wants. This change of the scale changes proportionally the frequencies of the sound in the nozzle.", "pages": [ 5, 6 ] }, { "title": "5. Conclusion", "content": "We have considered the Laval nozzle as an acoustic analogue of the massive scalar field in the background of the Schwarzschild black hole. We presented the general method to determine the form of the Laval nozzle, such that the sound waves in it are described by a given effective potential, what can be used to study an analogue of black hole perturbations in a laboratory. The method can be used to obtain the forms of the /LParen1 b /RParen1 nozzles, which are acoustic analogs of other spherically symmetric black holes. The acoustic analogs for perturbations of Reissner-Nordstrom(-de Sitter) black holes and their higher-dimensional generalizations, black strings, and Gauss-Bonnet black holes are of special interest. For some set of the parameters the higher-dimensional black holes and black strings suffer instability [28, 29, 30], which in the corresponding nozzle can manifest itself as increasing of the sound amplitude. It is clear that for a large amplitude of the sound waves cannot be described within the linear approximation and the considered analogue between the linear perturbations cannot be applied. Nevertheless, we believe that the consideration of different physical systems, which have linear instability in the same parametric region, could help us to understand better its nature.", "pages": [ 6, 7, 8 ] }, { "title": "Acknowledgments", "content": "I would like to thank to my supervisor A. Zhidenko for his help in preparing this paper for publication. This work was supported by Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N'ıvel Superior (CAPES), Brazil.", "pages": [ 8 ] } ]

[ { "title": "Holographic anomaly in 3d f (Ric) gravity", "content": "Farhang Loran ∗ Department of Physics, Isfahan University of Technology, Isfahan, 84156-83111, Iran", "pages": [ 1 ] }, { "title": "Abstract", "content": "By applying the holographic renormalization method to the metric formalism of f (Ric) gravity in three dimensions, we obtain the Brown-York boundary stress-tensor for backgrounds which asymptote to the locally AdS 3 solution of Einstein gravity. The logarithmic divergence of the on-shell action can be subtracted by a non-covariant cutoff independent term which exchanges the trace anomaly for a gravitational anomaly. We show that the central charge can be determined by means of BTZ holography or in terms of the Hawking effect of a Schwarzschild black hole placed on the boundary.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Black hole physics is the essential ingredient of any quantum theory of gravity. In the context of AdS 3 /CFT 2 correspondence, the CFT partition function of a BTZ black hole [1, 2] can be identified via a modular transformation in terms of the free energy of the vacuum which corresponds to the thermal AdS 3 [3], and the Cardy formula [4] reproduces the BekensteinHawking black hole entropy [5]. The Virasoro algebra of the dual CFT is initially identified as the asymptotic symmetry algebra of the AdS 3 spacetime [6]. For Einstein gravity, the corresponding central charge can be determined in terms of the holomorphic Weyl anomaly [7]. In [8, 9] the holographic stress-energy tensor is identified in terms of the Brown-York tensor [10]. Inspired by the Brown-Henneaux approach to the AdS/CFT correspondence [6], it is natural to seek the extension of the duality to higher-derivative gravity in AdS 3 . Since in three dimensions, the Riemann tensor can be given in terms of the metric G µν and the Ricci tensor R µ ν , f ( R µ ν ) gravity, in which f is a polynomial in R µ ν , is quite interesting. Massive gravity studied in [11] is an example of such models. The first step towards holography is identifying the CFT stress-tensor. Following [12], the AdS/CFT correspondence implies that the expectation value of the stress-energy tensor of the dual CFT can be identified with the Brown-York tensor [9]. In order to obtain the Brown-York tensor, one needs to identify the surface terms which are needed to make the action stationary given only δ G µν = 0 on the boundary. For the two-derivative EinsteinHilbert action, the surface term is the Gibbons-Hawking term [13]. For f ( R ) models, in which R denotes the Ricci scalar, one needs to cancel surface terms that depend on δR . In [14], the authors argue that no such boundary terms exist in general. It is known that f ( R ) gravity is equivalent to Einstein gravity coupled to a scalar field. Of course this equivalence relies on a conformal transformation which can be in general singular [15]. More precisely, f ( R ) model in metric formalism is equivalent to ω = 0 BransDicke theory [16]. From this point of view, R carries the scalar degree of freedom and ψ ≡ f ' ( R ) is christened scalaron [17]. So it is reasonable to set δR = 0 on the boundary [18]. In the GR limit f ( R ) → R the scalar field decouples from the theory [19] and consequently there is no need to make any assumption on δR | B in GR. 1 In the more general case of f ( R µ ν ) gravity, different approaches have been considered. For example, in [20] the surface terms are determined for general Euler density actions; in [21] this terms are given in a first order formulation of the theory, and in [22], the surface terms are obtained in an on-shell perturbative approach, i.e. one considers the higher derivative terms as perturbations to the Einstein-Hilbert action, and uses the field equations to compute the necessary boundary term. In order to find the Brown-York tensor, one also needs to determine the counter-terms which holographically renormalize the action, i.e. make the action finite for asymptotically locally AdS backgrounds. For Einstein gravity, these terms are computed in [7, 8, 9]. In [23, 24], this method is generalized to R 2 models and in [25], the corresponding counter-terms are obtained in the second order formulation involving an auxiliary tensor field. We intend to generalize these results to arbitrary f ( R µ ν ) models in three dimensions. Actually, the Ostrogradski's theorem implies that f ( R µ ν ) models are in general instable [26]. This instability is explicitly shown e.g. in [27], and is extensively studied in the case of massive gravity [11]. We are not going to study the stability of f ( R µ ν ) models here. Our goal is to obtain the holographically renormalized Brown-York tensor for f ( R µ ν ) gravity in backgrounds which asymptote to locally AdS 3 solution of Einstein gravity, In principle, if AdS/CFT correspondence can be generalized to higher-derivative gravity, then the instability of the f ( R µ ν ) model can be realized in the dual CFT. So, in principle, the issue of stability could deepen our understanding of holography. The central charge of the dual CFT can be identified in terms of the Weyl anomaly [7]. In [28] a universal formula for the so-called type A anomalies is obtained for f ( R ) gravity. In particular, in three dimensions, the value of the central charge computed by this method equals the value obtained in [29, 30] which generalizes the results of [7] to higherderivative models of gravity. By using these methods, one can determine the central charge without necessarily obtaining the stress-tensor. The central charge appears to be given by the Brown-Henneaux formula, in which, the Newton's constant G is screened by Ω defined by [25, 29, 30], In this paper, we apply the holographic renormalization method to the f ( R µ ν ) model in backgrounds that asymptote to locally AdS 3 spacetimes (1.1). In the second-order formulation given by the action [27], in which, ∫ V stands for ∫ d d +1 x √ G and χ µ ν is an auxiliary tensor field, one can simply follow the method of [25]. In this formulation, δχ µ ν is assumed to be vanishing on the boundary, and the method of [7, 8] can be used, where, effectively, the Gibbons-Hawking term is given in terms of the screened Newton's constant. The higher-derivative formulation of the f ( R µ ν ) model is given by the action, In this case, one needs to add a counter-term to compensate for the δR -dependent surface terms. As we discuss in section 4.2, such a boundary term is accessible in asymptotically locally AdS 3 backgrounds, where, the traditional Fefferman-Graham expansion [31] is available. We show that the resulting stress-tensor is essentially equivalent to the one obtained in the second-order formulation. We then turn to the on-shell value of the action, which, following [12] is an essential ingredient of holography, as it gives the leading term in the CFT partition function. It is known that there is a logarithmic divergence in the on-shell value of the action, which, can be subtracted by a cut-off dependent covariant counter-term [7, 8]. Here, we examine a cutoff independent term which appears to be not covariant. After adding this term, the trace anomaly disappears and a gravitational anomaly materializes instead. It is known that in two-dimensions, gravitational anomaly and trace anomaly can be switched by adding a local counter-term [32]. Here, we show that the value of the central charge can be determined in terms of the gravitational anomaly, by means of the holography of BTZ black holes or in terms of the Hawking effect of a Schwarzschild black hole placed on the boundary. The organization of the paper is as follows. In section 2, following [28] we compute the Weyl anomaly in f ( R µ ν ) model by studying bulk diffeomorphisms corresponding to the Weyl transformation of the boundary metric. In section 3, we review the holographic renormalization in Einstein gravity [7, 8], and extend it to f ( R µ ν ) gravity in section 4. In section 5, we study the gravitational anomaly that appears when the logarithmic divergence is subtracted by means of a cut-off independent counter-term. Section 6 is devoted to a short discussion about the CFT dual to f ( R µ ν ) gravity. Some technical details are relegated to appendices.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 Weyl anomaly in f ( R µ ν ) -model", "content": "Assume a general gravitational action, We are considering f ( R µ ν ) as a function of R µ ν = G µρ R ρν , with all contractions made between raised and lowered indices so that the metric does not enter explicitly [27]. Under a bulk diffeomorphism, this action is invariant up to a boundary term [33], in which, ∫ B stands for ∫ d d x √ γ , where, γ is the induced metric on the boundary and n α is the inward pointing unit normal to the boundary. For an asymptotically locally AdS solution G µν = ¯ G µν , the Weyl anomaly is given by this boundary term for a PBH (Penrose-BrownHenneaux) transformation [28, 33]. Details of this transformation is not important for us. What we are going to show is that, the Weyl anomaly of f ( R µ ν ) model for an asymptotically locally AdS solution G µν = ¯ G µν , equals the Weyl anomaly of the Einstein-Hilbert action with a cosmological constant term corresponding to the AdS background G AdS describing the asymptotic geometry of ¯ G µν , and a screened Newton's constant G/ Ω. To see this, one needs to compute the Taylor expansion of f ( R µ ν ) around ¯ R µν , the Ricci tensor corresponding to ¯ G µν , Thus, in which, Ω is given by Eq.(1.2), and, In other words, where, This result confirms that the Weyl anomaly in f ( R µ ν ) gravity on asymptotically locally AdS backgrounds is given by the Brown-Henneaux formula [6] with a screened Newton's constant [29].", "pages": [ 5 ] }, { "title": "3 Holographic renormalization in pure Einstein gravity", "content": "In this section, we review the holographic renormalization of Einstein gravity in asymptotically locally AdS 3 spacetimes [7, 8]. The AdS 3 solution of the Einstein field equation with a negative cosmological constant Λ = -/lscript -2 , is given by, in which, t = /lscript -1 t AdS . An asymptotically locally AdS solution in normal coordinates is given by, where, using the traditional Fefferman-Graham asymptotic expansion [31], In these coordinates, the boundary is located at r = 0. The extrinsic curvature of the boundary is given by, in which, ∇ µ denotes the covariant derivative with respect to the Levi-Civita connection corresponding to the metric (3.3), and, is the inward pointing surface-forming normal vector. The components of the extrinsic curvature are, Eq.(3.1) implies that [8], where, γ ij and its inverse are used to lower and raise the Latin indices, while the trace operator 'tr' is defined in terms of g (0) ij . The covariant derivative D i is defined with respect to the Levi-Civita connection corresponding to γ ij , in which, D (0) i is defined with respect to g (0) ij , and R (0) is the corresponding scalar curvature. Finally, For example, Eq.(3.9) is given by the field equation Π ri = 0, which implies that, The last equality is obtained by noting that, where, in order to obtain the first equality, we have used Lemma 10.2.1 in [34]. The second equality is obtained by noting that n. ∇ n ν = n µ K µ ν = 0. The Einstein-Hilbert action is given by, in which, κ 2 = 8 π G . The variation of the action with respect to δ G µν is given by, where, we have used Eqs.(D.2), (D.3), (B.13) and (B.14). The second term gives the Einstein field equation (3.1) and is vanishing on-shell. Henceforth, we drop this term. The first term depends on n. ∇ δγ µν and can be removed by adding the Gibbons-Hawking term, Thus, in which, S = S EH + S GH . The Brown-York tensor is defined by, where, δγ µν is the variation of the induced metric on the boundary, which obeys the constraint n µ δγ µν = 0. Furthermore, one assumes that δn µ = 0. The minus sign in Eq.(3.19) reflects the fact that, one defines the energy-momentum tensor in terms of δγ ij . Here, noting that δγ ij ∼ O ( r ) and √ γ ∼ O ( r -1 ), we have given the Brown-York tensor in terms of δγ ij . The idea is to identify T ij , after renormalization, with the expectation value of the stress-energy tensor of the dual CFT [9], where, on the boundary, the indices are raised and lowered by g (0) ij = r γ ij | B . Using Eq.(3.18), one obtains, The first term is singular on the boundary and can be removed by adding a counter-term to the Gibbons-Hawking term [9], Thus, the action is, and the regularized Brown-York tensor is, We still need to remove a logarithmic divergence in the on-shell value of the action [8]. Recall that the on-shell value of the action gives the tree-level contribution to the free-energy of the boundary CFT [12]. Since, one verifies that, The regularized Gibbons-Hawking term S reg GH , removes the /epsilon1 -1 term. Thus, using Eq.(3.10), one obtains [7, 8], in which, χ is the Euler-characteristic of the boundary, Thus, the counter-term is a topological term and do not contribute to the Brown-York stress tensor [8]. Eq.(3.9) implies that, and Eq.(3.10) gives, in which, c = 3 /lscript/ 2 G is the Brown-Henneaux central charge [6]. It is important to note that the logarithmic divergence of the on-shell action (3.27) is given by the central charge [8, 29],", "pages": [ 6, 7, 8 ] }, { "title": "4 Holographic renormalization of f ( R µ ν ) -model", "content": "In the previous section, we studied renormalization of the on-shell Einstein-Hilbert action and the corresponding Brown-York stress tensor for asymptotically locally AdS 3 spacetimes. In this section, we study this problem in the f ( R µ ν ) model of gravity. In section 4.1, we discuss the generalization of the Gibbons-Hawking term in the second-order formulation and in the higher-derivative formulation of f ( R µ ν ) gravity. In section 4.2, we obtain the surface terms for asymptotically locally AdS 3 spacetimes, and study holographic renormalization of the corresponding Brown-York tensor.", "pages": [ 9 ] }, { "title": "4.1 Surface terms", "content": "The higher-derivative formulation of f ( R µ ν ) gravity is given by the action (1.4) which is classically equivalent to a second order action given by Eq.(1.3) [35]. The field equation for χ gives, /negationslash In the both formulations, one supplements the action with a boundary term, implying that χ ν µ = R ν µ whenever det df µ ν dχ α β = 0 [27]. It should be noted that this field equation does not depend on δχ ν µ ∣ ∣ B . As far as the auxiliary field is considered as an independent field, one can assume that δχ µ ν is vanishing on the boundary [25]. in which, L GH is the Gibbons-Hawking term and L ct is a counter-term that subtracts the infinite terms in the on-shell action and the Brown-York tensor. We will discuss the counterterm later. The Gibbons-Hawking term is added in such a manner that δS does not depend on the normal derivative of γ µν . We begin by studying the higher-derivative formulation. In this case, where [27], in which, /square = ∇ µ ∇ µ . Henceforth, we drop the first term on the right hand side of Eq.(4.3). The surface terms are: where, inspired by Eq.(B.2), we have defined, and, Thus, on-shell, in which, and, The generalized Gibbons-Hawking term should be added such that it remove the n α ∂ α δγ µν -dependent terms in δ ˜ S 1 . A covariant choice is, This term has been derived in [25] for D = 3 massive gravity. Using the normal coordinates, the Brown-York tensor is defined by, T ij ct comes from the counter-terms, to be discussed later, and, in which, where D r = n µ ∂ µ . Furthermore, γ ij,r = 2 n r K ij and consequently, So far, our results are valid in both the second-order and the higher-derivative formulations. If one assumes that δf µ ν | B = 0 which is a legitimate assumption in the second-order formulation, then, In this case, and T 1 + T 2 reproduces the stress-tensor derived in [25] for the massive gravity. On the contrary, the assumption δf µ ν | B = 0 can not be taken for granted in the higherderivative formulation of the f ( R µ ν ) model, and the contribution from δR µ ν | B has to be taken into account [22, 23]. Since we are interested in asymptotically locally AdS spacetimes, we simplify the problem by assuming that, where, Ω is a constant. In this case, in the both formulations, δ ˜ S 2 does not contribute in the Brown-York tensor, i.e. T ij 2 = 0, as can be verified by evaluating Eq.(4.16). Furthermore, the Gibbons-Hawking term (4.12) simplifies to where, we have added a counter-term similar to Eq.(3.22). Noting that for such backgrounds, which follows from Eq.(A.8), one verifies that, Thus, in which, t ij is defined in Eq.(3.12). δR is given by Eq.(B.17) and depends on the normal derivative of δγ ij . In principle, one seeks a surface term which removes this term. In [14] it is argued that no such surface term exists in general. In the next section we obtain the corresponding surface term for the asymptotically locally AdS 3 spacetimes given by Eq.(1.1).", "pages": [ 9, 10, 11 ] }, { "title": "4.2 Asymptotically locally AdS Einstein solutions", "content": "Henceforth, we restrict ourselves to backgrounds which asymptote to locally AdS 3 solution, and use the traditional Fefferman-Graham asymptotic expansion of the metric given by Eqs.(3.3) and (3.4). 2 In the Fefferman-Graham coordinates, δR is given by Eq.(C.10). In this case, the unwanted term in Eq.(4.26) is encapsulated in δ P . Furthermore, i.e. K is constant on the boundary located at r = 0. Consequently, one can use the following counter-term in order to remove δ P in Eq.(4.26), where α ∈ R is arbitrary, and Υ is defined in Eqs.(4.24) and (4.25). Note that P α ∼ O ( r ) and P α, on -shell = -αr R (0) + O ( r 2 ). This changes Eq.(4.26) to, Eqs.(C.7) and (C.10) imply that P α δ Υ ∼ O ( r 2 ) and consequently, the last term in Eq.(4.29) is vanishing. 3 Therefore, no further counter-term is needed in order to make the variational principle well-defined. The second term in Eq.(4.29) is vanishing on-shell because, and, In summary, we have verified that the variational principle is well-defined for the action, and the corresponding Brown-York tensor is, We still need to determine another counter-term which subtracts the logarithmic divergence in the on-shell value of the action (4.32) [8]. Using Eq.(3.25) one obtains, where, f 0 denotes the (asymptotic) on-shell value of f ( R µ ν ). Furthermore, and, ∣ where, χ is the Euler characteristic of the boundary given by Eq.(3.28). For an AdS 3 solution, Ω in Eq.(4.22) is a constant, and the equation of motion Ξ µν = 0 implies that Consequently, the /epsilon1 -1 -terms in Eqs.(4.34) and (4.35) cancel out, and, The parameter α in Eq.(4.28) remains arbitrary. This reflects the fact that classically, one can arbitrarily add or remove the Euler characteristic to the action. Since this is a finite term, holographic renormalization is also ignorant of it. In principle, α can be determined by AdS/CFT correspondence, since the on-shell value of the action gives the leading term in the CFT partition function [12]. The formula (4.29) is obtained in the higher-derivative formulation given by the action (1.4). By simply omitting the Υ-terms, one obtains the corresponding formula in the secondorder formulation (1.3).", "pages": [ 12, 13 ] }, { "title": "Brown-York stress-tensor", "content": "Since the log-counter-term is a topological term, it will not contribute to the Brown-York stress tensor (4.33). Using Eq.(3.9) one verifies that, Furthermore, Thus, which is the central charge obtained in [25, 29]. Eq.(4.38) implies that, similar to Eq.(3.31), tr T is given by the logarithmic divergence of the action [7, 8, 29],", "pages": [ 13 ] }, { "title": "5 A non-covariant cut-off independent counter-term", "content": "By the AdS/CFT correspondence, the leading term in the CFT partition function is given by the finite term of the classical gravity action [12] in which φ (0) denotes the boundary value of the classical field φ cl , and the expectation value of the stress-energy tensor of the dual CFT is identified with the Brown-York tensor [9]. The finite term in the gravity action (4.38) is the sum of the finite terms in the bulk term (4.34) and the boundary terms (4.35) and (4.36). The contribution from the boundary terms is given by, where, χ is the Euler characteristic of the boundary. It is a topological term and consequently, the boundary data g (0) ij is obscured in this term. Aclosely related problem is the value of the divergence of the stress-tensor. The argument in [28] reviewed in section 2, as well as the method of [29] can not determine the divergence of the stress-tensor. Since f ( R µ ν ) gravity is parity-preserving, there is no room for a gravitational anomaly in the dual CFT given by, i.e. β = 0. Nevertheless, one can still add boundary local terms which induce a gravitational anomaly given by, The holographic renormalization can produce such an anomaly, depending on the counterterm one uses to subtract the logarithmic divergence in the on-shell value of the action given by Eqs.(3.31) and (4.42). The prescription in [7, 8] is subtracting the 'covariant' cut-off dependent counter-term S ct log = -S log -term given in Eq.(4.42). This results in Eq.(5.2). One can instead use another counter-term which is independent of the cut-off, This counter-term is not covariant. Its contribution to the on-shell value of the classical action is which, unlike the topological term (5.2) inherits the boundary data. Furthermore, it adds a new term to the Brown-York tensor, In this scenario, the renormalized Brown-York tensor is, Consequently, which is similar to the case studied in [32]. This observation motivates us to consider a more general situation, where, which gives, For the covariant subtraction ( a, b ) = ( c, 0), and for the cut-off independent subtraction ( a, b ) = (0 , -c/ 2).", "pages": [ 14, 15 ] }, { "title": "5.1 Hawking effect of a 2d Schwarzschild black hole", "content": "In the following, we show that the true value of the central charge c = a -2 b can be recognized via the Hawking effect of an asymptotically flat two-dimensional black hole located on the boundary [37]. Consider a Schwarzschild black hole, where u ( x ) has a simple zero at x h indicating the event-horizon and, The non-vanishing Christoffel symbols are, and R (0) = -u '' ( x ). Eq.(5.11) reads, These equations can be solved and the integration constants can be determined by requiring that: ( a ) T t t and T xt are finite at the horizon [37], and ( b ) asymptotically, in which, T H = g ' ( x + ) / 4 π is the Hawking temperature of the black hole and c ± = ( c L ± c R ) / 2. Finiteness of T xt at the horizon implies that c -= 0 and consequently no gravitational anomaly is detected by the Hawking effect, i.e. c L = c R . Finiteness of T t t at the horizon gives,", "pages": [ 15, 16 ] }, { "title": "5.2 BTZ-black hole", "content": "It is interesting to note that the true value of the central charge can also be recognized by studying BTZ black holes. The boundary of a static BTZ black hole is a flat torus, i.e. both the trace-anomaly and the gravitational anomalies (5.11) are vanishing in this case. Thus, the BTZ-black hole can be used to verify, via holography, whether c + defined by Eq.(5.17) is the correct central charge or not. The BTZ geometry, in the Fefferman-Graham coordinates is given by, in which, η ij = diag( -1 , 1), δ ij = diag(1 , 1) and the Hawking temperature β -1 gives the torus complex structure τ = iβ/ 2 π . Since R (0) = -2 /lscript -2 tr g (2) = 0, Eq.(5.10) gives, To see why this result is important recall that the CFT free-energy of a BTZ black hole can be obtained by a modular transformation τ →-τ -1 from the the free-energy of the vacuum which, corresponds to the thermal AdS [3, 29], Consequently, the corresponding CFT weights are, Thus, ∆ + ¯ ∆ is equivalent to the Brown-York mass of the black hole, Note that the time coordinate t in Eq.(5.18) equals, /lscript -1 t BTZ , and consequently, M BY = /lscriptM BTZ . The Cardy formula gives [5], where A BTZ is the area of the event horizon,", "pages": [ 16, 17 ] }, { "title": "6 Discussion", "content": "For backgrounds in which, the traditional Fefferman-Graham expansion is available, we found the Gibbons-Hawking term in the higher-derivative formulation of f ( R µ ν ) gravity, and determined the corresponding counter-terms. The resulting Brown-York tensor appeared to be equivalent to the one obtained in the second-order formulation, in which, an auxiliary field is used. We also verified that the logarithmic divergence of the on-shell action can be subtracted either by a cut-off dependent covariant counter-term quite similar to the one used in [7, 8], or by a cut-off independent non-covariant counter-term. In the former case, one obtains a trace anomaly equivalent to the one obtained in [29, 30]. In the later case, the Weyl anomaly is vanishing and one encounters a gravitational anomaly instead, which can be exchanged for the familiar Weyl anomaly by adding a local surface term. We verified that, keeping the gravitational anomaly, one can determine the value of the central charge in term of the Hawking effect of a Schwarzschild black hole placed on the boundary, or by means of BTZ holography. The CFT dual to f ( R µ ν ) gravity should address various phenomena which are absent in General Relativity. For example, the Ostrogradski's theorem implies that f ( R µ ν ) theories are in general instable [26]. From this point of view, f ( R ) models in which f is an algebraic function of undifferentiated Ricci scalar are viable models [26]. Of course, in these models positivity of the screened Newton's constant requires that Ω ∼ f ' > 0. This condition is also necessary for the unitarity of the boundary CFT as it implies that the central charge given by the holographic Weyl anomaly is positive. Unitary f ( R µ ν ) gravities in three dimensions and their CFT duals are widely studied, see e.g. [38] and references therein. In the context of f ( R ) gravity, Ricci stability also imposes Υ ∼ f '' ( R ) > 0 [39], which should be addressed in the dual CFT. Furthermore, there is vDVZ discontinuity [40] in f ( R ) gravity models [41] since f ( R ) gravity models are essentially equivalent to GR with an additional scalar. Thus, it is necessary to realize the vDVZ discontinuity in the CFT dual. We could not trace these effects in the holographic renormalization of the theory, since both the Brown-York stress-tenor and the on-shell action appeared to be insensitive to such details.", "pages": [ 17 ] }, { "title": "A f ( R µ ν ) as a Polynomial in R µ ν", "content": "In this appendix, we compute f µ ν and d f α β /dR µ ν . Assuming that f is a polynomial in R µ ν , where, one verifies that, in which, Thus, where, the term with a hat is replaced by 1, e.g. x ̂ yz = xz . In order to compute δf β α one needs to compute, which is given by Eq.(A.4) and, Consequently, /negationslash", "pages": [ 18 ] }, { "title": "B Induced geometry on the boundary", "content": "In this paper, we assume that the spacetime given by the metric G µν is surrounded by a space-like boundary B given by a continuous and surface-forming vector field n µ [14], Furthermore, we assume that this vector field is 'inward' pointing normal to the boundary. The induced metric on the boundary is given by, where, The extrinsic curvature of the boundary is defined by It is useful to recall that in the ADM decomposition, n ρ = (0 i , N ) and n ρ = ( -N -1 N i , N -1 ). Furthermore, δ G ij = δγ ij and δ G ri = N j δγ ij . One defines the Brown-York tensor with respect to δγ ij , assuming that δN = 0. It is clear that δn ρ = 0 and n ρ δ G ρi = 0. See also appendix A of [25]. Following section 10 of [34] and noting that here, γ µν = g µν -n µ n ν i.e. n 2 = 1 , one verifies that, where, R l ijk denotes the Riemann tensor defined with respect to γ ij , the metric induced on the boundary. Similar to [34], our curvature convention is [ ∇ ρ , ∇ σ ] A µ = R µ νρσ A ν and R µν = R ρ µρν . Consequently, Since, using Eq.(B.1), one verifies that, This gives, in particular [14], where, we have used, In order to obtain the Gibbons-Hawking term, one needs to compute the surface terms that appear in the variation of the action with respect to the metric. Assuming that the boundary B is fixed [14], i.e. δn ρ = 0 and δγ µν is tangential, one obtains, using Eq(D.2), Consequently, Therefore,", "pages": [ 18, 19, 20 ] }, { "title": "C Curvature in Fefferman-Graham coordinates", "content": "The asymptotically locally AdS 3 backgrounds, where, the boundary is located at r = 0, can be given in terms of the Fefferman-Graham expansion [7, 31], For d = 2 one obtains, where, the trace operator is defined with respect to g (0) ij ; e.g. tr h = g (0) ij h ij , and we have used the equality, in which, the i and j indices are raised and lowered by g (0) ij . Some other useful identities are: which, using Eq.(B.10) give, Here, R (0) denotes the scalar curvature defined with respect to g (0) ij . One verifies that, where, R (0) ij is the Ricci tensor corresponding to g (0) ij and consequently, Thus, Since tr h = 0 on-shell [8], one obtains,", "pages": [ 20, 21 ] }, { "title": "D Some useful identities", "content": "In sections 3 and 4, we have used the following identities, where, Γ αβρ = G ασ Γ σ βρ . Consequently, This identity can be used to show that,", "pages": [ 21 ] } ]

[ { "title": "Consistent probabilities in loop quantum cosmology", "content": "David A. Craig ∗ Perimeter Institute for Theoretical Physics Waterloo, Ontario, N2L 2Y5, Canada and Department of Chemistry and Physics, Le Moyne College Syracuse, New York, 13214, USA", "pages": [ 1 ] }, { "title": "Parampreet Singh †", "content": "Department of Physics, Louisiana State University Baton Rouge, Louisiana, 70803, USA A fundamental issue for any quantum cosmological theory is to specify how probabilities can be assigned to various quantum events or sequences of events such as the occurrence of singularities or bounces. In previous work, we have demonstrated how this issue can be successfully addressed within the consistent histories approach to quantum theory for Wheeler-DeWitt-quantized cosmological models. In this work, we generalize that analysis to the exactly solvable loop quantization of a spatially flat, homogeneous and isotropic cosmology sourced with a massless, minimally coupled scalar field known as sLQC. We provide an explicit, rigorous and complete decoherent histories formulation for this model and compute the probabilities for the occurrence of a quantum bounce vs. a singularity. Using the scalar field as an emergent internal time, we show for generic states that the probability for a singularity to occur in this model is zero, and that of a bounce is unity, complementing earlier studies of the expectation values of the volume and matter density in this theory. We also show from the consistent histories point of view that all states in this model, whether quantum or classical, achieve arbitrarily large volume in the limit of infinite 'past' or 'future' scalar 'time', in the sense that the wave function evaluated at any arbitrary fixed value of the volume vanishes in that limit. Finally, we briefly discuss certain misconceptions concerning the utility of the consistent histories approach in these models. PACS numbers: 98.80.Qc,04.60.Pp,03.65.Yz,04.60.Ds,04.60.Kz", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "When do statements about the behavior of a physical system constitute a prediction, in the probabilistic sense, of the corresponding quantum theory? The answer, according to the consistent histories approach to quantum theory pioneered by Griffiths [1], Omnes [2, 3], Gell-Mann and Hartle [4], Halliwell [5] and others [6], is when - and only when - the quantum interference between the histories corresponding to those statements vanishes. A framework of this kind is essential to the quantum theory of gravity applied to the universe as a whole because the universe is a closed quantum system. The usual formulation of quantum theory in which measurement by an external classical observer fixes whether a quantum amplitude determines a quantum probability is therefore not available [4]. Investigation of real measurement-type interactions [7-9] shows that a key feature of measurements is that they destroy the interference between alternative outcomes. 1 The consistent or decoherent histories approach to quantum theory formalizes this observation by supplying an objective, observer-independent measure of quantum interference between alternative histories called the decoherence functional . The decoherence functional, constructed from the system's quantum state, both measures the interference between histories in a complete set of alternative possibilities, and, when that interference vanishes between all members of such a set, determines the probabilities of each such history. This framework reproduces the ordinary quantum quantum mechanics of measured subsystems in situations to which it applies, but generalizes it to situations in which it does not, such as when applying quantum theory to a closed system such as the universe as a whole. In previous work [10-12], we have developed the consistent histories framework for a model quantum gravitational system, a Wheeler-DeWitt quantization [13] of a spatially flat Friedmann-Robertson-Walker (FRW) cosmology sourced by a free, massless, minimally-coupled scalar field. In this paper, we give the consistent histories formulation of the corresponding model [13-15] in loop quantum cosmology (LQC). (See Ref. [16] for a review of LQC.) A key prediction of LQC is the existence of a bounce of the physical volume (or the scale factor) of the universe when the energy density of the matter content (in the present case, the scalar field φ ) reaches a universal maximum ρ max = 0 . 41 ρ Planck in the isotropic models. 2 The existence of a bounce was first obtained for the model under consideration [13, 14, 18], and since then has been confirmed for a variety of matter models, using sophisticated numerical simulations. 3 These numerical simulations show that semi-classical states peaked at late times on classical expanding trajectories, bounce in the backward evolution (in 'internal time' φ ) to a classical contracting branch. Since the inner product, physical Hilbert space and a set of Dirac observables are completely known, the detailed physics can be extracted and reliable predictions can be made. Interestingly, the spatially flat isotropic model with a massless scalar field can be solved exactly in LQC [15]. This model, dubbed sLQC, serves as an important robustness check of various predictions in loop quantum cosmology. In particular, it has been shown that the bounce occurs for all the states in the physical Hilbert space, and the energy density is bounded above by the same universal maximum ρ max [15, 20]. All of these studies, though, address in practice only questions concerning individual quantum events, for example, the density or volume (of a fiducial spatial cell of the universe) at a given value of internal time. However, as discussed in detail in Refs. [10-12], conclusions drawn from such individual quantum events can be in certain situations badly misleading as a guide to probabilities for sequences of quantum occurrences histories of the universe - precisely the kind of physical questions in which we are most interested in the context of the physics of cosmological history. The question is, when does the amplitude for a sequence of quantum events correspond to the probability for that particular history? The answer is, when, and only when, the interference between the alternative histories vanishes just as in the two-slit experiment - as determined by the system's decoherence functional. In this paper we construct the decoherence functional for sLQC and employ it to study quantum histories of physical observables, concentrating on the physical volume of the fiducial cell. We examine both semiclassical and generic quantum states. We work within a complete predictive framework for the quantum mechanics of history to study the physics of the quantum bounce, showing that the corresponding quantum histories decohere, and that the probability of a cosmological bounce in these models is unity for generic quantum states (not just semiclassical ones). This stands in stark contrast to the predictions for the Wheeler-DeWitt quantization of the same model, which is shown in Refs. [12, 15] to be certain to be singular for generic quantum states. We close this introduction with a note on the role played in quantum cosmology by larger issues in the interpretation of quantum mechanics. It is perhaps an understatement to observe that the philosophical challenges presented by the effort to apply quantum theory to closed systems - particularly, the universe as a whole - do not end with questions of consistency of histories or decoherence. A fundamental challenge to the program is to offer a coherent account of the physical meaning of the probabilities at which one consistently arrives [4, 21, 22]. This profound question is not the subject of this paper. Indeed, there is little agreement on the 'true' nature of probability even in classical physics, never mind quantum mechanics more broadly [23, 24] or the quantum theory of closed systems in particular. Here we adopt the pragmatic attitude fairly typical in physics. When multiple instantiations of the 'same' physical system are available, probability is interpreted through 'for all practical purposes' operational definitions based on relative frequencies of outcomes [25, 26]. For single systems (such as the whole universe), a frequentist interpretation is not so easily accessible. 4 Even though we do not shy away from writing down probabilities in this paper, we recognize the interpretational challenges and therefore concentrate particularly on a class of quantum predictions for which the interpretation of probabilities might be hoped to be less controversial: those which are certain i.e. have probabilities equal to 0 or 1 - or very close thereto [4, 21, 27]. The plan of the paper is as follows. In Sec. II we briefly summarize the framework of loop quantum cosmology and discuss the quantization of sLQC. Starting from the classical theory formulated in Ashtekar variables, we show the way inner product, physical Hilbert space and Dirac observables are constructed, and an evolution equation in the emergent 'internal time' φ is obtained. In Sec. III, we summarize generalized decoherent (or consistent) histories quantum theory in the context in sLQC, by rewriting the standard approach in proper time in terms of the internal time φ . We describe the construction of the generalized quantum theory for sLQC, including definitions of its class operators (histories), branch wave functions, and decoherence functional. (More details of the classical theory of the model considered and the standard consistent histories approach can be found in our previous work [10-12].) In Sec. IV we apply these constructions to quantum predictions concerning histories of the cosmological volume by using some important properties of the eigenfunctions of the quantum Hamiltonian constraint derived recently [20]. We first introduce class operators for the volume observable, and discuss the way probabilities can be computed for histories involving single and multiple instants of internal time φ . We evaluate the probability for occurence of a quantum bounce for semi-classical states, as well as generic states. We show that the probability of occurence of a bounce in sLQC turns out to be unity for all states in the theory. Sec. V closes with some discussion.", "pages": [ 2, 3 ] }, { "title": "II. LOOP QUANTIZATION OF FLAT, HOMOGENEOUS AND ISOTROPIC COSMOLOGY", "content": "In this section, we briefly outline the quantization of a spatially flat, homogeneous and isotropic spacetime in loop quantum cosmology. 5 A complete loop quantization of this model sourced with a massless, minimally coupled scalar field φ was first provided in Refs. [13, 14, 18], and the model was demonstrated in Ref. [15] to be exactly solvable in the 'harmonic gauge' N = a ( t ) 3 , where a ( t ) denotes the scale factor of the universe described by the Friedmann-Lemaˆıtre-Robertson-Walker metric Here ˚ q ab is a flat fiducial metric on the spatial slices Σ. In loop quantum cosmology the quantization procedure parallels that of loop quantum gravity (LQG). The gravitational phase space variables in loop quantum cosmology are the symmetric connection c and its conjugate triad p , obtained by a symmetry reduction of the gravitational phase space variables in LQG, the Ashtekar-Barbero SU(2) connection A i a , and the densitized triad E a i . These are related by Here V o denotes the volume with respect to ˚ q ab of a fiducial cell introduced in order to define a symplectic structure on Σ, 6 and ˚ e a i and ˚ ω i a respectively denote a fiducial triad and co-triad compatible with the fiducial metric. (In these variables the physical volume of the fiducial cell is V = a 3 V o = | p | 3 / 2 .) For the massless scalar field model, the matter phase space variables are φ and its conjugate momentum p φ . In terms of these phase space variables, the classical Hamiltonian constraint C cl can be written as where b and ν are related to c and p by Here l p = √ G glyph[planckover2pi1] is the Planck length. (We have set c = 1.) Note that ν , though a measure of the physical volume of the fiducial cell, has dimensions of length. The modulus sign arises due to the two physically equivalent orientations of the triad. We will choose the orientation to be positive without any loss of generality. Hamilton's equations for Eq. (2.3) yield the classical trajectories via the Poisson brackets { b, ν } = 2 glyph[planckover2pi1] -1 and { φ, p φ } = 1. These yield p φ = V o a 3 ˙ φ as a constant of motion, and relate φ and ν by where ν o and φ o are constants of integration. In the classical theory, for ν ≥ 0 and regarding φ as an emergent internal physical 'clock', there exist two disjoint solutions, one expanding and the other contracting, with a fixed value of p φ . In the limit φ →-∞ the expanding branch encounters a big bang singularity in the past evolution, whereas in the limit φ →∞ the contracting branch encounters a big crunch singularity in the future evolution. These singularities are reached in a finite proper time, and all the classical solutions are singular. We now summarize the quantization procedure for this model in loop quantum cosmology in brief. As in LQG, the fundamental variables for quantization of the gravitational sector are the holonomies of the connection and the fluxes of the triads. Due to spatial homogeneity, the fluxes turn out to be proportional to the triads themselves [28], whereas the holonomies of the connection, along straight edges labelled by µ , are given by where the σ k are the Pauli spin matrices. The matrix elements of the holonomies generate an algebra of almost periodic functions of the connection, the representation of which, found via the Gel'fand-Naimark-Segal contruction, supplies the kinematical Hilbert space. It turns out that even at the kinematical level, the quantization of this model in LQC is strikingly different from that of the Wheeler-DeWitt theory. The gravitational sector of the kinematical Hilbert space in loop quantum cosmology is H (kin) grav = L 2 ( R Bohr , dµ Bohr ) where R Bohr is the Bohr compactification of the real line, and µ Bohr is the Haar measure on it. In contrast, the kinematical Hilbert space in the Wheeler-DeWitt theory is L 2 ( R , dc ). Unlike the Wheeler-DeWitt theory, a generic state in the kinematical Hilbert space of LQC can be expressed as a countable sum of orthonormal eigenfunctions (matrix elements of holonomies). The matrix elements of the holonomies act on states in the volume (or the triad) representation as translations. If | ν 〉 denotes the eigenstates of the volume operator, which has the action ˆ V | ν 〉 = 2 πγl 2 p | ν || ν 〉 , then elements of the holonomies act as ̂ exp( iλb ) | ν 〉 = | ν -2 λ 〉 . Here λ is a parameter determined by the underlying quantum geometry, and is given by λ 2 = 4 √ 3 πγl 2 p [29]. A consequence is that the action of the Hamiltonian constraint operator, expressed in terms of the holonomies, on the states in the volume representation does not lead to a differential equation, but rather to a difference equation in which the discreteness scale is determined by the parameter λ . For the total Hamiltonian constraint ˆ C grav +16 πG ˆ C matt ≈ 0, the resulting difference equation is given by where the gravitational part of the constraint Θ is a self-adjoint, positive definite operator. 7 The similarity of this equation to the Klein-Gordon equation is compelling. Since φ is monotonic, it may be treated as an emergent internal time. Solutions of the constraint equation can then be divided into orthogonal, physically equivalent positive and negative frequency subspaces. As in the Klein-Gordon theory, it suffices to consider only one of these subspaces to extract physics. We consider states lying in the positive frequency subspace, satisfying which are normalized with respect to the inner product Note the inner product so defined is independent of the choice of φ o . Eq. (2.9) defines the Hilbert space of sLQC. Physical states have a support on the lattices ν = (4 n ± glyph[epsilon1] ) λ , with n ∈ Z and glyph[epsilon1] ∈ [0 , 4). Thus, there is super-selection among lattices with different glyph[epsilon1] . In this manuscript, we will focus on the glyph[epsilon1] = 0 sector, which allows the states to have support on zero volume - the big bang singularity in the classical theory. An additional requirement on the physical states Ψ( ν, φ ) arises by noting that in the absence of fermions, physics should be independent of the orientation of the triad. We can thus choose physical states to be symmetric under this change, which therefore satisfy Ψ( ν, φ ) = Ψ( -ν, φ ). Because of the symmetry of the physical states and observables under changes of orientation of the triad, we will as applicable treat ν as positive in the sequel. In order to extract physics, we introduce a set of Dirac observables. These are the volume of the fiducial cell at time φ ∗ , and the conjugate momentum p φ , which have the following action (consistent with the inner product): Using these observables, it is straightforward to also introduce an energy density observable, which turns out to have expectation values bounded above by a critical density ρ max for all the states in the physical Hilbert space [15, 20]. Analysis of these observables in sLQC, in confirmation with the earlier results in LQC obtained using numerical simulations [13, 14, 18], show that the expectation value of the volume observable has a minimum which is reached when the energy density reaches its maximum value. This is the quantum bounce in sLQC. Our goal is now to understand the occurrence of a bounce in sLQC using the consistent histories approach, which is addressed in the following.", "pages": [ 3, 4, 5 ] }, { "title": "III. CONSISTENT HISTORIES FORMULATION OF SLQC", "content": "In this section we apply the ideas of the consistent histories approach to quantum mechanics (also known as 'generalized quantum theory' a la Hartle [4]) to the sLQC model discussed in Sec. II. The formalism will then be used in Sec. IV, to make quantum-mechanically consistent predictions concerning the behavior of the physical universe by employing the decoherence functional to measure the quantum interference between possible alternative histories. Our definitions will naturally directly mirror those for the Wheeler-DeWitt quantization of the same model [1012], facilitating easy comparison of the sometimes divergent predictions of the two models. Moreover, as noted, the construction of class operators, branch wave functions, and the decoherence functional for sLQC precisely mirrors that of the Wheeler-DeWitt theory. In this section, therefore, we restrict ourselves to a concise summary of the definitions and main formulæ, referring the reader to Ref. [12] for a more in depth discussion and commentary. The three essential ingredients of a generalized quantum theory are: (i) The fine-grained histories , the most refined descriptions of a system it is possible to give. (These might be individual paths in a path integral formulation of the theory, for example.) (ii) The coarse-grained histories , a specification of the allowed partitions of the fine-grained histories into physically meaningful subsets. (Only diffeomorphism invariant partitions might be allowed in a covariant quantization of gravity, for example.) Since most all physical predictions concern highly coarse-grained descriptions of the universe, it is the coarse-grained histories which correspond to physically meaningful questions, and for which quantum theory must be able to determine probabilities - and indeed, if those probabilities are meaningful at all. (iii) The decoherence functional provides an objective, observer-independent measure of the quantum interference between alternative coarse-grained histories of a system. When that interference vanishes among all members of a coarse-grained family, that set is said to 'decohere', or to 'be consistent'. In that case, and in that case only, does the decoherence functional assign logically consistent probabilities - in the sense that probability sum rules are satisfied - to the members of each consistent set of histories. Any specific implementation of a generalized quantum theory must realize these elements in a coherent and mathematically consistent way. In formulations of quantum theory in Hilbert space, fine-grained histories can be specified by (for example) time-ordered products of Heisenberg projections onto eigenstates of physical observables, representing the history in which the system assumes those particular values of those particular observables at those particular times. Coarse-grained histories are represented by sums of such fine-grained histories. 'Branch wave functions' corresponding to the state of a system that has followed a particular coarse-grained history are defined by the action of these history (or 'class') operators on the quantum state. The decoherence functional, which measures the interference between alternative histories, and also the probabilities of histories in consistent or decoherent families as determined by the absence of such interference, can be defined by the physical inner product between branch wave functions. In the consistent histories approach to ordinary non-relativistic quantum theory, histories are defined using coordinate time t . As discussed in the previous section, in sLQC, the role of time is naturally played by the massless scalar field φ . Indeed, using Eq. (2.8) we see that the states | Ψ 〉 evolve unitarily in φ , U ( φ ) is thus the propagator for evolution in 'time' φ . Using φ as the internal time, we can define Heisenberg projections in analogy with non-relativistic quantum mechanics and obtain class operators, branch wave functions, and a decoherence functional. 8 In this set up, the class operators provide predictions concerning histories of values of the Dirac observables. Our strategy here directly parallels the one we followed for the quantization of the WheelerDeWitt model with a massless scalar field [12].", "pages": [ 5, 6 ] }, { "title": "A. Class operators", "content": "Class operators correspond to the physical questions that may be asked of a given system. All such questions come in exclusive, exhaustive sets - at the most coarse-grained level, simply 'Does the universe have property P , or not?' The sum of all the class operators in such an exclusive, exhaustive set must therefore be, in effect, the identity, up to an overall unitary factor. Homogeneous class operators describe possible sequences of (ranges of) values of observable quantities, with sums of them corresponding to coarse-grainings thereof. We will often refer to class operators simply as 'histories'. In quantum cosmology relevant physical questions include 'What is the physical volume of the fiducial cell when the scalar field has value φ ∗ ?' 'Does the volume of the cell ever drop below a particular value, let us say ν ∗ ?' 'Is the momentum of the scalar field conserved during evolution?' Does the density exceed ρ ∗ ?' - and so forth. In the present model, which possesses a physical clock - the monotonic (unitary) internal time supplied by the scalar field φ - class operators for questions of this kind may be constructed similarly to those of non-relativistic quantum theory, in which fine-grained class operators correspond to predictions concerning the values of physical observables at given moments of time. From a physical point of view, in quantum cosmology, class operators constructed in a similar manner correspond to physical questions concerning the correlation between values of various observable quantities and the value of the scalar field. It is no surprise, then, that class operators of this type naturally correspond to predictions concerning the values of relational observables, as noted in Sec. III D. 9 Stated in this way, it is clear that the interpretation of the scalar field φ as a background physical clock is an inessential, if useful, feature of this particular model. In sLQC, we have states | Ψ 〉 with a unitary evolution in φ given by Eq. (3.1). As noted, among the physical questions of interest are the values of volume and scalar momentum at given values of φ . To extract physical predictions concerning quantities of this kind, we proceed as in ordinary quantum theory. We consider a family of observables A α , labelled by index α , with eigenvalues a α k in the physical Hilbert space H phys of sLQC. We denote the ranges of eigenvalues as ∆ a α k . Projections onto the corresponding eigensubspaces will be denoted P α a k and P α ∆ a k , respectively. For a given choice of observable A α i at each time φ i , an exclusive, exhaustive set of fine-grained histories in sLQC may be regarded as the set of sequences of eigenvalues { h } = { ( a α 1 k 1 , a α 2 k 2 , . . . , a α n k n ) } , corresponding to the family of histories in which observable A α i has value a α i k i at time t i . (Each k i for fixed i therefore runs over the full range of the eigenvalues a α i k i .) A different choice of observables ( α 1 , α 2 , . . . , α n ) leads to different exclusive, exhaustive families of histories { h } . Using the propagator we define 'Heisenberg projections' where φ o is a value of the scalar field at which the quantum state is defined. 10 The fine-grained history h may then be conveniently represented by the class operator 11 Since ∑ k P α a k = 1 for each observable α , the class operator C h satisfies corresponding to the fact that the set of fine-grained histories { h } represents a mutually exclusive, collectively exhaustive description of the possible fine-grained histories in sLQC. The coarse-grained history in which the variable α 1 takes values in ∆ a α 1 k 1 at φ = φ 1 , variable α 2 takes values in ∆ a α 2 k 2 at φ = φ 2 , and so on, then has the class operator where we suppress the superscripts on the eigenvalue ranges to minimize notational clutter. It is straightforward to see that the class operators for the coarse grained histories satisfy ∑ h C h = 1 , the identity on the physical Hilbert space H phys .", "pages": [ 6, 7 ] }, { "title": "B. Branch wave functions", "content": "Class operators capture the physical questions that may be asked of a system, as specified by an exclusive, exhaustive set of histories { h } . The amplitude for a quantum state | Ψ 〉 specified at φ = φ o to 'follow' one of the histories h -i.e. for the universe to have the properties described by h - is given by the branch wave function | Ψ h 〉 . 12 The branch wave function for a history h in the physical Hilbert space of sLQC is defined in a manner parallel to non-relativistic quantum mechanics. Defining the branch wave function is given by This branch wave function is, by construction, a solution to the quantum constraint everywhere. The propagator U simply evolves the branch wave function to any convenient choice of φ . All inner products will of course be independent of this choice. The projections implement, in the standard Copenhagen interpretation, 'wave function collapse'. From the consistent histories point of view, however, the branch wave function is viewed merely as an amplitude from which one may ultimately construct the probabilities of individual histories - the likelihoods that the universe possesses these particular sequences of physical properties. In particular, the 'collapse' is not to be regarded as a physical process in this framework.", "pages": [ 7 ] }, { "title": "C. The decoherence functional", "content": "Given a complete exclusive, exhaustive set of histories { h } and a quantum state | Ψ 〉 in the physical Hilbert space, the decoherence functional measures the interference among the branch wave functions | Ψ h 〉 , and, if that interference vanishes, determines also the probabilities of each of the | Ψ h 〉 - in other words, the probability that a universe in the state | Ψ 〉 has the physical properties described by the history h . If the interference does not vanish, then quantum theory can make no predictions concerning the particular set of physical questions { h } , in just the same way the question of which slit a particle passed through cannot be coherently analyzed when it is not recorded. The decoherence functional in non-relativistic quantum mechanics is defined by and from Eq. (3.5) is normalized, ∑ h,h ' d ( h, h ' ) = 1. In quantum cosmology the decoherence functional may be constructed from the branch wave functions in essentially the same manner [6]. 13 'Decoherent' or 'consistent' sets of histories are by definition exclusive, exhaustive sets of histories { h } which satisfy glyph[negationslash] among all members of the set. Here p ( h ) is the probability for the history h . The physical meaning of this expression is the following. When interference between all the members of an exclusive, exhaustive set of coarse-grained histories { h } vanishes, d ( h, h ' ) = 0 for h = h ' , that set of histories is said to decohere, or be consistent. In such sets, the probabilities of the individual histories are then simply the diagonal elements of the decoherence functional, p ( h ) = d ( h, h ). It is easily verified that this is simply the standard Luders-von Neumann formula for probabilities of sequences of outcomes in ordinary quantum theory, when such probabilities may be defined - typically in measurement situations or interactions with an external 'environment' that leads to decoherence in the now-conventional sense [8]. In the framework of generalized (decoherent histories) quantum theory, however, no external notion of observers or measurement or environment is required. It is the objective, observer independent criterion of Eq. (3.11) that determines when probabilities may be defined, and which ensures these probabilities are meaningful in the sense that probability sum rules are obeyed when histories are coarse-grained: p ( h 1 + h 2 ) = p ( h 1 ) + p ( h 2 ), with ∑ h p ( h ) = 1. If histories do not decohere, then the diagonal elements of the decoherence functional do not sum to unity and cannot therefore be interpreted as probabilities. For such families of histories, quantum theory is silent: it simply has no logically consistent predictions at all. Note that as constructed, the decoherence functional for sLQC involves an inner product of branch wave functions on a minusuperspace slice of fixed φ . The unitary evolution in φ , and the fact that the branch wave functions | Ψ h ( φ ) 〉 are by construction everywhere solutions to the quantum constraint, makes the specific choice of φ irrelevant in the definition of the branch wave functions and decoherence functional, and therefore may be chosen as is convenient.", "pages": [ 8 ] }, { "title": "D. Relational observables", "content": "Class operators constructed according to Eq. (3.7) express questions concerning the correlation between the values of the quantities A α i at the specified values φ i of the scalar field. Probabilities computed in this way therefore naturally correspond to predictions concerning the corresponding relational observables, such as for the volume Dirac observable ˆ ν | φ defined in terms of the propagator, Eq. (3.2), by (See, for example, Eq. (4.5).) In other words, as discussed in detail in Ref. [12], probabilities for histories of values of an observable ˆ A , which does not commute with the constraint, are naturally expressed in terms of the corresponding relational Dirac observable ˆ A | φ ∗ ( φ ), which does. In this sense the notion of relational observables arises naturally, indeed, almost inevitably, in the framework of consistent histories when predictions concerning correlations between values of observables are concerned.", "pages": [ 8 ] }, { "title": "IV. APPLICATIONS", "content": "We now apply the generalized 'consistent histories' quantum theory of loop-quantized cosmology we have constructed to a series of predictions concerning histories of physical quantities of interest. In each case the approach is the same. Within the decoherent histories framework for quantum prediction, for any physical question that is to be investigated the corresponding class operators and branch wave functions must be constructed. If the interference between these branch wave functions disappears - i.e. if the family of branch wave functions corresponding to the question decoheres - then quantum probabilities may be assigned to the alternatives according to the diagonal elements of the decoherence functional, the norms of the corresponding branch wave functions. It should be noted that in general there are a number of distinct reasons decoherence might occur. Predictions concerning physical quantities at a single value of the scalar field (moment of 'time') always decohere, because the corresponding family of class operators are simply orthogonal projections. (Compare for example Eq. (4.4). This is essentially the reason the need for decoherence is not so evident in simple applications of quantum mechanics that do not concern predictions for sequences of quantum events.) More generally, decoherence might obtain because of symmetries or selection rules; because of individual properties of the histories in question; or because of properties of the particular quantum state. In the applications we consider, we encounter examples in which decoherence occurs for each of these reasons. Predictions concerning histories of values of the scalar momentum p φ in sLQC - making precise the sense in which it is a conserved quantity in the quantum theory - follow precisely the same pattern as in the Wheeler-DeWitt theory, for which see Ref. [12]. Decoherence in this case is essentially a consequence of the fact that the scalar momentum commutes with the constraint. Our primary goal in this paper, however, will be to demonstrate how probabilities for the quantum bounce of the volume observable can be computed. We begin with the construction of class operators for the volume of the fiducial cell of the universe at an instant of 'time' φ , and also a sequence of values of φ . Predictions concerning histories of values of the volume are interesting because in this case decoherence is no longer trivial, and indeed will frequently not obtain. We will nonetheless exhibit several physical examples in which it does, and use these to study two important physical problems: quasiclassical behavior of the universe; and the quantum 'bounce'.", "pages": [ 9 ] }, { "title": "A. Class operators for volume", "content": "We begin with the class operator for the history in which the volume ν is in ∆ ν when the scalar field has value φ ∗ . It is simply given by where the projection P ν ∆ ν is Note that we employ here projections onto ranges of values of the volume operator ˆ ν , not the Dirac observable ˆ ν | φ ∗ . These ranges form a collection of disjoint sets that cover the full range of discrete volume eigenvalues, 0 ≤ | ν | = 4 λn < ∞ , such that ∑ i C ∆ ν i | φ ∗ = 1 . If | Ψ 〉 denotes a quantum state of the universe at φ = φ o , the branch wave functions for these histories are Because in this instance the class operators are simply projections, the branch wave functions for these histories are orthogonal, This implies that for this family of histories decoherence is automatic. One can thus meaningfully compute the quantum probabilities. Using Eqs. (4.1) and (4.2), the probability that the universe has volume in the range ∆ ν when φ = φ ∗ is then given by By way of example, Fig. 1 shows probabilities calculated from Eq. (4.5) that a state which is quasiclassical at large volume 14 takes on volumes in the range ∆ ν for two choices of that range. For example, Fig. 1a shows the probability that the universe takes on small volume. Specifically, the plot shows the probability as a function of the scalar field φ that the volume of the fiducial spatial cell has volume less than or equal to ν ∗ i.e. that | ν | ∈ ∆ ν ∗ , where ∆ ν ∗ = [0 , ν ∗ ]: The quantum bounce is clearly visible in the plot, the probability the universe has small volume becoming zero as | φ | becomes large. We now consider the more interesting case when a sequence of 'time' instants is involved. In contrast to the class operator representing the volume of the universe at an instant φ = φ ∗ (Eq.(4.1)), the class operator for the volume to take particular values in ranges ∆ ν i at a sequence of different instances of internal time { φ 1 , ..., φ n } is not a simple projection. It is given by where the sets of ranges ( { ∆ ν 1 } , { ∆ ν 2 } , . . . , { ∆ ν n } ) partition the allowed range of volumes. As remarked earlier, in general it is neither obvious nor trivial that the corresponding branch wave functions (Eq. (3.9)) decohere. Nevertheless, in the following we will exhibit several important (and typical) examples for which they do, and extract the corresponding quantum probabilities.", "pages": [ 9, 10, 11 ] }, { "title": "B. Decoherence for semiclassical states", "content": "We first apply this framework to states which are semi-classical at late times in this loop quantized model. Analysis of such states in loop quantum cosmology using sophisticated numerical simulations was first performed in Refs. [13, 14, 18] for the spatially flat homogeneous and isotropic model sourced with a massless scalar field. 15 The states are chosen such that they are initially peaked on classical trajectories in a macroscopic universe, and evolved using the quantum gravitational Hamiltonian constraint (see Eq. (2.7)). Numerical simulations show that such states remain peaked on classical trajectories until the spacetime curvature reaches almost a percent of its value at the Planck scale. As the Planck scale is approached, significant departures arise between the classical trajectory, Eq. (2.5), and the trajectory obtained from the expectation value of the volume observable. Instead of reaching the classical big bang singularity, such states bounce when the energy density of the universe reaches a maximum value ρ max ≈ 0 . 41 ρ Planck . After the bounce, states are found to be peaked on an expanding classical solution (disjoint in the classical theory from the one where the initial state was peaked) [13, 14]. This result, initially obtained using numerical simulations for a class of semiclassical states, can be generalized to all the states in the physical Hilbert space. It turns out that in sLQC the expectation value of the volume observable has a minimum irrespective of the choice of state [15]. Further, all states in the physical Hilbert space reach arbitrarily large volume in the infinite past and future ( φ → ±∞ ). The minimum of the expectation value of volume translates to an upper bound on the expectation values of the energy density observable [15]. Recently, this result has also been understood from an analytical study of the properties of the eigenfunctions of the gravitational constraint, Eq. (2.7) [20]. An important result of various numerical investigations in loop quantum cosmology is that states which are semi-classical at late times follow an effective trajectory throughout their evolution [16]. The effective trajectory is derived using an effective Hamiltonian constraint obtained via geometrical methods of quantum mechanics [36, 37]. As with the expectation value of the volume observable, significant departures exist between the effective and the classical trajectories at the Planck scale, whereas at spacetime curvatures much smaller than the Planck value, the effective and classical trajectories coincide. How is the question of whether or not a given state follows a classical or an effective trajectory posed within the framework of generalized quantum theory? A state 'follows a trajectory' in minisuperspace when it exhibits a correlation between φ and ν given by that trajectory with a high probability. The fidelity of this correlation may be specified with varying degrees of precision. To accomplish this, we consider a coarse-graining of minisuperspace on a set of slices { φ 1 , φ 2 , · · · , φ n } by positive ranges of volume { ∆ ν i k , k = 1 . . . n } on each slice φ k , so that ∪ i k ∆ ν i k = [0 , ∞ ) on each slice k . To track a particular minisuperspace trajectory γ , choose the partitions { ∆ ν i k } such that one range ∆ ν γ k from each partition encloses γ at each φ k . To the degree of precision specified by this coarse-graining, a state | Ψ 〉 may be said to 'follow' γ with near certainty if the only branch wave function that is not essentially zero is If indeed the branch wave function for the complementary history ¯ γ ('does not follow γ ') vanishes, then the partition ( γ, ¯ γ ) - i.e. ('follows γ ','does not follow γ ') - decoheres, and | Ψ 〉 may be said to follow the trajectory γ with probability 1. Put another way, the state | Ψ 〉 may be said to exhibit the pattern of correlation between volume and scalar field specified by the trajectory γ with a high probability. Even for a state centered on γ , whether or not | Ψ ¯ γ 〉 ≈ 0 will depend on the width of the intervals ∆ ν γ k relative to the width of the state Ψ( ν, φ k ) at each φ k . Trying to specify the path too narrowly will lead to a partition which fails to decohere and must be further coarse-grained (by combining some of the intervals surrounding the ∆ ν γ k ) to regain decoherence, and therefore the means to define probabilities consistently. Thus, as is usual in quantum theory, attempting to specify a path too precisely leads to a loss of predictability. For further discussion, see Ref. [12]. In loop quantum cosmology, as noted above, numerical simulations show that states | Ψ sc 〉 which are semi-classical 16 at early times on a contracting branch are peaked on classical solutions at large volume and connect to the expanding branch smoothly through a 'bounce' in the Planck regime. Such states are peaked on a trajectory which is a solution to the modified Friedmann and Raychaudhuri equations of LQC noted above for the entire evolution. If γ sc is chosen to be such an effective trajectory in Fig. 2, and the widths ∆ ν γ k chosen to be wider than the width of Ψ sc ( ν, φ k ) at each φ k , 17 then essentially the only non-zero branch wave function will be the state | Ψ γ sc 〉 of Eq. (4.8), and | Ψ sc 〉 follows the trajectory γ sc with probability 1. The origin of this behavior can also be analytically understood via the dynamical eigenstates e ( s ) k ( ν ) [20]. All states in sLQC - whether semiclassical or not - approach a particular symmetric superposition of expanding and contracting Wheeler-DeWitt universes at large volume. If the state is chosen in such a way that it is peaked on a collapsing classical trajectory at large volume as φ → -∞ (say), then this state will be peaked on a corresponding expanding classical trajectory as φ → + ∞ . (See Ref. [38] for further details.) The asymptotic behavior of the eigenfunctions dictates the symmetric nature of the bounce.", "pages": [ 11, 12, 13 ] }, { "title": "C. Singularity avoidance in loop quantum cosmology", "content": "We have already discussed the manner in which semi-classical states which are peaked on classical trajectories in a large macroscopic universe at early times bounce at a finite volume in LQC, connecting collapsing and expanding classical solutions. In this way, such states avoid the classical singularity at zero volume. As first shown analytically in Ref. [15], this behavior is generic: the expectation value of the volume is bounded below for all states (in the domain of the physical operators) in sLQC. In this subsection we discuss the quantum bounce for generic states from the perspective of consistent histories. In Ref. [12], the problem of the singularity in a Wheeler-DeWitt quantized flat scalar Friedmann-Lemaˆıtre-RobertsonWalker cosmology was addressed through a study of the volume observable. There it was shown that for any choice of fixed volume V ∗ of the fiducial cell, the volume of the quantum universe would invariably fall below it with unit probability. The Wheeler-DeWitt universes are therefore inevitably singular in the sense that they assume arbitrarily small volume at some point in their history. 18 In this analysis, the role of a proper understanding of quantum history proved crucial. As noted, the loop quantization of this model yields states which are a symmetric superpositions of expanding and contracting cosmologies at large volume. Ref. [12] therefore analyzed a superposition of expanding and contracting Wheeler-DeWitt universes with an eye toward the question of whether this superposition itself could in some sense be the reason for the bounce. Calculation of the probability that the universe is found at small volume for such a superposition reveals that at any given value φ of the scalar field, the probability that the universe has volume less than V ∗ | φ is in general between 0 and 1. The probability that the universe is not at arbitrarily small volume at φ = -∞ or φ = + ∞ is therefore in general not 0. Naively this suggests the possibility that a superposition of expanding and contracting Wheeler-DeWitt universes has a non-zero probability of being at non-zero volume at both φ = -∞ and φ = + ∞ i.e. that there is a non-zero probability of a quantum bounce. A more careful consistent histories analysis showed that this naive possibility is not realized. The physical statement that the universe 'bounces' is the statement that the volume of the universe is large at both φ = -∞ and φ = + ∞ . A proper characterization of the bounce is therefore a statement about the volume of the universe at a sequence of values of φ - a history. Ref. [12] shows that for generic initial states the histories corresponding to the alternatives { bounce , singular } decohere in the limit | φ | → ∞ so that probabilities may be consistently assigned to them, and that the probability for the bouncing history p bounce = 0: even superpositions of expanding and contracting WheelerDeWitt universe cannot bounce for any choice of state. 19 We now show in detail that, in sharp contrast to the case of the Wheeler-DeWitt quantization, the probability that generic quantum states in sLQC are at small volume as φ → ±∞ is zero. In fact, for any choice of volume V ∗ , we show in a sense to be made precise below that the probability the volume of the universe is larger than V ∗ is unity as | φ | → ∞ : all states in this model achieve arbitrarily large volume in both limits. In this sense every state retains some flavor of the striking 'bounce' of the narrowly peaked quasi-classical ones. Next, we address histories of the volume with evolution in φ . We show that for arbitrary quantum states the family of coarse-grained alternative histories { bounce , singular } decoheres, as in the Wheeler-DeWitt case. However, in contrast to the Wheeler-DeWitt case, the probability that the universe is singular in the scalar past or future is zero, and the probability that the universe bounces, unity. All states in sLQC bounce from arbitrarily large volume in the 'past' ( φ →-∞ ) to arbitrarily large volume in the 'future' ( φ → + ∞ ). As in the Wheeler-DeWitt theory, it is worth emphasizing the role of the limit φ →±∞ . One may expect that for wide classes of states such as localized states with certain peakedness properties decoherence obtains to a high degree of approximation at finite φ . Nonetheless, it is only in the limit φ →±∞ that we are guaranteed decoherence, and hence a bounce with probability 1, for all states, and therefore - in that limit - that a bounce is a universal prediction of the theory. REMARK: In Refs. [43] it is argued that the consistent histories approach to quantum theory is insufficient to address questions such as whether a quantum bounce takes place because histories involving 'genuine' quantum states are inconsistent when more than two moments of (scalar) time are involved, or in other words, that in this case only histories for semiclassical states decohere. We do not agree. and where since φ ∗ is arbitrary we have set φ ∗ = φ in the expression for the probability. In order to compute this probability, we will use some key properties of the symmetric eigenfunctions of the gravitational constraint operator ˆ Θ of Eq. (2.7), labelled by k ∈ ( -∞ , ∞ ): where ω k is related to p φ and k by p φ = ± glyph[planckover2pi1] ω k and ω k = √ 12 πG | k | , respectively. The symmetric eigenfunctions are real and satisfy e ( s ) -k ( ν ) = e ( s ) k ( ν ) and e ( s ) k ( -ν ) = e ( s ) k ( ν ) [14, 20]. A notable property of these eigenfunctions is that they decay exponentially to zero for volumes smaller than a cutoff value proportional to the value of ω k . This result was first obtained in numerical simulations [14], and subsequently derived analytically in Ref. [20], in which it was shown that the cutoff occurs along the lines | k | = | ν | / 2 λ . Thus, one can consider | k | = | ν | / 2 λ as an ultra-violet | ν |∈ ∆ ν ∗ The basis for this argument is a nice calculation (in the Wheeler-DeWitt quantization) along the lines of the one we perform in Ref. [12] and below of the interference between histories characterized by the alternatives { bounce , singular } in both the infinite scalar past and future, but with a third projection onto these alternatives at an arbitrary intermediate φ . The authors calculate a representative off-diagonal (interference) matrix element of the decoherence functional and argue that it is zero if and only if the corresponding state is semiclassical in the sense that it is sharply peaked on a classical trajectory. Unfortunately, it is easy to generate a wide range of counter-examples to the claim that the calculated matrix element is zero only for semiclassical states. Therefore, it is simply not the case that it is only for semiclassical states that families of histories that study the bounce at more than two values of scalar time decohere. We do, however, expect the calculation the authors of Refs. [43] give of the decoherence functional itself for such three 'time' histories to be useful. Moreover, it is probably worth a certain emphasis that there are many instances in which one would not expect decoherence of a family of histories, and indeed, would be suspicious of a quantum theory that purports to do so. Far from being a defect of the theory, it is a necessary requirement of a theory that reproduces the predictions (or absence thereof) of quantum theory without the introduction of e.g. non-local hidden variables. (For the purposes of this remark we include the de Broglie-Bohm formulation in this class of theories.) The two-slit experiment is the classic example: any theory which assigns observationally verifiable probabilities to the individual paths the electron follows when the physical setup is not such that which-path information is gathered is not quantum mechanics, and indeed will have a difficult time reproducing the predictions of quantum mechanics absent such non-local modifications. However, when there are additional degrees of freedom (such as a gas of air molecules in the two-slit apparatus) which might carry a record of which-path information, decoherence is to be expected and probabilities for individual paths may be assigned. In a similar way, in cosmological models with realistic inhomogeneous matter degrees of freedom (for example), one would expect decoherence of histories for bulk variables like the volume for most quantum states.", "pages": [ 13, 14 ] }, { "title": "1. Probability for zero volume in sLQC", "content": "Following Ref. [12], one way to approach the question of whether a quantum universe is in some sense singular is to ask whether it achieves zero volume at any point in its evolution. 20 In Sec. IV A we showed how to calculate the probability that the volume falls in a range specified by ∆ ν = [ ν 1 , ν 2 ]. To ask whether the volume of the universe is ever small we choose a reference volume ν ∗ and partition the volume into the range ∆ ν ∗ = [0 , ν ∗ ] and its complement, ∆ ν ∗ = ( ν ∗ , ∞ ). The universe then has small volume at scalar time φ ∗ if | ν | ∈ ∆ ν ∗ at φ ∗ and not if | ν | ∈ ∆ ν ∗ . (See Fig. 3.) The class operators for these alternatives are simply the (Heisenberg) projections given by Eq. (4.1) with ∆ ν = ∆ ν ∗ , ∆ ν ∗ and corresponding branch wave functions, Eq. (4.3). The probabilities are given by Eq. (4.5). Thus = U ( φ - φ ∗ ) ∑ | ν 〉 Ψ( ν, φ ∗ ) (4.10b) momentum space cutoff in sLQC. The exponential decay of the eigenfunctions coincides with the volume at which the energy density attains a maximum value and the universe bounces; the linear scaling of the cutoff with volume is what leads to a universal maximum matter density that is independent of the quantum state. The symmetric eigenfunctions e ( s ) k ( ν ) satisfy and Physical states Ψ in sLQC can be constructed using the eigenfunctions e ( s ) k ( ν ), where we have set φ o = 0 for convenience. As a consequence of the ultra-violet cutoff on the eigenfunctions, Note in Eq. (4.10) that ν is bounded by ν ∗ . Further, the e ( s ) k ( ν ) are well-behaved functions of k for all values of ν . For any fixed value of ν , their rate of oscillation in k is fixed by ν . 21 φ →-∞ Thus, for large | φ | - meaning at a minimum ω k | φ | glyph[greatermuch] 1 - rapid oscillation of the factor exp( iω k φ ) will according to the Riemann-Lebesgue lemma eventually suppress the integral, and we find for any fixed value of ν ≤ ν ∗ . In other words, for any fixed ν , | φ | eventually becomes large enough to suppress the state, driving the state to larger volume as | φ | increases. Thus Ψ( ν, φ ) for ν < ν ∗ will always be suppressed for large enough | φ | for arbitrary states in the theory. On the other hand, for the complementary branch wave function | Ψ ∆ ν ∗ | φ ∗ ( φ ) 〉 corresponding to large volume universes, ν can be arbitrarily large. As ν becomes larger a wider range of k 's can contribute nontrivially to the integral in Eq. (4.16). For ν glyph[greatermuch] 2 λ | k | the rate of oscillation of the e ( s ) k ( ν ) with k is fixed by the asymptotic limit of the symmetric eigenfunctions, increasing in proportion with ln | ν | [20]. Again, for any fixed ν the state is suppressed as | φ | → ∞ , so that the region of support of this branch wave function in the ( ν, φ ) plane must have ln | ν | increasing in proportion with | φ | - just the behavior of Wheeler-DeWitt quantized states. We find, therefore, that since Ψ( ν, φ ) at any fixed ν vanishes in the limit | φ | → ∞ , As the intervals ∆ ν ∗ and ∆ ν ∗ are complementary, this implies As a consequence, one finds for the probabilities lim p ∆ ν φ → + ∞ ∆ ν We can see already from this that loop quantum states invariably bounce: the probability the universe is found at small volume as | φ | → ∞ is zero, regardless of the state. Eqs. (4.20) say that all states in sLQC achieve arbitrarily large volume in each of the limits φ →-∞ , φ → + ∞ . States in sLQC don't merely refrain from becoming singular. They inevitably grow to large volume, no matter how non-classical the state. (This result complements that of Ref. [15] that the expectation value of the volume becomes infinite in those limits for all states.) In the next section we will use this to show that the family of histories describing a quantum bounce decoheres, and that indeed all states in the theory bounce from arbitrarily large volume to arbitrarily large volume. Finally, we observe that as the state | Ψ 〉 was arbitrary and Eqs. (4.18-4.19) may be conveniently expressed in terms of volume projections as on all states in the theory. 22 Note we have not so far addressed the question of whether the universe ever assumes volumes in ∆ ν ∗ with non-zero probability. In fact, examination of Eq. (4.11) should be sufficient to show that so long as ν ∗ > 0, there always exist states for which it will. (See, for example, Fig. 1.) However, this is not sufficient to show that the universe might become singular in sLQC. Recall that the eigenfunctions e ( s ) k ( ν ) decay exponentially for volumes smaller than ∗ ( φ ) = 1 lim p ∗ ( φ ) = 1 . (4.20b) | ν | = 2 λ | k | and vanish at ν = 0, and thus, from Eq. (4.11) the probability that any state in sLQC assumes precisely zero volume is zero , This result stands in sharp contrast with the situation in Wheeler-DeWitt theory, where the rapid oscillations in the eigenfunctions as ν → 0 inevitably 'draw in' Wheeler-DeWitt states to zero volume and infinite density, and the probability for a singularity turns out to be non-vanishing - and indeed, is unity for all states in the limits | φ | → ∞ [12].", "pages": [ 14, 15, 16, 17 ] }, { "title": "2. Quantum bounce", "content": "It is tempting to conclude that Eqs. (4.20) are sufficient to demonstrate that all states in sLQC 'bounce' from large volume as φ →-∞ to large volume as φ → + ∞ . However, as emphasized in Refs. [10-12], statements concerning a quantum bounce are inherently assertions concerning the volume at a sequence of values of φ , and, as in the two-slit experiment, it is in precisely such situations that decoherence becomes critical in order to arrive at consistent quantum predictions. Indeed, in Ref. [12] it is shown that consideration of the singleφ volume probability p ∆ ν ∗ ( φ ) alone for Wheeler-DeWitt states which are superpositions of expanding and contracting universes may lead one to the incorrect conclusion that a bounce is possible in that model. However, a proper analysis of the histories describing a quantum bounce shows that this naive conclusion based on singleφ probabilities is misleading, and that indeed the probability for a bounce is zero. How, then, is a 'bounce' characterized within quantum theory? The assertion that a universe bounces is the statement that the universe assumes large volume at both 'early' ( φ →-∞ ) and 'late' ( φ → + ∞ ) values of φ . A (highly coarse-grained) description of a bounce may therefore be obtained by making a choice of φ -slices φ 1 and φ 2 and volume partitions (∆ ν ∗ 1 , ∆ ν ∗ 1 ) and (∆ ν ∗ 2 , ∆ ν ∗ 2 ) on them. The class operator for the history in which the universe 'bounces' between φ 1 and φ 2 - i.e. is at large volume at both φ 1 and φ 2 - is then On the other hand, the class operator for the alternative history that the universe is found at small volume at either or both of φ 1 , φ 2 is It is clear from Eq. (4.25b) that C sing ( φ 1 , φ 2 ) encodes the various ways the universe can be at small volume at φ 1 and/or φ 2 . We now demonstrate that the only branch wave function which is non-vanishing in the limits φ 1 →-∞ and φ 2 →∞ is the one corresponding to the bounce. Using Eqs. (4.22), one finds, Using Eqs. (4.25) and (4.26), we find that the branch wave function for an sLQC quantum universe to encounter the singularity vanishes, On the other hand, the branch wave function for the history corresponding to a bounce in sLQC is Thus, the family of histories (bounce , singular) in sLQC decoheres, and a bounce is predicted with probability 1, Note that in this analysis, no assumption has been made on the the choice of state | Ψ 〉 , and thus this result holds for all states in the theory. Thus, we have shown in the consistent histories approach that the bounce is a universal feature of all states in sLQC. We finally note that the existence of bounce at a non-zero volume is tied to the existence of an upper bound on the expectation values of the energy density operator of the scalar field: 〈 ˆ ρ | φ 〉 = 〈 p φ 〉 2 / 2 〈 V | φ 〉 2 . For more discussion, see Refs. [15, 20]. Unlike the Wheeler-DeWitt theory, in sLQC the spacetime curvature thus never diverges during the evolution.", "pages": [ 17, 18 ] }, { "title": "V. DISCUSSION", "content": "The essence of quantum superposition is that independent reality cannot be assigned to the elements of that superposition unless interference among them vanishes. In the language of consistent or decoherent histories 'generalized' quantum mechanics, physical probabilities cannot be inferred from the transition amplitudes unless the corresponding family of histories is consistent, as emphasized in [10, 11]. For a closed quantum system, therefore, it is essential to have available an internally consistent measure of quantum interference in order to be able to arrive at meaningful quantum predictions. In a closed system such as the universe as a whole, an objective measure of quantum interference is provided by the system's decoherence functional. Construction of the decoherence functional is therefore an essential component of any quantum theory of gravity in which one intends to apply the theory to the whole universe, as in quantum cosmology. The point of view of the decoherent or consistent histories framework as applied here 23 may be sufficiently unfamiliar to some that it is important to emphasize that, in almost every respect, it is simply 'quantum mechanics as usual'. The single - but crucial - new concept is the addition of the decoherence functional to the technical and interpretational apparatus of quantum theory. The decoherence functional is essentially an extension of the concept of quantum state 24 to provide an objective, internally consistent measure of quantum interference, thus replacing the vague criterion of measurement by external classical observers with a rigorously formulated measure that reproduces the results of classical measurement theory when it is applicable, but extends it to situations when it is manifestly, and profoundly, not applicable - crucially, to closed systems for which the notion of external measurements is clearly meaningless. Simple examples of the necessity for such an extension are easy to come by in quantum gravity. For example, how is one to assign probabilities to the quantum density fluctuations that putatively lead to the large scale cosmological structure we observe today when no classical systems existed to 'observe' them? The focus on 'histories' may also give the framework an unfamiliar feel. However, it is precisely for making predictions concerning sequences of quantum outcomes that ordinary measurement-based formulations of quantum mechanics have no answers, no predictions, for closed quantum systems. Yet, patterns of correlations between observable quantities - paths, or 'histories' of those observables - are precisely the kind of quantities in which one is principally interested in cosmology. The decoherent histories framework provides a consistent and rigorous foundation for calculating quantum probabilities, whether for single quantum events, or sequences thereof, in terms of quantum amplitudes given by the system's state and physical inner product. In a properly formulated generalized quantum theory of cosmology, the physical meaning of the 'wave function of the universe' is unambiguous; there is no need to rely on heuristic arguments [46] to extract physical predictions [4, 6, 12]. The methodology for quantum prediction is precise and clear. We noted in the introduction that the interpretation of the meaning of probability in quantum theory quite generally - not only in the quantum mechanics of closed systems - remains controversial. Notwithstanding, we are not reluctant to write down expressions for quantum probabilities such as those found in Sec. IV with the expectation that their interpretation is as clear in context as it ever is in quantum mechanics. 25 In recognition nonetheless of the special status of closed systems, we focus particular attention on quantum predictions which are certain, those for which the probabilities are 1 or 0 - or close to it. For such predictions, the meaning of the probabilities is unambiguous: the universe either will (or will not) exhibit the property (or history) in question. If observation contradicts this prediction, the theory is simply incorrect. (Hartle [4, 27] and Sorkin [21] in particular have emphasized the special role played by such certain predictions in the quantum theory of closed systems.) In an exactly solvable model of loop quantum cosmology (sLQC) [15], for example, we have shown in a quantum mechanically consistent way that all states in the theory - whether peaked on classical trajectories at large volume or not - 'bounce' in the sense that the universe must have a large volume in the limit of large | φ | . In stark contrast to the corresponding classical model, and also to its older Wheeler-DeWitt-type quantization, these quantum universes are not drawn into an infinite density singularity, and indeed, cannot be. The role of the large-| φ | limit in our predictions is worth comment. As noted in passing in Sec. IV C, for a given state | Ψ 〉 it is certainly possible that the histories described by C bounce ( φ 1 , φ 2 ) and C sing ( φ 1 , φ 2 ) will decohere to a high degree of approximation at finite φ 1 , 2 . Indeed, wide classes of states such as states which are semi-classical at late times, i.e. peaked on classical trajectories at large volume, will do so. However, it is only in the limit that | φ | becomes large that all states, no matter how 'quantum' (with no peakedness properties), are guaranteed to decohere and exhibit a quantum bounce. This is the role of the limit: it is in this limit we arrive at a universal , and certain, prediction of sLQC - including both decoherence and unit probability - valid for all states. The bounce is a robust, universal prediction of sLQC. 26 It is of course the case that it was rigorously shown in Ref. [15, 20] that the matter density remains bounded above for all states in the theory (in the domain of the relevant physical operators.) Here, we complement this result and add a little more. First, as emphasized in Refs. [10, 11], the physical question of whether a quantum universe exhibits a 'bounce' is fundamentally a prediction about the correlation of the volume with (at least) two different values of the scalar field - at two different emergent 'times'. It is precisely for such predictions that the question of the decoherence of the corresponding histories becomes critical. Here we have shown that the coarse-grained histories describing a bounce do indeed decohere and predict a bounce with probability 1. This actually goes further than the assertion that the matter density remains bounded above. In fact, we showed that for arbitrary choice of volume ν ∗ , the branch wave function | Ψ ∆ ν ∗ 〉 describing a universe with volume | ν | in ∆ ν ∗ = [0 , ν ∗ ] becomes 0 as | φ | → ∞ , complementing the result of Ref. [15] that the expectation value of the volume becomes infinite in those limits for all states. The fact that ν ∗ is completely arbitrary implies that all quantum states in sLQC will eventually end up at large volume, and therefore, be described by a superposition of Wheeler-DeWitt quantum states in that regime [20, 38]. (Of course, that does not mean all states behave quasiclassically if the state is highly quantum. It only means that sLQC passes over to the Wheeler-DeWitt quantum theory at large volume, and that that limit obtains for every loop quantum state for some range of large | φ | . For a detailed analysis of the precise sense in which sLQC approximates the Wheeler-DeWitt theory, see Ref. [15].) This rules out, for example, the possibility of a highly quantum state that lingers indefinitely near the 'big bang' (i.e. at large matter density). Consistent with the prediction that all states 'bounce', as noted all states in sLQC look like a particular symmetric superposition of expanding and collapsing Wheeler-DeWitt universes at large volume [20]. While this is certainly a necessary condition for a theory in which a bounce is a generic feature, one may be led to inquire whether the presence of this superposition is in some sense the reason for the bounce. The answer is definitely 'NO'. In fact, it was shown in Ref. [10-12] that a superposition of expanding and contracting universes in the Wheeler-DeWitt quantization of this same physical model does not, and indeed cannot bounce: all states are sucked in to the singularity at large | φ | . If a physical reason for the bounce is to be sought, it is in the 'quantum repulsion' generated at small volume in loop quantum gravity, 27 and manifested in this model in the ultraviolet cutoff in the dynamical eigenfunctions [20], not in the superposition of large expanding and contracting classical universes. We have so far exhibited the construction of a generalized decoherent-histories quantum theory in two mathematically complete quantizations of cosmological models, sLQC in this analysis, and the corresponding Wheeler-DeWitt model in an earlier work [10-12], demonstrating how these theories may be used to arrive at quantum mechanically consistent physical predictions. In both of these models, the presence of an internal time, a monotonic physical clock provided by the scalar field in the canonical quantization of the models, simplified the construction of the generalized quantum theories through the structural analogy with ordinary quantum particle mechanics. Is this feature essential to the construction of a generalized quantum theory? The answer is no. A path integral formulation for a closed Bianchi model was studied in detail in Ref. [6] and references cited therein, and upon which much of the present work is based. More rigorously, we have now constructed along the same lines the generalized quantum theory for a path-integral (spin foam) quantization [30, 31, 48] of the same physical model (flat scalar FRW) as studied here [32]. This will again be employed to study the same physical questions examined here, in particular, the quantum behavior near the classical singularity. This example shows that the presence of an internal time in the model, while convenient, is not essential to the formulation of its class operators, branch wave functions, and decoherence functional. 28 Taken together, these examples lay the foundation for the construction and application of generalized quantum theories the quantum theory of closed physical systems - in quantum cosmology and quantum gravity more broadly.", "pages": [ 18, 19, 20 ] }, { "title": "ACKNOWLEDGMENTS", "content": "Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. D.C. would like to thank the Department of Physics and Astronomy at Louisiana State University, where portions of this work were completed, for its hospitality. D.C. was supported in part by a grant from FQXi. P.S. is supported by NSF grant PHYS1068743. [17] Brajesh Gupt and Parampreet Singh, 'Contrasting features of anisotropic loop quantum cosmologies: The role of spatial curvature,' Phys. Rev. D85 , 044011 (2012), arXiv:1109.6636 [gr-qc]. A54 , 182-203 (1996); John T. Whelan, 'Spacetime alternatives in relativistic particle motion,' Phys. Rev. D50 , 6344 (1994), arXiv:gr-qc/9406029 [gr-qc]; Richard Micanek and James B. Hartle, 'Nearly instantaneous alternatives in quantum mechanics,' Phys. Rev. A54 , 3795 (1996), arXiv:quant-ph/9602023 [quant-ph].", "pages": [ 20, 21, 22 ] } ]

[ { "title": "Connection dynamics of a gauge theory of gravity coupled with matter", "content": "Jian Yang, 1, ∗ Kinjal Banerjee, 2, 3, † and Yongge Ma ‡ 2, § 1 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China. 2 Department of Physics, Beijing Normal University, Beijing 100875, China. 3 BITS Pilani, K.K. Birla Goa Campus, NH 17B Zuarinagar, Goa 403726, India. (Dated: June 15, 2021) We study the coupling of the gravitational action, which is a linear combination of the HilbertPalatini term and the quadratic torsion term, to the action of Dirac fermions. The system possesses local Poincare invariance and hence belongs to Poincare gauge theory with matter. The complete Hamiltonian analysis of the theory is carried out without gauge fixing but under certain ansatz on the coupling parameters, which leads to a consistent connection dynamics with second-class constraints and torsion. After performing a partial gauge fixing, all second-class constraints can be solved, and a SU (2)-connection dynamical formalism of the theory can be obtained. Hence, the techniques of loop quantum gravity can be employed to quantize this Poincare gauge theory with non-zero torsion. Moreover, the Barbero-Immirzi parameter in loop quantum gravity acquires its physical meaning as the coupling parameter between the Hilbert-Palatini term and the quadratic torsion term in this gauge theory of gravity. PACS numbers: 04.50.kd, 04.20.Fy, 04.60.Pp", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "General Relativity(GR) has been very successful in describing universe at large scales. However, it is believed that we have to develop a quantum theory of gravity for a consistent description of nature. One of the reasons that classical GR cannot be consistent can be seen from the Einstein's equations which relate gravitational and matter degrees of freedom. While the gravitational part is classical and is encoded in the Einstein tensor, since matter interactions are very well described by quantum field theory, we need to use some quantum version of the stress energy tensor for the matter part. This would imply that a consistent coupling of matter and gravity for all energy scales requires both of them to be quantized. Einstein's equations can be obtained via an action principle starting from the first-order Hilbert-Palatini action. However, if we consider fermionic matter sources, the equations of motion from this action will not provide the torsionfree condition of vacuum case. Hence, we have to either allow for torsion or make some suitable modification of the action. (See [1] and references therein for a comprehensive account of torsion in gravity). So, if one wants to start with first-order action, it is very possible that quantum theory of gravity would incorporate torsion in its formalism in order to consistently couple gravity to fermions. Among various attempts to look for a quantum gravity theory, gauge theories of gravity are very attractive since the idea of gauge invariance has already been successful in the description of other fundamental interactions. Local gauge invariance is a key concept in Yang-Mills theory. Together with Poincare symmetry, it lays the foundation of standard model in particle physics. Localization of Poincare symmetry leads to Poincare Gauge Theory(PGT) of gravity. One of the key features in PGT is that, in general, gravity is not only represented as curvature but also as torsion of space-time. GR is a special case of PGT when torsion equals zero. PGT provides a very convenient framework for studying theories with torsion. A number of actions which satisfy local Poincare symmetry have been analyzed by various researchers (Refs.[2, 3] provide the comprehensive review and bibliography of the progress made in PGT). However, one of the drawbacks of PGT is that its Hamiltonian formulation is usually very complicated. Although Hamiltonian analysis is performed for many models in PGT, the results are at a formal level without explicit expressions of the additional required second-class constraints. From the point of view of canonical quantization, it is essential to have a well-defined consistent Hamiltonian theory at the classical level. Such an ingredient is missing if we want to incorporate torsion into candidate quantum gravity models constructed from PGT. Moreover, the internal gauge group in PGT is in general non-compact, while most of the standard tools developed in quantum field theory apply to gauge theories with compact gauge groups. There exists a well-known SU (2) gauge theory formulation of canonical GR [4, 5], where the basic variables are the densitized triad and Ashtekar-Barbero connection. A candidate canonical quantum gravity theory known as Loop Quantum Gravity (LQG) [6-9] can be constructed starting from the connection dynamical formulation. Moreover, LQG can also be extended to some modified gravity theories such as, f ( R ) theories [10, 11] and scalar-tensor theories [12]. However, the action of GR from which the connection dynamics can be derived is not the standard HilbertPalatini action. An additional term known as the Holst term has to be added to the standard Hilbert-Palatini action in order to rewrite GR as a SU (2) gauge theory [13, 14]. It is customary to multiply the additional Holst term with a coupling constant γ known as the Barbero-Immirzi parameter. Classically these two actions are equivalent in vacuum case, since the additional Holst term does not affect the equations of motion although it is not a total derivative. The parameter γ does not appear in the classical equations of motion. This is because the Holst term differs from a total derivative known as the Nieh-Yan term [15] by a term quadratic in torsion (for the exact relations between them see [16, 17]). Since the torsion term is zero when there is no fermionic matter, the Nieh-Yan term and the Holst term are same, and hence the connection dynamics obtained from adding either term to the Hilbert-Palatini action would be equivalent. It has been shown that a SU (2) gauge theory can also be constructed from an action containing the standard Hilbert-Palatini term and the Nieh-Yan term [16]. However, when there are fermions, the T 2 term is not zero and the the difference in the Holst term and the Nieh-Yan term shows up. In Ref.[18] it was found that adding the standard fermion action along with the Holst term leads to equations of motion which depend on γ and are therefore not equivalent to standard GR with fermions. The difference arises because the Holst term is not a total derivative. In Ref.[16] it was shown that there is no such issue if the full Nieh-Yan term is used. An alternative possibility of modifying the fermion action to be non-minimally coupled has been analyzed in detail in Refs.[19, 20] and also in Refs.[21, 22]. The additional piece in fermion action cancels the contribution of the Holst piece if the coupling constants are chosen accordingly (see [23] for a recent account of these issues). In the absence of direct experimental or observational evidence of quantum gravity and of torsion, it is not clear which action should be the appropriate starting point for quantization, particularly from the perspective of LQG. It is therefore very important to study all the different possibilities. However to apply the LQG techniques, it is essential to first reformulate these candidates as gauge theories with a compact gauge group. In this series of works, instead of the Holst piece of the Nieh-Yan term, we consider the T 2 piece. In Ref.[24] we considered the vacuum case, i.e. an action with only this T 2 term along with the standard Hilbert-Palatini term. An arbitrary coupling constant α between the Hilbert-Palatini and T 2 terms was employed. There it was shown that, although we started from an action with explicit torsion dependence, the constraint equations imply that torsion is zero, and hence we go back to standard GR. This is consistent with the results that there is no torsion in the absence of spinors. The variables we choose are motivated by PGT. But unlike other analysis in PGT we obtain explicit expressions of the second-class constraints. In this paper, we add Dirac fermions to the action and apply the techniques developed in Ref.[24] to carry out the Hamiltonian analysis. We consider the fermions to be non-minimally coupled, because the T 2 term is not a total derivative and indeed, by proper choice of the two coefficients, the contribution of the additional non-minimal piece is canceled by the contribution of the torsion piece. Also the relation between torsion and the fermions we obtain is the same as the one obtained in Ref.[16] with Nieh-Yan term and minimally coupled fermion action. To the best of our knowledge, this is the first action with explicit torsion terms which has been reformulated as a Hamiltonian SU (2) gauge theory. The new connection we obtain is algebraically same as the standard Ashtekar-Barbero connection but is valid even in the presence of explicit torsion dependent terms of the form we have chosen. This is unlike the standard derivation of the Ashtekar-Barbero formalism[5] which was done for the torsion-free case. The coupling parameter α in our action plays the role of Barbero-Immirzi parameter. The classical system we obtain in this paper can subsequently be loop quantized using the tools already developed in LQG. Also, Hamiltonian formulation of theories with torsion are usually very complicated. We think that the techniques developed in this and the previous paper [24] can be used for analyzing other similar actions with torsion terms. If that is possible, then the general programme of loop quantization can be applied to a much wider class of theories which include torsion. The paper is organized as follows. In section II we give the explicit expression of the action with which we start and derive the equations of motion for the coupled system. It is shown that under certain ansatz on the coupling parameters, the dynamical system we obtain is equivalent to the standard Palatini formulation of GR minimally coupled to fermions. In section III we perform a 3 + 1 decomposition of this action and perform the Hamiltonian analysis under the ansatz but without fixing time gauge. Having obtained a consistent Hamiltonian system, we fix time gauge and then solve the second class constraints in section IV. Fixing the time gauge also breaks the SO (1 , 3) gauge invariance to SU (2). Then in section V a new connection which is conjugate to the densitized triad is derived, and thus we obtain a SU (2) gauge theory. Our analysis has several novel and peculiar features. We conclude with a discussion of these and some comparison of our results with those obtained by using the Holst and Nieh-Yan terms in [ µ section VI. We will restrict ourselves to 4 dimensions. The Greek letters µ, ν . . . refer to space-time indices while the uppercase Latin letters I, J . . . refer to the internal SO (1 , 3) indices. Our spacetime metric signature is ( -+ ++). Later when we do the 3 + 1 decomposition of spacetime, we will use the lowercase Latin letters from the beginning of the alphabet a, b, . . . to represent the spatial indices. After we reduce the symmetry group to SU (2), the internal indices will be represented by lowercase Latin letters from the middle of the alphabet i, j . . . .", "pages": [ 1, 2, 3 ] }, { "title": "II. THE ACTION", "content": "In this paper we consider an action which has three pieces, a Hilbert-Palatini term, a term quadratic in torsion and a term for the massless fermionic matter. It reads where Here e µ I is the tetrad, e denotes the absolute value of the determinant of the co-tetrad, ω IJ µ is the spacetime spinconnection which is not torsion-free, /epsilon1 µνρσ denotes the 4-dimensional Levi-Civita tensor density, and the covariant derivatives in the fermion action read, Note that we denote γ µ = γ I e µ I with 4-dimensional Dirac matrices γ I , σ IJ := 1 4 [ γ I , γ J ] and γ 5 := iγ 0 γ 1 γ 2 γ 3 . Our conventions regarding the Dirac matrices and their properties are given in Appendix (A). Note also that λ and λ := λ † γ 0 , representing the fermionic degrees of freedom, are 4-dimensional row and column vector respectively. Further, T I µν = ∂ e I ν ] + ω [ µ I | J | e J ν ] (3) are the definitions for curvature and torsion respectively 1 . It should be noted that the boundary terms of the action (1) are neglected. This means that we either consider a compact spacetime without boundary or assume suitable boundary conditions for the fields configuration such that there is no boundary term. It is obvious that this action is invariant under local Poincare transformations [24]. We will be working in the first-order formalism and hence both the co-tetrad e I µ and the spin connection ω IJ µ are treated as independent fields. Our covariant derivative D µ acts in the following way: Note that the coupling parameter α in action (1) is a non-zero real number. The parameter ε in the matter action denotes nonminimal coupling and with ε = 0 we get back minimally coupled Fermion action. Let us consider the Lagrangian equations of motion. The variations of action (1) yield The parameter ε , in general, has no relation with the parameter α . However if we choose the ansatz ε = α 2 the equations of motion would be simplified. Let us consider the equations of motion of the spin connection. If we choose ε = α 2 , Eq. (4) is reduced to Denoting Eq. (8) implies s µ [ KL ] = 0, which is α -independent. Hence the α term in Eq.(8) will disappear from the equations of motion of ω IJ µ . Using this result and the identity /epsilon1 µρνσ /epsilon1 IJKL e K ν e L σ = 2 ee µ [ I e ρ J ] , it can be shown, after some calculation, that for the case ε = α 2 , the equations of motion of e K α reduce to the standard form given by Further, using the fact that D µ ( ee µ I ) = 0, it can be easily shown that the ε dependence drops out from the equations of motion of the fermion degrees of freedom λ and λ [20], leaving So if we impose the relation ε = α 2 , the dynamical system we obtain is equivalent to the standard Palatini formulation of GR minimally coupled to fermions. We therefore adopt that relation between the two parameters from here onwards. In Ref.[24], the Hamiltonian analysis of the action (1) without the matter part was carried out. In that case, the Lagrangian equations of motion showed that torsion was zero on-shell although the action has explicit torsion terms. In the next section we will carry out a complete Hamiltonian analysis with action (1) where the torsion is expected to be non-zero.", "pages": [ 3, 4 ] }, { "title": "III. HAMILTONIAN ANALYSIS", "content": "We shall perform the Hamiltonian analysis of action (1) similar to what was done in Ref.[24] for the action without the matter term. Recall that in the Hamiltonian formulation of Hilbert-Palatini theory the basic variables are the SO (1 , 3) spin connection ω IJ a and its conjugate momentum. It is well known that this formulation contains secondclass constraints. Since our action (1) contains the other term which explicitly depends on torsion, we expect that there will be another pair of conjugate variables and the second-class constraints will be somehow different from the Hilbert-Palatini case. It is also well known that in the absence of fermionic matter, torsion is zero. In the analysis of Ref.[24], this was obtained after we identified all the constraints. Owing to the presence of the fermion term in the action, here torsion will not be zero. In this section we will show how the torsion and the spinorial degrees of freedom are related.", "pages": [ 4 ] }, { "title": "A. 3+1 Decomposition", "content": "To seek a complete Hamiltonian analysis, we assume the spacetime be topologically Σ × R with some compact spatial manifold Σ without boundary so that the surface terms can be neglected. We first perform the 3 + 1 decomposition 5 of our fields without breaking the internal SO (1 , 3) symmetry and also without fixing any gauge. To identify our configuration and momentum variables for performing Hamiltonian analysis, we can rewrite the three pieces in the action as: We can read off the momenta with respect to ω IJ a , e I a , λ and λ respectively as Π := ieλ (1 + i α 2 γ ) e t I γ , Π := - iee t I γ (1 + i α 2 γ 5 ) λ, (17) where /epsilon1 abc denotes the 3-dimensional Levi-Civita tensor density, and we have used the relation γ µ = γ I e µ I . For our analysis we shall use a standard parametrization of the tetrad and the co-tetrad fields as in Ref.[25]. This is the same parametrization used in the Hamiltonian analysis of the first two terms of our action in Ref.[24]. Since the parametrization which we are using is standard, its details and some related identities are given in Appendix B. After some manipulation and neglecting the total derivatives, the pieces (13), (14), and (15) of the action can be written in this parametrization respectively as", "pages": [ 4, 5 ] }, { "title": "B. Primary and Secondary Constraints", "content": "Let us now consider the constraints in the theory. At this stage we have the following constraints From Eq.(21) we get 12 constraints, while Eq.(22) gives 18 because of the antisymmetry in IJ . I I These are the primary constraints of our theory. By performing Legendre transformation, the Hamiltonian corresponding to the action (1) can be expressed as where Subsequently we will drop the subscript t from G tIJ and denote it as G IJ . Including all of above primary constraints we can write the total Hamiltonian as where ρ , ρ a , λ IJ t , γ I a , λ IJ a , u and u are the Lagrangian multipliers. At this point they are completely arbitrary. In order to preserve primary constraints Π N ≈ 0, Π N a ≈ 0 and Π t IJ ≈ 0, one has to impose the following secondary constraints: which are called scalar,vector and Gaussian constraints respectively. We now need to check whether the Hamiltonian system is consistent. To ensure the consistency of the Hamiltonian system, the constraints have to be preserved under evolution. Note that the primary constraints Π N , Π N a and Π t IJ are preserved in evolution respectively by the secondary constraints H , H a and G IJ . Note also that the Gaussian constraint G IJ generates the SO (1 , 3) transformations, and hence the Poisson bracket of any constraint with G IJ is weakly equal to zero. However, as shown in Ref.[24] the constraint which actually generates the spatial diffeomorphisms for the gravitational variables is a combination given by This can be easily demonstrated as: where ˜ H a ( ν a ) ≡ ∫ Σ d 3 xν a ˜ H a denotes the smeared constraint. From now on, we will keep this convention to denote the smeared version of a constraint with a smearing function, e.g., Ψ( u ) ≡ ∫ Σ d 3 xu Ψ. Also we will continue using the same notation ω IJ t and γ I a for the Lagrange multipliers of G IJ and C a I respectively. For the matter variables the constraint (29) acts as Clearly this combination ˜ H a , acting on all the variables, generates Lie derivatives [23] and can therefore be identified as the diffeomorphism constraint. Using the property of Lie derivatives (or by explicit calculation) it can be shown that the Poisson bracket of any constraint with ˜ H a vanishes on the constraint surface. In fact we have Note that the smeared scalar constraint reads H ( M ) ≡ ∫ Σ d 3 xMH with M as a smearing function. Now the H a term in the total Hamiltonian (28) can be replaced by ˜ H a . Thus we can rewrite our total Hamiltonian as", "pages": [ 5, 6, 7 ] }, { "title": "C. Consistency Conditions", "content": "The terms in the constraint algebra which are not weakly zero are respectively For a consistent Hamiltonian system, the constraints should be preserved under evolution, i.e., for all the constraints C m , we require ˙ C m := { C m , H T } ≈ 0. Our analysis will be along the lines of Ref.[24]. However, owing to presence of fermions, it will turn out that torsion is not zero. As a consequence, the calculations are much more complicated. Let us first consider the consistency of constraint Φ a IJ . From Eqs. (33) and (34) we need where σ IJ a is an arbitrary smearing function. Using Eqs. (33) and (34), and after some calculation, Eq.(39) implies Multiplying Eq.(40) with /epsilon1 abc , we have Multiplying Eq.(41) with V b J and using the properties (B3) we get By multiplying this equation with N I , V c I and V I d respectively and using the relations (B2) and (B3), we obtain the following relations where we have used Eq. (44) to obtain Eq. (45). Finally from Eqs. (43) and (45) we get a solution for the Lagrangian multiplier γ I c as Note that, all these equations differ from the corresponding equations in Ref.[24] only by the fermion-dependent terms. So, we have obtained 12 components of γ I a from the 18 equations in Eq.(40). Consequently there are 6 constraints remaining. By inserting the solutions (46) back into Eq.(40) and after some calculation, we get the following secondary constraint: Since χ ab is symmetric in ( a ↔ b ), it contains just the 6 required constraints. As seen above, the condition ˙ Φ a IJ ≈ 0 fixed the Lagrange multipliers γ I a of the constraint C a I to the form given by Eq. (46). This can however be further simplified. For this and for subsequent calculations, we now derive some useful identities using the constraint equations. All these identities hold weakly, i.e., they are true only when the constraints are used. From the definition of σ IJ and using the properties of gamma matrices (A2), the Gaussian constraint can also be written as From the constraints (21) and (22) we can easily obtain the relation: Using this and the Gaussian constraint (48) we get, after some algebra, Multiplying this equation with N J and then with V b I , and using the properties (B2) and (B3), we get By multiplying relation (50) with V b I and then with V c J , and again using the properties (B2) and (B3), we get Plugging Eq. (52) in the constraint (47) we get the relation These identities can be used to greatly simplify the subsequent calculations. Note that because of the identity (52), the second term on the RHS of Eq. (46) drops out and the Lagrangian multiplier of C c I in H T becomes This leads to further simplification of our problem. Moreover, let us consider the identity (53) again. Multiplying it by V bI and using Eq.(51), properties (B3) and (B5) , we get This equation relates the torsion degrees of freedom encoded in Π a I with the spin degrees of freedom λ and λ . Note that we have used only constraint equations and not equations of motion in deriving Eq.(55). This is a weak relation since it has been derived by using the constraints G IJ , C a I , Φ a IJ , χ ab . When there is no matter, this equation would indicate that torsion is zero [24]. Note also that relation (55) is as same as the one obtained in Ref.[16]. Now let us consider the constraints Ψ and Ψ. For the consistency conditions for constraints Ψ and Ψ, we need Note that since χ ab is a secondary constraint, we do not add it in H T . As proved beforehand, the condition that Φ a IJ be preserved under evolution has fixed γ I a to the specific form given by Eq. (54). Now recall from Eq. (37), for an arbitrary smearing function ξ I a we have When ξ I a = γ I a , which is of the form given in Eq. (54), using Eqs. (B5) and (B2) we get Therefore, once the Lagrange multiplier γ I a is fixed to the value required for a consistent Hamiltonian system, Eq.(56) becomes: By using Eqs. (B1) and (B2), we can obtain γ I N I = γ µ e µI N I = -γ t N . Since both γ t and N are nonzero, one has γ I N I = 0. It is obvious that the only solution for Eq.(57) is u = 0. Similarly, we need /negationslash Its only solution is u = 0. We now turn to the additional secondary constraint χ ab (see Eq.(47)). We now have to check its contribution to the constraint algebra. Obviously, χ ab commutes with primary constraints Π N , Π N a and Π t IJ . Moreover one has The additional non-zero terms in the constraint algebra are The consistency conditions of constraints C a I and χ ab read respectively where η I a and σ ab are arbitrary smearing functions, H T is still given by Eq.(32). It turns out that we can indeed solve the 18 independent equations (64) and (65) to fix the 18 independent components of the Lagrangian multiplier λ IJ a . This calculation is slightly lengthy and complicated and has been given in Appendix (C). We are finally left with the scalar constraint. We now need to prove ˙ H ( M ) ≈ 0. The time evolution of scalar constraint reads where γ I a and λ IJ a are given by Eq.(54) and Eq.(C4) respectively. By using Eq.(C2), we have Using the solution (54) of γ I a and also using Eq.(55), it can be shown that the first two terms in Eq.(67) cancel each other. For the last term of above equation, by using Eq.(C7) and properties (B2) and (B3) we find it is exactly equal to zero. Therefore we get ˙ H ( M ) ≈ 0. We have now exhausted all the consistency conditions. We have also proved that the constraints are preserved under evolution, i.e., for all the constraints C m , we have shown ˙ C m := { C m , H T } ≈ 0. We have therefore obtained a consistent Hamiltonian system. Now all the constraints have been identified, we can classify them into first-class constraints and second-class ones. It is obvious that ˜ H a and G IJ are first class. Since none of constraints contain N , N a or ω IJ t , primary constraints Π N ≈ 0, Π N a ≈ 0 and Π t IJ ≈ 0 are first class. In this sense, N , N a and ω IJ t are arbitrary Lagrangian multipliers. We may eliminate configuration N , N a and ω IJ t as well as their conjugate momenta Π N , Π N a and Π t IJ from dynamical variables[6, 26]. The term ρ Π N + ρ a Π N a + λ IJ t Π t IJ in the total Hamiltonian (32) can be eliminated. Thus we get where ω IJ t is replaced by Λ IJ . In light of the argument given above, Λ IJ , N and N a are just Lagrangian multipliers. At this stage, γ I a and λ IJ a in the above total Hamiltonian (68) are given by Eq.(54) and Eq.(C4) respectively. Also we have removed the second-class constraints Ψ and Ψ from H T as we have proved that the Lagrange multipliers for these two constraints, u and u respectively, are zero. Recall that all the non-zero terms of the constraint algebra are given in Eqs. (33-38) and (59-63). It can be easily seen that Φ a IJ , C a I , χ ab , are second class. Although the Poisson brackets of some constraints with the scalar constraint H are still not weakly equal to zero, it can be shown that we can construct a new first-class Hamiltonian constraint by the combination: Since we have already identified all the constraints, we can now count the degrees of freedom. The gravitational degrees of freedom are incorporated in the pair ( Π a IJ , ω IJ a ) , which have 36 degrees of freedom and in the pair ( Π a I , V I a ) , which have 24. The matter degrees of freedom are in the two pairs ( Π , λ ) and ( Π , λ ) each with 8 degrees of freedom. The total number of degrees of freedom without considering the constraints is therefore 76. Clearly G IJ , ˜ H a and ˜ H contribute (6 + 3 + 1) = 10 first-class constraints removing 20 degrees of freedom. The constraints C a I , Φ a IJ , Ψ and Ψ are primary second class removing (12 + 18 + 4 + 4) = 38 degrees of freedom. Finally the secondary constraint χ ab turns out to be second class and thus removes 6 degrees of freedom. Thus the number of independent degrees of freedom in our system is 12. In these 12 degrees of freedom, 4 represent gravity and 8 denote Dirac fermions.", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "IV. SOLVING THE SECOND-CLASS CONSTRAINTS", "content": "Second-class constraints are problematic because the flows generated by them do not lie on the constraint surface. Having obtained a consistent Hamiltonian system in the previous section, we now proceed to solve all the second-class constraints and eliminate spurious degrees of freedom. We will do this after performing a partial gauge fixing. Since we have already proved the consistency of the Hamiltonian system, we can be sure that making a gauge choice now will not lead to any inconsistency. Our goal is to reduce the internal SO (1 , 3) gauge symmetry to SU (2). So we break the SO (1 , 3) symmetry by fixing the internal timelike vector N I = (1 , 0 , 0 , 0), i.e.,we fix a specific timelike direction in the internal space. This is a standard gauge choice and is known as time gauge . From Eqs. (B1), (B2) and (B3) it is easy to see that For consistency, this gauge fixing condition has to be preserved, i.e., Hence in time gauge we get The Lagrangian multiplier Λ 0 i of G 0 i gets fixed. This is expected because, by fixing N I , we have broken the SO (1 , 3) gauge invariance. The preservation of this gauge fixing condition implies that the boost part of the Gaussian constraint does not generate gauge transformations. We first solve the constraint (22), which can be written as Thus in time gauge we have So, after solving this constraint only the Π a 0 i part of Π a IJ remains a basic dynamical variable. Consequently, only the ω 0 i a part of the SO (1 , 3) connection remains basic dynamical variable. For convenience we define K i a := 2 ω 0 i a which will be conjugate to E a i . The ω ij a is the remaining part of the connection which will get solved in terms of other variables while solving the remaining constraints. Since our gauge group is now reduced to SU (2), we will expand SO (1 , 3) connection components ω ij a in the adjoint basis of SU (2) as ω ij a := -/epsilon1 ijk Γ ak . The quantity Γ ak is the SU (2) spin connection. Note that we had started with a SO (1 , 3) spin connection ω IJ a which was not torsion-free. Therefore the variables K i a and Γ i a that we define above will contain information about torsion implicitly. Also, from Eq.(B3) it is clear that, in time gauge, V a i is the inverse of V i a . Using the properties of inverses and determinants of matrices, it is easy to see from Eq.(72) that E a i = √ qV a i is the densitized triad. We can also determine its inverse E i a = 1 √ q V i a . 2 Next we consider the constraint (47). It can be easily seen that once we substitute (55) into the expression of χ ab , it is identically satisfied. Using the above solution (72), in time gauge the equation (55) simplifies to The torsion degrees of freedom are solved in terms of the densitized triad E a i and the fermionic fields λ and λ . Using above results we can now solve the second-class constraint (21) as Equation (75) can be used to solve the spin connection Γ i a in terms of the other variables. After some algebra we obtain where we have denoted by ̂ Γ i a the first four terms in the RHS of Eq.(76), which do not depend on the fermions. It turns out that ̂ Γ i a is exactly the SU (2) spin connection which we would have obtained, had there been no fermionic matter [6]. So, when there is no matter we go back to the standard GR formulation. Also note that the spin connection Γ i a is independent of the arbitrary coupling parameter α . So far, we have reduced our original phase space by consistently imposing time gauge and then solving some second-class constraints. As a result, some basic variables in the original phase space have been eliminated in terms of the others. To obtain the basic variables in this phase space we need to find the symplectic structure after all these reductions. Recall that, we started with the symplectic structure given by: Using Eq. (72) and our definition K i a := 2 ω 0 i a , the first term in expression (78) becomes E a i ∂ t K i a . For the second term, recall that ∂ t V 0 a = 0 in time gauge. Then Using the constraint equation (21) we can calculate the second term: Let us define where we also used the solution (72) and neglected total derivative terms. For the last two terms, recall that λ := λ † γ 0 . Also Π and Π can be read off from the constraints (23). Then in time gauge, using the properties of the γ matrices given in appendix (A) we get Since ∂ t √ q = 1 2 √ qE i a ∂ t E a i , we have where we have again neglected the total time derivative term. Putting everything together, expression (78) becomes The second-class constraints Ψ and Ψ now become where, following Refs.[20, 27], we have defined half-densities of the fermionic variables: ζ := 4 √ qλ , ζ † := 4 √ qλ † and identified Π ζ = -iζ † , Π ζ † = iζ . The two constraints ψ and ˜ ψ can be solved quite easily. The two pairs of fermionic variables can be reduced to one pair. The symplectic structure (78) is then finally reduced to: where we define A i a := ̂ Γ i a -1 α K i a and have also neglected the total time derivative terms. All the second-class constraints have now been solved and we have finally obtained the basic phase space variables on the 'reduced phase space'. Note that A i a , the variable conjugate to E a i , is exactly same as the Ashtekar-Barbero connection obtained in standard analysis. We have not yet shown that it is a connection. We will do so in the next section.", "pages": [ 11, 12, 13 ] }, { "title": "V. SU (2) GAUGE THEORY", "content": "We have obtained a consistent Hamiltonian system which is invariant under local SU (2) rotations. However this is not a SU (2) gauge theory yet. The basic variables in the gravitational sector are the densitized triad E a i and its conjugate A i a . The spin connection Γ i a , given by Eq. (76), is a function of E a i , ζ and ζ † . In this section, we shall rewrite the remaining first class constraints in terms of the new variables. First, let us consider the Gaussian constraint (48). In time gauge, using the constraint equation (21) and the solutions of Φ a IJ we can rewrite it as Recall that the Gaussian constraint was used in getting Eq.(55). Then in the gauge fixed theory, G 0 i is explicitly resolved together with the second class constraint (21) by Eqs. (74) and (75). Hence, in terms of the new variables, it can be easily seen that G 0 i is identically zero as expected. Also comparing Eq. (85) with Eq. (74) it is easy to see that /epsilon1 jkl G jk ≈ 0 implies C a 0 ≈ 0 (assuming E a i = 0). /negationslash Therefore A i a ≡ ̂ Γ i a -1 α K i a is the new connection, and using this connection we have obtained the Gaussian constraint in the standard SU (2) gauge theory form. Tensorially, the new connection A i a which we have defined is in the same form as the standard Ashtekar-Barbero connection without torsion. The coupling parameter α plays the role of the Barbero-Immirzi parameter of the standard treatment. Hence the Barbero-Immirzi parameter in loop quantum gravity acquires its physical meaning as the coupling constant between the Hilbert-Palatini term and the quadratic torsion term through our formulation. Let us briefly recap what we have done. The basic variable Π a I which encoded the torsion has been solved in terms of the fermionic degrees of freedom using the constraints G IJ , C a I , Φ a IJ , χ ab . We had started with a SO (1 , 3) connection ω IJ a which is not torsion free. That fact is reflected in our expression of the SU (2) spin connection Γ i a in Eq. (77). But in the new connection A i a which we define above, we remove exactly that additional piece. However, since we had defined K i a := 2 ω 0 i a , the variable K i a implicitly contains information about the torsion, though this is not obvious from Hamiltonian formulation itself. When there is no matter, torsion goes to zero and the S T term in our action (1), and therefore, the terms originating from it in the Hamiltonian analysis vanish [24]. Then we go back to the standard formalism with a torsion-free SO (1 , 3) spin connection. We have obtained a SU (2) gauge theory formulation of our system. The remaining constraints H a , H can also be rewritten in terms of the new basic variables. Using K i a = -α ( A i a -̂ Γ i a ) and the Gaussian constraint (86), the vector constraint (26) can be written as where F i ab := ∂ [ a A i b ] + /epsilon1 i jk A j a A k b is the curvature of A i a . This is exactly the standard form of the vector constraint. The Hamiltonian constraint (25) is more complicated. After some calculation, we get where τ i = -i 2 σ i with σ i being Pauli matrices. This expression goes over to the standard expression when the fermions are set to zero. Thus we complete our task of obtaining a SU (2) gauge theory. Note that, since we have not split the connection into torsion dependent and torsion free parts, it is not obvious how to directly compare the expressions of our constraints with those obtained in Ref.[20].", "pages": [ 13, 14 ] }, { "title": "VI. CONCLUSION", "content": "Let us briefly summarize what we have achieved in this paper. We started with the action (1) containing a torsionsquared term and fermionic matter apart from the standard Hilbert-Palatini term. This T 2 term is just the difference between the total derivative Nieh-Yan term and the Holst term. Since an SU (2) gauge theory formulation can be derived from actions containing either [13, 16], it seemed possible that such a formulation can also be obtained from our action containing only the T 2 term. We also need to add fermionic matter because the vacuum case is torsion free [24] and we are left with only the well-known Hilbert-Palatini part. We take non-minimally coupled fermionic matter so that the classical equations of motion for the fermions may not depend on the coupling constant α multiplying the torsion term under certain condition. The equations of motion of the coupled system are derived. It is confirmed that, under the ansatz ε = α 2 , the dynamical system we obtain is equivalent to the standard Palatini formulation of GR minimally coupled to fermions. We do a 3 + 1 decomposition of our action, do a constraint analysis with the above ansatz and finally obtain a consistent Hamiltonian system with second-class constraints. All the second-class constraints are solved after breaking the SO (1 , 3) invariance by fixing time gauge. As far as we know, such Hamiltonian analysis on an action with nonzero torsion term with explicit expressions of all the second-class constraints is new in literature. Similar analysis with the Holst term (with non-minimally coupled fermions) [18-20] and the Nieh-Yan term (with minimally coupled fermions) [16] has already been attempted before. Apart from the crucial fact that the gravitational part of our action is different from those studied in literature so far, there are several other differences in our approach. Since we are motivated by PGT where the initial action is invariant under local Poincare transformations, our starting variables are different from those used in Refs.[18, 19]. Unlike the treatment in Ref.[20] we do not break up our variables into the torsion dependent and independent pieces. Moreover, since we do not have the Holst term, the techniques developed in Ref.[25] for dealing with second-class constraints and used in Refs.[16, 20] are not available to us. Also, unlike the treatments in Refs.[18-20], we fix the time gauge after we have found all the second-class constraints and obtained a consistent Hamiltonian system. Furthermore, the Barbero-Immirzi parameter in loop quantum gravity obtains a new understanding in our formulation. On solving the basic variable Π a I , torsion gets related to the fermionic degrees of freedom via Eq.(55) which is as same as the one obtained in Ref.[16]. Further, solution of the second-class constraint C a i gives the SU (2) spin connection Γ i a in terms of the densitized triad. This differs from the spin connection in GR [6] only by a term which depends on the fermions. In the final step we obtain the connection dynamics by defining a new connection A i a which is algebraically in the same form as the Ashtekar-Barbero connection without torsion. However, unlike the torsion-free case, the K i a part comes from the ω 0 i a part of the SO (1 , 3) connection which is not torsion free. As a result it is not obvious if K i a can be directly related to the extrinsic curvature K ab on shell. While the vector constraint (87) is standard, the additional terms in our Hamiltonian constraint (88) are somehow different from the ones obtained in literature. Although these constraints can be loop quantized using existing techniques, it may be possible to rewrite them in a form more convenient for loop quantization. We leave this issue for future research. The present work at least opens the door to extending loop quantization techniques from standard GR to more general PGT of gravity. It should be remarked that, as a result of non-minimally coupled fermionic matter and the ansatz ε = α 2 , the coupling constant α in our starting action totally disappears from the Lagrangian equations of motion. However, in the Hamiltonian connection formalism, the new connection A i a and thus the constraints depend on the parameter α explicitly. This is somehow required by the SU (2) gauge theory formulation. A similar case happens also in the connection dynamics derived from the generalized Palatini action. It is still interesting to consider the general case when the two coupling parameters are not related to each other and thus the gauge theory is different from Palatini theory. We leave this open issue for future study. Nevertheless, an enlightening result of our formulation is that the Barbero-Immirzi parameter in loop quantum gravity acquires its physical meaning as the coupling constant between the Hilbert-Palatini term and the quadratic torsion term. In fact, this parameter measures the relative contribution of torsion in comparison with curvature in the action for this Poincare gauge theory of gravity. However, it should be noted that, the SU (2) gauge theory which we obtain is based on the particular choice of basic canonical variables by Eq.(83). An alternative choice is to cancel all the α -dependent terms in Eq.(78). Then it is still possible to obtain, via a canonical transformation, a SU (2) gauge theory in which the connection does not depend on the coupling parameters of the starting action but on an arbitrary constant appearing in the canonical transformation.", "pages": [ 14, 15 ] }, { "title": "Acknowledgments", "content": "This work is supported in part by NSFC (Grant Nos. 10975017 and 11235003) and the Fundamental Research Funds for the Central Universities. JY would like to acknowledge the support of NSFC (Grant No. 10875018). KB would also like to thank China Postdoctoral Science Foundation (Grant No.20100480223) as well as DST-Max Planck Partner Group of Dr. S. Shankaranarayanan, IISER, Thiruvananthapuram for financial support.", "pages": [ 15 ] }, { "title": "Appendix A: Gamma Matrix", "content": "In this section we collect some of the standard properties of Dirac matrices which we have used in previous sections. The γ matrices, in any dimension, satisfy the Clifford algebra where η IJ is the flat Minkowski metric. We shall restrict ourselves to 4 dimensions and choose the signature ( -+++) which is different from the signature usually used in QFT. In this signature the above relation can be decomposed as This implies that γ 0 is anti-Hermitian while γ i is Hermitian. Note that all the γ matrices are unitary. We also define the commutator σ IJ := 1 4 [ γ I , γ J ] and another standard combination γ 5 := iγ 0 γ 1 γ 2 γ 3 . It is easy to check that ( γ 5 ) 2 = I and ( γ 5 ) † = γ 5 . In the Weyl representation, commonly used for massless fermions, the Dirac matrices can be explicitly written as In this paper we have used the following standard identities", "pages": [ 15, 16 ] }, { "title": "Appendix B: 3 + 1 Decomposition", "content": "In this section we give the parametrization of the tetrad and the co-tetrad fields which we use in this paper. They read [25] What we have done is that we have reparametrized the 16 degrees of freedom of e µI into 20 fields given by (B1) subject to the 4 relations (B2). Note that this is just a convenient reparametrization of the initial variables. From these definitions, the following identities also hold: In terms of these fields the metric takes the standard form ( - N 2 + N N a a N a N aI a V ) It is easy to see that Using the definitions given above we can also prove the following two identities which have been used in our analysis,", "pages": [ 16 ] }, { "title": "Appendix C: Determination of λ IJ a", "content": "In this section we show how to obtain λ IJ a from Eqs. (64) and (65). First let us consider ˙ C a I . Using Eqs.(34-37), we get For convenience, we define g µν = V I b . Thanks to this definition, it is easy to see from Eq.(C1) that Eq.(C2) can easily be inverted to express λ IJ a in terms of X IJ a as Now let us consider ˙ χ ab . We have where we have defined ∫ Σ d 3 xσ ab Σ ab := { χ ab ( σ ab ) , ( C a I ( γ I a ) + H ( N ) )} . The explicit form of Σ ab is very complicated and can be calculated using Eqs. (54), (61) and (63). However we do not need the explicit form of Σ ab . We are interested in solving for λ IJ a which only comes from the first part of Eq. (C5). After some more algebra we obtain the equation in terms of X IJ a as Using Eqs.(C3) and (C6), after a long calculation we get where, for brevity of notation, we have defined Putting Eq.(C7) into Eq.(C4), we get λ IJ a .", "pages": [ 16, 17 ] } ]

[ { "title": "Looking for empty topological wormhole spacetimes in F ( R ) -modified gravity", "content": "R. Di Criscienzo ∗ , R. Myrzakulov 1 , † and L. Sebastiani 1 , ‡ 1 Eurasian International Center for Theoretical Physics and Department of General Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan", "pages": [ 1 ] }, { "title": "Abstract", "content": "Much attention has been recently devoted to modified theories of gravity the simplest models of which overcome General Relativity simply by replacing R with F ( R ) in the Einstein-Hilbert action. Unfortunately, such models typically lack most of the beautiful solutions discovered in Einstein's gravity. Nonetheless, in F ( R ) gravity, it has been possible to get at least few black holes, but still we do not know any empty wormhole-like spacetime solution. The present paper aims to explain why it is so hard to get such solutions (given that they exist!). Few solutions are derived in the simplest cases while only an implicit form has been obtained in the non-trivial case.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In the last years, much attention has been paid to the so-called modified gravity theories in the attempt to unify the early-time inflation with the late-time acceleration of the universe (for recent review, see [1]). The simplest class of such theories is given by F ( R ) -gravity, where the action is given by a general function F ( R ) of the the Ricci scalar R . If some modified theory lies behind our universe, it may be interesting to explore its mathematical structure and the possibility to recover features and solutions of General Relativity in its framework. It is in this sense that, for example, we have to interpret current investigations of black holes in modified theories of gravity [2]. Among the amenities of General Relativity there are topologically non-trivial spacetimes like wormholes (cf. Ref. [3] for an exhaustive introduction and Ref. [4] and references therein for recent works). One can imagine a wormhole as a three-dimensional space with two spherical holes in it, connected one to another, by means of a 'handle' [5]. What likes of such objects is the possibility of entering the wormhole and exiting into external space again: a property which sensibly distinguishes (traversable) wormholes from black holes. Interest in this field arose twentyfive years ago or so, as it was found that stable wormholes could be transformed into time machines [6]. It is worth to note that even the Schwarzschild metric with an appropriate choice of topology, describes a wormhole, but not a traversable one. In fact, in order to prevent the wormhole's throat to pinch off so quickly that it cannot be traversed in even one direction, it is necessary to fill the wormhole with non-zero stress and energy of unusual nature. By 'unusual', we mean here that a stable, traversable wormhole is supported against (attractive) gravity by matter which violates some energy condition (typically, the weak energy condition)... or better, this is what occurs in General Relativity! In F ( R ) -theories of gravity new scenarios are possible: as shown in [7], it is always possible to get a wormhole with matter preserving at least one energy condition (e.g. the strong energy condition) just choosing in an opportune way the metric components. If this procedure may look quite artificial, then following [8, 9] it is possible to show that models as F ( R ) = ∑ n a n R n for suitable integer n display a matter behavior close to the wormhole throat such to respect the energy conditions and to prevent large anisotropies - typical features of Einstein's wormholes. The existence of necessary conditions for having wormholes which respect the weak energy condition and possibly the strong energy condition has been studied in [10] at least in polynomial models of the third order or higher. The interplay between F ( R ) -gravity and scalar-tensor theories has been studied in [11] with respect to the case of wormhole solutions. Lobo and Oliveira construct in [12] traversable wormhole geometries in the context of F ( R ) modified theories of gravity imposing that the matter threading the wormhole satisfies the energy conditions, so that it is the effective stress-energy tensor containing higher order curvature derivatives that is responsible for the null energy condition violation. In particular, by considering specific shape functions and several equations of state, exact solutions for F ( R ) are found. However, what still lacks in the physics of F ( R ) wormholes is an empty solution where the role of repulsive gravity is played by geometry and no matter is necessary to support the solution. As we shall see, giving an explicit empty wormhole solution is all but easy. Few new solutions will be found in the case R = 0 ; the highly non-trivial case being still inaccessible with respect to analytic techniques. Still we are confident that, also thanks to this work, numeric solutions are at disposal in the near future. The organization of the paper is as follows. In Section 2 we will derive the field equations of topological static spherical symmetric solutions in F ( R ) -gravity by using a method based on Lagrangian multipliers which permits to deal with a system of ordinary equations and we will see how is possible to reconstruct the models by starting from the solutions. Important classes of F ( R ) -black hole solutions can be found in this way. In Section 3 the structure of the metric able to realize traversable wormholes is introduced in two equivalent forms. In Section 4 , effective energy conditions are investigated and it is shown that the equivalent description of modified gravity as an effective fluid violates the weak energy condition on the throat. In Section 5 some wormhole solutions in empty space are found in the framework of F ( R ) -modified gravity. These solutions are characterized by null or constant Ricci scalar and can be realized by a large class of models. In Section 6 , solutions with non zero curvature are discussed and the implicit form of F ( R ) -models which realizes these solutions is derived. Final remarks are given in Section 7 . We use units where k B = c = /planckover2pi1 = 1 and denote the gravitational constant κ 2 = 8 πG N ≡ 8 π/M 2 Pl with the Planck mass of M PL = G -1 / 2 N = 1 . 2 × 10 19 GeV.", "pages": [ 1, 2 ] }, { "title": "2 Topological SSS vacuum solutions in F ( R ) -gravity", "content": "In this Section we derive the equations of motion (EoMs) for topological static spherical symmetric (SSS) solutions in F ( R ) -gravity. We will write the metric in a general form in order to use it to investigate vacuum wormholes. To derive our equations we use a method based on Lagrangian multipliers, which permits to deal with a system of ordinary differential equations (see Ref. [13] and Ref. [14] for its application in FRW case). The action of modified F ( R ) -gravity in vacuum reads where g is the determinant of the metric tensor, g µν , M is the space-time manifold and F ( R ) is a generic function of the Ricci scalar R . Let the metric assume the most general static spherically symmetric topological form, where V ( r ) and B ( r ) are functions of r > 0 only and represents the metric of a topological two-dimensional surface parametrized by k , such that the manifold will be either a sphere S 2 ( k = 1 ), a torus T 2 ( k = 0 ) or a compact hyperbolic manifold Y 2 ( k = -1 ). With this ansatz , the scalar curvature reads After integration over the transverse metric dσ 2 k , introducing the Lagrangian multiplier λ and thanks to Eq. (4), the action may be re-written as Making the variation with respect to R , one gets Substituting Eq. (6) in (5) and making a partial integration, the effective Lagrangian L of the system assumes the form where the prime index ' denotes derivative with respect to r . Therefore, the equations of motion are The EoMs with Eq. (4) form a system of three ordinary differential equations in the three unknown quantities V ( r ) , B ( r ) and R ( r ) . Starting from these equations, we may try to reconstruct SSS solutions realized in F ( R ) -gravity. In particular, combining equations (8) and taking the derivative of the first equation in (8) with respect to r , we end with In this way, we deal with equations that depend on the model only through F R . Of course, one might replace F R by F ' /R ' , but to the price of dealing with much more involved equations. In principle, given the model, that is F R as a function of r , equations (9) let us to derive both V ( r ) and B ( r ) , i.e. the explicit form of the metric. /negationslash A remark is still in order about Eq. (9): also if we have assumed from the beginning that V ( r ) = B ( r ) , in case V ( r ) = B ( r ) the equation becomes F '' R = 0 recovering the important class of black hole solutions already discussed in Ref. [13] and in Ref. [15]. Let us see how the reconstruction method works with a simple class of solutions. We can consider for example F R ( r ) ∝ ( r/ ˜ r ) q , q being a fixed parameter and ˜ r a dimensional constant. From Eq. (9) one has By taking into account that we can solve Eq. (9) and obtain ( C 0 , C 1 being constants of integration) where and and from Eq. (10), This is in fact the class of Liefshitz solutions discussed in Ref. [16]. To explicitly reconstruct the model F = F ( R ) which generates these solutions, we use Eq. (4) to find r as a function of R and one of the EoMs (8). For example, for q = 4 , one has that F ( R ) ∝ √ k/R generates the SSS solution B ( r ) = -k/ 7 + C 0 /r 2 + C 1 /r 7 and V ( r ) = V 0 ( r/ ˜ r ) 7 B ( r ) , V 0 being a constant. Despite the fact that following this procedure we can generate a large number of vacuum F ( R ) -SSS solutions, the possible choices of F R are limited by those simple cases where Eqs. (9) are not transcendental. In particular, this procedure is useful to describe black hole solutions with V ( r ) = e α ( r ) B ( r ) , α ( r ) being a suitable function of r . In the next Section, we will introduce the metric form for traversable wormholes.", "pages": [ 2, 3, 4 ] }, { "title": "3 Traversable wormhole parametrization", "content": "To describe a time independent, non rotating, traversable wormhole, we introduce the line element where l ∈ ( -∞ , + ∞ ) and r ( l ) is supposed to have at least one minimum r 0 which, without loss of generality, can occur at l = 0 . In order to avoid event horizons of sort, we shall assume φ ( l ) to be finite everywhere and metric components at least twice differentiable with respect to l . These are merely the minimal requirements to obtain a wormhole that is 'traversable in principle'. By this expression, we mean that we look for wormhole solutions with no event horizons and not based on naked singularities. It is understood that for 'realistic traversable' models one should add technical features we are not going to discuss at this point. It is worth to notice that reference to the asymptotic behavior of the solution is not addressed here. In General Relativity one is typically concerned with asymptotic flatness where instead F ( R ) -solutions hardly meet this requirement. Many viable F ( R ) -gravity models representing a realistic scenario to account for dark energy have been proposed in the last years. These models must satisfy a list of viability conditions (positive definiteness of the effective gravitational coupling, matter stability condition, Solar-system constraints etc.). Some simple examples of F ( R ) -viable models can be found in Ref. [17], where a correction term to the Hilbert-Einstein action is added as F ( R ) = R + f ( R ) , being f ( R ) a generic function of the Ricci scalar which plays the role of an effective cosmological constant. However, since the interest in modified gravity is not limited to the possibility of reproducing the dark energy epoch, but we may find many other applications (for example, related to inflation, quantum corrections in the early stage of the universe, string-inspired gravities, ...), we will carried out our analysis without any particular restriction on the feature of the models out of the wormhole solutions. The Ricci scalar corresponding to (15) reads where, the prime ' means derivative with respect to l . The EoMs become An alternative parametrization of the wormhole in terms of the line element (2) can be obtained by replacing V ( r ) and B ( r ) with four new functions φ ± ( r ) and b ± ( r ) , according to  This corresponds to the fact that the two pairs ( φ ± ( r ) , b ± ( r )) actually describe two different universes joined together at the throat of the wormhole, r 0 . In this case, the Ricci scalar is and the EoMs become  with the requirements that [3] 1. φ ± ( r ) , b ± ( r ) are well defined for all r ≥ r 0 Although written in a different fashion, these requirements simply reproduce the physical conditions of a traversable wormhole given after equation (15). For sake of simplicity, one could add also the condition that the time coordinate is continuous across the wormhole throat, i.e. φ + ( r 0 ) = φ -( r 0 ) ≡ φ ( r 0 ) .", "pages": [ 4, 5, 6 ] }, { "title": "4 Effective energy conditions", "content": "It is well known that in Einstein's gravity (namely, F ( R ) = R ) vacuum wormhole solutions do not exist and the weak energy condition (WEC) must be violated by static wormholes near to the throat [6, 18]. It means that, by introducing the stress energy tensor of matter T µν such that T ν µ = diag ( -ρ, p r , p θ , p ϕ ) , ρ and p r,θ,ϕ being the energy density and the pressure components of matter, respectively, the following relation is violated near to the throat namely In a different theory of gravity, the region around the throat may respect this condition, as it has been explicitly demonstrated in Ref. [9], where a wormhole solution supported by matter which satisfies the WEC has been presented. In such a case, the geometry plays the role of repulsive gravity necessary to construct the traversable wormhole. In principle, any modification to Einstein's gravity of the type under investigation (1) may be considered as an effective fluid (see also [19] for the coupling of General Relativity with a nonlinear fluid), and in the case of vacuum solutions we expect that such effective fluid violates the WEC near to the throat. The field equations of F ( R ) -modified gravity theories may be rewritten in the form where G µν is the Einstein tensor and T (MG) µν is a suitable 'modified gravity' tensor which encodes the gravity modification. Since for the metric (2) with relations (18) the null zero-components of the Einstein tensor are one has that the EoMs (20) correspond to as a consequence of the summation of the two EoMs. We can now define an effective energy density ρ eff and effective pressures p r,/rho1,ϕ eff given by modified gravity in the form T (MG) ν µ = diag( -ρ eff , p r eff , p /rho1 eff , p ϕ eff ) . Let us suppose to have found a wormhole solution characterized by b ( r ) and φ ( r ) . At the throat r 0 , such that b ( r 0 ) = r 0 , from the EoMs we get We use the suffix '0' for all quantities evaluated on the throat. Then, by taking the derivatives of the EoMs, we also have By combining Eq. (4) and Eq. (28) we get From above relations (in particular, by using Eq. (28)) we get and and, as a consequence, according with Ref. [12] in the topological case k = 1 . Now, by using (28), we have and, due to the fact that for traversable wormholes b ' ( r 0 ) < 1 , the WEC (21) is violated. Some important remarks on energy conditions in F ( R ) -gravity theories can be found in Ref. [20].", "pages": [ 6, 7, 8 ] }, { "title": "5 Solutions with constant curvature", "content": "In this Section, at first we will consider the simple case of solutions with R = 0 with the possibility to reproduce vacuum wormhole solutions. Then, a generalization to the case of constant Ricci scalar different to zero will be investigated. If F ( R ) = Rg ( R ) such that lim R → 0 g ( R ) = 0 , it is easy to see that the EoMs trivially are satisfied for R = 0 . We remark that this ansatz includes solutions of a a large class of interesting F ( R ) -gravity models, in particular Lagrangian of the type F ( R ) ∝ R n , n ≥ 2 . This kind of terms based on the power law of the Ricci scalar has been often studied in literature (some important features are related to the possibility to support the early time inflation, to protect the theory against divergences and singularities...). In addition, this form of the Lagrangian possesses the Schwarzshild solution and the important class of black hole Clifton-Barrow solutions. In some sense we can say that these solutions are trivial in the measure that is the Schwarzschild black hole.", "pages": [ 8 ] }, { "title": "Solution n.1:", "content": "  where ˜ r is a scale, q a dimensionless parameter and C 0 a positive integration constant. It is not difficult to show that requirements from 1. to 4. above are fulfilled by this solution for any k if and only if q > -4 . If this is the case, the wormhole throat is located at", "pages": [ 8 ] }, { "title": "Solution n.2:", "content": " where ˜ r is a dimensional parameter and C 1 the integration constant of the solution. Solution n.3: only if k = 1 , referring to the metric form (15), where r 0 explicitly represents the wormhole throat, Φ 0 > 0 is an opportune dimensionless constant of integration which basically determine the flow of time at the throat l → 0 and at infinity l →±∞ . Let us consider now a generalization of solution (36) to the case of constant R = R 0 , but with R = 0 . /negationslash Solution n.4: By putting R = R 0 in (16), it is easy to see that for the topological case k = 1 one possibility is given by where Φ 0 , 1 are constants and for R 0 = 0 one recovers solution (36). Here, one important remark is in order. Since the metric parameter exp[ φ ( l )] must be different to zero for any point of the space time, we must require R 0 < 0 , such that It represents a wormhole solution with constant and negative curvature R 0 . Lagrangians of the type F ( R ) ∝ ( R -R 0 ) n , n ≥ 2 possess this kind of solution (in this case, F ( R 0 ) = F ' ( R 0 ) = 0 and the EOMs are trivially satisfied). It is interesting to note that this models also have the topological Schwarzschild-de Sitter solution (2) with V ( r ) = B ( r ) = k -C/r -Λ r 2 / 3 , where C is an integration constant and Λ is given by Λ = R 0 / (4 -2 n ) > 0 , since in this case the two EOMs (8) are equal and satisfied (remember that R = 4Λ on Schwarzschild-de Sitter solution).", "pages": [ 8, 9 ] }, { "title": "6 Solutions with non constant curvature", "content": "In this Section, we will furnish the formalism and the implicit form of the F ( R ) -gravity models where vacuum wormhole solutions with non constant curvature can be realized. Despite to the fact that the explicit forms of the models are hard to be derived, they may be accessible via numerical analysis. With reference to equations (15), (16) and (17), let us re-write the EoMs more explicitly, as:  where it is understood that φ and r are functions of l . Subtracting one from the other and integrating, we get 0 F R being an integration constant. Summing up the two EoMs, we have that is expression that can be assumed as the implicit definition of ∆( l ) . On the other hand, and then where 0 F is an integration constant. Combining (40), (43) and (44), we find that Note that [ π ( l )] = [1 /l ] and [∆( l )] = [1 /l 2 ] . Thus, because of (44) and (45), one has and also 0 ∆ is an integration constant. It is possible to verify that, by using (45), this expression is consistent with (40) provided that the integration constants 0 F R , 0 F/ 0 ∆ be equal. We stress that up to this point the description is completely model independent. To describe a wormhole, we need to consider some well-defined, twice differentiable function r ( l ) with at least one minimum. Mimicking the General relativity standard wormhole, we could think to look for a similar function among the two-parameter family with n integer and greater than 1 and a m positive rational number satisfying the EoMs (we recover (36) for n = 1 ). This solution would represent a wormhole with the throat placed at l = 0 and an asymptotic behavior strongly dependent on the values of ( n, m ) but presumably far from asymptotic flatness. It turns out that The scalar curvature (16) reads Then, for π ( l ) , ∆( l ) one derives and in principle one can solve Eq. (45). Let us summarize the result. The topological wormhole solutions (15) in empty space-time with r ( l ) in the (appropriate) form of (47) can be realized in F ( R ) -gravity consistently with Eq. (45), which is in fact a differential equation for φ ( l ) . Thus, the implicit form of the model is given by (46). In principle, given the Ricci scalar as a function of l , one can try to reconstruct the models possessing these solutions. However, it is not possible to solve Eq. (45) in an analytical way, and some specific choice of ( n, m ) in (47) and of the topology must be done. Thus, by introducing some boundary conditions, numerical calculations might be implemented in this formalism.", "pages": [ 9, 10 ] }, { "title": "7 Conclusions", "content": "In this paper, we have considered topological wormhole solutions in F ( R ) -gravity, motivated by the popularity of this kind of modifications to Einstein's gravity and by the importance to recover such a kind of objects in their framework. Since in General Relativity traversable wormholes have to been supported by a matter source which violates the energy conditions in a region around the throat, it may be interesting to see if empty traversable wormholes exist in modified theories of gravity, where the geometry plays the role of repulsive gravity. At this regard, we explicitly show that effective energy conditions are violated on the throat in F ( R ) -gravity, generalizing the results already present in literature to the topological case. The formalism of traversable wormhole metric in F ( R ) -gravity has been derived by using a method based on the Lagrangian multipliers, which permits to deal with a system of ordinary differential equations. The metric is presented in two suitable equivalent forms. Despite to the fact that F ( R ) -black hole solutions can be easily reconstructed by starting from the metric, vacuum wormhole solutions are much more difficult to be found, except for the case of null (or constant) Ricci scalar. For this case, we have found several solutions. We would like to stress that this kind of solutions can be realized by a large class of F ( R ) -models: for example, this is the case of Lagrangians of the type F ( R ) ∝ R n , n ≥ 2 , which are used in inflationary scenario in the attempt to reproduce the early-time acceleration, and as a consequence it may be interesting to investigate the possibility of having wormholes in primordial universe. In the last part of the paper, we had a look to wormhole solutions with non-zero curvature and implicit form of F ( R ) -models which realize these solutions is derived. Here, exact solutions cannot be found in analytical way. However, numerical calculations might be implemented in the formalism, an interesting task for future works. We end with the following consideration. F ( R ) -gravity not only represents a possible alternative to cosmological constant to explain current acceleration of the universe, but it may be understood also as an effective action coming from a still unknown quantum theory of gravity. In this sense, F ( R ) would take into account quantum corrections to classical theory, corrections which probably should also account for wormhole production (e.g. [21]). In this respect, the question posed in this paper is all but useless, and it might be of interest to study what is the most important contribution in primordial wormholes creation: whether modified gravity or quantum effects of GUTs, as investigated in Ref. [22].", "pages": [ 11 ] }, { "title": "Acknowledgments", "content": "We thank K. A. Bronnikov for valuable comments and suggestions. RDC wishes to acknowledge with gratitude the financial support and friendly hospitality of the Interdisciplinary Laboratory for Computational Science (LISC), FBK-CMM, Trento - Italy where part of this work has been done.", "pages": [ 11 ] }, { "title": "References", "content": "(2008) [arXiv:0708.0432 [hep-th]]; E. Bellini, R. Di Criscienzo, L. Sebastiani and S. Zerbini, Entropy 12 , 2186 (2010) [arXiv:1009.4816 [gr-qc]]; T. Clifton and J. D. Barrow, Phys. Rev. D 72 , 103005 (2005) [gr-qc/0509059]; T. Clifton, Class. Quant. Grav. 23 , 7445 (2006) [gr-qc/0607096]; R. -G. Cai, L. -M. Cao, Y. -P. Hu and N. Ohta, Phys. Rev. D 80 , 104016 (2009) [arXiv:0910.2387 [hep-th]]; G. J. Olmo and D. Rubiera-Garcia, arXiv:1301.2091 [gr-qc]; A. Sheykhi, Phys. Rev. D 86 , 024013 (2012) [arXiv:1209.2960 [hep-th]]; S. G. Ghosh and S. D. Maharaj, arXiv:1208.3028 [gr-qc]. M. De Laurentis and S. Capozziello, arXiv:1202.0394 [gr-qc]; S. H. Hendi and D. Momeni, Eur. Phys. J. C 71 , 1823 (2011) [arXiv:1201.0061 [gr-qc]]; S. H. Mazharimousavi and M. Halilsoy, Phys. Rev. D 84 , 064032 (2011) [arXiv:1105.3659 [gr-qc]]; T. Moon, Y. S. Myung and E. J. Son, Gen. Rel. Grav. 43 , 3079 (2011) [arXiv:1101.1153 [gr-qc]]; V. Faraoni, arXiv:1005.5397 [gr-qc]; V. Faraoni, arXiv:1005.2327 [gr-qc]; A. Aghamohammadi, K. Saaidi, M. R. Abolhasani and A. Vajdi, Int. J. Theor. Phys. 49 , 709 (2010) [arXiv:1001.4148 [gr-qc]]; C. S. J. Pun, Z. Kovacs and T. Harko, Phys. Rev. D 78 , 024043 (2008) [arXiv:0806.0679 [gr-qc]]; T. Multamaki, A. Putaja, I. Vilja and E. C. Vagenas, Class. Quant. Grav. 25 , 075017 (2008) [arXiv:0712.0276 [gr-qc]]; S. i. Nojiri and S. D. Odintsov, arXiv:1301.2775 [hep-th]; L. Sebastiani, D. Momeni, R. Myrzakulov and S. D. Odintsov, arXiv:1305.4231 [gr-qc].", "pages": [ 12 ] } ]

[ { "title": "Ricardo E Gamboa Sarav'ı 1 , 2 , Marcela Sanmartino 3 and Philippe Tchamitchian 4", "content": "[email protected], [email protected], E-mail: [email protected] Abstract. We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general ( n +2)-dimensional static and spherically symmetric spacetimes. They are related to properties of the underlying spatial part of the wave operator, one of which being the standard essentially selfadjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially selfadjoint, but it does satisfy a weaker property that we call here quasi essentially self-adjointness , which is enough to ensure the desired wellposedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Hawking and Penrose have shown that, according to general relativity, there must exist singularities of infinite density and space-time curvature in many physically reasonable situations. This phenomenon occurs in the big bang scenery at the very beginning of time, and it would be an end of time for sufficiently massive collapsing bodies (see, for example, [1] and references therein). At these singularities all the known laws of physics and our ability to predict the future would break down. However, in the case of black holes, any observer who remained outside the event horizon would not be affected by this failure of predictability, because neither light nor any other signal could reach him from the singularity. This notable feature led Penrose to propose the weak cosmic censorship hypothesis : all singularities produced by gravitational collapse occur only in places, like black holes, where they are hidden from outside view by an event horizon [2]. The strong version of the cosmic censorship hypothesis states that any physically realistic spacetime must be globally hyperbolic [3]. The concept of global hyperbolicity was introduced for dealing with hyperbolic partial differential equations on a manifold [4]. A spacetime is said to be globally hyperbolic if, given any two of its points, the set of of all causal curves joining these points is compact (in a suitable topology). Only in this case there is a Cauchy surface whose domain of dependence is the entire spacetime. This is a reasonable condition to impose, for example, to ensure the existence and uniqueness of solutions of hyperbolic differential equations [4, 5]. Nevertheless, the relevant physical condition to assure predictability is not global hyperbolicity, but the well-posedness of the field equations. Indeed, there are many examples of spacetimes that are not geodesically complete and violate cosmic censorship, but where there is still a well-posed initial-value problem for test fields. Global hyperbolicity is sufficient, but not necessary for this. This suggests that, in more general situations, we could find a weaker condition to replace the notion of global hyperbolicity by making direct reference to test fields [6, 7, 8]. The above considerations motivate a deeper study of the well-posedness of the initial-value problem for fields in more general singular spacetimes. This paper is a continuation of a previous one [9], tackling the well-posedness of Cauchy problem for waves in static spacetimes. This subject has been launched by Wald in [6], and further developed by, among others, the authors of references [7, 10, 11]. The propagation of waves is, in such spaces, described by a classical equation of the form where A is a selfadjoint extension of a given symmetric and positive operator A which reflects the underlying geometry. Our motivation relies on the following observation: although A may not be essentially selfadjoint ( e.s.a. ), boundary conditions are not necessary to construct A in some geometries of interest. Such a situation arises when, even if A has many selfadjoint extensions, only one has its domain included in the energy space naturally associated to A . Here we call quasi essentially selfadjoint ( q.e.s.a. ) this property. We have shown in [9] that operators A given by propagation of massless scalar fields in static spacetimes with naked timelike singularities may be q.e.s.a. but not e.s.a. . Thus, in such situations, demanding the finiteness of the energy is enough to select one selfadjoint extension of A , and only one; in addition, we proved that the solutions of the wave equation may have a non trivial trace at the boundary of the geometrical domain, even though this trace is not imposed by any boundary condition at all. This phenomenon never happens with e.s.a. operators. Here we deeply examine the case of general ( n +2)-dimensional static and spherically symmetric spacetimes. More precisely, the concrete setting is the following. The domain is of the form I × M , where I ⊂ (0 , + ∞ ) is an open interval and M is a compact, oriented Riemannian manifold without boundary. The operator A is defined on C ∞ 0 ( I × M ) as where ∆ M is the Laplace-Beltrami operator on M , and a , b , c and d are suitable positive coefficients only depending on the radial variable z ∈ I . No condition is prescribed on the coefficients at the boundary of the domain. For this class of operators we fully characterize e.s.a. and q.e.s.a. properties. More precisely, under rather general conditions on the coefficients, we give a necessary and sufficient condition for q.e.s.a. depending only on the integrability of the function ( 1 b ( z ) + d ( z ) + a ( z ) ) at the boundary of I . We also give a necessary and sufficient condition for e.s.a. , in this case the condition depends also on the integrability of the functions a ( z ) and β ( z ) 2 a ( z ) at the boundary of I , where β ( z ) is a particular solution of the ordinary differential equation -( b ( z ) β ' ( z ) ) ' + d ( z ) β ( z ) = 0. We then apply this analysis to scalar fields propagating in static spherically symmetric spacetimes of arbitrary dimension, solutions of the Einstein equations with cosmological constant and matter satisfying the dominant energy condition or vacuum. The criteria for e.s.a. and q.e.s.a. on the coefficients of the operator A are then translated into criteria on the components of the metric tensor. This provides a systematic procedure to analyze the situations where boundary conditions are, or are not, necessary for the Cauchy problem to be well-posed. A significant physical result is stated in theorem 5.5: in the outer region of a static, spherically symmetric and asymptotically flat spacetime where the dominant energy condition holds, the operator A is essentially selfadjoint, i.e. the Cauchy problem is well-posed without any boundary conditions, if, and only if, an observer at infinity measures that it takes an infinite time to a photon to reach the boundary. Finally, we directly apply the developed theory to the discussion of some exact vacuum solutions as explicit examples. We discuss the ( n +2)-dimensional Minkowski spacetime with a removed spatial point and the higher-dimensional generalization of Schwarzschild and Reissner-Nordstrom geometries; we systematically describe the situations where boundary conditions are, or are not, necessary for the Cauchy problem to be well-posed. The outline of the paper is as follows. Section 2 is devoted to abstract results on e.s.a. and q.e.s.a. properties. In section 3 we completely characterize e.s.a. and q.e.s.a. properties of the operator given in (1). We show, in section 4, the well-posedness of the Cauchy problem when the operator A is q.e.s.a. but not necessarily e.s.a. . In section 5 we apply our results to the study of propagation of scalar fields in general ( n + 2)-dimensional static and spherically symmetric spacetime with n ≥ 1. We close by discussing the examples in section 6.", "pages": [ 1, 2, 3 ] }, { "title": "2. Quasi essentially and essentially selfadjointness", "content": "Let Ω ⊂ R n +1 be a Lipschitz domain ‡ and H a Hilbert space such that C ∞ c (Ω) is dense in H , where C ∞ c (Ω) is the space of the restrictions to Ω of C ∞ 0 ( R n +1 ). We consider an unbounded symmetric definite positive operator A , whose domain is C ∞ 0 (Ω). We assume the existence of a Hilbert space E , continuously embedded in H , and a related bilinear symmetric form b with domain E having the following properties: The reader should note that A is defined only on C ∞ 0 (Ω), and that consequently the relation between the form b and the operator A is only stated for functions in C ∞ 0 (Ω) as well, although C ∞ c (Ω) is dense in both spaces H and E . This is motivated by the difficulties arising with boundary conditions: whether they must be specified in advance or not is the question we consider in the subsequent theorem 2.2. We will show that there is a 'natural' self-adjoint extension of A , defined without specifying any boundary condition, if and only if C ∞ 0 (Ω) is dense in E . We will also show that this density property is always true when A is essentially self-adjoint, but may occur even when A is not. Various examples are given at the end of the paper. Definition 2.1 We shall say that A , any given selfadjoint extension of A , is of finite energy when D ( A ) ⊂ E , with continuous injection. Calling E 0 the closure of C ∞ 0 (Ω) in E , we have the following result: Theorem 2.2 Under these hypotheses we have:", "pages": [ 4 ] }, { "title": "Proof:", "content": "hand, if T ( r → ∞ r - m √ a ( r ) b ( r ) ∞ lemma 5.4 we have ∫ r -m a ( z ) dz = + ∞ . Therefore, it follows from theorem 3.24 that the operator A is q.e.s.a and from theorem 3.25 (i) that the operator A is e.s.a . ∫ + ∞ . And it immediately follows from lemma 5.4 that r -m a ( r ) dr < + ∞ and ∫ r -m 1 b ( r ) dr < + ∞ . On the other hand, since F ( r ) G ( r ) is bounded by lemma 5.1, ∫ r -m d ( r ) dr = m 2 ∫ r -m r n √ F ( r ) G ( r ) dr < + ∞ . Therefore, it follows from theorem 3.24 that the operator A is not q.e.s.a . ∫ r - m ) = + , it follows by (34) that dr = + , and then from Remark 5.6 Note that the boundedness of F ( r ) G ( r ) is only used in the proof of the sufficiency of the condition T ( r → r -m ) = + ∞ , to guarantee that d ( r ) is integrable at r -m . Therefore, for massless fields, since in this case d ( r ) ≡ 0 the theorem follows without invoking any energy condition. Similar results also follow from remark 3.23 and lemma 5.4 at internal horizons.", "pages": [ 30, 31 ] }, { "title": "(ii) Recall that", "content": "We assume first that C ∞ 0 (Ω) is dense in D ( A F ). It is enough to see that D ( A ∗ ) ⊂ D ( A F ). Taking φ 0 ∈ D ( A ∗ ) and η 0 = ( A ∗ + I ) φ 0 , we have for all ψ ∈ C ∞ 0 (Ω) and then, since C ∞ 0 (Ω) is dense in D ( A F ), for all ϕ ∈ D ( A F ) Taking into account that ( A F + I ) -1 is defined on all H , by calling ϕ 0 = ( A F + I ) -1 η 0 ∈ D ( A F ) we have and then Since Im( A F + I ) = H , we have ϕ 0 = φ 0 . It implies D ( A ∗ ) ⊂ D ( A F ) and so A ∗ = A F . Then A is essentially selfadjoint. On the other hand, if C ∞ 0 (Ω) is not dense in D ( A F ), there exists ϕ ∈ D ( A F ) such that A F ϕ = 0 and /negationslash Let us call η = A F ϕ . If η ∈ E , then b ( η, ψ ) = 〈 η, Aψ 〉 = 〈 η, A F ψ 〉 = 0 for all ψ ∈ C ∞ 0 (Ω) and then by density of C ∞ 0 (Ω) in E , b ( η, η ) = 0. Since by hypothesis η = 0, we have η / ∈ E . /negationslash Therefore, we have proved that there exists η ∈ H , such that η ∈ ker( A ∗ ) but η / ∈ E , so A cannot be essentially self adjoint. /square Definition 2.3 Under the preceding hypotheses, the operator A is quasi essentially selfadjoint (q.e.s.a.) if it has only one extension with finite energy. Lemma 2.4 If A is a q.e.s.a. operator, then D ( A F ) = D ( A ∗ ) ∩ E .", "pages": [ 5 ] }, { "title": "3. A characterization of some q.e.s.a. and e.s.a. divergence type operators", "content": "Let M be a Riemannian manifold of dimension n with a metric ( g ij ). We also assume that M is compact, connected, without boundary and with a given orientation. In local coordinates, for u ∈ C ∞ ( M ) the Laplace-Beltrami operator is where g is the determinant of the metric. Let us consider in Ω = (0 , + ∞ ) × M , the operator A given by for all ϕ ∈ C ∞ 0 (Ω), where the functions a, b, c and d satisfy the following hypotheses: Examples will be presented in the two last sections. Let us state in advance that the coefficient d is non vanishing only in the massive case. This is why we will call massless the case d = 0. We define the Hilbert spaces and the energy space where we denote ω M the natural measure in M , and Thus, H and E are Hilbert spaces, equipped with their canonical norms: ‖ ϕ ‖ 2 H = ∫ Ω | ϕ ( z, x ) | 2 a ( z ) dω M dz and ‖ ϕ ‖ 2 E = ‖ ϕ ‖ 2 H + b ( ϕ, ϕ ). The operator A is well defined on C ∞ 0 (Ω) and it is symmetric in H by definition. for ϕ, ψ ∈ C ∞ 0 (Ω). We shall explore when A is a q.e.s.a. operator by using Theorem 2.2. Then the question is to determine under which conditions on the coefficients of A , C ∞ 0 (Ω) is dense in E . A related one is whether C ∞ c (Ω) ∩E is dense in E . positive and small enough ε . And ∫ < + ∞ means that their exists z > 0 such that Theorem 3.2 Let A be the operator defined in (3). Then", "pages": [ 6, 7 ] }, { "title": "Step 1: reduction to the one dimensional case.", "content": "Let { λ k , k ≥ 0 } be the spectrum of -∆ M , with λ 0 = 0 and λ k an increasing sequence, and let ( ψ k ) k ≥ 0 be an associated orthonormal basis of L 2 ( M ). We define, for each k ≥ 0, for u ∈ C ∞ 0 ( (0 , + ∞ ) ) , with the underlying Hilbert space H 0 = L 2 ( (0 , + ∞ ) , a ( z ) dz ) and energy spaces E k = u ∈ H 0 ∩ H 1 loc (0 , + ∞ ) : b k ( u, u ) < + ∞ , where Then we consider the Hilbert spaces E k with their natural norms Lemma 3.3 C ∞ c (Ω) ∩ E (respectively C ∞ 0 (Ω) ) is dense in E if and only if C ∞ c ( [0 , ∞ ) ) ∩ E k (respectively C ∞ 0 ( (0 , + ∞ ) ) is dense in E k for all k ≥ 0 . Proof: Given ϕ ∈ E , it can be decomposed into a sum ϕ = ∑ k ≥ 0 u k ⊗ ψ k , where u k ∈ E k and So, density in E implies density in each E k . For the reciprocal, given ϕ ∈ E we first approximate it by the functions ϕ m = m ∑ k =0 u k ⊗ ψ k , and density in E k for all k ≥ 0 implies that each ϕ m can be approximate by functions of C ∞ (Ω) (respectively C ∞ (Ω)). /square", "pages": [ 7, 8 ] }, { "title": "Step 2: density of compactly supported functions in E 0 .", "content": "Here, for convenience we shall restrict our attention at first to the case k = 0 and d ( z ) ≡ 0. We define E 0 ,c = E 0 ∩ { functions with compact support in [ 0 , + ∞ ) } ,", "pages": [ 8 ] }, { "title": "Proof.", "content": "By (33) we have that a ( r ) b ( r ) = r 2 n . For r ∗ < r < r ∗ < + ∞ , we readily get the inequalities Now, by integrating these expressions between r ∗ and r , we get the result. Observe that by the properties of the functions F and G , under the hypotheses of lemma 5.4 we have Then, if we consider the operator defined by (32) in Ω = ( r -i , r + i ) × S n , we have: Theorem 5.5 For 0 < r -m < ∞ , let A be the operator corresponding to the propagation of a scalar field in Ω = ( r -m , ∞ ) × S n in a static, spherically symmetric and asymptotically flat spacetime where the dominant energy condition holds. The three following statements are equivalent: Or, in other words, A is e.s.a. if and only if a radial photon needs an infinite amount of time to get r -m .", "pages": [ 30 ] }, { "title": "Proof of theorem 3.2", "content": "Let us now prove theorem 3.2 (ii) : if C ∞ 0 (Ω) is dense in E , by lemma 3.3 C ∞ 0 ( (0 , + ∞ ) ) is dense in E k for all k ≥ 0, in particular for k = 0, then by lemma 3.6, we have ∫ + ∞ ( 1 b ( z ) + d ( z ) + a ( z ) ) dz = ∫ 0 ( 1 b ( z ) + d ( z ) + a ( z ) ) dz = + ∞ . Conversely, if ∫ + ∞ ( 1 b ( z ) + d ( z ) + a ( z ) ) dz = ∫ 0 ( 1 b ( z ) + d ( z ) + a ( z ) ) dz = + ∞ , we also have then C ∞ 0 ( (0 , + ∞ ) ) is dense in E k for all k , we can see it changing a ( z ) by d ( z ) + a ( z ) + λ k c ( z ) in all the previous results, and again by lemma 3.3, C ∞ 0 (Ω) is dense in E . The proof of (i) analogously follows. Theorem 3.2 is completely proved. /square Remark 3.7 Under different hypotheses, when the coefficients of the operator A depend on ( z, x ) we have given a characterization of q.e.s.a. operators in [9]. Warning: in page 21 of that reference, the integrand of (43) was mistakenly written as 1 M n +1 ,n +1 ( z, x ) instead of ( M -1 ) n +1 ,n +1 ( z, x ).", "pages": [ 13 ] }, { "title": "Essentially selfadjointness characterization", "content": "The characterization of e.s.a. for the operator A defined in (3) will rely on the realvalued solutions of the O.D.E. on (0 , z ' ) and on ( z ' , + ∞ ). A typical case is when ∫ 0 a ( z ) dz < + ∞ , but ∫ + ∞ a ( z ) dz = + ∞ . Then since we may assume A to be q.e.s.a. (otherwise it cannot be e.s.a. ), we have ∫ 0 ( 1 b ( z ) + d ( z ) ) dz = + ∞ . In such a case, we will show that there is a unique solution of (9), denoted by α , such that  Then, we define β ( z ), z ∈ (0 , z ' ), by Note that, by construction, β is another solution of (9) in (0 , z ' ). We shall prove that: A is e.s.a. if and only if ∫ 0 β ( z ) 2 a ( z ) dz = + ∞ . In the case where the role of 0 and + ∞ are exchanged, the result is similar. We will show that there exists a unique function α such that  Then, we define β ( z ), z ∈ ( z ' , + ∞ ), by and we shall prove that: A is e.s.a. if and only if ∫ + ∞ β ( z ) 2 a ( z ) dz = + ∞ . Note that, when d ( z ) ≡ 0 the problem considerably simplifies since, in this case, α ≡ 1 and β ( z ) turns out to be either β 0 ( z ) = ∫ z ' z 1 b ( z ) dz or β 0 ( z ) = ∫ z z ' 1 b ( z ) dz respectively. Notation 3.8 We denote ( α ( z ) , β ( z ) ) the above couples of solutions of (9); the context will indicate whether z ∈ (0 , z ' ), in which case ( α ( z ) , β ( z ) ) are given by (10) and (11), or z ∈ ( z ' , + ∞ ), where ( α ( z ) , β ( z ) ) are given by (12) and (13). With this notation, the result is the following. Theorem 3.9 Assume the operator A given in (3) to be q.e.s.a., that is to say There are four cases: On well-posedness of the Cauchy problem . . . Remark 3.10 Take care of the uniqueness of α (and thus the meaningfulness of the definitions above): it holds when ∫ 0 ( 1 b ( z ) + d ( z ) ) dz = + ∞ or ∫ + ∞ ( 1 b ( z ) + d ( z ) ) dz = + ∞ , according to where the variable z lives.", "pages": [ 13, 14, 15 ] }, { "title": "Preliminary step: study of solutions of (9)", "content": "Lemma 3.11 Let u ( z ) be a solution of (9) in some interval I ⊂ (0 , + ∞ ) . Then the function b ( z ) u ( z ) ' u ( z ) is increasing in I .", "pages": [ 15 ] }, { "title": "(9), we obtain", "content": "Since β is positive by construction, it turns out to be decreasing in (0 , z ' ), with Hence, there exists a constant C such that β ( z ) ≥ C if z ≤ z ' / 2, and we obtain The lemma is proved. Lemma 3.13 has an analogous counterpart near + ∞ , which is the following. and /square", "pages": [ 18 ] }, { "title": "Lemma 3.14 Let z ' > 0 be chosen.", "content": "(i) There exists at least one solution α ( z ) of (9), in the interval ( z ' , + ∞ ) , such that and This solution is positive and decreasing in ( z ' , + ∞ ) , satisfying", "pages": [ 19 ] }, { "title": "Main step: e.s.a. characterization in dimension one", "content": "Let us consider now the operator defined as in (4) with Lemma 3.16 If ∫ 0 a ( z ) dz = ∫ + ∞ a ( z ) dz = + ∞ , A 0 is an e.s.a. operator.", "pages": [ 19 ] }, { "title": "Final step: reduction to the one-dimensional case", "content": "Defining the operators A k as in (4), i.e., we have the following result: Lemma 3.20 A is an e.s.a. operator if and only if for all k ≥ 0 A k is an e.s.a. operator.", "pages": [ 22 ] }, { "title": "Proof of theorem 3.9", "content": "and ∫ + ∞ ( a ( z ) + λ k c ( z ) ) dz = + ∞ , for all k ≥ 0. Therefore A k is e.s.a. by lemma 3.16 with a changed in a + λ k c ( z ), and by lemma 3.20 A is e.s.a. . In the cases (ii) , (iii) and (iv) if A is e.s.a. it follows by lemma 3.20 that in particular A 0 is e.s.a. . Then lemmas 3.17, 3.18 and 3.19 give the result. For the converse, let us take the case (ii) . If A is not e.s.a. , by lemma 3.20 there exists k ≥ 0 such that A k is not e.s.a. . Then by lemma 3.17 where β k is the solution of ( ' ) ' Decompose - b ( z ) u ( z ) +( c ( z ) λ k + d ( z )) u ( z ) = 0 /square on (0 , z ' ) with Cauchy data u ( z ' ) = 0 and u ' ( z ' ) = -1 b ( z ' ) . A classical comparison principle, applied to the functions β k and β , defined in (11), give us 0 ≤ β ≤ β k on (0 , z ' ) . Then (17) implies as desired. The other cases are analogous. Theorem 3.9 is completely proved. /square Remark 3.21 The precise definition of the function β ( z ) is needed only for the sufficiency of the condition for A to be e.s.a. . This is not used in the reciprocal, where the 'masslessβ ' would have worked as well (see (15)). But, for the sufficiency, if we choose u ( z ) = β 0 ( z ) η ( z ) in lemma 3.17, with η ∈ C ∞ ([0 , + ∞ )), η = 1 near 0 and η = 0 for z ≥ z ' 2 , then and this belongs to H 0 only when This gives a necessary and sufficient condition for e.s.a. in terms of β 0 ( z ) only, not β ( z ), when d ( z ) a ( z ) is bounded: Corollary 3.22 When d ( z ) a ( z ) is bounded near 0 , ∫ 0 a ( z ) dz < + ∞ and ∫ + ∞ a ( z ) dz = + ∞ , A is e.s.a. if and only if ∫ 0 β 0 ( z ) 2 a ( z ) dz = + ∞ . There are similar statements in the other cases.", "pages": [ 22, 23 ] }, { "title": "Remark 3.23 The previous results in the domain ( z 0 , z 1 ) × M", "content": "In some relevant examples one is lead to consider Ω = ( z 0 , z 1 ) × M , 0 ≤ z 0 ≤ z 1 ≤ ∞ , and a differential operator A defined as in (3) by for all ϕ ∈ C ∞ 0 (Ω), where the functions a , b , and c satisfy the following hypotheses: On well-posedness of the Cauchy problem . . . The previous results straightforwardly generalize to such a case. For the convenience of the reader, we state the two main theorems. Theorem 3.25 We assume A is a q.e.s.a. operator, There are four cases: z Atypical situation where these results apply is when ∫ 1 ( 1 b ( z ) ( a ) + ∞ but ∫ z 0 ( 1 b ( z ) + d ( z ) + a ( z ) ) dz < + ∞ . Then C ∞ 0 (Ω) is not dense in E , but the only non trivial linear forms continuous on E , vanishing on C ∞ 0 (Ω), are supported on { z 0 }× M . This means that a boundary condition must be chosen at z = z 0 , but not at z = z 1 . + d z ) + ( z ) dz = Moreover if we have, for example, ∫ z 1 a ( z ) dz = + ∞ , the selfadjoint extension ˜ A , defined from A with an appropriate boundary condition at z = z 0 , will be unique. In particular, considering null Dirichlet boundary condition, ˜ A will be the selfadjoint extension of A constructed from the restriction of the bilinear form to E 0 .", "pages": [ 23, 24 ] }, { "title": "4. Well-posedness of the Cauchy problem", "content": "Let A and Ω be as in the previous section. We assume A to be at least q.e.s.a. but not necessarily e.s.a.; we denote in the same way its unique selfadjoint extension with finite energy. We take functions f ∈ E and g ∈ H and consider the Cauchy problem  Theorem 4.1 Under the hypotheses above, the problem (P) has a unique solution and there exists a constant C > 0 such that In this case, the energy is well-defined and conserved:", "pages": [ 25 ] }, { "title": "(P) and we have", "content": "We consider each term separately, obtaining for the first one and for the second one (see for instance [12]) Now, by (19), adding (20) and (21), we have for all t > 0 Again, by a density argument as before, this result remains true when f ∈ E and g ∈ H . /square", "pages": [ 26 ] }, { "title": "5. Propagation of classical scalar fields in static spherically symmetric spacetimes", "content": "We consider a ( n +2)-dimensional static and spherically symmetric spacetime with n ≥ 1 and metric signature ( -+ . . . +). Due to the required isometries the more general line element can be written as where d/lscript 2 S n is the metric on the unit n -sphere S n and r in (0 , + ∞ ). For a nondegenerate Lorentzian metric g ab , (22) makes sense only for those values of r such that 0 < F ( r ) G ( r ) < + ∞ . On the other hand, since g ab ( ∂ t ) a ( ∂ t ) b = -F , the Killing vector field ∂ t is timelike only in the region F ( r ) > 0, and so spacetime is static only in this region. Therefore, without loss of generality, from now on we shall restrict ourselves to the region where F ( r ) and G ( r ) are both finite and positive. In addition we shall assume that F and G are such that the condition 0 < F ( r ) , G ( r ) < + ∞ holds in a finite union of disjoint non empty open subintervals ( r -i , r + i ) of (0 , + ∞ ) and ( r -m , + ∞ ), we can find coordinates such that lim r → + ∞ F ( r ) = lim r → + ∞ G ( r ) = 1. F, F ' , G ∈ C 1 ( m ⋃ i =1 ( r -i , r + i )). If the spacetime is asymptotically flat , in the outer region Due to the required symmetries the more general energy-momentum tensor can be written as where ρ ( r ) is the energy density, and p r ( r ), p θ ( r ) are the principal pressures. We shall assume that ρ ( r ) is bounded and the dominant energy condition + is satisfied, which, in this case, is equivalent to From (22) and (23) we get that Einstein's equations, i.e., G ab + Λ g ab = 8 πT ab , become where Λ is the cosmological constant. Furthermore, the local energy-momentum conservation ( ∇ a T ab = 0) gives Of course, due to Bianchi's identities, (25)-(28) are not independent. These are a system of three linear independent ODE 's and, in order to find the five unknown functions F ( r ), G ( r ), ρ ( r ), p r ( r ) and p θ ( r ), we have to provide equations of state relating the functions ρ ( r ), p r ( r ) and p θ ( r ). From (25) and (26) we can write down a more handleable set of two equivalent independent equations which in the vacuum cases, leads readily to the solution. Indeed, if we for instance set ρ ( r ) = -p r ( r ) = p θ ( r ), from (28) we immediately get that where the constant C 1 must be positive by (24). Then, we find from (29) that where C 2 is a new arbitrary constant. And (30) immediately gives F ( r ) G ( r ) = C 3 , and we can always set the constant C 3 = 1 by scaling the time. This family of solutions, depending on three parameters, includes the higher-dimensional generalization of Schwarzschild, de Sitter and Reissner-Nordstrom geometries. For future use, we shall prove the following result. Lemma 5.1 If 0 < F ( r ) , G ( r ) < + ∞ in some interval ( r -i , r + i ) , then", "pages": [ 26, 27, 28 ] }, { "title": "6.1. ( n +2) -dimensional punctured Minkowski spacetime", "content": "Here we consider the flat ( n + 2)-dimensional Minkowski spacetime with a removed spatial point. We chose the origin of coordinates at this point and then the line element can be written as where -∞ < t < + ∞ and 0 < r < + ∞ . This spacetime has a time-like singular boundary along the t axis. In this case, Ω = (0 , ∞ ) × S n and F ( r ) = G ( r ) = 1, so the coefficients in (33) are a ( r ) = b ( r ) = r n , c ( r ) = r n -2 and d ( r ) = m 2 r n . The operator A in (32) turns out to be which formally is nothing but -∆+ m 2 . We turn now to explore whether A is an e.s.a. operator too. Taking into account that d ( z ) /a ( z ) = m 2 , ∫ 0 a ( z ) dz = ∫ 0 r n dz < + ∞ and ∫ + ∞ a ( z ) dz = + ∞ , we can apply corollary 3.22. Now, for n ≥ 1, we have that ∫ + ∞ a ( r ) dr = + ∞ and ∫ 0 dr b ( r ) = + ∞ . Then it immediately follows from theorem 3.2 that A is a q.e.s.a. operator for every m 2 ≥ 0 and every n ≥ 1. Now, for 0 < r 1 < + ∞ , we have Thus, if and only if n = 1 , 2. Therefore, it immediately follows from corollary 3.22 that A is an e.s.a. operator only if n ≥ 3. This is a well known result, see for instance [13, 14]. . 6.2. ( n +2) -dimensional anti-Schwarzschild ( M < 0 ) spacetime Here we consider the ( n +2)-dimensional spacetime with line element where -∞ < t < + ∞ , 0 < r < + ∞ , r s is a positive constant and n ≥ 2 ∗ . This spacetime has a naked timelike singularity at r = 0 where some components of the Weyl tensor diverge. In this case, Ω = (0 , ∞ ) × S n and we get from (33) that the coefficients of the operator A are We get therefore For m = 0 and n = 2, we have already proved in [9] that A is not an e.s.a. operator. Here, we shall analyze the general case. Then it immediately follows from theorem 3.2 that A is a q.e.s.a. operator for every m 2 ≥ 0 and every n ≥ 2. We first consider the case m = 0. Taking into account that we can apply corollary 3.22. For 0 < r < r s , we have and Thus, in the massless case, A is not an e.s.a. operator for every n ≥ 2 thanks to the corollary 3.22. For m 2 > 0 we cannot apply corollary 3.22 since d ( z ) /a ( z ) is not bounded near 0. Nevertheless, the ordinary differential equation (9), satisfied by the function α ( z ) of lemma 3.13, becomes in this case and a straightforward computation shows that near 0. Furthermore, since by lemma 3.13 α ( r ) is positive and increasing in (0 , r s ), and by definition α ( r s ) = 1, we get that 0 < α (0) < 1. Therefore and It follows from theorem 3.9 (ii) that A is not an e.s.a. operator for every n ≥ 2 and m 2 ≥ 0.", "pages": [ 31, 32, 33 ] }, { "title": "Remark 6.1 Note that the estimate", "content": "/negationslash when α (0) = 0 , also gives a necessary and sufficient condition for e.s.a. in terms of β 0 ( z ) only. /negationslash For analytic b ( z ) and d ( z ) , as in our example, α (0) = 0 if one of the roots of the indicial polynomial of (9) is zero and the other non positive, which requires that 6.3. ( n +2) -dimensional Schwarzschild-Tangherlini spacetime Here we consider the ( n +2)-dimensional spacetime with line element where r s is a positive constant, -∞ < t < + ∞ , 0 < r < r s or r s < r < + ∞ and n ≥ 2. This spacetime has a spacelike irremovable singularity at r = 0 where some components of the Riemann tensor diverge and an event horizon at r = r s , the latter may be removed by introducing suitable coordinates and extending the manifold to obtain a maximal analytic extension [15]. As already mentioned, our wave formulation only makes sense in the static region ( r s < r < + ∞ ), and we will use it to explore the properties of the wave equation (31) in this region. Thus, we consider the operator A given by (32) in Ω = ( r s , ∞ ) × S n , and we see from (33) that Now, we get from (34) that Therefore, it immediately follows from theorem 5.5 that A is an e.s.a. operator in Ω = ( r s , ∞ ) × S n for every n ≥ 2 and any m 2 ≥ 0, and the Cauchy problem is wellposed without requiring any boundary condition at the event horizon. On well-posedness of the Cauchy problem . . . 6.4. ( n +2) -dimensional Reissner-Nordstrom spacetime Here we consider the ( n +2)-dimensional spacetime with line element where r s and q 2 are positive constants and n ≥ 2 /sharp . If q 2 > r 2 s the metric is non-singular everywhere except for the timelike irremovable repulsive singularity at r = 0. If q 2 ≤ r 2 s , the metric also has singularities at r + and r -, where r n -1 ± = ( r n -1 s ± √ r 2 n -2 s -q 2 n -2 ) / 2; it is regular in the regions defined by ∞ > r > r + , r + > r > r -and r -> r > 0 (if q 2 = r 2 s only the first and the third regions exist). As in the Schwarzschild case, these singularities may be removed by introducing suitable coordinates and extending the manifold to obtain a maximal analytic extension [16, 17]. The first and the third regions are both static, whereas the second region (when it exists) is spatially homogeneous but not static. We shall study the properties of the wave equation in the static regions. For convenience we shall analyze separately the three cases. Note that, in the three cases this spacetime is asymptotically flat. 6.4.1. Case q 2 > r 2 s This spacetime has only a naked timelike irremovable repulsive singularity at r = 0. In this case, we consider the operator A given by (32) in Ω = (0 , ∞ ) × S n , and from (33) we have Then Hence it follows from theorem 3.2 (ii) that A is not even a q.e.s.a. operator in this case, for every n ≥ 2 and any m 2 ≥ 0. Therefore, in contrast to the anti-Schwarzschild case, in order to have a well-possed Cauchy problem a boundary condition at the singularity must be given. 6.4.2. Case r 2 s = q 2 (extreme case) This spacetime also has a removable singularity at r ∗ = 2 -1 n -1 r s . In this case, we consider the operator A given by (32) in two regions Ω = (0 , r ∗ ) × S n or Ω = ( r ∗ , ∞ ) × S n . We get from (33) that We first consider the outer region ( r ∗ < r < + ∞ ). In this case, we get from (34) that ( ) Therefore, it follows from theorem 5.5 that A is an e.s.a. operator in Ω = ( r ∗ , ∞ ) × S n for every n ≥ 2 and any m 2 ≥ 0, and the Cauchy problem is well-posed without requiring any boundary condition at the event horizon. Regarding the inner region 0 < r < r ∗ , we get that However, we have Hence it follows from theorem 3.24 that A is not even a q.e.s.a. operator, for every n ≥ 2 and any m 2 ≥ 0. ( ) so it follows from remark 3.23 that in order to have a well-posed Cauchy problem in Ω = (0 , r ∗ ) × S n a boundary condition at the singularity ( r = 0) must be given but not at the horizon ( r = r ∗ ). 6.4.3. Case r 2 s > q 2 This spacetime has, besides the timelike irremovable repulsive singularity at r = 0, two removable singularities at r + and r -. In this case, we consider the operator A given by (32) in two regions Ω = (0 , r -) × S n or Ω = ( r + , ∞ ) × S n , by abuse of notation we call A these two different operators. From (33) we can write We first consider the outer region ( r + < r < + ∞ ). In this case, we get from (34) that ( ) ( ) Therefore, it follows from theorem 5.5 that A is an e.s.a. operator in Ω = ( r + , ∞ ) × S n for every n ≥ 2 and any m 2 ≥ 0, and the Cauchy problem is well-posed without requiring any boundary condition at the event horizon. Regarding the inner region 0 < r < r -, we get Hence it follows from theorem 3.24 that A is not even a q.e.s.a. operator, for every n ≥ 2 and any m 2 ≥ 0. On well-posedness of the Cauchy problem . . . However, we have so it follows from remark 3.23 that in order to have a well-posed Cauchy problem in Ω = (0 , r -) × S n a boundary condition at the singularity ( r = 0) must be given but not at the horizon ( r = r -).", "pages": [ 33, 34, 35, 36 ] } ]

[ { "title": "John Schliemann", "content": "Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany E-mail: [email protected] Abstract. It is shown that the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator. This result relies on the fact that (i) the volume operator couples only neighboring states of its standard basis, and (ii) its matrix elements show a unique maximum as a function of internal angular momentum quantum numbers. These quantum numbers, considered as a continuous variable, are the coordinate of the oscillator describing its quadratic potential, while the corresponding derivative defines a momentum operator. We also analyze the scaling properties of the oscillator parameters as a function of the size of the tetrahedron, and the role of different angular momentum coupling schemes.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The quantum volume operator is one of the most studied objects in the field of loop quantum gravity and of crucial importance for the construction of dynamics within this approach [1, 2, 3]. In the literature, one finds traditionally two versions of such an operator, due to Rovelli and Smolin [4], and to Ashtekar and Lewandowski [5], respectively. Their properties and interrelations have been intensively investigated [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], including a third proposal for a volume operator by Bianchi, Dona, and Speziale [20]. The latter one is closer to the concept of spin foams [3] and relies on an older geometric theorem due to Minkowski [24]. Volume operators are usually considered in connection with polyhedra. The most elementary objects of this kind are tetrahedra consisting of four faces which are represented by angular momentum operators coupling to a total spin singlet [11, 25]. Here all three definitions of the volume operator coincide. Among the most recent developments, Bianchi and Haggard have performed a Bohr-Sommerfeld quantization of the volume using an appropriate parameterization of the classical phase space of a tetrahedron, and the obtained semiclassical eigenvalues agree amazingly well with exact numerical data [21, 22]. The purpose of the present communication is to point out that, in the sector of large eigenvalues, the volume operator of such a quantum tetrahedron is accurately described by a quantum harmonic oscillator. Our presentation will continue as follows: After briefly summarizing important features of the quantum tetrahedron and its volume operator in section 2, we derive our central result, starting from numerical observations, in section 3. We give explicit formulae for the large-eigenvalue sector of the (square of the) volume operator and also analyze its scaling behavior as a function of the tetrahedron size. In section 4 we discuss the role of different angular momentum coupling schemes, and in section 5 we close with an outlook.", "pages": [ 1, 2 ] }, { "title": "2. The Quantum Tetrahedron", "content": "A quantum tetrahedron consists of four angular momenta glyph[vector] j i , i ∈ { 1 , 2 , 3 , 4 } representing its faces and coupling to a vanishing total angular momentum [11, 12, 25, 21, 22] , i.e. the Hilbert space consists of all states | k 〉 fulfilling In what follows we will adopt the coupling scheme where both pairs glyph[vector] j 1 , glyph[vector] j 2 and glyph[vector] j 3 , glyph[vector] j 4 couple first to two irreducible SU(2) representations of dimension 2 k +1 each, which are then added to give a singlet. Thus, the quantum number k ranges as k min ≤ k ≤ k max with leading to a total dimension of d = k max -k min + 1. The volume operator can be formulated as where the operators i ∈ { 1 , 2 , 3 , 4 } represent the faces of the tetrahedron with glyph[lscript] 2 P = ¯ hG/c 3 and γ being the Immirzi parameter. As seen form Eq. (3) it is useful to consider the operator which, in the basis of the states | k 〉 , can be represented as [12, 22, 26, 27, 28] with Here ∆( a, b, c ) is the area of a triangle with edges a, b, c expressed via Heron's formula, Note that ˜ Q couples only basis states | k 〉 with neighboring labels. In the following it will be convenient to readjust the phases of these states via the unitary matrix u ± = diag(1 , ± i, -1 , ∓ i, 1 , . . . ) such that the resulting operator becomes real, Since ˜ Q is antisymmetric, the spectrum of ˜ Q and, in turn, Q consists for even d of pairs of eigenvalues q, ( -q ) differing in sign. Moreover, because of with u = ( u ± ) 2 = diag(1 , -1 , 1 , . . . ), the corresponding eigenstates | φ q 〉 , | φ -q 〉 fulfill For odd d an additional zero eigenvalue occurs whose eigenvector (with respect to ˜ Q ) has the unnormalized form [13] which is, as it must be, an eigenstate of u .", "pages": [ 2, 3 ] }, { "title": "3. Large-Volume Limit", "content": "We denote by | n 〉 , n ∈ { 0 , 1 , 2 , . . . } , the eigenstates of Q in descending order of eigenvalues with | 0 〉 being the state with the largest eigenvalue. In the above basis they can be expanded as where the coefficients 〈 k | n 〉 can be viewed as the 'wave function' of the state | n 〉 with respect to the 'coordinate' k . Fig. 1 shows this data for small n and a typical choice of angular momentum quantum numbers (all being of order a few ten). As seen there, the functions 〈 k | n 〉 show the characteristic features of wave functions of the harmonic oscillator for low-lying states. Indeed, the solid lines in Fig. 1 are gauss-hermitian oscillator wave functions for parameters to be determined a few lines below. Such properties of the functions 〈 k | n 〉 occur for arbitrary sufficiently large angular momentum quantum numbers j i and sets in when all j i exceed a value of about five. For illustration, Fig. 2 displays the data for the case j i ≡ 4 where the oscillator-like features of the wave functions gradually disappear with increasing n . The observation made in Figs. 1 and 2 can be explained as follows: Fig. 3 shows the matrix elements α ( k ) as a function of k for several arbitrary choices of angular momentum lengths including the situation of Fig. 1. In all cases, minima occur at k ∈ { k min +1 , k max } with a unique maximum in between at k = ¯ k determined by where we have considered k as a continuous variable. The above features can also be established by a detailed analytical discussion of the function α ( k ). Now, since the operator Q couples only states with neighboring label k , the wave functions with large eigenvalues will have predominantly support around the maximum of α ( k ). We therefore expand the matrix elements of Q between arbitrary states | Φ 〉 , | Ψ 〉 (lying predominantly in the sector of large eigenvalues) around ¯ k , i.e. k = ¯ k -1 / 2 + x , where the decrement of (1 / 2) accounts for the fact that α ( ¯ k ) couples states of the form | ¯ k -1 〉 and | ¯ k 〉 . In doing so, we obtain Here we have introduced the notations Φ( x ) = 〈 ¯ k + x | Φ 〉 , Ψ( x ) = 〈 ¯ k + x | Ψ 〉 , and additionally performed a continuum approximation to the latter function according to From Eq. (15) one easily reads off an effective operator having the form of a harmonic oscillator, with and The eigenstates ψ n ( x ) = 〈 x | n 〉 of Q eff are just the well-known wave functions diagonalizing the harmonic oscillator in real-space representation, where H n ( x ) are the usual Hermite polynomials. These functions ψ n ( x ; ω ) = ψ n ( k -¯ k + 1 / 2; ω ) are plotted as solid lines in Fig. 1 and are remarkably accurate approximations to the coefficients 〈 k | n 〉 . The corresponding eigenvalues are Table 1 compares the largest eigenvalues of Q obtained via exact numerical diagonalization with the approximate results Eq. (21). Both data coincide within a few per mille, in accordance with our previous findings regarding the corresponding eigenvectors. Note that under a rescaling of all four angular momenta, j i ↦→ uj i , ¯ q scales in leading order as u 3 , while for the frequency one finds ω ∝ 1 /u . In summary, we have constructed an effective operator describing the sector of large eigenvalues of the square of the volume operator of a quantum tetrahedron. This operator has the form of a harmonic oscillator with a 'coordinate' x and a 'momentum' fulfilling the commutation relation which is part of the bedrock of quantum theory. The approximate data shown in Fig. 1 (solid lines) and table 1 was generated by first finding numerically the maximum position ¯ k of α ( k ) and inserting this value into an analytical expression of ( d 2 α/dk 2 ) to obtain ω via Eq. (19). Thus, no adjustable parameter is involved. Closed analytical results for ¯ k are possible if the four angular momenta come in two pairs of equal length, and the expressions become particularly simple in the case of a regular tetrahedron, j 1 = j 2 = j 3 = j 4 =: j . Here one has such that the parameters entering the effective operator (17) are given, to leading orders in j , by Thus, the eigenvalues (21) of the effective operator read to the first leading orders in j In particular, from Eq. (28) we see that the width 1 / √ ω of the wave functions (20) is proportional to √ j ( j +1). Moreover, for the largest eigenvalue of the volume operator, one finds from Eq. (3) Here the leading term ( ∝ j 3 / 2 ) is exactly the classical volume of a regular tetrahedron whose faces have area j , and the subleading correction is, after a redefinition of the face area, identical to the one found in Ref. [22] using Bohr-Sommerfeld quantization. Furthermore, our findings here suggest that the classical volume of a general tetrahedron with face areas j 1 , j 2 , j 3 , j 4 is, to leading order in all j i , given by As already discussed in section 2, the large eigenvalues q n have counterparts q ' n = -q n with the the same modulus but negative sign, and according to Eq. (11), the pertaining eigenvectors can be obtained from the previous ones by changing the sign of any other component. Regarding the wave functions (20) one could try to mimic this behavior by attaching an appropriate phase factor, However, the above functions are clearly not eigenfunctions of the effective operator (17). In fact, an operator having ψ ' n as eigenstates with eigenvalues ( -q osc n ) can be constructed as follows: This operator is not invariant under a 'time reversal' p ↦→ -p which corresponds to the fact that the eigenfunctions (32) cannot be chosen to be real. Moreover, the operators Q osc and Q ' osc are, along with their eigenfunctions, obviously just related by a U(1) gauge operation, apart from the global minus sign on the r.h.s of Eqs. (33) and (34). However, since Q ' osc is merely a consequence of the rather phenomenological ansatz (32), a more rigorous effective description of eigenstates with negative eigenvalue is desirable. Work in this direction could possibly build upon ideas of Ref. [23] where the quantity ± 2 α ( k ) was considered as an effective potential for states with eigenvalues of both sign.", "pages": [ 3, 4, 5, 6, 7, 8 ] }, { "title": "4. Recoupling of Angular Momenta", "content": "There are obviously alternatives to the coupling scheme of angular momenta we have used so far. For instance, instead of the previous procedure, glyph[vector] j 1 , glyph[vector] j 3 and glyph[vector] j 2 , glyph[vector] j 4 could first be coupled to two irreducible representations of dimension 2 l +1 each, which are then combined to a total singlet. The operator Q is then expressed in a form analogous to Eq. (9) with matrix elements β ( l ) given by the r.h.s of Eq. (7) and obvious interchanges of labels. As seen before, β ( l ) has a unique maximum at some l = ¯ l . Thus, putting again l = ¯ l + y -1 / 2, the eigenstates with large eigenvalues will again be accurately approximated by oscillator wave functions ψ n ( y ; ν ) according to Eq. (20) with and the corresponding approximate eigenvalues read with ¯ r = 2 β ( ¯ l ). Since the exact spectrum of Q is of course independent of the coupling scheme used, this holds as well, to an excellent degree of approximation, for the approximate eigenvalues, as it is easily checked by numerics. For instance, for the parameters of Fig. 1 and table 1 we find ¯ q = 13649 . 6, ¯ r = 13650 . 4 and ω = 0 . 075206, ν = 0 . 075198. Thus, we have, as an again excellent approximation, which in particular means that the wave functions ψ n ( x ; ω ) and ψ n ( y ; ν ) can be taken as identical. Moreover, since switching to another coupling scheme implies just a change of basis in the Hilbert space, the above two gauss-hermitian wave functions should be related by a unitary transformation, with some phase factor η n , | η n | = 1. An obvious solution is given by η n ≡ 1 and U ( y, x ) = δ ( y -x ), while another possibility follows from the well-known fact that the Fourier transform of a gauss-hermitian function is a function of that same type: Here one has η n = ( -i ) n and Thus, up to the scale factor ω occurring in Eq. (39), changing from one coupling scheme to another just corresponds to a Fourier transform of the approximating oscillator wave functions. This observation is of course strongly reminiscent of switching from real space to momentum representation in standard quantum mechanics. On the other hand, treating k and l again as discrete state labels, the basis states in both coupling schemes are related by using Wigner 6 j -symbols in the standard convention of prefactors [28]. For the case of all angular momenta being large compared to unity, Ponzano and Regge [29] have devised the following asymptotic expression for such quantities (for more recent developments, see also Refs. [30, 31, 32]), Here V is the volume of a tetrahedron having edge lengths ( j i + 1 / 2), i ∈ { 1 . . . 6 } where edges occurring in the same column of the 6 j -symbol are opposite to each other, i.e. do not have a common vertex, and θ i is the external dihedral angle between faces joining at edge j i . The cosine occurring in the above equation bears some similarity to the exponential in the transformation (39). Moreover, under a rescaling of all angular momenta, j i ↦→ uj i , k ↦→ uk, l ↦→ ul , V scales obviously as u 3 , such that the prefactor of | uk 〉 in Eq. (40) is proportional to 1 / √ u , which is the same scaling behavior as in (39). However, we leave it to future studies to more deeply investigate the possible relationship between the transformations (39) and (40).", "pages": [ 8, 9 ] }, { "title": "5. Conclusions and Outlook", "content": "We have shown that the (square of the) volume operator of a quantum tetrahedron is, in the sector of large egenvalues, accurately desribed by a quantum harmonic oscillator. This finding is a consequence of the fact that (i) the volume operator couples only neighboring states of its standard basis, and (ii) its matrix elements show a unique maximum as a function of state labels. The ingredients of the harmonic oscillator constructed here are an appropriate coordinate variable and a momentum operator defined by the corresponding derivative. These two quantities fulfill the canonical commutation relation. We give explicit formulae for the large eigenvalues of the volume operator in terms of the equidistant harmonic oscillator spectrum. It is an interesting speculation whether or not these linear excitations of space are related to gravitational waves. Moreover, in this limit the quantum tetrahedron is naturally described semiclassically by oscillator coherent states, in contrast to other approaches where tensor products of SU(2) coherent states projected onto the singlet subspace are used [33, 3]. We have also analyzed the scaling properties of the oscillator parameters as a function of the size of the tetrahedron. For a regular tetrahedron we reproduce recent findings [22] on the largest volume eigenvalue and generalize them to the next smaller eigenvalues. In terms of classical geometry, our approach here also suggests an interesting expression given in Eq. (31) for the volume of a general tetrahedron. To further investigate this conjecture might be, from a more mathemaitcal perspective, a route for future studies (possibly starting from numerical tests). Here we have shown the result only for the very special case of a regular tetrahedron. Finally, we have discussed the role of different angular momentum coupling schemes. One might argue that the findings here on the tetrahedral volume operator are in fact very general: Expanding a classical system described by just one pair of canonical variables [21, 22] around an extremum will generically lead to an effective harmonic oscillator. An interesting point here is that this oscillator-like behavior sets in at already quite moderate lengths of the involved angular momenta (being about five). Moreover, the quantum number resulting from the coupling of angular momenta has an immediate interpretation in terms of the oscillator coordinate. The present work exclusively deals with tetrahedra, i.e., in the language of spin networks, 4-valent nodes [3]. An obvious and interesting question is how the results found here translate to higher nodes. Recent work, in a similar spirit as here, on the semiclassical properties of pentahedra includes Refs. [34, 35].", "pages": [ 10 ] }, { "title": "Acknowledgements", "content": "I thank Hal Haggard for useful correspondence.", "pages": [ 10 ] } ]

[ { "title": "A Homogeneous Model of Spinfoam Cosmology", "content": "Julian Rennert 1 , 2 ∗ and David Sloan 3 † Philosophenweg 16, D-69120 Heidelberg Cambridge University, Cambridge CB3 0WA, UK We examine spinfoam cosmology by use of a simple graph adapted to homogeneous cosmological models. We calculate dynamics in the isotropic limit, and provide the framework for the anisotropic case. We calculate the transition amplitude between holomorphic coherent states on a single node graph and find that the resultant dynamics is peaked on solutions which have no support on the zero volume state, indicating that big bang type singularities are avoided within such models. PACS numbers: 04.60.Pp, 04.60.Kz,98.80.Qc", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Cosmological models within the spinfoam framework serve a dual purpose [1]: Their primary function is to form a proposal for extracting cosmological predictions from a full theory of quantum gravity. These models also perform a useful secondary role in forming a bridge between the canonical [2] and covariant [3] formulations of Loop Quantum Gravity (LQG). The covariant, or spinfoam, approach is a 'bottom up' construction - one predicates a quantum model and thence derives dynamics. As such the existence of a semi-classical limit and its agreement with the predictions of General Relativity are not a foregone conclusion, but rather must be examined within physical scenarios. This situation contrasts that of Loop Quantum Cosmology (LQC) [4], the application of the principles of LQG to cosmological mini-superspaces. Another important question that has to be clarified by spinfoam cosmology is whether physical predictions such as the resolution of cosmological singularities can also be derived within this approach. The resolution of the big bang singularity, and its replacement with a deterministic bounce, is a key success of the canonical theory. The absence of strong singularities is a well trusted result in LQC in the k = 0 [5, 6], and k = ± 1 [7] FLRW models, and has been extended to include Bianchi I spacetimes [8]. It forms the basis of investigation of observable consequences of the theory [9, 10]. It is therefore a crucial test of the spinfoam approach that it reproduces these features in the ultraviolet sector. In [1] it was shown how to calculate the transition amplitude between two quantum states of gravity in the homogeneous and isotropic cosmological regime using a simple two-node graph (the dipole graph) at the first order in the vertex expansion. The main result of this work was to demonstrate that the used new spinfoam vertex amplitude (EPRL/KKL), [11-14], together with some other ingredients, are adequate to derive a classical limit which can be identified with the Friedmann dynamics of an empty flat, homogeneous and isotropic universe, i.e. (static) Minkowski space. This result was further strengthened in [15], where a slight modification of the spinfoam vertex was utilized to implement a cosmological constant and derive a de Sitter universe as the classical limit. However, despite these original results being interesting, they exhibit a considerable deficiency, namely they fail to reproduce the curvature term k a 2 , which appears in the Friedmann equation. Such a term is expected, since the chosen graph is dual to a (degenerate) triangulation of the three sphere, (the closed topology, in which k = 1). The authors of [1] argue that this term might be recovered by taking higher orders of the spin approximation into account. However, this term appears in a more natural manner, as we will show in section II C. Our approach is the following: We will examine flat ( k = 0) Friedmann-LemˆaitreRobertson-Walker (FLRW) models by use of a simple graph. This 'Daisy' graph consists of a single node which is both the source and target of three links. This graph can be thought of in two equivalent ways: In the first instance one has tessellated space by identical cubes, and so by symmetry opposite faces of a cube are identified, thus an outgoing edge dual to a given face is an incoming edge dual to the opposite face. The second instance is to consider the spatial slice to be a flat three-torus, with each link transcribing a compact direction. This is what separates this approach from the cubulation used in [16] and also in [17]. The second motivation is to provide the framework for investigating the more complicated case of Bianchi I cosmologies. Since the inclusion of matter within the spinfoam paradigm has not yet been fully realized, one can only investigate FLRW models with trivial classical dynamics. In the anisotropic homogeneous systems comprising the Bianchi models there is a rich physical evolution even in the absence of any matter. These models have been examined extensively in LQC, both within the quantum framework [18-20] and the semiclassical effective framework [21-29] and it would be a strong evidence for the validity of the spinfoam cosmology approach if one could derive these models from the full quantum theory.", "pages": [ 2, 3 ] }, { "title": "II. RECAP OF THE THEORETICAL FRAMEWORK", "content": "Let us briefly review the necessary theoretical input to make our ideas and calculations tractable, and fix our notation. Since we rely heavily on the theory as introduced in [1] we refer the reader to the original source or [3, 30] for a more detailed discussion.", "pages": [ 3 ] }, { "title": "A. LQG and spinfoams", "content": "The kinematical Hilbert space of LQG is defined as the direct sum of subspaces H Γ over all graphs Γ, embedded in a three dimensional manifold Σ. Since we want to work in a cosmological regime, describing just a finite number of degrees of freedom, it is sufficient for us to consider just one of these subspaces. This Hilbert space H Γ is defined on a graph Γ with L links and N nodes. Its elements are the spin network functions; gauge invariant, square integrable functions Ψ : SU (2) L → C , ( holonomy representation ). Since gauge transformations act on the nodes N the Hilbert space H Γ is given by The name holonomy representation results from the circumstance that the SU (2) elements h l are the holonomy of the Ashtekar-Barbero connection along the link l , i.e. with A = A i a τ i d x a . The components of A are given by A i a = Γ i a + γK i a with Γ i a being the spin-connection, K i a the extrinsic curvature of Σ and γ ∈ R > 0 is the Barbero-Immirzi parameter. Thus the SU (2) elements h l contain the geometrical information of the quantum state Ψ( h l ). Another representation, related to the former one via the Peter-Weyl transformation [3, 31], is the spin-intertwiner representation. In this representation the graph Γ carries spins j l ∈ N 2 at each link and invariant tensors i n , called intertwiners, at each node. Those spins correspond to the spins of the unitary irreducible representations of SU (2) and the intertwiners belong to the SU (2)-invariant subspace K n = Inv SU (2) [ H n ], where H n is the tensor product of the representation spaces carried by the links meeting at the node n , H n = ⊗ l ∈ n H j l . A general state in H Γ has the following structure in the spin-intertwiner representation where D ( j l ) ( h l ) is the 2 j l +1 dimensional Wigner matrix of the holonomy h l and the dot indicates contraction of indices. There are now two interpretations of the 3-dim. manifold Σ in which Γ lives. First one can imagine Σ to be a spacelike slice at some coordinate time t . The spinfoam model would then define an amplitude from H Γ(Σ t ) to H Γ(Σ t +1 ) which allows us to interpret this amplitude as a transition amplitude between two states of geometry on the spatial slice. The second interpretation holds Σ to be a 3-dim. boundary of a 4-dim. spacetime region. The states in H Γ(Σ) are thus not thought of as 'states at some time', but rather as boundary states , [32-34]. We note that the first case is a special case of the second one for disconnected spatial boundaries. The dynamics of these quantum states can be defined via the spinfoam formalism. Think again of a boundary state Ψ ∈ H Γ with Γ ⊂ Σ. A spinfoam lives on a 2-complex made up of vertices, edges and faces. A 2-complex can be seen as a discretization of 4-dim. spacetime and heuristically may be thought of as resulting from a canonical spin network evolving in time. Even if one deals with the diffeomorphism invariant s-knot states one has to consider an explicit embedding in order to calculate holonomies and fluxes for a given (patch of) spacetime. Thus, we work with embedded graphs to facilitate contact with the canonical formulation (i.e. LQC) in which most work to date has been performed. This embedding enables the direct projection of established holonomies and fluxes representing the FRW geometry onto our network. Overall, this picture leads us to view also our spinfoam to be embedded in spacetime which contrasts the viewpoint of abstract non-embedded spinfoams. Either way a spinfoam model assigns an amplitude to the state Ψ in the following way where W ( h l ) is given by the EPRL/FK spinfoam model [11-14, 31] and is given by The sum ranges over spin network histories, the h f are the holonomies around a face and A v ( h vl ) is called the vertex amplitude. We will present its precise structure in section III where we again will follow closely [1]. As explained above, for a disconnected boundary Σ this amplitude is interpreted as a transition amplitude and captures the probability for a state on Σ t 1 to evolve into another (or the same) state on Σ t 2 >t 1 . Thus, we are not going to calculate the explicit time evolution of states nor the full time evolution operator but probability amplitudes for single coherent states.", "pages": [ 3, 4 ] }, { "title": "B. Coherent states", "content": "Coherent states are an important tool for the examination of the classical limit of any quantum theory. In this section we will summarize a few definitions about the coherent states for LQG. In particular we will use the Livine-Speziale coherent intertwiners [35] as well as the coherent states in the holomorphic representation [31, 36] later in this work. The Livine-Speziale coherent intertwiners make use of the Perelomov coherent states for SU (2) such that the intertwiner i n in (2.3) is replaced by a coherent intertwiner. A Perelomov coherent state for SU (2) takes the highest weight state | j, j 〉 ∈ H ( j ) , which is a coherent state along ˆ e z , and rotates it with a Wigner matrix D ( j ) ( h glyph[vector]n ) such that it is coherent along another axis glyph[vector]n . The element h glyph[vector]n ∈ SU (2) corresponds to the SO (3) element R glyph[vector]n that rotates ˆ e z into glyph[vector]n . Thus we obtain the coherent state | j, glyph[vector]n 〉 ≡ D ( j ) ( h glyph[vector]n ) | j, j 〉 . Consider a node n which joins E links e together. A coherent intertwiner at this node n is now given by the tensor product of the coherent states coming from each single link. The gauge invariance of these states is achieved via group integration. The holomorphic coherent states are characterized by an element H l ∈ SL (2 , C ) given at each link of the graph Γ. They are defined by where K t is the analytic continuation of the SU (2) heat kernel to SL (2 , C ) and the group integration again ensures gauge invariance. The heat kernel is given by with a ∈ SU (2), B ∈ SL (2 , C ) and α, t ∈ R > 0 . The SL (2 , C ) label H l now allows for two different decompositions [31]. The first one is the polar decomposition and shows clearly that H l determines a point in classical phase space on which the coherent state is peaked. h l ∈ SU (2) is the holonomy of the Ashtekar connection A i a and E l ∈ su (2) is the flux of the densitized triad E a i . Thus, a coherent state with label (2.9) corresponds to a classical configuration ( A i a , E a i ). The second decomposition of H l uses two SU (2) elements h glyph[vector]n l and h glyph[vector]n ' l which, analogously to the SU (2) elements of the Perelomov coherent states, correspond to the transformation of ˆ e z into glyph[vector]n l and glyph[vector]n ' l . Furthermore, a complex number z l is used whose real part is associated to the extrinsic curvature and its imaginary part is related to the area that is pierced by the link l [31]. We denote the real and the imaginary part of z l as z l = c l + ip l and σ 3 is the third Pauli matrix. The relation between the two decompositions becomes clear by writing (2.10) in the polar decomposition. One finds that [31] Where f l is the face dual to the link l with area A l = 8 πG glyph[planckover2pi1] γ p l /t l . Other coherent states, based on the so called flux representation for LQG were introduced in [37, 38]. These states posses a slighly different peakedness behaviour for the mean value of the flux operator which derives from a modified heat equation on SU (2) using a different Laplacian, [39]. For future research it might be interesting to consider these states instead of the above presented ones. However, in order to be able to compare our results with [1] we stick to the holomorphic coherent states in this work.", "pages": [ 4, 5 ] }, { "title": "C. Classical preliminaries", "content": "We are interested in the applicability of spinfoam cosmology to homogeneous models, both in the isotropic and anisotropic cases. In this section we will establish the holonomies and the fluxes for such models. We assume our spacetime to be of the form M = R × Σ, with Σ being a homogeneous 3-space. Under the additional assumption of isotropy the metric of M can be given by with d Ω 2 = dr 2 / (1 -kr 2 ) + r 2 dθ 2 + r 2 sin 2 θ dφ 2 and k ∈ { 0 , ± 1 } . The parameter k distinguishes three different spaces with constant curvature, where we are interested in the closed ( k = 1) and the flat ( k = 0) case. The flat and closed universes are special cases of the Bianchi I and IX universes respectively, in which all scale factors have been identified. If we consider a universe without matter but just a cosmological constant Λ, the metric (2.13) evolution obeys the Friedmann equation glyph[negationslash] In the case of vanishing cosmological constant the only possible solution is a static spacetime a ( t ) = const. , where for k = 0 one recovers Minkowski space. If Λ = 0 one obtains for k = 0, and under the assumption that a ( t ) glyph[greatermuch] 1 also for k = 1, the de Sitter solution , a ( t ) = exp( ± √ Λ / 3 t ). If we drop the restriction to isotropic models we obtain a Bianchi I universe in the flat case, which is described by the following line element Considering again a vacuum spacetime (with Λ = 0), the three directional scale factors a 1 , a 2 , a 3 have to satisfy This equation is solved by the so called Kasner universe and is given by a i ( t ) = t κ i . The Kasner exponents have to fulfill the conditions ∑ i κ 2 i = ∑ i κ i = 1. From those conditions one deduces that one exponent has to be negative, while the other two are positive which leads to a contraction in one direction and an expansion in the other two (the standard choice is κ 1 = -1 / 3 and κ 2 = κ 3 = 2 / 3). Now, in order to specify the holonomy and the flux, we need the Ashtekar connection and the corresponding densitized triad. For that we will use the results provided in [5, 40-42]. In a general, i.e. non-cosmological, setting the Ashtekar connection is given by A i a = Γ i a + γK i a , with K i a beeing related to the extrinsic curvature where the e i a are co-triads, such that the spatial metric can be expressed as h ab = δ ij e i a e j b . The connection coefficients Γ i a are calculated via contraction of the spin connection Γ i a = -1 2 ε ijk θ ajk , which is given by Γ c ab is the Levi-Civita connection compatible with h ab , expressed in terms of the cotriads. However, using the framework of invariant connections on principal fibre bundles, as explained in [40], simplifies the tedious calculation of A i a via (2.17) and (2.18) enormously. Now, a Bianchi model is a symmetry reduced model of general relativity by a symmetry group S , which acts freely and transitively on Σ. If Σ is invariant under the action of S it is an homogeneous 3-space and a connection can be decomposed as A i a = φ i I ω I a , with left invariant 1-forms ω I a and constant coefficients φ i I . A further reduction, which leaves us for example with the three gauge invariant degrees of freedom in (2.16), is achieved by diagonalizing φ i I . This has the effect that we can write with Λ i K ∈ SO (3), [5]. Using the left-invariant vector fields X a I , dual to the 1-forms ω I a , allows us to decompose also the densitized triad as where the second equality results again from a diagonalization of p I i . These six coefficients ( c K , p K ), K = 1 , 2 , 3 now span the phase space of our reduced homogeneous model with the symplectic structure [18] If we expand the co-triads as e i a = e ( K ) Λ i K ˜ ω K a , with arbitrary e K ∈ R , we get the following relations (no summation) With these simplifications the connection components Γ i a = Γ ( I ) Λ i I ˜ ω I a are given by, (no summation, even permutation of { 1,2,3 } ) The n I characterize our Bianchi model, we have for example n I = 0 for Bianchi I and n I = 1 for Bianchi IX. Thus, we see that the Bianchi I models have vanishing spin connection Γ i a . The extrinsic curvature is given by K I = 1 2 ˙ e I , [5], where the dot indicates a derivative with respect to the coordinate time t . Now, we find the following results for the Ashtekar connection in the homogeneous setting In the isotropic case we have p 1 = p 2 = p 3 and (2.23) gives us Γ I = 0 in the flat case (Bianchi I), whereas we get Γ I = 1 2 in the model with positive curvature (Bianchi IX). We can thus write c = 1 2 ( k + γ ˙ e ), k ∈ { 0 , 1 } . For the anisotropic (Bianchi I) model we get c I = γ 2 ˙ e I . Before we apply this formalism to our one-node graph let us make the following observation. In [1] it was shown that the holomorphic transition amplitude between two homogeneous and isotropic quantum states, which are supposed to correspond to a curved geometry ( k = 1), is given by Following the reasoning in [15] the main contribution of W ( z ) is obtained when the real part of z 2 vanishes and its imaginary part is proportional to πl , l ∈ Z . Now, if we use the correct relation (which was already noted in [43]) between c and the metric variables, i.e. c = Re( z ) = 1 2 ( k + γ ˙ a ), instead of just c = γ ˙ a we can reproduce the correct Hamiltonian constraint. Therefore, we require that the real and the imaginary part (which doesn't contribute anyway, if we consider | W ( z ) | ) vanish. Thus, we get from z 2 = ( c + ip ) 2 However, the p 2 term will disappear if we consider the proper normalized amplitude as done in [15] or [44]. Thus, we find c 2 = 0 and Scaling of ˙ a and multiplication by a gives us which is the correct Hamiltonian constraint for a curved FLRW universe [45]. In this paper we are interested in flat spatial slices but it is imaginable to include curvature analogously in our model (i.e. using the correct Ashtekar connection). However, one should note that the graph structure must also support the topology under consideration and thus one might be forced to choose a different graph to probe a curved spacetime, e.g. as done in [16, 17]. We have already mentioned the definition of the holonomy in (2.2). Now, let us define the flux. If we denote the link along which we evaluate the holonomy by l then S l denotes a surface pierced by l . One says S is dual to l . The flux of the electric field E = E a i τ i X a through a surface S l is given by where ∗ denotes the Hodge dual, which converts our vector E into a 2-form, (dim(Σ) = 3), and n j = n i τ i is a su (2) valued scalar smearing function [46]. We will use the definition with These definitions will become necessary especially for the anisotropic case, when we explicitly have to calculate the Ashtekar connection and the flux for our model.", "pages": [ 6, 7, 8 ] }, { "title": "III. OUR MODEL", "content": "In this section we want to calculate the transition amplitude between two flat, homogeneous and isotropic universes using the spinfoam formalism. As is customary in (quantum) cosmology, we are interested in the largest wavelength modes and ignore shorter scale fluctuations. Our model can be interpreted in two ways: Either as probing the universe on the largest scales, in which only the largest wavelength is relevant, or equivalently as tessellating space with cubes and restricting the geometry to homogeneity thereupon. Ideally one should take a large number of such cubes and consider all fluctuations away from homogeneity in the calculation of transition amplitudes and then coarse-grain for large scale behaviour. However, in practice this is highly impractical and therefore we follow the usual philosophy applied in cosmology and symmetry reduce before establishing dynamics. Despite the inherent shortcomings of such a simplification, this has proven highly effective in classical cosmological approaches, and is the basis of all quantum cosmologies. In the spinfoam cosmology approach this means that on the one hand we have to identify certain homogeneous states which are presumably characterized by a certain subclass of all possible graphs and a certain (homogeneous) coloring. On the other hand, given that the spinfoam formalism is considered as a non-perturbative framework for quantum gravity, we have to employ a truncation of the full quantum dynamics. Therefore, we think of a cubical partition of 3-space Σ. Homogeneity then allows us to restrict our considerations to a single cube whose dual graph (with toroidal topology) is given by the Daisy graph 1 , see FIG.(1). It is not hard to see that the restriction to a smaller graph corresponds to a truncation of degrees of freedom at the kinematical level. But it is also true that this does not automatically imply a cosmological setting. In fact, one can certainly build homogeneous states by using a larger lattice and keeping all holonomies and fluxes the same. Homogeneity then allows us to identify all lattice points, and thus the simplification made is appropriate. Note further, that the identification with cosmology also arises because of the particular holonomies and fluxes which we are using: In this sense we identify our simple graph with a cosmological setting. The original motivation for this graph, especially the use of three closed links, was its potential applicability to anisotropic cosmological settings and therewith a physically more complex situation, a problem we will tackle in a follow up paper [47]. In this paper we will restrict our attention mostly to the isotropic case and see that this one node graph is already sufficient to reproduce the original result of [1]. Let us furthermore point out that, unless one is dealing with a symmetry reduced dynamics, the regime in which a graph provides the basis for a good homogeneous state, may depend on the full quantum dynamics. This means that by allowing for larger quantum fluctuations, i.e. a more complicated dynamics, two different graphs, which were originally thought to describe the same homogenous state, may lead to different results 2 . This closely relates to our use of the one-vertex spinfoam expansion and the objective of finding an effective dynamics from the full, non-perturbative quantum dynamics as pursued for example in [48]. In spinfoam cosmology calculations to date were all performed using a single spinfoam vertex to specify the dynamics. The rationale behind this approxmation, next to its calcula- bility, is beautifully elaborated on in [49] and [50]. There it is shown, in a simple discretized parametrized model, how an expansion in a small number of vertices allows one to achieve good agreement with the continuum model, both in the classical and the quantum regime. Note that approximating the number of spinfoam vertices is not to be confused with a semiclassical approximation in terms of a dimensionful parameter such as glyph[planckover2pi1] . Note furthermore, that in TQFT the result of transition amplitudes does not depend on the underlying triangulation. Now quantum gravity is certainly not a topological theory, however, the point is, that in certain regimes it may behave similarly and thus is not sensitive whether one uses a finer or more complicated bulk triangulation ('Ditt-invariance'). Of course, the goal has to be to gradually increase the number of spinfoam vertices. Another interesting approach towards the extraction of a cosmological scenario was recently obtained within the Group Field Theory approach to quantum gravity [48]. The GFT approach offers some promising features concerning the identification of general homogeneous states, independent of the underlying graph structure, and the mentioned interplay with the full quantum dynamics. This may allow for an inclusion of inhomogeneities and may also provide a possibility to calculate corrections to the Friedmann equation. Eventually, one would like to compare predictions coming from both models.", "pages": [ 9, 10 ] }, { "title": "A. The Setting", "content": "First, let us recall the definition for the vertex amplitude to specify the dynamics. Despite it being shown in [51, 52] that there exist additional 2-complexes which contribute at the one vertex level (for the dipole graph) we will consider just the single spinfoam history which corresponds to our boundary graph in the sum in (2.5). We consider the spinfoam that simply connects the two graph vertices with a single spinfoam vertex. The one vertex spinfoam expansion ( v = 1) leads to the factorization of our amplitude 〈 W | Ψ 〉 3 . The face amplitude δ ( h f ) in Eq.(2.5) peaks the h vl onto the h l and the coherent states Ψ H l ( h l ) are peaked on the H l . With these simplifications and following [1] the transition amplitude between an initial and a final geometry is given by where the vertex amplitude is given by [31] where G is SO (4) for the Euclidean theory and SL (2 , C ) for the Lorentzian theory, respectively. As was explained in [53] for the Lorentzian case we will neglect one integration so that W ( z ) does not diverge. Furthermore, since our graph has just one node we find that source and target node of each link are the same, i.e. s ( l ) = t ( l ), thus leading to G s ( l ) G -1 t ( l ) = I . A clear advantage of using this graph is its simple application in the homogeneous case. It allows us to explicitly calculate the SL (2 , C ) elements for our coherent states and with that provides helpful insights also for more complicated structures. We begin by calculating the SL (2 , C ) elements H l ( z ) using the decomposition (2.10) where u l and ˜ u l are elements of SU (2). We have three links, l 1 , l 2 , l 3 and six normal vectors n 1 = ˆ e x and ˜ n 1 = -ˆ e x , n 2 = ˆ e y and ˜ n 2 = -ˆ e y and n 3 = ˆ e z and ˜ n 3 = -ˆ e z . The normal vectors n l and ˜ n l are obtained via a SO (3) transformation of ˆ e z and the SU (2) elements u l and ˜ u l are related to these SO (3) transformations, cf. appendix A. Now, we have to bring the three SL (2 , C ) elements in the following form This means that we have to find the angles α 1 , α 2 and β . Given the SL (2 , C ) elements H l ( z ) in this form we are then able to represent their Wigner matrices for all j if we recall that the angular momentum operators ˆ J x , ˆ J y , ˆ J z are given by ˆ J x = σ 1 2 , ˆ J y = σ 2 2 and ˆ J z = σ 3 2 in the j = 1 2 representation. Hence, we get One finds the following angles (cf. equation (A10), (A11), (A12) in the appendix) We can now calculate the transition amplitude and for l = 3 We will now use the large volume approximation, i.e. for Im( z ) glyph[greatermuch] 1 the term with m = j dominates. Now, how do we treat the links l = 1 and l = 2? One can argue, that due to the highly symmetric setting we should also use m = j for those cases. If we do so we can make use of the following asymptotic relation [54] So lets start with l = 1. For m = j we get where we have defined d j = 2 j +1 and ˜ G ≡ Y † D ( j + ,j -) ( G s G -1 t ) Y in the case of Euclidean gravity, ( G ∈ SO (4)), or ˜ G ≡ Y † D ( γj,j ) ( G s G -1 t ) Y in the Lorentzian case, ( G ∈ SL (2 , C )). For detail cf. [3]. (Despite GG -1 = I , because s ( l ) = t ( l ) as mentioned earlier , we keep ˜ G for completeness.) Lets start by calculating the trace for l = 1 where we have inserted a unit operator. This leads to a separation of the geometrical information, stored in H l ( z ), and the gauge invariant contribution, given by 〈 j, k | ˜ G | j, m 〉 . For simplicity we will neglect the gauge contribution later on. For a detailed way of dealing with the trace cf. [44]. For l = 2 we get analogously By making use of and (3.14) we can analyse d ( j ) jk ( π -z ) and get For the trigonometric functions we get for Im( z ) glyph[greatermuch] 2 which gives us the following expression We can use this to obtain If we take the e -i π 2 j term inside the sum we get e -i π 2 ( k + j ) ( -1) j + k i j + k , which is equal to ( -1) 2 j +2 k i 2 j +2 k , so we get Now let us make a few comments about the last expression. First, notice that we can reproduce the factor e -izj , which appears also in the original work [1] and is crucial for the derivation of the transition amplitude. The second point is that we can proceed by approximating also the second part, namely the sum over k , as k = j , which simplifies the result and the important thing is, that by doing this approximation we are not doing worse then the projection onto m = j in the original work. The calculation for l = 2 is identical to the case l = 1. If we now use the following two relations and apply the approximation k = j we get (neglecting factors like ( -1) 2 j ) Now, let us compare these results with the calculations of the original paper [1]. There the authors used a projection onto the highest spin state m = j and got a factor exp( -izj ) for all links. Our calculation gives the same result using a simpler graph. Furthermore, due to our explicit calculation we see in a more precise way where this factor comes from. Now, the term 〈 j, j | ˜ G | j, j 〉 can be neglected in our case because s ( l ) = t ( l ) and thus where we have used the projection property of the unitary Y -map, cf. [1]. In [1] the authors replaced the whole gauge contribution ( 〈 j, j | ˜ G | j, j 〉 ) from all four links by a factor N 0 j 3 based on a calculation done in [35]. So far we have assumed that all our z -labels are the same for each link, which is a consequence of the isotropic configuration we are considering. If we investigate the anisotropic case these labels are going to be different z = z l .", "pages": [ 10, 11, 12, 13, 14 ] }, { "title": "B. Results", "content": "Inserting the results for the traces given by (3.23) and (3.24) into the transition amplitude (3.9) we get Furthermore, we have assumed that Im( z ) = p glyph[greatermuch] ln(1 / 4) ≈ -1 . 38, which is justified since p > 0, and thus all three links give the same contribution. Now we can either apply a gaussian approximation as was done in [1], we can investigate the amplitude numerically or we calculate (3.25) explicitly using the Cauchy product, all of which yield the same result. The gaussian approximation gives the result where j 0 is given by In order to get meaningful results we now have to normalize the amplitude. For this we use the following expression for the norm of a heat kernel coherent state given on a single link [44] √ where we have taken just the leading order term with n = 0, cf. [44]. The small g in the above formula corresponds to our SL (2 , C ) element H l ( z ) and the heat kernel time ˜ t is related to our t via ˜ t = 2 t glyph[planckover2pi1] . A detailed analysis reveals furthermore that the ˜ p in (3.28) corresponds our p 2 . We will start with a numerical analysis of (3.25). Therefore, we plot A ( z ), which we define as the absolute value of W ( z ) divided by the norm to the third, because our graph has three links. We set glyph[planckover2pi1] = 1 2 and truncate the sum (3.25) at j max = 150 where one has to make sure that j 0 < j max , (3.27) holds. If we set the heatkernel time t = 1 we get FIG.(2) and with a heatkernel time t = 0 . 1 we see that the peak becomes sharper FIG.(3). Now, recall that Re( z ) corresponds to the extrinsic curvature of our model and thus we find Re( z ) ∝ ˙ a = 0 for all volumes Im( z ) ∝ a 2 . Hence, we find that our universe is static. As we would expect. We don't have any matter or a cosmological constant, nor anisotropies. What is remarkable now is the decrease of the amplitude towards small Im( z ). To see this more clearly insert (3.27) into (3.26) and calculate A ( z ) using (3.28). The result is Plotting A (0 , p ) we get the shape in the p -direction which shows us the drop off for small scale factors FIG.(4). What does this tell us about the quantum dynamics of our model? Recall the definition of the transition amplitude between two quantum states of geometry Ψ i and Ψ f , (3.1). One finds that the main contributions come from those configurations corresponding to classical geometries, namely ˙ a = 0 and large scale factors a . The remarkable result is now, that the transition amplitude decreases for small scale factors and has zero support on a = 0. This is a statement about the occurrence of singularities in our model in that it tells us that a transition to a singularity is ruled out dynamically. The transition amplitude now gives us the possibilities for quantum fluctuations from one scale factor to another one. It doesn't give us the dynamical evolution of our universe. The classical notion of dynamics, i.e. ˙ a , is encoded in the phase space coordinates. In the same sense as a transition amplitude in QFT does not give us a temporal information of the time evolution of a certain process. Certainly this result has to be strengthened by future investigations, where it remains to show that one can circumvent the large spin approximation. In fact, one can calculate the traces in (3.9) in our model explicitly, thanks to the fact that our graph has just one node and thus 〈 j, k | ˜ G | j, m 〉 = 〈 j, k | Y † D ( γj,j ) ( G s ( l ) G -1 t ( l ) ) Y | j, m 〉 = 〈 ( γj, j ) , j, k | D ( γj,j ) ( I ) | ( γj, j ) , j, m 〉 = δ km holds. This way one can in principle avoid the large spin approximation and indeed finds that the amplitude is still peaked on c = Re( z ) = 0. However, the shape along the p = Im( z ) direction changes, a problem which probably has to be solved by the use of a different normalization.", "pages": [ 14, 15, 16 ] }, { "title": "IV. DISCUSSION", "content": "We showed in this paper that within the spinfoam cosmology approach the Daisy graph is sufficient to reproduce the vacuum Friedmann equation for a flat 3-space in the isotropic setting and thus may also be useful for the investigation of anisotropic models. The Daisy graph is perfectly suited to the relaxation of the restriction to isotropic models just by using three different holomorphic labels z i at each link. This will be our setting in the following paper [47]. Furthermore, we showed how one can reproduce the missing curvature term in the curved model, described by the Dipole graph, without using higher terms in the spin expansion. The right dependence of the real and imaginary part of the holomorphic labels z will also be important for the description of the anisotropic model. It has been suggested that the curvature term arises as a 'higher order' effect in the prior models. However, this derivation applies even in the flat case, which would contradict the agreement with classical dynamics at large scale factor. The main result of this work, however, is the statement about the avoidance of singularities in our model. The dynamics of the Daisy graph show zero support for a transition amplitude from a finite scale factor to zero, therefore the singularity itself is not accessed by dynamics. Since the model under consideration does not include matter terms, there can be no direct comparison made with the bouncing models of LQC. It remains to be seen at this stage whether this result is an artefact of approximations made, or is a deeper feature of the full covariant dynamics. A singularity resolution theorem or results analogous to those of [6] would be a strong achievement for the theory. Since the interpretation of a transition amplitude between an in and out state in this model is asymptotic, and we only consider the first order in perturbation theory (in terms of graph expansion, vertex expansion etc) one cannot make any strong claim of singularity avoidance. However, at this order there is a hint of a resolution in the manner of a bounce: Consider a sequence of transitions between scale factors each described with a single transition amplitude. This will be a random walk in the space of scale factors with transition probabilities as described by the distribution in FIG.(3). As the scale factor tends to zero, the probability of moving to a larger scale factor increases, being unity in the limit of zero scale factor. Thus we see that the probability of a collapsing universe continuing to collapse tends to zero, and the probability of expansion tends to one, as we approach the classical singularity, and thus the universe will undergo a bounce. As we have noted, this is a preliminary result, and may not survive extension of the model beyond the simple expansion used here, but nonetheless is encouraging in its similarity to the singularity resolution seen in LQC. Obviously this isn't the complete picture, as we are moving within points of configuration space at which ˙ a = 0 without giving the detailed dynamics between. This behaviour in which the zero volume state is excluded from solutions hints that singularity resolution may well be a feature of these models, as has recently been argued in [55]. Needless to say, that there are numerous open questions that need to be investigated within the spinfoam cosmology approach, such as the treatment of higher orders in the vertex expansion and of course the inclusion of further spinfoam histories. It is hoped that this will shed some light on the connection with the (quantum reduced) canonical approach as put forward by [17, 56] and also the GFT cosmology in [48]. In the long term perspective the coupling of matter has also high priority if one aims to seriously do quantum cosmology using such models.", "pages": [ 17 ] }, { "title": "Acknowledgments", "content": "The authors would like to thank Carlo Rovelli and Francesca Vidotto for useful comments and discussion. DS gratefully acknowledges support from a Templeton Foundation grant. The authors are indebted to the referees whose extensive input has greatly improved this paper.", "pages": [ 18 ] }, { "title": "Appendix A: Details of the calculations", "content": "In this section we present the calculation of the SL (2 , C ) elements H l ( z ), (3.3). As explained in section 2.1 the normal vectors n l and ˜ n l are obtained via a SO (3) transformation of ˆ e z and the SU (2) elements u l and ˜ u l are related to these SO (3) transformations. We start with ˆ e z ↦→ ˆ e x and ˆ e z ↦→ -ˆ e x . We calculate the two corresponding SU (2) elements for the SO (3) rotation matrix R with the formula [57, 58] We get for R ( x ) y and R ( -x ) y Analogously one calculates ˆ e z ↦→ ˆ e y and ˆ e z ↦→ -ˆ e y with the resulting SU (2) elements Finally, we have to calculate ˆ e z ↦→ ˆ e z and ˆ e z ↦→ -ˆ e z but it is clear that R ( z ) y is given by and thus the corresponding SU (2) elements is Now we have to connect each two SU (2) elements with one link. We connect those vectors who are co-linear. Furthermore, we need The inverse matrices are given by where we have chosen (A9) corresponding to the inverse elements of u ( x ) and u ( y ) resp. The reason is that formula (A3) is not valid for a rotation of π about the x -or y -axis. We get for the SL (2 , C ) elements and for l = 2 and l = 3 we get analogously", "pages": [ 18, 19 ] } ]

[ { "title": "The electromagnetic spike solutions", "content": "Ernesto Nungesser ∗ 1,2,4 and Woei Chet Lim † 3,4 1 School of Mathematics, Trinity College, Dublin 2, Ireland 2 Department of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden 3 Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand 4 Max-Planck-Institute for Gravitational Physics, Am Muhlenberg 1, 14476 Golm, Germany April 20, 2022", "pages": [ 1 ] }, { "title": "Abstract", "content": "The aim of this paper is to use the existing relation between polarized electromagnetic Gowdy spacetimes and vacuum Gowdy spacetimes to find explicit solutions for electromagnetic spikes by a procedure which has been developed by one of the authors for gravitational spikes. We present new inhomogeneous solutions which we call the EME and MEM electromagnetic spike solutions.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "According to Belinskii, Khalatnikov and Lifshitz (BKL) [1, 2, 3], a generic spacelike singularity is characterized by asymptotic locality. Asymptotically towards the initial singularity each spatial point evolves independently from its neighbors in an oscillatory manner that is represented by a sequence of Bianchi type I and II vacuum models. In [4] Berger and Moncrief studied T 3 -Gowdy spacetimes numerically and observed the development of large spatial derivatives near the singularity, which they called 'spiky features'. These structures where found to occur in the neighborhood of isolated spatial surfaces, cf. [5]. Further numerical investigations (see [6] for an overview) seemed to indicate that the BKL conjecture is correct generically, but certain difficulties arose in simulating these spikes. An important step was made in [7] where a solution generating technique and Fuchsian methods developed in [8, 9] where used to produce asymptotic expansion for spikes, which where classified in 'true' and 'false' spikes, where the latter are only a rotation artifact. Based on these transformations, in [10] an explicit spike solution was found in terms of elementary functions. The explicit spike solution suggests a new way to simulate spikes numerically, and in [11] it was confirmed that the explicit spike solution indeed describes the spiky structures, and does so remarkably accurately. The numerical results provide very strong evidence that apart from local BKL behavior, there also exist formation of spatial structures at and in the neighborhood of certain spatial surfaces, thus breaking asymptotic locality. Moreover the complete description of a generic spacelike singularity should involve spike oscillations [12], which are described by sequences of spike solutions and rotated Kasner solutions. We are interested in investigating the nature of BKL behaviour at spacetime singularities in the presence of electromagnetic fields. The aim of this paper is to use the known relation between polarized/diagonal electromagnetic Gowdy spacetimes and Gowdy spacetimes, as given in [13], to find spike solutions for the electromagnetic case. According to the BKL-picture a generic spacelike singularity is vacuum dominated. This project thus will help to clarify whether the introduction of a Maxwell field changes the picture or not. The sign conventions of [14] are used. In particular, we use metric signature -+++ and geometrized units. We use the sign convention /epsilon1 0123 > 0 for the Levi-Civita tensor, but because we will switch to using a time variable that increases towards the past, the component /epsilon1 0123 with respect to that coordinate system will be negative.", "pages": [ 1, 2 ] }, { "title": "2 Basic equations and metric", "content": "If ( ¯ P, ¯ Q, ¯ λ ) is a solution of the vacuum Einstein equations with Gowdy symmetry (or orthogonally transitive G 2 isometry), i.e. with line element then ( P, χ, λ ) with (9)-(10) of [13] [with the following correspondence t = e -τ and ( θ, x, y ) = ( x, y, z )]: will be a solution of the Einstein-Maxwell equations with polarized Gowdy symmetry (or diagonal G 2 isometry) with line element: and a Maxwell field described by the vector potential with only one non-zero component, namely A 3 = χ ( x, τ ). We will assume from now on that we are in the second case, i.e. our metric is described via (3). The Einstein-Maxwell equations with P and χ are given by (19)-(24) of [15]: We will use the non-vanishing β -normalized variables [16]. We take this opportunity to clarify the sign confusion in Eq. (9) of [10]. β and other kinematic variables are defined with respect to the future-pointing congruence. Therefore a positive β describes expansion towards the future (and contraction towards the past). We also take this opportunity to correct the error in the β expression in Eq. (2) of [11]. These β -normalized variables refer to the β -normalized commutation functions associated with an orthonormal frame: The electric and magnetic fields are similarly normalized: The 3-by-3 Σ αβ and N αβ matrices in our case are while the non-zero electric and magnetic components are The β -normalized variables Y = (Σ + , Σ -, N × , E , B ) are related to the partial derivatives of P , λ and χ as follows. The evolution equations for the β -normalized variables can be derived from the above Einstein-Maxwell equations and (7): We do not use the evolution equation for Σ + , instead using the Gauss constraint to find Σ + :", "pages": [ 2, 3 ] }, { "title": "3 Explicit solutions", "content": "We now apply the vacuum-to-electromagnetic transformation (2) to the explicit solutions in Section 4 of [10].", "pages": [ 4 ] }, { "title": "3.1.1 Reparameterized Kasner solution", "content": "Applying the transformation (2) to the Kasner seed solution (17) of [10] yields where v = 2 w -1 and P 0 , χ 0 and λ 0 are arbitrary constants. The result is trivial - this solution is just a re-parametrization of the Kasner solution. Nevertheless, we will use this parametrization of the Kasner solution in Figure 3 later. The β -normalized variables Y K have the same form as (18) of [10]:", "pages": [ 4 ] }, { "title": "3.1.2 Electric Rosen solution", "content": "Applying (2) to the rotated Kasner solution (22) of [10] with the simplifying choice P 0 +ln χ 0 = 0 yields In terms of the β -normalized variables, This describes an electric Bianchi I model. We refer to [17] for a presentation of this solution in a broader context. Up to the arbitrary constant and choosing χ 0 = 1, ξ = e -τ we have exactly the same χ as described in [17] with a minus sign where now w = α 0 2 and γ 0 = λ 0 . This solution was first found by Gerald Rosen [18] and we will refer to it as the electric Rosen solution with subscript ER.", "pages": [ 4 ] }, { "title": "3.1.3 Magnetic Rosen solution", "content": "Applying (2) to the Taub solution (24)-(26) of [10] yields We assume now that χ 1 = 0 for simplicity. In terms of the β -normalized variables, We will refer to it as the magnetic Rosen solution and use the subscript MR. In particular we will use in the following the variables (Σ -) MR = 1 √ 3 (2 w tanh wτ -1) and B MR = -2 w sech wτ . The solution (17) is sometimes called the pure magnetic or magnetovacuum solution since later it has been generalized to include, e.g. a perfect fluid. It is not surprising that this solution appears here instead of the Bianchi II vacuum solution since there is a natural correspondence between heteroclinic chains consisting of Bianchi type II solutions in the vacuum case and heteroclinic chains in the case with a magnetic field which include orbits corresponding to both solutions of the vacuum Einstein equations of Bianchi type II and solutions of the Einstein-Maxwell equations of Bianchi type I. For recent work on oscillatory singularities in Bianchi models with magnetic fields we refer to [19].", "pages": [ 4, 5 ] }, { "title": "3.2 Inhomogeneous solutions", "content": "We will now proceed to the derivation of the inhomogeneous solutions. It is convenient to introduce the following definitions: where it holds that c 2 + s 2 = 1.", "pages": [ 5 ] }, { "title": "3.2.1 EME electromagnetic spike solution", "content": "Applying (2) to the rotated Taub solution (28)-(30) of [10] 1 we obtain a new, inhomogeneous solution: The spike occurs at x = 0, and the electric field is zero there. We will refer to the solution as the EME spike solution, because (for | w | > 1 cases) worldlines along large x experience a sequence of three Rosen transitions: electric-magneticelectric. Its β -normalized variables are given by where we have denoted ¯ Σ -= (Σ -) MR -1 √ 3 , and the subscripts MR and EME refer to the magnetic Rosen and the EME spike solution.", "pages": [ 5 ] }, { "title": "3.2.2 MEM electromagnetic spike solution", "content": "Applying (2) to the spike solution (33)-(35) of [10] yields We will refer to it as the MEM spike solution and use the subscript MEM. Its β -normalized variables are given by This completes the transformation of the explicit solutions in Section 4 of [10]. These spike solutions are new. The solutions above can also be generated by starting with the Kasner solution (8) above and applying the transformations in the Subsection 3.2.4 successively.", "pages": [ 6 ] }, { "title": "3.2.3 Properties of the spike", "content": "Both electromagnetic spike solutions have non-trivial electric and magnetic fields, and the names 'EME' and 'MEM' indicate what Rosen transitions occur at large x for | w | > 1. Note that the spatial dependence of the different variables lies, as in the vacuum case, in c and s , which depends on x in a spiky way. The electromagnetic spike solutions have the same radius as the vacuum spike solution. For more discussions on c and s , and on the radius of the spike, see Sections 4.5 and 4.6 of [10]. In the vacuum case there is a distinction between the false spike solution and true spike solution, with the false spike solution being a rotated Taub solution. There is no such distinction here because the frame rotation transformation is absent in polarized/diagonal case. Both the EME and MEM spike solutions are two different spiky solutions.", "pages": [ 6 ] }, { "title": "3.2.4 Alternative derivation", "content": "The solution-generating transformations in polarized electromagnetic Gowdy spacetimes are and These transformations correspond to transformations (6) and (7) in [10], but curiously the role of solution-generating transformation has switched. Transformation (6) in [10] is merely a frame rotation, but here transformation (27) is a solution-generating transformation. On the other hand, transformation (7) in [10] is a solution-generating transformation, but here transformation (28) is merely a 90-degree duality rotation for the electromagnetic field (see e.g. [20] about duality rotation) followed by a switch of the coordinates y and z . Transformation (28), like transformation (7) in [10], is also very simple when expressed in β -normalized variables: Thus here we do not have 'false' spikes, since both spikes represent real inhomogeneous solutions, although quite similar. Transformation (27) maps the Kasner solution (8) to the electric Rosen solution (10)-(11); transformation (28) then maps it to the magnetic Rosen solution (14)-(15); (27) then maps it to the EME spike solution (19)-(20); and (28) maps it to the MEM spike solution (23)-(24). At each step, λ is obtained by quadrature.", "pages": [ 6, 7 ] }, { "title": "4 Visualization", "content": "We have already seen that the vacuum transformations have a different interpretation in the electromagnetic case. Now we proceed to visualize the dynamics of the different solutions and we compare the dynamics of the solutions found here corresponding to polarized Gowdy spacetimes with a electromagnetic field with the ones found in [10] corresponding to Gowdy spacetimes. For this purpose we use Hubble-normalized variables [21, 22], denoted with a superscript H . For the spatially homogeneous background dynamics, it is best to use the Hubble-normalized variables which are related to the β -normalized ones via The Hubble-normalized energy density Ω of the electromagnetic field is given by and satisfy while the Hubble-normalized spatial curvature Ω k is given by The dynamics of the electric and magnetic Rosen solutions can be described by their orbits in the state space. When projected on the (Σ + , Σ -) H plane, these orbits form straight lines emanating from one corner of a triangle superscribing the Kasner circle (see Figure 1). The two transition sets combine to describe the dynamics during an electromagnetic equivalence of a Kasner era, which consists of long Kasner epochs (described by the Kasner equilibrium points), punctuated by brief periods of transitions. Either the electric or the magnetic component becomes significant during these periods of transitions. Compare with Figure 5 of [10]. The coincidence of the projected Rosen orbits with the projected Taub orbits compels one to compare the Rosen solutions with the Taub solution. Consider the Taub solution (24)-(26) of [10] with a particular value for w (call it w T ) and the corresponding magnetic Rosen solution whose orbit starts and ends at the same Kasner points as this Taub orbit. Then one finds that the w -parameter for the magnetic Rosen solution takes the value w MR = w T / 2. Consequently the rate of change for the magnetic Rosen transition from one Kasner point to the next is only one half of that for the Taub transition. The implication of this is that in more general models where both modes are present, gravitationally-driven Taub transitions would dominate electromagneticallydriven Rosen transitions towards the singularity, and that the Hubble-normalized w=1.25 energy density of the electromagnetic field would tend to zero towards the singularity, with the caveat that this occurs almost everywhere, except possibly at some 'spiky' worldlines where the Taub mode has a local zero. Along these spiky worldlines, whether (vacuum) spike transitions would dominate Rosen transitions is unknown. In Figure 2 we visualize the orbits of the MEM spike solution (23)-(25) along various worldlines x = const, projected on the (Σ + , Σ -) H plane. The projected orbits of the EME spike solution (19)-(21) differ only in the sign of Σ -. Along worldlines far away from the spike worldline, the orbits approximate the electric and magnetic Rosen orbits. Along the spike worldline, the projected orbit is a straight line. For the case | w | ≥ 1, all these orbits end at the same Kasner points. For the case 0 < | w | < 1, the orbit along the spike worldline ends at a different Kasner point from all others. Compare with Figure 6 of [10]. In Figure 3 we visualize two families of the solutions (one with w = 0 . 75 and the other with w = 1 . 25). Note that the Kasner solution plotted here uses the parametrization in (8). Compare with Figure 9 of [10]. We now compare the orbits of MEM spike solution with the orbits of vacuum spike solution. Call the corresponding parameters w MEM and w S . We set w S = 2 w MEM + 1 so that both solutions start at the same Kasner point. But the solutions will not end at the same Kasner points. See Figure 4.", "pages": [ 7, 8, 9 ] }, { "title": "5 Discussion and outlook", "content": "In this paper we have presented new solutions to the Einstein-Maxwell solutions with polarized Gowdy symmetry which generalize the known magnetic and electric Rosen solutions which are spatially homogeneous to the inhomogeneous case. These solutions represent spikes in the electromagnetic field as well as the gravitational field, and are building blocks of oscillatory behavior. In contrast to the vacuum case there are no false and true electromagnetic spikes, however the solutions are now linked via a duality rotation. From the analysis one can see that it is the gravitational solutions which dominate the electromagnetic solutions. Therefore we conjecture that the (vacuum) gravitational spike solution plays a larger role than the electromagnetic spike solutions in the oscillatory regime. The analysis represents also additional support to the analysis carried out in [23], where an inhomogeneous generalization of Bianchi VI 0 with a pure magnetic field was considered. In fact, the latter solution can be seen as coming from an electromagnetic field which does not come from a vector potential, cf. [24]. In [24] an inhomogeneous generalization of Bianchi VII 0 was also presented. Both these inhomogeneous generalizations are called of local or twisted Gowdy symmetry. It is of interest to investigate further these cases as a pre-step to the analysis of the Einstein-Maxwell system with full Gowdy symmetry which is considerably more complicated. It remains unclear whether pure magnetic spike solutions exist. In [25] numerical and analytical evidence was presented that (gravitational) spikes can generate matter perturbations and it was argued that this phenomenon might explain the formation of structure in the early Universe. Electromagnetic spikes should have a similar effect on matter as well. It remains to be seen how gravitational spikes interact with electromagnetic spikes, and whether such interactions amplify or suppress the electromagnetic field.", "pages": [ 9, 10 ] }, { "title": "Appendix", "content": "The equations in this section are taken from from [15] where -λ instead of λ was used. Note also that we have put ω = 0 and we have used a different sign convention for /epsilon1 αβγδ . The basic quantity in electromagnetism is the electromagnetic field tensor F αβ which is antisymmetric. The Maxwell equations in absence of charged matter are The second Maxwell equation can be solved by introducing a four potential such that In the Lorentz gauge , where by definition holds, the Maxwell equations in curved space time can be written as: The energy-momentum tensor for an electromagnetic field is which is trace-free. The dual electromagnetic field tensor is where where det g is the determinant of the matrix g αβ . Let n α be a unit futurepointing vector orthogonal to a spacelike hypersurface. Then we can define the electric and the magnetic fields as follows: The symmetry assumptions imply that vector potential has only the following components: A 2 is periodic of period 2 π with respect to x . Using (34) one can compute that the electromagnetic field tensor has the following non-trivial components: The dual electromagnetic field tensor has according to (37) the following nontrivial components: Choosing as the unit future-pointing vector we compute the non-vanishing components of the electric and the magnetic field with (38) and (39): E and B are β -normalized orthonormal frame components of the electric and magnetic fields. where e 3 3 = e ( -P -τ ) / 2 and e 2 2 = e ( P -τ ) / 2 are the frame coefficients.", "pages": [ 10, 11, 12 ] }, { "title": "Acknowledgments", "content": "We thank Alan A. Coley, Alan D. Rendall and Claes Uggla for helpful comments and suggestions. This work was initiated when both authors were still at the Max-Planck-Institute for Gravitational Physics, the first author being funded through the project SFB 647 of the German Research Foundation. E.N. is grateful to the Goran Gustafsson Foundation for Research in Natural Sciences and Medicine and the Irish Research Council for their financial support.", "pages": [ 12 ] } ]

[ { "title": "Entanglement entropy from the holographic stress tensor", "content": "Arpan Bhattacharyya and Aninda Sinha 1 Centre for High Energy Physics, Indian Institute of Science, Bangalore, India. November 20, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider entanglement entropy in the context of gauge/gravity duality for conformal field theories in even dimensions. The holographic prescription due to Ryu and Takayanagi (RT) leads to an equation describing how the entangling surface extends into the bulk geometry. We show that setting to zero the time-time component of the Brown-York stress tensor evaluated on the co-dimension one entangling surface, leads to the same equation. By considering a spherical entangling surface as an example, we observe that Euclidean action methods in AdS/CFT will lead to the RT area functional arising as a counterterm needed to regularize the stress tensor. We present arguments leading to a justification for the minimal area prescription. Entanglement entropy of a local quantum field theory is a useful concept featuring in diverse areas ranging from black holes in general relativity [1, 2] to Fermi surfaces in condensed matter systems [3, 4]. Entanglement entropy in conformal field theories in even d dimensions [5, 6, 7] takes the form Here l is a length scale parametrizing the size of the entangling region and /epsilon1 is a short-distance cutoff. The leading l d -2 term gives the famous area law with a non-universal proportionality constant-when d = 2 the leading term is the log term. The coefficient of the log term is a universal quantity typically related to a function of the conformal anomalies in the theory [5, 8, 9]. Entanglement entropy has also proved useful in quantifying the number of degrees of freedom in quantum field theories [10, 11]. In the context of quantum field theories, a direct computation of entanglement entropy is hard and has been possible only in very specific examples. Typically numerical techniques and the so-called replica trick are used [12] . Owing to its diverse applications [13], it is of crucial importance to probe other computational tools available to us. One useful computational prescription originally proposed by Ryu and Takayanagi (RT) comes from the gauge/gravity correspondence [14]. The correspondence demands the existence of a duality between a quantum field theory in d dimensions and a theory of gravity (possibly string theory) in one dimension higher. For computational purposes one typically uses Einstein gravity in a weakly curved anti-de Sitter (AdS) background which corresponds to a strongly coupled conformal field theory (CFT). According to this prescription [5], in order to derive the holographic entanglement entropy for a d dimensional quantum field theory, one has to minimize the following entropy functional on a d -1 dimensional hypersurface (a co-dimension 2 surface), where /lscript P is the Planck length and h is the induced metric on the hypersurface. The minimal surface extending into the bulk coincides with the entangling surface in the CFT at the AdS boundary. The gravity dual theory is simply Einstein gravity with a negative cosmological constant where g is the determinant of the bulk metric, R is the scalar curvature for the bulk space time and L is the AdS radius. The generalization of the RT prescription to a class of higher derivative theories of gravity called Lovelock theories has been proposed recently [15]. The hypersurface is a co-dimension 2 surface and extends into the extra dimension. The minimization of S leads to an equation which gives the way the entangling surface extends into the bulk spacetime. In this paper, for definiteness we will consider d = 4 and take the entangling surface to be either a sphere or a cylinder. For the AdS 5 metric in Euclidean signature , Here z is the radial coordinate of the AdS space corresponding to the extra dimension. The field theory lives on the surface parametrized by ( t, x i ). The z = 0 slice corresponds to the boundary of AdS and according to the AdS/CFT dictionary corresponds to the ultraviolet (UV) regime of the field theory. Now we can choose, either ∑ 3 i =1 dx 2 i = dr 2 + r 2 d Ω 2 2 corresponding to spherical coordinates for the boundary or ∑ 3 i =1 dx 2 i = dv 2 + dr 2 + r 2 dφ 2 corresponding to cylindrical coordinates for the boundary. Choosing the surface t = 0 , r = f ( z ) one has to evaluate the entropy functional S , then find the equations of motion for f ( z ) for the corresponding geometry of the entangling surface. Since we want the entangling surface in the field theory on the z = 0 slice to be a sphere ( S 2 ) or a cylinder ( R × S 1 ) we demand that f ( z ) satisfies where f 0 gives the radius of the S 2 or the S 1 . Let us review how this works. We first put r = f ( z ) , t = 0 in (4). Then, 1 where n = 1 for the cylinder and n = 2 for the sphere. The volume form d 3 x = sin( θ ) dzdθdφ for the sphere and d 3 x = dzdvdφ for the cylinder. From here we get the following Euler-Lagrange equation for f ( z ) , Solving this equation as an expansion around z = 0 fixes f 1 , f 2 in terms of f 0 . It leads to for the cylinder. For the sphere one can get an exact solution of the form When one evaluates the on-shell action eq.(6) and expands around z = 0, then one gets a result exactly of the form in eq.(1). The RT prescription satisfies the strong subadditivity condition that entanglement entropy is known to satisfy and also passes some other nontrivial consistency checks [6]. However, attempted derivations [16] of this prescription are plagued with problems [17]. For instance, the implementation of the Replica trick needs the introduction of a conical deficit in the spacetime that the field theory lives. This leads to an introduction of the conical singularities in the bulk and it is not known how to deal with such singularities consistently. Until recently, the only case where a derivation exists is in the case where the entangling surface is a sphere [18]-the situation has changed with a proposed derivation by Lewkowycz and Maldacena [39]. However, this derivation uses the replica trick in order to derive entanglement entropy. It is important to know if there are ways to derive entanglement entropy in holography that does not use the replica trick. As such it is important to explore the RT prescription to find clues that may shed light on an alternative way to a derivation. Let us begin by calculating the stress tensor for the field theory living on a r = f ( z ) co-dimension 1 surface using holography. The Brown-York (holographic) stress tensor is given by [19] where K ab = e α a e δ b p β δ ∇ α n β is the extrinsic curvature, K is the trace of it, p α γ = g α γ -n α n γ is the projection operator, n β is the normal for the surface r = f ( z ), α, β, γ, δ and a, b are bulk and boundary indices respectively. e α a 's are the pullbacks. S ct is the counterterm action needed to regularize the stress tensor. We will address this contribution separately in a moment. Conventionally, the stress tensor is evaluated on z = /epsilon1 with /epsilon1 → 0, corresponding to the UV of the field theory. However, we will compute the stress tensor on the slice r = f ( z ). Furthermore, note that we will not set t = 0 when calculating the stress tensor, unlike the RT prescription. As a result we will be computing the tensor on a 4d slice. One could have equivalently chosen the slice z = ρ ( r ). In this case for each fixed z , on the dual CFT side, we would have to consider the entangling surface at different RG scales. It may be possible to set up the RG equations leading to the entangling surface along these lines similar in spirit to [20]. For this paper, we will compute on the r = f ( z ) slice. Another point that we should emphasise is that we are assuming Dirichlet boundary conditions throughout. Neumann boundary conditions on the r = f ( z ) slice appear in the considerations of Boundary Conformal Field Theory as in [21]. Now observe the following. Since the time direction is a direct product with the rest, the trace of the extrinsic curvature satisfies (4) K a a = (4) K t t + (3) K i i . Thus if we demand that T t t = 0 = (4) K t t -h t t (4) K a a leads to (3) K i i = 0 which is the same as the minimal surface condition for the 3d slice used in the RT calculation. Of course, if we considered higher curvature gravity, the T t t = 0 condition may be used to fix the surface f ( z ). At this stage the stress tensor could still have divergences and we need to add a suitable S ct to remove them. If the ( K tt -h tt K ) piece of the stress tensor is zero, then the counterterm should not affect this result. As such the relevant counterterm will turn out to live on the t = 0 slice as we will explicitly show below. One can easily calculate K ab and hence T ab using standard methods. We find Here n = 2 would correspond to a sphere and n = 1 to a cylinder respectively. Thus we find that Setting this to zero will lead to exactly the same equations as the RT prescription as we argued. This also seems consistent with the fact that we are after all interested in the entanglement entropy of the ground state [22]. We will now determine the counterterm needed to get a finite stress tensor. Let us focus on the sphere case. Using the solution for f ( z ) we can calculate the stress tensor as an expansion around the boundary z = 0. Let us start with the total Euclidean gravitational action. We will use the intuition gained from computing black hole entropy where the total action is evaluated by integrating from the horizon to infinity, I bulk is given by eq.(3) where the integration limits for r go from f ( z ) to Λ with Λ →∞ is a radial cut-off, I r = f ( z ) GH is the surface term for the r = f ( z ) slice which is used in the calculation of eq.(12). I z = /epsilon1 GH , I z = /epsilon1 ct are the usual surface and counterterm [19] evaluated on the z = /epsilon1 surface with /epsilon1 → 0 corresponding to the usual AdS boundary. We will rescale the time coordinate as ˆ t = t f 0 . The stress tensor contribution coming from r = f ( z ) slice is given by, Note that there are 1 z divergences in the stress tensor in all components except T tt which is zero. Thus to regularize this stress tensor we will need a three dimensional counterterm on a co-dimension 2 surface. Remarkably, if we add the RT-functional in eq.(2) as I r = f ( z ) ct = -S , the stress-tensor becomes free of divergences. Now let us evaluate various pieces of the total action as an expansion around z = /epsilon1 : Firstly notice that the log /epsilon1 pieces arising from I bulk and I r = f ( z ) GH cancel. How does one get entropy from here? When we have a black hole, we identify the periodicity of the Euclidean time as inverse temperature. In this case if we identify the periodicity of the time coordinate t with the inverse Unruh temperature [23, 24] 1 / (2 πf 0 ) which can be shown using independent arguments as in [18], then after identifying the total action as the free energy we find where F = I tot T is the free energy functional and a = π 2 L 3 /lscript 3 P . We identify S EE as the entanglement entropy and is precisely what arises as the log term in the RT prescription. Notice that in this way of calculating the entropy, the power law divergences have cancelled out. We have also checked that the results of Gauss-Bonnet gravity [15] are reproduced using this approach [25]. At this stage one can give an argument as to why the area minimization prescription of RT leads to the same result as above. In an Euclidean path integral, the total action considered above would be a functional of f ( z ). The way that f ( z ) gets fixed in the path integral is as follows. In the total action considered above, f ( z ) appears in I bulk + I r = f ( z ) GH + I r = f ( z ) ct . If we consider varying f ( z ) then since in I bulk + I r = f ( z ) GH , such variations correspond to variations in the g zz component of the metric, these will vanish on using the equations of motion for the background. However, δI r = f ( z ) ct /δf ( z ) has to be set to zero independently. Thus we are led to the minimization prescription. The fact that T tt = 0 gave rise to the same equation as what follows from the minimization is a consistency check of Lastly, this argument. This argument is similar to the one used by Fursaev [16] except that in our approach we did not have conical singularities in the bulk. This points to the possibility that there may be methods that do not rely on the replica trick for computing entanglement entropy. Now one can ask if this observation can be used to extract f ( z ) from the field theory. Firstly note that in the holographic calculation we put r = f ( z ) and the resulting slice was a 4d one. We will consider field theory on this 4d slice-the idea is to see if we can construct f ( z ) by demanding the vanishing of the tt component of the field theory stress tensor in this geometry. This stress tensor is not the usual stress tensor that is evaluted on the z = 0 surface. We can give a heuristic motivation for our calculation as follows (see [25] for some more details). It has been argued in [38] that there is a connection between the entanglement renormalization scheme of MERA and the way that entanglement entropy is calculated in AdS/CFT. In this connection, the more coarse grained one makes the quantum system, the deeper one is in the IR. The coarse graining is done in a specific way using unitary operators. As such if one starts with the ground state, we expect to be in the ground state after any number of coarse-graining steps with respect to a new hamiltonian that is a unitary transformation of the orginal hamiltonian. Imposing T tt = 0 along the radial evolution in AdS/CFT will ensure that we are in the ground state. Roughly speaking this is the reason why we are interested in doing the field theory calculation in this seemingly unusual way. It is well known that when one considers the expectation value of the stress tensor of any quantum field theory on a curved background [27], then there are UV divergences. These UV divergences depend on the local geometry and involve the local Riemann tensor and its contractions. This is expected since the divergences arise due to short wavelength modes which probe local geometry. Crucially the geometric feature of the divergences is independent of the global features of the spacetime as well as the actual quantum state involved [27, 28]. Once we regularize, the finite stress tensor will depend not only on geometric pieces but also the long wavelength features such as the global properties of the manifold as well as the actual quantum state involved. We will focus on cases where the stress tensor has divergences since here they are governed (upto overall constants) by local geometry [29]. This is also to be consistent with the fact that in the holographic calculation we worked with the unrenormalized stress tensor. Let us begin by considering for definiteness a massless scalar whose divergent stress tensor in dimensional regularization about d = 4 is given by [27, 30, 31] where, Following the holographic calculation, if we demand 〈 T tt 〉 div = 0, then we find [32] that for the sphere case, f ( z ) = √ f 2 0 -z 2 while for the cylinder case the expansion agrees with what arises from the RT prescription upto O ( z 2 ). Specifically, for the sphere, when evaluated on-shell R ab = -3 L 2 h ab . Using this it is easy to see that (1) H ab and (2) H ab will vanish. For the cylinder we first assume a f ( z ) of the form, f ( z ) = f 0 + f 2 z 2 . Then, Setting the O ( z 2 ) term to zero gives ( f 2 = 0 as otherwise it leads to inconsistency at the next order), /negationslash which is the same as what follows from the RT prescription. In the cylinder case, we note that it is precisely upto O ( z 2 ) order that is needed to extract the universal log term in the entanglement entropy while O ( z 4 ) terms gave rise to subleading non-universal terms. We find that the agreement for the cylinder case f ( z ) breaks down at O ( z 4 ). This is not surprising since the O ( z 4 ) terms will also be sensitive to the finite pieces of the stress tensor. What is perhaps surprising is the exact agreement for the sphere. Let us conclude with some observations. First, we found that the RT area functional arises as a counterterm to get a regularized stress tensor on the r = f ( z ) slice. As a strong cross-check this result extends easily to Gauss-Bonnet gravity and reproduces the area functional considered in [15, 26]. Wald entropy evaluated on the entangling surface gives wrong results for the cylinder case as noted in [15]. The difference is due to extrinsic curvature terms in the Wald entropy. Our derivation explains why the area functional does not have terms depending on the 3-extrinsic curvatures as observed in [15]. The reason is that we want a well defined Dirichlet problem and the presence of 3-extrinsic curvatures in the counterterm action would lead to variations in directions normal to the surface. This way of thinking also suggests a systematic way of deriving the area functional in more complicated examples. For example, while we have shown that the RT functional arises as a counterterm at a CFT fixed point, our approach would also suggest some differences from the RT prescription in the context of RG flows in case there are contributions to the co-dimension two counterterm action from the matter inducing the flow. It would be interesting to study examples where this happens. Second, we have tried to extract information about the way the entangling surface extends into the bulk by using the UV divergences in the field theory. We assumed a background that is given by the r = f ( z ) slice of AdS. As stated earlier, ideally, we would like to start by considering r = f 0 and a UV cutoff and see how r changes as we change the cutoff-this is an important future problem. In fact, in the field theory calculation, although we used a massless scalar for concreteness, the same result would hold for any massless field [27]. This strongly suggests that the form for f ( z ) at least upto O ( z 2 ) is independent of the actual gravity dual to conformal field theories. This is borne out in the higher derivative calculations with Gauss-Bonnet terms performed in [15]. Further this procedure was crucially tied to even dimensions since in odd dimensions the analogous divergences are absent in dimensional regularization. In odd dimensions, one would need to use the regularized stress tensor which would then become sensitive to global properties of spacetime as well as the state being used. It will be interesting to extend this to d = 3 [33]. It will also be interesting to see if there is any connection with the covariant generalizations of the RT proposal [34]. Curiously, eq.(20) is what would arise from the 'pole' type counterterms in AdS/CFT. These encode the conformal anomaly. In fact the so-called dilaton action [37] can be derived by carefully regularizing the pole term [36]. This leads us to suspect that it may be possible to derive information about f ( z ) from the dilaton action itself by appropriately identifying the dilaton with f ( z ). Another point that we wish to emphasise is that constructing f ( z ) from entanglement renormalization is a part of the program [38] for connecting AdS with continuous Multiscale Entanglement Renormalization Ansatz (cMERA). The fact that we have been able to get some information of f ( z ) using a field theory calculation gives credence to the RT prescription as well as hope that f ( z ) construction from AdS/cMERA should be possible. Acknowledgments : We thank Sayantani Bhattacharyya, Janet Hung, Chethan Krishnan, Gau- tam Mandal, Rob Myers, Miguel Paulos, Suvrat Raju and Tadashi Takayanagi for useful discussions. We especially thank Rob Myers, Miguel Paulos and Tadashi Takayanagi for useful comments on the draft. AS acknowledges support from a Ramanujan fellowship, Govt. of India.", "pages": [ 1, 2, 3, 4, 5, 6, 7, 8 ] } ]

[ { "title": "On the properties of the irrotational dust model", "content": "Jędrzej Świeżewski ∗ Faculty of Physics, University of Warsaw, Hoża 69, 00-681 Warszawa, Poland In this note we analyze the model of the irrotational dust used recently to deparametrize gravitational action. We prove that the remarkable fact that the Hamiltonian is not a square root is a direct consequence of the time-gauge choice in this model. No additional assumptions or sign choices are necessary to obtain this crucial feature. In this way we clarify a point recently debated in the literature.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The possibilty of obtaining a simpler, e.g., avoiding the problem of time, description of gravitational interactions by the means of a coupling to specific matter fields has been present in the literature for some time [1-4]. Some of those models gained more attention due to new possibilities they provided for the quantization of gravitational interaction especially in the context of Loop Quantum Gravity. One of the models considered is the irrotational dust model first introduced in [3]. It has recently been analyzed by the authors of [5-7], where it is claimed that the surprising fact that the Hamiltonian of this theory is not a square root, together with the kinematical structure of Loop Quantum Gravity, provides a complete theory of quantum gravity.[10] However, the lack of the square root in this model has been questioned in [8], where it is claimed that the square root may not explicitly appear in the action due to an artificial sign choice, but removing it in this way leads to serious problems on the quantum level. Hence, it is stated there that the square root is present in the theory in the form of an absolute value (a square root of a square). The aim of this note is to clarify the issue of whether the square root (or the absolute value) is absent from the model of irrotational dust and whether any artificial sign choices are necessary for this important feature to take place.", "pages": [ 1 ] }, { "title": "II. THE IRROTATIONAL DUST MODEL", "content": "The theory analyzed here consists of the irrotational dust field T coupled to gravity. It is described by the following action where the first term is the Hilbert-Einstein action, the second describes any type of standard matter content and the last term is the dust action. The dust Lagrangian is of the form where the non-dynamical field M plays a role of a Lagrange multiplier. In the following we will discuss the deparametrization scheme of the action (1) with respect to the dust field T , in which the rest of the matter content plays no role, hence we disregard it in the remainder of this work. After introducing ADM [9] variables to describe the geometric degrees of freedom, namely one finds the dust Lagrangian (2) in the form T Introducing the momentum conjugate to the dust field and rewriting the dust Lagrangian with the use of the momentum, the Legendre transformation can be completed. The dust action has the form Now one can proceed along two paths.", "pages": [ 1, 2 ] }, { "title": "A. Previous treatment", "content": "The authors of [5-7] and the authors of [8] turn to the analysis of the equations of motion implied by that action. The equation for M reads This condition links the values of M and p T , but not their signs. In order to express M as a function of p T the authors of [5-7] invoke an argument about the role of energy density played by M in the stress-energy tensor of the dust. Using this additional requirement they guarantee positivity of M .[11] Hence they can solve (7) and obtain This expression is then plugged into the dust action and the following expression is obtained Note that an absolute value of p T is present in the action. To dispose of the absolute value the authors of [5-7] fix the sign of p T to be positive by hand and proceed with their treatment. This sign choice is of crucial importance to the present note and we will come back to it later. At this stage the authors introduce the time-gauge namely they choose coordinates such that Two simple consequences of this choice are ˙ T = 1 and ∂ a T = 0 . Moreover, the dynamical preservation of this gauge enforces the condition Implementing these facts, they obtain the full constraints of the theory in the form C = C + p = 0 , (12) where C and C a are the standard Hamiltonian and vector constraints of canonical gravity. Solving the first constraint for p T , they complete the deparametrization of the action with respect to the dust ending up with a theory given by the action where C plays a role of a true, nonvanishing, Hamiltonian of the theory generating evolution in the dust time. This result has been questioned in [8], where the authors argue that the sign choice of p T imposes a sign choice of C (due to the constraint (12)), hence although no absolute value appears explicitly in (14), it is there implicitly since the sign of C is limited. This is then argued to be a source of problems in the process of quantization of the considered theory, since imposing the sign condition on the quantum level requires a detailed knowledge of the spectrum of C , which is not currently available. tot", "pages": [ 2 ] }, { "title": "B. Alternative treatment", "content": "Arriving at (6), one can follow a different route. Instead of analyzing stationarity of the action with respect to variations of M to express it as a function of p T , one can realize one more consequence of the choice of the time-gauge. As already noticed in [5] the choice t = T leads to ˙ T = 1 , ∂ a T = 0 and N = 1 . These results implemented into the definition of the dust momentum, given by (5), imply Note that here we can express M as a function of p T without any choices of signs. The M obtained from this equality satisfies the stationarity condition (7), and when plugged into (6) leads to where again the spatial derivatives of T vanish due to the gauge choice. After solving the constraints as it was done previously, we end up with a theory given by the action functional apparently identical with (14), with the crucial difference that the sign of C , being linked with the sign of p T by (12), is no longer limited since the latter sign is not fixed in the presented treatment. If one wishes to impose the condition limiting M to be positive introduced in [5-7] then from (15) we see that it limits p T to be positive and from (12) also C to be negative. We now see clearly that it is actually the stress-energy condition imposed on M which limits the sign of C and that the strongest result is obtained if no sign choices are introduced.", "pages": [ 3 ] }, { "title": "III. SUMMARY", "content": "To summarize, we analyzed the question whether gravitional action deparametrized with the use of the irrotational dust possesses the feature of its Hamiltonian not being a square root. The importance of this feature has been underlined in both the works of [5-7] and [8], however, the latter work criticizes the price that is paid to obtain it in the former treatment. Here we showed that this crucial feature is a direct consequence of the time-gauge in this model. Hence, no additional input, like the stress-energy tensor argument or the artificial sign choice is necessary. Therefore, at least some of the problems of the model pointed out in [8] can be avoided. We leave the evaluation of the full consequences of the new treatment on the quantum level for future research.", "pages": [ 3 ] } ]

[ { "title": "Andrew King 1 ,/star &Chris Nixon 1 , 2 , 3", "content": "Abstract. We review recent progress in studying accretion flows on to supermassive black holes (SMBH). Much of this removes earlier assumptions of symmetry and regularity, such as aligned and prograde disc rotation. This allows a much richer variety of e ff ects, often because cancellation of angular momentum allows rapid infall. Potential applications include lower SMBH spins allowing faster mass growth and suppressing gravitational-wave reaction recoil in mergers, gas-assisted SMBH mergers, and near-dynamical accretion in galaxy centres.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "It is now widely accepted that almost all reasonably large galaxies have supermassive black holes (SMBH) in their centres, and further that these holes grow predominantly through luminous accretion of gas (Soltan 1982, Yu & Tremaine 2002). Any realistic model of gas flows around black holes must take into account the angular momentum of the gas. Radiation processes can relieve the gas of orbital energy, but there is no equivalent process which can intrinsically remove angular momentum. As a result, gas flows tend to form rotationally supported discs with characteristic radii given by the angular momentum of the flow (Pringle 1981). Inward gas flow - accretion - would be impossible without some process (usually called 'viscosity') transporting angular momentum outwards in the disc, allowing mass to flow inwards (Lynden-Bell & Pringle 1974) and release its gravitational binding energy. Near the black hole this becomes very large, approaching a significant fraction of the rest-mass energy. For this reason, accretion discs probably power the most luminous objects in the universe, and understanding the nature of their angular momentum transport has been a major goal of modern astrophysics. The most promising candidate is the magnetorotational instability (MRI) (Balbus & Hawley 1991) which injects turbulence into the gas by stretching magnetic field lines. The viscosity coe ffi cient in accretion discs is often assumed isotropic and parameterised as (Shakura & Sunyaev 1973) where c s is the sound speed, H is the disc angular semi-thickness and α < 1 is a dimensionless parameter. In principle α is a function of position within the disc gas. This formalism characterises the maximum viscosity feasible in an accretion disc through an e ffi ciency parameter: any turbulent gas velocities above the sound speed would quickly shock and dissipate, and any turbulent length scales longer than H would lead to a disc thicker than H . The maximum viscosity is then ≈ c s H (Pringle 1981). In practice much of the physics of accretion discs depends only on low powers of α , allowing simple insights by taking α as a global constant. From (1) we can write down the accretion timescale t visc = R 2 /ν for a disc as This is evaluated here for typical active galactic nuclei (AGN) disc parameters: α ∼ 0 . 1 from King et al. (2007), and H / R ∼ 10 -3 from e.g. Collin-Sou ff rin & Dumont (1990). This form already allows an important conclusion about AGN accretion. From our remarks above, this is how SMBH grow, and so must occur on timescales significantly shorter than the age of the universe. Requiring t visc to be shorter than a Hubble time, we see that the disc scale radius must be R /lessmuch 1 pc. We reach a similar conclusion by considering the e ff ects of self-gravity on an AGN disc. At a radius of only ∼ 0 . 01 pc King et al. (2008) (see also Levin 2007, Goodman 2003) find that discs become gravitationally unstable. For SMBH this instability is catastrophic and results in a complete fragmentation. Most of the disc gas forms stars, starving the inner disc (Goodman 2003, Levin 2007, King et al. 2008). This suggests that discs feeding SMBH must either be fed at mass rates many orders of magnitude below Eddington, or form as small scale 'shots'. Observations of Eddington accretion suggest the latter. These simple arguments already point to a new picture for SMBH accretion which we discuss here.", "pages": [ 2, 3 ] }, { "title": "2. SMBH accretion", "content": "Astrophysical black holes have two important parameters: mass and angular momentum (spin). The spin controls the specific binding energy of matter accreting on to the hole, and so the accretion luminosity for a given black hole mass. Accretion grows the SMBH mass, but also a ff ects its spin. The huge disparity in scales between the black hole ( R g = 5 × 10 -6 M 8 pc) and the galaxy ( ∼ 10 -100 kpc) strongly suggests that the angular momenta of the black hole spin and of the gas trying to accrete on to it cannot be correlated in direction, at least initially. In particular the orbital plane of most of the disc mass is unlikely to be aligned with the SMBH spin plane. (This is the first example of several we shall encounter where strong symmetry assumptions [here alignment], originally made for simplicity, turn out to have a distorting e ff ect in suppressing various important e ff ects.) The first work on the behaviour of misaligned discs was by Bardeen & Petterson (1975). This was actually in the context of stellar-mass black hole binary discs where for example an asymmetric supernova kick may have significantly misaligned the black hole spin and the binary orbit (e.g. Roberts 1974). The physics here is set by the Lense-Thirring e ff ect. Lense & Thirring (1918) showed that the dragging of inertial frames makes misaligned test particle orbits precess around the angular momentum of a gravitating body at a rate where J h is the angular momentum (here of the black hole). This precession is strongly di ff erential - much faster for gas close to the black hole. The precession time is ∣ ∣ where J h = | J h | = aGM 2 / c (Kumar & Pringle 1985). From (4) it is easy to see that for gas orbiting close to the black hole horizon ( R ≈ R g) the precession time can be as short as the dynamical time (at which point the orbit is no longer near-circular). However, the precession is strongly dependent on radius, so for gas far from the black hole the precession is entirely negligible: Thus on scales /greaterorsimilar 1 pc, precession induced by the SMBH can be ignored. Bardeen & Petterson (1975) considered a disc of gas subject to a strongly di ff erential precession of this type (modelled in Newtonian gravity, as everything we shall consider). These authors suggested that dissipation in the disc between the di ff erentially precessing rings causes the disc to align to the black hole spin, more quickly in the centre than in the outer parts. So after some time the inner disc is aligned, the middle disc is warped and the outer disc is still misaligned. For an isolated disc-hole system, the warp propagates outwards until the entire disc is aligned. Later, Papaloizou & Pringle (1983) showed that these early investigations into warped discs (e.g. Petterson 1977, Petterson 1978, Hatchett et al. 1981) did not properly handle the internal fluid dynamics in a warped disc. In particular their equations did not fully conserve angular momentum, although the conclusions of Bardeen & Petterson (1975) still hold qualitatively. Papaloizou & Pringle (1983) investigated the Navier-Stokes equations for a warped disc in the linear (small warp) regime. They discovered that the warp can propagate in two distinct ways. In the first of these, viscosity dominates ( α > H / R ), and things behave di ff usively, as in Bardeen & Petterson (1975). The second mode of propagation is wave-like, and pressure dominates ( H / R > α ). For black hole discs, the di ff usive mode is expected to dominate as the discs are generally thin ( H / R ∼ 10 -3 ) and viscous ( α ≈ 0 . 1). We therefore focus on this case. For wave-like discs see e.g. Papaloizou & Lin (1995); Lubow & Ogilvie (2000) & Lubow et al. (2002). Global solutions in full 3D hydrodynamics were hard to achieve, so to make progress Pringle (1992) used conservation equations to derive an equation governing the evolution of a twisted disc composed of circular rings interacting viscously. Using the equations of Pringle (1992), Scheuer & Feiler (1996) calculated the secular evolution of warped discs. A subtle error in their calculation led them to conclude that discs always coaligned with the black hole spin - even if they began close to counteralignment. This led to the conclusion that all black hole discs align on timescales ∼ α 2 t visc. As α < 1, with observations suggesting α ≈ 0 . 1 (King et al. 2007), this disc-BH coalignment occurred long before significant accretion could take place. This largely removed the motivation for studying misaligned or warped discs. The alignment error had serious consequences: since alignment would always occur rapidly whatever the initial orientation of disc and hole spin, SMBH would always gain almost all their mass from prograde accretion discs, thus spinning up to near-maximal values after doubling their masses. This in turn made the specific binding energy release maximal (about 40% of rest-mass energy). Since radiation pressure inhibits accretion (via the Eddington limit), this implied that only rather low mass accretion rates were possible. The discovery of SMBH with masses /greaterorsimilar 10 9 M /circledot at redshifts z /greaterorsimilar 6 (allowing accretion only for a timescale of order 10 9 yr) then appeared to require that these holes must have started from initial 'seeds' with significant masses /greaterorsimilar 10 6 M /circledot . Almost a decade passed (with the belief in rapid coalignment well entrenched in SMBH models) before King et al. (2005) discovered the error in Scheuer & Feiler (1996) - the implicit assumption of infinite disc angular momentum. With this restriction lifted King et al. (2005) showed that counteralignment simply requires where θ is the angle between the disc and hole angular momentum vectors with magnitude J d & J h respectively. This condition ensures that the the total angular momentum vector (sum of disc and hole) is shorter than the hole angular momentum. Since the latter is subject only to precessions, its length cannot change during the alignment process, meaning that hole and disc must end up opposed to account for the shorter total angular momentum. With retrograde discs now plausible - indeed likely, King & Pringle (2006) & King & Pringle (2007) began to explore the e ff ect of misaligned discs on the evolution of SMBH. The larger lever-arm of retrograde accretion flows near the black hole horizon o ff ers an obvious way of keeping their spins low, and so making SMBH growth e ffi cient, particularly if accretion disc events have no preferred direction with respect to the host galaxy. This scenario (often called 'chaotic accretion'; King & Pringle 2006, King & Pringle 2007) is consistent with many observational e ff ects, such as the observed lack of correlation of the directions of AGN jets (thought to be orthogonal to the plane of the accretion disc close to the hole) with large-scale properties of the host (King et al. 2008). There was renewed interest in warp propagation through accretion discs. Lodato & Pringle (2006) used the numerical method of Pringle (1992) to explore the evolution predicted by King et al. (2005), confirming that the criterion (6) correctly determines the alignment history (co, counter, or more complex) of accreting black holes. In the chaotic accretion picture, a natural question is what happens when a misaligned accretion event occurs on to a pre-existing disc. Nixon, King & Price (2012) explored this with numerical simulations. As partially opposed gas flows interact through viscous spreading, significant amounts of angular momentum can be cancelled, leading to gas infall and strong accretion. Perhaps surprisingly, this cancellation can occur even in a single disc event, as su ffi ciently inclined discs may change their planes almost discontinuously, rather than in the smooth warps envisaged by BP (Nixon & King 2012). There had been sporadic evidence that discs could 'break' in this way (Larwood et al. 1996, Fragner & Nelson 2010, Lodato & Price 2010). The theoretical possibility of this idea was further studied by Ogilvie (1999): for a locally isotropic viscosity coe ffi cient, he showed that the stresses in a strongly warped disc would evolve in such a way that the forces trying to bring the disc back into a single plane would actually weaken as the warp grew. The first systematic investigation of this possibility was by Nixon & King (2012), who used the Pringle (1992) method to explore warp propagation with the full e ff ective viscosity coe ffi cients derived by Ogilvie (1999). This revealed modified Bardeen-Petterson (BP) behaviour where a sharp break in the disc occurs between the aligned inner disc and the misaligned outer disc. However, the method of Pringle (1992) forces the disc to respond viscously (following a di ff usion equation) to the Lense-Thirring precession and excludes other possible hydrodynamical e ff ects. The sharp disc break found in Nixon & King (2012) suggested that this assumption was too restrictive, and that a full 3D hydro numerical approach was needed. Accordingly Nixon, King, Price & Frank (2012) made SPH (smoothed particle hydrodynamics) simulations of misaligned discs around a spinning black hole. SPH was already known to reproduce the behaviour derived by Ogilvie in modelling the communication of a warp in a fluid disc (Lodato & Price 2010), and therefore suitable for modelling such discs. Nixon, King, Price & Frank (2012) confirmed the expected BP behaviour of lowinclination discs, but showed that misaligned discs can break. Further, a break can promote cancellation of disc angular momentum. Separated disc annuli precess independently of each other, and so inevitably become partially opposed. In e ff ect each annulus borrows angular momentum from the central black hole to achieve the cancellation, so that in some way the black hole is complicit in feeding itself more rapidly than simple viscous evolution would allow. This tearing behaviour di ff ers radically from smooth BP evolution, and we are only beginning to understand its implications. Fig. 1 shows the 3D disc structure for a small and large inclination disc around a spinning black hole (Nixon, King, Price & Frank 2012). Importantly, it is clear that although tearing was first noted for accretion on to spinning black holes, the crucial element is that disc orbits precess di ff erentially. This always happens if the e ff ective potential for the disc flow has a quadrupole component. We can therefore expect tearing to occur quite generically for disclike flows on all scales, not merely around spinning black holes, but also around SMBH binaries (see below) and on the scale of an entire galaxy. The potential objection to the ideas of breaking and tearing is that current treatments rely on a scalar viscosity (stress proportional to strain), whereas a more general tensor relation might conspire to hold the vertical disc structure together. Intuitively this seems unlikely for the currently-favoured MRI picture: where the 'horizontal' viscosity driving angular momentum transport and accretion is constantly pumped by azimuthal winding up of magnetic fields, the vertical relative motion of the two sides of a warp is periodic and bounded, suggesting that if anything an MRI viscosity might be still weaker in the vertical direction. Current MRI simulations are very far from being able to answer such questions, not least because of the unphysical di ff usive e ff ects of having a disc plane inclined with respect to the grid symmetries.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3. SMBH binaries", "content": "Most galaxies have supermassive black holes, and galaxy mergers are common. Dynamical friction drives the SMBHs of a merging galaxy pair close together in the centre of the merged galaxy: binary angular momentum is lost to surrounding stars. This forms an SMBH binary with a typical separation of order 1 pc, but cannot coalesce the holes, since all the suitable stars have been driven away by the dynamical friction process itself. Since very few SMBH binaries are actually observed, some as yet unknown process must drive the binary to coalescence. This is the last parsec problem (Begelman et al. 1980, Milosavljevi'c & Merritt 2001). There have been various suggested solutions, generally of two types. One type invokes collisionless matter to arrange that stellar orbits passing close to the SMBH binary are constantly refilled, and so available to remove it angular momentum. Recent attempts along these lines have met with some success, e.g. with triaxial dark matter (DM) haloes refilling the stellar orbits (Berczik et al. 2006), or using the Kozai mechanism in triple SMBH systems (Blaes et al. 2002), or non-axisymmetric potentials (Iwasawa et al. 2011). However it seems clear that at some level gas must play a role, and attempts to invoke it constitute the second type of proposed solution of the last parsec problem. Shocks in the galaxy gas flows can rob the gas of the angular momentum supporting its orbit so that it falls towards the binary with a random inclination. Quite separately, to observe SMBH binaries in action we need to understand how they interact with gas. All early studies of gas interacting with an SMBH binary were, as in Section 2, limited to discs which were both coplanar and prograde wrt the binary. But then two physical e ff ects severely limit the e ff ect on binary evolution. First, the disc mass is limited by self-gravity to where H / R /lessmuch 1. So any disc with enough mass to a ff ect the binary quickly fragments into stars. Second, an infinite family of prograde disc orbits are resonant with the binary rotation, holding the gas far out. In principle this shrinks the binary by transferring angular momentum to the gas, but much too slowly to be helpful. Rather as in the case of dynamical friction, the e ff ect which removes angular momentum from the binary tends to weaken itself by gradually pushing away the agency (here the prograde disc gas) which e ff ects this removal in the first place. But there is no compelling reason to suppose that the circumbinary gas disc is prograde. And the reasoning of the last paragraph strongly suggests that things would work much better with retrograde discs, as happened with SMBH growth in Section 2. Retrograde discs do not su ff er resonances (Papaloizou & Pringle 1977), so disc gas can instead accrete freely on to the binary with negative angular momentum. Nixon, Cossins, King & Pringle (2011) show that once a retrograde moving mass ∼ M 2 has interacted with the binary, its eccentricity approaches unity (here 'interacted' means gravitationally - the gas need not accrete for example). Even before this happens, gravitational wave losses complete the SMBH coalescence. So the timescale for this phase of the merger is given by ∼ M 2 / ˙ M , where ˙ M is the accretion rate through the retrograde disc. However, (7) tells us that any individual accretion event cannot have a mass /greaterorsimilar M 2 unless the mass ratio is very small ( /lessorsimilar H / R ). Therefore we must consider multiple, randomlyoriented events. As prefigured at the end of the last Section, Nixon, King & Pringle (2011) showed that much of the disc alignment and breaking phenomena derived for a disc around a single spinning black hole hold for a circumbinary disc. Here the quadrupole part of the binary potential induces a similar (but stronger) precession to the Lense-Thirring e ff ect. In particular Nixon, King & Pringle (2011) conclude that counter-alignment of the circumbinary disc occurs if and only if exactly as (6) for the black hole case. Nixon (2012) showed that counteralignment of a circumbinary disc is stable (contrary to previous reports), and that for any reasonable set of parameters the binary dominates the angular momentum of the system. So approximately half of all randomly oriented accretion events are retrograde. All this means that a sequence of ∼ 2 qR / H randomly oriented accretion events, each limited by self-gravity, can drive the binary eccentricity close to unity and allow gravitational wave losses to complete the black hole merger. But there is a subtlety here. If we assume that collisionless processes can drive the binary in to 0.1 pc, the viscous time (appropriate for retrograde circumbinary discs) is This time is so long that at first glance it appears unlikely that gas accretion can help on these scales. The merger time is given by where taking t visc from (9) gives an upper limit. For typical numbers, qR / H ≈ 10 -100, and so (10) would be longer than a Hubble time. However (Nixon et al. 2013) recently showed that misaligned circumbinary discs can tear and cancel angular momentum (see Fig. 2), and therefore the accretion rates may be increased by factors up to 10 4 for sustained periods (see Fig. 3). So depending on the mass supply from the galaxy, it is possible that the gas-driven merger timescale can be as short as ∼ 10 6 -10 7 yr from 0.1 pc, and considerably shorter still if collisionless dynamics puts the starting point even further in. It appears then that collisionless (e.g. Berczik et al. 2006, Khan et al. 2011) and collisional processes (e.g. those considered here) conspire together to solve the last parsec problem with collisionless processes driving the binary to the scales ( /lessorsimilar 0 . 1 pc) where gas can have a strong e ff ect. Then, if collisionless processes can go no further, gas completes the merger provided the galaxy can supply enough mass ( ∼ M 2) on the right (chaotic) orbits.", "pages": [ 6, 8, 9, 10 ] }, { "title": "4. Conclusion", "content": "The common thread in most of the work reported here is the gradual removal of assumptions of regularity and symmetry of gas flows near supermassive black holes. These assumptions included those of disc flows aligned with the spin plane of a single black hole, or the orbital plane of an SMBH binary, as well as prograde rotation in most cases. Although probably initially made on grounds of simplicity, these assumptions have little basis in reality. We have seen that they arbitrarily rule out a rich variety of phenomena and actually create artificial di ffi culties in many cases. In particular by keeping the sense of rotation of everything parallel, they make angular momentum barriers to infall and mergers formidable. The central regions of galaxies are not in general strongly coherently rotating, so cancellations between opposed gas flows must be a common phenomenon. So we have seen that in general SMBH do not inevitably have high spins - indeed if accretion events have no preferred direction there is a slow but persistent statistical trend towards lower spins (King et al. 2008), something that was already known to be the outcome of repeated SMBH coalescences (Hughes & Blandford 2003). Several problems are greatly eased by this - we have already noted that the Eddington limit of a slowly spinning hole is much less of a barrier to black hole mass growth, and in addition note that low spins make strongly anisotropic gravitational wave reaction unlikely, allowing galaxies to retain the merged black holes. For completeness we should add that the same physics makes the often-invoked idea of jet precessions seem unlikely (Nixon & King 2013). This paper also points out that SMBH spins barely move under the e ff ect of individual accretion events, and instead perform very slow random walks in direction. Perhaps the most spectacular consequence of removing symmetry assumptions is that disc flows may generically be subject to breaking and tearing (Nixon & King 2012, Nixon, King, Price & Frank 2012, Nixon et al. 2013). By temporarily borrowing angular momentum from the source of the potential, gas rings arrange to cancel it with neighbours to allow accretion to occur on near-dynamical timescales. This may potentially add considerable new insight to our current picture of many gas-dynamical e ff ects in astrophysics.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "Research in theoretical astrophysics at Leicester is supported by an STFC Consolidated Grant. CN acknowledges support for this work, provided by NASA through the Einstein Fellowship Program, grant PF2-130098.", "pages": [ 11 ] }, { "title": "References", "content": "Balbus S A & Hawley J F 1991 ApJ 376 , 214-233. Bardeen J M & Petterson J A 1975 ApJ 195 , L65 + . Begelman M C, Blandford R D & Rees M J 1980 Nature 287 , 307-309. Berczik P, Merritt D, Spurzem R & Bischof H P 2006 ApJ 642 , L21-L24. Blaes O, Lee M H & Socrates A 2002 ApJ 578 , 775-786. Collin-Sou ff rin S & Dumont A M 1990 A & A 229 , 292-328. Fragner M M & Nelson R P 2010 A & A 511 , A77. Goodman J 2003 MNRAS 339 , 937-948. Hatchett S P, Begelman M C & Sarazin C L 1981 ApJ 247 , 677-685. Hughes S A & Blandford R D 2003 ApJ 585 , L101-L104. Iwasawa M, An S, Matsubayashi T, Funato Y & Makino J 2011 ApJ 731 , L9. Lense J & Thirring H 1918 Phys. Z. 19 , 156. Levin Y 2007 MNRAS 374 , 515-524. Lodato G & Price D J 2010 MNRAS 405 , 1212-1226.", "pages": [ 11 ] } ]

[ { "title": "Modified Ricci flow and asymptotically non-flat spaces", "content": "Shubhayu Chatterjee ∗ 1 and Narayan Banerjee † 2 ∗ Department of Physics, Indian Institute of Technology, Kanpur; Kanpur 208016; India. IISER - Kolkata, Mohanpur Campus, P.O. BCKV Main Office, District Nadia, † West Bengal 741252, India.", "pages": [ 1 ] }, { "title": "Abstract", "content": "The present work extends the application of a modified Ricci flow equation to an asymptotically non flat space, namely Marder's cylindrially symmetric space. It is found that the flow equation has a solution at least in a particular case. PACS Nos.:02.40.-k; 04.70.Dy", "pages": [ 1 ] }, { "title": "Introduction:", "content": "The Ricci Flow (RF) is an evolution equation for a Riemannian metric g µν with respect to a scalar parameter, say τ , in terms of the Ricci tensors R µν . The equation reads as RF equation, or some suitably modified form of it, finds a lot of application in gravitation theories[1]. The Ricci flow equation has also been discussed in a cosmological scenario[2]. Recently Ricci flow is used in black hole physics, particularly in the investigation of the stability of a black hole[3]. In the context of Supergravity vacuum solutions, Ricci flow has been discussed by Hu et al[4]. Ricci flow in the connection with a braneworld scenario has been investigated by Das etal[5]. The reason for using the Ricci flow equation is the following. For three-dimensional manifolds, if we expand the metric around a flat space given by the equation g µν = η µν + h µν , we find that the equation (1), to linear order, yields This looks like a 'heat conduction equation' for h µν . This is indeed a parabolic equation and thus the solution tends to lose memory of the initial condition. It is well known in physics that a heat flow finally leads to a state with maximum entropy irrespective of the initial conditions. It is thus interesting to see if the Ricci flow equation can also lead to a state of maximum entropy for the gravitating system given by h µν , with the system forgetting the initial conditions when it reaches its final state. In order to bring out the correspondence between the statistical mechanis and gravity, Perelman proposed a modification of Ricci flow, known as the gradient formulation[6]. The equation looks like where f is a scalar on the manifold. However, as shown by Samuel and Roy Chowdhury [7], there exists no fixed point solution to these equations in the Schwarzschild space for any choice of fand thus does not give any extremum. They therefore concluded that Perelman's entropy function is not connected to the Bekenstein-Hawking entropy for black holes. They have also postulated a modification of Perelman's gradient formulation as supplemented by an evolution equation for f, which we can relate to the scalar diffusivity in thermodynamics, The only difference here from Perelman's formulation being the appearence of the function f in the first term of the right hand side. The equations for the fixed points of the above flow are For the asymptotically flat exterior Schwarzschild space, the geometry for a static spherically symmetric mass distribution, the scalar function f ( r ) = ( 1 -2 M r ) 1 2 satisfies both the equations, where M is the total mass of the spherical body. Thus the modified Ricci flow might have some connection with geometric entropy as its fixed points are extrema of the Bekenstein-Hawking entropy. We try to extend this concept of a scalar diffusivity function to asymptotically non-flat spaces, using Marder's solution to Einstein's field equations for cylindrically symmetric space, and thereby to check if thermodynamical considerations could be dealt with for asymptotically non-flat spaces also.", "pages": [ 1, 2 ] }, { "title": "The Scalar Diffusivity function:", "content": "Marder [8] gave the following exterior metric for a static cylindrically symmetric matter distribution Here c is a parameter roughly twice the mass per unit length of the cylinder, and B is another parameter related to c . For some t = constant hypersurface, the spatial part of the metric is The corresponding non-zero Ricci tensors are We now want a solution to the equations (6) and (7). Consistent with cylindrical symmetry, we assume that f is a function of the radial coordinate r only. For µ = ν = r , we get For µ = ν = z or φ , we get The equation for the covariant laplacian of f gives Equation (12) readily integrates to yield (subject to a simple choice for the arbitrary constants of integration) which satisfies both (10) and (11), and is hence a solution. Thus we have found a solitonic solution to the modified RF equations. The existence of such a scalar diffusivity indicates that we can pursue thermodynamics even for an asymptotically non-flat space. Equation (13) indicates that the scalar diffusivity function f ( r ) attains the significance of the lapse function, as f 2 is equal to g 00 . This is indeed consistent with the result obtained by Samuel and Roy Chowdhury[7].", "pages": [ 2, 3 ] }, { "title": "Application of the modified RF to the exterior Marder space:", "content": "We try to study the evolution of area of a cylinder of fixed height z and radius r with the parameter τ using the modified RF equation[7] in exterior Marder space. This is an extension of the work of Samuel and Roy Chowdhury [9]. The difference is that in the present case we extend the formalism for an asymptotically non-flat space. The curved surface area is The area of each of the caps is Therefore, the total area is The compactness C of the surface is given by where, R is the curvature scalar, and K is the trace of the extrinsic curvature. For the cylindrical surface under consideration, R = 0. K = D a ˆ n a , where ˆ n is the unit normal to the surface we are considering. For the curved surface of the cylinder, For the top surface, the unit normal is To calculate dA dτ , we need to use the Ricci flow equations (4)with the metric (9). For µ = ν = r , we get whereas, µ = ν = z or φ gives us In order to make equations (23) and (24) consistent, it is easy to see that c = 1 2 . Using (23) in (16) one obtains For c = 1 2 , we can clearly see that the following inequality holds As C is negative (equation (22)), the area under the modified Ricci flow is a decreasing function of τ .", "pages": [ 3, 4 ] }, { "title": "Conclusion:", "content": "We started with the modified Ricci flow [7], and proved that a scalar diffusivity function exists for asymptotically non-flat spaces by finding such a solution for the exterior Marder space. We also used the Ricci flow technique to study the evolution of a cylindrical surface Therefore, with fixed height, and found an upper bound to the rate of change of area of the surface in terms of the compactness. However, the Ricci flow could be related to the existence of a fixed point solution only for c = 1 2 , which is the limiting case leading to the RaychaudhuriSom distribution [10]. It deserves mention that for the Marder space, 1 2 c indicates the mass per unit length of the cylinder, and for c > 1 2 a photon cannot escape to infinity[11] but rather has a turning point at a finite r . This indicates that c = 1 2 is the limiting case that the cylindrical mass distribution indeed has an event horizon. At this stage one is not sure if an entropy could be defined in the case of a Marder space, but the modified Ricci flow equation indeed has a fixed point, and the scalar diffusivity has the significance of the lapse function for the spacetime. This indicates that such flow equations need attention in the asymptotically non-flat spaces as well.", "pages": [ 4, 5 ] }, { "title": "Acknowledgement:", "content": "One of the authors (SC) gratefully acknowledges the warm hospitality provided by IISERKolkata, where a major part of the work was done under the Summer Research Programme. We thank the anonymous referee for the suggestions on the earlier draft which indeed resulted in an improvement of the paper.", "pages": [ 5 ] } ]