The Halo Boundary of Galaxy Clusters in the SDSS
Eric Baxter, Chihway Chang, Bhuvnesh Jain, Susmita Adhikari, Neal Dalal, Andrey Kravtsov, Surhud More, Eduardo Rozo, Eli Rykoff, Ravi K. Sheth
MMNRAS , 1–17 (2017) Preprint 8 February 2017 Compiled using MNRAS L A TEX style file v3.0
The Halo Boundary of Galaxy Clusters in the SDSS
Eric Baxter (cid:63) , Chihway Chang , Bhuvnesh Jain , Susmita Adhikari ,Neal Dalal , , Andrey Kravtsov , , , Surhud More , Eduardo Rozo ,Eli Rykoff , , Ravi K. Sheth , Center for Particle Cosmology, Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, IL 60637, USA Department of Astronomy, University of Illinois at Urbana-Champaign, Champaign, IL 61801, USA Department of Physics, University of Illinois at Urbana-Champaign, Champaign, IL 61801, USA Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL 60637, USA Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA Kavli Institute for the Physics and Mathematics of the Universe (WPI), Tokyo Institutes for Advanced Study,The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan Department of Physics, University of Arizona, Tucson, AZ 85721, USA Kavli Institute for Particle Astrophysics & Cosmology, P.O. Box 2450, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
Last updated 8 February 2017
ABSTRACT
Mass around dark matter halos can be divided into “infalling” material and “col-lapsed” material that has passed through at least one pericenter. Analytical modelsand simulations predict a rapid drop in the halo density profile associated with causticsin the transition between these two regimes. Using data from SDSS, we explore theevidence for such a feature in the density profiles of galaxy clusters and investigate theconnection between this feature and a possible phase space boundary. We first estimatethe steepening of the outer galaxy density profile around clusters: the profiles showan abrupt steepening, providing evidence for truncation of the halo profile. Next, wemeasure the galaxy density profile around clusters using two sets of galaxies selectedbased on color. We find evidence of an abrupt change in the galaxy colors that coin-cides with the location of the steepening of the density profile. Since galaxies are likelyto be quenched of star formation and turn red inside of clusters, this change in thegalaxy color distribution can be interpreted as the transition from an infalling regimeto a collapsed regime. We also measure this transition using a model comparison ap-proach which has been used recently in studies of the “splashback” phenomenon, butfind that this approach is not a robust way to quantify the significance of detecting asplashback-like feature. Finally, we perform measurements using an independent clus-ter catalog to test for potential systematic errors associated with cluster selection. Weidentify several avenues for future work: improved understanding of the small-scalegalaxy profile, lensing measurements, identification of proxies for the halo accretionrate, and other tests. With upcoming data from the DES, KiDS and HSC surveys, wecan expect significant improvements in the study of halo boundaries.
Key words: galaxy: clusters: general – cosmology: observations
In the standard cosmological model, gravitational collapsecauses small perturbations in an initially smooth dark mat- (cid:63)
E-mail: [email protected] ter density field to collapse into dense clumps known as ha-los. The matter distribution in and around halos can bedivided into two components, which we will refer to as “in-falling” and “collapsed.” Infalling material is in the processof falling towards the halo, but has not yet passed throughan orbital pericenter. Such material has experienced a first c (cid:13) a r X i v : . [ a s t r o - ph . C O ] F e b Baxter, Chang et al. turnaround at the point when gravity halted its motion awayfrom the halo due to the expansion of the universe, but hasnot yet experienced a second turnaround after passing by thehalo. Collapsed material, on the other hand, has experiencedat least one orbital pericenter passage and is in orbit aroundthe halo (Gunn & Gott 1972; Fillmore & Goldreich 1984;Bertschinger 1985). Close to the halo center collapsed ma-terial dominates the mass distribution, while far away fromthe halo center infalling material dominates. The transitionbetween these two regimes happens near the halo virial ra-dius; the scale of first turnaround, on the other hand, isabout five times larger.Using N-body simulations, Diemer & Kravtsov (2014,hereafter DK14) determined that stacked dark matter halodensity profiles exhibit a sharp decline near the transitionfrom the infalling regime to the collapsed regime. DK14 as-sociated this feature with the second turnaround of darkmatter particles, which results in a density caustic in theaccreted material. A caustic here refers to a narrow, lo-calized region of enhanced density; just beyond the sec-ond turnaround caustic, the density declines rapidly, pro-ducing the feature observed by DK14. Owing to the con-nection between the observed steepening of the profile andsecond turnaround, this feature has recently been termedthe splashback feature . Subsequently, Adhikari et al. (2014)developed a simple model for the location of the featureand confirmed the results from DK14. The idea that sec-ond turnaround is associated with a caustic in the densityprofile dates back to work by Fillmore & Goldreich (1984)and Bertschinger (1985). However, it was not obvious thata clear feature resulting from second turnaround would per-sist in realistic simulations and after averaging across manyhalos.A significant steepening of the profile followed by a flat-tening as one moves outward from the center is clearly seenin the clustering signal of galaxies and clusters measuredfrom Sloan Digital Sky Survey (SDSS) data (Abbas & Sheth2007; Sheldon et al. 2009) and in weak lensing measurementsaround SDSS clusters (Johnston et al. 2007; Sheldon et al.2009). Recently, Tully (2015) presented evidence for a steepdecline in the galaxy density profile and a discontinuity inthe velocity dispersion around galaxy groups in a contextsimilar to that considered here. Related investigations intocluster density profiles in the infalling and collapsed regimesusing spectroscopic data have been performed by Rines et al.(2013), Gifford et al. (2017) and references therein.More et al. (2016, hereafter M16) measured a steepen-ing of the galaxy density profile around redMaPPer clus-ters (Rykoff et al. 2014; Rozo & Rykoff 2014) identified indata from the SDSS eighth data release (DR8) (Aihara et al.2011) and identified this steepening with the splashback fea-ture seen by DK14. By fitting the DK14 model for the ra-dial density profile — which accounts for the rapid steepen-ing of the profile around the splashback radius — to theirSDSS measurements, M16 determined that the model witha splashback feature provided a good fit to the data, whilea model without a splashback feature did not ( χ of 60–140for 9 degrees of freedom). M16 then compared the location Note that infalling material could include collapsed subhalosthat are falling into a larger halo. of the feature inferred from the data to expectations fromN-body simulations, finding evidence of tension.Identifying the splashback feature in data is challengingfor many reasons. First, observers typically measure only theprojected density profile of a halo rather than the 3D radialprofile. Projection smears out the otherwise sharp splash-back feature, making it harder to distinguish from a pro-file that does not have a splashback feature. Second, whilemeasuring the mass profile of halos is possible with gravita-tional lensing, such measurements currently have relativelylow signal-to-noise. Measurements of galaxy density can beused as a high signal-to-noise proxy for the matter density,but doing so introduces additional uncertainties as the re-lation between the galaxy density and the matter density isnot known precisely. Third, to increase the signal-to-noise ofdensity profile measurements, one typically stacks measure-ments across halos of a range of mass, redshift and accretionrate. Stacking can broaden the sharp splashback feature,making it more difficult to detect. Finally, effects such ashalo miscentering can introduce significant systematic un-certainties into measurements of the halo profiles.The main goal of this work is to carefully examine thetransition from the infalling to collapsed regimes aroundgalaxy cluster halos using data from SDSS. In particular,we are interested in whether the data provide evidence fora truncation of the halo density profile consistent with thatseen in simulations and whether such truncation can be con-nected to the phase space behavior of the matter around thehalo. These findings together would imply the existence aphysical halo boundary. We employ the same SDSS-derived redMaPPer cluster catalog and galaxy catalog as used inM16. The large number of redMaPPer clusters and galax-ies detected in SDSS make this data set the best currentlyavailable for measuring the galaxy density profile aroundclusters. We extend the modeling of M16 to include an im-portant source of systematic error: the miscentering of halosin the cluster catalogs (George et al. 2012; van Uitert et al.2015; Hoshino et al. 2015; Rykoff et al. 2014). Using theseimproved models, we explore whether the data favor thetruncated Einasto model introduced by Diemer & Kravtsov(2014) to describe the splashback feature over a pure Einastomodel (S´ersic 1963; Einasto 1965). We present constraintson the steepening of the collapsed component of the haloprofile near the splashback region and compare with exist-ing literature in both data and simulations. Additionally, weinvestigate the relative abundance of red galaxies around thesame clusters as a signature of the transition from the in-falling to the collapsed regimes. Finally, we perform similargalaxy profile measurements using a cluster catalog derivedfrom the same SDSS data but independent of the redMaP-Per cluster catalog. This test is important since it is con-ceivable that some feature of the redMaPPer algorithmcould lead to the appearance of an artificial splashback-likefeature. Concerns about potential systematic biases affect-ing measurements of the splashback feature are well mo-tivated: recent work by Zu et al. (2016) suggests that thequantity (cid:104) R mem (cid:105) used by M16 to split their cluster samplecan be significantly contaminated by projection effects. Acloser examination of the splashback feature in the absenceof (cid:104) R mem (cid:105) splitting is therefore warranted.As a brief aside, we note that the dark matter mass dis-tribution is commonly described using the halo model (for MNRAS , 1–17 (2017) he Boundaries of Halos in SDSS a review see Cooray & Sheth 2002). In the simplest ver-sion of this model, all of the dark matter in the Universeis assumed to live inside of halos. The matter distributionas measured by the halo-matter cross-correlation can thenbe divided into two components: the ‘one-halo’ term, whichdescribes the distribution of matter within halos, and the‘two-halo’ term, which describes the distribution of the ha-los themselves. In the language of the halo model, the “col-lapsed” material can be associated with the one-halo termand the “infalling” material can be associated with the two-halo term. Halo models can also be written down in a waythat make the connection to phase space more explicit (e.g.Sheth et al. 2001). In this work, however, we will generallyuse the terminology of the collapsed and infalling material.The paper is organized as follows. In § § § § §
6. Throughout this anal-ysis we will assume a flat-ΛCDM cosmological model with h = 0 . M = 0 . We use data from SDSS in our analysis. The main datasetis the same as that used by M16: the redMaPPer galaxyclusters described in Rykoff et al. (2014) and the SDSSDR8 photometric galaxies (Aihara et al. 2011). We selectgalaxy clusters with richness 20 < λ <
100 and redshifts0 . < z < .
33, resulting in a catalog of 8649 clusters. Thephotometric galaxies are selected by requiring the galaxyto have an i -band magnitude brighter than 21.0 (after dustextinction correction), a magnitude error smaller than 0.1,and none of the following flags: saturated , satur center , bright , and deblended . The Landy-Szalay estimator usedin § redMaPPer algorithm; these incorporatethe redshift and richness distribution of the redMaPPer clusters. The galaxy randoms were generated by distribut-ing points uniformly inside the footprint of the i <
21 galaxysample.To select the red and blue galaxies used in §
5, we per-form an additional color cut in the rest-frame g − r color. Wecompute this quantity using the K -corrected absolute mag-nitudes in the SDSS database, absMagG and absMagR .We define two subsamples: the quartile with the largest g − r (the “red” sample) and the quartile with the smallest g − r (the “blue” sample). For the purposes of this study, a moresophisticated selection based on e.g. a red-sequence selec-tion in color-magnitude-redshift space is not necessary. Thesimple selection defined here is sufficient to demonstrate theconnection between galaxy color and features of interest inthe galaxy density profile. We calculate the projected number density of galaxiesaround clusters, Σ g ( R ), as a function of the projected co-moving cluster-centric distance, R , by cross-correlating theclusters and the galaxies. We compute the cluster-galaxy an-gular correlation using the Landy-Szalay estimator (Landy& Szalay 1993) in redshift bins of ∆ z = 0 .
05. We only binthe clusters in redshift bins; the galaxy photometric red-shift information is not used because of the large associateduncertainties. For each redshift bin centered at ¯ z i , we as-sume that clusters with ¯ z i − ∆ z < z < ¯ z i + ∆ z are locatedat z = ¯ z i . We then calculate the i -band absolute magni-tude, M i , for all galaxies assuming that they are locatedat ¯ z i . Following M16, we then restrict the galaxy sampleto M i − h ) < − .
43, corresponding to an apparentmagnitude cut of m i <
21 at the redshift limit of the clus-ter catalog, z = 0 .
33. For each redshift bin, we measurethe cluster-galaxy correlation function in 15 comoving ra-dial bins from 0.1 to 10.0 h − Mpc. The correlation functionmeasurements in a given redshift bin are then converted toΣ g by multiplying by the mean galaxy density in that red-shift bin. Finally, we average the measurements in all red-shift bins, weighting by the number of cluster-galaxy pairsin each bin. Similar to M16, we use a jackknife resamplingapproach with 100 subregions to estimate the covariance ofour Σ g ( R ) measurement.M16 also measured the galaxy density profiles aroundtwo subsamples of clusters split on the parameter (cid:104) R mem (cid:105) ,defined as the average of the cluster member distances fromthe cluster center, weighted by the probability of clustermembership. (cid:104) R mem (cid:105) was first introduced by Miyatake et al.(2016), where it was shown that a sample of redMaPPer clusters split on this parameter exhibited similar masses (asinferred from weak lensing observations), but different largescale clustering biases, with the larger (cid:104) R mem (cid:105) sample havinga larger bias. M16 showed that the location of the splashbackradius inferred from their density profiles measurements wascorrelated with (cid:104) R mem (cid:105) . Given the connection between thesplashback radius and cluster accretion rate established byDK14, it was argued that (cid:104) R mem (cid:105) could therefore provide ameasure of the cluster accretion rate. However, recent workby Zu et al. (2016) suggests that (cid:104) R mem (cid:105) is strongly affectedby projection effects which are in turn correlated with thesurrounding density field. Given these concerns, we do notrely on (cid:104) R mem (cid:105) splits in this analysis. DK14 measured the stacked density profile of dark matterhalos in simulations. They fit an Einasto model (Einasto1965; Navarro et al. 2004) to the inner halo profile (radii r < . R vir ) while using the relation of Gao et al. (2008) to fixthe Einasto parameter α as a function of halo peak height, ν .Extending these fits to the outer profile ( r > . R vir ), DK14found that the stacked density profiles exhibited a sharp de-cline relative to the Einasto fit just outside the halo virialradius; this decline was associated with the caustic producedby splashback of dark matter particles. To model this behav-ior, DK14 introduced simple fitting formulae. They model MNRAS000
33. For each redshift bin, we measurethe cluster-galaxy correlation function in 15 comoving ra-dial bins from 0.1 to 10.0 h − Mpc. The correlation functionmeasurements in a given redshift bin are then converted toΣ g by multiplying by the mean galaxy density in that red-shift bin. Finally, we average the measurements in all red-shift bins, weighting by the number of cluster-galaxy pairsin each bin. Similar to M16, we use a jackknife resamplingapproach with 100 subregions to estimate the covariance ofour Σ g ( R ) measurement.M16 also measured the galaxy density profiles aroundtwo subsamples of clusters split on the parameter (cid:104) R mem (cid:105) ,defined as the average of the cluster member distances fromthe cluster center, weighted by the probability of clustermembership. (cid:104) R mem (cid:105) was first introduced by Miyatake et al.(2016), where it was shown that a sample of redMaPPer clusters split on this parameter exhibited similar masses (asinferred from weak lensing observations), but different largescale clustering biases, with the larger (cid:104) R mem (cid:105) sample havinga larger bias. M16 showed that the location of the splashbackradius inferred from their density profiles measurements wascorrelated with (cid:104) R mem (cid:105) . Given the connection between thesplashback radius and cluster accretion rate established byDK14, it was argued that (cid:104) R mem (cid:105) could therefore provide ameasure of the cluster accretion rate. However, recent workby Zu et al. (2016) suggests that (cid:104) R mem (cid:105) is strongly affectedby projection effects which are in turn correlated with thesurrounding density field. Given these concerns, we do notrely on (cid:104) R mem (cid:105) splits in this analysis. DK14 measured the stacked density profile of dark matterhalos in simulations. They fit an Einasto model (Einasto1965; Navarro et al. 2004) to the inner halo profile (radii r < . R vir ) while using the relation of Gao et al. (2008) to fixthe Einasto parameter α as a function of halo peak height, ν .Extending these fits to the outer profile ( r > . R vir ), DK14found that the stacked density profiles exhibited a sharp de-cline relative to the Einasto fit just outside the halo virialradius; this decline was associated with the caustic producedby splashback of dark matter particles. To model this behav-ior, DK14 introduced simple fitting formulae. They model MNRAS000 , 1–17 (2017)
Baxter, Chang et al. the halo density profile as the sum of an Einasto profile thateffectively describes the collapsed material and a power lawprofile that effectively describes the infalling material . Theuse of an Einasto profile to model the collapsed materialis well motivated by many studies using N-body simulations(Navarro et al. 2004; Merritt et al. 2005, 2006; Navarro et al.2010). The use of a power law term to describe the infallingmaterial is motivated by e.g. the self-similar collapse mod-els of Gunn & Gott (1972). For a single peak, self-similarcollapse models predict a power law profile with index -1.5.However, for CDM halos forming as a result of gravitationalcollapse around intially Gaussian perturbations, the infallingmaterial is not expected to follow a pure power law profile atlarge scales. Furthermore, non-linear dynamics can modifythe profile of infalling material within the halo. The pre-cise form of the infalling material profile must therefore becalibrated using e.g. N-body simulations. The simple powerlaw model, however, was shown to provide a good fit to thestacked profiles of simulated halos out to ∼ R vir in DK14.To model the observed steepening of the density profile near R vir , DK14 multiplied the Einasto profile by the function f trans ( r ), which is unity for small r , but declines rapidly ina narrow region near the radius r t .The complete profile introduced by Diemer & Kravtsov(2014) that provides good fits to the stacked 3D densityprofile of simulated halos from small scales out to ∼ R vir has the form: ρ ( r ) = ρ coll ( r ) + ρ infall ( r ) , (1) ρ coll ( r ) = ρ Ein ( r ) f trans ( r ) (2) ρ Ein ( r ) = ρ s exp (cid:18) − α (cid:20)(cid:18) rr s (cid:19) α − (cid:21)(cid:19) , (3) f trans ( r ) = (cid:34) (cid:18) rr t (cid:19) β (cid:35) − γ/β , (4) ρ infall ( r ) = ρ (cid:18) rr (cid:19) − s e , (5)where ρ coll and ρ infall represent the profiles of the collapsedand infalling material, respectively. Note that ρ coll and ρ infall correspond to the ρ inner and ρ outer used by DK14. Since r iscompletely degenerate with ρ , we will fix r = 1 . h − Mpcthroughout.The profile of Eqs. 1–5 contains eight free parameters.DK14 first fit density profile measurements from simulationsallowing all eight parameters to vary freely, and found thatthe profile provided a good fit to these measurements. Be-cause some of the parameters in their fits were correlated,DK14 also explored how the number of free parameters couldbe reduced by fixing various parameter combinations. In thisanalysis, we will allow all eight model parameters (after fix-ing r ) to vary independently for two reasons. First, theparameter combinations constrained by DK14 depend onquantities such as the halo peak height and the virial ra-dius, both of which cannot be measured precisely from thedata. Second, it is not necessarily true that parameter com-binations that can be fixed when fitting the dark matter The DK14 model also includes a constant term equal to themean density of the Universe. Here, since the measurements areeffectively mean-subtracted, we do not include such a constantterm. alone can also be fixed when fitting the galaxy distribution,given the uncertain relation between galaxies and mass. Al-lowing all eight parameters to vary simultaneously was alsothe approach taken by M16. As we will discuss below, how-ever, allowing all eight parameters to vary freely (with someweak priors) can make distinguishing between models thathave a truncation caused by f trans and models that have f trans = 1 difficult.Another common parameterization for modeling thedensity profiles of dark matter halos is the Navarro-Frenk-White (NFW) profile of Navarro et al. (1996). The NFWprofile is also known to be a good fit to simulated dark mat-ter halos, although it may not be as successful as the Einastomodel at capturing the behavior of the inner halo profile(Navarro et al. 2004; Merritt et al. 2005, 2006; Navarro et al.2010). Since we do not have a very strong theoretical priorto prefer the Einasto profile over the NFW profile in thisanalysis of galaxy density profiles, we will also consider theimpact on our splashback fits of replacing the Einasto profilewith the generalized NFW model (gNFW): ρ gNFW ( r ) = ρ i (cid:16) rr s (cid:17) α gNFW (cid:16) rr s (cid:17) − α gNFW , (6)where ρ i sets the normalization of the profile and α gNFW sets its shape.Since we measure projected densities on the sky, it isnecessary to integrate ρ ( r ) along the line of sight to obtainthe projected density Σ( R ):Σ( R ) = (cid:90) h max − h max dh ρ ( (cid:112) R + h ) , (7)where R is the projected distance to the halo center. Toavoid divergence of the profiles, we restrict the line ofsight integration to − h max < h < h max . We set h max =40 h − Mpc, but find that our results are quite robust to thischoice.The above equations for ρ ( r ) and Σ( R ) were found toaccurately describe the mass distribution around simulateddark matter halos in simulations by DK14. In this work,however, we will follow M16 and apply the same models tothe measured galaxy distributions, which we label with sub-script ‘g’s: ρ g ( r ) and Σ g ( R ) (note that these functions mea-sure number densities rather than mass densities). That is,we are assuming that any differences between the galaxy dis-tribution and the dark matter mass distribution (i.e. galaxybias) can be absorbed into the fitting parameters. In thelimit of constant galaxy bias, this assumption is certainlytrue. However, at small scales, galaxy bias is expected to bescale-dependent (e.g. Seljak 2000; Peacock & Smith 2000)and as a result, this assumption may break down. M16 testedthis assumption using subhalo profiles around cluster-sizehalos in dark matter simulations, showing that it is robust.However, the galaxy density profile is not expected to followthe subhalo profile at small scales, and the precise relationbetween the galaxy profile and the matter profile on smallscales is still an active research area (e.g. Nagai & Kravtsov2005; Guo et al. 2011; Budzynski et al. 2012).In the model testing parts of this work, we will adoptan operational definition and define the splashback radiusas the location of the steepest slope in the model densityprofiles. To differentiate between the splashback radius inthe 2D and 3D profiles, we define R D sp as the location of MNRAS , 1–17 (2017) he Boundaries of Halos in SDSS steepest slope in the three-dimensional galaxy density ( ρ g ),and R D sp as the analogous quantity in the projected galaxydensity (Σ g ). Note that alternate ways of identifying thesplashback radius exist in the literature (e.g. Mansfield et al.2016). Our definition has the benefit of being well-definedand relatively easy to measure in observational data. To measure the cluster-centric distance R in the data weuse the cluster centers computed by the redMaPPer algo-rithm. redMaPPer assigns cluster centers in a probabilisticfashion: each cluster member galaxy is assigned a probabilityof being the cluster center, P cen based on its color, magni-tude, redshift and local density (Rykoff et al. 2014). Thegalaxy with the highest P cen is then considered to be thecluster center. The model for the cluster density profile in-troduced in § Rockstar halo finder (Behroozi et al.2013).It is possible for the redMaPPer cluster center to differfrom the centers of
Rockstar -identified halos in two ways.First, it is possible that no cluster galaxy lies at the true cen-ter of the dark matter halo. This can happen stochastically,or if observational effects such as masking prevented the cen-tral galaxy from being observed. A second possibility is thata cluster member galaxy does lie at the true center of thedark matter halo, but it is not the galaxy with the highest P cen . We refer to both of these effects as miscentering . Mis-centering can significantly alter the measured density pro-file at scales below the typical miscentering distance (i.e.the distance between the assumed and true halo centers).Although the transition between the infalling and collapsedregimes occurs at scales greater than the miscentering dis-tance, we will see below that changes to the small-scale haloprofile can significantly alter how models fit the profile in thetransition region. We note that M16 tested for the effects ofmiscentering on their determination of the splashback ra-dius by selecting clusters with high P cen and repeating thedensity profile measurements; however, they did not includea prescription for miscentering in their model for the galaxydensity.We model the effects of miscentering following the ap-proaches of Melchior et al. (2016) and Simet et al. (2016).The miscentered density profile, Σ g , can be related to theprofile in the absence of miscentering, Σ g, , viaΣ g = (1 − f mis )Σ g, ( R ) + f mis Σ g, mis ( R ) , (8)where f mis is the fraction of clusters that are miscentered,and Σ g, mis is the galaxy density profile for the miscenteredclusters. For clusters that are miscentered by R mis from thetrue halo center, the corresponding density profile is (Yanget al. 2006; Johnston et al. 2007)Σ g, mis ( R | R mis ) = (cid:90) π dθ π Σ g, (cid:18)(cid:113) R + R + 2 RR mis cos θ (cid:19) . (9)The profile averaged across the distribution of R mis values is thenΣ g, mis ( R ) = (cid:90) dR mis P ( R mis )Σ g, mis ( R | R mis ) , (10)where P ( R mis ) is the probability that a cluster is miscen-tered by a (comoving) distance R mis . Following Simet et al.(2016), we assume that P ( R mis ) results from a miscenter-ing distribution that is a 2D Gaussian on the sky. The 1Dprobability distribution P ( R mis ) is then given by a Rayleighdistribution: P ( R mis ) = R mis σ R exp (cid:20) − R σ R (cid:21) , (11)where σ R controls the width of the distribution. Follow-ing Simet et al. (2016), we set σ R = τ R λ , where R λ =( λ/ . h − Mpc and we adopt the mean value of ¯ R λ =0 . h − Mpc for our sample. The miscentering model is thencompletely specified by the parameters f mis and τ . To de-termine how uncertainty on miscentering propagates to un-certainty on evidence for splashback we will consider severaldifferent (reasonable) priors on f mis and τ below.The weak lensing analysis of redMaPPer clusters bySimet et al. (2016) assumed Gaussian priors of f mis =0 . ± .
07 and τ = 0 . ± .
1, derived using results from Rykoffet al. (2014). Rykoff et al. (2014) quantified the miscenter-ing of the SDSS redMaPPer clusters based on 82 and 54X-ray selected clusters in the XCS (Mehrtens et al. 2012)and ACCEPT (Cavagnolo et al. 2009) data sets, respec-tively. Follow up studies from Hoshino et al. (2015) examineddata from stripe 82, finding that the visually-determinedcentroid of the redMaPPer clusters agree fairly well withthe redMaPPer -determined centroid. Our fiducial analysisuses the Simet et al. (2016) priors, but we will also considervariations on these priors, including a model without mis-centering and a model where the widths of the priors onthe miscentering parameters are doubled. Existing data setsused to infer the amount of cluster miscentering are quitelimited and systematics in identifying/selecting the X-raycluster centers can introduce additional uncertainties in theinferred miscentering priors (George et al. 2012).
We fit the model described in § § L , vialn L ( (cid:126)d | (cid:126)θ ) = − (cid:16) (cid:126)d − (cid:126)m ( (cid:126)θ ) (cid:17) T C − (cid:16) (cid:126)d − (cid:126)m ( (cid:126)θ ) (cid:17) , (12)where (cid:126)d is the data vector of Σ g measurements, (cid:126)m ( (cid:126)θ ) isthe model for these measurements evaluated at parametervalues (cid:126)θ , and C is the covariance matrix of the data. Thefree parameters of the model are ρ , ρ s , r t , r s , α , β , γ , s e ,and the miscentering parameters τ and f mis .To compute the posteriors on the model parametersgiven the likelihood of Eq. 12 we run a Markov ChainMonte Carlo (MCMC) analysis using emcee (Foreman-Mackey et al. 2013). Following M16, when computing theposterior we impose Gaussian priors on several model pa-rameters: log α = log 0 . ± .
6, log β = log 4 . ± .
2, log γ =log 6 . ± .
2. Note that the prior on α adopted here andin M16 is quite weak. The central value of α = 0 . MNRAS , 1–17 (2017)
Baxter, Chang et al. estimates of the redMaPPer clusters, simulations predictthat dark matter halos of the same mass should have α ∼ . ∼ h − Mpc.Simulations also show that α is dependent on the halo peakheight, ν , which leads to a dependence on halo mass andredshift (Gao et al. 2008). Finally, Dutton & Macci`o (2014)have shown that there is significant scatter in α betweendifferent halos, with σ log α = 0 .
16 + 0 . z . Our prior on α is wide enough that it has little effect on our parameterconstraints. One could imagine, however, imposing a tighterprior on α motivated by simulations; as we will discuss be-low, such a prior can significantly impact model fits to themeasured density profiles. The central values of the priorson β and γ are motivated by the analysis of DK14. Finally,we restrict r s ∈ [0 . , . h − Mpc, r t ∈ [0 . , . h − Mpc and s e ∈ [1 . , . R D sp . Note that whencomputing R D sp , we use the profile without the modifica-tions for miscentering since we are interested in the truesplashback radius of the halo.Throughout this analysis, we set the upper limit of thescales we fit to be 8 . h − Mpc, since the model introduced in § ∼ R vir ,where R vir is the halo virial radius (DK14), and for thecluster sample considered here R vir ∼ h − Mpc. At smallscales, systematics in the galaxy density measurements areexpected to become significant. This is especially importantin this analysis as cluster fields are inherently crowded. Is-sues such as detection incompleteness, photometry inaccu-racy and blending can be important at these scales (Melchioret al. 2015, 2016). In addition, the relation between galaxydistribution and dark matter distribution become more com-plicated at small scales, and the model described in § h − Mpc was used. In this analysis, we will consider twochoices of the minimum scale.As splashback corresponds to a steepening of the outerhalo density profile, it is worthwhile to consider model-independent approaches to measuring this steepening. Thelogarithmic derivative of the density profile can be con-strained in model-independent ways (e.g. the Savitzky-Golay approach taken by M16 and Adhikari et al. 2016).However, a steep feature in the 3D profile will appear signif-icantly less steep in the 2D projected profile. Furthermore,miscentering can change the shape of the splashback fea-ture in the 2D profile. These effects present a challenge fornon-parametric methods, since such methods can only beapplied to the measured 2D profile and cannot be used toinfer the 3D profile or the profile in the absence of miscen-tering. Additionally, near the splashback radius the infallingmatter and collapsed matter make roughly equal contribu-tions to the total density profile, making inferences aboutthe collapsed material alone difficult with non-parametricmethods.
We first consider a model comparison approach to determinewhether the data support the existence of a splashback fea-ture. In such an approach, one must take care to define themodels with and without the feature. For the model with asplashback feature, we adopt the model of DK14. Defininga model without a splashback feature is more complicated.Since f trans was introduced by DK14 to describe this fea-ture, it makes sense to consider f trans = 1 as a splashback-free model as was done by M16. However, in the simulationsof DK14, the Einasto parameters were fixed by fitting onlythe inner halo ( R < . R vir ). We fit the Einasto profile aspart of the global fit using all radii, and in this case thesteep outer profile around the splashback radius may drivethe best fit values of r s and α to smaller and larger values,respectively. This can both compromise the quality of the fitat small radii and allow the Einasto profile in the f trans = 1model to become sufficiently steep at R ∼ R D sp to describesteepening due to a splashback feature. For now, we will per-form the model comparison between models with f trans freeand f trans = 1, but we will return to the subtleties of thiscomparison in § f trans = 1 and f trans = free models, weuse a Bayesian odds ratio. The odds ratio is defined as theratio of the posteriors for two models, M and M , given thedata D and prior information I (see e.g. Ivezi´c et al. 2014): O ≡ p ( M | D, I ) p ( M | D, I ) = P ( D | M , I ) P ( M | I ) P ( D | M , I ) P ( M | I ) . (13)Assuming we have no prior reason to prefer M over M ,the above reduces to O = P ( D | M , I ) P ( D | M , I ) (14)= (cid:82) p ( D | M, (cid:126)θ , I ) p ( (cid:126)θ | M , I ) d(cid:126)θ (cid:82) p ( D | M, (cid:126)θ , I ) p ( (cid:126)θ | M , I ) d(cid:126)θ , (15)where (cid:126)θ represents the parameter spaces of the two models.The terms in the numerator and denominator of Eq. 14 aresometimes referred to as the evidence for M and M , re-spectively. The evidence measures the probability that thedata would be observed if a particular model is correct.The integrals in Eq. 15 are high-dimensional, so to evalu-ate them we use a method that relies on the MCMC pa-rameter chains for both models . To interpret the odds ra-tios that we compute we use the Jeffreys’ scale (Jeffreys& Lindsay 1963). The Jeffreys’ scale identifies the differentregimes of ln( O ) with ‘weak’ (0 < ln( O ) < . . < ln( O ) < . . < ln( O ) < . . < ln( O )) preference for one modelover another. In addition to the odds ratio we also reportthe ∆ χ between the two models in order to compare withM16. The construction of a cluster catalog like redMaPPer re-quires many non-trivial choices that can potentially lead to see for an example.MNRAS , 1–17 (2017) he Boundaries of Halos in SDSS poorly understood selection effects that could alter the mea-sured Σ g . To make any claim of a splashback detection morerobust, it is therefore important to test the measurementson alternative cluster catalogs. A large number of cluster-finding algorithms exist in the literature (van Breukelen &Clewley 2009; Soares-Santos et al. 2011; Wen et al. 2012;Rykoff et al. 2014).To this end, we repeat the measurements of Σ g usingthe group catalog of Yang et al. (2007, hereafter Y07). TheY07 catalog contains groups of galaxies identified in SDSSDR7 data across a wide range of masses, including clusters asmassive as the redMaPPer clusters. Unlike the redMaP-Per algorithm, the Y07 algorithm does not require groupinggalaxies in color space (see Yang et al. 2005, for a detaileddescription of the algorithm). Instead, the Y07 algorithmuses spectroscopic redshift information to iteratively assigngalaxies to groups. Briefly, this iterative assignment assumesthat all groups live in dark matter halos whose masses are es-timated from an assumed mass-to-light relationship. Giventhe estimated halo mass, galaxy assignment is performed as-suming that the galaxy distribution is described by an NFWprofile. The halo centers and the mass-to-light ratio are thenadjusted and group assignment is iterated until convergenceis achieved. The scatter in the halo masses, M h , assigned inthis fashion is expected to be roughly twice as large as thescatter in the mass estimates obtained from the redMaP-Per mass-richness relation .We select groups in the Y07 catalog with halo massesin the range 10 . < M h < . h − M (cid:12) . Accountingfor the different mass definitions used by Y07, this massrange corresponds roughly to the mass of the 20 < λ < redMaPPer , assuming the mass-richness rela-tion derived in Simet et al. (2016). We impose a redshiftselection of 0 . < z < .
2, where the upper cutoff comesfrom Y07 and the lower bound is to match the redMaP-Per catalog. The redshift range is slightly different fromthe redMaPPer cluster sample, but we do not expect sig-nificant redshift-dependences of the splashback feature re-sulting from this difference. This selection yields a total of3292 groups, which is roughly 2.5 times fewer than in the redMaPPer cluster sample. This is mainly due to the dif-ferent redshift ranges of the two samples.We measure the Σ g profile around the Y07 groups us-ing the galaxy sample described in § § f mis and τ are not well constrained forthe Y07 catalog, we adopt the loose miscentering priors ofModel D in Table 1. The projected galaxy density profile, Σ g ( R ), measuredaround redMaPPer clusters is shown as the blue pointswith error bars in Fig. 1. Our measurement appears to be The most massive groups in Y07 are estimated to have a massscatter of ∼ . redMaPPer mass scatter is expected to be roughly ∼ . in excellent agreement with that of M16. We now explorevarious model fits to these measurements and the results ofour model comparison tests.We first consider as a check on our measurements thecase where miscentering is not included in the model and wefit the data over a range of scales from 0.1 to 8.0 h − Mpc.This is essentially identical to the analysis of M16. The in-ferred ∆ χ and Bayesian odds ratio between the modelswith f trans = 1 and f trans = free are listed in Table 1( redMaPPer Model A). In this case, we find significantpreference for the model with f trans = free, with ∆ χ ∼ f trans = 1 model. This value is consistent withthe ∆ χ reported by M16. The evidence ratio also indicatesstrong preference for the f trans = free model, having a valueof ln O ∼
76, amounting to “decisive” evidence on Jeffreys’scale. The best-fit model parameters for the two models areshown in Table 2. Also listed in Table 2 are the constraintson R D sp determined from the f trans = free model fits. Wenote that our determination of R D sp is consistent with thatof M16. We do not list R D sp here, but note that it is smallerthan R D sp due to projection.Next, we consider how miscentering affects the prefer-ence for the f trans = free model. We first explore the casewhere the miscentering parameters f mis and τ are fixed tothe central values from the priors of Simet et al. (2016)(Model B in Table 1). In this case, the ∆ χ and the ev-idence ratio are decreased significantly relative to the no-miscentering model, although both still indicate strong pref-erence for the f trans = free model. Comparing the ModelsA and B for redMaPPer in Table 2, we see that includingmiscentering in the model fits changes the best-fit r s un-der the model with f trans = free from 0.85 h − Mpc to 0.32 h − Mpc.When we allow the miscentering parameters to varywhile imposing the priors from Simet et al. (2016) (ModelC), we find that the ∆ χ and odds ratio are reduced evenfurther. As shown in Table 2, the f trans = 1 model fitsthe data in this case with a large miscentering fraction, f mis = 0 .
47. This value is disfavored at roughly 3 σ by themiscentering priors of Simet et al. (2016). As we discuss be-low, we find that it is precisely the miscentering prior thatis largely driving the f trans = 1 model to be disfavored.To help understand how miscentering is affecting thefits to Σ g ( R ) and the results of the model comparison, weshow the model fits with the Simet et al. (2016) miscenteringpriors (Model C) in Fig. 1. The left panels of the figure illus-trate the fits with f trans = 1, while the right panels illustratethe fits when f trans = free (note that in the left panel, thegreen and red curves are identical since f trans = 1). Compar-ing the grey curves (which represent the total model without miscentering) to the black dashed curves (which representthe total model with the preferred miscentering), we see thatmiscentering has the effect of flattening the inner galaxy pro-file of the clusters. This makes sense: the offsets caused bymiscentering mean that the density profile is effectively av-eraged across scales on the order of the miscentering radius,causing an otherwise sharp inner profile to be flattened. Thisredistribution of the density at small scales also has the ef-fect of narrowing the minimum in the logarithmic derivativeof the profile.Fig. 1 makes it clear that the models with f trans = freeand with f trans = 1 can both fit the data very well (as fur- MNRAS , 1–17 (2017)
Baxter, Chang et al. . . Σ g ( R ) [ h M p c − ] f trans = 1 . . − . − . − . − . . d l og Σ g / d l og R . . R [ h − Mpc] . . . Σ g / Σ g , m o d e l f trans = free ρ Ein ρ Ein f trans ρ Ein f trans + ρ infall + miscenteringData − . − . − . − . . without miscenteringwith miscentering . . R [ h − Mpc] . . . Figure 1.
The measured galaxy profile Σ g around redMaPPer clusters in SDSS and the corresponding best-fitting models. The toppanels show Σ g measurements and model fits, the middle panels show the logarithmic derivative of the Σ g models, and the bottom panelsshow the ratio of the Σ g data points to the model. The left panels shows the model fits with no steepening function (i.e. f trans = 1),while the right panels show the fits with additional steepening beyond an Einasto profile (i.e. f trans is allowed to vary). The red curvesin the upper panels show contribution to the projected galaxy density from the collapsed component ( ρ coll g ( r ) = ρ Ein ( r ) f trans ( r )). Thegreen curve in the right panel shows the contribution from the Einasto term of the model ( ρ Ein ( r )). The grey curves are the total profilewithout miscentering, while the dashed black curves are the profiles with miscentering. Comparing the left and the right panels revealsthat a model with large miscentering and f trans = 1 can produce very similar total profile as a model with small miscentering and f trans free. Table 1.
Results of model comparison with various modeling and data choices. RM indicates the redMaPPer catalog, Y07 indicates thecatalog of Y07. ∆ χ and ln( O ) values indicate the results of the model comparison between the f trans = 1 and f trans = free models,and are computed as described in § h − Mpc] ∆ χ ln ( O )A: no miscentering RM f mis = 0 . τ = 0 . . < R < . f mis = 0 . τ = 0 . . < R < . f mis = 0 . ± . τ = 0 . ± . . < R < . f mis = 0 . ± . τ = 0 . ± . . < R < . f mis = 0 . ± . τ = 0 . ± . . < R < . f mis = 0 . ± . τ = 0 . ± . . < R < . α Y07 f mis = 0 . ± . τ = 0 . ± . α = log(0 . ± . . < R < . , 1–17 (2017) he Boundaries of Halos in SDSS Table 2.
Best-fit model parameters with f trans free (number preceding semicolon in each column) and f trans = 1 (number followingsemicolon) and under different modeling assumptions. Modeling choices are described in Table 1. We have excluded some parameters inthis table for clarity. The remaining parameters are given in Table A1.Model Catalog r s [ h − Mpc] r t [ h − Mpc] α β γ f mis τ R D sp [ h − Mpc]A RM 0.85 ; 0.36 1.25 ; — 0.10 ; 0.42 3.83 ; — 6.26 ; — 0.0 ; 0.0 — ; — 1 . ± .
05B RM 0.32 ; 0.29 1.31 ; — 0.16 ; 0.41 3.71 ; — 6.42 ; — 0.22 ; 0.22 0.32 ; 0.32 1 . ± .
05C RM 0.27 ; 0.20 1.38 ; — 0.17 ; 0.41 3.98 ; — 6.73 ; — 0.22 ; 0.47 0.34 ; 0.40 1 . ± .
08D RM 0.24 ; 0.19 1.42 ; — 0.19 ; 0.44 4.11 ; — 6.82 ; — 0.25 ; 0.51 0.34 ; 0.41 1 . ± .
09E RM 0.35 ; 0.44 1.34 ; — 0.23 ; 0.93 3.66 ; — 6.45 ; — 0.20 ; 0.22 0.42 ; 0.43 1 . ± .
07F RM 0.79 ; 0.10 1.23 ; — 1.54 ; 0.74 3.65 ; — 6.23 ; — 0.21 ; 0.50 0.45 ; 0.33 1 . ± .
17G Y07 0.35 ; 0.28 1.30 ; — 0.21 ; 0.38 3.75 ; — 6.20 ; — 0.51 ; 0.48 0.16 ; 0.20 1 . ± . ther evidenced by the low value of ∆ χ in this case). Theresiduals for both fits (shown in the bottom panels of the fig-ure) appear almost identical between the two model fits. Thepreference for one model over the other, then, is driven bythe priors on the model parameters, in particular the prioron f mis , as we will discuss below. The minimum of the loga-rithmic derivative of Σ g occurs in roughly the same locationin both fits, as can be seen in the middle panels of Fig. 1.However, while the two models generate very similar totalprofiles (and similar total logarithmic derivatives), they fitthe data in significantly different ways. To see this, considerthe red curves in Fig. 1, which show profile of the collapsedcomponent when f trans = 1 (left) and when f trans = free(right). We see that the f trans = 1 model fits the outer profile( R (cid:38) . h − Mpc) with a large value of α . In general, larger α results in a shallower inner profile for r < r s . However, inthe case of the f trans = 1 fit, the value of r s is decreased,which results in a steep profile at R (cid:46) . h − Mpc; a large f mis then flattens the inner profile somewhat. The modelwith f trans = free, on the other hand, prefers a shallowerinner profile at the same radii (as a result of larger r s ), doesnot require as much miscentering, and is steepened substan-tially by the f trans term at R (cid:38) . h − Mpc.The model with f trans = 1 prefers a miscentering frac-tion of f mis ∼ .
45, in tension with the miscentering prior,which prefers f mis = 0 .
2. This tension between the preferredmiscentering fraction and the miscentering prior, when com-bined with the behavior of the Einasto profile, drives thepreference for the model with f trans = free evaluated usingthe evidence ratio. In support of this conclusion, when weallow more freedom in the miscentering model by doublingthe widths of the Simet et al. (2016) miscentering priors(Model D in Table 1), we find that the evidence ratio in fa-vor of f trans = free is weakened by roughly a factor of 300relative to the case with Model C miscentering priors. Solelyby going from a model without miscentering (Model A) toa model with weak miscentering (Model D), the log oddsratio has been reduced from ln O = 69 to ln O = 3 . O = 0 .
86, amounting to only “weak evidence.”As noted previously, M16 also considered the effects ofmiscentering on their analysis, but took a very different ap-proach than that taken here. M16 repeated their measure- ments of the galaxy density using only clusters of low mis-centering probability ( P cen > . R D sp was within measurement uncertainty. We have re-peated this test using our measurements, finding similar re-sults. However, we do find that the galaxy density profile forthe high P cen clusters is somewhat steeper on small scalesthan for the full sample, as is expected for a sample withbetter centering. While miscentering may not significantlyimpact the location of R D sp , as we have shown above, it canstill have a significant impact on the inferred model param-eters and the shape of the logarithmic derivative of the pro-file. Furthermore, the P cen parameter in redMaPPer doesnot fully encapsulate all possible mechanisms of cluster mis-centering, such as the intrinsic scatter between the centerof the dark matter halo and the BCG (since the redMaP-Per center is constrained to be on top of one of the clustergalaxies). For these reasons it is important to include mis-centering when modeling the halo profile as we have donehere.Given the impact of systematics such as miscentering,non-linear galaxy bias, detection incompleteness, photome-try inaccuracy and blending on the inner density profile, itmakes sense to consider removing the innermost scales whenfitting the galaxy density measurements. We perform sucha fit by excluding scales below 0.3 h − Mpc; the results areshown as Model E in Table 1 and Table 2. We find that whenscales below 0.3 h − Mpc are excluded, the data no longerexhibit a statistically significant preference for f trans (cid:54) = 1:∆ χ = 0 . f trans = 1 and f trans =freemodels.The above analysis highlights the fact that allowing ad-ditional freedom in the inner galaxy density profile signifi-cantly affects the ability of the data to distinguish betweenmodels with f trans = 1 and f trans (cid:54) = 1. This behavior canbe understood in the following way. The Einasto model ofEq. 2 couples the inner profile and the outer profile: as α is increased, the inner profile becomes shallower while theouter profile becomes steeper. If one ignores miscentering inthe modeling of the galaxy density profile (as was done inM16) then the value of α is strongly constrained by the innerprofile to be α ∼ .
2. In this case, fitting the data at inter-mediate scales requires truncation of the Einasto profile bythe f trans term, and f trans = 1 will be excluded at high sig- MNRAS000
2. In this case, fitting the data at inter-mediate scales requires truncation of the Einasto profile bythe f trans term, and f trans = 1 will be excluded at high sig- MNRAS000 , 1–17 (2017) Baxter, Chang et al. . . . . . r s ( h − M p c ) . . . . f m i s . . . . α . . . . . τ . . . . . r s ( h − Mpc) .
15 0 .
30 0 .
45 0 . f mis . . . . . τ Figure 2.
Posteriors on the galaxy profile parameters recoveredfrom the MCMC analysis of the galaxy profile measurements.Black curves show results of analysis that allows the parametersin the f trans term of Eq. 4 to be free, while red curve shows resultswhen f trans = 1. Both analyses use the Model C miscenteringpriors from Table 1. nificance. If, on the other hand, one allows for miscentering(or removes the innermost scales), the inner density profilecan be fit by larger α , smaller r s and larger f mis . Since amodel with larger α already has a steep outer profile, thepreference for additional steepening in the outer halo profile(as parameterized with f trans ) is reduced. We compare theposteriors on α , r s and f mis for the two model fits in Fig. 2.We note that the high values of α preferred by the modelfits with f trans = 1 are disfavored by other studies. As shownin Table 2, these fits generally prefer α in the range 0.3–0.4.These values are well within our prior of log α = log 0 . ± . α from a combination of simulations anddata would mean less sensitivity to the uncertainties in themiscentering parameters, and would therefore help improveour ability to make a more robust case for f trans (cid:54) = 1 usingthe model comparison approach explored in this section.Unlike the Einasto profile, the gNFW profile of Eq. 6forces the slope of the outer halo profile to asymptote to − f trans (cid:54) = 1 willbe greater when assuming an gNFW profile than when us-ing the Einasto model. Indeed, as shown in Table 1 (ModelF), the evidence for f trans = 1 is increased when using thegNFW model. In some sense, the gNFW analysis providesa better measure of the detection significance of f trans (cid:54) = 1because the outer slope is essentially fixed. However, the sen-sitivity of the model comparison results to the parameteri-zation of the profile of the collapsed component is certainlya drawback to this approach to detecting a splashback-likefeature. . . r [ h − Mpc] − − − − − − − d l n ρ g ( r ) / d l n r ρ g ρ coll g Figure 3.
Constraints on the 3D logarithmic derivative of thecollapsed component ( ρ coll g ( r )) and total galaxy density ( ρ g ( r ))from our model fits to the measured galaxy density profile around redMaPPer clusters. The best fit value of the splashback radius, R D sp , is shown as the vertical line. The data prefer a profile whichexhibits a steepening to slopes significantly steeper than − − − − − − ∂ ln ρ g coll ∂ ln r (cid:0) R D sp (cid:1) f trans = free f trans = 1 − . − . − . − . ∂ ln ρ g ∂ ln r (cid:0) R D sp (cid:1) Figure 4.
Constraint on the logarithmic derivative of the threedimensional profile of collapsed component (top panel) and thetotal galaxy profile (bottom panel) evaluated at the splashbackradius, R D sp , inferred from model fits to the measured galaxydensity profiles. Solid (red) curves show results for fits when theparameters describing the profile of the splashback feature (i.e. f trans ) are allowed to vary, while dashed (green) curves showresults for fits with f trans = 1. As discussed in the text, the f trans = free model provides a better description of the data. Wefind significant evidence for slopes of the collapsed material profilesignificantly steeper than − , 1–17 (2017) he Boundaries of Halos in SDSS R sp We have seen that once miscentering is introduced into the f trans = 1 model from DK14, the resultant model is flexi-ble enough to produce a steep outer profile while still be-ing consistent (at moderate significance) with our fiducialpriors. Consequently, the ability of the data to distinguishbetween this model and the f trans = free model is reduced.An alternative to the model comparison approach is to usethe model fits to directly constrain the logarithmic deriva-tive of the halo profile in the infalling-to-collapsed transitionregime, as shown in Fig. 3. Using simple collapse models,Dalal et al. (2010) argued the power law index of the outerhalo profile is set by the profile of the initial perturbationthat gave rise to the halo. For power law initial profiles ofarbitrary steepness, the steepest logarithmic slope of the col-lapsed outer halo profile is −
3. Departures from the powerlaw form can give rise to logarithmic slopes slightly steeperthan −
3, and for an NFW profile the slope asymptotes to − − To constrain the logarithmic slope of the galaxy den-sity profile, we draw sample profiles from our MCMC chainsfor the fits with f trans = free and compute the logarithmicderivatives of these profiles. Fig. 3 shows the resultant con-straints on the logarithmic derivatives of ρ g ( r ) (grey band)and ρ coll g ( r ) (red band). The figure shows that we obtain afairly tight constraint on the slope of ρ coll g out to radii atleast as large as R D sp (i.e. where the logarithmic derivativeof the total profile has a minimum, marked with a verti-cal line in the figure). This is a non-trivial finding since ρ coll g and ρ infall g have roughly equal magnitudes in this regime andtherefore make degenerate contributions to the total profile.It is clear from Fig. 3 that over a narrow range of radius(roughly 0 . h − Mpc to 1 . h − Mpc) the profile of the col-lapsed component exhibts a rapid steepening from logarith-mic slopes shallower than − − ρ coll g (top panel) and ρ g (bottom panel) evaluated at R sp in Fig. 4. We show theposteriors for model fits with f trans = free (red curves) andfor f trans = 1 (green curves). Note, though, that since the f trans = 1 model is a special case of the f trans = free model Halos that are simulated into the future and left to relax exhibittruncated Hernquist profiles (Hernquist 1990) that reach slopes of −
4, however this steepening happens gradually rather than overa narrow region around R sp (Busha et al. 2005). and since we have shown that the latter model generallyprovides a better fit to the data, the f trans = free modelshould provide a more accurate representation estimate ofthe profile slope than the f trans = 1 model.Fig. 4 makes it clear that the data prefer a logarithmicslope of ρ coll g at R sp that is quite steep, indeed significantlysteeper than the − R sp is − . ± .
4. The estimatedslope of the total profile is also significantly steeper than − − . ± .
15. Therefore, even al-lowing for considerable uncertainty in the profile of infallingmatter, these results suggest that the profile of collapsedcomponent reaches slopes significantly steeper than − R sp . Again, these findings can be taken as evidence for trun-cation of the halo profile similar to that seen in simulationsby DK14. Unlike the model comparison results discussed in § f trans = 1 (green dashed curves). These fits yieldsignificantly shallower slopes than the model fits where f trans is allowed to vary. This may be surprising given that the log-arithmic slopes of the projected profiles have almost iden-tical steepness (see Fig. 1). The explanation is that pro-jection and miscentering act to smooth out a steep featurein the 3D profile, making it appear significantly less steepin the 2D profile. Even these fits, however, prefer slopes of ρ coll g at R D sp that are significantly steeper than -3. Note,though, that R D sp inferred from the f trans = 1 fits tends tobe smaller than R D sp inferred from the f trans = free fits (seeFig. 1). Consequently, the distributions shown in Fig. 4 arenot coming from the same physical radius. Since the slopeof the Einasto profile gets more negative with increasing ra-dius, evaluating the f trans = free model slope at the R D sp ofthe f trans = free fits would make the value of the slope morenegative. We now consider the results of analyzing the measurementsof Σ g around the groups identified in the Y07 catalog. Asnoted above, the use of an alternative cluster catalog pro-vides an important systematics test for the existence of asplashback-like feature in the data.Since the miscentering parameters appropriate for theY07 groups are not known precisely, we use the wide miscen-tering priors of Model D in Table 1. However, given thesewide priors and the lower signal-to-noise of the Σ g mea-surement for the Y07 groups, we find that for our fiducialanalysis the model fits with f trans = free prefer values of r t that are at the edges of the prior on this parameter. We findthat this can be prevented by imposing a somewhat tighterprior on α : log α = log(0 . ± .
1. This prior is still fairlyloose relative to expectations from simulations, and is con-sistent with the values of α recovered from the analysis of redMaPPer clusters when f trans is allowed to vary.The Σ g measurements around the Y07 groups are shownin Fig. B1 and the results for the model fits are summarizedin Table 1 and Table 2. In general, we find that the pa-rameter values recovered from the Y07 fits agree quite wellwith those from the analysis of redMaPPer clusters. Therecovered splashback radius is also in good agreement. The MNRAS , 1–17 (2017) Baxter, Chang et al.
Y07 analysis exhibits a large evidence ratio in support of themodel with f trans = free, but this is somewhat misleading(at least in comparison to the redMaPPer results) becauseof the prior we have imposed on α for this analysis.Fig. B2 shows the distribution of profile slopes for theY07 groups, analogous to Fig. 4 for the redMaPPer clus-ters. The slope distributions shown in Fig. 4 are quite consis-tent with those from the redMaPPer measurement. Theyindicate logarithmic slopes of ρ coll g at R sp of − . ± .
7, signif-icantly steeper than the slope expected for an NFW profileand consistent with expectations for a splashback feature.We note that this test does not completely exclude thepossibility of systematic effects introduced by the clusterfinder ( redMaPPer or Y07) which could bias the loca-tion of R sp or the steepness of the collapsed componentat R sp . Nevertheless the fact that both redMaPPer andY07 strongly prefer logarithmic slopes that are significantlysteeper than − R sp is fairly convincing evidence that thefinding of truncation of the halo profile is robust. Our analysis has until now focused on examining the totalgalaxy density profile near the transition between the infallregime and the collapsed regime. Another approach to prob-ing this transition is to examine galaxy colors. The passageof a galaxy through a cluster is expected to quench star for-mation in the galaxy. This process can happen through sev-eral channels: gravitational interactions with other galaxiesor the cluster potential itself (e.g. Moore et al. 1996), strip-ping of the gas in the galaxy as a result of ram pressure fromcluster gas (e.g. Gunn & Gott 1972), and stripping of gasfrom the galaxy’s gaseous halo, thereby preventing replen-ishment of gas used to form stars (i.e. “strangulation”, seee.g. Larson et al. 1980; Kawata & Mulchaey 2008). Regard-less of how it happens, quenching of star formation will causea galaxy to appear redder than galaxies with active starformation. Measurements of a transition in galaxy clustersnear the cluster virial radius have been performed in severalprevious studies including Dressler et al. (1997), Weinmannet al. (2006) and references therein. Here, we focus on theshape of this transition and its connection to the phase spaceboundary between the infalling and collapsed regimes.In these scenarios, the typical time scales for quench-ing are comparable to the time taken to move across theextent of the cluster, roughly 2-4 Gyr (Wetzel et al. 2013).Therefore, if interactions within the cluster are responsiblefor quenching, a galaxy that has undergone a single passagethrough a cluster will appear redder than a galaxy that hasnot yet passed through the cluster. Since galaxies outsidethe splashback radius are significantly more likely to still beon their first infall, we expect a sharp increase in the fractionof red galaxies near the splashback radius.Another possibility is that galaxy color simply corre-lates with formation time or the time of accretion of thegalaxy onto the cluster and is not affected by processes in-side the cluster (e.g. Hearin et al. 2015). In this case a sharpincrease in the red fraction at R D sp would still be expectedbecause within this radius the infalling galaxies suddenlystart to be mixed with galaxies which were accreted 2-4 R [ h − Mpc] − Σ g ( R ) [ h M p c − ] All galaxiesRed galaxiesBlue galaxies . . r [ h − Mpc] − . − . − . − . − . − . − . d l og ρ g / d l og r Figure 5.
The top panel shows the Σ g ( R ) measurements for thefull sample (black data points), the reddest quartile of galaxies(red data points) and bluest quartile of galaxies (blue points);best fit models to the different measurements are shown as solidlines. The bottom panel shows the corresponding log-derivativesof ρ g ( r ) inferred from model fiting. Gyrs ago, which would also result in a sharp change in thered fraction. The main point in the context of this paperis that in both scenarios the sharp increase in the red frac-tion is associated with the transition from the infalling tocollapsed regimes at the splashback radius.To investigate this, we consider two galaxy subsamplesselected based on their rest-frame colors as described in § g for the full galaxysample. The Σ g and corresponding model fits for the twogalaxy color subsamples and the full sample are shown inthe top panel of Fig. 5; the inferred logarithmic derivativesof the 3D profile from the model fits are shown in the bottompanel. For this analysis we use the Model C miscenteringpriors, as these reflect recent constraints from analysis of redMaPPer clusters.Note that the profile of the blue galaxies approachesa power law with index close to − . R D sp that are falling towards acluster which dominates the local mass distribution. In theabsence of shell crossing, the mass interior to the particlesremains constant as they fall, and so their free fall velocityscales as v ∼ r − . , where r is the distance between the clus-ter center and the particles. The mass contributed by suchparticles to a radial shell at r and with thickness dr will be MNRAS , 1–17 (2017) he Boundaries of Halos in SDSS . . R [ h − Mpc] . . . . . . R e d o r B l u e F r a c t i o n Red galaxiesBlue galaxies
Figure 6.
The fraction of red and blue galaxies relative to allgalaxies around redMaPPer clusters as a function of the pro-jected distance from the cluster center. proportional to the time the particles spends in the shell, so( dM ( r ) /dr ) dr ∼ dr/v , where M ( r ) is the mass enclosed atradius r . Assuming M ( r ) follows a power law and substitut-ing the radial dependence of v , we have M ( r ) ∼ r . . Thedensity profile, then, scales as ρ ∼ M/r ∼ r − . .The blue galaxy sample appears to be quite consis-tent with a purely infalling component at large r . In “pre-processing” models, quenching occurs in the dense environ-ment surrounding the cluster, but prior to falling into thecluster (Fujita 2004). Quenching in this way should removegalaxies from the blue sample as r decreases and add galax-ies to the red sample, leading to a departure from the ex-pected slope of − .
5. The fact that we observe logarithmicderivatives close to -1.5 constrains the degree to which pre-processing contributes to quenching. Note, however, that wedo observe slightly steeper slopes for the red galaxies thanfor the blue galaxies, which could be consistent with someamount of quenching due to pre-processing. This picture iscomplicated somewhat by the presence of so-called “back-splash” galaxies that have passed through the cluster (andmay therefore have been quenched by processes inside thecluster) but have been ejected as a result of gravitationalslingshot to several virial radii (Wetzel et al. 2014). We post-pone more in depth modeling of these scenarios to futurework.Next, we show the red/blue fraction measurements inFig. 6. It is clear from Fig. 6 that the red fraction showsan abrupt steepening at around 1 . h − Mpc. In a scenariowithout a phase space boundary between the infalling andcollapsed regimes, it is hard to imagine how a sharp upturnin the red fraction could arise at such large scales. The gasdensities at these radii are quite low; how would a galaxypassing through the cluster outskirts know to become red atthis particular radius? The picture of phase space causticsand quenching by the cluster (or at some time after accre-tion onto the cluster) provide a natural explanation for theobserved red fraction behavior. In this picture, the galaxyquenches after one or more passages through the cluster,and the transition from outside R sp to inside R sp marks the transition from a regime for which most galaxies havenever undergone a passage through the cluster to a regimefor which most galaxies have undergone passage through thecluster. In support of this picture, the location of the upturnin the red fraction is in excellent agreement with the R D sp in-ferred from the galaxy density measurements (shown as thegrey band in Fig. 6). Note that the three dimensional R D sp isthe relevant radius of comparison since it is this radius thatmarks the physical phase space boundary in projection. Theagreement between the projected red fraction measurementsand the 3D splashback radius is non-trivial. Secondary infall models have predicted caustics in the phasespace distribution of particles being accreted onto halossince the early studies of Fillmore & Goldreich (1984) andBertschinger (1985). It was not clear however whether thedisruptive processes in the formation of cold dark matterhalos would smear out caustic-like features, especially whenstacking across many halos. Recently, using simulations andanalytic arguments, DK14 and Adhikari et al. (2014) identi-fied a rapid steepening of the density profile of stacked halosin N-body simulations that they associated with a densitycaustic arising from the second turnaround of matter par-ticles, also known as splashback. Recent work by M16 haspresented evidence of a narrow steepening of the galaxy den-sity around redMaPPer clusters detected in SDSS; such afinding is consistent with expectations for a splashback fea-ture.In this work, we attempt to determine to what extentavailable data support the existence of a halo boundary re-lated to the presence of a phase space boundary betweeninfalling and collapsed contributions to the total density pro-file. Two analyses are presented here: • We decompose the profile into “infalling” and “col-lapased” (or 1-halo) components and use a model fittingapproach to estimate the slope of the collapsed componentnear the transition between these two regimes. Near thelocation where the steepest slope of the total profile occurs,the 1-halo profile reaches a logarithmic slope of about -5over a narrow range of radius. This is significantly steeperthan the expectation of an NFW-like profile, and supportsthe idea of a truncated halo profile. • The second evidence is that the location of steepestslope of the total profile coincides with an abrupt increase inthe fraction of red galaxies. Presumably a fraction of galaxiesinside the halo are quenched due to one or more passagesthrough the cluster. In the infalling regime, on the otherhand, galaxies have (for the most part) never been insidethe cluster and are therefore likely to have at most a gradualtrend in red fraction.The results of these two analyses, shown in Fig. 3 and Fig. 6,lend support to the presence of a phase-space halo boundaryassociated with a sharp decline in the halo density profile. Incontrast with common definitions of halo boundaries, suchas R or R vir , a phase-space halo boundary at R sp would MNRAS000
5. The fact that we observe logarithmicderivatives close to -1.5 constrains the degree to which pre-processing contributes to quenching. Note, however, that wedo observe slightly steeper slopes for the red galaxies thanfor the blue galaxies, which could be consistent with someamount of quenching due to pre-processing. This picture iscomplicated somewhat by the presence of so-called “back-splash” galaxies that have passed through the cluster (andmay therefore have been quenched by processes inside thecluster) but have been ejected as a result of gravitationalslingshot to several virial radii (Wetzel et al. 2014). We post-pone more in depth modeling of these scenarios to futurework.Next, we show the red/blue fraction measurements inFig. 6. It is clear from Fig. 6 that the red fraction showsan abrupt steepening at around 1 . h − Mpc. In a scenariowithout a phase space boundary between the infalling andcollapsed regimes, it is hard to imagine how a sharp upturnin the red fraction could arise at such large scales. The gasdensities at these radii are quite low; how would a galaxypassing through the cluster outskirts know to become red atthis particular radius? The picture of phase space causticsand quenching by the cluster (or at some time after accre-tion onto the cluster) provide a natural explanation for theobserved red fraction behavior. In this picture, the galaxyquenches after one or more passages through the cluster,and the transition from outside R sp to inside R sp marks the transition from a regime for which most galaxies havenever undergone a passage through the cluster to a regimefor which most galaxies have undergone passage through thecluster. In support of this picture, the location of the upturnin the red fraction is in excellent agreement with the R D sp in-ferred from the galaxy density measurements (shown as thegrey band in Fig. 6). Note that the three dimensional R D sp isthe relevant radius of comparison since it is this radius thatmarks the physical phase space boundary in projection. Theagreement between the projected red fraction measurementsand the 3D splashback radius is non-trivial. Secondary infall models have predicted caustics in the phasespace distribution of particles being accreted onto halossince the early studies of Fillmore & Goldreich (1984) andBertschinger (1985). It was not clear however whether thedisruptive processes in the formation of cold dark matterhalos would smear out caustic-like features, especially whenstacking across many halos. Recently, using simulations andanalytic arguments, DK14 and Adhikari et al. (2014) identi-fied a rapid steepening of the density profile of stacked halosin N-body simulations that they associated with a densitycaustic arising from the second turnaround of matter par-ticles, also known as splashback. Recent work by M16 haspresented evidence of a narrow steepening of the galaxy den-sity around redMaPPer clusters detected in SDSS; such afinding is consistent with expectations for a splashback fea-ture.In this work, we attempt to determine to what extentavailable data support the existence of a halo boundary re-lated to the presence of a phase space boundary betweeninfalling and collapsed contributions to the total density pro-file. Two analyses are presented here: • We decompose the profile into “infalling” and “col-lapased” (or 1-halo) components and use a model fittingapproach to estimate the slope of the collapsed componentnear the transition between these two regimes. Near thelocation where the steepest slope of the total profile occurs,the 1-halo profile reaches a logarithmic slope of about -5over a narrow range of radius. This is significantly steeperthan the expectation of an NFW-like profile, and supportsthe idea of a truncated halo profile. • The second evidence is that the location of steepestslope of the total profile coincides with an abrupt increase inthe fraction of red galaxies. Presumably a fraction of galaxiesinside the halo are quenched due to one or more passagesthrough the cluster. In the infalling regime, on the otherhand, galaxies have (for the most part) never been insidethe cluster and are therefore likely to have at most a gradualtrend in red fraction.The results of these two analyses, shown in Fig. 3 and Fig. 6,lend support to the presence of a phase-space halo boundaryassociated with a sharp decline in the halo density profile. Incontrast with common definitions of halo boundaries, suchas R or R vir , a phase-space halo boundary at R sp would MNRAS000 , 1–17 (2017) Baxter, Chang et al. constitute a real, physical boundary to the halo (More et al.2015).As a systematics test of these findings, we repeat themeasurements of galaxy density around groups identified inthe Y07 catalog. We find that these measurements prefersimilar slopes of the halo profile in the infalling-to-collapsedtransition region as the redMaPPer clusters, as shown inFig. B2. Recent work by Zu et al. (2016) found that the (cid:104) R mem (cid:105) quantity used by M16 to split their cluster samplecan be severely impact by projections along the line of sight,which are in turn related to the local environment of thecluster. Since M16 found large changes in the inferred splash-back radius with (cid:104) R mem (cid:105) , one may worry about the effects ofprojections on the measurement of the splashback feature.Our analysis does not preclude the possibility that R sp is af-fected by projection effects or by selection effects inherent to redMaPPer . However, while projections might smooth outan otherwise sharp splashback feature or change the inferred R sp , it seems unlikely that they could be responsible for theartificial appearance of a splashback-like steepening. On theother hand, it is possible that some feature of the redMaP-Per algorithm could cause clusters that exhibit a sharp de-cline in their profiles to be preferentially selected, therebyleading to the appearance of a splashback feature. Our mea-surement of a rapid steepening around the Y07 groups, how-ever, disfavors the possibility of the observed steepening be-ing due solely to a redMaPPer artifact. Since every clusterfinder is in principle susceptible to spatially dependent se-lection effects, such a test cannot be definitive. However,extending the measurements performed here to cluster sam-ples selected based on e.g. Sunyaev Zel’dovich decrementwould provide an independent check on the optical clustersamples used in this work.Using the parameterized models for the splashback fea-ture introduced by DK14 we have also attempted to quantifywhether the data support a model that has such a featureover a model that does not. We perform a model compar-ison by computing a Bayesian odds ratio between a modelthat includes the steepening caused by a splashback feature(parameterized via f trans in Eq. 4) and a pure Einasto pro-file that does not have additional steepening. A similar ap-proach was taken by M16 to quantify the significance of theirsplashback detection, although only χ was reported there.We extend the modeling efforts of M16 to include miscen-tering, an important systematic affecting the galaxy densityprofile. We find, however, that given reasonably weak priorson the parameters of the pure Einasto model, this modelcan come close to matching the steepening of the modelwith additional steepening caused by splashback. We there-fore conclude that until better data are available or tighterpriors on the model parameters can be obtained, such a testis not a particularly useful way to quantify evidence for asplashback-like feature in the data.While basic uncertainties remain in connecting obser-vations of a projected density profile (further modulated bythe relation between galaxies and mass) with a boundaryin phase space, we believe the findings summarized aboveare a promising indication of a dynamically motivated haloboundary. They provide avenues for studying physical pro-cesses such as dynamical friction (Adhikari et al. 2016), darkenergy (Stark et al. 2016), self-interacting dark matter andmodifications of gravity (M16). In this analysis we have used the galaxy density profile andthe red fraction in an attempt to infer something about thephase space behavior near the boundaries of cluster halos.There may be alternate routes to probing this boundarythat go beyond these two observables. For instance, prox-ies for halo accretion rate would be of significant utility tostudies of splashback since the accretion rate is expected tocorrelate with the location of the splashback feature. Whilethe analysis of M16 attempted to use (cid:104) R mem (cid:105) as a proxy foraccretion, subsequent work by Zu et al. (2016) has shownthat this proxy is contaminated by projection effects. An-other avenue, namely the impact of dynamical friction onsplashback, has been explored by Adhikari et al. (2016) andwould be valuable to test with future datasets. Our mea-surement of the red fraction provides qualitative evidencefor a phase space boundary. However, more definitive state-ments about how the red fraction relates to splashback willlikely require further effort with models and simulations ofgalaxies in cluster environments. The present work moti-vates exploration of these various avenues for detecting thesplashback feature.Ongoing and future galaxy surveys such as the HyperSuprimeCam (HSC), the Dark Energy Survey (DES), theKilo Degree Survey (KiDS), and the Large Synoptic Sur-vey Telescope (LSST) will provide much higher statisticalpower to constrain splahback models than the SDSS data.Together with better understanding of priors for both themodel of the galaxy profile and systematic effects such asmiscentering, we should expect significant improvement inour ability to characterize the splashback feature in the nearfuture. The use of weak lensing measurements to more di-rectly measure the cluster mass profile and study splashback(e.g. Umetsu & Diemer 2016) may also be fruitful, particu-larly with upcoming weak lensing data sets from DES, HSCand KiDS.Finally, we note that this work has not addressed inany detail the location of the splashback feature. Althoughour model fits differ from those of M16, we do not find anysignificant difference in the recovered values of R sp (althoughwe have not performed any splits on (cid:104) R mem (cid:105) in this analysis).More exploration into the location of R sp and comparisonto simulations is a fruitful avenue for future work. ACKNOWLEDGEMENTS
We thank Kathleen Eckert, William Hartley, Rachel Man-delbaum, Philip Mansfield, Carles S´anchez, Risa Wechsler,Simon White, and Ying Zu for fruitful discussions whilepreparing this manuscript. CC and AK were supported inpart by the Kavli Institute for Cosmological Physics at theUniversity of Chicago through grant NSF PHY-1125897 andan endowment from Kavli Foundation and its founder FredKavli. EB and BJ are partially supported by the US De-partment of Energy grant DE-SC0007901. kids.strw.leidenuniv.nl MNRAS , 1–17 (2017) he Boundaries of Halos in SDSS REFERENCES
Abbas U., Sheth R. K., 2007, MNRAS, 378, 641Adhikari S., Dalal N., Chamberlain R. T., 2014, J. CosmologyAstropart. Phys., 11, 019Adhikari S., Dalal N., Clampitt J., 2016, J. Cosmology Astropart.Phys., 7, 022Aihara H., et al., 2011, ApJS, 193, 29Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013, ApJ, 762, 109Bertschinger E., 1985, ApJS, 58, 39Budzynski J. M., Koposov S. E., McCarthy I. G., McGee S. L.,Belokurov V., 2012, MNRAS, 423, 104Busha M. T., Evrard A. E., Adams F. C., Wechsler R. H., 2005,MNRAS, 363, L11Cavagnolo K. W., Donahue M., Voit G. M., Sun M., 2009, ApJS,182, 12Cooray A., Sheth R., 2002, Phys. Rep., 372, 1Dalal N., Lithwick Y., Kuhlen M., 2010, preprint,( arXiv:1010.2539 )Diemer B., Kravtsov A. V., 2014, ApJ, 789, 1Dressler A., et al., 1997, ApJ, 490, 577Dutton A. A., Macci`o A. V., 2014, MNRAS, 441, 3359Einasto J., 1965, Trudy Astrofizicheskogo Instituta Alma-Ata, 5,87Fillmore J. A., Goldreich P., 1984, ApJ, 281, 1Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013,PASP, 125, 306Fujita Y., 2004, PASJ, 56, 29Gao L., Navarro J. F., Cole S., Frenk C. S., White S. D. M.,Springel V., Jenkins A., Neto A. F., 2008, MNRAS, 387, 536George M. R., et al., 2012, ApJ, 757, 2Gifford D., Kern N., Miller C. J., 2017, ApJ, 834, 204Gunn J. E., Gott III J. R., 1972, ApJ, 176, 1 Guo Q., et al., 2011, MNRAS, 413, 101Hearin A. P., Watson D. F., van den Bosch F. C., 2015, MNRAS,452, 1958Hernquist L., 1990, ApJ, 356, 359Hoshino H., et al., 2015, MNRAS, 452, 998Ivezi´c ˇZ., Connelly A. J., VanderPlas J. T., Gray A., 2014, Statis-tics, Data Mining, and Machine Learningin AstronomyJeffreys H., Lindsay R. B., 1963, Physics Today, 16, 68Johnston D. E., et al., 2007, preprint, ( arXiv:0709.1159 )Kawata D., Mulchaey J. S., 2008, ApJ, 672, L103Landy S. D., Szalay A. S., 1993, ApJ, 412, 64Larson R. B., Tinsley B. M., Caldwell C. N., 1980, ApJ, 237, 692Mansfield P., Kravtsov A. V., Diemer B., 2016, preprint,( arXiv:1612.01531 )Masjedi M., et al., 2006, ApJ, 644, 54Mehrtens N., et al., 2012, MNRAS, 423, 1024Melchior P., et al., 2015, MNRAS, 449, 2219Melchior P., et al., 2016, preprint, ( arXiv:1610.06890 )Merritt D., Navarro J. F., Ludlow A., Jenkins A., 2005, ApJ, 624,L85Merritt D., Graham A. W., Moore B., Diemand J., Terzi´c B.,2006, AJ, 132, 2685Miyatake H., More S., Takada M., Spergel D. N., MandelbaumR., Rykoff E. S., Rozo E., 2016, Physical Review Letters, 116,041301Moore B., Katz N., Lake G., Dressler A., Oemler A., 1996, Nature,379, 613More S., Diemer B., Kravtsov A. V., 2015, ApJ, 810, 36More S., et al., 2016, ApJ, 825, 39Nagai D., Kravtsov A. V., 2005, ApJ, 618, 557Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563Navarro J. F., et al., 2004, MNRAS, 349, 1039Navarro J. F., et al., 2010, MNRAS, 402, 21Peacock J. A., Smith R. E., 2000, MNRAS, 318, 1144Rines K., Geller M. J., Diaferio A., Kurtz M. J., 2013, ApJ, 767,15Rozo E., Rykoff E. S., 2014, ApJ, 783, 80Rykoff E. S., et al., 2014, ApJ, 785, 104Seljak U., 2000, MNRAS, 318, 203S´ersic J. L., 1963, Boletin de la Asociacion Argentina de Astrono-mia La Plata Argentina, 6, 41Sheldon E. S., et al., 2009, ApJ, 703, 2232Sheth R. K., Hui L., Diaferio A., Scoccimarro R., 2001, MNRAS,325, 1288Simet M., McClintock T., Mandelbaum R., Rozo E., RykoffE., Sheldon E., Wechsler R. H., 2016, preprint,( arXiv:1603.06953 )Soares-Santos M., et al., 2011, ApJ, 727, 45Stark A., Miller C. J., Huterer D., 2016, preprint,( arXiv:1611.06886 )Tully R. B., 2015, AJ, 149, 54Umetsu K., Diemer B., 2016, preprint, ( arXiv:1611.09366 )Weinmann S. M., van den Bosch F. C., Yang X., Mo H. J., 2006,MNRAS, 366, 2Wen Z. L., Han J. L., Liu F. S., 2012, ApJS, 199, 34Wetzel A. R., Tinker J. L., Conroy C., van den Bosch F. C., 2013,MNRAS, 432, 336Wetzel A. R., Tinker J. L., Conroy C., van den Bosch F. C., 2014,MNRAS, 439, 2687Yang X., Mo H. J., van den Bosch F. C., Jing Y. P., 2005, MN-RAS, 356, 1293Yang X., Mo H. J., van den Bosch F. C., Jing Y. P., WeinmannS. M., Meneghetti M., 2006, MNRAS, 373, 1159Yang X., Mo H. J., van den Bosch F. C., Pasquali A., Li C.,Barden M., 2007, ApJ, 671, 153Zu Y., Mandelbaum R., Simet M., Rozo E., Rykoff E. S., 2016,preprint, ( arXiv:1611.00366 )van Breukelen C., Clewley L., 2009, MNRAS, 395, 1845MNRAS000
Abbas U., Sheth R. K., 2007, MNRAS, 378, 641Adhikari S., Dalal N., Chamberlain R. T., 2014, J. CosmologyAstropart. Phys., 11, 019Adhikari S., Dalal N., Clampitt J., 2016, J. Cosmology Astropart.Phys., 7, 022Aihara H., et al., 2011, ApJS, 193, 29Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013, ApJ, 762, 109Bertschinger E., 1985, ApJS, 58, 39Budzynski J. M., Koposov S. E., McCarthy I. G., McGee S. L.,Belokurov V., 2012, MNRAS, 423, 104Busha M. T., Evrard A. E., Adams F. C., Wechsler R. H., 2005,MNRAS, 363, L11Cavagnolo K. W., Donahue M., Voit G. M., Sun M., 2009, ApJS,182, 12Cooray A., Sheth R., 2002, Phys. Rep., 372, 1Dalal N., Lithwick Y., Kuhlen M., 2010, preprint,( arXiv:1010.2539 )Diemer B., Kravtsov A. V., 2014, ApJ, 789, 1Dressler A., et al., 1997, ApJ, 490, 577Dutton A. A., Macci`o A. V., 2014, MNRAS, 441, 3359Einasto J., 1965, Trudy Astrofizicheskogo Instituta Alma-Ata, 5,87Fillmore J. A., Goldreich P., 1984, ApJ, 281, 1Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013,PASP, 125, 306Fujita Y., 2004, PASJ, 56, 29Gao L., Navarro J. F., Cole S., Frenk C. S., White S. D. M.,Springel V., Jenkins A., Neto A. F., 2008, MNRAS, 387, 536George M. R., et al., 2012, ApJ, 757, 2Gifford D., Kern N., Miller C. J., 2017, ApJ, 834, 204Gunn J. E., Gott III J. R., 1972, ApJ, 176, 1 Guo Q., et al., 2011, MNRAS, 413, 101Hearin A. P., Watson D. F., van den Bosch F. C., 2015, MNRAS,452, 1958Hernquist L., 1990, ApJ, 356, 359Hoshino H., et al., 2015, MNRAS, 452, 998Ivezi´c ˇZ., Connelly A. J., VanderPlas J. T., Gray A., 2014, Statis-tics, Data Mining, and Machine Learningin AstronomyJeffreys H., Lindsay R. B., 1963, Physics Today, 16, 68Johnston D. E., et al., 2007, preprint, ( arXiv:0709.1159 )Kawata D., Mulchaey J. S., 2008, ApJ, 672, L103Landy S. D., Szalay A. S., 1993, ApJ, 412, 64Larson R. B., Tinsley B. M., Caldwell C. N., 1980, ApJ, 237, 692Mansfield P., Kravtsov A. V., Diemer B., 2016, preprint,( arXiv:1612.01531 )Masjedi M., et al., 2006, ApJ, 644, 54Mehrtens N., et al., 2012, MNRAS, 423, 1024Melchior P., et al., 2015, MNRAS, 449, 2219Melchior P., et al., 2016, preprint, ( arXiv:1610.06890 )Merritt D., Navarro J. F., Ludlow A., Jenkins A., 2005, ApJ, 624,L85Merritt D., Graham A. W., Moore B., Diemand J., Terzi´c B.,2006, AJ, 132, 2685Miyatake H., More S., Takada M., Spergel D. N., MandelbaumR., Rykoff E. S., Rozo E., 2016, Physical Review Letters, 116,041301Moore B., Katz N., Lake G., Dressler A., Oemler A., 1996, Nature,379, 613More S., Diemer B., Kravtsov A. V., 2015, ApJ, 810, 36More S., et al., 2016, ApJ, 825, 39Nagai D., Kravtsov A. V., 2005, ApJ, 618, 557Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563Navarro J. F., et al., 2004, MNRAS, 349, 1039Navarro J. F., et al., 2010, MNRAS, 402, 21Peacock J. A., Smith R. E., 2000, MNRAS, 318, 1144Rines K., Geller M. J., Diaferio A., Kurtz M. J., 2013, ApJ, 767,15Rozo E., Rykoff E. S., 2014, ApJ, 783, 80Rykoff E. S., et al., 2014, ApJ, 785, 104Seljak U., 2000, MNRAS, 318, 203S´ersic J. L., 1963, Boletin de la Asociacion Argentina de Astrono-mia La Plata Argentina, 6, 41Sheldon E. S., et al., 2009, ApJ, 703, 2232Sheth R. K., Hui L., Diaferio A., Scoccimarro R., 2001, MNRAS,325, 1288Simet M., McClintock T., Mandelbaum R., Rozo E., RykoffE., Sheldon E., Wechsler R. H., 2016, preprint,( arXiv:1603.06953 )Soares-Santos M., et al., 2011, ApJ, 727, 45Stark A., Miller C. J., Huterer D., 2016, preprint,( arXiv:1611.06886 )Tully R. B., 2015, AJ, 149, 54Umetsu K., Diemer B., 2016, preprint, ( arXiv:1611.09366 )Weinmann S. M., van den Bosch F. C., Yang X., Mo H. J., 2006,MNRAS, 366, 2Wen Z. L., Han J. L., Liu F. S., 2012, ApJS, 199, 34Wetzel A. R., Tinker J. L., Conroy C., van den Bosch F. C., 2013,MNRAS, 432, 336Wetzel A. R., Tinker J. L., Conroy C., van den Bosch F. C., 2014,MNRAS, 439, 2687Yang X., Mo H. J., van den Bosch F. C., Jing Y. P., 2005, MN-RAS, 356, 1293Yang X., Mo H. J., van den Bosch F. C., Jing Y. P., WeinmannS. M., Meneghetti M., 2006, MNRAS, 373, 1159Yang X., Mo H. J., van den Bosch F. C., Pasquali A., Li C.,Barden M., 2007, ApJ, 671, 153Zu Y., Mandelbaum R., Simet M., Rozo E., Rykoff E. S., 2016,preprint, ( arXiv:1611.00366 )van Breukelen C., Clewley L., 2009, MNRAS, 395, 1845MNRAS000 , 1–17 (2017) Baxter, Chang et al. Σ g ( R ) [ h M p c − ] f trans = free (Y07) f trans = 1 (Y07)Data (Y07) . . − . − . − . − . d l og Σ g / d l og R f trans = free (RM) f trans = 1 (RM) . . R [ h − Mpc] . . . . Σ g ( R ) / Σ g , m o d e l ( R ) Figure B1.
Measurement of the splashback feature using theY07 group catalog. The top panel shows the projected galaxydensity profile Σ g overlaid with models with f trans free (red solid)and f trans = 1 (green dashed). The middle panel shows the log-derivative of Σ g . We note that the two model fits are nearly iden-tical in both panels. We also overlay in grey the same measure-ments shown in Fig. 1, which is based on the redMaPPer (RM)cluster catalog. The feature around 1 h − Mpc in the redMaP-Per measurements appear slightly sharper than the Y07 groupmeasurement. The bottom panel shows the ratio of the Y07 mea-surements to the best-fit models.van Uitert E., Cacciato M., Hoekstra H., Herbonnet R., 2015,A&A, 579, A26
APPENDIX A: MODEL PARAMETERCONSTRAINTS
Table A1 shows the fit results for the remaining parametersnot shown in Table 2.
APPENDIX B: GALAXY DENSITYMEASUREMENT AROUND Y07 GROUPS
The measurement of galaxy density around the Y07 groupsis shown in Fig. B1. The distribution of logarithmic slopes at R sp corresponding to this measurement is shown in Fig. B2. − − − − − ∂ ln ρ g coll ∂ ln r (cid:0) R D sp (cid:1) . . . . . f trans = free f trans = 1 − . − . − . − . ∂ ln ρ g ∂ ln r (cid:0) R D sp (cid:1) Figure B2.
Same as Fig. 4, but for the measurement of galaxydensity around the groups in the Y07 catalog.MNRAS , 1–17 (2017) he Boundaries of Halos in SDSS Table A1.
Additional best-fit model parameters with f trans free (number preceding semicolon in each column) and f trans = 1 (numberfollowing semicolon) and under different modeling assumptions.Model Catalog ρ s [ h Mpc − ] ρ [ h Mpc − ] s e A RM 3.76 ; 22.58 0.43 ; 0.27 1.61 ; 1.34B RM 27.08 ; 37.15 0.43 ; 0.31 1.61 ; 1.43C RM 36.99 ; 103.92 0.43 ; 0.39 1.61 ; 1.55D RM 48.98 ; 121.73 0.43 ; 0.41 1.61 ; 1.58E RM 23.35 ; 14.86 0.43 ; 0.43 1.61 ; 1.60F RM 11.72 ; 2043.12 0.43 ; 0.12 1.61 ; 1.05G Y07 15.75 ; 29.53 0.32 ; 0.21 1.63 ; 1.41MNRAS000