MMNRAS , 1– ?? (2017) Preprint September 17, 2018 Compiled using MNRAS L A TEX style ﬁle v3.0
Assembly Bias and Splashback in Galaxy Clusters
Philipp Busch, (cid:63) and Simon D. M. White Max-Planck-Institut f¨ur Astrophysik, Postfach 1317, D-85741 Garching, Germany
Accepted XXX. Received YYY; in original form ZZZ
We use publicly available data for the Millennium Simulation to explore the impli-cations of the recent detection of assembly bias and splashback signatures in a largesample of galaxy clusters. These were identiﬁed in the SDSS/DR8 photometric data bythe redMaPPer algorithm and split into high- and low-concentration subsamples basedon the projected positions of cluster members. We use simpliﬁed versions of these pro-cedures to build cluster samples of similar size from the simulation data. These matchthe observed samples quite well and show similar assembly bias and splashback signals.Previous theoretical work has found the logarithmic slope of halo density proﬁles tohave a well-deﬁned minimum whose depth decreases and whose radius increases withhalo concentration. Projected proﬁles for the observed and simulated cluster samplesshow trends with concentration which are opposite to these predictions. In addition,for high-concentration clusters the minimum slope occurs at signiﬁcantly smaller ra-dius than predicted. We show that these discrepancies all reﬂect confusion betweensplashback features and features imposed on the proﬁles by the cluster identiﬁcationand concentration estimation procedures. The strong apparent assembly bias is notreﬂected in the three-dimensional distribution of matter around clusters. Rather itis a consequence of the preferential contamination of low-concentration clusters byforeground or background groups.
Key words: cosmology: theory – large-scale structure of Universe – galaxies: clusters:general
It has long been known that, according to our standardparadigm for the formation of cosmic structure, the clus-tering of dark matter haloes depends strongly on their mass(Kaiser 1984; Efstathiou et al. 1988; Mo & White 1996).At ﬁxed mass, large simulations of ΛCDM universes haveshown that halo clustering depends in addition on a hostof other properties such as formation time, concentration,spin, shape, substructure fraction and internal velocity dis-persion structure (Gao et al. 2005; Wechsler et al. 2006; Gao& White 2007; Li et al. 2008; Dalal et al. 2008; Faltenbacher& White 2010). This additional dependence is genericallycalled ’assembly bias’, It is sensitive to the speciﬁc deﬁni-tion of the property considered, and it varies with halo massin diﬀerent ways for diﬀerent properties. There is still nodetailed theoretical understanding of its origin, and our in-ability to measure the structure of individual dark haloesdirectly has made it diﬃcult to identify observationally.Until recently, attempts to detect an observational sig-nal of assembly bias were inconclusive (e.g Yang et al. 2006;Tinker et al. 2012; Wang et al. 2013; Hearin et al. 2014) (cid:63)
E-mail: [email protected] (MPA) and controversial (e.g. Lin et al. 2016). A strong indicationof assembly bias as a function of halo concentration wasidentiﬁed by Miyatake et al. (2016) in their study of weakgravitational lensing by a large sample of clusters identiﬁedin the SDSS/DR-8 photometric data. Their result was con-ﬁrmed at much higher signal-to-noise by More et al. (2016),who cross-correlated this same cluster sample with individ-ual SDSS galaxies. In both studies, the mean projected dis-tance of individual cluster members from cluster centre wasadopted as a measure of concentration and used to split thesample into equal high- and low-concentration subsamples.Diﬀerences at large radius in the mean projected mass andgalaxy number density proﬁles of these two subsamples thenprovided the evidence for surprisingly strong assembly bias, b lo /b hi ∼ . c (cid:13) a r X i v : . [ a s t r o - ph . C O ] N ov P. Busch & S. D. M. White
Our goal in this paper is to see whether the assembly biasand splashback signals detected by Miyatake et al. (2016)and More et al. (2016) are consistent with current modelsfor galaxy formation in a ΛCDM universe. In particular, wewould like to understand the origin of the strong observeddependence of bias on cluster concentration, of the unex-pectedly small scale of the detected splashback signal, andof the fact that this signal varies between high and low con-centration clusters in the opposite sense to that expectedboth in strength and in radius. For this purpose, we needa realistic simulation of the formation and evolution of thegalaxy population throughout a suﬃciently large volume forour analogue of redMaPPer to identify a large sample of richgalaxy clusters.
Our analysis is based on the
Millennium Simulation de-scribed in Springel et al. (2005). This followed structuredevelopment within a periodic box of side 500 h − Mpc as-suming a ﬂat ΛCDM cosmology with parameters from theﬁrst-year WMAP results. Although these parameters are notconsistent with more recent data, the oﬀsets are relativelysmall and are actually helpful for this paper since they en-hance the abundance of rich clusters in the mass range ofinterest. The dynamical N-body simulation followed the col-lisionless dark matter only, representing it with 2160 ∼ particles of individual mass 8 . × h − M (cid:12) and gravita-tional softening length 5 h − kpc.Haloes and their self-bound subhaloes were identiﬁed in64 stored outputs of this simulation using the subfind al-gorithm (Springel et al. 2001), and these were linked across MNRAS , 1– ?? (2017) ssembly Bias and Splashback time to build subhalo trees which record the assembly his-tory of every z = 0 halo and its subhaloes. These trees arethe basis for simulation (in post-processing) of the forma-tion and evolution of the galaxy population. Galaxies areassumed to form as gas cools, condenses and turns into starsat the centre of every dark matter halo and are carried alongas halos grow by accretion and merging. Both the subhalomerger trees and the speciﬁc galaxy formation simulationused in this paper (and discussed next) are publicly availablein the Millennium Database (Lemson & the Virgo Consor-tium 2006). The particular galaxy population used in this paper wascreated using the semianalytic model described in detail inGuo et al. (2011). These authors implemented their model si-multaneously on the Millennium Simulation and on the 125times higher resolution but smaller volume Millennium-IISimulation (Boylan-Kolchin et al. 2009). This allowed themto tune its parameters in order to reproduce the z = 0 galaxypopulation over a very wide mass range. In this paper we willonly need to consider relatively bright galaxies, well abovethe limit to which results for the two simulations converge.As a result we will only use data from the larger volume sim-ulation. We will analyse the simulation data from a singlesnapshot at z = 0 .
24. This is the mean redshift of the clus-ters in the SDSS sample we compare with and is the closestsnapshot to its median redshift of 0.25.For all galaxies, the simulated galaxy catalogue pro-vides positions, velocities and a range of intrinsic proper-ties, including estimated magnitudes in the SDSS photo-metric bands. We restrict ourselves to galaxies with i -bandabsolute magnitude, M i < − .
43 + 5 log h , which, forour adopted value h = 0 .
7, gives M i < − .
20. The chosenmagnitude limit is very close to the one corresponding tothe redMaPPer luminosity limit of 0 . L ∗ at z = 0 .
24, i.e. M i = − .
25 (see Rykoﬀ et al. 2012). This selection cri-terion leaves us with 2,239,661 galaxies and matches thatadopted by More et al. (2016) for their SDSS galaxies in or-der to achieve volume completeness over the redshift range,0 . ≤ z ≤ . . × − h yr − . This avoids usingmodel colour directly which would introduce a dependenceon the (uncertain) modelling of dust eﬀects. However, thetwo methods produce very similar results in practice, so thechoice has has no signiﬁcant eﬀect on the analysis of thispaper. 897,604 galaxies qualify as red by our criterion. Given the galaxy data described above, we wish to identifyclusters using a simpliﬁed version of the scheme applied tothe SDSS photometric data to generate the catalogue anal-ysed by Miyatake et al. (2016) and More et al. (2016). Weproject the simulated galaxy and mass distributions along each of the three principal axes of the Millennium simulationto obtain three ‘sky’ images, for each of which depth infor-mation is available for the galaxies either in real space or inredshift space. In the latter case, the line-of-sight peculiarvelocities of galaxies are added to their Hubble velocities toproduce redshift space distortions (RSD). These are impor-tant when considering how the use of photometric redshiftsaﬀects the assignment of galaxies to clusters (see 2.2.1). Thefollowing describes our cluster identiﬁcation scheme and ex-plains how we split the clusters into equal high- and low-concentration subsamples. Our cluster identiﬁcation algorithm, inspired by redMaPPer,ﬁnds clusters in the projected distribution of red galaxies.Every red galaxy in each of our three projections is consid-ered as the potential centre of a cluster. The algorithm growsclusters by adding new red galaxies (deﬁned as in 2.1.2) inorder of increasing projected separation until the richness λ and the cluster radius R c reach the largest values satisfyingthe relation given by Rykoﬀ et al. (2014), R c ( λ ) = 1 . (cid:18) λ (cid:19) . h − Mpc (1)in physical (rather than comoving) units. Initialising with λ = 1 and R c (1),(i) we consider as possible members the N g red galaxieswhich lie within R c and have a (redshift space) depth oﬀsetbelow ∆ z m ,(ii) we calculate ¯ N , the expected number of uncorrelated(’background’) galaxies within R c and ∆ z m ,(iii) we update λ = N g − ¯ N and R c ( λ ),(iv) we check whether the current central galaxy still hasa higher stellar mass than any other cluster member, other-wise we delete it as a potential central and move to the nextone,(v) we start the next iteration at (i) if λ has increased,otherwise we stop.This process usually converges quickly and only in afew cases is it unsuccessful in ﬁnding a cluster. Note that wechoose to require that the central galaxy should be the onewith the highest stellar mass. Only in ∼ i -band,and we have checked that choosing to require instead thatit should the most luminous has a negligible eﬀect on ourresults. In the following we will only consider clusters with20 ≤ λ ≤ z m =60 h − Mpc, 120 h − Mpc and 250 h − Mpc; the largest ofthese is equivalent to projecting through the full MillenniumSimulation. For comparison, the 1 σ uncertainty in the pho-tometric redshift of a single SDSS red galaxy is estimated byRykoﬀ et al. (2014) to be about 90 h − Mpc at the medianredshift of the observed cluster sample. The total number ofclusters found (summed over the three projections) is givenin Table 1.These numbers are similar to the number of clusters(8,648) in the observed sample we are comparing with. This
MNRAS , 1– ????
MNRAS , 1– ???? (2017) P. Busch & S. D. M. White
The size of simulated cluster samples for diﬀerent max-imal depth oﬀsets, ∆ z m .Sample Name ∆ z m No. Members h − MpcCS60 60 9196CS120 120 9213CS250 250 8930
The fractional overlap between diﬀerent cluster samples.Base sample Comparison sampleCS60 CS120 CS250CS60 1.0 0.876 0.736CS120 0.874 1.0 0.783CS250 0.758 0.808 1.0 is a coincidence since the volume of the Millennium Simu-lation is only about a tenth of that in the SDSS footprintover the redshift range 0 . ≤ z ≤ .
33, but the abundanceof rich clusters is enhanced by a factor of about three in thesimulation because it assumes σ = 0 .
9, signiﬁcantly abovecurrent estimates .There is, of course, a very substantial overlap betweenthese three cluster samples, but it is not perfect. In Table 2we give the fraction of clusters in a given sample that sharetheir central galaxy (in the same projection) with a clusterin a comparison sample and pass the richness ﬁlter in both.We see that most clusters are indeed duplicated. Those thatare not, fail because in one of the two samples either a moremassive potential member is included or the richness fallsoutside the allowed range. Such diﬀerences are a ﬁrst indi-cation of sensitivity to projection eﬀects, an issue that isdiscussed further in subsection 2.2.3.Notice that the algorithm described above allows agiven galaxy to be considered a member of more than onecluster. Although the majority of our simulated clusters donot have such overlaps, they are not negligible; the fractionof clusters which share at least one galaxy with another clus-ter in the same projection is 18.8, 21.8 and 26.7 per cent forCS60, CS120 and CS250, respectively. The average numberof galaxies in these overlaps is ∼
14, which should be com-pared with the mean number of galaxies per cluster which is37 to 46. In order to check the importance of the overlaps, wehave repeated our analysis using only the ∼ We checked the results of this paper using the public semian-alytic catalogue of Henriques et al. (2015) which is implementedon a version of the Millennium Simulation rescaled to the Planck2013 cosmology (Planck Collaboration 2014). We ﬁnd far fewerclusters: 2407, 2244 and 2307 for the equivalents of CS250, CS120,and CS60, respectively. This corresponds to 83.1%, 77.5% and79.6% of the expected number of clusters in three times the(rescaled) volume of the simulation. We decided to stay with theoriginal cosmology since the larger number of clusters providesmuch better statistics. cluster, albeit by assigning probabilities to each potentialmembership based on the galaxy’s photometric redshift, itsprojected separation from each cluster centre, and the rich-ness of the clusters. The consistent use of such probabilitiesis the principal diﬀerence between the actual redMaPPeralgorithm and the simpliﬁed version we use here.
At the core of the following analysis is the separation of eachcluster sample into two equal subsamples with identical rich-ness distributions, but disjoint distributions of concentration c gal as introduced by Miyatake et al. (2016). This concen-tration is based on the mean projected distance from clustercentre of red galaxy members, c gal = R c / (cid:104) R mem (cid:105) where inour case (cid:104) R mem (cid:105) = 1 N mem N mem (cid:88) i R mem ,i . (2)We classify a particular cluster as high or low concentration,depending on whether c gal lies above or below the medianfor all clusters of the same richness. For richness values withfewer than 200 clusters in a given sample, we bin togetherneighbouring richness bins to exceed this number before de-termining the median. For the observed clusters Miyatakeet al. (2016) binned clusters by both richness and redshiftbefore determining the median, but redshift binning is notnecessary for the simulated samples since they are all takenfrom the same simulation output. It is not straightforward to connect a galaxy cluster deﬁnedin projection with a speciﬁc three-dimensional cluster, inour case a speciﬁc subfind halo. The idealised model of aspherically symmetric cluster centred on its most massivegalaxy and containing all the cluster members identiﬁed inprojection corresponds poorly to most of the clusters iden-tiﬁed either in the simulation or, most likely, in the SDSS.In almost all cases, the galaxies identiﬁed as members in 2Dreside in multiple 3D objects distributed along the line-of-sight. This makes the cross-identiﬁcation between 2D and3D ambiguous.Here we consider two possibilities for deﬁning the 3Dcounterpart of each 2D cluster: the dark matter halo thathosts the central galaxy and the one that hosts the largestnumber of member galaxies. The former deﬁnition followsthe logic of the cluster centring, while the latter ensures thatthe richness of the 3D counterpart corresponds most closelyto that of the 2D system. It is interesting to see how oftenthese deﬁnitions coincide, i.e., how often the central galaxyis actually part of the dominant galaxy aggregation alongthe line-of-sight. We give in Table 3 the fraction of clustersin each of our three samples for which both methods lead tothe same FoF halo. These numbers show that that the twodeﬁnitions are generally in good agreement, and that this isbetter for smaller maximal depth oﬀsets and for more con-centrated clusters. These trends reﬂect the projection eﬀectsdiscussed in detail in Section 5.It is also interesting to see how many of the potentialcluster members identiﬁed in 2D are, in fact, part of the
MNRAS , 1– ?? (2017) ssembly Bias and Splashback Table 3.
The fraction of clusters where the central galaxy residesin the FoF halo contributing the largest number of potential 2Dmembers; the mean fraction of such members in this halo; themean fraction of such members in the FoF halo of the centralgalaxy; the mean membership fraction predicted by ’standard’background subtraction.Subs. Sample F centred (cid:104) F biggest (cid:105) (cid:104) F central (cid:105) (cid:104) λ/N g (cid:105) All CS60 0.93 0.826 0.803 0.922CS120 0.903 0.755 0.726 0.856CS250 0.848 0.635 0.595 0.743high c gal CS60 0.983 0.880 0.874 0.922CS120 0.973 0.819 0.812 0.855CS250 0.948 0.709 0.697 0.742low c gal CS60 0.876 0.772 0.732 0.923CS120 0.833 0.69 0.64 0.857CS250 0.749 0.561 0.494 0.744 same 3D object. For each of our clusters we ﬁnd the maximalfraction of its members contained in a single 3D FoF halo.The third column of Table 3 then gives the average of thesefractions. This can be compared with the average fraction ofits members contained in the FoF halo of its central galaxy(fourth column) and with the average expected as a result ofour background correction, (cid:104) λ/N g (cid:105) , given in the last column.The values for (cid:104) F biggest (cid:105) , (cid:104) F central (cid:105) and (cid:104) λ/N g (cid:105) in Ta-ble 3 show that we consistently ﬁnd more ’foreign’ galaxiesin our clusters than we would expect from contamination bya uniform background. The more concentrated clusters havecontamination ratios close to, yet still a few percent belowthe expected ones. The low-concentration clusters have con-tamination fractions more than twice the expected values.We therefore conclude that the identiﬁed clusters are biasedtowards arrangements of multiple objects along the LoS, es-pecially in the low c gal case. Again, this is very much in linewith our discussion on the preferential selection of alignedsystems in Section 5. We are now in a position to investigate whether the assem-bly bias and splashback features identiﬁed in SDSS data byMiyatake et al. (2016) and More et al. (2016) are reproducedwhen our simpliﬁed version of the redMaPPer algorithm isapplied to the public Millennium Simulation data . We beginby comparing the observed mean galaxy and mass proﬁles todirectly analogous proﬁles for CS250, ﬁnding that both thesurprisingly strong assembly bias and the unexpected prop-erties of the apparent splashback signal are reproduced well.Most diﬀerences can be ascribed to the ﬁnite size of the sim-ulation or to the simpliﬁcations of our cluster identiﬁcationscheme. We then use our three cluster catalogues to investi-gate the dependence of these successes on ∆ z m , the maximaldepth oﬀset allowed for potential cluster members, ﬁndingthat the assembly bias signal is sensitive to this parameterbut the splashback features are not. Finally we look in moredetail at the radial dependence of the ratio of the proﬁlesof low- and high-concentration clusters. Later sections em-ploy the full 3D information available for the simulation toexplore the origin of the observed features, and vary ourcluster identiﬁcation scheme to demonstrate how its imprint on the measured proﬁles can confuse identiﬁcation of thesplashback signal. We collect the main proﬁle results for the CS250 samplein Figures 1 to 3. Here and in the following, unless notedotherwise, the solid line represents the median value from10000 bootstrap resamplings of the relevant cluster sample.The shaded regions denote the 68 per cent (darker) and 95per cent (lighter) conﬁdence intervals around this median.We calculate the mean galaxy surface number densityproﬁle for each cluster sample as∆Σ g ( R ) = Σ g ( R ) − ¯Σ g (3)where we use all galaxies brighter than M i = − .
43 +5 log h , not just the red ones, and we impose no maximaldepth oﬀset from the cluster. Σ g ( R ) is then the mean over allclusters of the surface number density of such galaxies in anannular bin at projected distance R from the central galaxy,and ¯Σ g is the mean surface density over the full projectedarea of the simulation.Figure 1 shows that CS250 reproduces the ﬁndings ofMore et al. (2016) remarkably well. The deviation at largescales ( > h − Mpc) is expected and reﬂects a lack of large-scale power in the Millennium Simulation due to its ﬁnitesize. The oﬀset between the high- and low-concentration sub-samples at
R > h − Mpc shows that the simulation re-produces the strong assembly bias seen in the SDSS data.On small scales ( < h − kpc) the number density pro-ﬁle is slightly too steep for the high-concentration clus-ters, but shows otherwise very good agreement, while thereis an oﬀset of 0 . h − kpc. The most notable diﬀerences are onintermediate scales, especially in the range 1 h − Mpc ≤ R ≤ h − Mpc for the low-concentration case. For high-concentration clusters the agreement in this range is excel-lent and extends out to well beyond 10 h − Mpc. This is theradial range where splashback features are expected, but isalso comparable to the radius, R c , used operationally to de-ﬁne clusters. These diﬀerences are highlighted in the radialvariations of the proﬁle slope, which we look at next.In Figure 2 we plot the logarithmic derivatived log ∆Σ g / d log R over a restricted radial range for thesesame two CS250 subsamples, comparing with the samequantity for SDSS clusters as plotted by More et al. (2016).The simulated curves appear noisier than those observedThis is at least in part because of the more direct deriva-tive estimator used here. Nevertheless, we reproduce themain features highlighted by More et al. (2016), who iden-tiﬁed the position of the minimum of these curves (i.e. thesteepest proﬁle slope) as their estimate of the splashbackradius. The minima occur at similar radii in the observedand simulated data which, as More et al. (2016) pointedout, are smaller than expected given lensing estimates ofcluster mass. Further the minimum is deeper for the highconcentration sample and occurs at smaller radius, whereasthe opposite is expected from earlier work on the depen-dence of splashback radius on halo accretion history (andhence concentration, see Diemer & Kravtsov (2014)). In ad-dition, there are clear diﬀerences between the observed and MNRAS , 1– ?? (2017) P. Busch & S. D. M. White − R (cid:2) h − Mpc (cid:3) − − ∆ Σ g (cid:2) h M p c − (cid:3) Millenniumhigh c gal low c gal More et al.high c gal low c gal Figure 1.
Mean surface number density proﬁles ∆Σ g for galax-ies with M i < − .
20 surrounding clusters in the low- and high-concentration subsamples of CS250 are compared with observa-tional results from More et al. (2016). simulated curves. In particular, the proﬁles of simulated low-concentration clusters are clearly shallower than observed inthe range 200 h − kpc < R < . h − Mpc.We discuss these features in more detail in Section 6,showing them to result from the superposition of eﬀects in-duced by the cluster selection algorithms on the true splash-back signal.Mean mass density proﬁles can be computed much morestraightforwardly for our simulated cluster samples than ispossible observationally, where such proﬁles are obtainedfrom the correlated orientation of background galaxies in-duced by gravitational lensing. In order to compare withthe lensing results in Miyatake et al. (2016), we bin up theprojected mass distribution of the simulation around clus-ter central galaxies in exact analogy to the above procedurefor creating galaxy number density proﬁles, and we manip-ulate the resulting proﬁles to obtain the directly observablequantity,∆Σ m ( < R ) = Σ m ( < R ) − Σ m ( R ) . (4)Here, Σ m ( R ) is the surface mass density proﬁle analogousto ∆Σ g ( R ) above, while Σ m ( < R ) is the mean of this quan-tity over projected radii interior to R . Note that despite thesimilarity in notation (which we have inherited from earlierwork) ∆Σ m ( < R ) is not directly analogous to ∆Σ g ( R ) andwill diﬀer from it in shape even if the projected mass andgalaxy number density proﬁles parallel each other exactly.In Figure 3 we compare ∆Σ m ( < R ) obtained in this wayfor the high- and low-concentration subsamples of CS250to the proﬁles inferred by Miyatake et al. from their SDSSlensing data. Whereas the observational data show at mostsmall diﬀerences between the high- and low-concentrationsubsamples for R < h − Mpc, our simulated proﬁles diﬀer R (cid:2) h − Mpc (cid:3) − . − . − . − . − . d l og ∆ Σ g / d l og R Millenniumhigh c gal low c gal More et al.high c gal low c gal Figure 2.
The logarithmic derivatives of the ∆Σ g proﬁles forCS250 shown in Figure 1 are compared with those plotted fortheir SDSS clusters by More et al. (2016). signiﬁcantly in a way which is related to the diﬀerences seenin Figure 1. Indeed, we have plotted the surface mass den-sity proﬁles Σ m ( R ) directly, and ﬁnd they are very similarin shape and relative amplitude to the simulated galaxy sur-face density proﬁles of Figure 1. We note that the disagree-ment between simulation and observation is limited to low-concentration clusters – agreement is very good for the high-concentration systems on all scales below about 15 h − Mpc.We have found no explanation for this discrepancy. The un-certainties on the ∆Σ m ( < R ) inferred from lensing data aremuch larger than the purely statistical uncertainty in thesimulation results, but below 1 h − Mpc the simulation re-sults for low-concentration clusters lie systematically belowthe observations, while beyond 3 h − Mpc they tend to lieabove them. (Note that the coloured bands in Figure 3 showthe estimated 1 σ uncertainties in the observations.) This dis-agreement is in line with the stronger diﬀerences between theprojected galaxy proﬁles for the low-concentration subsam-ple. Our ﬁndings for the diﬀerences in the inner part areclose to the ﬁndings of Dvornik et al. (2017) who recentlyinvestigated the mass proﬁles of galaxy groups. These lessmassive objects were identiﬁed with a diﬀerent group ﬁnder(based on the FoF algorithm), but the same c gal projectedconcentration measure was used to divide the sample. Whilethey found a similar split at small scales in the lensing pro-ﬁles, they did not see a signiﬁcant signal of assembly bias onlarge scales. This is expected around the masses of groupswhen splitting by concentration.Miyatake et al. (2016) inferred almost equal meantotal masses, M m ∼ × h − M (cid:12) , for high- low-concentration clusters from their measured ∆Σ m ( < R ) pro-ﬁles. Processed in the same way, our simulated proﬁle forhigh-concentration clusters would give a very similar an-swer, whereas that for low-concentration clusters would give MNRAS , 1– ?? (2017) ssembly Bias and Splashback − R (cid:2) h − Mpc (cid:3) − ∆ Σ m (cid:2) M (cid:12) p c − h (cid:3) Millenniumhigh c gal low c gal Miyatake et al.high c gal low c gal Figure 3.
The mean lensing observable ∆Σ m for high- and low-concentration clusters in the CS250 sample is compared to obser-vational results for SDSS clusters from Miyatake et al. (2016). a lower value by a few tens of percent. (For M m =2 × h − M (cid:12) , R m = 1 . h − Mpc, in the middle of therange where simulated and observed ∆Σ m ( < R ) agree best.)Thus the overall mass-scale of the clusters identiﬁed in theGuo et al. (2011) galaxy catalogues by our redMaPPer-likealgorithm is close to that of the SDSS clusters studied byMiyatake et al. (2016) and More et al. (2016). The simulation results shown in the last section all referredto CS250 for which any red galaxy projected within R c isconsidered a potential cluster member, regardless of its dis-tance in depth (“redshift”) from the central galaxy. As notedpreviously, Rykoﬀ et al. (2014) estimate the 1 σ uncertaintyin the photometric redshift of an individual red SDSS mem-ber galaxy to increase with distance and to be 90 h − Mpcat z = 0 .
25, the median redshift of the SDSS cluster sample.Thus many of the observed clusters may be better localisedin depth than in the CS250 catalogue. In this section wecompare galaxy and mass proﬁles among our three simulatedcluster catalogues, CS250, CS120 and CS60, for which thedepth selection parameter ∆ z m = 250 ,
120 and 60 h − Mpc,respectively. This allows us to assess how strongly the eﬀec-tive selection depth of clusters aﬀects their apparent splash-back and assembly bias signals. We ﬁnd that eﬀects are smallon the former, but can be substantial on the latter.Figure 4 shows the overall shape of the projected galaxynumber density proﬁles to be very similar in the three clus-ter catalogues. The high concentration proﬁles diﬀer fromeach other by at most 10 per cent within R c and remainwithin the same bound out to ∼ h − Mpc. Beyond thispoint the uncertainties increase drastically and the ratiosof the proﬁles with smaller ∆ z m quickly depart from unity but stay within a less than the 68-percentile of the bootstrapdistribution of it. The variation is somewhat smaller for low-concentration clusters and is also below 10 per cent within R c , but also below 25 per cent all the way out ∼ h − Mpc.Beyond R c the proﬁle amplitude of low-concentration clus-ters decreases with decreasing ∆ z m at all separations whereit is reliably determined.This level of agreement is such that all three cataloguesagree almost equally well with observation. In the proﬁlesthemselves, systematic diﬀerences only start to become no-ticeable outside R c and the largest eﬀect is the shift in thelarge-scale amplitude of the proﬁle for the low-concentrationclusters, which, as we will see below (in Section 3.3) isenough to aﬀect the apparent level of assembly bias signiﬁ-cantly. At the intermediate radii relevant for splashback de-tection, the proﬁle shapes are suﬃciently similar that curveslike those of Figure 2 show almost no dependence on ∆ z m .The ∆Σ m proﬁles (shown in Figure 5) also vary onlyslightly as a function of eﬀective cluster depth, ∆ z m , withshifts of similar amplitude to those seen in the projectedgalaxy number density proﬁles. For high-concentration clus-ters these are even smaller than for the previous case, whilefor low-concentration clusters they are larger within R c andhave the eﬀect of increasingly smoothing the sudden changesin slope seen in the CS250 proﬁle as ∆ z m decreases. Forboth cases the amplitude of the proﬁles on large scales isdecreased for smaller ∆ z m , though by less than 25 per centout to ∼ h − Mpc.
By taking the ratio of the proﬁles discussed in the previ-ous section we can obtain a measure of the relative bias ofhigh- and low-concentration clusters at ﬁxed cluster rich-ness, hence of assembly bias . In Figure 6 we show this ratiofor the ∆Σ g proﬁles as a function of projected separation forour three catalogues of simulated clusters. In order to mea-sure the large-scale bias, More et al. (2016) only plotted thisratio at R ≥ h − Mpc (the orange points with error barsin Figure 6). However, since they give the individual proﬁlesfor high- and low-concentration clusters, it is straightforwardto reconstruct the observed ratio on smaller scale. We showthis as a dashed orange line in Figure 6.The observed and the simulated ratios show similar be-haviour which separates into three distinct radial regimes.At R ≥ h − Mpc, the relative bias varies little and theobserved value of 1 . ± .
07 matches very well that forCS250 outside of R = 8 h − Mpc. CS120 gives a somewhatsmaller value ﬁtting the observations well between 3 and10 h − Mpc, while at larger R it is still within about 1 σ .CS60 has even weaker relative bias barely within 1 σ . Boththese signals appear to decline with increasing R . The be-haviour at smaller scales diﬀers markedly on either side ofa sharp peak which, for the simulated clusters, occurs al-most exactly at (cid:104) R c (cid:105) ∼ h − Mpc, coinciding with that forthe observed clusters. At smaller R , the ratio of the pro-ﬁles increases smoothly and strongly with R , reﬂecting therequirement that the two cluster subsamples have similarrichness but systematically diﬀerent values of (cid:104) R mem (cid:105) . Thisalso enforces a ratio substantially above unity at R = R c .At intermediate radii, R c < R < h − Mpc, the ratio has todecline from the high value at the peak to the more mod-
MNRAS , 1– ?? (2017) P. Busch & S. D. M. White − ∆ Σ g (cid:2) h M p c − (cid:3) CS60high c gal low c gal CS120high c gal low c gal CS250high c gal low c gal − R (cid:2) h − Mpc (cid:3) . . . . . . ∆ Σ g , X / ∆ Σ g , C S Figure 4.
Comparison of the ∆Σ g proﬁles for the high c gal andlow c gal subsamples of our three simulated cluster catalogues (up-per panel) and ratios of the proﬁle amplitudes for CS120 andCS60 to that for CS250 (lower panel). − ∆ Σ m (cid:2) h M (cid:12) p c − (cid:3) CS60high c gal low c gal CS120high c gal low c gal CS250high c gal low c gal − R (cid:2) h − Mpc (cid:3) . . . . . . ∆ Σ m , X / ∆ Σ m , C S Figure 5.
Comparison of the ∆Σ m proﬁles for the high c gal andlow c gal subsamples of our three simulated cluster catalogues (up-per panel) and ratios of the proﬁle amplitudes for CS120 andCS60 to that for CS250 (lower panel). est value characteristic of the large-scale assembly bias. Inall three samples there is a noticeable change in slope justoutside 2 h − Mpc which appears to reﬂect true splashbackeﬀects (see Section 4.2).These properties demonstrate that the operational def-inition of clusters has a substantial eﬀect on the ratio of theproﬁles out to at least 3 h − Mpc. These eﬀects must there-fore be present also in the individual proﬁles, and hencemust aﬀect their use for identifying splashback features. Inaddition, the variation of the ratios at large R among ourthree cluster catalogues shows that the apparent assemblybias signal is signiﬁcantly aﬀected by projection eﬀects.The ratio of the ∆Σ m proﬁles for the high- and low con-centration subsamples of each of our three simulated clustercatalogues are shown in Figure 7 in exactly analogous for-mat to Figure 6. They are compared to observational resultstaken directly from Miyatake et al. (2016). The diﬀerence inshape between the simulation curves in Figures 7 and 6 isdue primarily to the conversion of Σ m ( R ) to ∆Σ m ( < R ). Aratio plot constructed using Σ m ( R ) directly is quite similarto Figure 6, although the peak at (cid:104) R c (cid:105) is less sharply de-ﬁned. The behaviour of the observational points in Figure 7is quite erratic and looks rather implausible when comparedwith the smooth variation predicted by the simulation. Overthe ranges 3 h − Mpc < R < h − Mpc and
R > h − Mpc the predicted assembly bias signal is almost constant, butover the ﬁrst range it is much larger than and apparentlyinconsistent with that observed, whereas over the second itis smaller than and again apparently inconsistent with thatobserved. It is our impression that the uncertainties of theseobservational points are too large for secure interpretationto be possible.The diﬀerences in large-scale assembly bias between ourthree simulated cluster catalogues are similar to those seenfor the cluster number density proﬁles of Figure 6, althoughpushed out to systematically larger radii. Again this is aconsequence of the conversion from Σ m ( R ) to ∆Σ m ( < R ).On small scales the simulation curves lie well below the ob-servational points. This is a restatement of the fact that thesimulated proﬁles in Figure 3 diﬀer much more at these radiithan the observed proﬁles. Miyatake et al. (2016) and More et al. (2016) interpret theirSDSS results under the implicit assumption that the fea-tures seen in the stacked 2D proﬁles correspond to similarfeatures in the ’true’ 3D proﬁles. In our simulations, it is pos-sible to test the extent to which this is the case, so in this
MNRAS , 1– ?? (2017) ssembly Bias and Splashback R (cid:2) h − Mpc (cid:3) . . . . . ∆ Σ g , l o w / ∆ Σ g , h i g h More et al.More et al., ExtendedCS60 CS120CS250 h R c i Figure 6.
The ratio of the projected galaxy number density pro-ﬁles of the low c gal and high c gal subsamples of our three sim-ulated cluster catalogues (solid lines surrounded by their 68 percent conﬁdence regions). Points with error bars are observationaldata taken directly from More et al. (2016), while the continua-tion of these data to smaller scales (the dashed orange line) wascalculated from the individual proﬁles in their paper. The dottedvertical line indicates (cid:104) R c (cid:105) for the simulated clusters. The hori-zontal orange band is the observed assembly bias signal quoted byMore et al. (2016) with its 68 and 95 per cent conﬁdence ranges. R (cid:2) h − Mpc (cid:3) . . . . . ∆ Σ m , l o w / ∆ Σ m , h i g h MiyatakeCS60 CS120CS250
Ratios of ∆Σ m for the high- and low-concentrationsubsamples of our three cluster catalogues (solid lines with their68 per cent conﬁdence ranges). Points with error bars are resultsderived from the gravitational lensing signal of SDSS clusters byMiyatake et al. (2016). section we compute stacked 3D proﬁles of mass density andof galaxy number density around the central galaxies of ourthree cluster catalogues, splitting them into high- and low-concentration subsamples as before using the 2D values of c gal = R c ( λ ) / (cid:104) R mem (cid:105) . This allows us to make plots directlyanalogous to those discussed above, and so to check the 2D– 3D correspondence. In this section all proﬁles are calcu-lated in true position space rather than in redshift space.Note that we here use a standard deﬁnition of the spher-ically averaged mass density proﬁle rather than some 3Danalogue of ∆Σ m . Note also that since each central galaxycan appear in one to three diﬀerent projections, we give itthe corresponding weight when constructing the 3D proﬁlesin order to keep as close a correspondence as possible to the2D results discussed previously. As was the case in 2D, we ﬁnd that plots of the 3D proﬁleslope, analogous to those of Figure 2, are very similar forour three cluster catalogues. In Figures 8 and 9 we thereforeshow results for CS250 only. Since recent theoretical workon splashback properties has concentrated on cluster massproﬁles (e.g. Diemer & Kravtsov 2014, hereafter DK14), westart with a discussion of Figure 8 which shows logarithmicslope (referred to as γ below) as a function of 3D radius r . These slope proﬁles show relatively smooth behaviourwith well-deﬁned minima at r ∼ . h − Mpc. The mean M m values in the two sub-samples correspond to R m ∼ . h − Mpc and R m ∼ . h − Mpc, so these minimaoccur at 1 . R m and 1 . R m for the high- and low-concentration samples, respectively. These values are veryclose to the expected values given in More et al. (2015) forthe expected mass accretion rates at the given masses andredshift. The slopes at minimum are signiﬁcantly shallowerfor our stacks ( γ ∼ − .
8) than DK14 found for halos ofsimilar mass ( γ ∼ − . R m and then take the median density at each r/R m ,whereas we take the mean density at each radius directly.The DK14 procedure typically produces deeper and sharperminima, hence better deﬁned splashback radii which occurat slightly smaller radii, but it is not easily implemented onobserved samples. For example, the redMaPPer samples aredeﬁned to have similar (and known) values of R c but theirindividual values of R m are unknown. In addition, weaklensing reconstructions of the mass distribution naturallyproduce mean rather than median mass proﬁles.The two slope proﬁles of Figure 8 diﬀer signiﬁcantly inshape. In the inner regions ( r < R c ) this reﬂects the fact MNRAS , 1– ????
8) than DK14 found for halos ofsimilar mass ( γ ∼ − . R m and then take the median density at each r/R m ,whereas we take the mean density at each radius directly.The DK14 procedure typically produces deeper and sharperminima, hence better deﬁned splashback radii which occurat slightly smaller radii, but it is not easily implemented onobserved samples. For example, the redMaPPer samples aredeﬁned to have similar (and known) values of R c but theirindividual values of R m are unknown. In addition, weaklensing reconstructions of the mass distribution naturallyproduce mean rather than median mass proﬁles.The two slope proﬁles of Figure 8 diﬀer signiﬁcantly inshape. In the inner regions ( r < R c ) this reﬂects the fact MNRAS , 1– ???? (2017) P. Busch & S. D. M. White that the two samples are separated by galaxy concentra-tion (in practice, by (cid:104) R mem (cid:105) /R c ) so that, by deﬁnition, thelow-concentration clusters have shallower 2D galaxy densityproﬁles within R c than the high-concentration clusters. Fig-ure 9 shows that this requirement carries over to the 3Dgalaxy proﬁles, and it is still very visible in Figure 8. Sim-ilar eﬀects are seen in Figure 14 of DK14 where they splittheir halo sample by 3D mass concentration. However, ourresults do not agree with the trend they ﬁnd for more con-centrated clusters to have a shallower minimum slope anda larger splashback radius. We have checked that if we fol-low their scaling and median stacking procedures, our high-concentration clusters still have a steeper minimum slopeand the same splashback radius as our low-concentrationclusters. The discrepancy must reﬂect the diﬀerence betweenselecting halos by 3D mass and mass concentration and se-lecting clusters by 2D richness and galaxy concentration.The shapes of the 3D slope proﬁles for the mass (Fig-ure 8) and for the galaxies (Figure 9) are very similar, inparticular, beyond the splashback minimum. At smaller radiithe features induced by cluster selection are stronger in thegalaxy proﬁle, with a secondary minimum just inside (cid:104) R c (cid:105) which is just visible as a slight inﬂection in the mass pro-ﬁle. Overall, however, the features in the galaxy proﬁle aremuch less dramatic than in its 2D analogue, Figure 2. Thisjust reﬂects the fact that clusters were selected and theirconcentrations estimated using the 2D data We now look at the ratios of stacked 3D mass overdensityproﬁles for our low- and high-concentration clusters, and atthe corresponding ratios of their galaxy number overdensityproﬁles. These are directly analogous to the ratios of 2Dgalaxy number overdensity proﬁles shown in Figure 6. As inthat ﬁgure, we here compare results for the three samples,CS60, CS120 and CS250. Ratios as a function of r are shownfor mass overdensities in Figure 10 and for galaxy numberoverdensities in Figure 11. The shapes of the curves and theirrelative positions for the three samples are very similar inthese two ﬁgures.In the inner regions, r < R c , all curves are rapidly andsmoothly rising, showing that the diﬀerence in 2D galaxyproﬁles resulting from our classiﬁcation by concentrationcarries over to the 3D galaxy and mass proﬁles. In thisregime and in both plots the ratio for CS60 is slightly largerthan that for CS120 and signiﬁcantly larger than that forCS250. This behaviour mirrors that of the ratio of the frac-tions of 2D potential members which are part of the centralgalaxy’s FoF group (see Table 3). Interestingly, this rank-ing of amplitudes for the three samples persists to muchlarger scales and is opposite to that seen in 2D (Figure 6).Clearly, with increasing ∆ z m , projection eﬀects contributemore strongly to low- than to high-concentration clustersnot only at R ∼ R c but also at much larger projected sepa-ration.In the range R c < r < h − Mpc, all curves continue torise to a sharp peak before dropping again to a value whichremains approximately constant over the interval 5 h − Mpc < r < h − Mpc. The peak corresponds to the crossingof the derivative curves for the low- and high-concentrationsubsamples in Figures 8 and 9. It thus reﬂects diﬀerences in the way the splashback feature merges into larger scalestructure in the two cases. As noted above, it appears to bevisible as a sharp change in slope in the proﬁles of Figure 6(see also Figure 15 below). Between R c and the peak, eﬀectsfrom sample deﬁnition clearly modulate galaxy overdensityproﬁle ratios more strongly than mass overdensity proﬁleratios but the diﬀerence is quite small.The constant proﬁle ratios seen over the range 5 h − Mpc < r < h − Mpc are a direct measurement of the 3D assem-bly bias for cluster samples split by 2D concentration. Thesevalues are signiﬁcantly smaller than the 2D values inferredfrom Figure 6. In addition, they rank in the opposite sensewith ∆ z m , they are consistent between Figures 8 and 9, andthey are similar to the values expected from previous workon assembly bias for cluster mass haloes split by concentra-tion (e.g. More et al. 2016). As we will see in the next section,a clue to the origin of this diﬀerence between the 2D and 3Destimates of assembly bias comes from the largest r bins inthese ﬁgures where, although noisy, the ratios of the proﬁlesrise to large values. In the preceding sections we found a number of diﬀerences inthe apparent splashback and assembly bias signals betweenthe 2D and the 3D proﬁles of our simulated galaxy clus-ters. These diﬀerences are present both in the mass and inthe galaxy number density proﬁles, and they aﬀect the low-and high-concentration subsamples to diﬀering degrees. Inthis section we focus speciﬁcally on galaxy number densityproﬁles, compiling them in the two dimensions of projectedseparation and line-of-sight depth so that we can compareresults for the two subsamples and isolate the distributionin depth of the galaxies which give rise to the diﬀerence inprojected proﬁles.Let R , as above, denote projected separation, and q > q = | d | ) or in redshift space ( q = | π | ). We deﬁnea set of cells of constant width in ln R and ln q and compilegalaxy counts in these cells around the central galaxies ofthe low- and high-concentration subsamples of each of ourcluster samples, N lo ( R, q ) and N hi ( R, q ) respectively.In Figures 12 and 13 we show the quantity β ( R, q ) = N lo ( R, q ) − N hi ( R, q ) (cid:80) q [ N lo ( R, q ) + N hi ( R, q ) − N c n gal V ( R, q )] , (5)for the real-space and redshift space cases respectively. Inthis equation, N c is the total number of clusters in the sam-ple, n gal is the mean space density of galaxies, and V ( R, q )is the volume of the cell at (
R, q ). Thus 2 (cid:80) q β ( R, q ) = b lo ( R ) − b hi ( R ), where the assembly bias factors b lo and b hi are the ratios of the stacked 2D galaxy number overdensityproﬁles of the low- and high-concentration subsamples tothat of the cluster sample as a whole. The distribution of β over q at ﬁxed R thus indicates the distribution in depthof the diﬀerence in galaxy counts which gives rise to theapparent 2D assembly bias signal.In the inner regions ( R < h − kpc) the projectedproﬁle of high c gal clusters lies above that of low c gal clustersfor all three samples (see Figure 6). Figure 12 shows that, MNRAS , 1– ?? (2017) ssembly Bias and Splashback r (cid:2) h − Mpc (cid:3) − . − . − . − . d l og δ m / d l og r high c gal low c gal h R m i Figure 8.
Logarithmic derivative proﬁles of the 3D mass overden-sity around the central galaxies of the high- and low-concentrationsubsamples of CS250. Vertical lines mark the R m values for thetwo samples calculated directly from their stacked mass proﬁles. − r (cid:2) h − Mpc (cid:3) − . − . − . − . d l og δ g / d l og r high c gal low c gal h R m ih R c i Figure 9.
Logarithmic derivative proﬁles of the 3D galaxy num-ber overdensity around the central galaxies of the high- and low-concentration subsamples of CS250 in identical format to Figure 8except that a solid vertical line indicates (cid:104) R c (cid:105) for the two samples. − r (cid:2) h − Mpc (cid:3) . . . . . . . δ m , l o w / δ m , h i g h CS60CS120 CS250 h R c i Figure 10.
Ratios of the 3D mass overdensity proﬁles of low- andhigh-concentration clusters for each of our three cluster samples.The vertical line indicates the mean cluster radius (cid:104) R c (cid:105) . − r (cid:2) h − Mpc (cid:3) . . . . . . . δ g , l o w / δ g , h i g h CS60CS120 CS250 h R c i Figure 11.
Ratios of the 3D galaxy number overdensity proﬁlesof low- and high-concentration clusters for each of our three clus-ter samples with a vertical line indicating the mean cluster radius (cid:104) R c (cid:105) . as expected, the additional galaxies which produce this ex-cess lie in the inner regions of the clusters, with a mediandepth oﬀset from the central galaxy of 150 h − kpc or less.In redshift space, the random motions within clusters movethis excess out to | π | ∼
700 km s − , as shown in Figure 13. Beyond R = 400 h − kpc the behaviour switches andthe projected proﬁle of low c gal clusters lies above that ofhigh c gal clusters (again see Figure 6). The galaxies whichproduce this excess lie in two diﬀerent ranges of depthwhose relative contribution varies both with R and with MNRAS , 1– ????
700 km s − , as shown in Figure 13. Beyond R = 400 h − kpc the behaviour switches andthe projected proﬁle of low c gal clusters lies above that ofhigh c gal clusters (again see Figure 6). The galaxies whichproduce this excess lie in two diﬀerent ranges of depthwhose relative contribution varies both with R and with MNRAS , 1– ???? (2017) P. Busch & S. D. M. White ∆ z m . At R < h − Mpc, a ’local’ component centred near R ∼ | d | ∼ (cid:104) R c (cid:105) contributes most of the excess low c gal counts in CS60, about half of them in CS120, and a minorityof them in CS250, producing much of the pronounced peakseen at these R in the proﬁle ratios of Figure 6. A secondcomponent, distributed relatively uniformly over ± ∆ z m , thefull allowed depth for potential cluster members, contributesexcess counts to the low c gal cluster proﬁles at all R > R c and is responsible for most of the large-scale assembly bias.It also dominates the excess counts near (cid:104) R c (cid:105) in CS250. Thesystematic change in the relative weight of these two compo-nents with increasing R results in a shift in the median depthoﬀset of the excess counts, indicated by the black solid linesin Figures 12 and 13. The increasing strength of the secondcomponent from CS60 to CS120 to CS250 is the cause of theincrease in 2D assembly bias with ∆ z m . Figure 13 shows thatredshift space distortions signiﬁcantly smear out these twocomponents and make them more diﬃcult to distinguish.These results explain why strong assembly bias is seenin 2D for CS250 and CS120 (see Figure 6) but only a muchweaker signal is seen in 3D (Figure 11). Many of the low-concentration clusters in these samples have signiﬁcant fore-ground/background groups projected on their outer regions.These groups are distributed over the full depth ± ∆ z m ,and are visible in Figures 12 and 13 as an excess in binsat large q and R ∼ R c . Galaxies correlated with theseforeground/background groups then produce excess galaxycounts at similar q for all R values shown in the plot. Sincethe fall-oﬀ in these counts with R at the q of the backgroundgroup is similar to that of galaxy counts at relatively small q correlated with the primary cluster, the induced apparentassembly bias is almost independent of R . The rise in 3Dassembly bias seen at the largest r in Figure 11 is a result ofbeginning to pick up this additional correlated componentin the counts around low-concentration clusters.The strength of this eﬀect clearly depends on the sen-sitivity of the cluster identiﬁcation algorithm to projectioneﬀects at R ∼ R c . This in turn depends both on the eﬀec-tive ∆ z m and on the weight assigned to potential membersnear the cluster edge. Hence, the apparent bias may diﬀerbetween the real redMaPPer sample and our simulated sam-ples. Nevertheless, the strong similarity seen in previous sec-tions between the behaviour of our CS250 and CS120 sam-ples and the SDSS sample analysed by More et al. (2016) andMiyatake et al. (2016) suggests that the assembly bias signalthey found has a similar origin to that in the simulation. Inthe next section we will explore further the dependence ofapparent splashback features on cluster deﬁnition and arguethat the unexpected properties of the features detected byMore et al. (2016) are a result of confusion with featuresimposed by the cluster selection procedure. We have argued above that the details of our redMaPPer-like algorithm leave an imprint on the stacked proﬁles ofour simulated clusters. Although this is most evident in thestrong peak at R c in the proﬁle ratios of Figure 6 and inthe steep gradient interior to this radius induced by our sep-aration of the two subsamples by concentration, c gal , it is also visible in the crossing at R c of the individual gradi-ent proﬁles of Figure 2 and in their minima close to and onopposite sides of this radius. In this section we investigatethese eﬀects further by varying the value of R c used to deﬁneclusters. Speciﬁcally, we set R c = 1 . η (cid:18) λλ n ( η ) (cid:19) . h − Mpc (6)and we change η .The variable normalisation λ n ( η ) in Equation 6 ac-counts for the fact that a given cluster will contain moregalaxies within a larger projected radius. In the following wewill consider η = 2 / , λ n (cid:0) η = (cid:1) = 74, λ n (1) = 100,as before, and λ n (cid:0) η = (cid:1) = 130. For each choice of η werepeat the cluster selection and concentration calculationprocedures of Sections 2.2.1 and 2.2.2. Since changing R c changes the richness value λ assigned to each cluster, weshift the richness range deﬁning our samples (20 ≤ λ ≤ η = 1) so that the total numbers of simulated clustersabove the upper and lower limits remain unchanged. In thefollowing we show results for ∆ z m = 250 h − Mpc only, sincethe two other cases behave very similarly.Figure 14 repeats the observational and CS250 resultsfrom Figure 6 and compares them with predictions for η = 2 / η and shifts to match (cid:104) R c (cid:105) in all three cases.Interestingly, the proﬁle ratio for η = 2 / η = 3 /
2, and the ratio isunity for η = 2 / η = 3 / R c , the secondary fea-ture noted in Section 3.3 and apparently associated with truesplashback eﬀects is clearest for η = 2 / η = 3 /
2. At large R , the strength of assembly bias increasesnoticeably with η . The stronger peak, the weaker splashbacksignal and the stronger large-scale assembly bias found withincreasing η are all consistent with the expectation that pro-jection eﬀects should increase in importance when clustersare identiﬁed within larger radii, hence at lower projectedoverdensities. Also as expected, overall the SDSS results ofMore et al. (2016) behave most similarly to the η = 1 curvesin Figure 14. Nevertheless the large scale ratios agree equallywell with the ones using η = 3 / g shows a strong and complex response to η . The mid-dle panel here is essentially a repeat of Figure 2, while theupper and lower panels show similar plots for η = 2 / η = 3 / R = (cid:104) R c (cid:105) and at a value of about -1.4. The cross-ing ’coincidence’ is mathematically equivalent to the factthat all the proﬁle ratios have a maximum at R ∼ R c inFigure 14, which itself is easily understood as a consequenceour creating subsamples with identical distributions of λ butdisjoint distributions of c gal , thus forcing the proﬁle ratio toincrease over the range 0 < R < (cid:104) R c (cid:105) . The uniform slopevalue at curve crossing reﬂects the fact that this value equalsthe slope for the sample as a whole, which is quite slowlyvarying and close to -1.4 at these projected radii. MNRAS , 1– ?? (2017) ssembly Bias and Splashback − | d | (cid:2) h − M p c (cid:3) CS6010 − | d | (cid:2) h − M p c (cid:3) CS12010 − R (cid:2) h − Mpc (cid:3) − | d | (cid:2) h − M p c (cid:3) CS250 − . − . . . . . . . C o n t r i bu t i o n t o D i ﬀ e r e n t i a l P r o j e c t e d P a i r C o un t s β Figure 12.
The quantity β ( R, q ) of Equation 5 for the case q = | d | . This shows the distribution over depth q of the fractionaldiﬀerence between the projected galaxy count proﬁles of the low c gal and high c gal subsets of each of our three simulated clustersamples. The black curves give the median oﬀset in depth of theexcess counts as a function of R . − | π | (cid:2) h − M p c (cid:3) CS6010 − | π | (cid:2) h − M p c (cid:3) CS12010 − R (cid:2) h − Mpc (cid:3) − | π | (cid:2) h − M p c (cid:3) CS250 − . − . . . . . . . C o n t r i bu t i o n t o D i ﬀ e r e n t i a l P r o j e c t e d P a i r C o un t s β Figure 13.
Identical to Figure 12 except for the redshift spacecase, q = | π | . Within the crossing point, the slope for low-concentration clusters rises rapidly to a maximum of about γ = − . R ∼ (cid:104) R c (cid:105) , while the slope for the high-concentration clusters drops to a minimum at approximatelythe same radius but with a value which decreases stronglywith increasing η . This behaviour is clearly a consequenceof our deﬁnition of c gal and our separation of clusters into subsamples according its value. On larger scale, the slopeproﬁles appear independent of η when R exceeds twice thelargest value of (cid:104) R c (cid:105) for the samples being compared. How-ever, the curves for high- and low-concentration clusters dif-fer both from each other and from those of More et al. (2016)in this regime. In the intermediate range, (cid:104) R c (cid:105) < R < (cid:104) R c (cid:105) ,the shape of the curves is set by the need to interpolate be- MNRAS , 1– ????
Identical to Figure 12 except for the redshift spacecase, q = | π | . Within the crossing point, the slope for low-concentration clusters rises rapidly to a maximum of about γ = − . R ∼ (cid:104) R c (cid:105) , while the slope for the high-concentration clusters drops to a minimum at approximatelythe same radius but with a value which decreases stronglywith increasing η . This behaviour is clearly a consequenceof our deﬁnition of c gal and our separation of clusters into subsamples according its value. On larger scale, the slopeproﬁles appear independent of η when R exceeds twice thelargest value of (cid:104) R c (cid:105) for the samples being compared. How-ever, the curves for high- and low-concentration clusters dif-fer both from each other and from those of More et al. (2016)in this regime. In the intermediate range, (cid:104) R c (cid:105) < R < (cid:104) R c (cid:105) ,the shape of the curves is set by the need to interpolate be- MNRAS , 1– ???? (2017) P. Busch & S. D. M. White R (cid:2) h − Mpc (cid:3) . . . . . ∆ Σ g , l o w / ∆ Σ g , h i g h More et al.More et al., Extended η = 2 / η = 1 η = 3 / h R c i Figure 14.
The ratio of the projected galaxy number densityproﬁles of the low c gal and high c gal subsamples of CS250, takenfrom Figure 6, is compared with those found for cluster samplesselected with the same value of ∆ z m but with η = 2 / η = 1. Points with error bars and theircontinuation to smaller scales are the same as in Figure 6. Verticallines indicate (cid:104) R c (cid:105) for the three samples. tween these two diﬀerent behaviours, causing a minimum ator just outside (cid:104) R c (cid:105) and a maximum at slightly larger radiusin the low- and high-concentration cases respectively.In none of these panels are the simulated curves a goodﬁt to the observed ones. The results for high c gal clustersmatch quite well for η = 3 /
2, but the best ﬁt for the low c gal clusters is for η = 1, and even here the overall depth andthe general shape of the features diﬀer signiﬁcantly. Giventhe strong sensitivity to the cluster identiﬁcation algorithmand to the splitting by c gal , it is likely that these discrepan-cies reﬂect detailed diﬀerences between the real redMaPPerand concentration deﬁnition procedures and the simpliﬁedversions used here. It is clear that it will be very diﬃcultto infer reliable information about splashback signals fromdata of this kind without a complete understanding of theseeﬀects. In their analysis of a volume-limited sample of 8648 clus-ters selected by applying the redMaPPer algorithm to theSDSS/DR8 photometric data, More et al. (2016) detectedstrong assembly bias as a function of cluster concentrationon projected scales 5 h − Mpc < R < h − Mpc, and sub-stantial variations in the slope of cluster projected galaxynumber density proﬁles in the range 500 h − kpc < R < h − Mpc which they attributed to splashback eﬀects. Theassembly bias signal had previously been seen at lowersignal-to-noise by Miyatake et al. (2016) in gravitational − . − . − . − . − . d l og ∆ Σ g / d l og R η = 2 / c gal low c gal Morehigh c gal low c gal h R c i − . − . − . − . − . d l og ∆ Σ g / d l og R η = 1 R (cid:2) h − Mpc (cid:3) − . − . − . − . − . d l og ∆ Σ g / d l og R η = 3 / Figure 15.
The logarithmic derivatives of simulated and ob-served ∆Σ g proﬁles from Figure 2 are repeated in the middlepanel and compared with results from simulated cluster cata-logues with the same value of ∆ z m but η = 2 / (cid:104) R c (cid:105) for the relevant sample.MNRAS , 1– ?? (2017) ssembly Bias and Splashback lensing data for the same cluster sample. By using a simpli-ﬁed version of the redMaPPer scheme on three orthogonalprojections of publicly available galaxy catalogues from theMillennium Simulation, we have been able to identify up to9196 clusters of similar richness, which we classify by con-centration in a similar way to the SDSS studies. This allowsus to carry out analyses directly analogous to those of Moreet al. (2016) and Miyatake et al. (2016) and to comparewith results obtained from the full 3D information availablefor the simulation. This gives considerable insight into thefeatures seen in the SDSS analysis.The mean projected proﬁles of mass and galaxy numberdensity which we ﬁnd for the simulation are very similar tothose found observationally, both for the cluster sample asa whole and for its low- and high-concentration subsamples.The apparent assembly bias on large scales agrees well withthat observed, as does the shape of the ratio of the low-and high-concentration proﬁles which rises with decreasingprojected radius R to a peak at the mean value of R c , thelimiting radius used to deﬁne clusters, before dropping pre-cipitously to smaller scales. The variation with R of the log-arithmic slope of the mean galaxy number density proﬁlesshows a more complex structure than in SDSS, but repro-duces the main features pointed out by More et al. (2016):the main minimum (the point where the proﬁle is steepest)occurs at smaller radius than expected from the splashbackstudies of Diemer & Kravtsov (2014) and in addition theminima for the low- and high-concentration subsamples rankoppositely to the splashback expectation both in depth andin radius.The observed large-scale assembly bias is best repro-duced when all red galaxies projected onto a cluster (hencewithin ± h − Mpc in depth) are considered as potentialmembers. The signal is slightly weaker if the maximal al-lowed depth oﬀset is reduced to 120 h − Mpc and signiﬁ-cantly weaker if it is reduced to 60 h − Mpc. Such changeshave negligible eﬀect on the logarithmic slope proﬁles ofstacked galaxy counts. Hence projection over relatively largedepths appear to be a signiﬁcant factor in apparent assemblybias but not in apparent splashback features.The above results, derived by stacking simulated clus-ters in projection, can be compared to results obtained froma directly analogous analysis of the full 3D data. This showssome striking diﬀerences. The 3D assembly bias for separa-tions between 3 and 30 h − Mpc is considerably smaller thanthat seen in 2D ( b ∼ .
15 rather than b ∼ .
5) and varies inthe opposite way with the maximum depth oﬀset allowed forcluster members. The peak in the ratio of the galaxy num-ber density proﬁles for low- and high-concentration clustersoccurs at a substantially larger radius in 3D than in 2D( r ∼ . h − Mpc rather than R ∼ h − kpc). The loga-rithmic derivatives of the 3D mass and galaxy overdensityproﬁles vary more smoothly than in 2D, and show a sin-gle minimum which is at larger radius than in 2D and atthe same position for the low- and high-concentration clus-ters. The ranking of the minima in depth remains oppositeto that expected from splashback theory. (See the Appendixfor a discussion of how cluster selection, scaling and stackingprocedures can aﬀect apparent splashback features).The eﬀects of projection and cluster deﬁnition onstacked cluster proﬁles can be clariﬁed by examining themin the two-dimensional space of projected separation and line-of-sight depth. This allows identiﬁcation of the depthranges which give rise to the diﬀerence in projected countsaround low- and high-concentration clusters. As expected,the galaxy excess at small projected radius which pro-duces the high central surface density of high-concentrationclusters is made up of objects which are close to thecluster centre also in 3D. In redshift space, these excesscounts appear at oﬀsets ∼
800 km s − , in the wings of thecluster velocity dispersion. At projected radii 500 h − kpc < R < h − Mpc, much of the projected count excessaround low-concentration clusters comes from galaxies oﬀ-set in depth by ∼ h − Mpc, apparently indicating that low-concentration clusters live in richer environments than theirhigh-concentration analogues. At larger projected separa-tion, the galaxies responsible for the strong assembly biassignal are distributed almost uniformly over the full depthaccessible to potential cluster members, showing that theyare correlated with background groups preferentially pro-jected onto the low-concentration clusters, rather than withthe clusters themselves. The overall eﬀect of projection on2D assembly bias clearly depends strongly both on the de-tails of cluster and concentration deﬁnition and on the ac-curacy of the available photometric redshifts.At projected radii 500 h − kpc < R < h − Mpc wheresplashback eﬀects are expected to be present, distant fore-ground and background galaxies contribute negligibly toprojected cluster proﬁles. These are, however, strongly af-fected by the speciﬁc algorithms used to identify clusters andto classify them according to concentration. We demonstratethis explicitly by changing the limiting radius R c withinwhich red galaxies are counted as cluster members. Eventhough we take care to adjust parameters so that the abun-dance and typical mass of clusters are matched for diﬀerentchoices of limiting radius, we ﬁnd that this radius is stronglyimprinted on the mean projected proﬁles of the resultingsamples. The eﬀects are dramatic, both on the ratio of theproﬁles for low- and high-concentration clusters and on theshape of the logarithmic derivative proﬁles for the individualsubsamples. It will be diﬃcult to obtain reliable informationabout splashback without detailed understanding of such ef-fects for the particular algorithms used to select an observedcluster sample. ACKNOWLEDGEMENTS
The authors thank Surhud More for useful discussions ofthis work.
Adhikari S., Dalal N., Chamberlain R. T., 2014, Journal of Cos-mology and Astro-Particle Physics, 11, 019Bertschinger E., 1985, The Astrophysical Journal Supplement Se-ries, 58, 39Boylan-Kolchin M., Springel V., White S. D. M., Jenkins A., Lem-son G., 2009, Monthly Notices of the Royal Astronomical So-ciety, 398, 1150Dalal N., White M., Bond J. R., Shirokov A., 2008, The Astro-physical Journal, 687, 12Diemer B., Kravtsov A. V., 2014, The Astrophysical Journal, 789,1MNRAS , 1– ????
Adhikari S., Dalal N., Chamberlain R. T., 2014, Journal of Cos-mology and Astro-Particle Physics, 11, 019Bertschinger E., 1985, The Astrophysical Journal Supplement Se-ries, 58, 39Boylan-Kolchin M., Springel V., White S. D. M., Jenkins A., Lem-son G., 2009, Monthly Notices of the Royal Astronomical So-ciety, 398, 1150Dalal N., White M., Bond J. R., Shirokov A., 2008, The Astro-physical Journal, 687, 12Diemer B., Kravtsov A. V., 2014, The Astrophysical Journal, 789,1MNRAS , 1– ???? (2017) P. Busch & S. D. M. White
Dvornik A., et al., 2017, Monthly Notices of the Royal Astronom-ical Society, 468, 3251Efstathiou G., Frenk C. S., White S. D. M., Davis M., 1988,Monthly Notices of the Royal Astronomical Society, 235, 715Faltenbacher A., White S. D. M., 2010, The Astrophysical Jour-nal, 708, 469Fillmore J. A., Goldreich P., 1984, The Astrophysical Journal,281, 1Gao L., White S. D. M., 2007, Monthly Notices of the RoyalAstronomical Society, 377, L5Gao L., Springel V., White S. D. M., 2005, Monthly Notices ofthe Royal Astronomical Society, 363, L66Guo Q., et al., 2011, Monthly Notices of the Royal AstronomicalSociety, 413, 101Hearin A. P., Watson D. F., Becker M. R., Reyes R., BerlindA. A., Zentner A. R., 2014, Monthly Notices of the RoyalAstronomical Society, 444, 729Henriques B. M. B., White S. D. M., Thomas P. A., Angulo R.,Guo Q., Lemson G., Springel V., Overzier R., 2015, MonthlyNotices of the Royal Astronomical Society, 451, 2663Kaiser N., 1984, The Astrophysical Journal Letters, 284, L9Lemson G., the Virgo Consortium 2006, ArXiv Astrophysics e-prints, pp arXiv:astro–ph/0608019Li Y., Mo H. J., Gao L., 2008, Monthly Notices of the RoyalAstronomical Society, 389, 1419Lin Y.-T., Mandelbaum R., Huang Y.-H., Huang H.-J., Dalal N.,Diemer B., Jian H.-Y., Kravtsov A., 2016, The AstrophysicalJournal, 819, 119Lithwick Y., Dalal N., 2011, The Astrophysical Journal, 734, 100Miyatake H., More S., Takada M., Spergel D. N., MandelbaumR., Rykoﬀ E. S., Rozo E., 2016, Physical Review Letters, 116,041301Mo H. J., White S. D. M., 1996, Monthly Notices of the RoyalAstronomical Society, 282, 347More S., Diemer B., Kravtsov A., 2015, The Astrophysical Jour-nal, 810, 36More S., et al., 2016, The Astrophysical Journal, 825, 39Planck Collaboration 2014, Astronomy and Astrophysics, 571, A1Rykoﬀ E. S., et al., 2012, The Astrophysical Journal, 746, 178Rykoﬀ E. S., et al., 2014, The Astrophysical Journal, 785, 104Shi X., 2016, Monthly Notices of the Royal Astronomical Society,459, 3711Springel V., White S. D. M., Tormen G., Kauﬀmann G., 2001,Monthly Notices of the Royal Astronomical Society, 328, 726Springel V., et al., 2005, Nature, 435, 629Tinker J. L., George M. R., Leauthaud A., Bundy K., FinoguenovA., Massey R., Rhodes J., Wechsler R. H., 2012, The Astro-physical Journal Letters, 755, L5Vogelsberger M., White S. D. M., Mohayaee R., Springel V., 2009,Monthly Notices of the Royal Astronomical Society, 400, 2174Wang L., Weinmann S. M., De Lucia G., Yang X., 2013, MonthlyNotices of the Royal Astronomical Society, 433, 515Wechsler R. H., Zentner A. R., Bullock J. S., Kravtsov A. V.,Allgood B., 2006, The Astrophysical Journal, 652, 71Yang X., Mo H. J., van den Bosch F. C., 2006, The AstrophysicalJournal Letters, 638, L55Zu Y., Mandelbaum R., Simet M., Rozo E., Rykoﬀ E. S., 2016,arXiv:1611.00366 [astro-ph]
APPENDIX A: THE EFFECT OF STACKINGPROCEDURES ON APPARENT SPLASHBACKSIGNAL
In Section 4.1 we noted that logarithmic derivative curvesfor the stacked 3D mass proﬁles of our clusters (Figure 8)diﬀer in shape, particularly in the depth of the minimum, from those shown for objects of similar mass by Diemer &Kravtsov (2014) (DK14). A general diﬀerence in behaviourbetween mean and median of proﬁle stacks was already men-tioned in DK14. Here we investigate how the shapes of suchproﬁles depend on the deﬁnition of the sample to be stackedand on the scaling and stacking procedures adopted.In Figure A1, the purple curve is taken directly fromDK14 where it is the one labelled z = 0 .
25 in the uppercentral panel of their Figure 4. It corresponds to haloes in arelatively narrow range of M m , selected at a redshift andwith a mean mass which are close to those of the clustersample analysed in this paper. DK14 scaled the 3D massproﬁle of each cluster to its individual R m and then con-structed the stack by taking the median value of density ateach r/R m . The logarithmic derivative of the resultingproﬁle is the quantity plotted. Note that it diﬀers from thequantity plotted in Figure 8 in that DK14 did not subtractthe mean background density from their stack. This has asigniﬁcant eﬀect beyond a few Mpc.The light blue curve in Figure A1 corresponds to our fullsample CS250, stacked in the same way as in Section 4.1, i.e.we constructed a spherically averaged mass proﬁle aroundthe central galaxy of each cluster, we averaged these pro-ﬁles directly to obtain the stack, we scaled the result by the (cid:104) R m (cid:105) of the stack, and we then plotted its derivative. Thecurve eﬀectively corresponds to an average of the two curvesshown in Figure 8, except for diﬀerences at large r/R m due to the inclusion of the cosmic mean density. Its mini-mum value is about -2.7, just above the average of the valuesfor the two curves in Figure 8 and considerably above thevalue found by DK14.The orange curve in Figure A1 shows what happens ifwe scale the proﬁle of each cluster in radius by its individualvalue of R m before stacking. This changes the shape ofthe curve, lowering its minimum slightly and moving it toslightly smaller radii. Not surprisingly, scaling before stack-ing results in a sharper transition between the one-halo andtwo-halo parts of the stacked proﬁle.If we stack these same scaled proﬁles by constructingtheir median at each r/R m , rather than their mean, weobtain the green curve. The minimum is now signiﬁcantlydeeper, although still not as deep as that found by DK14.The shape of the curve outside the minimum agrees verywell with their results.Finally, if we select halos directly from the MillenniumSimulation with a narrow range of M m at z = 0 .
24, andwe make a median stack after scaling each proﬁle to its indi-vidual R m value, then we should be reproducing the haloselection and stacking procedures of DK14 almost exactly.The result is shown as a red curve in Figure A1. It nowdiﬀers only slightly from the purple curve.We suspect that these small residual discrepancies re-ﬂect diﬀerences in the eﬀective smoothing associated withhalo proﬁle construction and diﬀerentiation. Overall, the re-sults described here indicate that curves of this type aresensitive to how the halos are scaled and whether a meanor median stack is constructed. The minimum logarithmicslope is particularly sensitive to these factors, and changesin shape can also shift the position of the minimum by 10or 20 per cent. We note that for individual observed clustersthe value of R m is unknown, the full 3D information isnot available, and the selection and deﬁnition eﬀects on 2D MNRAS , 1– ?? (2017) ssembly Bias and Splashback r/R m − . − . − . − . − . − . − . d l og ρ m / d l og r CS250 Mean, mean R m CS250 Mean, indiv. R m CS250 Median, indiv. R m M m sel. MS haloes, MedianDiemer et al. Figure A1.
Logarithmic derivative curves for diﬀerent deﬁni-tions of the radially rescaled 3D mass density proﬁle of simulatedclusters are compared to the z = 0 .
25, 2 < ν < . proﬁles which we discuss in the main body of our paper arelarge compared to the eﬀects described here. MNRAS , 1– ????