Detection of the Splashback Radius and Halo Assembly bias of Massive Galaxy Clusters

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by  Surhud More

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DETECTIONOFTHESPLASHBACKRADIUSANDHALOASSEMBLYBIASOFMASSIVEGALAXYCLUSTERS
SurhudMore1,HironaoMiyatake2,3,1,MasahiroTakada1,BenediktDiemer4,AndreyV.Kravtsov5,6,7,NealK.Dalal8,1,Anupreeta
More1,RyomaMurata1,9,RachelMandelbaum10,EduardoRozo11,EliS.Rykoff12,MasamuneOguri9,13,1,DavidN.Spergel3,1
1KavliInstituteforthePhysicsandMathematicsoftheUniverse(WPI),TokyoInstitutesforAdvancedStudy,TheUniversityofTokyo,
5-1-5Kashiwanoha,Kashiwa-shi,Chiba,277-8583,Japan;[email protected]
2JetPropulsionLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA91109,USA
3DepartmentofAstrophysicalSciences,PrincetonUniversity,PeytonHall,PrincetonNJ08544USA
4Harvard-SmithsonianCenterforAstrophysics,60GardenSt.,Cambridge,MA02138USA
5DepartmentofAstronomyandAstrophysics,TheUniversityofChicago,Chicago,IL60637USA
6 6KavliInstituteforCosmologicalPhysics,TheUniversityofChicago,Chicago,IL60637USA
1 7EnricoFermiInstitute,TheUniversityofChicago,Chicago,IL60637USA
0 8DepartmentofPhysics,UniversityofIllinoisUrbana-Champagne,1110WestGreenStreetUrbana,IL61801-3080USA
2 9DepartmentofPhysics,UniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-0033Japan
10McWilliamsCenterforCosmology,DepartmentofPhysics,CarnegieMellonUniversity,Pittsburgh,PA15213USA

n 11DepartmentofPhysics,UniversityofArizona,1118E4thSt,Tucson,AZ85721USA
a 12SLACNationalAcceleratorLaboratory,MenloPark,CA94025USA
J 13ResearchCenterfortheEarlyUniverse,UniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-0033,Japan

2
2 TobesubmittedtotheAstrophysicalJournal

ABSTRACT
]
O We show that the projected number density profiles of SDSS photometric galaxies around galaxy clusters
C displaysstrongevidenceforthesplashbackradius,asharphaloedgecorrespondingtothelocationofthefirst
orbitalapocenterofsatellitegalaxiesaftertheirinfall. Wesplittheclustersintotwosubsampleswithdifferent
.
h mean projected radial distances of their members, (cid:104)R (cid:105), at fixed richness and redshift, and show that the
mem
p samplewithsmaller(cid:104)R (cid:105)hasasmallerratioofthesplashbackradiustothetraditionalhaloboundaryR ,
mem 200m
- thanthesubsamplewithlarger(cid:104)R (cid:105),indicativeofdifferentmassaccretionratesforthetwosubsamples. The
o mem
same cluster samples were recently used by Miyatake et al. to show that their large-scale clustering differs
r
t despitetheirsimilarweaklensingmasses,demonstratingstrongevidenceforhaloassemblybias. Weexpand
s
a onthisresultbypresentinga6.6-σdetectionofhaloassemblybiasusingthecluster-photometricgalaxycross-
[ correlations. Our measured splashback radii are smaller, while the strength of the assembly bias signal is
stronger, than expectations from N-body simulations based on the Λ-dominated, cold dark matter structure
1 formationmodel. Dynamicalfrictionorcluster-findingsystematicssuchasmiscenteringorprojectioneffects
v
arenotlikelytobethesolesourceofthesediscrepancies.
3
6
0 1. INTRODUCTION assembly bias requires identifying samples of isolated halos
06 Darkmatterhaloswithmasseslargerthan1014h−1M(cid:12) col- whicharematchedintheirhalomassesbutdifferintheiras-
semblyhistories. Therehavebeenseveralclaimsofdetection
. lapse out of dense peaks in the primordial Gaussian density
1 ofhaloassemblybiasongalaxyscalesintheliterature(e.g.,
fluctuationsthatarebelievedtooriginatefromquantumfluc-
0 Yangetal.2006;Tinkeretal.2012;Hearinetal.2014).How-
tuations in cosmic inflation (see e.g., Kaiser 1984; Bardeen
6 ever,Linetal.(2015)investigatedthefirstoftheseclaimsand
et al. 1986, see Kravtsov & Borgani 2012 for a recent re-
1 found no strong evidence for halo assembly bias on galaxy
view). Clusters of galaxies form within such massive dark
: scales. Thedifferenceintheconclusionswasaresultofcon-
v matterhalos. Thelargescaleclusteringamplitudeoftheha-
taminationofthehalosamplesbysatellitegalaxies,orthedif-
i loshostinggalaxyclustersisthusheavilybiasedcomparedto
X ferencesinhalomassesofthesamplesusedtolookforhalo
theunderlyingmatterdistribution(Kaiser1984;Mo&White
r 1996;Shethetal.2001;Tinkeretal.2010). assemblybias(Linetal.2015).
a Recently,Miyatakeetal.(2015)presentedthefirstevidence
Althoughthelargescaleclusteringamplitudeofdarkmatter
ofhaloassemblybiasonclusterscales. Galaxyclustersoffer
halos is primarily expected to depend upon the height of the
two advantages: first, the probability of a cluster-sized halo
initial density peak out of which a halo collapses (therefore
beingasatelliteofanevenbiggerhaloismuchsmallerthan
its halo mass), it can have secondary dependencies on other
in the case of galaxies, and secondly, the weak gravitational
parameters related to the assembly history of the halo, such
lensingsignalcanbeusedtomatchgalaxyclustersubsamples
as the radial profile of the initial peak, especially on cluster
fortheirhalomasseswithagreateraccuracy.
scales(Dalaletal.2008). Thedependenceofthelargescale
The galaxy cluster subsamples used by Miyatake et al.
clustering amplitude on parameters other than the halo mass
(2015)weredrawnfromtheSDSSredMaPPergalaxycluster
has been broadly referred to as halo assembly bias, and has
catalog, and were matched in redshift and richness distribu-
been studied in great detail using cosmological simulations
tion, but differed in the compactness of the member galaxy
(Sheth&Tormen2004;Gaoetal.2005;Gao&White2007;
distribution. Thesesampleswereshowntohaveverysimilar
Wechsleretal.2006;Lietal.2008).
Halo assembly bias has however been difficult to estab- massesbasedonweaklensing,buthaddifferentlargescalebi-
ases. Themaingoalofthispaperistoobservationallyestab-
lish in astrophysical observations. A clean detection of halo
2 More,S.etal.
lish the connection between the member galaxy distribution r to denote three dimensional distances, and R for projected
and the mass assembly of these cluster subsamples without distances between galaxies or between galaxies and cluster
relying on proxies related to complicated baryonic physics, centers. Forcaseswherewewanttopreservenotationsfrom
suchasthestarformationrates. previouspapers, suchasusingR forthesplashbackradius,
sp
For this purpose, we use a unique probe of the mass as- wewillspecificallymention2dor3dtoavoidconfusion. The
sembly of galaxy clusters, which relies on the observational subscript 200m on halo mass M or radius R will refer to the
detection of the edges of galaxy clusters. Models of self- mass or radius corresponding to spherical overdensity halos
similarsecondaryinfallofmatterontoasphericaloverdensity suchthattheirboundariesenclose200timesthemeanmatter
predict the presence of a density jump at the location where densityoftheUniverse.
recently accreted material is reaching its first apocenter, as-
2. DATAANDMETHODS
sociated with the last density caustic (Fillmore & Goldreich
We start from the publicly available catalog of galaxy
1984; Bertschinger 1985). Although the collapse of matter
clusters identified from the SDSS DR8 photometric galaxy
ontorealisticdensitypeaksincolddarkmattermodelsiscon-
catalog by the red-sequence Matched-filter Probabilistic
siderablymorecomplexthanthatenvisionedinthesemodels,
Percolation(redMaPPer)clusterfindingalgorithm(v5.10,see
thelastdensitycausticmanifestsitselfasasharpsteepening
the website1 for details and Rykoff et al. 2014; Rozo et al.
ofthedensityprofileindarkmatterhalos(Diemer&Kravtsov
2014). Theclusterfinderusestheugrizmagnitudesandtheir
2014).
errors,toidentifyoverdensitiesofred-sequencegalaxieswith
Thelocationofthisdensitycaustic,alsocalledthesplash-
similarcolorsasgalaxyclusters. Foreachcluster,thecatalog
backradiusortheturnaroundradius, canbeusedtodefinea
containsanopticalrichnessestimateλ,aphotometricredshift
physical boundary for dark matter halos (More et al. 2015).
estimate z , as well as the position and centering probabili-
The splashback radius crucially depends upon the mass ac- λ
tiesof5candidatecentralgalaxies p . Aseparatemember
cretion rate of the collapsing halo (Vogelsberger et al. 2011; cen
galaxy catalog provides a list of members for each cluster,
Diemer&Kravtsov2014;Adhikarietal.2014). Forhalosof
eachofwhichisassignedamembershipprobability, p .
thesamemass,largeaccretionrateresultsinasmallersplash- mem
The parent cluster catalog used in Miyatake et al. (2015)
back radius. The physical reason is simple: the deeper the
consistsofanapproximatelyvolumelimitedsampleof8,648
halopotentialwellgetsduringtheorbitofadarkmatterpar-
redMaPPerclusterswith20 < λ < 100and0.1 ≤ z ≤ 0.33.
ticle,thesmalleristhevalueofitsapocenter. λ
The average and the median redshift of our subsamples are
As discussed in More et al. (2015), hints for the splash-
0.24and0.25,respectively. Throughoutthispaperweusethe
backradiusmayhavebeenseenbeforeforindividualclusters
positionofthemostprobablecentralgalaxyineachclusterre-
(Rinesetal.2013;Tully2015;Patej&Loeb2015).Inthispa-
gionasaproxyoftheclustercenter.However,wewilldiscuss
per,wewillharnessthepowerofstatisticstopresentthefirst
theeffectofmiscenteringonourconclusionsinSection4.3.
highsignal-to-noisedetectionofthesplashbackradiusforour
Inthispaper, wesubdivide thisgalaxyclustersampleinto
galaxyclustersubsamples. Wewillusethesplashbackradius
toestablishthatthesegalaxyclustersubsampleshavedifferent twosubsamplesfollowingthesameprocedureasinMiyatake
massaccretionrates,andhavedifferentlargescaleclustering etal.(2015). Briefly,weobtaintheaverageprojectedcluster-
centricseparationofmembergalaxies,(cid:104)R (cid:105),foreachclus-
amplitude,asignatureofhaloassemblybias. mem
ter,andcomputethemedian(cid:104)R (cid:105)asafunctionofrichness
Thepaperisorganizedasfollows: mem
andredshift2.Weusethismediantodividetheparentsample
• Section 2 describes the cluster subsamples and the into two subsamples. The large- and small-(cid:104)Rmem(cid:105) subsam-
Sloan Digital Sky Survey (SDSS) photometric galaxy ples,labelledaslow-andhigh-cgal,respectively,inthispaper,
data which form the basis of our study, and the meth- consistof4,235and4,413clusters,respectively.
odsweadoptinordertoobtainthemeasurementsofthe Inordertocomputegalaxysurfacenumberdensityaround
galaxynumberdensitiesaroundourclustersubsamples. these cluster subsamples, we make use of the photometric
galaxy catalog from SDSS DR8 (Aihara et al. 2011). We
• Section 3 presents our measurements of the galaxy exclude galaxies with any of the following flags: saturated,
numberdensitiesaroundourclustersubsamples,ourin- satur center, bright, deblended as moving. We correct the
ferencesforthelocationofthesplashbackradiusfrom magnitudes for galactic dust extinction using the maps of
these measurements, and our detection of halo assem- Schlegel et al. (1998), and use all photometric galaxies with
blybias. extinction corrected i-band model magnitudes brighter than
21.0andwithmagnitudeerrorslessthan0.1.
• Section 4 presents the predictions for the location of We compute the stacked surface number density of the
thesplashbackradiusandtheamountofhaloassembly SDSSphotometricgalaxysamplesaroundeachofourcluster
bias from numerical simulations in the context of the subsamples as a function of comoving projected separation,
standardcosmologicalmodel. Inparticular,wediscuss R, from the galaxy cluster center. Since our cluster subsam-
anumberofsystematiceffects, whichcouldaffectour plesspanawiderangeinredshift(0.1≤z≤0.33),thesurface
interpretation. densityprofiles aroundlower redshiftclusterswill systemat-
ically contribute galaxies from a fainter photometric galaxy
• The broad implications of our results are discussed in
population. Toavoidsuchbiases,forourfiducialanalysis,we
Section5,andconclusionsandasummaryispresented
onlycountcluster-galaxypairsifthephotometricgalaxyhas
inSection6.
an absolute magnitude of M −5logh < −19.433, assuming
i
Throughoutthispaper, weadoptaflatΛCDMcosmologi- 1http://risa.stanford.edu/redmapper/
cal model with matter density parameter Ωm = 0.27 and the 2 Whilecomputingtheaverage,weweighteachgalaxy’sclustercentric
Hubble parameter h = 0.7, unless otherwise stated. We use distancewithitsmembershipprobability(pmem).
logtodenotelogarithmswithrespecttobase10. Wewilluse 3 Notethatwedonotuseanyk-correctionsorcorrectionsforluminos-
SplashbackradiusandHaloAssemblybias 3
thatitislocatedattheredshiftofthecluster(thislimitcorre- bleenoughtoreproducethegalaxysurfacedensitymeasured
sponds to an apparent magnitude of mi = 21 at z = 0.33 for from the SDSS data.5 We chose the maximum projection
ourassumedcosmologicalmodel). Additionally,wewillalso length z = 40h−1Mpc as our default value, and we have
max
presentresultsforphotometricgalaxiesthatareoneandtwo checkedthatthelocationofthesplashbackradiusisinsensi-
magnitudesbrighterthanourfiducialmeasurement,toexplore tive to this choice, in particular reducing z to be even as
max
thedependenceofthesplashbackradiusonthemagnitudeof smallas10h−1Mpc.
photometricgalaxiesused. Giventhattheparameterr andρ areentirelydegenerate
out o
Weexpectthatthesurfacedensitymeasurementwillconsist witheachother,wefixr =1.5h−1Mpc. Wefindthatallow-
out
of galaxies correlated with the galaxy clusters under consid- ingαtovaryfreelyresultsinanalmostperfectdegeneracybe-
erationaswellasuncorrelatedgalaxiesintheforegroundand tweenρ andr ,withverylittleimpactonthelocationofthe
s s
the background. To determine this uncorrelated component, steepeningofthegalaxydensityprofiles. Thereforeweusea
we compute the galaxy surface density around a sample of prioronlogα = log0.2±0.6,centeredatthevalueexpected
random points. We use 100 times larger number of random forthedarkmatterhaloscorrespondingtoourmassestimates
pointsthanthenumberofclustersinoursubsamples4. These fromweaklensing(Gaoetal.2008). Forourfiducialmodel-
random pointsincorporate thesurvey geometry, depth varia- ing scheme, we also use priors on logβ = log4.0±0.2 and
tions,anddistributionsofclustersinredshiftandrichness.We logγ=log6.0±0.2centeredaroundthevaluesrecommended
subtractthebackgroundaroundrandompointsfromthetotal byDiemer&Kravtsov(2014)andconstraintheparametersr
s
toobtainthesurfacedensityofcorrelatedgalaxies,Σg(R).We andrttoliewithin[0.1,5.0]h−1Mpc.
use102jackkniferegionsinordertocomputethecovariance Forourdefaultmodelingscheme,weassumethatthemost
in the measurements of Σg(R) with typical size of 10 × 10 probable central galaxy for every cluster (one with the high-
sq.deg. whichcorrespondstoabout100×100(h−1Mpc)2 at est p ), assigned in the redMaPPer catalog, resides at the
cen
themedianredshiftofourclustersubsamples. Thejackknife true center of gravitational potential in each cluster region.
regionsarethussignificantlylargercomparedtothescalesof However,asstudiedinMiyatakeetal.(2015)(seealsoHikage
interest in this paper, justifying the assumptions behind the etal.2013),somefractionofthecentralgalaxiesinourcluster
jackknifeerrors. subsamplesmaybemis-centered,characterizedbyoffsetradii
Usingthemeasurementofthegalaxysurfacedensities,we rangingfrom400h−1kpcpossiblyupto800h−1kpc. Ifsuch
would like to infer the location of the splashback radius of mis-centeredclustersareindeedpresentinlargenumbers,our
ourclustersubsamples,i.e.,thesteepestlogarithmicslopeof measurementsofthesplashbackradiuswouldbebiasedhigh.
the galaxy number density distributions in three dimensions. We will present tests for the effects of mis-centering in Sec-
Given that the splashback radius is expected to be of the or- tion4.3below.
der of R of our halos, we fit the surface densities in the WewillusetheaffineinvariantMarkovChainMonteCarlo
200m
range [0.1, 5.0]h−1Mpc. The location of the steepening in samplerofGoodman&Weare(2010),asimplementedinthe
threedimensionsisexpectedtobedifferentfromthatinpro- softwarepackageemcee(Foreman-Mackeyetal.2013),inor-
jection (Diemer & Kravtsov 2014). Therefore, we will use der to sample from the posterior distribution of the param-
a 3-dimensional parameterization first proposed by Diemer eters, logρ , logr , logα, logr, logγ, logβ, logρ and s ,
s s t o e
&Kravtsov (2014)toforward modeltheprojected measure- giventhegalaxysurfacedensitymeasurementsandthestated
ments(seealsoMoreetal.2015).Themodelconsistsofinner priors.
andoutersurfacedensityprofileswithasmoothtransitionbe- Asatestofourfittingmethod,inAppendixAweapplyitto
tweenthetwo, projectednumberdensitydistributionsof(sub)-halosaround
galaxyclustersinnumericalsimulations(seebelowforthede-
ρ (r)=ρinnerf +ρouter,
g g trans g tailsofthesimulations)andshowthatweareabletorecover
(cid:32) 2 (cid:34)(cid:32)r(cid:33)α (cid:35)(cid:33) thelocationofthesteepeningofthethreedimensionaldensity
ρignner=ρsexp −α r −1 , distribution of subhalos quite accurately with our modelling
s scheme.
(cid:32) r (cid:33)−se
ρouter=ρ , 3. RESULTS
g o r
out 3.1. Splashbackingalaxynumberdensityprofiles
ftrans=(cid:104)1+(r/rt)β(cid:105)−γ/β, We begin by presenting how the stacked surface density
(cid:90) zmax (cid:18)(cid:112) (cid:19) profileofgalaxies,Σg(R),aroundtheentireparentsampleof
Σ (R)=2 ρ R2+z2 dz. (1) redMaPPerclusters,describedintheprevioussection,varies
g g
0 with galaxy samples of different absolute magnitude thresh-
Notethattheabovefittingformula, whichisanEinastopro- olds. These measurements are shown in the top panels of
file in the inner regions which transitions to a power law in Figure 1 using points with errorbars. The brightness of the
the outer regions, is able to reproduce the dark matter pro- photometric galaxy sample increases from left to right. The
file around halos, and is flexible enough to reproduce the solid lines in the bottom panels correspond to the profiles of
simulationresultscomparedtootherfittingformulaesuchas the logarithmic slope of the galaxy surface densities. These
theNavarro-Frenk-Whitemodelandthehalomodel(Navarro slopeprofileswereobtainedbyusingtheSavitzky-Golayal-
etal.1996;Oguri&Hamana2011;Hikageetal.2013). Here gorithm to smooth the observed measurements, by fitting a
wesimplyassumethatthesamefittingformulaisalsoflexi- third-order polynomial over a window of five neighbouring
points, and then using a cubic spline to interpolate between
ityevolutionhere,sincetheredshiftsofthephotometricgalaxiesarequite
uncertain. 5Testsusingsubhalodensityprofilesaroundclusterscalehalosfromsimu-
4WehavetestedthattheuseoftheimprovedrandomcatalogfromRykoff lationspresentedinAppendixA,aswellasthereasonablevaluesofχ2values
etal.(2016)doesnotchangeanyconclusionsinthispaper. weobtainfordescribingtheobservedmeasurementsjustifythischoice.
4 More,S.etal.
101
Mi-5logh<-19.43 Mi-5logh<-20.43 Mi-5logh<-21.43
2] 101
−
pc 101
M
2 100
h
[
)
R
( 100
g
Σ
100 Vpeak>135kms−1 Vpeak>175kms−1 Vpeak>280kms−1
Observed Observed 10−1 Observed
10−1 100 101 10−1 100 101 10−1 100 101
0.2
−
0.4
−
0.6
R −
g 0.8
o −
dl 1.0
/ −
Σg 1.2
−
g
o 1.4
l −
d 1.6 m m m
− 00 00 00
1.8 R2 R2 R2
−
2.0
− 10−1 100 101 10−1 100 101 10−1 100 101
R [h−1Mpc] R [h−1Mpc] R [h−1Mpc]
Figure1. Toppanels:Thesurfacenumberdensityprofiles,Σg(R),ofSDSSphotometricgalaxieswithdifferentmagnitudethresholdsaroundtheentireredMaP-
Perclustersamplewithz∈[0.1,0.33]andrichnessλ∈[20,100],areshownusingsymbolswitherrorbars. Thedashedlinescorrespondto(sub)-halosurface
densityprofilesintheMultidarkPlanckIIsimulationaroundclusterswiththemassthresholdsimilartooursampleatz = 0.248. Thethresholdonsubhalo
Vpeakvaluesroughlycorrespondtothemagnitudethresholdsineachpanelandwereobtainedbysubhaloabundancematching(seeAppendix B).Bottompanels:
Thelogarithmicslopeofthesurfacedensityprofilesareshownusingsolidanddashedlinesfortheobservedgalaxyandthesubhalosurfacedensityprofiles,
respectively. Theobservedslopeofthesurfacedensityprofilehasashapewhichissimilartothatexpectedfromsimulations. Notethatalthoughthesurface
densityprofilesbothinobservationsandsimulationsexhibitsimilarsteepening,thecorrespondingradiiofthesteepestslopeareatslightlydifferentlocations.
Table1
PosteriordistributionofparametersfromtheMCMCanalysis
Magnitude cgal logρ0 logα logrs logρo se logrt logβ logγ R2spd R3spd χ2/dof
-19.43 high 1.10+0.25 −0.95+0.22 −0.32+0.40 0.349+0.031 1.601+0.076 −0.082+0.049 0.762+0.119 0.66+0.14 0.778+0.015 0.971+0.025 6.0/8
-19.43 low −0.68−+00.7.370 −1.090−+00..30288 0.55+−00.1.113 0.545−+00..003555 1.600−+00..008608 0.058+−00.0.02430 1.10+−00..10295 0.64−+00..1123 1.153−+00..001249 1.378−+00..002216 13.2/8
-20.43 high 0.70+−00.3.210 −0.97+−00..20863 −0.27−+00.1.475 0.167−+00..006174 1.613−+00..008704 −0.098−+00.0.02418 0.82−+00..1113 0.69−+00..1114 0.756−+00..002114 0.938−+00..002264 2.9/8
-20.43 low −0.89−+00.8.368 −1.019−+00..31518 0.48+−00.1.156 0.276−+00..001168 1.655−+00..007572 0.072+−00.0.02368 1.10−+00..1112 0.80−+00..1142 1.128−+00..002192 1.352−+00..002266 12.4/8
-21.43 high 0.10+−00.3.298 −1.00+−00..30386 −0.25−+00.2.419 0.0385−+00..0020145 1.496−+00..005966 −0.087−+00.0.02531 0.85−+00..1114 0.78−+00..1135 0.754−+00..002242 0.938−+00..002356 10.5/8
-21.43 low −1.31−+00.9.435 −0.97−+00..3156 0.40+−00.2.200 0.0712−+00..00004683 1.624−+00..009793 0.087+−00.0.02464 1.12−+00..1124 0.90−+00..1135 1.132−+00..001493 1.361−+00..004304 23.6/8
−0.37 −0.12 −0.25 −0.0066 −0.073 −0.025 −0.13 −0.14 −0.035 −0.038
Note.— Thedifferentrowslistthe68%confidenceintervalsonthemodelparameters(seeEq.1)giventhesurfacenumberdensitydatashowninFigure2.
Theχ2perdegreeoffreedomaswellastheinferred2-dand3-dsplashbackradiusarealsoshowninthelastthreecolumns.
these smoothed measurements. In contrast with the tradi- observed surface density profiles with those expected from
tionalSavitzky-Golayalgorithm,weexplicitlyaccountforthe thestandardstructureformationmodel, weutilizeMultidark
covariant errors on these data points, as proposed by More Planck II (MDPL2), a 38403 particle cosmological N-body
(2016a)6. simulation with box size of 1h−1Gpc and mass resolution of
In Appendix B, we have used subhalo abundance match- 1.51×109h−1M(cid:12)(Klypinetal.2014). Wealsousetheassoci-
ing to obtain an estimate of the approximate V (the max- atedhalocatalogsfoundusingRockstar, ahalofinderwhich
peak
imum circular velocity of a halo throughout its entire his- groups particles into halos using their phase space informa-
tory) value of dark matter subhalos7 hosting our photomet- tion(Behroozietal.2013)8.
ricgalaxiesasafunctionoftheirmagnitude. Tocomparethe
aswell,notjustsatellitehalos. Wewillusethetermhalosexplicitlywhen
6https://github.com/surhudm/savitzky_golay_with_errors referringtoonlyisolatedhalos.
7 Our use of the term subhalos henceforth includes isolated host halos 8 These catalogs are publicly available at the website http://www.
SplashbackradiusandHaloAssemblybias 5
102
Low-cgal 0.4
−
0.6
101 −
] R −0.8
2
− g
pc dlo −1.0
M 100 /
2h Σg −1.2
[ g
g o
Σ l 1.4
d −
10−1 1.6 m
Mi−5logh<−19.43 − R200
Mi−5logh<−20.43 −1.8
Mi−5logh<−21.43
10−210−1 100 101 −2.0 100
R [h−1 Mpc] R [h−1 Mpc]
102
High-cgal 0.5
−
101
] R 1.0
2 −
− g
pc dlo
M 100 /
2h Σg 1.5
[ g − m
Σg lo 200
d R
10−1 2.0
−
10−210−1 100 101 −2.5 100
R [h−1 Mpc] R [h−1 Mpc]
Figure2. Leftpanels: Thesurfacenumberdensityprofiles, Σg(R), ofSDSSphotometricgalaxiesaroundthelow-cgal (top)andhigh-cgal (bottom)cluster
subsamples.Thethreedifferentpointtypeswitherrorbarsineachpanelcorrespondtothethreedifferentmagnitudelimitedsamplesofphotometricgalaxieswe
use.Rightpanels:ThelogarithmicdensityslopeofthesurfacenumberdensityprofilesobtainedaftersmoothingthedatapointswithanimprovedSavitzky-Golay
filter(order3,windowsize5). Thetwo-dimensionalsplashbackradiuscorrespondstothelocationofthesteepestslopeortheminimumofdlogΣg/dlogR.
Forcomparison,theshadedregionscorrespondtothetraditionalhaloboundary,R200m,estimatedusingtheposteriordistributionofhalomassesfromtheweak
lensingprofileforeachclustersubsamplefromMiyatakeetal.(2015)(bothareconsistentwitheachotherwithintheerrorbars).Notethatthesteepestslope(i.e.,
theminimumindlogΣg/dlogR)occursatdifferentlocationsforthetwoclustersubsamples.
The dashed lines in the top and bottom panels of Fig- splashback radius, is also seen in observations, as expected
ure1correspondtotheexpectedsubhalosurfacedensitypro- from subhalo surface density profiles in simulations. How-
filesaroundclustersinthecosmologicalsimulationMultidark ever,itisalsoclearthatthelocationswherethesurfaceden-
Planck II at z = 0.248. We have selected cluster sized halos sity profiles reach their steepest slopes are different between
above a mass threshold of 1014 h−1M , which results in the observations and simulations, especially for the left and the
(cid:12)
sameaveragehalomassasthatofoursample. Wehavenor- middle panels. This discrepancy between the observed and
malizedthesurfacedensityprofilesinthetoppanelstomatch expectedsplashbackradiiisalsoseenfortheclustersubsam-
theobservationsat∼ 11h−1Mpc. Therearemarkedsimilar- ples,whichweinvestigateatlengthnext. Wewillextensively
ities between the density profiles of subhalos in simulations quantify, comment and explore this discrepancy in the loca-
and the galaxies in observations. The surface number den- tionofsplashbackradiusaroundtheseclustersubsamples.
sities strongly deviate from a simple power law and show a Thesurfacedensityofphotometricgalaxiesaroundthelow-
clearbreakonscalesof∼1h−1Mpcinbothobservationsand and high-c cluster subsamples are shown in the upper and
gal
simulations. This is most clearly seen in the bottom panels, lower panels of Figure 2 using orange and purple symbols
where we see that the profiles reach their steepest slope on with errorbars, respectively. The lighter shades correspond
scales of ∼ 1h−1Mpc. This steepening, associated with the to photometric galaxies with brighter magnitude limits. The
solid lines in the upper and lower right hand panels of Fig-
cosmosim.org
6 More,S.etal.
102
high c
gal 0.5
−
low c
gal
2] R 1.0
− g −
pc 101 dlo
M /
2[h χ2highc=5.9 gΣg −1.5
Σg χ2lowc=13.1 dlo
Degreesoffreedom=8 2.0
− mm
100 0000
22
RR
2.5
100 − 100
R [h−1Mpc] R [h−1Mpc]
Figure3. Thesurfacenumberdensityprofiles,Σg(R),ofourfiducialsampleofSDSSphotometricgalaxiesaroundthetwoclustersubsamplesareshownin
thelefthandpanel. Theshadedregionsshowthe68and95percentconfidenceregionsofourmodelfittothedata. Therighthandpanelshowstheinferred
constraintsonthelogarithmicslopeofΣg(R)forthetwosubsamples. Thesplashbackradiusin2d,R2spd,correspondstothelocationofthesteepestslopeorthe
minimumofdlogΣg/dlogR.The68percentconstraintsonR2spdaremarkedwithverticalshadedregions.Theseminimaoccuratsignificantlydifferentlocations
forthetwoclustersubsamples.Thetraditionalhaloboundary,R200m,ismarkedbythegreydottedverticalline.
very little difference in the location of the steepest slope in
30
projection.
high-c 2d high-c 3d
gal gal Wefitthegalaxysurfacedensityprofileswiththemodelde-
low-c 2d low-c 3d
gal gal scribedintheprevioussection.Themedianandthe68percent
25
confidenceintervalsoftheposteriorsofeachoftheseparam-
eters,aswellasthebestfitχ2valuesarelistedinTable1.The
20 m numberofdegreesoffreedomforourmodelis8.
00 We show the 68 and 95 percent confidence regions from
2
R the fits to the surface density of the fiducial sample of pho-
tometric galaxies around both our cluster subsamples in the
15
left hand panel of Figure 3. The corresponding confidence
regions for the logarithmic slope, including marginalization
over other model parameters, are shown in the right hand
10
panel. We use the samples from the posterior of the model
parameter space to infer the location of the steepest slope of
5 the projected galaxy density profile, R2d, and its uncertainty.
sp
Thesenumbersarereportedforallofoursubsamplesandfor
thedifferentmodelsinTable1aswell.
0 Thelocationofthesplashbackradiuscanbecomparedwith
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 the traditional halo boundary definition, R for each sub-
200m
Radius of steepest slope [h−1 Mpc] sample. This is shown by the vertical shaded bands in the
rightpanelsofFigure2, asestimatedfromtheposteriordis-
Figure4. Theposteriordistributionsforthelocationofthesteepestslopeof tribution of the halo masses for our two subsamples inferred
thegalaxydensityprofilesaroundthehigh-andlow-cgalclustersubsamples fromtheweaklensingmeasurementinMiyatakeetal.(2015).
areshowninorangeandpurplecoloredhistograms,respectively. Thelight Wenowusethesamplesfromtheposteriordistributionof
shadedhistogramscorrespondtothelocationofthesteepestslopesofthe
model parameters to infer the constraints on the location of
surfacedensityprofiles(2-d),whilethedarkshadedhistogramscorrespond
tothelocationofsteepestslopeofthe3-dnumberdensityprofilesinferredby theminimumofthelogarithmicderivativeofthethreedimen-
ourfits. Thelocationsofthesteepestslopesforthetwoclustersubsamples sional galaxy density profile, dlogρ /dlogr. The resultant
aresignificantlydifferent,implyingadifferentmassaccretionrateontothese g
constraints on R3d are reported in the penultimate column of
clustersubsamples. sp
Table 1. The inferred value of R3d is always larger than the
ure 2 show the logarithmic slope of the surface density pro- sp
correspondingR2d forallphotometricgalaxysamplesaround
filesaroundthetwosubsamples. Theslopesforbothcluster sp
subsamplesreachvaluessteeperthan∼−1.6oneithersideof bothclustersubsamples,asshownexplicitlyinFigure4. The
∼1h−1Mpc. Thesurfacedensityofgalaxiesaroundthehigh- verticaldashedlinecorrespondstothetraditionalhalobound-
arydefinition,R ,forthetwosubsamples.
c cluster subsample reaches its steepest slope at a smaller 200m
gal Note that, for our model, a transition function f = 1,
radiuscomparedtothelow-c subsample. Thevalueofthe trans
gal
would correspond to a simple density profile: a sum of
steepest slope is considerably larger for the high-c cluster
gal
Einasto profile which describes well the inner regions and a
subsample than the low-c subsample. A comparison be-
gal
tweentheprofilesofgalaxiesasafunctionofdifferentmag- power law profile for the outer regions. However, the data
strongly disfavor such a model, with χ2 values ranging from
nitudethresholdsaroundanygivenclustersubsamplesshows
SplashbackradiusandHaloAssemblybias 7
2.5 7
1.48 0.07
±
6
2.0
al 5
g
c
h
higg 1.5 4 6.6−σ
Σ
/
cgal 1.0 3
w
loΣg 2
0.5
1
0.0 0
101 0.8 1.0 1.2 1.4 1.6 1.8 2.0
R [h−1Mpc] blow cgal/bhigh cgal
Figure5. Detectionofthehaloassemblybias.Leftpanel:Theratioofthesurfacenumberdensityprofilesofourfiducialsamplesofphotometricgalaxiesaround
thetwogalaxyclustersubsamples.Theshadedregionscorrespondtothe1-and2-sigmaconfidenceregionsforasingleconstantparameterfittothesedata.Right
panel:Theposteriordistributionoftheratiogiventhemeasurementsshownintheleftpanel. Wedetecttheassemblybias–differenceinthehalobiasesofthe
twosamples–at6.6σ.Thereisasignificantcovarianceintheerrors,hencethesmallpoint-to-pointvariationgiventheerrors.Thequotedsignificanceaccounts
forthecovariance.
3.2. DetectionofHaloAssemblybias
MDK 15 Typical Γ The mean number density profile of galaxies correlated
1.4
high-cgal high Γ with clustersat large separationsis proportionalto the prod-
uct of the biases of clusters and galaxies in the photometric
low-c low Γ
gal sample. Wehaveshownabovethattheseprofilesaredifferent
1.2 forthelow-andhigh-cgal clustersubsamples. Giventhatour
clustersampleshavethesameredshiftdistribution,thebiasof
m photometricgalaxiesshouldcanceloutintheratioofthenum-
0
20 ber density profiles, and we can use the ratio to test whether
R 1.0 thetwoclustersubsampleshavedifferentintrinsicclustering
/
dp biases,asrecentlyreportedbyMiyatakeetal.(2015).
3s
R
InthelefthandpanelofFigure5,weshowtheratioofthe
numberdensityprofilesforourfiducialsampleofphotomet-
0.8
ricgalaxiesaroundthetwosubsamplesofgalaxyclusterson
scalesof3−40h−1Mpc. Wefitaconstantratiotothesemea-
surementsaccountingforthecovariancedeterminedfromthe
0.6 jackknifetechnique. Thisassumesthatthetwosampleshave
similar scale dependence for their bias, and the data support
thisassumption. Theposteriordistributionoftheconstantra-
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
tio obtained using this procedure is shown in the right hand
Mass accretion rate Γ
panelofFigure5. Wefinda6.6σdeviationoftheratioofthe
twosurfacedensityprofilesfromunity: 1.48±0.07. Wehave
Figure6. ThedependenceofRsp/R200montheaccretionrateΓatz=0.24 thus detected halo assembly bias – the two cluster subsam-
predictedbyΛCDM,shownbyagraylinewith5%uncertainty(Moreetal.
2015).Thedarkandfaintshadedpurple(orange)regionsdisplaythe68and ples have the same halo mass based on weak lensing, but a
95percentconfidencelimitsonthesplashbackradiusin3dforourlow-(high- differentlargescalehalobias. Forcomparison,thedifference
)cgal sample. Thegray, orangeandpurplestarscorrespondtothetypical in the bias ratio that was obtained in Miyatake et al. (2015)
splashbackradiiforatypicalaccretionrate,aswellasslowandfastaccreting usingtheweaklensingsignalwas1.64+0.31,andthatfromthe
clustersfromnumericalsimulations(seetextfordetails). Theobservedval- −0.26
uesofthesplashbackradiiaresignificantlysmallerfromthepredictedvalues auto-correlationfunctionofgalaxyclusters was1.40±0.09.
theirhalomasses,evenifweconsiderhaloswithtypicalaccretionrates. The three different measurements give results which are sta-
tisticallyconsistentwitheachother.
60to140for9degreesoffreedomdependingupontheclus-
tersubsampleandthephotometricgalaxiesunderconsidera-
4. COMPARISONSWITHEXPECTATIONSFROMΛCDMMODEL
tion9.Therefore,ourmeasurementsimplyasteepeningofthe
numberdensityprofileofgalaxiesaroundbothofourcluster Usingtheprojectedgalaxynumberdensityprofilesaround
subsamplesbeyondthatpredictedbytheEinastoprofile. two cluster subsamples from the redMaPPer catalog, we
haveshownthatthesetwosubsampleshavedifferentprofiles,
splashback radii, as well as a different clustering bias. We
9Thereisonly1additionaldegreeoffreedomforthesemodels,aswelose
now compare these measurements to the predictions of the
tohnelyfidounclyiaolnmeofdreeleinpgarsacmheemteerarnt,dththeuostdhoernpoatrcahmaentgeersthγeadnedgrβeehsaovfefprerieodrosmin. concordancecosmologicalΛCDMmodel.
8 More,S.etal.
Γ=[1.0,1.5] Γ=[1.5,2.2] Γ=[2.2,5.0]
1
−
r
g 2
o −
l
d
/
g 3 135 km/s
ρ
−
g 175 km/s
o
dl 4 280 km/s
− Matter
Miscentered
5
−
100 100 100
r/r r/r r/r
200m 200m 200m
Figure7. ComparisonbetweenthelogarithmicslopeofthedensityprofileformatterandthatofsubhalosselectedusingdifferentVpeakthresholdsasindicated
inthelegend.Differentpanelscorrespondtohaloswithdifferentmassaccretionrates,Γ.Thearrowsindicatethelocationofthesteepestslope,orthesplashback
radiiforthecorrespondingpopulation,withthelongestarrowusedtorepresentdarkmatter. Thegreyverticalbandscorrespondstothefittingfunctionfor
Rsp/R200msimilartotheonebyMoreetal.(2015),butforthemeanprofiles,andincludesanuncertaintyof±5%.
4.1. Isthesplashbackradiusforthetwosubsamplesatthe cretion rates (see above). The data seem to prefer a much
expectedlocation? smallersplashbackradiusforeachofourclustersubsamples
Usingtheweaklensinginferredmassesforourclustersub- (R3spd/R200m = 0.675+−00..002241 and0.955±0.035forthehigh-and
samples,wecancomputethebaselineexpectationforthelo- the low-cgal subsamples, respectively), even when compared
cation of the splashback radius in the standard cosmologi- tothesplashbackradiuscorrespondingtohaloswithtypicalΓ
cal model. The ratio R3d/R is expected to depend upon forourmassscales.
sp 200m
the accretion rate of the halos as well as redshift (Diemer
4.2. Doesdynamicalfrictionresultinasmallersplashback
& Kravtsov 2014; Adhikari et al. 2014; More et al. 2015).
radius?
To compute the mean accretion rate on to halos, we use
halos from MDPL2 at z = 0.248, closest to the median So far in our analysis, we have identified the splashback
redshift of our redMaPPer subsamples and select all halos radius using the galaxy distribution around our cluster sub-
above a certain halo mass threshold10. We choose the halo samples. The splashback radius of galaxies could be differ-
mass threshold such that the average halo mass of the sam- entfromthatofdarkmatterduetodynamicalfrictionacting
ple is consistent with the M of the redMaPPer subsam- on the subhalos that host our galaxies, provided these sub-
200m
ples obtained by Miyatake et al. (2015). As a best case ex- halos are sufficiently massive (Adhikari & Dalal 2016). In
pectation, we divide the halo sample into two based on the whatfollows,weshowthatthesteepeningofthethreedimen-
dependence of the halo mass accretion rate on halo mass, sionaldensityprofilesforbothmatterandsubhalosthathost
Γ(M ) = ∆logM /∆loga. The derivative for Γ is com- our fiducial photometric sample of galaxies are expected to
200m vir
putedusingafinitedifferenceschemeusingthevirialmasses occuratsimilarlocations.
atredshift0.248and0.748(Diemer&Kravtsov2014;More Forthispurpose,weagainmakeuseofthehaloandsubhalo
etal.2015)11. catalogsfromtheMDPL2simulation. Wematchthecumula-
InFigure6,wecomparethelocationofthesplashbackra- tiveabundancesofdarkmattersubhalosasafunctionofV
peak
dius with respect to R observed for our cluster subsam- (the maximum circular velocity of a halo throughout its en-
200m
ples,tothebestcaseexpectationsimpliedbytheseaccretion tirehistory)andthatofourphotometricgalaxiesasafunction
rates. Thegreybandcorrespondstothefittingfunction oftheirmagnitude, toobtainanestimateoftheV ofsub-
peak
halos hosting our galaxies (see Appendix B). The subhalos
R3d (cid:32) (cid:34) Γ (cid:35)(cid:33)
sp =0.58[1+0.63Ω (z)] 1+1.08exp − (2) that host our fiducial subsample of photometric galaxies ap-
R200m m 2.26 proximatelycorrespondtosubhaloswithVpeak >135kms−1,
while the brighter subsamples correspond to subhalos with
with a 5 percent uncertainty. This fitting function is a good V >175kms−1andV >280kms−1,respectively.
fittothesplashbackradiiofdarkmatterhalosinsimulations pFeaokrthisanalysis,weuspeeakthez = 0particlesnapshotofthe
used in More et al. (2015)12, but corresponds to the mean simulation13.Weuseallhalosidentifiedbythe6dphasespace
profiles instead of the median. The grey star corresponds halo finder ROCKSTAR (Behroozi et al. 2013) in the z = 0
to the typical expected value of Γ for halos in the sample, snapshot with halo mass, M , above 8.5×1013 h−1M as
estimated from the simulations, while the orange and pur- 200m (cid:12)
oursampleofgalaxyclusters. Wesubdividetheseinbinsof
ple stars similarly correspond to the average Γ for the best- Γ = ∆logM /∆loga, and compute the three-dimensional
case simulation subsamples with the fastest and slowest ac- density profilveirof matter around them. The derivative for Γ
forthisparticularsnapshotwascomputedbetweenz = 0and
ber1s0fUorsitnhgeeaxcpoescmteodlomgaicsaslascicmreutliaotniornatwesit.hΩm=0.27,wegetsimilarnum- z=0.5(Diemer&Kravtsov2014;Moreetal.2015).Thelog-
11ThereisverylittledifferenceintheaveragevalueofΓifweuseahalo
masssamplewithathresholdonNsat 13Ideallywewouldhavelikedtoalsocarryoutthisexercisenearz=0.24,
12Thefittingfunctionwascalibratedintheredshiftrange[0,4]. butwehadonlythez=0particlesnapshotavailable.
SplashbackradiusandHaloAssemblybias 9
arithmicslopeofthematterdensityprofilearoundthecluster second, thecentralgalaxymaybephysicallydisplacedfrom
samples are shown in the different panels of Figure 7 using thepotentialminimumoftheclusteraroundwhichallgalax-
a black solid line. For reference we also show the expected iesorbit.Totestforthefirstkindofmis-centering,wehavere-
locationsofthesplashbackradiusforeachofthesubsamples, strictedourmodelfitstoscales>400h−1kpcortousingclus-
using Equation 2. The fitting function seems to capture the ters where the most probable central galaxy has p > 0.9.
cen
trend observed for the splashback radius of dark matter as Theserestrictionsproducefitparametersconsistentwiththose
a function of the accretion rate in the simulation reasonably listedinTable1,especiallythepositionofthesplashbackra-
well(within5percent). dius,withinthereporteduncertainties.
Inthesamefigureswealsoshowthelogarithmicslopesof To test for the second kind of mis-centering, we have also
subhalodistributionsaroundgalaxyclusterhalosfordifferent considered all halos from the MDLP2 simulation used in
V thresholdsobtainedfromoursimplesubhaloabundance the previous section, and displaced 40 percent of these ha-
peak
matchingmethod. Weobservethatthelocationsofthesteep- los in their positions with an offset drawn from a multivari-
estslopeforsubhaloswiththelowestV thresholdissim- ate Gaussian distribution with standard deviation equal to
peak
ilar to that in dark matter within 5 percent for all the Γ bins 400h−1kpc14. In each panel of Figure 7, we have addition-
shown in the figure. Thus the location of the splashback ra- allyincludedadashedlinewhichshowstheslopeofthelog-
diusisnotexpectedtobesignificantlydifferentforsubhalos arithmic density profiles around such a sample of halos. We
hosting our fiducial sample of photometric galaxies. As we findthat,asexpected,inallcasesthesplashbackradiuswould
consider V thresholds corresponding to our brighter sub- be overestimated by (cid:38) 20%, an effect which goes in the op-
peak
samples, we see effects of dynamical friction acting on the posite direction required to explain a smaller splashback ra-
subhalos(seealsoJiang&vandenBosch2014). Thesplash- dius. Moreover, the change of slope around the splashback
back radius of these larger subhalos systematically shifts to radiusismuchlesspronouncedandoverallshapeofthepro-
smallervalueswithincreasingV threshold. fileissignificantlymodified. Thisisincontrastwiththegood
peak
Wehavetriedtomaximisetheeffectofdynamicalfriction agreement we find between the shapes of the predicted and
in the above exercise by not considering scatter between the observedslopeprofiles.
luminosity of galaxies and the V of their subhalos while
peak
performingabundancematching. Wedonotseealargeshift 4.4. Couldaveragingeffectsresultinasmallersplashback
in the splashback radius of the photometric galaxies around radius?
any of our cluster subsamples as a function of their magni-
We have used the average halo mass of our subsamples
tude threshold. However, this does not imply that our data
as inferred from weak lensing to obtain the average R
rule out dynamical friction acting on the brighter sample of 200m
of our cluster halos to compare with the observed values of
photometricgalaxies. Itisquitelikelythatthereisareason-
thesplashbackradius,R3d. Couldthedifferencebetweenthe
ablylargescatterbetweenthemagnitudeandVpeak,whichcan sp
washoutthedynamicalfrictioneffect. R3spd/R200mseeninobservationsandthatpredictedfromsimu-
lationsariseduetothefinitewidthofthehalomassdistribu-
4.3. Backgroundsubtractionandmis-centeringuncertainties
tion? Weconsideredthedistributionofhalomassesresulting
We have used the number density profiles around random fromathresholdsamplewiththesameaveragehalomassas
pointstosubtracttheuncorrelatedgalaxiesinthebackground our cluster subsamples. For such samples, we find that the
and the foreground of our subsamples. We have also tested difference between (cid:104)R (cid:105) and that inferred from the aver-
200m
how residual uncertaintiesin background subtraction can af- age halo mass is different by only ≈ 3 percent, whereas the
fectourresults. Asaninitialtestofuncertaintiesintheback- discrepancyweobserveismuchlarger15.
ground which are constant with radius, we have added in a Wehavealsoverifiedthatthelocationofthesplashbackra-
constant parameter to our model for the projected number diusforathresholdmasssampledoesnotresultinasmaller
densityprofiles. Evenaftermarginalizingoversuchaparam- inferredsplashbackradiuscomparedtotheexpectationbased
eter, we obtain values for the splashback radii and its uncer- on using the average halo mass, and the average mass ac-
tainty which are very consistent in two dimensions, and vir- cretion rate onto the halo samples. These tests confirm that
tuallyidenticalinthreedimensions. However,therecouldbe thesmallervalueforthesplashbackradiusweobserveisnot
additionalbackgrounduncertaintieswhichvarywiththepro- likelytobearesultofsomeaveragingeffects.
jecteddistance.
For example, we expect that the clusters in our subsam- 4.5. Whatisthesystematicerrorintheweaklensinghalo
ples will cause the galaxies in the background to be magni- masses?
fied(seee.g.,Umetsuetal.2011). Weexplorethechangesto
Our conclusion that the observed splashback radius is
the background due to cluster magnification in Appendix D,
and find that the splashback radius is not affected even after smallerthantheexpectationfromsimulationsisbasedonthe
comparisonwiththevirialradiusforweaklensinghalomass
applying a conservative correction to the background due to
inferredbyMiyatakeetal.(2015). However,Miyatakeetal.
themagnificationoftheclusters. Theskysubtractionaround
bright or highly clustered objects can also potentially affect (2015) assumed a δ-function distribution in halo masses to
model the weak lensing measurements for each cluster sub-
thephotometryofgalaxiesandhencethebackgroundobjects
sample and infer the average halo mass. Such a simplified
in clusters (Aihara et al. 2011). This can also partly cancel
the magnification effect, as it reduces the number density of
backgroundgalaxiesinclusters. 14 This assumes that the redMaPPer centering probabilities are unreli-
ableandthecenteringalgorithmperformsasbadlyasselectingthebrightest
Mis-centering of central galaxies in redMaPPer clusters
galaxyinthecluster,whichcouldresultin40percentmis-centeringfraction
could affect the profiles and our estimates of the splashback (Skibbaetal.2011).
radius. There are two kinds of mis-centering: first, a galaxy 15UsingahalomasssamplewithathresholdinNsatasinourdata,also
may be mis-classified as a central by the cluster finder and doesnotaffectthisconclusion.
10 More,S.etal.
101
c]
Mp 100
1
−
h m
[ cd
i
m
3dme
R
h
1014 1015 1014 1015
M [h 1M ] M [h 1M ]
200m − 200m −
(cid:12) (cid:12)
2.0 0.5
Autocorrelation low-cgal
1.8 high-cgal
Cross-correlation
0.4
1.6
Observed
highb]cgal 1.4 Γ) 0.3
lowb/cgal 1.2 P( 0.2
[
1.0
0.8 M200m>1.0×1014M(cid:12)/h 0.1
DivisionbasedonhRm3demi(M200m)
0.6
0.0
101 0 1 2 3 4 5 6
r [h 1 Mpc] Mass accretion rate Γ
−
Figure8. Expectationforhaloassemblybiasfromsimulations.Topleftpanel:Theaverageclustercentricdistanceofmembersubhalosinclustersizedhalosin
MDPL2simulationasafunctionofhalomass.Weusetheradiustodividetheclustersampleintotwoatfixedmass.Thetwosubsamplesareshowninorangeand
purplecolors.Toprightpanel:Thesamesubsamplesasinthetopleftpanelbutinthedarkmatterconcentration-halomassplane.Thesubhalodistributionseems
tohaveverylittlecorrelationwiththedarkmatterdistribution. Bott(cid:68)omlef(cid:69)tpanel:Theratioofthehalobiasesofsubsamplesofhaloswithmasses1014h−1M(cid:12)
splitusingtheaveragedistanceoftheirsubhalosfromtheircenters, R3mdem (M200m).Thebiasratioobtainedfromcluster-clusterauto-correlationsisshownwith
asolidline,whilethedashedlinecorrespondstotheratioobtainedfromthecross-correlationsofcluster-scalehaloswithallsubhaloswithVpeak >135kms−1
thatareselectedtomimicthefidualphotometricgalaxiesusedinouranalysisasinFigure1.Bottomrightpanel:ThedistributionofthemassaccretionratesΓ
forthetwosubsamples.
fittingtotheweaklensingsurfacemassdensityprofileisex- halobias. Isthesenseandtheamplitudeofthehaloassembly
pectedtounderestimatethevirialhalomassby∼ 10%com- bias signal we see consistent with expectations from cosmo-
paredtothemeanhalomassofclustersinthesample(Man- logicalsimulationsofcolddarkmatter?
delbaumetal.2005;Becker&Kravtsov2011;Niikuraetal. Various proxies such as formation time scales of halos or
2015), or ∼ 3% in radius. The difference between the mea- theirdarkmatterconcentrationshavebeenusedinthelitera-
suredandexpectedsplashbackradiiismuchlargerthansuch turetoquantifyhaloassemblybias. Thesenseofthehaloas-
systematic error, and if at all will increase the inconsistency semblybiaseffectvariesdependingupontheassemblyproxy
ratherthandecreaseit. Thestatisticalerrorintheweaklens- used. For example, Gao et al. (2005) find that halo assem-
ing masses is less 10 percent, so even 2-σ deviations will blybiasisstrongestforlowmassgalaxyscalehalos,andthat
result in only ∼ 7 percent change in the expectation in the the earliest forming halos cluster more strongly than the av-
splashbackradius.Theerrorsintheweaklensingmasseshave erage for their halo masses. However, they find that the ef-
alsobeenfoldedinwhencomputingtheerrorontheobserved fectalmostdisappearsonthemassscalesweconsiderinour
R /R showninFigure6. paper. On the contrary, when the concentration of halos is
sp 200m
used as a proxy, halo assembly bias manifests itself at both
4.6. Isthehaloassemblybiassignalconsistentwith
galaxyscalesandgalaxyclusterscales(Wechsleretal.2006).
expectations?
Atgalaxyscales(massesbelow1012h−1M )highconcentra-
(cid:12)
Our observational results indicate that the galaxy cluster tionhalos(whichformearlier)havealargerhalobias,butthe
subsamplewithlowerconcentrationofmembergalaxieshasa trend reverses on galaxy cluster scales, as expected from the
largersplashbackradius(loweraccretionrateΓ),andalarger