The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: modeling of the luminosity and colour dependence in the Data Release 10

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by  Hong Guo

Mon.Not.R.Astron.Soc.000,000–000(0000) PrintedMay8,2014 (MNLATEXstylefilev2.2)
The clustering of galaxies in the SDSS-III Baryon Oscillation
Spectroscopic Survey: modeling of the luminosity and colour
dependence in the Data Release 10
4
1
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2 Hong Guo1,2⋆, Zheng Zheng1, Idit Zehavi2, Haojie Xu1, Daniel J. Eisenstein3, David

y H. Weinberg4,5, Neta A. Bahcall6, Andreas A. Berlind7, Johan Comparat8, Cameron K.
a
M McBride3, Ashley J. Ross9, Donald P. Schneider10,11, Ramin A. Skibba12, Molly E. C.
Swanson3, Jeremy L. Tinker13, Rita Tojeiro9, David A. Wake14,15
7
1DepartmentofPhysicsandAstronomy,UniversityofUtah,UT84112,USA
O] 2DepartmentofAstronomy,CaseWesternReserveUniversity,OH44106,USA
3Harvard-SmithsonianCentreforAstrophysics,60GardenSt.,Cambridge,MA02138,USA
C 4DepartmentofAstronomy,OhioStateUniversity,Columbus,OH43210,USA
. 5CentreforCosmologyandAstro-ParticlePhysics,OhioStateUniversity,Columbus,OH43210,USA
h 6DepartmentofAstrophysicalSciences,PrincetonUniversity,PeytonHall,Princeton,NJ08540,USA
p 7DepartmentofPhysicsandAstronomy,VanderbiltUniversity,Nashville,TN37235,USA
o- 8AixMarseilleUniversite´,CNRS,LAM(Laboratoired’AstrophysiquedeMarseille)UMR7326,13388,Marseille,France
r 9InstituteofCosmology&Gravitation,DennisSciamaBuilding,UniversityofPortsmouth,Portsmouth,PO13FX,UK
t 10DepartmentofAstronomyandAstrophysics,ThePennsylvaniaStateUniversity,UniversityPark,PA16802,USA
s
a 11InstituteforGravitationandtheCosmos,ThePennsylvaniaStateUniversity,UniversityPark,PA16802,USA
[ 12CentreforAstrophysicsandSpaceSciences,UniversityofCalifornia,9500GilmanDrive,SanDiego,CA92093,USA
13CentreforCosmologyandParticlePhysics,NewYorkUniversity,NewYork,NY10003,USA

2 14DepartmentofAstronomy,UniversityofWisconsin-Madison,475N.CharterStreet,Madison,WI,53706,USA
v 15DepartmentofPhysicalSciences,TheOpenUniversity,MiltonKeynesMK76AA,UK
9
0
0
May8,2014
3
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1
0 ABSTRACT
4 WeinvestigatetheluminosityandcolourdependenceofclusteringofCMASSgalaxiesinthe
1 Sloan Digital Sky Survey-IIIBaryon Oscillation SpectroscopicSurveyTenth Data Release,
v: focusing on projected correlation functions of well-defined samples extracted from the full
i catalog of ∼ 540,000 galaxies at z ∼ 0.5 covering about 6,500 deg2. The halo occupa-
X
tiondistributionframeworkisadoptedtomodelthemeasurementsonsmallandintermediate
r scales (from0.02to 60h−1Mpc), infer the connectionof galaxiesto darkmatter halosand
a
interprettheobservedtrends.WefindthatluminousredgalaxiesinCMASSresideinmassive
halos of mass M∼1013–1014h−1M⊙ and more luminous galaxies are more clustered and
hostedbymoremassivehalos.Thestrongsmall-scaleclusteringrequiresafractionofthese
galaxies to be satellites in massive halos, with the fraction at the level of 5–8 per cent and
decreasingwithluminosity.Thecharacteristicmassofahalohostingonaverageonesatellite
galaxyabovealuminositythresholdisaboutafactor8.7largerthanthatofahalohostinga
centralgalaxyabovethesamethreshold.Atafixedluminosity,progressivelyreddergalaxies
aremorestronglyclusteredonsmallscales,whichcanbeexplainedbyhavingalargerfrac-
tionofthesegalaxiesintheformofsatellitesinmassivehalos.Ourclusteringmeasurements
onscalesbelow0.4h−1Mpcallowustostudythesmall-scalespatialdistributionofsatellites
insidehalos.Whiletheclusteringofluminosity-thresholdsamplescanbewelldescribedby
aNavarro-Frenk-White(NFW)profile,thatofthereddestgalaxiesprefersasteeperormore
concentratedprofile.Finally,we also usegalaxysamplesof constantnumberdensityatdif-
ferentredshiftstostudytheevolutionofluminousredgalaxies,andfindtheclusteringtobe
consistentwithpassiveevolutionintheredshiftrangeof0.5.z .0.6.
Key words: galaxies: distances and redshifts—galaxies: halos—galaxies: statistics—
cosmology:observations—cosmology:theory—large-scalestructureofuniverse
2 Guo etal.
1 INTRODUCTION intheCMASSsamplewasalsofoundtoberoughlyconsistentwith
passiveevolutionpredictions.
Galaxy luminosity and colour, the two readily measurable quan-
While G13 presented the clustering measurements based on
tities, encode important information about galaxy formation and
theDR9sample,inthispaperwemoveastepforwardtoperform
evolutionprocesses.Theclusteringofgalaxiesasafunctionoflu-
theHODmodelingtoinfertheconnectionbetweengalaxiesandthe
minosity and colour helps reveal the role of environment in such
hostingdarkmatterhalos,usingtheclusteringmeasurementsfrom
processes.Thedependenceofclusteringonsuchgalaxyproperties
theDR10data.Whiteetal.(2011)presentedthefirstHODmodel-
isthereforeafundamentalconstraintontheoriesofgalaxyforma-
ingresultforanearlyCMASSsample(fromthefirstsemesterof
tion,anditisalsoimportantwhenattemptingtoconstraincosmo-
data).Now withDR10, thesurvey volumeismore than11times
logicalparameterswithgalaxyredshiftsurveys,sincethedifferent
larger than that in Whiteetal. (2011), which allows us to study
typesofgalaxiestracetheunderlyingdarkmatterdistributiondif-
thedetailedrelationbetweentheproperties(specificallyluminos-
ferently. Inthispaper, wepresent themodeling of theluminosity
ityandcolour)oftheCMASSgalaxiesandtheirdarkmatterhalos.
and colour dependent clustering of massive galaxies in theSloan
WebuildonsimilarstudiesfortheSDSSMAINsamplegalaxies
DigitalSkySurvey-III(SDSS-III;Eisensteinetal.2011).
atz 0.1(Zehavietal.2011)andluminousredgalaxies(LRGs)at
A fundamental measure of clustering is provided by mea- z 0∼.3(Zhengetal.2009),extendingthemnowtohigherredshifts
suring galaxy two-point correlation functions (2PCFs). Galaxy (z∼0.5) andforgalaxiesatthehigh-massendofthestellarmass
clustering provides a powerful approach to characterize the fun∼ction.GiventhekeyroleoftheCMASSgalaxiesasalarge-scale
distribution of galaxies and probe the complex relation be- structureprobe,itisalsoimportanttounderstandindetailhowthe
tween galaxies and dark matter. More luminous and red- CMASSgalaxiesrelatetotheunderlyingdarkmatterhalosforop-
der galaxies are generally observed, in various galaxy sur- timallyutilizingthemforconstrainingcosmologicalparameters.
veys, to have higher clustering amplitudes than their fainter With about a factor of two increase in survey volume from
and bluer counterparts (e.g. Davis&Geller 1976; Davisetal. DR9toDR10,theDR10dataproducemoreaccuratemeasurements
1988; Hamilton 1988; Lovedayetal. 1995; Benoistetal. 1996; ofthe2PCFs,andthusbetterconstraintsonHODparameters.After
Guzzoetal. 1997; Norbergetal. 2001, 2002; Zehavietal. 2002, applying a fibre-collision correction, with the method developed
2005, 2011; Budava´rietal. 2003; Madgwicketal. 2003; Lietal. andtestedinGuo,Zehavi,&Zheng(2012),weobtaingoodmea-
2006; Coiletal. 2006, 2008; Meneuxetal. 2006, 2008, 2009; surementsofthe2PCFsdowntoscalesof 20h−1kpc,withthe
Wangetal. 2007; Wakeetal. 2008, 2011; Swansonetal. 2008; helpofthelargersurveyareaofDR10.This∼leadstothepossibility
Ross&Brunner2009;Ross,Percival,&Brunner2010;Rossetal. of determining the small-scale galaxy distribution profiles within
2011; Skibba&Sheth 2009; Skibbaetal. 2013; Lohetal. 2010; halos,andwealsopresenttheresultsofsuchastudy.
Christodoulouetal.2012;Bahcall&Kulier2013;Guoetal.2013, The paper is organized as follows. In Section 2, we briefly
2014). describe the CMASS DR10 sample and the clustering measure-
Theclusteringdependenceofgalaxiesontheirluminosityand mentsfortheluminosityandcoloursamples.TheHODmodeling
colourcanbetheoreticallyunderstoodthroughthehalooccupation methodispresentedinSection3.Wepresentourmodelingresults
distribution(HOD)modeling(seee.g.Jing,Mo,&Boerner1998; inSection4andgiveasummaryinSection5.
Peacock&Smith 2000; Seljak 2000; Scoccimarroetal. 2001; Throughoutthepaper,weassumeaspatiallyflatΛCDMcos-
Berlind&Weinberg 2002; Berlindetal. 2003; Zhengetal. 2005, mology (the same as in G13), with Ω = 0.274, h = 0.7,
m
2009; Miyatakeetal. 2013) or the conditional luminosity func- Ωbh2 =0.0224,ns=0.95,andσ8 =0.8.
tion(CLF)method(Yang,Mo,&vandenBosch2003;Yangetal.
2005). In HOD modeling, two determining factors that affect the
clusteringarethehostdarkmatterhalomass,M,andthesatellite
fractionfsat.Theemergingexplanationfortheobservedtrendsis
thatmoreluminousgalaxiesaregenerallylocatedinmoremassive 2 DATAANDMEASUREMENTS
halos,whileforgalaxiesofthesameluminosity,redderonestend
2.1 BOSSGalaxiesandLuminosityandColourSubsamples
tohave ahigher fraction inthe formof satellitegalaxies inmas-
sivehalos(e.g.Zehavietal.2011).Residinginmoremassivehalos The SDSS-III BOSS selects galaxies for spectroscopic observa-
leads to a stronger clustering of galaxies on large scales, whilea tionsfromthefive-bandSDSSimagingdata(Fukugitaetal.1996;
highersatellitefractionresultsinstrongersmall-scaleclustering. Gunnetal.1998,2006;Yorketal.2000).A detailedoverview of
In this paper, we investigate the colour and luminosity de- theBOSSsurveyisgivenbyBoltonetal.(2012)andDawsonetal.
pendent galaxy clustering measured from the SDSS-III Baryon (2013), and the BOSS spectrograph is described in Smeeetal.
Oscillation Spectroscopic Survey (BOSS; Dawsonetal. 2013) (2013). BOSS is targeting 1.5 million galaxies and 150,000
Data Release 10 (DR10; Andersonetal. 2013; Ahnetal. 2014). quasars covering about 10,000deg2 of the SDSS imaging area.
The SDSS-IIIBOSS survey is providing a large sample of lumi- About 5 per cent of the fibres are devoted to more than 75,000
nousgalaxiesthatwillallowastudyofthegalaxy-haloconnection ancillary targets probing a wide range of different types of ob-
andtheevolutionof massivegalaxies(withatypical stellarmass jects(Dawsonetal.2013).InoneBOSSancillaryprogram,fibre-
of1011.3h−1M ).Bycarefullyaccountingfortheeffectofsam- collided galaxies in the BOSS sample were fully observed in a
pleselectionsto⊙constructnearlycompletesubsamples,Guoetal. small area. Wewill present their clustering resultsinanother pa-
(2013)(hereafterG13)investigatedtheluminosityandcolourde- per(Guoetal.,inpreparation).
pendence of galaxy 2PCFs from BOSS DR9 CMASS sample We focus on the analysis of the CMASS sample
(Andersonetal. 2012) in the redshift range of 0.43 < z < 0.7. (Eisensteinetal. 2011; Andersonetal. 2012, 2013) selected
Itwasfoundthatmoreluminousandreddergalaxiesaregenerally fromSDSS-IIIBOSSDR10.Thesamplecoversaneffectivearea
moreclustered,consistentwiththepreviouswork.Theevolutionof ofabout6,500deg2,almosttwiceaslargeasinDR9.Theselec-
galaxyclusteringonlargescales(characterizedbythebiasfactor) tion of CMASS galaxies is designed to be roughly stellar-mass
ModelingtheLuminosityand ColourDependentClustering 3
limitedatz>0.4.Thedetailedselectioncutsaredefinedby, 2.2 MeasurementsoftheGalaxy2PCFs
17.5<icmod < 19.9 (1) Approximately 1.5 per cent of CMASS galaxies in DR10 were
d⊥ > 0.55, (2) previously observed in SDSS-II whose angular distribution dif-
fersfromotherBOSSgalaxies(seemoredetailsinAndersonetal.
icmod < 19.86+1.6(d⊥−0.8) (3) 2012). The redshift measurements of these SDSS-II ‘Legacy’
ifib2 < 21.5 (4) galaxies are, by construction, 100 per cent complete, while the
r i < 2.0 (5) redshift and angular completeness of BOSS galaxies vary with
mod mod
−
sky position. The different distributions of these ‘Legacy’ galax-
where all magnitudes are Galactic-extinction corrected
ies and the newly observed BOSS galaxies need to be carefully
(Schlegel,Finkbeiner,&Davis 1998) and are in the observed
taken into account for clustering measurement. In previous work
frame.WhilethemagnitudesarecalculatedusingCMODELmagni-
(e.g. Andersonetal. 2012,G13), this is achieved by subsampling
tudes(denotedbythesubscript‘cmod’),thecoloursarecomputed
the SDSS-II galaxies to match the sector completeness of BOSS
using MODEL magnitudes (denoted by the subscript ‘mod’). The
survey.Here,topreservethefullinformationinthe‘Legacy’galax-
magnitude i corresponds to the i-band flux within the fibre
fib2 ies, weadopt an alternative method, withadecomposition of the
aperture (2′′ in diameter). The quantity d⊥ in Equations (2) and total3Dcorrelationfunctionsasfollows(Zuetal.2008;Guoetal.
(3)isdefinedas
2012)
d⊥ =(rmod imod) (gmod rmod)/8. (6)
− − −
n2 2n n n2
SincethebluegalaxiesaregenerallyfarfromcompleteinCMASS ξ = Lξ + L Bξ + Bξ , (8)
T n2 LL n2 LB n2 BB
duetotheselectioncutsofEquations(2)and(3)(seealsoFigure1 T T T
ofG13),wefocusinthispaperontheclusteringandevolutionof
the red galaxies. The red galaxies in this paper are selected by a where nL, nB and nT are the number densities of the Legacy,
luminosity-dependentcolourcut(G13), (uniquely) BOSS, and all galaxies, respectively, ξLL is the auto-
correlationfunctionofLegacygalaxies,ξ istheauto-correlation
BB
(r−i)>0.679−0.082(Mi+20) (7) function of BOSS galaxies, and ξLB is the cross-correlation of
where theabsolute magnitude M and r icolour are both k+e LegacyandBOSSgalaxies.Thedecompositioncanbeunderstood
i
correctedtoz =0.55(Tojeiroetal.2012)−. intermsofgalaxypaircounts–thetotalnumberofgalaxypairsis
Inthispaper,wefocusonmodelingtheluminosityandcolour composed of Legacy-Legacypairs(relatedtoξLL), BOSS-BOSS
dependence of the CMASS red galaxies. We therefore construct pairs(relatedtoξBB),andtheLegacy-BOSScross-pairs(relatedto
suitable luminosity and colour subsamples of galaxies. Three lu- ξLB).Therandomsamplesareseparatelyconstructed forLegacy
minosity threshold samples of red galaxies are constructed, with andBOSSgalaxiestoreflectthedifferentangularandredshiftdis-
M < 21.6,M < 21.8,andM < 22.0,inthesamered- tributions.
i i i
shiftran−ge,0.48<z<−0.55.Weuselumin−ositythresholdsamples In galaxy surveys using fibre-fed spectrographs, the precise
tofacilitateamorestraight-forwardHODmodelingofthemeasure- small-scaleauto-correlationmeasurementsarehinderedbytheef-
ments.Theredshiftrangeisselectedtoensurethattheredgalaxies fectthattwofibresonthesameplatecannotbeplacedcloserthan
inthesamplesarenearlycompleteandminimallysufferfromthe certainangularscales,whichis62′′ inSDSS-III,correspondingto
selectioneffects(G13).DetailsofthesamplesaregiveninTable1 about 0.4h−1Mpcatz 0.55. Suchfibre-collisioneffectscanbe
∼
(together withthe best-fittingparameters fromHOD modeling to correctedbyusingthecollidedgalaxiesthatareassignedfibresin
be presented in Section 3). The left panel of Figure 1 shows the thetileoverlapregions,asproposedandtestedbyGuoetal.(2012)
selectionof thethreeluminosity-threshold samples inthecolour- andimplementedinG13.
magnitudediagram(CMD).Thecontoursrepresentthedensitydis- WeapplythesamemethodheretoBOSSgalaxiestocorrect
tributionof theCMASS galaxies intheCMD. Thesolid linede- for the fibre-collision effect in measuring the 2PCFs. We divide
notesthecolourcutofEquation(7)andtheshadedregionandthe the BOSS sample into two distinct populations, one free of fibre
three dashed lines show the selection of the luminosity threshold collisions(labelledbysubscript‘1’)andtheotherconsistingofpo-
samples. tentiallycollidedgalaxies(labelledbysubscript‘2’).Withsucha
For studying thecolour dependence of clustering, we divide division,Equation (8)isfurtherdecomposedintosixtermsas,
theCMASSgalaxiesinto‘green’,‘redseq’,and‘reddest’subsam-
n2 2n n 2n n
ples, using the colour cuts in Table 2 of G13. In order to have ξ = Lξ + L B1ξ + L B2ξ
thebestsignal-to-noiseratioanddecouplethecolour dependence T n2T LL n2T LB1 n2T LB2
from the luminosity dependence, we only select galaxies in the n2 2n n n2
+ B1ξ + B1 B2ξ + B2ξ . (9)
redshift range of 0.48 < z < 0.55 and luminosity range of n2 B1B1 n2 B1B2 n2 B2B2
T T T
22.2 < M < 21.6.TherightpanelinFigure1showsthese-
i
− −
lectionofthethreesubsamples.Thethreecolouredsolidlinesare Inactualmeasurements,thecorrelationfunctionsξ ,ξ ,and
LB2 B1B2
thethreecutsforthefinecoloursamples(G13):thegreen,magenta, ξ involving thecollided galaxiesare estimatedusing there-
B2B2
andredlinesdividethegalaxiesinto‘blue’,‘green’,‘redseq’,and solvedcollidedgalaxiesintileoverlapregions(asdetailedinGuo
‘reddest’ subsamples, respectively. The ‘reddest’ sample has the etal.2012).
reddest colour, while the ‘redseq’ sample represents the galaxies Wefirstmeasuretheredshift-space2PCFξ(r ,r )inbinsof
p π
occupying the central part of the red sequence in the CMD. The transverseseparationr andline-of-sightseparationr (r inloga-
p π p
‘green’ sampleisselectedtorepresent thetransitionfromblueto rithmicbinsfrom 0.02to 63h−1Mpcwith∆logr =0.2and
p
redgalaxies.The‘blue’galaxysampleisnotconsideredherebe- r inlinearbinsfr∼om0to1∼00h−1Mpcwith∆r =2h−1Mpc),
π π
causeofitslowcompleteness.Moreinformationonthecoloursub- usingtheLandy&Szalay(1993)estimator.Wethenintegratethe
samplescanbefoundinTable2. 2PCFalongtheline-of-sightdirectiontoobtaintheprojected2PCF
4 Guo etal.
Table1.Samplesofdifferentluminositythresholdsintheredshiftrange0.48<z<0.55
Mimax Ngal n¯(z) χ2/dof logMmin σlogM logM0 logM1′ α fsat(percent)
21.6 114417 2.18 10−4 19.74/14 13.37 0.05 0.58 0.05 0.57 2.09 14.30 0.02 1.56 0.03 7.91 0.43
−21.8 65338 1.25×10−4 28.83/14 13.57±0.05 0.59±0.05 3.67±4.25 14.46±0.02 1.64±0.07 6.30±0.40
−22.0 33964 0.65×10−4 21.19/14 13.80±0.06 0.61±0.06 2.81±3.07 14.59±0.03 1.82±0.09 5.04±0.36
− × ± ± ± ± ± ±
Themeannumberdensityn¯(z)isinunitsofh3Mpc−3.Thehalomassisinunitsof h−1M .Thesatellitefactionfsatisthederivedparameter
⊙
from the HOD fits. The best-fitting χ2 and the degrees offreedom (dof) with the HOD modeling are also given. The degrees offreedom are
calculatedasdof =Nwp+1−Npar,wherethetotalnumberofdatapoints(Nwp+1)isthatofthewp(rp)datapointsplusonenumberdensity
datapoint,andNparisthenumberofHODparameters.
Figure1.Left:theselectionoftheluminositythresholdsamplesinthecolour-magnitudediagram(CMD).Thecontoursrepresentthedensitydistributionof
theCMASSgalaxiesintheCMD.Theshadedregionsshowthegalaxiescoveredintheluminositythresholdsamples.Thesolidlinedenotesthecolourcutof
Equation(7),andthethreedashedlinesrepresentthethreeluminositythresholds.Right:theselectionofthe‘green’,‘redseq’and‘reddest’coloursamples
intheluminosityrangeof−22.2<Mi <−21.6andredshiftrangeof0.48<z <0.55.Thethreecoloursolidlinesarethethreecutsforthefinecolour
samples.Thegreenlineisforthecutbetweenthe‘blue’and‘green’samples.Themagentalineisthecutbetweenthe‘green’and‘redseq’samples.Thered
lineisthecutbetweenthe‘redseq’and‘reddest’samples.Theshadedregionsrepresentourselectionofthecorrespondingcoloursamples.
3 HODMODELING
Table2.HODparametersforthecoloursamplesin−22.2<Mi<−21.6
We perform the HOD fits to the projected two-point auto-
Sample Ngal n¯(z) χ2/dof logM1′ fsat(percent) correlation functions wp(rp), measured in different luminosity
‘green’ 28835 0.54 20.81/16 14.57 0.04 3.68 0.47 and colour bins. In the HOD framework, it is helpful to sepa-
‘redseq’ 32221 0.61 15.62/18 14.42±0.03 6.15±0.46 ratethecontribution tothemean number N(M) of galaxiesin
± ± h i
‘reddest’ 34670 0.66 35.26/18 14.30 0.02 9.35 0.46 halos of mass M into those from central and satellite galaxies
± ±
(Kravtsovetal.2004;Zhengetal.2005).
The mean number density n¯(z) is in unit of 10−4h3Mpc−3. The halo
For luminosity-threshold samples, we follow
mass is in units of h−1M . The redshift range of colour samples are Zheng,Coil,&Zehavi (2007) to parameterize the mean oc-
limitedto0.48<z<0.55.⊙ cupationfunctionsofcentralandsatellitegalaxiesas
wp(rp) hNcen(M)i = 21(cid:20)1+erf(cid:18)logMσ−logloMgMmin(cid:19)(cid:21) (11)
wp(rp)=2Z0rπ,maxξ(rp,rπ)drπ, (10) hNsat(M)i = hNcen(M)i(cid:18)MM−1′M0(cid:19)α (12)
with rπ,max = 100h−1Mpc. This projected 2PCF is what we whereerf istheerrorfunction.Intotal,therearefivefreeparame-
present and model in this paper. The covariance error matrix for tersinthisparametrization.TheparameterMmindescribesthecut-
wp(rp)isestimatedfrom200jackknifesubsamples(Zehavietal. offmassscaleofhaloshostingcentralgalaxies( Ncen(Mmin) =
h i
2002,2005,G13).Weprovidethemeasurementsfortheprojected 0.5). The cutoff profile is step-like but softened to account for
2PCF, w (r ), for all the subsamples used in this paper in the thescatterbetweengalaxyluminosityandhalomass(Zhengetal.
p p
AppendixB.WiththeadvantageoflargerskycoverageinDR10, 2005),andischaracterizedbythewidthσ .Thethreeparam-
logM
the correlation function measurements have much smaller errors etersfor the mean occupation function of satellitesarethe cutoff
comparedwiththoseinDR9.ThemeasurementsinDR9andDR10 massscaleM0,thenormalizationM1′,andthehigh-massendslope
aregenerallyconsistentwithintheerrors. α of Nsat(M) . In halos of a given mass, the occupation num-
h i
ModelingtheLuminosityand ColourDependentClustering 5
Figure2.Projectedtwo-point correlation functions measuredinDR10andthecorresponding best-fitting HODmodelsforthethreeluminositythreshold
samples.Topleft:themeasurementsofwp(rp)fromDR10(squares)comparedwiththebest-fittingHODmodels(lines).Thebluedottedlinesrepresentthe
one-haloandtwo-halotermsforthesampleofMi<−21.6.Theχ2perdegreeoffreedom(dof)forthethreebest-fittingsarealsoshown.Topright:themean
occupationnumberdistributionsofthethreesamples.Thetotalmeanhalooccupationfunction(solidlines)isdecomposedintocontributions fromcentral
galaxies(dashedlines)andsatellitegalaxies(dottedlines).Bottomleft:Theprobabilitydistributionoffsat.BottomRight:Theprobabilitydistributionofthe
hosthalomass.
bersofthecentralandsatellitegalaxiesareassumedtofollowthe withintheHODframework,wefollowtheprocedures laidoutin
nearestintegerandPoissondistributionswiththeabovemeans,re- Zheng (2004) and Tinkeretal. (2005). When computing the pro-
spectively.Inourfiducialmodel,thespatialdistributionofsatellite jected 2PCF from the real-space 2PCF, we also incorporate the
galaxiesinhalosisassumedtofollowthatofthedarkmatter,i.e. effect of residual redshift-space distortions to improve the mod-
theNavarro-Frenk-White(NFW)profile(Navarro,Frenk,&White elingonlargescales.Thisisdonebydecomposingthe2PCFinto
1997),withhaloconcentrationparameter monopole, quadrupole, and hexadecapole moments and applying
themethodofKaiser(1987)(alsoseevandenBoschetal.2013).
c(M)=c0(M/Mnl)β(1+z)−1, (13) WeuseaMarkovChainMonteCarlo(MCMC)methodtoex-
ploretheHODparameterspaceconstrainedbytheprojected2PCF
where Mnl is the non-linear mass scale at z = 0, c0 = 11 and
β = 0.13 (Bullocketal. 2001; Zhaoetal. 2009). Later in this wp(rp)andthenumberdensityng ofeachgalaxysample.Theχ2
paper,−wewillalsoconsiderageneralizedNFWprofileandusethe isformedas
2PCFmeasurementstoconstrainit.Halosherearedefinedtohave
ameaTnodtehnesoitryet2ic0a0lltyimceosmthpautteoftthheerbeaaclk-sgproacuend2uPnCivFerosfe.galaxies χ2 =(wp−wp∗)TC−1(wp−wp∗)+ (ngσ−2n∗g)2, (14)
ng
6 Guo etal.
where wp is the vector of wp at different values of rp and C Thebottom-rightpanelofFigure2showstheprobabilitydis-
is the full error covariance matrix determined from the jackknife tributionsofhosthalomassforthethreesamples,generatedfrom
resampling method (as detailed in Guoetal. 2013). The mea- the product of the mean occupation function and the differential
sured values are denoted with a superscript ‘ ’. The error σ halomassfunction(Wakeetal.2008;Zhengetal.2009).Thehost
∗ ng
on thenumber density isdetermined fromthe variation of n (z) halos refer to the main halos, i.e. we do not consider the subha-
g
in the different jackknife subsamples. Finally, in order to ac- los as the host halos. Most of the central galaxies in these sam-
countforthebiasintroducedwheninvertingthecovariancematrix ples reside in halos of about a few times 1013h−1M , while
(Hartlap,Simon,&Schneider2007),wemultiplytheaboveχ2by the satellitegalaxies are mostly found in halos of masse⊙s around
afactor(n n 2)/(n 1),whichisabout0.9inourcase. 1014h−1M .Moreluminousgalaxieshaveahigherprobability
jk d jk
Heren isth−enum−berofjack−knifesamplesandn isthedimen- ∼tobefoundin⊙moremassivehalos.Forcentralgalaxies,inthenar-
jk d
sionofthedatavector.InAppendixA,wedemonstratetherobust- rowluminosityrangeinoursamples,thepeakhosthalomassvaries
nessandaccuracyofourfittingwithjackknifecovariancematrices from1.1 1013to3.3 1013h−1M forthethreesamples.
bycomparingwithresultsfromusingmockcovariancematrices. Figu×re3displays t×herelationof⊙the HODparameters Mmin
andM1 withthethresholdluminosityMi.Notethatthequantity
Mmin is the characteristic mass of halos hosting central galaxies
4 MODELINGRESULTS atthethresholdluminosity(withhNcen(Mmin)i = 0.5),andM1
is the characteristic mass of halos hosting on average one satel-
4.1 HODfortheLuminosity-thresholdSamples lite galaxy above the luminosity threshold ( Nsat(M1) = 1),
which has a subtle difference from M′. Cleahrly, the tigiht corre-
Thebest-fittingHODparametersforthethreeluminositythreshold 1
lation between galaxy luminosity and halo mass scales persists
samplesarelistedinTable1.Figure2showsthemodelingresults.
for massive galaxies in massive halos at z 0.5. The scal-
(Tshqeuatoreps-)lecfotmpapnaerelddiwspitlhaytshethbeesmt-efiatstuinrgemHeOntDsomfowdepl(srp(l)iniens)D.RT1h0e ing relation between Mmin and M1 in our sam∼ples roughly fol-
top-rightpanelshowsthemeanoccupationfunctionsfromthethree lows M1∼8.7Mmin. The large gap between M1 and Mmin im-
best-fitting models. Overall, the trend of stronger clustering for pliesthatahalowithmassbetweenMmin andM1 tendstohosta
moremassivecentralgalaxyratherthanmultiplesmallergalaxies
moreluminousCMASSsamplesisexplained intheHODframe-
workasashifttowardhigher massscaleofhosthalos, similarto (Berlindetal.2003).ThisM1-to-Mmin ratioiscomparabletothe
one found for SDSS LRGs (Zhengetal. 2009) and significantly
thatfortheSDSSMAINgalaxies(Zehavietal.2011).
smaller thanthe scaling factor found for theSDSSMAIN galax-
The mean occupation functions in the top-right panel also
ies( 17,Zehavietal.2011).Theratiodecreasessomewhatwith
show that a fraction of the CMASS luminous red galaxies must ∼
increasingluminosity–forthemostluminoussampleweanalyse
be satellites in massive halos. This is required to fit the small-
(M < 22.0),theratiois6.4+0.5,smallerthanthe 8.7inferred
scaleclustering.Theprobabilitydistributionoffsatisshowninthe fromi the−lowerluminosity-thre−sh0o.4ldsamples.Sucha∼trendwithlu-
bottom-leftpanelofFigure2.Moreluminousgalaxieshavealower
minosityisalsofoundinotherSDSSanalyses(Zehavietal.2005,
fraction of satellites, consistent with the trend found for MAIN
2011; Skibba,Sheth,&Martino 2007; Zhengetal. 2009). These
sample galaxies (Zehavietal. 2011) and luminous red galaxies
behaviorsarelikelyrelatedtothedominanceofaccretionofsatel-
(Zhengetal. 2009). The peak fsat varies from 8 per cent for the litesoverdestructioninmassivehalos.Moremassive,cluster-sized
M < 21.6sampleto5percentfortheM < 22.0sample.
i − i − halos form late and accrete satellites more recently, leaving less
timeforsatellitestomergeontothecentralgalaxiesandthuslow-
eringthesatellitethresholdmassM1(Zentneretal.2005).
In Figure 4 we compare the measurements of Mmin
and M1 of our samples with those from the literature of
various surveys (Mandelbaumetal. 2006; Phlepsetal. 2006;
Kulkarnietal. 2007; Zheng,Coil,&Zehavi 2007; Brownetal.
2008; Blake,Collister,&Lahav 2008; Wakeetal. 2008;
Padmanabhanetal. 2009; Zhengetal. 2009; Whiteetal. 2011;
Zehavietal. 2011; Couponetal. 2012; Beutleretal. 2013;
Miyatakeetal. 2013; Parejkoetal. 2013). We plot them as a
function of galaxy number density n¯ (note that a lower n¯
g g
corresponds to a higher threshold in galaxy luminosity or stellar
mass).Themassscalesareallcorrectedtothecosmologyadopted
inthispaperaccordingtotheirproportionalitytoΩ (Zhengetal.
m
2002,2009).TheleftpanelshowsMmin (opensymbols)andM1
(solid symbols) as a function of (decreasing) number density of
thedifferent samples. The right panel displays the corresponding
ratios M1/Mmin. Our results of the three luminosity threshold
samples(blackstars)areingoodagreementwiththetrendshown
inothersamples.
Figure 3. Two mass scales in HOD models as a function of threshold Brownetal. (2008) noted that Mmin and n¯g approximately
luminosity of the galaxy samples, where hNcen(Mmin)i = 0.5 and followapower-lawrelationwithapower-lawindex 1.Such
hNsat(M1)i = 1.ThesquaresandsolidlinesaretheHODmodelingre- a power-law relation can be largely explained from∼the−the halo
sultsofthethreeluminositythresholdsamples,whilethedashedlineshows
massfunction.Intheleftpanel,weplotthecumulativehalomass
therelationofM1 8.7Mmin.
∼ functions n (> M) at three typical redshifts, z = 0, 0.5, and
h
ModelingtheLuminosityand ColourDependentClustering 7
Figure4.Leftpanel:HODparametersMmin(opensymbols)andM1(solidsymbols)asafunctionoftheaveragenumberdensityn¯gofthesamples.Different
symbolsrepresentthemeasurementsfromtheliterature,aslabelledinthefigure.Ourmeasurementsofthethreeluminositythresholdsamplesaredisplayed
bytheblackstars,whichareingoodagreementwiththeliterature.Thehalomassfunctionsatthethreetypicalredshiftsz=0,0.5,and1arealsoshownas
thesolidlines.Rightpanel:theratiobetweenM1andMminforallthemeasurementsinliterature.
1, assolid curves. The halomass functions areanalytically com-
puted for the assumed cosmological model. The low-mass end
(M < 1012h−1M ) of the halo mass function closely follows
n (> M) M−1⊙andevolvesslowlywithredshift.Atthehigh-
h
∝
mass end, the halo mass function drops more rapidly than the
power-law at thelow massend, and shows stronger redshift evo-
lution.Thehalomassfunctionnh(M > Mmin)canberegarded
as resulting from a simple form of HOD — one galaxy per halo
andasharpcutoffatMmin,i.e. N(M) =1forM >Mminand
h i
0 otherwise, where Mmin is determined by matching the galaxy
number density. Thus,anydeviation fromthehalo massfunction
curvescould onlybecausedby theexistence ofsatellitegalaxies
andthesoftenedmasscutoffaroundMminforcentralgalaxies.For
highnumberdensitygalaxysamples,thedeviationarisesfromthe
satellitegalaxies,sincethesehaloshavelargesatellitefractions(see
Figure5).Forlownumber densitysamples, theprominent devia-
tion is mainly a result of the wide softened cutoff in the central
galaxy mean occupation function. Such awide softened cutoff is
a manifestation of the large scatter between central galaxy lumi-
nosityandhalomass(Zhengetal.2007).Itisinterestingthatthe Figure5. Satellitefractionfsatasafunctionoftheaveragenumberden-
deviationsatboththelow-andhigh-massenddrivetheMmin–n¯g sthiteylni¯tegraotfutrhee,assamlapbleelsle.dDiinffethreenfitgsuyrme.bOolusrremperaessuenretmtheenmtseoafstuhreemtherneetslfurmomi-
relationtowardsapowerlaw.
nositythresholdsamplesaredisplayedbytheblackstars.
The M1–n¯g relation also roughly follows a power law with
4.2 HODfortheColourSamples
a slightly shallower slope than the Mmin–n¯g relation. As a con-
sequence, thereisatrendthattheratioM1/Mmin decreaseswith To model the colour dependence of the 2PCFs for the CMASS
decreasingn¯g,albeitwithalargescatter,asshownintherightpanel galaxies in the luminosity bin of 22.2 < Mi < 21.6, we
ofFigure4.Thisresultisconsistentwithwhatwefindinthelumi- formthemean occupation function−of thecentral galax−iesinthis
nositydependenceofM1/Mmin. luminositybinasthedifferencebetween Ncen(M) oftheMi <
h i
21.6 sample and that of the M < 22.2 sample. Following
i
− −
Figure 5 presents the satellite fraction fsat as a function of Zehavietal. (2011), we fix the slope α of the satellite mean oc-
thenumberdensityn¯ fromourluminosity-thresholdsamplesand cupation function to be 1.56, the value from the fainter luminos-
g
thosefromtheliterature.Thesatellitefractionappearstofollowa ity threshold sample that dominates the number density of the
well-definedsequence,especiallytowardlownumberdensity,de- luminosity-bin sample. The cutoff mass of the mean occupation
cliningwithdecreasingnumberdensityandcanbewelldescribed functionofsatellitegalaxiesisalsosettobethesmallerofthetwo
byapowerlaw,fsat 0.1[n¯g/(10−3h3Mpc−3)]1/3. valuesfromthetwothresholdsamples.Thus,wemodelthe2PCF
≃
8 Guo etal.
Figure6.SimilartoFigure2,butforthedifferentcoloursamplesintheluminosityrange−22.2<Mi <−21.6andredshiftrange0.48<z<0.55.The
red,magenta,andgreenlinesareforthe‘reddest’,‘redseq’and‘green’samples,respectively.
ofeachcoloursubsamplewithonlyonefreeparameter,M′.Inthis resultof alargerfractionof thembeing satellites(inmassive ha-
1
simplemodel,theshapeofthecentralorsatellitemeanoccupation los),asshowninthetop-rightandbottom-leftpanels.Thesatellite
functions for different colour samples remains the same, and the fractionfsat variesfrom3.7percentinthe‘green’sampleto9.4
relativenormalizationbetweenthecentralandsatellitemeanoccu- percentinthe‘reddest’sample,consistentwiththetrendinMAIN
pationfunctionsisgovernedbyM′ andconstrainedbythesmall- sample(Zehavietal.2005,2011).Compared withtheluminosity
1
scale clustering. The overall normalizations of the mean occupa- dependenceofgalaxyclustering,wheremoreluminousgalaxiesre-
tion functions are determined from the relative number densities sideinmoremassivehalosandhavesmallersatellitefractions,the
ofthecoloursamplestothetotalnumberdensityinthisluminos- trendinthecolour dependence indicatesthatthesatellitefraction
ity bin. By construction, the sum of the mean galaxy occupation mostly affects the small-scale clustering, while halo mass scales
functionsofallcoloursamples(includingthe‘blue’galaxiesthat affecttheoverallclusteringamplitudes(especiallyonlargescales
arenotmodeledinthispaper)equalsthatofthefullluminosity-bin dominatedbythetwo-haloterm).
sample.
Figure 6 shows the modeling result (also in Table 2) of the Thetop-leftpanelofFigure6demonstratesthattheprojected
threecolour samples,inasimilarformat tothatof Figure2.Itis 2PCFofthe‘reddest’sampleonsmallscales(r <0.2h−1Mpc)
p
evidentfromthefigurethatreddergalaxieshaveahigherclustering isnotwellfittedbythesimpleHODmodel,leadingtoahighvalue
amplitude,especiallyonsmallscales(one-haloterm).Withinour ofbest-fittingχ2.Evenifweallowtheslopeαofthemeansatellite
model, thehigher clusteringamplitudeintheredder galaxiesisa occupationfunctiontovary,thesituationdoesnotsignificantlyim-
ModelingtheLuminosityand ColourDependentClustering 9
Table3.χ2/dofingeneralizedNFWmodels
Sample χ2/dof ∆AIC exp(∆AIC/2) fsat(percent)
Mi<−21.6 19.34/12 3.60 6.05 7.76±0.55
Mi<−21.8 24.75/12 -0.08 0.96 5.77±0.55
Mi<−22.0 18.93/12 1.74 2.39 4.66±0.56
‘green’ 17.84/14 1.03 1.67 4.11 0.53
±
‘redseq’ 11.76/16 0.14 1.07 6.50 0.61
±
‘reddest’ 28.94/16 -2.32 0.31 9.73 0.62
±
∆AIC=AICGNFW AICNFW
−
prove.Weexploretheimplicationforthesmall-scalegalaxydistri-
butioninsidehaloswithageneralizedNFWprofileinSection4.3.
A close inspection of thebest-fittingprojected 2PCFsinthe
top-left panel shows that the model predicts a narrower range of
clusteringamplitudethanthatfromthedataonscalesaboveafew
h−1Mpc.Sincethelarge-scaleamplitudeinthe2PCFismainlyde- Figure8.ThesamemeasurementsofFigure6,butnowwiththeGNFW
terminedbycentralgalaxies,thisresultimpliesthatthehalomass best-fittingmodels.Thered,magenta,andgreenlinesareforthe‘reddest’,
scalesforcentralgalaxiesintheluminositybincanvarywiththe ‘redseq’and‘green’samples,respectively.Forcomparison,wealsoshow
colourtosomedegree(inthesenseofhighermassscalesforred- thebest-fittingNFWmodelforthe‘reddest’sampleastheblackline.
dercentralgalaxies)inamannerthatisnotcapturedinoursimple
model.
data of the luminosity-threshold samples do not require a profile
differentthantheNFWprofile.
TopresentadetailedexaminationoftheeffectoftheGNFW
4.3 GeneralizedNFWprofile
profile,weshowinthetoppanelsofFigure7theratiosofw (r )
p p
In our fiducial HOD model, we assume that the spatial distribu- predictedfromthebest-fittingGNFWHODmodels(redlines)and
tionofsatellitegalaxiesinsidehalosfollowsthesameNFWprofile the measured wp(rp) (squares) to those of the NFW model pre-
as the dark matter. The best-fitting small-scale clustering ampli- dictions.BoththeNFWandGNFWmodelsfitthedatareasonably
tudeforthe‘reddest’sampleshowsdeviationsfromthedata(see well on scales of rp < 10h−1Mpc. Their predictions are sim-
Figure6),implyingapossibledepartureofthesatellitedistribution ilar on large scales and they only differ slightly on scales below
fromtheNFWprofile.Toexploresuchapossibility,wealsocon- 1h−1Mpc.Onlargerscales,themodelsappeartooverestimate
∼
siderageneralizedNFW(hereafterGNFW)profiletodescribethe thelargescalebiasforallsamples,whichmightbecausedbythe
distributionofsatellitegalaxiesinsidehalosbyallowingtwomore samplevariance.
free parameters in the HOD model, the normalization parameter Themarginalized joint distributionof the concentration nor-
c0 for thehalo concentration inEquation (13) andthe slopeγ in malization c0 and the slope γ for the three luminosity threshold
thedensityprofile(Watsonetal.2010,2012;vandenBoschetal. samples are shown in the bottom panels of Figure 7. The best-
2013), fittingmodelsaredisplayedastheredcrossesandtheNFWmodel
isrepresentedbythebluecircles.Whilethereisaweaktrendthat
cr γ cr 3−γ −1 theprofileformoreluminoussamplespreferstodeviatefromthe
ρ(r) 1+ (15)
∝"(cid:18)rvir(cid:19) (cid:18) rvir(cid:19) # tNhFeWconptrooufirsle.,WthaetsoNnFeWtalp.r(o2fi0l1e0i)sanstdilWl watistohninetthael.∼(2021σ2)rfianndgethoaft
where rvir is the virial radius of the halo. As a special case, the thedistributionsofsatelliteLRGsandsatellitesofluminousgalax-
NFWprofilehasc0 = 11(Equation13)andγ = 1.Anotherspe- iesinSDSSMAINsamplehavesignificantlysteeperinnerslopes
cialcaseisthesingularisothermalsphere(SIS)distribution,which thantheNFWprofile.FromFigure1inbothWatsonetal.(2010)
hasc0 0andγ =2. and Watsonetal. (2012), weinfer that constraining the deviation
→
WefirstapplytheGNFWmodel totheluminosity-threshold from the NFW profile requires accurate measurements on scales
samples. From Figure 2, the HOD model using the NFW profile r < 0.03h−1Mpc. However, at such small scales, the effect
p
canfitthe2PCFsof theluminosity-threshold samplesreasonably of photometric blending of close pairs may be important inclus-
well. By including the two additional free parameters, the best- teringmeasurements(Masjedietal.2006;Jiang,Hogg,&Blanton
fitting χ2 values only decrease slightly, as shown in Table 3. To 2012), which is not corrected for in our samples. Moreover, the
compare the goodness of fits between the generalized and origi- measurement errors at these scales are large in our samples. We
nal NFW profiles, we make use of the Akaike information crite- thereforecanonlyconcludethatourmeasurementsandmodeling
rion (AIC; Akaike 1974), defined as AIC = χ2 +2k for each resultsshownostrongdeviationsfromtheNFWprofileinthedis-
model,wherekisthenumberofHODparameters.Thedifference tributionofsatellitesinsidehalosforluminosity-thresholdsamples.
∆AIC AICGNFW AICNFW betweentheAICvaluesofthe WethenapplytheGNFWmodeltothefinecoloursamplesin
≡ −
GNFW and NFW models reveals that the model with the NFW Section4.2.SignificantimprovementovertheNFWprofilemodel
profileisexp(∆AIC/2)timesasprobableasthatwiththeGNFW isfoundinfittingthesmall-scale2PCFofthe‘reddest’sample,as
profile.AsshowninTable3,onlytheMi < 21.8sampleshowsa showninFigure8.Withoutthevariationinc0andγ,thepureNFW
−
marginalpreferencefortheGNFWprofile.Overall,theclustering modelcannot fitwellthesmall-scaleclusteringbyonlyadjusting
10 Guoet al.
Figure7.Toppanels:ratiosofwp(rp)predictedfromthebest-fittingGNFWHODmodelsandthosemeasuredfromthedatatothoseoftheNFWmodel
predictions.Theblacksquaresrepresentthedatameasurements,whiletheredlinesarethepredictionsoftheGNFWmodels.Bottompanels:marginalized
jointdistributionoftheconcentrationparameterc0andtheslopeγinthegeneralizedNFWmodelforthethreeluminositythresholdsamples.Thecontours
showthe68percentand95percentconfidencelevelsforthetwoparameters.Theredcrossesrepresentthebest-fittingGNFWHODmodels,whiletheblue
circlesarethepredictionsoftheNFWmodel.
thesatellitefractionortheslopeαofthesatellitemeanoccupation
function. The best-fitting χ2 value for this colour sample is also
reduced by adding thetwo freeparameters, as shown in Table3.
FromthedifferenceintheAIC,theNFWmodel ismuchlessfa-
vorable than the GNFW model for the ‘reddest’ sample, while it
stillprovidesreasonablefitstothe‘green’and‘redseq’samples.
The ratiosof the HOD model predictions of GNFW models
andthemeasuredw (r )tothoseoftheNFWmodelareshownin
p p
thetoppanelsofFigure9.Thesmall-scaleclusteringofthe‘red-
dest’sampleisbetterfitbytheGNFWmodel,whichconfirmsthe
importanceofthesmall-scale(r < 0.1h−1Mpc)measurements
p
indistinguishingtheNFWandGNFWmodels.Themarginalized
jointdistributionofc0 andγ ispresentedinthebottompanelsof
Figure9.Fromthe1σcontoursofFigure9,thereisavisibletrend
thatreddergalaxiesfavorsmallerc0 andlargerγ.TheSISprofile
(correspondingtoc0 0andγ = 2)seemstoprovidebetterfits
→
thantheNFWprofile,consistentwiththepreviousfindings(Grillo
2012;Watsonetal.2012).Thistrendisclearlymanifestedforthe
‘reddest’sample.
Figure 10. Probability distributions of the effective slope γeff at r =
0.1h−1Mpcinhalosof2 1014h−1M forthethreecoloursamples.
Sincec0andγarecorrelated,differentcombinationsofthem Theverticalblacklinedeno×tesγeff =1.37⊙,thevaluefortheNFWprofile
canleadtosimilarshapeoftheprofile.Abetterquantitytorepre- atthisradius.
sent theshape ofthedensity profileistheeffectiveslopedefined