Cosmology and the Halo Occupation Distribution from Small-Scale Galaxy Clustering in the Sloan Digital Sky Survey

0.23 MB·English

by  Kevork Abazajian

LA-UR 04-1121
FNAL-PUB-04-137-A
astro-ph/0408003
DRAFTVERSIONFEBRUARY2,2008
PreprinttypesetusingLATEXstyleemulateapjv.6/22/04
COSMOLOGYANDTHEHALOOCCUPATIONDISTRIBUTIONFROMSMALL-SCALE
GALAXYCLUSTERINGINTHESLOANDIGITALSKYSURVEY
KEVORKABAZAJIAN1,2,ZHENGZHENG3,IDITZEHAVI4,DAVIDH.WEINBERG3,JOSHUAA.FRIEMAN2,5,
ANDREASA.BERLIND6,MICHAELR.BLANTON6,NETAA.BAHCALL7,J.BRINKMANN8,DONALDP.SCHNEIDER9,
MAXTEGMARK10,11
DraftversionFebruary2,2008
ABSTRACT
5 Weusetheprojectedcorrelationfunctionwp(rp)ofavolume-limitedsubsampleoftheSloanDigitalSkySur-
0 vey(SDSS)maingalaxyredshiftcataloguetomeasurethehalooccupationdistribution(HOD)ofthegalaxies
0 ofthe sample. Simultaneously,we allow thecosmologytovarywithincosmologicalconstraintsimposedby
2 cosmicmicrowavebackgroundexperimentsinaΛCDMmodel.Wefindthatcombiningw (r )forthissample
p p
alonewiththeobservationsbyWMAP,ACBAR,CBIandVSAcanprovideoneofthemostprecisetechniques
n
availabletomeasurecosmologicalparameters. Foraminimalflatsix-parameterΛCDMmodelwithanHOD
a
J with three free parameters, we find Ωm =0.278-+00..002277, σ8 =0.812-+00..002278, and H0 =69.8-+22..66kms- 1Mpc- 1; these
errorsaresignificantlysmallerthanfromCMBaloneandsimilar tothoseobtainedbycombiningCMBwith
8
the large-scale galaxy power spectrum assuming scale-independent bias. The corresponding HOD parame-
1
tersdescribingtheminimumhalomassandthenormalizationandcut-offofthesatellitemeanoccupationare
2 Mmin=(3.03+- 00..3366)×1012h- 1M⊙,M1=(4.58-+00..6600)×1013h- 1M⊙,andκ=4.44-+00..6591.TheseHODparametersthus
v havesmallfractionaluncertaintywhencosmologicalparametersareallowedtovarywithintherangepermit-
3 tedbythedata. WhenmoreparametersareaddedtotheHOD model,theerrorbarsontheHOD parameters
0 increase because of degeneracies, but the error bars on the cosmologicalparametersdo not increase greatly.
0 Similarmodelingforothergalaxysamplescouldreducethestatisticalerrorsontheseresults,whilemorethor-
8 oughinvestigationsofthecosmologydependenceofnonlinearhalobiasandhalomassfunctionsareneededto
0 eliminateremainingsystematicuncertainties,whichmaybecomparabletostatisticaluncertainties.
4
Subjectheadings:cosmology:observations—cosmology:theory—galaxies:formation—galaxies:halos
0
/
h
p 1. INTRODUCTION 1997;Bensonetal.2000;Berlindetal. 2003;Kravtsovetal.
- 2004). GivencosmologicalparametersandaspecifiedHOD,
o Over the last several years, halo occupation models of
onecancalculateanygalaxyclusteringstatistic,onanyscale,
r galaxy bias have led to substantial progress in characteriz-
t either by populating the halos of N-body simulations (e.g.,
s ingtherelationbetweenthedistributionsofgalaxiesanddark
Jingetal. 1998, 2002; Berlind&Weinberg 2002) or by us-
a matter. Gravitationalclusteringofthedarkmatterdetermines
: the population of virialized dark matter halos, with essen- inganincreasinglypowerfularrayofanalyticapproximations
v
(e.g.,Peacock&Smith2000;Seljak2000;Scoccimarroetal.
i tiallynodependenceonthemorecomplexphysicsofthesub-
X dominantbaryoncomponent. Galaxy formationphysics de- 2001;Takada&Jain2003;seeCooray&Sheth2002forare-
centreview).
r terminesthehalooccupationdistribution(HOD),whichspec-
a The2dFGalaxyRedshiftSurvey(2dFGRS)(Collessetal.
ifiestheprobabilityP(N M)thatahaloofvirialmassMcon-
| 2003) and the Sloan Digital Sky Survey (SDSS) (Yorketal.
tains N galaxies of a given type, together with any spatial
2000; Abazajianetal. 2003, 2004b) allow galaxy clustering
andvelocitybiasesofgalaxieswithinhalos(Kauffmannetal.
measurementsofunprecedentedprecisionanddetail,making
1TheoreticalDivision,MSB285,LosAlamosNationalLaboratory,Los them ideal data sets for this kind of modeling. Zehavietal.
Alamos,NM87545 (2004a) (hereafter Z04a) show that the projected correlation
2TheoreticalAstrophysicsGroup,FermiNationalAcceleratorLaboratory, functionw (r )ofluminous(M <- 21)SDSSgalaxiesex-
p p 0.1r
P.O.Box500,Batavia,IL60510 hibits a statistically significant departure from a power law,
3DepartmentofAstronomy,OhioStateUniversity,Columbus,OH43210
and that a 2-parameter HOD model applied to the prevail-
4StewardObservatory,UniversityofArizona,933N.CherryAve.,Tuc-
son,AZ85721 ing ΛCDM (cold dark matter with a cosmologicalconstant)
5KavliInstituteforCosmologicalPhysics,DepartmentofAstronomyand cosmologyaccountsnaturallyforthisdeparture,reproducing
Astrophysics, TheUniversity ofChicago, 5640 S.Ellis Ave., Chicago, IL theobservedw (r ). Here,M istheabsolutemagnitudein
60637 p p 0.1r
6CenterforCosmologyandParticlePhysics,DepartmentofPhysics,New theredshiftedrband,withobservedmagnitudesK-corrected
YorkUniversity,4WashingtonPlace,NewYork,NY10003 to rest frame magnitudes for the SDSS bands blueshifted
7ApachePointObservatory,P.O.Box59,Sunspot,NM88349 by z=0.1, the median redshift of the survey (Blantonetal.
8DepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton, 2003a). Magliocchetti&Porciani(2003)haveappliedasim-
NJ08544
ilar type of analysis to w (r ) for a fixed cosmology in the
9 Department of Astronomy and Astrophysics, The Pennsylvania State p p
University,UniversityPark,PA16802 2dFGRS. The halo model plus HOD has also been used to
10 Department ofPhysics, University ofPennsylvania, Philadelphia, PA successfully describe the clustering of Lyman-break galax-
19104 ies (Porciani&Giavalisco 2002), high-redshift red galaxies
11 Department ofPhysics, Massachusetts Institute ofTechnology, Cam-
(Zheng 2004), as well as 2dF quasars (Porcianietal. 2004).
bridge,MA02139
2 Abazajianetal.
Recently, it was shown thatlarge-scale overdensitiesare not isdoneinFourierspace,
correlated with galaxy color or star formation history at a 1 ∞ sinkr
fixedsmall-scaleoverdensity,supportingtheHODansatzthat ξ (r)= P2h(k)k2 dk, (2)
a galaxy’s properties are related only to the host halo mass 2h 2π2Z0 gg kr
andnotthelarge-scaleenvironment(Blantonetal.2004a). where
In this paper, we go beyond the Z04a analysis by bring-
ing in additional cosmological constraints from cosmic mi- 1 Mmax dn 2
P2h(k)=PNL(k) dM N(M) b (M)y (k,M) .
crowavebackground(CMB)measurementsandallowingthe gg mm (cid:20)n¯gZ0 dMh i h g (cid:21)
HOD and cosmological parameters to vary simultaneously. (3)
Thisinvestigationcomplementsthat ofZehavietal. (2004b) ThemeanoccupationofhalosofmassMis N(M) ,y (k,M)
g
(hereafterZ04b), who examine the luminosityand color de- isthenormalizedFouriertransformofthegahlaxydiistribution
pendenceof galaxyHOD parametersfor a fixed cosmology. profilein a haloofmassM. We approximatehaloexclusion
It also complements analyses that combine CMB data with effects in two-halo correlation separations of r by choosing
thelarge-scalepowerspectrummeasurementsfromthe2dF- theupperlimitoftheintegralinEq.(3)suchthatM isthe
max
GRSor SDSS(e.g.,Percivaletal.2001;Spergeletal. 2003; massofahalowithvirialradiusr/2,asincorporatedinZheng
Tegmarketal. 2004). Such analyses use linear perturbation (2004), Z04a, and Z04b. The importance of the nonlinear
theory to predict the dark matter power spectrum, and they matterpowerspectrumandhaloexclusioninaccuratelymod-
assume that galaxy bias is scale-independent in the linear elingthetwo-halogalaxycorrelationfunctionwasalsofound
regime. It also complements HOD and cosmological pa- byMagliocchetti&Porciani(2003)andWangetal.(2004).
rameter determinationapproachesusing galaxy-galaxylens- In order to accurately include the dependence of the halo
ing in the SDSS (Seljaketal. 2004a) and their combination modeling of galaxy clustering for a varying cosmology, we
with Lyman-α forest clustering in the SDSS quasar sample includecosmologicallygeneral(withinΛCDM)formsofthe
(Seljaketal. 2004b). Our analysis draws on data that ex- nonlinearmatterpowerspectrum,halobias, halomassfunc-
tendintothehighlynon-linearregime,andinplaceofscale- tion,anddarkmatterhaloconcentration. Weusethenonlin-
independentbiasitadoptsaparameterizedformoftheHOD earmatterspectrumPNL(k)ofSmithetal.’s(2003)halofit
mm
motivatedbytheoreticalstudiesofgalaxyformation. code, modified to utilize a numerically calculated transfer
function from the Code for Anisotropies in the Microwave
2. THEORY Background (CAMB, Lewisetal. 2000), based on CMB-
FAST (Seljak&Zaldarriaga 1996). We use halo bias fac-
We explore spatially flat, “vanilla” cosmological models
that have six parameters, (Ω h2,Ω h2,Θ ,ln(A),n,τ), where tors bh(M) determined in the high-resolution simulations of
b c s
Ω and Ω are fractions of the critical density in baryons Seljak&Warren (2004), along with its given cosmological
anbd cold dcark matter; Θ is the angular acoustic peak scale dependence, which provides a better fit (lower χ2) to our
s
observational data than halo bias models based on the peak
of the CMB, a useful proxy for the Hubble parameter, H =
0
100hkms- 1Mpc- 1;Aandnaretheamplitudeandtiltofthe background split (Sheth&Tormen 1999; Shethetal. 2001).
WeusetheJenkinsetal.(2001)sphericaloverdensityof180
primordial scalar fluctuations; τ is the optical depth due to
[SO(180),Eq.(B3)]halomassfunction,andincludeinitsin-
reionization.
terpretationofthedefinitionofhalomassthevariationinthe
In the halo model of galaxy clustering, the two-pointcor-
virialoverdensitywithcosmology(Bryan&Norman1998)
relation function of galaxies is composed of two parts, the
1-halo term and the 2-halo term, ξ(r) = 1+ξ1h(r)+ξ2h(r), 18π2+82x- 39x2
whichrepresentcontributionsbygalaxypairsfromsameha- ∆v= 1+x , (4)
los and different halos which dominate at small scales and
and its effect in relating the Jenkinsetal. (2001) mass
largescales,respectively.Here,thecorrelationfunctioniscal-
culatedattheeffectiveredshiftofourobservedSDSSsample function to varying cosmologies (see, e.g., White 2001;
Hu&Kravtsov 2003). Here, x Ω (z)- 1. The variation
at z=0.1, which is a nontrivialmodification since errorson m
≡
ofthevirialoverdensityalsochangesthehaloexclusionscale
theamplitudeofthepowerspectrumatsmallscales(i.e.,σ )
8
ofR (M )usedinEq.(3). SinceourluminousSDSSsub-
arefoundbecomparableto thegrowthfactorshiftatz=0.1 vir max
(seebelow). Thecalculationofthe1-halotermisstraightfor- samplepopulateshalosofM>1012M⊙,thebreakdownofthe
ward(e.g.,Berlind&Weinberg2002): Jenkinsetal. (2001) mass-functionfitatM.1010M⊙ isnot
important.
1 ∞ dn N(N- 1) We assume thattheaveragespatialdistributionof satellite
1+ξ1h(r)= 2πr2n¯2Z dMh 2 iM galaxieswithinahalofollowsaNavarro-Frenk-White(NFW)
g 0 densityprofileofthedarkmatter(Navarroetal.1996),moti-
1 F′ r dM, (1) vatedbyhydrodynamicsimulationresults(Whiteetal.2001;
×2R (M) (cid:18)2R (cid:19) Berlindetal. 2003) and N-body simulation galaxy cluster-
vir vir
ingpredictionswithhalospopulatedbysemi-analyticmodels
where n¯g is the mean number density of galaxies calculated (Kauffmannetal.1997;Kauffmannetal.1999;Bensonetal.
fromtheHODandhalomodel,dn/dMisthehalomassfunc- 2000;Somervilleetal.2001).However,asatest,wedropthis
tion, N(N- 1) M/2istheaveragenumberofgalaxypairsin assumption of no spatial bias within halos between galaxies
h i
ahaloofmassM,andF(r/2Rvir)isthecumulativeradialdis- and dark matter and find it is not important (see §4 below).
tribution of galaxy pairs (Berlind&Weinberg 2002; Zheng Inthecaseofnospatialbias,eachhaloisassumedtohavea
2004). cosmologically-dependentconcentration
For the 2-halo term term, in order to reach the accuracy
β
neededtomodeltheSDSSdata,weincludethenonlinearevo- M
c=c , (5)
lutionofmatterclusteringandthehaloexclusioneffect. This 0(cid:18)M∗(cid:19)
CosmologyandtheHODfromSmall-ScaleClustering 3
where
Ω - 0.35 n - 1.6
c =11 m eff , (6)
0 (cid:18)0.3(cid:19) (cid:16)- 1.7(cid:17)
β=- 0.05, (7)
as foundin fits to numericalresultsfor varyingcosmologies
byHuffenberger&Seljak(2003). Here,
dlnP (k)
n lin , (8)
eff
≡ dlnk (cid:12)k∗
(cid:12)
wherek isthenonlinearscalesuchth(cid:12)at∆2 (k )=1,Misthe
∗ lin ∗
virial mass of the halo and M is the nonlinear mass scale.
∗
There is a scatter about any mean concentration value and
thiscouldchangethepredictionoftheshapeofagivenhalo.
However,aswedescribebelow,ourresultsarelargelyinsen-
sitive to the exact form of the concentration of the galaxies - 2F1IGg.al1a.x—iesSfhroowmnthaereStDheSSprLojSeSctesdacmoprlreela1ti2on(pfouinnctstiownitwhp1(σrpd)ioafgoMn0a.1lre<r-
withrespecttothedarkmatter.
rors)andthebestfitthreeparameterHODmodel(solid). Onlypointswith
Our HOD parameterization for a luminosity-threshold r<20h- 1Mpcareusedinthefit. Alsoshownarepredictedwp(rp)models
galaxy sample (M0.1r < - 21 in this paper) is motivated by withΩc (dotted)andσ8 (dashed)at±3σ fromtheirbestfitvalues. HOD
results of substructures from high-resolution dissipationless parametersandothercosmologicalparametersareheldfixed. Asseenhere,
simulationsofKravtsovetal.(2004). TheHODhasasimple thesensitivity toσ8 andΩc comesfromboththeamplitudeandcombined
shapeofthe1-haloand2-haloregimesofwp(rp).
form when separated into centraland satellite galaxies. The
meanoccupationnumberof centralgalaxiesismodeledasa
stepfunctionatsomeminimummass,smoothedbyacomple- galaxiesover2500deg2ofsky(Blantonetal.2004b). Weuse
mentaryerrorfunctionsuchthat the M <- 21 sample with the full w (r ) data covariance
0.1r p p
matrixfromthejackknifeestimatesofZ04b.Thereare26,015
1 ln(M /M)
hNcen(M)i= 2Erfc(cid:20) √2mσin (cid:21), (9) galaxiesintheM0.1r<- 21sample.
cen The observed projected correlation function is obtained
to account for scatter in the relation between the adopted fromthe2-dcorrelationfunctionξ(rp,π)byintegratingalong
magnitudelimit andthe halo mass limit(Zhengetal. 2004). thelineofsightinredshiftspace:
(Note that the number of central galaxies is always Ncen =0 πmax
orN =1bydefinition.) Theoccupationnumberofsatellite w (r )=2 ξ(r ,π)dπ, (11)
cen p p p
galaxiesis wellapproximatedbya Poisson distributionwith Z0
themeanfollowingapowerlaw, wherer andπ areseparationstransverseandparalleltothe
p
M- κM α lineofsight.Weadoptπmax=40h- 1Mpc(inthemeasurement
N (M) = min , (10) and modeling), large enoughto include nearly all correlated
sat
h i (cid:20) M (cid:21)
1 pairsandthusminimizeredshift-spacedistortionwhilekeep-
where we introduce a smooth cut-off of the average satel- ing backgroundnoise from uncorrelated pairs low. Because
lite number at a multiple κ 1 of the minimum halo mass. oursample is volume-limited, we are measuringthe cluster-
Guzik&Seljak(2002)andB≥erlindetal.(2003),usingsemi- ingofa homogeneouspopulationofgalaxiesthroughoutthe
analyticmodelcalculations,andKravtsovetal.(2004),using surveyvolume,whichgreatlysimplifiesHODmodeling.Fur-
high resolution N-body simulations, found α 1. The gen- therdetailsofthesampleandmeasurementaregiveninZ04b.
eralHODaboveischaracterizedbyfivequanti≈ties:Mmin,M1, In our analysis, we use 11 wp(rp) data points in the range
σcen, κ, and α, and we refer to this as the 5p model. It pro- 0.1h- 1Mpc<rp <20h- 1Mpc, sufficiently below the projec-
vides an excellent fit to predictions of semi-analytic models tion scale πmax to avoid contaminationof redshiftspace dis-
and hydrodynamic simulations (Zhengetal. 2004), in addi- tortions, though we have found that including points up to
tiontodescribingsubhalopopulationsinN-bodysimulations r 40h- 1Mpc, which have low statistical weight, does not
≈
(Kravtsovetal.2004). alterourresults.
We also require our models to reproduce the measured
3. OBSERVATIONS mean comoving space density of our sample, n¯obs =1.17
g ×
The SDSS uses a suite of specialized instruments and 10- 3h3Mpc- 3. Thisquantityhasanuncertaintyduetosample
datareductionpipelines(Gunnetal.1998;Hoggetal.2001;
Pieretal. 2003; Stoughtonetal. 2002) to image the sky in variancethatcanbewrittenasσn¯/n¯ogbs= δg2 ,where δg2
q
five passbands (Fukugitaetal. 1996; Smithetal. 2002) and isthevarianceofthegalaxyoverdensity.We(cid:10)est(cid:11)imate δ2(cid:10) by(cid:11)
g
obtainspectraofwelldefinedsamplesofgalaxiesandquasars integratingthetwo-pointcorrelationfunctionoverthe(cid:10)volu(cid:11)me
(Eisensteinetal. 2001; Richardsetal. 2002; Straussetal. of the SDSS M <- 21 sample. To compute this integral
0.1r
2002;Blantonetal.2003b). Forouranalysis,weuseZ04b’s we generate a large number of independentrandom pairs of
measurementof the projected correlation function w (r ) of pointswithinthesample volumeandsumξ(r) overallthese
p p
a volume-limitedsample of galaxies with M <- 21. This pairs. We use the Z04bcorrelationfunctionand extendit to
0.1r
sample is in turn selected from a well characterized subset largerscales withthe linear theorycorrelationfunctionmul-
ofthe maingalaxysample asofJuly, 2002, knownasLarge tiplied by b2, where b=1.4 is the large-scalebias factor for
ScaleStructure(LSS)sample12,whichincludes 200,000 M <- 21 galaxies. We vary the number of random pairs
0.1r
∼
4 Abazajianetal.
109M M 5 1013M
⊙ min ⊙
1010M ≤ M ≤4×1014M
⊙ 1 ⊙
≤ ≤ ×
1 κ 30
≤ ≤
0 σ 10
cen
≤ ≤
0.2 α 2.5. (13)
≤ ≤
Here, A is related to the amplitude of curvature fluctuations
at horizon crossing, ∆R2 = 2.95 10- 9A at the scale k =
0.05Mpc. Theangula|rac|ousticpea×kscale Θ is theratio of
s
thesoundhorizonatlastscatteringtothatoftheangulardiam-
eterdistancetothesurfaceoflastscattering(Kosowskyetal.
2002).
To measure the likelihood space allowed by the data, we
use a Metropolis MCMC method with a modified version
of the Lewis&Bridle (2002) CosmoMC code. We use the
WMAP team’s code to calculate the WMAP first-year ob-
servations’ likelihood, and CosmoMC to calculate that for
ACBAR, CBI and VSA. After burn-in, the chains typically
sample 105 points, and convergenceand likelihoodstatistics
arecalculatedfromthese.Sinceitisnotknownaprioriwhich
HODparametersaremostconstrainedbythew (r )measure-
p p
ment, we use the Akaike and Bayesian Information Criteria
(AICandBIC)todeterminewhichparametersarestatistically
relevant to describing w (r ) (Akaike 1974; Schwarz 1978;
p p
seealsoLiddle2004).Moreparametersmightwellbeneeded
once we have more data to constrain the HOD, but w (r )
FIG.2.—Shownarethemarginalizedposteriorlikelihoodsforthecosmo- p p
logicalparametersdeterminedfromCMB+wp(rp)[M0.1r<- 21]withathree- alonedoesn’tprovideenoughinformationtodemandit.
parameterHODinsolidblue,thatforCMB+SDSS3DPg(k)inreddashed, Likelihood analyses were performed for several cases
andthatfortheCMBaloneinyellow(gray)shaded. wheresomeparameterswerekeptfreeandotherswerefixed
to a physical limit, i.e. where the scatter in the mass-
luminosityrelationisunimportant(σ 0),thecut-offscale
cen
used and find that the integral converges at 105 pairs. Us- of the satellite galaxiesis exactly thato≡f the minimummass
ingtwomillionpairs,wefindthatthenumberdensityuncer- Mmin(κ 1),ortoavalue(α 1)predictedinthenumerical
tainty due to sample variance is σn¯/n¯ogbs =0.0377. There is simulatio≡nsandsemi-analytic≡modelsof satellite halo distri-
also a shot noise Poisson uncertainty in the number density butionsinBerlindetal.(2003)andKravtsovetal.(2004). If
that, for this number of galaxies, is 0.0062. We add these weadoptallthreeoftheseconstraintsandallowonlytwopa-
twocomponentsinquadraturetoobtainatotaluncertaintyof rameters,M andM ,tovarytofitw (r )andn¯obs,thenwe
min 1 p p g
σn¯tot/n¯ogbs=0.0382,andtherefore obtaina poorfit. Thismodelis aninadequatedescriptionof
thedataaccordingtotheinformationcriteria(∆BIC=7.2and
n¯obs=(1.17 0.05) 10- 3h3Mpc- 3. (12) ∆AIC=12.5) relative to the three-parameterM , M , and
g ± × min 1
κ model(3p). We also investigated a four parameter model
4. RESULTS (4p),varyingM , M , κ, andσ withα 1, aswellasa
min 1 cen
≡
ForagivencosmologyandHODparameterchoice,weuse fiveparametermodel(5p)varyingallparametersinthisHOD.
thepredictedw (r )tocalculatethelikelihoodtoobservethe Relative to the 3p model, the 4p and 5p models introduce
p p
M <- 21sample’sw (r )andn¯obs. We combinethislike- newparametersthatarenotjustifiedbytheinformationcrite-
0.1r p p g ria(∆BIC>6,cf. Table1),sincethesemodelsaddfreedom
lihood with that for the model’s prediction for the cosmic
butyieldonlyaverysmallreductioninχ2.
microwave background anisotropy temperature correlation
Toassess theimportanceofone aspectof thehalo model-
andtemperature-polarizationcross-correlationtoproducethe
ing,weperformedatestonthe3pmodelwherebytheNFW
WMAP (first year), ACBAR (ℓ > 800), CBI (600 < ℓ <
concentrationc [Eq. (7)] of dark matter is replaced by that
2000)andVSA(ℓ>600)observations(Hinshawetal.2003; 0
Verdeetal. 2003; Kogutetal. 2003; Dickinsonetal. 2004; forthegalaxies,cgal,andisalsoleftfreeintheMCMCwithin
0
Readheadetal. 2004). We vary the six parameters for the 0.01<cgal<200andindependentofthedarkmatterconcen-
“vanilla”ΛCDMcosmologicalmodelplusthefiveHODpa- tration o0f the halos. We find that the derived cosmological
rameters:p=(Ωbh2,Ωch2,Θs,ln(A),n,τ,Mmin,M1,κ,σcen,α). parameters and their uncertainties remain nearly unchanged
The ranges allowed in our Markov Chain Monte Carlo from a model with no spatial bias, and the constraints on
(MCMC)samplingofparametersarechosentoavoidanyar- the galaxy concentration are consistent with no spatial bias:
tificialcut-offofthelikelihoodspaceandare cgal=11.1+0.7. ThemarginalizedvaluesoftheHODparame-
0 - 5.3
0.005 Ω h2 0.1 tersM andM remainunchangedwithvaryingcgal,though
0.01≤Ωbh2≤0.99 the errmoinr on the1cut-off scale of the satellite gala0xies κ in-
c
0.005≤ Θ ≤0.1 creases (κ=4.71+0.60). This increase is expected since it is
- 0.68≤ln(As)≤0.62 precisely the cent-r1a.4l1distribution of satellite galaxies that is
≤ ≤
0.5 n 1.5 positivelycorrelatedto theone-halogalaxydistributioncon-
≤ ≤
0.01 τ 0.8 centration,withacorrelationcoefficientofr=0.72.Asalarge
≤ ≤
CosmologyandtheHODfromSmall-ScaleClustering 5
TABLE1
COSMOLOGICALPLUSHODPARAMETERS,MARGINALIZEDCONSTRAINTSWITH68.3%C.L.ERRORS.
Parameter CMB+wp(rp)3p+Pg(k) CMB+wp(rp)3p CMB+wp(rp)4p CMB+wp(rp)5p CMB+Pg(k) CMB
A 0.731+- 00..005573 0.749+- 00..006538 0.768+- 00..007648 0.803+- 00..100933 0.747+- 00..007771 0.79+- 00..1152
n 0.947+- 00..001178 0.953+- 00..001199 0.958+- 00..002201 0.968+- 00..002277 0.956+- 00..002201 0.974+- 00..003376
τ 0.100+- 00..001271 0.115+- 00..001293 0.131+- 00..002217 0.155+- 00..002367 0.105+- 00..001278 0.158+- 00..009834
h 0.687+- 00..002233 0.698+- 00..002266 0.708+- 00..002289 0.721+- 00..003346 0.697+- 00..002288 0.755+- 00..005598
Ωch2 0.1148+- 00..00003399 0.1126+- 00..00004434 0.1107+- 00..00005409 0.1088+- 00..00005587 0.1176+- 00..00006699 0.104+- 00..001133
Ωbh2 0.02234+- 00..0000007890 0.02247+- 00..0000008844 0.02263+- 00..0000008867 0.0229+- 00..00001100 0.0227+- 00..00000099 0.0230+- 00..00001133
M1[1013h- 1M⊙] 4.79+- 00..6633 4.58+- 00..6600 4.52+- 00..6633 3.31+- 11..6614 - -
Mmin[1012h- 1M⊙] 3.23+- 00..3365 3.03+- 00..3366 3.26+- 00..4468 3.08+- 00..4591 - -
κ 4.44+- 00..4683 4.44+- 00..5619 3.85+- 00..4556 6.31+- 11..3934 - -
σcen 0 0 0.41+- 00..1118 0.39+- 00..1108 - -
α 1 1 1 0.83+- 00..2223 - -
bg∗ 1.48+- 00..008888 1.47+- 00..009933 1.43+- 00..007777 1.40+- 00..009955 - -
Ωm 0.292+- 00..002254 0.278+- 00..002277 0.268+- 00..002299 0.256+- 00..003343 0.291+- 00..003344 0.231+- 00..005546
σ8 0.809+- 00..002288 0.812+- 00..002287 0.816+- 00..003300 0.829+- 00..003399 0.834+- 00..004590 0.802-+00..007723
χ2 /DOF 1483.4/1391 1458.5/1373 1458.4/1372 1458.1/1371 1477.2/1383 1452.5/1365
eff
AIC 1503.4 1476.5 1478.4 1480.1 1491.2 1464.5
BIC 1555.9 1523.5 1530.7 1537.7 1527.9 1495.8
NOTE.—∗Thelargescalegalaxybias(k→0).
FIG. 3.—Plottedarethe∆χ2<1rangefromthebestfitfortheHOD
dinragwrenefnro(mmetdhieumMCgrMayC),cahnadin5spfomrothdeel3ipnmyeoldloewlin(lirgehdt(gdraaryk).gray),4pmodel inFσI8G.vs4..Ω—mSfhoorwthneaWreMthAePm+AarCgBinAalRiz+eCdB6I8+.V3%SAan(CdM95B.4)%datCa.Lal.ocnoen(tgoruarys
shaded),fromtheCMB+SDSS3DPg(k)(orange/light-graylines)andCMB
+SDSSwp(rp)(blue/dark-graylines)fromthe3pHODanalysispresented
here.
cgal makesthedistributionofgalaxiesinsidehalosmorecon-
0
centrated, to maintainthe small-scale clustering,κ increases
regimesandthereforetheoverallshapeofw (r ).
toallowrelativelymoregalaxiestobeputinhaloswithlarger p p
Thecosmologicalparameters’marginalizedposteriorlike-
virialradiiandlowerconcentrations.
lihoodsforthe3pmodelareshowninFigure2. Alsoshown
Figure 1 illustrates the way that w (r ) constrains cosmo-
p p
forcomparisonarethemarginalizedlikelihoodsfortheCMB
logical parameters. Data points show the Z04b measure-
plusSDSS3DP(k)[updatedfromTegmarketal.(2004)with
ments,andthesolidlineshowsthepredictionofthebest-fit3p g
new CMB results], and that from the CMB data alone. We
model.Dashedcurvesshowthepredictionofthew (r )after
p p
also combine the CMB+w (r )(3p) measurement with the
σ is perturbed by 3σ relative to its best-fit value given in p p
8
± SDSS3DP (k)forajointconstraintoncosmologicalparam-
column2ofTable1, withallothercosmologicalparameters g
eters. SincetheP (k)datapointsincludedintheanalysisare
(andthereforetheshapeofthelinearmatterpowerspectrum) g
aswellastheHODparametersheldfixed.Dottedcurvesshow at wavelengthsλ=2π/k>30h- 1Mpc, the informationthey
thepredictionofwp(rp)afterchangingΩc,andthustheshape containislargelyindependentofthatinthewp(rp)datapoints
ofthetransferfunctioninthematterpowerspectrum,by 3σ, atr <20h- 1Mpc. Allparameters’bestfitvaluesanderrors
p
±
withall othercosmologicalandHOD parametersfixed. The are listed in Table 1. The resulting range of the HOD mea-
strength of the constraints derived from w (r ) stems from suredforallmodelshereareshowninFigure3.
p p
the combined relative dependence of the 1-halo and 2-halo Two-dimensionalcontoursofΩ andσ areshowninFig-
m 8
6 Abazajianetal.
thatthehalomassfunction,halobiasandhaloprofileisthatof
thecolddarkmatteralonesinceneutrinoclusteringisavery
smalleffectonthesequantities(Abazajianetal.2004a). The
presenceofmassiveneutrinosisconstrainedtom <0.27eV
ν
(95%C.L.)foreachof3neutrinoswithdegeneratemass. The
statisticalerrorsfromouranalysisoftheCMBplusthisw (r )
p p
measurement on dn/dlnk and m are comparable to those
ν
from other cosmological parameter analyses, being smaller
thanthosefromtheshapeoftheSDSS3DP (k)plusWMAP
g
(Tegmarketal. 2004), comparable to the WMAP plus 2dF-
GRS 3D P(k) plus modeled bias constraints (Spergeletal.
g
2003),butnotasstringentasthosefrommodelingthegalaxy
bias in the SDSS from galaxy-galaxylensing and clustering
oftheLyman-αforestintheSDSS(Seljaketal.2004a,b).
5. DISCUSSION
Theremaininguncertaintiesincosmologicalparametersin-
troduce relatively little uncertainty in the HOD parameters,
i.e., we now know the underlyingcosmologywith sufficient
precisiontopindowntherelationbetweengalaxiesandmass.
FIG.5.—Plottedarethe68.3%and95.4%C.L.contoursforthemarginal-
ized likelihoods for Ωm vs. Mmin. The strong degeneracy (correlation The strongest expected degeneracy is between the value of
of r=0.92) roughly follows Ωm ∝Mmin, as expected (Zhengetal. 2002; Ωm and the mass scale parameters Mmin and M1, since one
Rozoetal.2004)
can compensate a uniform increase in halo masses by sim-
ply shifting galaxies into more massive halos (Zhengetal.
2002;Rozoetal. 2004). TheerrorcontoursforΩ vs.M
m min
ure 4 and are comparedto those obtained using CMB alone are shown in Figure 5. The degeneracy between these pa-
orCMB + P(k). The anticorrelationof Ω and σ fromthe rametersis strong,with a correlationof r=0.96. While this
g m 8
w (r )constraintseeninFig.4arisesfromtheanticorrelated degeneracy would cause large uncertainties in the values of
p p
degeneracy in these parameters in the one-halo component M andΩ ifweusedthegalaxyclusteringdataalone,the
min m
duetoitsdependenceonthehalomassfunctionwhichneeds combination of CMB and w (r ) data constrains Ω fairly
p p m
tomaintainitsamplitudeathighhalomasses,andtheOmega tightly, leaving limited room to vary the mass scale param-
m
and σ anticorrelationin the two-halocomponentdue to the eters. Incorporating SDSS clustering measures that are di-
8
amplitude-shape degeneracy of the dark matter power spec- rectlysensitivetohalomasses, suchasredshift-spacedistor-
trum(ordarkmattercorrelationfunction). tions(Zehavietal.2002)andgalaxy-galaxylensingmeasure-
ImportantresultstonotefromtheFiguresandTablearethe ments(Sheldonetal. 2004; Seljaketal. 2004a), may further
following.CosmologicalconstraintsobtainedusingCMBand improvetheΩ constraints.
m
w (r ) are substantially tighter than those from CMB alone, As discussed by Berlind&Weinberg (2002), the galaxy
p p
and theyare similar in valueand tightnessto those obtained correlationfunctionplacesimportantconstraintsonHODpa-
from CMB + P (k) despite the introduction of new parame- rameters, but it still allows tradeoffs between different fea-
g
ters to represent the HOD. The σ constraints using w (r ) tures of P(N M) and (to a lesser degree) between P(N M)
8 p p
| |
aretighterthanthoseusingP(k);notethatthelatterestimate and the assumed spatial bias of galaxies within halos. Ad-
g
hasdroppedrelativetothatofTegmarketal.(2004)because ditional clustering statistics such as the group multiplicity
ofthe smallerscale CMB data. Ifwe incorporateP (k) con- function, higher order correlation functions, and void prob-
g
straints in addition to w (r ), then parameter values change abilities impose complementary constraints that can break
p p
by less than 1- σ and error bars improveslightly. Our cos- these degeneracies. Our analysis should thus be seen as a
mological parameter results also agree, within errors, with first step in a broader program of combining galaxy clus-
therecentresultsfromSDSSgalaxybiasandLyman-αforest tering measurements from the SDSS and other surveyswith
(Seljaketal. 2004b). The HOD parametersare partially de- other cosmological observables to derive simultaneous con-
generateamongthemselves,soaddingparameterstotheHOD straintsoncosmologicalparametersandthegalaxyHOD[see
modelworsenstheconstraintonanyoneofthem. However, Berlind&Weinberg (2002); Weinberg (2002); Zhengetal.
withintherangeofmodelsexaminedhere,addingparameters (2002) for further discussion]. VandenBoschetal. (2003)
to the HOD only slightly increasesthe errorbars on cosmo- havebeencarryingoutasimilarprogramusingthecloselyre-
logicalparameters. lated conditional luminosity function (CLF) method applied
Since the small scales of the primordial power spectrum tothe2dFGRSluminosityandcorrelationfunctions(seealso
probed by w (r ) could be useful in constraining any devi- vandenBoschetal. 2004). They find σ =0.78 0.12 and
p p 8
ationsfroma simplepower-lawprimordialspectrumaswell Ω = 0.25+0.10 in their analysis combined with C±MB data,
m - 0.07
asamodelincludingthesuppressioninpowerspectrumand withbotherrorsat95%C.L.asgiveninthatwork.Ourresults
mass function due to the presence of massive neutrinos, we are in agreement, within errors, with their determinationsof
performed an MCMC analysis including a running of the σ andΩ .
8 m
spectrum dn/dlnk about the scale k=0.05Mpc for the 3p Besidesthestatisticalerrorbars,therearetwomainsources
HOD model as well a model including massive neutrinos. ofsystematicuncertaintyinourcosmologicalparameteresti-
We find little evidence for running, dn/dlnk=- 0.062+0.026, mates. ThefirstisthepossibilitythatourHODparameteriza-
- 0.027
comparable to the results of Spergeletal. (2003). The halo tiondoesnothaveenoughfreedomtodescribetherealgalaxy
model in the presence of massive neutrinos is applied such HOD, and that we are artificially shrinking the cosmologi-
CosmologyandtheHODfromSmall-ScaleClustering 7
calerrorbarsbyadoptingarestrictivetheoreticalpriorinour HOD,notthecosmologicalparameterdeterminations. With-
galaxy bias model. For the 3p model, this is arguably the outacomprehensivenumericalstudyoftheseissues,itisdif-
case,sinceiteffectivelyassumesperfectcorrelationbetween ficulttoassesshowlargethesystematiceffectsonourparam-
the mass of a halo and the luminosity of its central galaxy. eter determinationscould be, butwe would not be surprised
However,our5pmodelis able to givean essentially perfect to find that they are comparableto our statistical errors. We
descriptionofthepredictionsofsemi-analyticgalaxyforma- plantocarryoutsuchastudytoremovethissourceofsystem-
tionmodelsandhydrodynamicsimulations(Zheng2004;see aticuncertaintyinfuturework;thepapersofSeljak&Warren
alsoGuzik&Seljak2002;Berlindetal.2003;Kravtsovetal. (2004)andTinkeretal.(2004)presentstepsalongthispath.
2004), so there is good reason to think that the error bars Analysesofmultipleclassesofgalaxieswillallowconsis-
quotedforthiscaseareconservative. Thismodelstillmakes tency checks on any cosmologicalconclusions, since differ-
theassumptionthatsatellitegalaxieshavenospatialbiaswith entclasseswillhavedifferentHODsbutshouldyieldconsis-
respecttodarkmatterwithinhalos,buttheconcentrationtest tent cosmologicalconstraints. By drawing on the full range
in §4 shows that dropping this assumption has minimal im- of galaxy clustering measurements, joint studies of galaxy
pactoncosmologicalconclusions.InplaceofanHODmodel, biasandcosmologicalparameterswillsharpenourtestsofthe
traditional analyses based on the large-scale galaxy power leadingtheoriesofgalaxyformationandthe leadingcosmo-
spectrum assume that the galaxy power spectrum is a scale- logicalmodel. With thiscurrentanalysisalone, we findthat
independentmultipleofthelinearmatterpowerspectrum,so thecombinationofCMBanisotropiesandsmall-scalegalaxy
thattheirshapesareidentical. Scale-independenceinthelin- clusteringmeasurementsprovides,simultaneously,tightcon-
earregimeisexpectedonfairlygeneralgrounds(Coles1993; straintsontheoccupationstatsticsofgalaxiesindarkmatter
Fry&Gaztanaga 1993; Weinberg 1995; Mannetal. 1998; halos, and some of the best available constraints on funda-
Scherrer&Weinberg 1998; Narayananetal. 2000). How- mentalcosmologicalparameters.
ever,itisnotclearjusthowwellthisapproximationholdsover
thefullrangeofscalesusedinthepowerspectrumanalyses,
soalthoughourHODmodelsareconsiderablymorecomplex WethankSalmanHabib,KatrinHeitmann,WayneHu,An-
thanlinearbiasmodels,ourapproachisarguablynomorede- dreyKravtsov,Chung-PeiMa,PederNorberg,RomanScoc-
pendentontheoreticalpriors. Infuturework,wecanusethe cimarro, Uroš Seljak, Erin Sheldon, Lu´ıs Teodoro, Jeremy
HOD modeling to calculate any expected scale-dependence Tinker,RobertoTrotta,FrankvandenBosch, Risa Wechsler
of the power spectrum bias, thus improving the accuracy of andMartinWhiteforfruitfuldiscussions.
thepowerspectrumanalysesandallowingthemtoextendto Funding for the creation and distribution of the SDSS
smallerscales. Archive has been provided by the Alfred P. Sloan Foun-
Thesecondsourceofsystematicuncertaintyisthepossibil- dation, the Participating Institutions, the National Aeronau-
ity thatour approximationfor calculating w (r ) for a given ticsandSpaceAdministration,theNationalScienceFounda-
p p
cosmologyandHODisinaccurateinsomeregionsofourpa- tion,theU.S.DepartmentofEnergy,theJapaneseMonbuka-
rameter space. The ingredients of this approximation have gakusho,andtheMaxPlanckSociety. TheSDSSWebsiteis
beencalibratedortestedonN-bodysimulationsofcosmolog- http://www.sdss.org/.
ical models similar to the best fitting models found here, so TheSDSSismanagedbytheAstrophysicalResearchCon-
wedonotexpectlargeinaccuracies. However,therearesev- sortium (ARC) for the Participating Institutions. The Par-
eralelementsofthe halomodelcalculationthatcouldbe in- ticipatingInstitutionsareThe Universityof Chicago, Fermi-
accurateorcosmologydependentatthe10%levelthatisnow lab,theInstituteforAdvancedStudy,theJapanParticipation
ofinterest,includingdeparturesfromtheJenkinsetal.(2001) Group, The Johns Hopkins University, the Korean Scientist
mass function, scale dependenceof halo bias, and details of Group, Los Alamos National Laboratory, the Max-Planck-
haloexclusion. Uncertaintiesinthehalomass-concentration InstituteforAstronomy(MPIA),theMax-Planck-Institutefor
relationandtheimpactofscatterinhaloconcentrationscome Astrophysics(MPA), New Mexico State University, Univer-
in at a similar level, though the test in §4 again indicates sity of Pittsburgh, Princeton University, the United States
thattheseuncertaintiesmainlyaffectthedetailsofthederived NavalObservatory,andtheUniversityofWashington.
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