The effects of assembly bias on cosmological inference from galaxy-galaxy lensing and galaxy clusters

1.3 MB·

by  Joseph E. McEwen

MNRAS000,1–??(2015) Preprint13January2016 CompiledusingMNRASLATEXstylefilev3.0
The effects of assembly bias on cosmological inference from
galaxy-galaxy lensing and galaxy clustering.
Joseph E. McEwen,1,3(cid:63) David H. Weinberg2,3
1 Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
2 Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA
3 Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA
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0 AcceptedXXX.ReceivedYYY;inoriginalformZZZ
2

n
ABSTRACT
a
Thecombinationofgalaxy-galaxylensing(GGL)andgalaxyclusteringisapromising
J
route to measuring the amplitude of matter clustering and testing modified gravity
1
theories of cosmic acceleration. Halo occupation distribution (HOD) modeling can
1
extend the approach down to nonlinear scales, but galaxy assembly bias could intro-

] duce systematic errors by causing the HOD to vary with large scale environment at
O fixed halo mass. We investigate this problem using the mock galaxy catalogs created
C by Hearin & Watson (2013, HW13), which exhibit significant assembly bias because
galaxy luminosity is tied to halo peak circular velocity and galaxy colour is tied to
.
h halo formation time. The preferential placement of galaxies (especially red galaxies)
p inolderhalosaffectsthecutoffofthemeanoccupationfunction N (M ) forcen-
cen min
- (cid:104) (cid:105)
tral galaxies, with halos in overdense regions more likely to host galaxies. The effect
o
r of assembly bias on the satellite galaxy HOD is minimal. We introduce an extended,
t environment dependent HOD (EDHOD) prescription to describe these results and
s
a fit galaxy correlation measurements. Crucially, we find that the galaxy-matter cross-
[ correlation coefficient, r (r) ξ (r) [ξ (r)ξ (r)]−1/2, is insensitive to assembly
biasonscalesr >1h−1Mgmpc,e≡vengmthoug·hmξm (r)gagndξ (r)arebothaffectedindivid-
1 gm gg
v ually. We can th∼erefore recover the correct ξ (r) from the HW13 galaxy-galaxy and
mm
3 galaxy-matter correlations using either a standard HOD or EDHOD fitting method.
9 For M 19 or M 20 samples the recovery of ξ (r) is accurate to 2% or bet-
r r mm
6 ter. For≤a s−ample of re≤d−M 20 galaxies we achieve 2% recovery at r >2h−1Mpc
2 r ≤−
with EDHOD modeling but lower accuracy at smaller scales or with a sta∼ndard HOD
0
. fit.
1
0 Key words: cosmological parameters – dark energy – gravitational lensing
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1
:
v
i 1 INTRODUCTION recent estimates of low redshift matter clustering are lower
X
than predicted from cosmic microwave background (CMB)
r A central challenge of contemporary cosmology is to deter- anisotropies evolved under a ΛCDM framework (see, Mor-
a minewhetheracceleratingcosmicexpansioniscausedbyan
tonson et al. 2014; Aubourg et al. 2014; Planck Collabora-
exotic“dark energy”component acting within General Rel-
tion 2015, ΛCDM = inflationary cold dark matter universe
ativity (GR) or whether it instead reflects a breakdown of
with a cosmological constant). If this discrepancy is con-
GRoncosmologicalscales.Onegeneralroutetodistinguish-
firmed,itcouldbethefirstclearindicationthatΛCDMisan
ingdarkenergyfrommodifiedgravityistotestwhetherthe
incompletedescriptionofcosmology,anditwouldhintinthe
growth of structure (measured through redshift space dis-
directionofmodifiedgravityexplanationsApromisingroute
tortions, gravitational lensing, or galaxy clustering) is con-
to measuring matter clustering is to combine galaxy clus-
sistentwithGRpredictionsgivenconstraintsontheexpan-
tering with galaxy-mass correlations inferred from galaxy-
sion history from supernovae, baryon acoustic oscillations
galaxy lensing (e.g. Mandelbaum et al. 2013; More et al.
(BAO), and other methods (see reviews by Frieman et al.
2015).Thisapproachnecessarilyrequiresamodelforgalaxy
2008;Weinbergetal.2013).Intriguingly,many(butnotall)
bias, i.e. for the relation between galaxy and dark-matter
distributions. In this paper we examine how the theoretical
(cid:63) E-mail:[email protected] uncertaintiesassociatedwithmodelinggalaxybiasinfluence
(cid:13)c 2015TheAuthors
2 McEwen et al.
the matter clustering inferred from combinations of galaxy h = H /100 kms−1 Mpc−1. To a first approximation, it
0
and galaxy matter clustering. is the z = 0 value of Ω that is constrained, though for
m
Galaxy-galaxy lensing (GGL) is a direct probe of the high-redshiftlensandsourcesamplesthenatureofthecon-
total matter content around a galaxy and provides a statis- strainedparametersbecomesmorecomplexanddependson
tical relationship between galaxy and matter distributions. whatauxiliaryobservationalconstraintsarebeingimposed.
Specifically GGL produces a tangential shear distortion of Under fairly general conditions, one expects r to ap-
gm
background galaxy images around foreground galaxies or proach unity on large scales, where ξ (r) ≤ 1 (see Bal-
gg
clusters(seeBartelmann&Schneider2001foradetailedre- daufetal.2010).However,because∆Σ(R)isanintegrated
view and §2 of Mandelbaum et al. 2013 for details that are quantity, it is affected by small scale clustering even at
relevanttothispaper).Withadequatephotometricredshifts large projected separation R and is therefore potentially
of background and foreground objects, the mean tangential susceptible to uncertainties in non-linear galaxy bias. To
shearcanbeconvertedtotheprojectedexcesssurfacemass mitigate this problem, Baldauf et al. 2010, constructed a
density∆Σ(R),whereRisthe2-dimensionalradialdistance filtered GGL estimator that eliminates small scale contri-
transversetothelineofsight.Theexcesssurfacemassden- butions. This approach was put in practice by Mandel-
sitycanberelatedtoanintegralofthegalaxy-mattercross baum et al. (2013), who applied the SDSS GGL at R >
correlation function (Sheldon et al. 2004) 2 and 4h−1Mpc to derive constraints on σ and Ω . They
8 m
(cid:104) 2 (cid:90) R(cid:90) ∞ found σ8(Ωm/0.25)0.57 = 0.80±0.05, about 2σ below the
∆Σ(R)=ρ¯ R2 r(cid:48)ξgm(r(cid:48),z)dzdr(cid:48) Planck+ΛCDM prediction.
0 −∞ (1) WhilethemethodinBaldaufetal.(2010)andMandel-
(cid:90) ∞ (cid:105)
− ξ (r(cid:48),z)dz , baumetal.(2013)isalreadycompetitivewithotherprobes
gm
−∞ of low-z structure, one could do better by incorporating
with ρ¯=Ω ρ (1+z)3. With sufficiently good measure- smaller scales, and thus increasing the signal-to-noise ra-
m crit,0
ments,∆Σ(R)inEqn.1canbeinvertedtoyieldtheproduct tio of the GGL measurement. This requires a description
of the mass density and galaxy-mass correlation function, of the relation between galaxies and mass that extends to
Ω ξ (r). Here we will assume that this inversion can be non-linear scales.
m gm
carried out and also that the projected galaxy correlation Halo occupation distribution (HOD) modeling offers
function can be inverted to yield the 3-dimensional, real- one approach to tie galaxies and dark-matter distributions
space correlation function ξ (r) of the same galaxies for downtonon-linearscales(Jingetal.1998;Peacock&Smith
gg
which Ω ξ (r) is measured by GGL. In practice, cosmo- 2000; Seljak 2000; Ma & Fry 2000; Scoccimarro et al. 2001;
m gm
logicalanalysesmayproceedbyforwardmodelingtopredict Berlind & Weinberg 2002). The HOD specifies P(N|Mh),
projectedquantitiesratherthaninversionto3-d(e.g.Man- theconditionalprobabilitythatahaloofmassMh hostsN
delbaum et al. 2013; More et al. 2015; Zu & Mandelbaum galaxies of a specified class, as well as the spatial and ve-
2015). For our present purpose of understanding the com- locity distribution of galaxies within host halos. Yoo et al.
plications (potentially) caused by complex galaxy bias, it is (2006)showedthatifonechoosesHODparameterstomatch
most straightforward to focus on the 3-d quantities them- galaxy clustering measurements, then the predicted GGL
selves. signal depends on the adopted cosmological model, increas-
The correlation functions ξgg(r) and ξgm(r) are related ing with σ8 and Ωm in both the large scale linear regime
to the matter auto-correlation function ξ (r) by and on smaller scales. Several variants of the HOD mod-
mm
eling approach to GGL have been described in the litera-
ξ (r)=b2(r)ξ (r) , (2)
gg g mm ture (Leauthaud et al. 2011; Yoo & Seljak 2012; Cacciato
et al. 2013). More et al. (2015) measured GGL by SDSS-
ξ (r)=b (r)r (r)ξ (r) , (3) III BOSS galaxies (Dawson et al. 2013) using imaging from
gm g gm mm
the CFHTlens survey (Heymans et al. 2012), and applying
where
the methods of van den Bosch et al. (2013) they obtained
ξ
rgm = (cid:112) gm (4) σ8 =0.785+−00..004444forΩm =0.310+−00..001290atthe68%confidence
ξmmξgg interval. Recently, Zu & Mandelbaum 2015 have applied a
is the galaxy-matter cross-correlation coefficient. Equations modified HOD method to the SDSS main galaxy sample
2, 3, and 4 are general and may be taken as definitions of (Straussetal.2002),obtaininganexcellentjointfittoclus-
the scale dependent bias factor bg(r) and cross correlation tering and GGL for a cosmological model with σ8 = 0.77
coefficient rgm(r). We note that the quantity rgm(r) in real and Ωm =0.27.
space is not constrained to be less than or equal to one in The philosophy of deriving cosmological constraints
magnitude, unlike the shot-noise corrected counterpart in fromsuchmodelingistotreatHODquantitiesas“nuisance
Fourier space (Guzik & Seljak 2001). parameters”thatallowonetomarginalizeoveruncertainties
Using Eqns. 2, 3, 4 one can combine observations of associated with galaxy formation physics (Zheng & Wein-
ξ (r) and Ω ξ (r) to determine berg 2007). Standard HOD modeling assumes P(N|M ) is
gg m gm h
[Ω ξ (r)]2 1 uncorrelatedwiththehalo’slargescaleenvironmentatfixed
Ω2mξmm(r)= mξgggm(r) · [rgm(r)]2 . (5) hroanlomemnat,sst.hIifs Pw(ilNl |cMhahn)gedotehsedperpeednidcteodnglaalragxeysccalulesteernivnig-
Thus, given a theoretical model for rgm(r), one can in- and galaxy-mass correlation for given set of HOD parame-
fer the product Ω ξ1/2, with an overall amplitude pro- ters. The risk is then that modeling with an environment-
m mm
portional to Ω σ (z), where σ (z) is the rms matter fluc- independent HOD may leave systematic bias in the cosmo-
m 8 8
tuation amplitude in 8h−1Mpc spheres at redshift z and logical inferences and/or underestimate the derived cosmo-
MNRAS000,1–??(2015)
3
logical parameter uncertainties associated with galaxy for- AM recipes typically use M or V at time of accretion,
h max
mation physics. withthe expectationthattidalstrippingwillaffectsubhalo
Thesimplestformulationofexcursionsettheory(Bond mass but not its stellar content (Conroy et al. 2006; Red-
et al. 1991) predicts that halo environment is correlated dick et al. 2013). With a subhalo mass at accretion recipe,
with halo mass but uncorrelated with formation history at AM is fairly successful at reproducing the galaxy content
fixed mass (White 1999), motivating the idea of an envi- of halos in hydrodynamic cosmological simulations (Simha
ronment independent HOD. However, N-body simulations et al. 2009, 2012; Chaves-Montero et al. 2015). Abundance
show that the clustering of halos of fixed mass varies sys- matching can easily be extended to incorporate scatter be-
tematically with formation time or concentration (Sheth & tweenbetweenhalomassandgalaxypropertiesandhasbeen
Tormen 2004; Gao et al. 2005; Wechsler et al. 2006; Harker shown to be remarkably successful at reproducing observed
etal.2006;Jingetal.2007).Thedependenceofhalocluster- evolution of galaxy clustering and other aspects of galaxy
ing on formation time or concentration is strongest for old evolution.
halos well below M . For halos above M there are indica- RecentlyHearin&Watson(2013;hereafterHW13)have
∗ ∗
tionsthatthesituationisreversed(Wechsleretal.2006).In extended the AM idea to galaxy color. Their age matching
generalthedependenceofhaloclusteringonhaloproperties technique monotonically maps a measure of halo formation
other than mass is termed assembly bias (Gottl¨ober et al. timetogalaxycoloratfixedstellarmass.AppliedtotheBol-
2001; Sheth & Tormen 2004; Gao et al. 2005; Avila-Reese shoi ΛCDM N-body simulation (Klypin et al. 2011), this
et al. 2005; Harker et al. 2006; Wechsler et al. 2006; Wang prescription produces good agreement with observed lumi-
etal.2007;Crotonetal.2007;Maulbetschetal.2007;Bett nosityandcolordependentclusteringandGGLobservations
et al. 2007; Wetzel et al. 2007; Angulo et al. 2008; Dalal ofSDSSgalaxies,despitehavingessentiallynofreeparame-
et al. 2008; Fakhouri & Ma 2009; Faltenbacher & White ters(Hearinetal.2014).(However,Zu&Mandelbaum2015
2010).Thephysicaloriginofassemblybiasremainsunclear, show that age-matching at fixed stellar mass over predicts
though a number of explanation have been proposed. Some the GGL signal of the most luminous blue galaxies.)
level of assembly bias may arise from correlated effects of BycomparingclusteringintheHW13galaxycatalogsto
long wavelength modes on halo formation times, breaking “scrambled”catalogs, that eliminate correlations with halo
theuncorrelatedrandomwalkassumptionthatunderliesthe formationhistory,Zentneretal.(2014)showthattheHW13
minimal excursion set model. Assembly bias can also arise catalogsexhibitsignificantgalaxyassemblybias.Forstellar
in the non-linear regime from tidal truncation of low mass mass threshold samples, this assembly bias arises because
halo growth in the environment of high mass halos. HW13 assign stellar mass based on V , and at fixed M
max h
If galaxy properties are tightly coupled to halo forma- the halos that form earlier tend to have higher concentra-
tion history, then a galaxy population can inherit assembly tions and higher Vmax. For color selected samples, the di-
bias from its parent halos. Such galaxy assembly bias im- rect mapping between formation time and color imprints a
plies that P(N|M ) depends on halo environment (or halo stronger assembly bias signature.
h
clustering)atfixedmass.Limitedworkhasbeencarriedout The HW13 catalogs adopt a physically plausible
measuringthegalacticassemblybiassignalinhydrodynamic and empirically successful description of galaxy formation
simulations. In some simulations, the HOD has shows little physics,soeveniftheyarenotcorrectinalldetails,wewould
tonodependenceonhaloenvironment(Berlindetal.2003; like cosmological inference methods based on HOD models
Mehta2014),whichsuggeststhatstochasticityinthegalaxy to be insensitive to galaxy assembly bias at this level.
formation physics in these simulations erases signatures of In this paper we examine the degree to which galaxy
haloassemblybias.However,recentworkofChaves-Montero assembly bias can affect matter clustering inference results
et al. (2015) has shown a galactic assembly bias signal in from GGL + galaxy clustering analysis. We begin by ex-
the EAGLE simulation (Schaye et al. 2015), boosting the amining the HOD and its environmental dependence in the
galaxy-correlationfunctionby∼25%onscalesgreaterthan HW13catalogs,confirmingfindingsofZentneretal.(2014)
∼ 1h−1Mpc. Although the analysis themselves are differ- but recasting them in a more HOD-specific form. We then
ent, the differing conclusions of Mehta (2014) and (Chaves- turn to the implications of GGL modeling, focusing our at-
Montero et al. 2015) in simulations of volume suggest that tention on the cross-correlation coefficient rgm(r), which is
the presence of galaxy assembly bias in hydrodynamic sim- thequantityneededtorecoverΩmξmm.WeshowthatHOD
ulationsdependsontheadoptedphysicaldescriptionofstar models fit to the galaxy correlation function of the HW13
formationandfeedback.Semi-analyticmodelspredictasig- catalogsyieldaccuratepredictions(atthe2−5%levelofpre-
nificant assembly bias effect in galaxy clustering for some cisionallowedbytheBolshoisimulationvolume)forrgm(r),
galaxy populations (Croton et al. 2007), particularly red even though they are incomplete descriptions of the bias in
galaxies of low stellar mass. thesegalaxypopulations.Weconcentratemainlyongalaxy
Abundance matching (AM) is an alternative route to samples defined by luminosity thresholds. However, we also
populating dark-matter halos with galaxies (e.g. Kravtsov considerasampleofredgalaxiesabovealuminositythresh-
et al. 2004; Vale & Ostriker 2004; Tasitsiomi et al. 2004; old, in part to examine a case with near-maximal assembly
Conroy & Wechsler 2009; Guo et al. 2010; Simha et al. bias effects, and in part because red galaxy samples allow
2009; Neistein et al. 2011; Watson et al. 2012; Rodr´ıguez- accurate photometric redshifts, which make them more at-
Puebla et al. 2012; Kravtsov 2013; Chaves-Montero et al. tractive for observational GGL studies. Our bottom line,
2015). Simple versions of abundance matching monotoni- illustrated in Fig.17, is that HOD modeling of galaxy clus-
cally tie galaxy luminosity or stellar mass to some proxy tering and GGL allows for accurate recovery of Ω2mξmm(r)
for the halo or subhalo gravitational potential well, such on scales r > 1h−1Mpc, even in the presence of galaxy as-
∼
as halo mass or maximum circular velocity. For subhalos, sembly bias as predicted by HW13.
MNRAS000,1–??(2015)
4 McEwen et al.
103 104 104 104
HW13
HW13 scrambled
103 103 103
102
) 102 102 102
r
( 101
g
ξg 101 101 101
100
100 100 100
M -19.0 M -20.0 M -21.0 M -21.5
r r r r
10-1 10-1 10-1 10-1
1.20 1.40 2.2 4.0
1.35
2.0 3.5
1.15
1.30
1.8 3.0
) 1.10 1.25
r
( 1.20 1.6 2.5
g
b 1.05 1.15
1.4 2.0
1.10
1.00
1.2 1.5
1.05
0.95 1.00 1.0 1.0
0.1 1.0 10.0 0.1 1.0 10.0 0.1 1.0 10.0 0.1 1.0 10.0
r [h 1 Mpc] r [h 1 Mpc] r [h 1 Mpc] r [h 1 Mpc]
− − − −
Figure 1. The impact of galaxy assembly bias on the galaxy correlation function, for samples defined by four thresholds in absolute
magnitude. Top panels compare ξgg from the HW13 abundance matching catalog (solid) to that of a scrambled catalog (dashed) in
which the effect of galaxy assembly bias is erased by construction. Bottom panels plot the corresponding galaxy bias factor bg(r) =
(cid:112)
ξgg(r)/ξmm(r).
2 HALO OCCUPATION DISTRIBUTION OF n (> L) = n (> V ). Specifically HW13 uses the peak
g h max
THE HW13 CATALOGS circular velocity V (Reddick et al. 2013), which is the
peak
largest V that the halo or subhalo obtains throughout
max
2.1 Galaxy assembly bias in the HW13 Catalogs itsassemblyhistory.Thesecondstep,age-matching,assigns
colors by imposing a monotonic relation between galaxy
The abundance and age-matching catalogs of HW13 are
colour and halo age at fixed luminosity, matching to the
built from the Bolshoi N-body simulation (Klypin et al.
observed colour distribution in SDSS. The redshift defining
2011),whichusestheAdaptiveRefinementTree(ART)code
halo age is set to the maximum of (1) the highest redshift
(Kravtsov et al. 1997; Gottloeber & Klypin 2008 ) to solve
fortheevolutionof20483 particlesina250h−1Mpcperiodic at which the halo mass exceeds 1012h−1M(cid:12), (2) the red-
box. The mass of each particle is m ≈ 1.9×108h−1M . shift at which the halo becomes a subhalo, (3) the redshift
p (cid:12)
The force resolution is (cid:15)≈1h−1kpc. The cosmological pa- at which the halo’s growth transitions from fast to slow ac-
cretion, as determined by the fitting function of Wechsler
rameters are: Ω =0.27, Ω =0.73, Ω =0.042, n =0.95,
m Λ b s
σ = 0.82, and H = 70 km s−1 Mpc−1. Bolshoi catalogs et al. (2002). Criterion (3) determines the age for most ha-
8 0
los and subhalos. These catalogs are publicly available at
and snapshots are part of the Multidark Database and are
http://logrus.uchicago.edu/∼aphearin/.
available at http:www.multidark.org. Halos are identified
bythe(sub)halofinderROCKSTAR(Behroozietal.2013), When abundance matching is based on halo mass e.g.
which uses adaptive hierarchical refinement of friends-of- Conroy et al. (2006), then the resulting population of cen-
friends groups in six phase-space dimensions and one time tral galaxies, has no assembly bias by construction. How-
dimension. Halos are defined within spherical regions such ever, at fixed halo mass, halos that form earlier are more
that the average density inside the sphere is ∆ ≈ 360 concentrated and thus have higher V , so even luminos-
vir max
times the mean matter density of the simulation. ity thresholded samples exhibit galaxy assembly bias in
To create galaxy catalogs HW13 follow a two step pro- the HW13 catalogs (Zentner et al. 2014). We consider four
cess. First galaxies of a particular luminosity are assigned samples defined by absolute magnitude thresholds M −
r
to (sub)halos based on an abundance matching scheme. 5logh≤−19,−20,−21,−21.5(hereafterweomitthe5logh
Their abundance matching algorithm requires that the cu- for brevity). The -20 and -21 samples bracket the char-
mulative abundance of SDSS galaxies brighter than lu- acteristic galaxy luminosity L , with -21 thresholds yield-
∗
minosity L is equal to the cumulative abundance of ha- ing the overall best clustering measurements in the SDSS
los and subhalos with circular velocities larger than V , main galaxy sample (Zehavi et al. 2011). The -19 thresh-
max
MNRAS000,1–??(2015)
5
old corresponds to fairly low luminosity galaxies with high 104
space density, while -21.5 corresponds to rare, high lumi- 103 HW13
HW13 scrambled
nosity galaxies. We also consider a sample of red galaxies
with Mr ≤−20 and g−r ≥0.8−0.3(Mr+20.0). Because ξ(r)gg110012
coloristiedmonotonicallytohaloformationtimeinHW13,
Red
thisselectionyieldsanear-maximaldegreeofgalaxyassem- 100 M -20.0
bly bias. In addition to testing our methods under extreme 10-110-1 r 100 101
conditions, this sample is observationally relevant because 1.8
1.7
redgalaxiesallowrelativelyaccuratephotometricredshifts,
making them attractive for galaxy-galaxy lensing measure- r)1.6
(1.5
ments in large imaging surveys such as the Dark Energy bg1.4
Survey (Rozo et al. 2015). The number of galaxies in the 1.3
Bolshoisimulationvolumeis244766,96595,17250,3954for 1.02.1 1.0 10.0
the M ≤ −19,−20,−21, and −21.5 samples, respectively, r [h−1 Mpc]
r
and 56591 for the red M ≤−20 samples. Of these, a frac-
r
tion f = 0.75,0.77,0.81,.85,0.77 are central galaxies of
cen Figure 2. Galaxy assembly bias in Mr ≤ −20 red samples. As
their host halos, and a fraction fsat = 1−fcen are satellite in Fig. 1, the top panel compares the measured galaxy correla-
galaxies located in subhalos. tionfunctioninHW13toascrambledversionofHW13,andthe
Fig. 1 compares galaxy correlation functions measured bottompanelcomparesresultsforthegalaxybiasfactor.
fromtheluminosity-thresholdHW13catalogstothosefrom
scrambledversionsofthesamecatalogs.Lowerpanelsshow
(cid:112) ual impact of assembly bias on the galaxy population is no
galaxy bias defined by b (r)= ξ (r)/ξ (r). Scrambled
g gg mm longer discernible.
catalogsareconstructedbybinningcentralandsatellitesys-
In the lower luminosity samples, b (r) becomes scale-
tems in host halo mass, randomly reassigning centrals to g
dependent(atthe10−20%level)atthetransitionbetween
other halos within the mass bin, then randomly reassigning
the 2-halo regime of ξ (r), where galaxy pairs come from
satellite systems to these centrals. By construction scram- gg
separate halos, and the 1-halo regime dominated by galaxy
blingremoves any galaxy assembly bias present in the orig-
pairswithinasinglehalo.Anyhalomassiveenoughtocon-
inal HW13 catalog, i.e., any correlation between the galaxy
tain two galaxies is far above the minimum mass threshold
content of a halo and any halo property other than mass.
for a central galaxy, so any assembly bias effects in the 1-
For full details of the scrambling process see Zentner et al.
halo regime will arise from the satellite galaxy population.
(2014), who present a similar clustering analysis.
Theconvergenceofcorrelationfunctionsatsmallrsuggests
For the Mr ≤ −19 sample, bg(r) is about 10% higher thatassemblybiaseffectsintheHW13catalogaredrivenby
in the HW13 catalog relative to the scrambled catalogs at thecentralgalaxypopulationratherthansatellites,apoint
r > 3 h−1Mpc. Both catalogs show a drop in bg(r) as r we demonstrate explicitly in below.
decreases from 3h−1Mpc to 0.5h−1Mpc, then a rise on still Figure2showsgalaxycorrelationfunctionsandgalaxy
smallerscales.Thetwocorrelationfunctionsconvergeatr< biasresultsforredM ≤−20galaxies(ourmaximalgalaxy
r
0.5h−1Mpc. assemblybiassample),againcomparingHW13toscrambled
The M ≤−20 sample shows similar behavior, but the catalogs. The large difference in bias factors in the bottom
r
differences between scrambled and unscrambled correlation panel of Fig. 2 is indicative of the strong galaxy assembly
functions are somewhat smaller. For luminous, M ≤ −21 bias for low luminosity red galaxies in HW13. As with the
r
galaxies, the difference in the large scale bias is only ∼3%, luminosity threshold case, ξgg(r) for colour selected HW13
andthetwocorrelationfunctionsareessentiallyconvergedat andscrambledcatalogsconvergesonsmallscales,indicating
r<1h−1Mpc.Forthemostluminoussample,M ≤−21.5, the assembly bias is also primarily due to central galaxy
∼ r
anydifferencesaresmallerstill,andconsistentwithnoisein populations.
ξ (r).
gg
ThetrendsinFig.1makesenseinlightoftheprevious
2.2 HOD Analysis of M ≤−19 Galaxies
r
studies of halo assembly bias, which show that the depen-
dence of clustering on formation time is strongest for low The HOD specifies the probability P(N|Mh) that a halo of
mass halos and declines as the halo mass approaches the mass Mh contains N galaxies of a specified class, together
characteristicmassM ofthehalomassfunction(Gaoetal. with auxiliary prescriptions that specify the spatial and ve-
∗
2005; Harker et al. 2006; Wechsler et al. 2006). The mini- locity distributions of galaxies within halos (Benson et al.
mum halo mass for M ≤−19 galaxies is low, and halos in 2000; Berlind & Weinberg 2002). Following Guzik & Seljak
r
denser environments are more likely to host HW13 galaxies (2001) and Kravtsov et al. (2004), we separate P(N|Mh)
because they have earlier formation times and higher circu- into contributions from central and satellite galaxies,
lar velocities at fixed mass. This preferential formation in
P(N|M )=P(N |M )+P(N |M ) . (6)
dense environments accounts for the higher large scale bias h cen h sat h
factor of the HW13 catalog relative to the scrambled cat- WeadopttheparameterizationofZhengetal.(2005),which
alog. As the luminosity threshold and minimum host halo providesagoodfittothetheoreticalpredictionsofhydrody-
mass increase, the bias factor grows but the impact of as- namic simulations and semi-analytic models and a good fit
semblybiasdiminishes.ForM ≤−21.5theminimumhalo toobservedgalaxycorrelationfunctions(Zehavietal.2005,
r
massisM ≈1013h−1M (seeFig.8below),andanyresid- 2011; Coupon et al. 2012, 2015).
h (cid:12)
MNRAS000,1–??(2015)
6 McEwen et al.
4500 700 ofenvironmentandfoundthatouroverallresultsareinsen-
4233050500000000 465000000 sinitcilvueditnog, teh.ge.,cecnhtarnagli1nhg−t1hMeprca,doiiroinfctohreposrpahteinlroiggc(Madlhis)at∼na1nn1.cu25elutso, log(Mh)∼12.25
2000 300 nearest large halo as an environmental measure.
1500 200
1500000 100 Fig.4illustratestheHODdependenceonhosthaloen-
0 0 vironmentintheHW13catalog.Thesolidgreycurveshows
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 log(M ) 14.25
120 10 (cid:104)N(Mh)(cid:105) for the global HOD computed by counting galax- h ∼
100 8 iesin0.2dexbinsofMh withoutreferencetoloegn(Mvhir)o∼n13m.25ent.
80 6 Solid(dashed)curvesshow(cid:104)N(Mh)(cid:105)computedforthe20%
60
4 of halos with the highest (lowest) density environments in
40
20 2 each mass bin. The shape of the measured HOD curves in
0 0 Fig. 4 is similar to the functional form predictions of Eqns.
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
7and8:asharprisein(cid:104)N(cid:105)fromzerotooneassociatedwith
log(δ ) log(δ )
1−5 1−5 centralgalaxies,andashallowplateaubetween(cid:104)N(cid:105)=1−2,
followedbyasteepeningtoapowerlaw.Theenvironmental
dependence is visually evident, primarily for low mass host
Figure 3. Distribution of halo environments for halos in
four 0.1-dex bins of mass, centered at logMh/h−1M(cid:12) = halos where (cid:104)N(cid:105) < 1. In the language of HOD parameters,
11.25,12.25,13.25,and14.25(toplefttobottomright).Theden- theHODforhalosinhigherdensityenvironments(top20%)
sity contrast δ is measured in a spherical annulus of 1 < r < hasalowerMmin andalargerσlogMh thantheglobalHOD,
5h−1Mpc. and the reverse is true in low density environments. Halos
with M ∼1−2×1011h−1M are therefore more likely to
h (cid:12)
hostacentralgalaxywithM ≤−19iftheyresideinahigh
Themeanoccupationofcentralgalaxiesisdescribedby r
density environment, giving rise to the higher bias factor
seen in Fig. 1, relative to the scrambled catalog which has
1(cid:104) (cid:18)logM −logM (cid:19)(cid:105)
(cid:104)N (cid:105)= 1−erf h min . (7) theenvironmentindependent,globalHODbyconstruction.
cen 2 σlogMh The satellite (cid:104)N(Mh)(cid:105) shows little dependence on environ-
The parameter M sets the scale where (cid:104)N (cid:105) = 1/2 ment,thoughthehighestmasshalosareonlypresentinthe
min cen
while σ controls the sharpness of the transition from dense environments.
logMh
(cid:104)N (cid:105)=0 to (cid:104)N (cid:105)=1. Physically σ represents the HowmuchoftheassemblybiasintheHW13catalogis
scactetner between hcaelno mass andcentral gloaglMaxhy luminosity. A explained by this dependence of (cid:104)N(Mh)(cid:105) on the 5h−1Mpc
large scatter corresponds to a soft transition and a small environment? To answer this question, we construct cata-
scatter to a sharp transition. logs with an environment-dependent HOD (EDHOD) and
Satellite occupancy is determined by compare their clustering to that of the HW13 galaxies. As
afirststep,wemeasure(cid:104)N(M )(cid:105)in30binsofenvironment
(cid:18)M (cid:19)α (cid:18) M (cid:19) h
(cid:104)Nsat(cid:105)= Mh exp − Mcut . (8) δ1−5 (and the same 0.2-dex mass bins). This bin-wise ED-
1 h HODautomaticallyincorporatesenvironmentaldependence
TheHODparameterM isapproximatelythemassscaleat forbothcentralandsatellitegalaxies.Intheterminologyin-
1
which halos have an average of one satellite. At larger halo troducedbyHearinetal.(2015)onecanregardourEDHOD
massesthesatelliteoccupancyincreasesasapower-lawwith as a“decorated HOD”with δ1−5 as the additional control
slope α. M controls the scale at which the power law is variable.
cut
truncated at low mass. Afterchoosingthenumberofcentralandsatellitegalax-
The total mean occupancy is the sum of central and ies in each halo by drawing from P(N|Mh,δ1−5), we must
satellite mean occupancies determine the positions of galaxies within the halos. A
standard approach is to place the central galaxy at the
(cid:104)N(cid:105)=(cid:104)N (cid:105)+(cid:104)N (cid:105) . (9)
cen sat halo center-of-mass and distribute satellite galaxies with a
A host halo is assigned a central galaxy by Bernoulli sam- Navarro et al. (1997) type profile (hereafter NFW) so that
plingwithEqn.7servingastheprobabilityforsuccess.Ifa n(r)∝ρNFW(r). However, we find that the satellite galaxy
host halo is determined to contain a central, the number of distributionintheHW13catalogsdifferssubstantiallyfrom
satellitesassignedtothecentralisdonebyPoissonsampling an NFW profile, a consequence of satellites being placed
with Eqn. 8 serving as the average. withinsubhalosintheAMscheme(Nagai&Kravtsov2005;
WewanttoexaminethedependenceoftheHODonthe Zentneretal.2005).Figure5illustratesthisdifference,com-
largescaleenvironmentofhalosatfixedM .Wedefinehalo paringtheHW13satelliteprofilesintwonarrowbinsofhalo
h
environmentbythedarkmatterdensitycontrastδ mea- mass to an NFW profile. The radial profile of HW13 satel-
1−5
sured in a spherical annulus of 1<r <5h−1Mpc. Figure 3 litesismuchflatterthananNFWprofile,anditextendsbe-
showsthedistributionofδ infournarrowbinsoflogM . yond the viral radius because Rockstar halos are aspherical
1−5 h
As expected, the higher mass halos tend to reside in higher whilerv isdefinedwithasphericaloverdensity.Wetherefore
density regions, giving rise to the well known mass depen- usethemeasuredHW13radialprofilesratherthananNFW
denceofhalobias.Toremovethetrendfromouranalysis,we formtocreateourEDHODcatalogs.Wefoundthatcluster-
rankthehalosbyδ innarrow(0.2-dex)massbins,sowe ing on scales r≤2h−1Mpc would be substantially different
1−5
can compare the HOD of halos in, e.g., the 20% highest or if we imposed an NFW profile for the satellite distribution.
lowest density environment relative to other halos of nearly In Fig. 6 we compare galaxy auto-correlation results
equalmass.Wehaveexperimentedwithdifferentdefinitions taken from our measured (ED)HOD catalogs and from
MNRAS000,1–??(2015)
7
103 104 104
global HOD
log(M ) 12.025 log(M ) 14.025
102 bottom 20 % h ∼ h ∼
top 20 % 103 103
) 1fi01 )
h r
M (
n
N( 100 102 102
› M 19.0
r −
10-1 Mmin=1011.5h−1Mfl 101 Hρ(Wr)1-N3FW 101
0.1 1.0 0.1 1.0
1011 1012 1013 1014 1015
r/r r/r
M [h−1M ] v v
fl
Figure 5. Radial distributions of HW13 Mr ≤ −19 satellite
galaxies(solidblackcurves),inhaloswithlogMh/h−1M(cid:12)=12−
Figure 4. The Measured HOD in the HW13 catalogs for the 12.05(left)andlogMh/h−1M(cid:12)=14−14.05(right).Greycurves
Mr ≤−19sample.Thegreylineshowstheglobalmeanoccupa- show an NFW profile with the mean concentration expected for
tion function (cid:104)N(Mh)(cid:105) for halos in all environments. Solid and thishalomasstruncatedattheviralradius.
dashed black curves show (cid:104)N(M )(cid:105) for halos in the 20% high-
h
est and lowest density environments, respectively, as measured
byδ1−5.FortheglobalHOD,(cid:104)N(Mh)(cid:105)=0.5atMh =Mmin = 103
1011.5h−1M(cid:12). HW13
102 HW13 + isotropic sat. dist.
EDHOD
ξ(r)gg101 HEDOHDOD-cen. + HOD-sat.
100 M 19.0
r −
10-1
HW13 for M ≤ −19 samples. The top panel plots ξ (r)
r gg 0.3
while the bottom panel shows the fractional difference in 0.2
gg 0.1
ξgg(r) compared to HW13. As seen in the lower panel, ran- nξ 0.0
domizingthehost-centricsatelliteanglesintheHW13cata- ∆l 0.1
0.2
log,thusremovingtheeffectsofhaloellipticityandsubstruc- 0.3
0.1 1.0 10.0
ture,depressesξgg(r)byupto10%below1h−1Mpcbuthas r [h−1Mpc]
negligible effect at larger separations. The standard HOD
modelunderpredictstheHW13ξ (r)evenatlargesepara-
gg Figure 6.Galaxy-correlationfunctionfortheMr ≤−19HW13
tions, an indication of the impact of galaxy assembly bias, catalog compared to several HOD realizations. The grey curve,
as already seen in Fig. 1. The EDHOD model, on the other obscuredintheupperpanel,showsξgg(r)fromtheHW13catalog.
hand, matches HW13 almost perfectly at r > 7 h−1Mpc. Dot-dashedandsolidblackcurvesshowξgg(r)fromcatalogscre-
However, the EDHOD ξ (r) rises 5% above HW13 at r ≈ atedusingtheglobalHODandenvironmentallydependentHOD
gg
5h−1Mpcandfalls10%belowatr≈1−2h−1Mpc,before (EDHOD), respectively, measured from the HW13 catalog. The
converging to the isotropized satellite case at still smaller bottompanelshowsfractionaldeviationsfromtheHW13ξgg(r).
Additionalcurvesshowtheeffectofisotropizingthesatellitedis-
scales. The dashed curve shows the effect of imposing an
tributions in the HW13 catalog (heavy dashed) or of combining
EDHOD for central galaxies but using the global HOD for
theenvironmentallydependentHODforcentralswiththeglobal
satellites. These results are nearly identical to those of the
HODforsatellites(light dashed).
full EDHOD model, demonstrating that for this sample it
iscentralgalaxyenvironmentdependencethatmatters.We
conclude that incorporating the environmental dependence
2.3 Results for other Galaxy Samples
ofcentralgalaxyoccupationsreproducesthelargescalebias
of the HW13 catalog but leaves a 5-10% residual in ξgg(r) Figure 8 plots the measured HOD for HW13’s Mr ≤ −20
on non-linear scales. (top),M ≤−21(middle),andM ≤−21.5(bottom)galaxy
r r
Figure 7 plots the M ≤ −19 cross correlation coeffi- samples, comparing the global HOD to that of halos in the
r
cient r (r) for the HW13, HOD, and EDHOD catalogs. top 20th and bottom 20th percentile in environmental den-
gm
The HW13 curve remains close to unity (within 0.5%) at sity,asinFig.4.LiketheM ≤−19sample,theM ≤−20
r r
r>1h−1Mpc,withadropandriseinside0.4h−1Mpc.The sampleshowsanincrease(decrease)in(cid:104)N(cid:105)forlowmasshost
EDHODpredictionisstrikinglysimilar,matchingHW13to halos residing in higher (lower) density environments. For
1% or better at r > 0.4 h−1Mpc and showing similar form brighter samples, the environmental dependence is weaker,
at smaller scales. Even the global HOD prediction is simi- and essentially indiscernible for M ≤−21 or M ≤−21.5.
r r
lar,deviatingby1.5%atr>0.4h−1Mpc,despitethemuch These results are consistent with the weakening impact of
larger deviation in ξ (r) seen in Fig. 4. These results are galaxy assembly bias at higher luminosities seen in Fig. 1.
gg
our first indication that using a standard HOD versus an Figure9,analogoustothelowerpanelofFig.6,plotsthe
environment-dependent HOD has little impact on matter fractionaldifferenceinξ (r)measuredfromtheHW13cat-
gg
clustering inferences. alogandfromcatalogsconstructedusingtheglobalHODor
MNRAS000,1–??(2015)
8 McEwen et al.
1.10 103
HW13 global HOD
HOD 102 bottom 20 %
1.05
EDHOD top 20 %
) 1fi01
r(r)gm1.00 N(Mh100
› M 20.0
r −
0.95 Mr −19.0 10-1 Mmin=1012.0h−1Mfl
0.90 1011 1012 1013 1014 1015
0.1 1.0 10.0
r[h−1 Mpc] M [h−1M ]
fl
Figure 7. Galaxy-matter cross-correlation coefficient (Eqn. 4)
computed from the HW13 Mr ≤ −19 catalog (grey) or from
catalogscreatedbyusingtheglobalHOD(blacksolid)orEDHOD 103
(dot-dashed)ofthissample. global HOD
102 bottom 20 %
top 20 %
theenvironmentallydependentHOD.ResultsforMr ≤−20 ) 1fi01
are similar to those for Mr ≤ −19: incorporating environ- Mh
mental dependence removes the ∼ 10% offset in the large N( 100
scale bias factor found for the global HOD, but there are › M 21.0
r −
sFtoilrlM5-r10≤%−2d1ifftheererencisesoninlyξaggs(mr)alflobriarso≈ffs0e.t5fo−r5thhe−1gMlobpacl. 10-1 Mmin=1012.9h−1M
fl
HODmodel,anddeviationsinξ (r)fortheEDHODmodel
gg
are consistent with random fluctuations. For M ≤ −21.5, 1011 1012 1013 1014 1015
r
all three models give consistent results. Fig. 10 shows re- M [h−1M ]
fl
sultsfortheEDHODandHODcross-correlationcoefficient
comparedtoHW13forourbrightersamples.ForM ≤−20
r
and−21,EDHODandHODr (r)resultstracktheHW13
results well on scales greater tghman ∼ 1h−1Mpc. Results for 103
global HOD
M ≤−21.5 are dominated by noise.
r 102 bottom 20 %
Color selection has the potential to introduce stronger
top 20 %
galaxy assembly bias because of the direct connection that ) 1fi01
the HW13 age-matching prescription introduces between h
M
colour and halo formation time. Figure 11 shows the HOD (
N 100
tehnevirHoWnm1e3nctaatlavloagri.atCioonmpfoarritshone rteodFMigr. 8≤sh−o2w0sgtahlaatxieensviin- › Mr −21.5
ronmental dependence is indeed stronger than that of the 10-1 Mmin=1013.7h−1M
fullM ≤−20sample;inparticular,theincreased(cid:104)N(M )(cid:105) fl
r h
in dense environments continues up to 1012.5h−1M(cid:12) halos. 1011 1012 1013 1014 1015
However,thereisstillnoindicationofanenvironmentalde- M [h−1M ]
pendence of the satellite HOD. fl
Figure12showsthedeviationsinthegalaxycorrelation
functionandgalaxy-masscrosscorrelationcoefficientforour Figure 8. Mean occupation functions of the HW13 catalogs for
red Mr ≤ −20 galaxy sample. As expected, differences be- Mr ≤ −20 (top), −21 (middle), −21.5 (bottom) for all halos
tween the global HOD model and the HW13 catalog are and for halos in the 20% highest or lowest density environment
largerthanthoseforthefullM ≤−20sample,witha20% measuredbyδ1−5,asinFig.4.Galaxyassemblybiaseffectsare
r
smallerformoreluminoussamples.
differenceinthelargescalebiasfactor.TheEDHODmodel
again removes this large scale offset but leaves significant
deviations in the 0.5−5h−1Mpc range. Crucially, however, luminositytohaloV ,whichiscorrelatedwithformation
max
thevaluesofrgm(r)computedfromthesethreecatalogsstill time at fixed Mh. For colour selected samples the connec-
match, at the 2% level or better for r≥1h−1Mpc. tion to halo assembly is imposed directly by HW13’s age
matching procedure.
In HOD terminology, the assembly bias manifests it-
2.4 Summary
self as a increase in (cid:104)N(M )(cid:105) for central galaxies of halos
h
As shown previously by Zentner et al. (2014), the HW13 in denser than average environments, and a correspond-
galaxy catalogs exhibit substantial impact of assembly bias ing decrease of (cid:104)N(M )(cid:105) in low density environments. We
h
on galaxy clustering, particularly for low luminosity or find no evident effect of halo environment on the satellite
colour selected samples. In the case of luminosity thresh- galaxy occupation. Constructing HOD mock catalogs that
old samples, galaxy assembly bias arises because HW13 tie incorporate the environmental dependence measured in the
MNRAS000,1–??(2015)
9
0.2 104
M < 20.0
0.1 r − 103 HW13
00..01 r) 102 HOD
( EDHOD
0.2 gg101
r) 0.2 ξ 100 Red M <=-20.0
ξ(gg 00..01 Mr <−21.0 10-1 r
n 0.1 ξgg 00..42
∆l 00..22 ∆ln 000...042
00..01 Mr <−21.5 r)11..1005
0.1 (m1.00
g0.95
0.2 r0.90
0.1 1.0 10.0 0.1 1.0 10.0
r [ h−1 Mpc] r[h−1Mpc]
Figure 9. Fractional deviations of ξgg(r) from global (dashed) Figure 12.Comparisonofthegalaxycorrelationfunctions(top
and environmentally dependent (solid) HOD catalogs compared and middle) and the galaxy-matter cross correlation coefficient
to the HW13 catalogs for the Mr ≤ −20,−21,−21.5 samples. (bottom) for red Mr ≤ −20 galaxies computed from the HW13
SimilartothebottompanelofFig.6. catalog (grey) and catalogs created using the global (dashed) or
environmentallydependent(dot-dashed)HODs.ComparetoFigs.
6,7,9.
1.10
1.05 M < 20.0
1.00 r − r > 1 h−1Mpc, typically within the statistical fluctuations
0.95
arising from the finite size of the simulated catalogs. This
0.90
1.15 similarity suggests that the impact of galaxy assembly bias
(r)gm0111....90105005 Mr <−21.0 opnoinξgtgw(re)aadnddreξsgsmm(ro)rewdililrceactnlcyelininthceosnmexotlosgeicctailonan.alysis, a
r0.90
0.85
1.15
1.10
11..0005 3 MATTER CLUSTERING INFERENCE
0.95 M < 21.5
0.90 r −
0.85 Inanobservationalanalysis,onedoesnotknowtheHODor
0.1 1.0 10.0
r [ h−1 Mpc] EDHOD of a galaxy sample a priori but infers it by fitting
theobservedgalaxyclustering.InajointGGL+clustering
Figure 10. Cross correlation coefficients rgm(r) from global analysis,thegoalistosimultaneouslyinferthevaluesofthe
(dashed) and environmentally dependent (solid) HOD cata- cosmologicalparametersthatdetermine∆Σ(R).Acomplete
logs compared to the HW13 catalogs (grey) for the Mr ≤ versionofsuchananalysiswouldlikelyinvolveforwardmod-
−20,−21,−21.5samples.SimilartoFig.7. elingoftheprojectedclusteringandGGLobservables,with
details that depend on the data sets being analyzed and on
theexternalconstraintsadoptedonthecosmologicalparam-
103 eters (e.g., from CMB measurements). Here we consider an
global HOD
idealized analysis in which de-projection has been used to
102 bottom 20 % translate w (R) and ∆Σ(R) into the 3-d quantities ξ (r)
p gg
top 20 % and Ω ξ (r). HOD or EDHOD parameters are inferred
M)h1fi01 byfittimngmξmgg(r),andequation(5)isusedtoinferΩ2mξmm(r)
fromtheGGLmeasurement.Wewanttoknowwhetherthis
(
N 1›00 RedM 20.0 approachwouldyieldunbiasedestimatesofrgm(r),andthus
r − ofΩ2 ξ (r),giventhegalaxyassemblybiaspresentinthe
10-1 Mmin=1012.5h−1M HW1m3mabmundance matching model. Our approach is inher-
fl entlynumerical,asweareusingpopulatedN-bodyhalosto
1011 1012 1013 1014 1015 calculate ξgg(r), ξgm(r) and rgm(r) on all scales
M [h−1M ] WeconstructanEDHODmodelbyallowingthecentral
fl galaxyHODparametersM andσ tohaveapower-
min logMh
Figure 11.MeanoccupationforredgalaxieswithMr ≤−20in law dependence on δ1−5, in equations:
theHW13catalog,inthesameformatasFigs.4and8.
logM =A+γlog(δ ) , (10)
min 1−5
HW13 catalog removes the large scale offset in ξ (r) that
gg logσ =B+βlog(δ ) . (11)
arises with an environment-independent HOD model. How- logMh 1−5
ever, deviations of ξ (r) at the 5-20 % level (depending on Note that our EDHOD model has seven parameters,
gg
scaleandgalaxysample)remainbetweentheHW13catalogs {A,B,γ,β,M ,M ,α}, whereas the HOD presented in §2
1 cut
andcatalogsconstructedfromanEDHOD.Nonetheless,the hasfive.(ED)HODparametersareinferredbyusingadown-
HW13 catalogs, EDHOD catalogs, and (to a lesser extent) hill simplex method to minimize a sum of squares func-
HOD catalogs yield similar predictions for r (r) at scales tion,(cid:80) (Di −Di )2/(Di )2.ThedatavectorD(cid:126) is
gm i model HW13 HW13
MNRAS000,1–??(2015)
10 McEwen et al.
103 103 104 104
HW13 HW13
102 HOD fit 102 103 HOD fit 103
EDHOD fit EDHOD fit
ξ(r)gg101 ξ(r)gm101 ξ(r)gg110012 ξ(r)gm110012
100 M < 19.0 100 100 M < 20.0 100
r − r −
10-1 10-1 10-1 10-1
ξ(r)gg 000...012 ξ(r)gm 0000....01010055 ξ(r)gg 000...012 ξ(r)gm 0000....01010055
∆ln 00..12 ∆ln 000...110055 ∆ln 00..12 ∆ln 000...110055
0.1 1.0 10.0 0.1 1.0 10.0 0.1 1.0 10.0 0.1 1.0 10.0
r [ h−1 Mpc] r [ h−1 Mpc] r [ h−1 Mpc] r [ h−1 Mpc]
Figure 13.HODandEDHODfittingresultsfortheMr ≤−19
sample. Left and right panels show ξgg(r) and ξgm(r), respec-
tively.(ED)HODparametersareinferredbyfittingtoξgg(r)over 104 104
the range 0.1−30h−1Mpc and including the total number of 103 HHWOD1 3fit 103
galaxies in HW13 as an additional fitting point. Lower panels EDHOD fit
sthheowHWfra1c3tioconrarledlaetvioiantifounnsctoifotnhse. best-fit(ED)HODmodelsfrom ξ(r)gg110012 ξ(r)gm110012
100 M < 21.0 100
r −
1.10 10-1 10-1
1.05 HHEDWOHD1O 3fDit fit ∆lnξ(r)gg 00000.....01212 ∆lnξ(r)gm 0000000.......01011100055055
) 0.1 1.0 10.0 0.1 1.0 10.0
(rm1.00 r [ h−1 Mpc] r [ h−1 Mpc]
g
r
0.95
104 104
M 19.0
r − 103 HHWOD1 3fit 103
0.900.1 r[1h.0−1 Mpc] 10.0 ξ(r)gg110012 EDHOD fit ξ(r)gm110012
Figure14.Galaxy-mattercrosscorrelationcoefficient(eq.4).for 100 M < 21.5 100
the Mr ≤ −19 HW13 catalog (thick grey) and for the EDHOD 10-1 r − 10-1
model (solid black) and HOD (dot-dashed) models that best fit
theHW13ξgg(r)asshownintheleftpanelsofFig.13. ξ(r)gg 000...012 ξ(r)gm 0000....01010055
galaxy-correlationfunctionintheranger=0.1−30h−1Mpc ∆ln 00..012.1 1.0 10.0 ∆ln 000...1100055.1 1.0 10.0
in30equallogarithmicbins,withthetotalnumberofgalax- r [ h−1 Mpc] r [ h−1 Mpc]
ies included as an additional fitting point for each sample
selection.
Figure 13 shows the results of fitting the M ≤ −19 Figure 15. Galaxy correlation function fitting results for the
r
galaxy sample. The EDHOD model achieves a good overall Mr ≤−20,−21,−21.5samplesinthesameformatasinFig.13.
fit to the HW13 ξ (r), with fluctuating deviations up to
gg
∼ 5% in the region of the 1-halo to 2-halo crossover. The
best-fitHODmodelhasalargescaleoffsetof10%inξ (r) alogs. In each case we use the simulation’s true ξ (r)
gg mm
(5%inb );whileseveralHOD parameterscanbeadjusted and the ξ (r) and ξ (r) computed numerically from the
g gg gm
toincreasethelargescalebias,doingsowouldalterthesmall corresponding catalog, calculating r (r) from Eqn. 4. At
gm
scaleξ (r)inawaythatworsenstheoverallfit.Thiskindof r > 1 h−1Mpc, the EDHOD and HOD models reproduce
gg
offsetcouldbeadiagnosticforgalaxyassemblybias,butwe the HW13 result to 1 % or better, despite the 5-10 % de-
have not explored whether it can be erased by giving more viations in ξ (r). For this sample, all three models predict
gg
freedom to the assumed radial profile of satellites within r (r) very close to one on these scales. Within the 1-halo
gm
halos. The right panel of Fig. 13 shows ξ (r) predicted by regime, the EDHOD and HOD fits continue to track the
gm
theHODorEDHODmodelthatbestfitsξ (r).Deviations HW13 result, with deviations of a few percent.
gg
from the HW13 ξ (r) are similar to those for ξ (r), but Figure 15 shows similar results for the M ≤
gm gg r
reduced in magnitude by a factor of two. −20,−21, and −21.5 samples. Results in § 2 show that
Figure14comparesthecross-correlationcoefficientsfor the impact of galaxy assembly bias decreases with increas-
M ≤ −19 samples in the HW13, HOD, and EDHOD cat- ingluminositythreshold.Consistentwiththisbehavior,the
r
MNRAS000,1–??(2015)