Populating Dark Matter Haloes with Galaxies Comparing the 2dFGRS with Mock Galaxy Redshift

a r X i v :a s t r o -p h /0303524v 3 20 F e

b 2004Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 2February 2008(MN L A T E X style ?le v1.4)

Populating Dark Matter Haloes with Galaxies:Comparing the 2dFGRS with Mock Galaxy Redshift Surveys

Xiaohu Yang 1,2,H.J.Mo 2,3,Y.P.Jing 4,Frank C.van den Bosch 3,YaoQuan Chu 1?

1Center

for Astrophysics,University of Science and Technology of China,Hefei,Anhui 230026,China

2Department of Astronomy,University of Massachusetts,Amherst MA 01003-9305,USA

3Max-Planck-Institut f¨u r Astrophysik Karl-Schwarzschild-Strasse 1,85748Garching,Germany

4Shanghai Astronomical Observatory;the Partner Group of MPA,Nandan Road 80,Shanghai 200030,China ABSTRACT In two recent papers,we developed a powerful technique to link the distribution of galaxies to that of dark matter haloes by considering halo occupation numbers as func-tion of galaxy luminosity and type.In this paper we use these distribution functions to populate dark matter haloes in high-resolution N -body simulations of the standard ΛCDM cosmogony with ?m =0.3,?Λ=0.7,and σ8=0.9.Stacking simulation boxes of 100h ?1Mpc and 300h ?1Mpc with 5123particles each we construct Mock Galaxy Redshift Surveys out to a redshift of z =0.2with a numerical resolution that guaran-tees completeness down to 0.01L ?.We use these mock surveys to investigate various clustering statistics.The predicted two-dimensional correlation function ξ(r p ,π)re-veals clear signatures of redshift space distortions.The projected correlation functions for galaxies with di?erent luminosities and types,derived from ξ(r p ,π),match the ob-servations well on scales larger than ～3h ?1Mpc.On smaller scales,however,the model overpredicts the clustering power by about a factor two.Modeling the “?nger-of-God”e?ect on small scales reveals that the standard ΛCDM model predicts pairwise veloc-ity dispersions (PVD)that are ～400km s ?1too high at projected pair separations of ～1h ?1Mpc.A strong velocity bias in massive haloes,with b vel ≡σgal /σdm ～0.6(where σgal and σdm are the velocity dispersions of galaxies and dark matter particles,respectively)can reduce the predicted PVD to the observed level,but does not help to resolve the over-prediction of clustering power on small scales.Consistent results can be obtained within the standard ΛCDM model only when the average mass-to-light ratio of clusters is of the order of 1000(M /L)⊙in the B -band.Alternatively,as we show by a simple approximation,a ΛCDM model with σ8?0.75may also reproduce the observational results.We discuss our results in light of the recent WMAP results and the constraints on σ8obtained independently from other observations.

Key words:dark matter -large-scale structure of the universe -galaxies:haloes -

methods:statistical

1INTRODUCTION

The distribution of galaxies contains important information

about the large scale structure of the matter distribution.On

large,linear scales the galaxy power spectrum is believed to

be proportional to the matter power spectrum,therewith

providing useful information regarding the initial conditions

of structure formation,i.e.,regarding the power spectrum

of primordial density ?uctuations.On smaller,non-linear

scales the distribution and motion of galaxies is governed

by the local gravitational potential,which is cosmology de-

pendent.One of the main goals of large galaxy redshift sur-

?E-mail:xhyang@0998eca3b0717fd5360cdc6c

veys is therefore to map the distribution of galaxies as ac-curately as possible,over as large a volume as possible.The Sloan Digital Sky Survey (SDSS;York et al.2000)and the 2degree Field Galaxy Redshift Survey (2dFGRS;Colless et al.2001)are two of the prime examples.These surveys,which are currently being completed,will greatly enhance and improve our knowledge of large-scale structure and will become the standard data sets against which to test our cosmological and galaxy formation models for the decade to come.However,two e?ects complicate a straightforward inter-pretation of the data.First of all,the distribution of galaxies is likely to be biased with respect to the underlying mass density distribution.This bias is an imprint of various com-c 0000RAS

2Yang,Mo,Jing,van den Bosch&Chu

plicated physical processes related to galaxy formation such as gas cooling,star formation,merging,tidal stripping and heating,and a variety of feedback processes.In fact,it is expected that the bias depends on scale,redshift,galaxy type,galaxy luminosity,etc.(Kau?mann,Nusser&Stein-metz1997;Jing,Mo&B¨o rner1998;Somerville et al.2001; van den Bosch,Yang&Mo2003).Therefore,in order to translate the observed clustering of galaxies into a measure for the clustering of(dark)matter,one needs to either un-derstand galaxy formation in detail,or use an alternative method to describe the relationship between galaxies and dark matter(haloes).One of the main goals of this pa-per is to advocate one such method and to show its poten-tial strength for advancing our understanding of large scale structure.

Secondly,because of the peculiar velocities of galaxies, the clustering of galaxies observed in redshift-space is dis-torted with respect to the real-space clustering(e.g.,Davis &Peebles1983;Kaiser1987;Regos&Geller1991;van de Weygaert&van Kampen1993;Hamilton1992).On small scales,the virialized motion of galaxies within dark mat-ter haloes smears out structure along the line-of-sight(i.e., the so-called“?nger-of-God”e?ect).On large scales,coher-ent?ows induced by the gravitational action of large scale structure enhance structure along the line-of-sight.Both ef-fects cause an anisotropy in the two-dimensional,two-point correlation functionξ(r p,π),with r p andπthe pair sepa-rations perpendicular and parallel to the line-of-sight,re-spectively.The large-scale?ows compress the contours of ξ(r p,π)in theπdirection by an amount that depends on β≡?0.6m/b.The small-scale peculiar motions implies that ξ(r p,π)is convolved in theπ-direction by the distribution of pairwise velocities,f(v12).Thus,the detailed structure of ξ(r p,π)contains information regarding the Universal matter density?m,the(linear)bias of galaxies b,and the pairwise velocity distribution f(v12).

From the above discussion it is obvious that understand-ing galaxy bias is an integral part of understanding large scale structure.One way to address galaxy bias without a detailed theory for how galaxies form is to model halo occu-pation statistics.One simply speci?es halo occupation num-bers, N(M) ,which describe how many galaxies on average occupy a halo of mass M.Many recent investigations have used such halo occupation models to study various aspects of galaxy clustering(Jing,Mo&B¨o rner1998;Peacock&Smith 2000;Seljak2000;Scoccimarro et al.2001;White2001;Jing, B¨o rner&Suto2002;Bullock,Wechsler&Somerville2002; Berlind&Weinberg2002;Scranton2002;Kang et al.2002; Marinoni&Hudson2002;Zheng et al.2002;Kochanek et al.2003).In two recent papers,Yang,Mo&van den Bosch (2003;hereafter Paper I)and van den Bosch,Yang&Mo (2003;hereafter Paper II)have taken this halo occupation approach one step further by considering the occupation as a function of galaxy luminosity and type.They intro-duced the conditional luminosity function(hereafter CLF)Φ(L|M)d L,which gives the number of galaxies with lumi-nosities in the range L±d L/2that reside in haloes of mass M.The advantage of this CLF over the halo occupation function N(M) is that it allows one to address the clus-tering properties of galaxies as function of luminosity.In ad-dition,the CLF yields a direct link between the halo mass function and the galaxy luminosity function,and allows a straightforward computation of the average luminosity of galaxies residing in a halo of given mass.Therefore,Φ(L|M) is not only constrained by the clustering properties of galax-ies,as is the case with N(M) ,but also by the observed LFs and the halo mass-to-light ratios.

In Papers I and II we used the observed LFs and the luminosity-and type-dependence of the galaxy two-point correlation function to constrain the CLF in the standard ΛCDM cosmology.In this paper,we use this CLF to pop-ulate dark matter haloes in high-resolution N-body simula-tions.The‘virtual Universes’thus obtained are used to con-struct mock galaxy redshift surveys with volumes and appar-ent magnitude limits similar to those in the2dFGRS.This is the?rst time that realistic mock surveys are constructed that(i)associate galaxies with dark matter haloes,(ii)are independent of a model for how galaxies form,and(iii)au-tomatically have the correct galaxy abundances and correla-tion lengths as function of galaxy luminosity and type.In the past,mock galaxy redshift surveys were constructed either by associating galaxies with dark matter particles(rather than haloes)using a completely ad hoc bias scheme(Cole et al.1998),or by linking semi-analytical models for galaxy formation(with all their associated uncertainties)to the merger histories of dark matter haloes derived from numeri-cal simulations(Kau?mann et al.1999;Mathis et al.2002).

We use our mock galaxy redshift survey to investigate a number of statistical measures of the large scale distribu-tion of galaxies.In particular,we focus on the two-point cor-relation function in redshift space,its distortions on small and large scales,and the galaxy pairwise peculiar veloci-ties.Where possible we compare our predictions with the 2dFGRS and we discuss the sensitivity of these clustering statistics to several details regarding the halo occupation statistics.We show that the halo occupation obtained ana-lytically can reliably be implemented in N-body simulations. We?nd that the standardΛCDM model,together with the halo occupation we have obtained,can reproduce many of the observational results.However,we?nd signi?cant dis-crepancy between the model predictions and observations on small scales.We show that to get consistent results on small scales,either the mass-to-light ratios for clusters of galaxies are signi?cantly higher than normally assumed,or the linear power spectrum has an amplitude that is signi?cantly lower than its‘concordance’value.

This paper is organized as follows.In Section2we re-view the CLF formalism developed in papers I and II.Sec-tion3introduces the N-body simulations and describes our method of populating dark matter haloes in these simula-tions with galaxies of di?erent type and luminosity.Section4 investigates several clustering statistics in real-space and fo-cuses on the accuracy with which mock galaxy distributions can be constructed using our CLF formalism.In Section5 we use these mock galaxy distributions to construct mock galaxy redshift surveys that are comparable in size with the 2dFGRS.We extract the redshift-space two-point correla-tion function from this mock redshift survey,investigate its anisotropies induced by the galaxy peculiar motions,and compare our results to those obtained from the2dFGRS by Hawkins et al.(2003).In section6we discuss possible ways to alleviate the discrepancy between model and observations on small scales,and we summarize our results in Section7.

c 0000RAS,MNRAS000,000–000

Populating Dark Matter Haloes with galaxies3 2THE CONDITIONAL LUMINOSITY

FUNCTION

In Paper I we developed a formalism,based on the condi-

tional luminosity functionΦ(L|M),to link the distribution

of galaxies to that of dark matter haloes.We introduced a

parameterized form forΦ(L|M)which we constrained using

the LF and the correlation lengths as function of luminos-

ity.In Paper II we extended this formalism by constructing

separate CLFs for the early-and late-type galaxies.In this

paper we use these results to populate dark matter haloes,

obtained from large numerical simulations,with both early-

and late-type galaxies of di?erent luminosities.For com-

pleteness,we brie?y summarize here the main ingredients of

the CLF formalism,and refer the reader to papers I and II

for more details.

The conditional luminosity function is parameterized by

a Schechter function:

Φ(L|M)d L=?Φ?

?L? ?αexp(?L/?L?)d L,(1)

where?L?=?L?(M),?α=?α(M)and?Φ?=?Φ?(M)are all functions of halo mass M?.Following Papers I and II,we write the average total mass-to-light ratio of a halo of mass M as

M2 M M1 ?γ1+ M

?L?(M)=

1

L 0f(?α)

M M2 γ3 ,(3)

with

f(?α)=Γ(?α+2)

g(L)h(M)

if g(L)h(M)>1(10)

is to ensure that f late(L,M)≤1.We adopt

g(L)=

?Φ

late

(L) ∞0Φ(L|M)h(M)n(M)d M(11)

where n(M)is the halo mass function(Sheth&Tormen

1999;Sheth,Mo&Tormen2001),?Φlate(L)and?Φ(L)corre-

spond to the observed LFs of the late-type and entire galaxy

samples,respectively,and

h(M)=max 0,min 1, log(M/M a)

4Yang,Mo,Jing,van den Bosch&Chu

observed correlation lengths as function of both luminosity and type?

3POPULATING HALOES WITH GALAXIES 3.1Numerical Simulations

The main goal of this paper is to use the CLF described in the previous Section to construct mock galaxy redshift sur-veys,and to study a number of statistical properties of these distributions that can be compared with observations from existing or forthcoming redshift surveys.The distribution of dark matter haloes is obtained from a set of large N-body simulations(dark matter only).The set consists of a total of six simulations with N=5123particles each,that have been carried out on the VPP5000Fujitsu supercomputer of the National Astronomical Observatory of Japan with the vectorized-parallel P3M code(Jing&Suto2002).Each sim-ulation evolves the distribution of the dark matter from an initial redshift of z=72down to z=0in aΛCDM‘concor-dance’cosmology.All simulations consider boxes with peri-odic boundary conditions;in two cases L box=100h?1Mpc while the other four simulations all have L box=300h?1Mpc. Di?erent simulations with the same box size are completely independent realizations and are used to estimate errors due to cosmic variance.The particle masses are6.2×108h?1M⊙and1.7×1010h?1M⊙for the small and large box simulations, respectively.One of the simulations with L box=100h?1Mpc has previously been used by Jing&Suto(2002)to derive a triaxial model for density pro?les of CDM haloes,and we re-fer the reader to that paper for complementary information about the simulations.In what follows we refer to simula-tions with L box=100h?1Mpc and L box=300h?1Mpc as L100and L300simulations,respectively.

Dark matter haloes are identi?ed using the standard friends-of-friends(FOF)algorithm(Davis et al.1985)with a linking length of0.2times the mean inter-particle separa-tion.Haloes obtained with this linking length have a mean overdensity of～180(Porciani,Dekel&Ho?man2002),in good agreement with the de?nition of halo masses used in our CLF analysis.For each individual simulation we con-struct a catalogue of haloes with10particles or more,for which we store the mass(number of particles),the position of the most bound particle,and the halo’s mean velocity and velocity dispersion.Note that the FOF algorithm can sometimes select poor systems(those with small number of ?Note that the parameters listed here are slightly di?erent from those given in the orignal version of Paper II,as they are based on a corrected version of the galaxy luminosity function.As shown in Paper I,a change in the overall amplitude of the luminosity function in the?tting has some e?ect on the best-?t values of the correlation lengths.This is due to the combination of the following two e?ects.First,our model assumes a?xed mass-to-light ratio for massive haloes and so a change in the amplitude of the luminosity function leads to a change in the relative number of galaxies in small/large haloes.Second,although the correlation length as a function of luminosity was used as input in our?tting of the conditional luminosity function,there is some freedom for the‘best-?t’values of the correlation lengths to change in the ?tting,because the errorbars on the observed correlation lengths are quite large.particles)that are spurious and have abnormally large ve-

locity dispersions.We therefore have made a check to make sure that the particles assigned to a system according to the FOF algorithm are gravitationally bound.Our test showed

that this correction is important only for low-mass haloes, and that it has almost no e?ect on our results.The left panel of Fig.1plots the z=0halo mass functions for one of the

L100simulations and for one of the L300simulations(his-tograms),with all spurious haloes erased.For comparison, we also plot(solid line)the analytical halo mass function

given in Sheth&Tormen(1999)and Sheth,Mo&Tormen (2001)§.The agreement is remarkably good,both between the two simulations and between the simulation results and the theoretical prediction.

Note that our choice for box sizes of100h?1Mpc and 300h?1Mpc is a compromise between high mass resolution and a su?ciently large volume to study the large-scale struc-ture.The impact of mass resolution is apparent from con-sidering the conditional probability function

P(M|L)d M=

Φ(L|M)

Populating Dark Matter Haloes with galaxies

5

Figure 1.The left-hand panel plots the halo mass functions of the numerical simulations discussed in the text (histograms).The mass function with a low mass cut at about 2×1011h ?1M ⊙corresponds to a simulation with L box =300h ?1Mpc,while the other corresponds to a L 100simulation with L box =100h ?1Mpc.The solid curve is the Sheth,Mo &Tormen (2001)mass function which is shown for comparison.Note the excellent agreement,both between the two simulations and between the simulation results and the theoretical prediction.The right-hand panel plots the conditional probability distributions P (M |L )for four di?erent luminosities as indicated.L ?=1.1×1010h ?2L ⊙is the characteristic luminosity of the Schechter ?t to the 2dFGRS LF of Madgwick et al.(2002).Combining these conditional probability distributions with the halo mass functions shown in the left-hand panel gives an indication of the completeness level that can be obtained with both the L 100and L 300simulations (see text).

P (N |M )(with N an integer)of which N (M ) gives the

mean,i.e.,

N (M ) =∞ N =0

N P (N |M )(15)

As a simple model we adopt

P (N |M )= N ′+1? N (M ) if N =N ′ N (M ) ?N ′

if N =N ′+10otherwise (16)Here N ′is the largest integer smaller than N (M ) .Thus,

the actual number of galaxies in a halo of mass M is ei-

ther N ′or N ′+1.This particular model for the distribution

of halo occupation numbers is supported by semi-analytical

models and hydrodynamical simulations of structure forma-

tion (Benson et al.2000;Berlind et al.2003)which indicate

that the halo occupation probability distribution is narrower

than a Poisson distribution with the same mean.In addition,

distribution (16)is successful in yielding power-law correla-

tion functions,much more so than for example a Poisson

distribution (Benson et al.2000;Berlind &Weinberg 2002).3.3Assigning galaxies their luminosity and type

Since the CLF only gives the average number of galaxies

with luminosities in the range L ±d L/2in a halo of mass

M ,there are many di?erent ways in which one can assign

luminosities to the N i galaxies of halo i ,and yet be consis-

tent with the CLF.The simplest approach would be to sim-

ply draw N i luminosities (with L >L min )randomly from Φ(L |M ).Alternatively,one could use a more deterministic approach,and,for instance,always demand that the j th brightest galaxy has a luminosity in the range [L j ,L j ?1].Here L j is de?ned such that a halo has on average j galax-ies with L >L j ,i.e., ∞L j

Φ(L |M )dL =j.(17)We adopt an intermediate approach in most of our dis-cussion,giving special treatment only to the one bright-est galaxy per halo.The luminosity of this so-called “cen-tral”galaxy,L c ,is drawn from Φ(L |M )with the restriction L >L 1and thus has an expectation value of L c (M ) = ∞L 1

Φ(L |M )L d L =?Φ??L ?Γ(?α+2,L 1/?L ?),(18)The remaining N i ?1galaxies are referred to as “satel-lite”galaxies and are assigned luminosities in the range L min

6Yang,Mo,Jing,van den Bosch &

Chu

Figure 2.Projected dark matter/galaxy distributions of a 100×100×10h ?1Mpc slice in one of the L 100mock galaxy distributions.The panels show (clockwise from top-left)the dark matter particles,all galaxies (early plus late),early-type galaxies,and late-type galaxies.Galaxies are weighted by their luminosities.Note how the galaxies trace the large scale structure of the dark matter,and how early-type galaxies are more strongly clustered than late-type galaxies.

3.4Assigning galaxies their phase-space

coordinates

Once the population of galaxies has been assigned luminosi-

ties and types,they need to be assigned a position within

their halo as well as a peculiar velocity.The central galaxy is

assumed to be located at the “center”of the corresponding

dark halo,which we associate with the position of the most

bound particle,and its peculiar velocity is set equal to the

mean halo velocity (cf.Yoshikawa,Jing &B¨o rner 2003).For

the satellite galaxies we follow two di?erent approaches.In

the ?rst,we assign the N i ?1satellites the positions and peculiar velocities of N i ?1randomly selected dark matter particles that are part of the FOF halo under consideration.

This thus corresponds to a scenario in which satellite galax-ies are completely unbiased with respect to the density and velocity distribution of dark matter particles in FOF haloes.We refer to satellite galaxies populated this way as “FOF satellites”.We also consider a more analytical model for the satel-lite distribution.This allows us ?rst of all to assess whether a simple analytical description can be found to describe the population of satellite galaxies,and secondly,provides us with a simple framework to investigate the sensitivity of various clustering statistics to details regarding the density and velocity bias of satellite galaxies.We assume that the number density distribution of satellite galaxies follows a NFW density distribution (Navarro,Frenk &White 1997):

c 0000RAS,MNRAS 000,000–000

Populating Dark Matter Haloes with galaxies

7

Figure3.Same as Fig.2,but for a300×300×20h?1Mpc slice taken from one of the L300mock galaxy distributions.ρ(r)=

ˉδˉρ

3c3√

2σ2

gal .(21)

Here v j is the velocity relative to that of the central galaxy

along axis j,andσgal is the one-dimensional velocity dis-

persion of the galaxies,which we set equal to that of the

dark matter particles,σdm,in the halo under consideration.

We refer to satellite galaxies populated this way as“NFW

satellites”.

c 0000RAS,MNRAS000,000–000

8Yang,Mo,Jing,van den Bosch &

Chu

Figure 4.The luminosity functions of the mock galaxies constructed from the L 100(left)and L 300(right)halo catalogues (solid lines).For comparison,we also plot the LFs obtained by Madgwick et al.(2002)for all galaxies (circles),for late-type galaxies (triangles)and for early-type galaxies (stars).For clarity,the latter two LFs have been shifted down by one and two orders of magnitude in the y -direction,respectively.Except for incompleteness e?ects due to the sampling of the halo mass function (see text for details),the mock galaxy distributions have LFs that are in excellent agreement with the data.

4RESULTS IN REAL SPACE Fig.2and 3show slices of mock galaxy distributions (here-

after MGDs)constructed from L 100and L 300simulations,

respectively.Satellite galaxies are assigned positions and ve-

locities using the NFW scheme outlined above.Results are

shown for all galaxies (upper right panels),and separately

for early types (lower right panels)and late types (lower left

panels).For comparison,we also show the distribution of

dark matter particles in the upper left panels.Note how the

large scale structure in the dark matter distribution is de-

lineated by the distribution of galaxies,and that early-type

galaxies are more strongly clustered than late-type galaxies.

In this section we discuss the general,real-space prop-

erties of these MGDs.In Section 5below we construct mock

galaxy redshift surveys to investigate the impact of redshift

distortions.The main goal of this section,however,is to in-

vestigate with what accuracy the combination of numerical

simulations and our CLF analysis can be used to construct

self-consistent mock galaxy distributions.In particular,we

want to examine to what accuracy these MGDs can recover

the input used to constrain the CLFs.Note that this is not

a trivial question.The CLF modeling is based on the halo

model,which only yields an approximate description of the

dark matter distribution in the non-linear regime (see dis-

cussions in Cooray &Sheth 2002and Hu?enberger &Seljak

2003).In addition,as described in Section 3,the CLF alone

does not yield su?cient information to construct MGDs,

and we had to make additional assumptions regarding the

distribution of galaxies within individual haloes.A further goal of this section,therefore,is to investigate how these

assumptions impact on the clustering statistics.

4.1The luminosity function The CLFs used to construct the MGDs shown in Fig.2and 3are constrained by the 2dFGRS luminosity functions for early-and late-type galaxies obtained by Madgwick et al.(2002).Therefore,as long as the halo mass function is well sampled by the simulations,the LFs of our MGDs should match those of Madgwick et al.(2002).Fig.4shows a comparison between the 2dFGRS LFs (symbols with er-rorbars)and the ones recovered from the MGDs (solid lines).To emphasize the level of agreement between the recov-ered LFs and the input LFs,Fig.5plots the ratio between the two.Over a large range of luminosities,the recovered LFs match the observational input extremely well.In the L 300simulation,however,the LFs are under-estimated for L <～3×109h ?2L ⊙(M b J ?5log h >～?18.4).This owes to the absence of haloes with M <～2×1011h ?1M ⊙(see Fig.1).Note how this discrepancy sets in at higher L for the late-type galaxies than for the early-types,because the latter are preferentially located in more massive haloes.For the early-types the L 300mock is virtually complete down to M b J ?5log h ??17(see Fig.10of Paper II),re?ecting the fact that only a very small fraction of the early-type galaxies brighter than this magnitude reside in haloes below the mass resolution limit.In the L 100simulations,on the other hand,the LFs accurately match the data down to the faintest lu-minosities,but here the MGD underestimates the LFs at

c 0000RAS,MNRAS 000,000–000

Populating Dark Matter Haloes with galaxies

9

Figure 6.Two point correlation functions for dark matter particles (left panel)and mock galaxies (right panel).The dotted and dashed lines correspond to results from the L 300and L 100simulations,respectively.The solid line in the left panel corresponds to the evolved,non-linear correlation function for the dark matter obtained by Smith et al.(2003),and is shown for comparison.Due to the limited box-sizes,the L 300(L 100)simulations slightly over (under)predict the correlation power on large scales with respect to Smith et al.’s model.The 2PCFs in the right panel are calculated for galaxies with absolute magnitudes M b J ?5log h

the bright end (M b J ?5log h <～?22).This owes to the lim-ited boxsize,which causes the number of massive haloes (the

main hosts of the brightest galaxies)to be underestimated

(cf.Fig.1).Note that even the LFs of the L 300simulations

underestimate the observed number of bright galaxies.This,

re?ects a small inaccuracy of our CLF to accurately match

the observed bright end of the LFs (see paper II).4.2The real-space correlation function In addition to the LFs of early-and late-type galaxies,the CLFs used here to construct our MGDs are also con-strained by the luminosity and type dependence of the cor-relation lengths as measured from the 2dFGRS by Norberg et al.(2002a).Here we check to what degree this “input”is recovered from the MGDs.The left panel of Fig.6plots the real-space two-point correlation functions (2PCFs)for dark matter particles in the L 100(dashed line)and L 300(dotted line)simulations.The solid line corresponds to the evolved,non-linear dark matter correlation function of Smith et al.(2003)and is shown for comparison ?.As one can see,on large scales (r >～6h ?1Mpc)the correlation amplitude obtained from the L 100simulations is systematically lower than both that obtained from the L 300simulations and that obtained from ?In ?tting the CLF we have used this function to compute the

correlation length of the dark matter (see Paper II).

the ?tting formula of Smith et al.,suggesting that the box-size e?ect is non-negligible in the L 100simulations.Note also that the large scale correlation amplitude given by the L 300simulations is slightly higher than Smith et al.’s model.It is unclear if this discrepancy is due to the inaccuracy of the ?tting formula,or due to cosmic variance in the present simulations.As we will see below,this discrepancy limits the

accuracy of model predictions.

The right-hand panel of Fig.6plots the 2PCFs for the

galaxies in the L 100(dashed line)and L 300(dotted line)

MGDs.Note how the galaxies reveal the same trend on large

scales as the dark matter particles,with larger correlations

in the L 300than in the L 100MGD.

Fig.7shows the correlation lengths r 0as function of

luminosity for all (upper panel),early-type (middle panel)

and late-type (lower panel)galaxies.These have been ob-

tained by ?tting ξ(r )with a power law relation of the form

ξ(r )=(r/r 0)?γover the same range of scales as used by

Norberg et al.(2002a).Solid squares and open stars cor-

respond to correlation lengths obtained from the L 300and

L 100MGDs,respectively.Note that the errobars on the pre-

dicted correlation lengths are based on the scatter among in-

dependent simulations boxes.They are signi?cantly smaller

than the errorbars on the observational data,because the

model predictions are based on real-space correlation func-

tions,while the observational results are based on projected

correlation functions in redshift space.The agreement with

the data (open circles)is reasonable,even though several

systematic trends are apparent.In particular,the correla-c 0000RAS,MNRAS 000,000–000

10Yang,Mo,Jing,van den Bosch &

Chu Figure 5.The ratio of the luminosity function of mock galax-ies,Φmock (L ),to that of the 2dFGRS,Φ2dFGRS (L )(taken from Madgwick et al.2002).The thin errorbars indicate the errors on Φ2dFGRS (L ).The thick solid (dashed)lines correspond to the LFs obtained from the L 100(L 300)simulations.The errorbars for the mock galaxies are obtained from the 1-σvariance of the two L 100and the four L 300simulations,respectively.See text for discussion.

tion lengths obtained from the L 300simulation are slightly higher than the observations while the opposite applies to the L 100simulation.These discrepancies are due to two ef-fects.First of all,as shown in Fig.6the dark matter on large scales is more strongly clustered in the L 300simulations than in the L 100simulations.That this can account for most of the di?erences between the scale-lengths obtained from the L 300and L 100simulations,is illustrated by the dotted and solid horizontal lines,which indicate the correlation lengths of the dark matter particles in the L 300and L 100simulations,respectively.Secondly,the measured correlation lengths cor-respond to a non-zero,median redshift which is larger for the more luminous galaxies.In determining the best-?t param-eters for the CLF this redshift e?ect is taken into account (see Papers I and II).However,in the construction of our MGDs,we only use the dark matter distribution at z =

0.Figure 7.The real space correlation length,r 0,as a function of galaxy luminosity and type.Top panel shows the results for the combined sample of early-plus late-type galaxies,while the middle (bottom)panel shows results for the early (late)type galaxies only.Solid squares and stars correspond to the corre-lation lengths obtained from the L 300and L 100simulations,re-spectively.The errorbars correspond to the 1-σvariance from the two (four)independent realizations for L 100(L 300).We also indi-cate (open circles with errorbars)the correlation lengths obtained from the 2dFGRS by Norberg et al.(2002a).In the upper panel,we also plot the correlation lengths for dark matter particles for L 100(solid line)and L 300(dotted line)simulations.Although the agreement between data and MGDs is reasonable there are small but signi?cant di?erences.The reason for these discrepancies is discussed in the text.As discussed in Paper I,this can over-estimate the correla-tion length by about 10%.Given these sources of systematic errors,one should be careful not to over-interpret any dis-crepancy between the correlation lengths in the mock survey and those obtained from real redshift distributions.In order to investigate the sensitivity of the 2PCF in the MGDs to the way we assign luminosities and phase-

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Populating Dark Matter Haloes with galaxies

11

Figure 8.The ratio of the 2PCF ξ(r )in three MGDs compared to that of our ?ducial MGD.The only di?erence among these various MGDs is the way that we assign luminosities and phase-space coordinates to the galaxies.Solid (dotted)lines correspond to a MGD in which we use a deterministic (random)method to assign galaxies their luminosities (see Section 3.3for de?nitions).In the MGD corresponding to the dashed line we use the inter-mediate,?ducial method to assign luminosities,but here we use ‘FOF satellites’rather than ‘NFW satellites’(see Section 3.4for de?nitions).Results are shown for galaxies in three di?erent mag-nitude bins (as indicated)in one of the L 300simulations.However,results for the L 100simulations look virtually identical.

space coordinates to the galaxies within the dark matter haloes,we construct MGDs using one of the L 300simula-tions with di?erent models for the luminosity assignment and spatial distribution of satellite galaxies within haloes.We have con?rmed that using one of the L 100simulations in-stead yields the same results.We ?rst test the impact of the luminosity assignment.Here,instead of the ?ducial model for the luminosity assignment (the intermediate approach discussed in Section 3.3),we use both the deterministic and random assignments (see Section 3.3for de?nitions)to con-struct the MGDs.In Fig.8we shown the ratios between the correlation functions obtained from these MGDs and those obtained from the ?ducial MGD.For bright galaxies,the deterministic model gives the lowest amplitudes on small scales (r <～1h ?1Mpc),while the random model gives the highest amplitudes.This is expected.The mean number of bright galaxies in a typical halo is not much larger than 1and so not many close pairs of bright galaxies are expected in the deterministic model.More such pairs are expected in the random model because more than one galaxies in a typical halo can be assigned a large luminosity due to ran-dom ?uctuations.The dashed lines in Fig.8correspond to a MGD with FOF satellites (see Section 3.4).The agreement of the 2PCFs between this MGD with ‘FOF satellites’and our ?ducial MGD indicate that the spherical NFW model is a good approximation of the average density distribution of dark matter haloes.We have also tested the impact of changing the concentration of galaxies,c g ;increasing (de-creasing)c g with respect to the dark matter halo concen-tration,c ,increases (decreases)the 2PCFs on small scales (r <～1h ?1Mpc).However,even when changing the ratio c g /c by a factor of two,the amplitude of this change is smaller than the di?erences resulting from changing the luminosity assignment.All in all,changes in the way we assign luminosities and phase-space coordinates to the galaxies only have a mild im-pact on the 2PCFs,and only at small scales <～1h ?1Mpc.This is in good agreement with Berlind &Weinberg (2002)who have shown that these e?ects are much smaller than changes in the second moment of the halo occupation dis-tributions.For example,assuming a Poissonian P (N |M ),rather than equation (16)has a much larger impact on the 2PCFs than any of the changes investigated above.As we show in Section 5below,with the P (N |M )of equation (16)we obtain correlation functions that are in better agreement with observations,providing empirical support for this par-ticular occupation number distribution.It is interesting to note that although small changes in the way we assign luminosities and phase-space coordinate do not have a big impact on the statistical measurements we are considering here,such changes can lead to quite di?erent results for other statistical measures.As shown in van den Bosch et al.(2004),various statistics of satellite galaxies around bright galaxies can be used to distinguish models that make similar predictions about the clustering on large scales.4.3Pairwise velocities The peculiar velocities of galaxies are determined by the ac-tion of the gravitational ?eld,and so are directly related to the matter distribution in the Universe.Observationally,the properties of galaxy peculiar velocities are inferred from dis-tortions in the correlation function.We defer this discussion to Section 5.Here we derive statistical quantities directly from the simulated peculiar velocities of galaxies.We de?ne the pairwise peculiar velocity of a galaxy pair as v 12(r )≡[v (x +r )?v (x )]·?r ,(22)with v (x )the peculiar velocity of a galaxy at x .The mean pairwise peculiar velocity and the pairwise peculiar velocity dispersion (PVD)are v 12(r ) and σ12(r )≡ [v 12(r )? v 12(r ) ]2 1/2,(23)c 0000RAS,MNRAS 000,000–000

12Yang,Mo,Jing,van den Bosch &

Chu

Figure 9.The mean pairwise velocities (upper panels)and pairwise velocity dispersions (lower panels)estimated from the three-dimensional (real-space)velocities of the mock galaxies and dark matter particles.All results correspond to the L 300simulations only.Left-hand panels compare dark matter particles (solid circles)with galaxies either with NFW satellites (open circles)or with FOF satellites (open stars).Right-hand panels display the galaxy-type dependence for a model with NFW satellites (errorbars indicate the rms-scatter among the four independent L 300simulations).See text for detailed discussion.

where ··· denotes an average over all pairs of separation

r .

In order to gain insight,we compute v 12(r ) and σ12(r )

from the L 300simulations for both dark matter particles and

for galaxies with M b J ?5log h ≤?18.4(which corresponds to the completeness limit of these simulations,see Fig.4).

Results are shown in Fig.9.The upper left panel com-

pares the mean pairwise peculiar velocities of the dark mat-

ter particles (solid circles)with those of two realizations of

the galaxies:one with ‘NFW satellites’(open circles)and

the other with ‘FOF satellites’(stars).At su?ciently small

separations,one probes the virialized regions of dark matter

haloes,and one thus ?nds v 12 =0.At larger separations,one starts to probe the infall regions around the virialized

haloes,yielding negative values for v 12(r ) .Finally,at su?-

ciently large separations v 12(r ) →0due to the large scale

homogeneity and isotropy of the Universe.

Both the dark matter particles and the galaxies from

our MGDs indeed reveal such a behavior,with v 12(r ) peak-

ing at ～3h ?1Mpc.However,there is a markedly strong

di?erence between the v 12(r ) of galaxies in the MGD with

NFW satellites and that of the dark matter.In this particu-

lar MGD,the galaxies experience signi?cantly smaller infall velocities than the dark matter particles.However,this dif-ference between dark matter and galaxies is almost absent in the MGD with FOF satellites.This is due to the fact that in the NFW model,we populate satellites with isotropic velocity dispersions within a sphere of radius r 180.We are thus assuming that the entire region out to r 180is virial-ized in that there is no net infall.However,simple collapse models predict that for our concordance cosmology only the region out to r 340(i.e.,the radius inside of which the aver-age overdensity is 340)is virialized (Bryan &Norman 1998).The di?erence between the MGDs with NFW satellites and FOF satellites indicates that the regions between r 340and r 180are still infalling,resulting in non-zero v 12 .In the lower-left panel,we compare the PVDs for galax-ies and dark matter particles.Here the MGDs with FOF satellites and NFW satellites are fairly similar,and signi?-cantly lower than for the dark matter.This can be under-stood as follows.At small separations,the PVD is a pair weighted measure for the potential well in which dark mat-ter particles (galaxies)reside.For the galaxies in our MGDs the halo occupation number per unit mass,N/M ,decreases with the mass of dark matter haloes (see Paper II).There-fore,the massive haloes (with larger velocity dispersions)

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Populating Dark Matter Haloes with galaxies

13

Figure 10.Distribution of pairwise velocities,f (v 12),for dark matter particles (solid curves),and for mock galaxies in the L 300simulation.Results are shown for four separations r as indicated,and for all galaxies (dot-dashed lines),early-type galaxies (dotted lines)and late-type galaxies (dashed lines).On small scales (r <1h ?1Mpc)the pairwise velocity distributions are symmetric and reveal an obvious exponential form.On larger scales,however,f (v 12)reveals clear asymmetries:for v 12<0the distribution is still close to an exponential,while for v 12>0the distribution more resembles a normal distribution.

contribute relatively less to the PVDs of galaxies.Although

the di?erence between the σ12(r )of the MGDs with FOF

and NFW satellites shows that the PVDs have some de-

pendence on the details regarding the infall regions around

virialized haloes,these e?ects are typically small.

The upper-right and lower-right panels of Fig.9show

how v 12(r ) and σ12(r )depend on galaxy type.Results are

shown for the MGD based on NFW satellites.The mean

velocities for early-type galaxies are larger than those for

late-type galaxies on large scales,but smaller on small scales.

In addition,the PVD of early-type galaxies is higher than

that of late-type galaxies on all scales.All these di?erences

are easily understood as a re?ection of the fact that early-

type galaxies are preferentially located in the larger,more

massive haloes which have larger velocity dispersions and

larger infall velocities.

Fig.10shows the pairwise velocity distributions for

four di?erent separations r ,within a logarithmic interval of

?log r =0.125.On small scales,the distribution is well ?t

by an exponential for both dark matter particles and galax-

ies.This validates the assumption made in earlier analyses

about this distribution (Davis &Peebles 1983;Mo,Jing &

B¨o rner 1993;Fisher et al.1994;Marzke et al.1995).It is also consistent with earlier results obtained from theoreti-cal models and numerical simulations based on dark matter particles (Diaferio &Geller 1996;Sheth 1996;Mo,Jing &B¨o rner 1997;Seto &Yokoyama 1998;Efstathiou et al.1988;Magira,Jing &Suto 2000).For larger separations f (v 12)is skewed towards negative values of v 12,because galaxies tend to approach each other due to gravitational infall.Clearly,a single exponential function is no longer a good approxima-tion to the pairwise peculiar velocity distribution at large separations.Although for v 12<0(infall)the exponential remains remarkably accurate,for v 12>0the pairwise ve-locity distribution reveals a more Gaussian behavior.This may have important implications for the derivation of PVDs (especially at large separations),which typically is based on the assumption of a purely exponential f (v 12).We shall re-turn to this issue in more detail in Section 5.2.5RESULTS IN REDSHIFT SPACE The statistical quantities of galaxy clustering discussed in the previous section are based on real distances between c 0000RAS,MNRAS 000,000–000

14Yang,Mo,Jing,van den Bosch&Chu

Figure11.The stacking geometry of the L100and L300sim-ulation boxes used to construct the MSB mock galaxy redshift surveys.The virtual observer is located at the center of the stack, indicated by a thick solid dot.Note that for MGRSs in the MB set,the stack of6×6×6L100boxes is replaced by a stack of 2×2×2L300boxes.

galaxies in our MGDs.However,because of the peculiar ve-locities of galaxies,such quantities cannot be obtained di-rectly from a galaxy redshift survey.On small scales the viri-alized motion of galaxies within dark matter haloes cause a reduction of the correlation power,while on larger scales the correlations are boosted due to the infall motion of galaxies towards overdensity regions(Kaiser1987;Hamilton1992). As discussed in the introduction,these distortions contain useful information about the Universal density parameter, the bias of galaxies on large(linear)scales,and the pairwise velocities of galaxies.

In this section,we use the MGDs presented above to construct large mock galaxy redshift surveys(hereafter MGRSs).The main goals are to compare various clustering statistics from these mock surveys with observational data from the2dFGRS,and to investigate how the details about the CLF and the distribution of galaxies within haloes im-pact on these statistics.For the model-data comparison we use the large scale structure analysis of Hawkins et al.(2003; hereafter H03),which is based on a subsample of the2dF-GRS consisting of all galaxies located in the North Galactic Pole(NGP)and South Galactic Pole(SGP)survey strips with redshift0.01≤z≤0.20and apparent magnitude b J<19.3.This sample consists of～166,000galaxies cov-ering an area on the sky of～1090deg2.

In order to carry out a proper comparison between model and observation,we aim to construct MGRSs that have the same selections as the2dFGRS.First of all,the survey depth of z max=0.2implies that we need to cover a volume with a depth of600h?1Mpc,i.e.,twice that of our big L300simulations.In principle,we could stack4×4×4identical L300boxes(which have periodic boundary condi-

tions),so that a depth of600h?1Mpc can be achieved in all directions for an observer located at the center of the stack.However,there is one problem with this set-up;as

we have shown in Figs.1and4the L300MGD is only complete down to M b

J?5log h??18.4.Taking account of the apparent magnitude limit of the survey,this implies

that our MGRSs would be incomplete out to a distance of ～350h?1Mpc.We can overcome this problem by using the higher resolution L100simulation,which is complete down to M b

J?5log h??14.We therefore replace the central2×2×2 L300boxes with a stack of6×6×6L100boxes.The?nal lay-

out of our virtual universe is illustrated in Figure11.Unless

stated otherwise,satellite galaxies are assigned to dark mat-ter haloes based on our standard NFW method described in Section3.4.

Observational selection e?ects,which are modelled ac-cording to the?nal public data release of the2dFGRS(see also Norberg et al.2002b),are taken into account using the following steps:

(i)We place a virtual observer at the center of the stack of boxes(the solid dot in Figure11),de?ne a(α,δ)-coordinate frame,and remove all galaxies that are not located in the areas equivalent to the NGP and SGP regions of the2dF-GRS.

(ii)Next,for each galaxy we compute the redshift as

‘seen’by the virtual observer.We take the observational velocity uncertainties into account by adding a random ve-locity drawn from a Gaussian distribution with dispersion 85km s?1(Colless et al.2001).

(iii)We compute the apparent magnitude of each galaxy according to its luminosity and distance.Since galaxies in the2dFGRS were pruned by apparent magnitude before a k-correction was applied,we proceed as follows:We?rst ap-ply a negative k-correction,then select galaxies according to the position-dependent magnitude limit(obtained using the apparant magnitude limit masks provided by the2dFGRS team),and?nally k-correct the magnitudes back to their rest-frame b J-band.Throughout we use the type-dependent k-corrections given in Madgwick et al.(2002).

(iv)To mimic the position-and magnitude-dependent completeness of the2dFGRS,we randomly sample each galaxy using the completeness masks provided by the2dF-GRS team.The incompleteness of the2dFGRS parent sam-ple is taking into account by randomly discarding9%of all mock galaxies(Norberg et al.2002b).

(v)Finally,we mimic the actual selection criteria of the 2dFGRS sample used in H03by restricting the sample to galaxies within the redshift range0.01≤z≤0.20and with completeness≥0.7.

Each MGRS thus constructed contains,on average,

169000galaxies,with a dispersion of～5000due to cosmic variance.The number of galaxies in our mock catalogues are consistent with the observations at the1σlevel.Note that the correlation functions presented by H03have been cor-rected for the observational bias due to?ber collisions,and we therefore do not mimic these e?ects in our MGRSs.

Since we have2L100simulations and4L300simula-tions,we construct2×4=8mock catalogues with di?er-ent combinations of small-and big-box simulations.In what follows,we refer to this set of mock catalogues as MSBs

c 0000RAS,MNRAS000,000–000

Populating Dark Matter Haloes with galaxies

15 Figure12.The distribution of a sub-set of galaxies in one of the MSB mock samples.For clarity,we plot galaxies only in two3-degree slices,one in the‘North Galactic Pole’region(NGP)and the other in the‘South Galactic Pole’region(SGP).Only galaxies with redshifts in the range0.01

(for Mock Small/Big).As an example,Fig.12shows the distribution of a sub-set of galaxies in one of these mock

catalogues.Although each of our MSB catalogs covers an

extremely large volume,and should thus not be very sensi-tive to cosmic variance,it is constructed using simulations

with box sizes of100and300h?1Mpc only.If,for instance,

the L100simulation contains a big cluster,the6×6×6re-production of this box in our MGRSs might introduce some

unrealistic features.Furthermore,as shown in Section4.2

the L100box underestimates the amount of clustering on large scales.Therefore,this set of MGRSs,which replicate

this box27times,might underestimate the clustering on

large scales as well.In order to test the sensitivity of our results to these potential problems,and to have a better

handle on the impact of cosmic variance in our mock sur-

veys,we construct four alternative MGRSs.Each consists of a4×4×4stack of one of the four L300simulations(i.e.,we

replace the6×6×6stack of L100boxes by a2×2×2stack

of L300boxes).In what follows we refer to this set of mock

catalogues as MBs(for Mock Big).These MGRSs,although incomplete for M b

J?5log h>?18.4,should not su?er from the lack of clustering power on large scales.The MSB set,

on the other hand,does not su?er from incompleteness,but instead lacks some large scale power.As we will see below, both the MSB and MB mocks give similar results on large scales,suggesting that the box-size e?ect does not have a signi?cant in?uence on our results.5.1Two-Point Correlation Functions

From our MGRSs we computeξ(r p,π)using the estimator (Hamilton1993)

ξ(r p,π)= RR DD

|l|,r p=

2

(s1+s2)is the line of sight intersecting the pair, and s=s1?s2.Random samples are constructed using two di?erent methods.The?rst uses the mean galaxy num-ber density at redshift z calculated from the2dFGRS LF. The second randomizes the coordinates of all mock galax-ies within the simulation box.Both methods yield indistin-guishable estimates ofξ(r p,π)and in what follows we only use the former.Following H03each galaxy in a pair with redshift separation s is weighted by the factor

w i=

1

16Yang,Mo,Jing,van den Bosch &

Chu

Figure 13.The upper panels show the projected 2PCFs w p (r p )/r p for galaxies of di?erent luminosity and type.The errorbars correspond to the 1-σvariance among distinct MGRSs (i.e.among the 8MSBs for the faintest subsamples,and among the 4MBs for the brightest subsamples).For clarity,the error bars are only plotted for the brightest and faintest subsamples.The lower panels plot the ratios of

these w p (r p )to that of a reference sample.The reference sample contains all galaxies within the magnitude range ?19.5>M ′b J >?20.5(with M ′b J

=M b J ?5log h ).Note that the faintest subsamples,which are impacted by the boxsize e?ect of the L 100simulation,reveal

a ‘break’at r p ?10h ?1Mpc.with n (z )the number density distribution as function of red-shift and J 3(s )= s 0ξ(s ′)s ′2d s ′.Hence each galaxy-galaxy,

random-random,and galaxy-random pair is given a weight

w i w j .We substitute ξ(s ′)with a power law using the same

parameters as in H03.This redshift dependent weighting

scheme is designed to minimize the variance on the esti-

mated correlation function (Davis &Huchra 1982;Hamilton

1993).

Since the redshift-space distortions only a?ect π,the

projection of ξ(r p ,π)along the πaxis can get rid of these

distortions and give a function that is more closely related

to the real-space correlation function.In fact,this projected

2PCF is related to the real-space 2PCF through a simple

Abel transform

w p (r p )= ∞

?∞ξ(r p ,π)d π=2

∞r p

ξ(r )r d r r 2?r 2p (27)(Davis &Peebles 1983).Therefore,if the real-space 2PCF

is a power-law,ξ(r )=(r 0/r )γ,the projected 2PCF w (r p )

can be written as

w p (r p )=√Γ(γ/2) r 0

Populating Dark Matter Haloes with galaxies

17

Figure 14.The correlation lengths,r 0,and slopes,γ,of the power-laws that best ?t the projected correlation functions over the range 2≤r p ≤15h ?1Mpc (solid squares).The results for the 2faintest luminosity bins are based on the mean and variance of the sample of 8MSB mocks,while results for the other bins are based on the mean and variance of the sample of 4MB mocks.Open circles with errorbars correspond to the 2dFGRS data of Norberg et al.(2002a),and are shown for comparison.Except for a systematic overestimate of the correlation lengths,the cause of which has been discussed in Section 4.2,there is good agreement between our MGRSs and the 2dFGRS.

M b J ?5log h >?18.5(solid lines)are obtained from the MSB set.As discussed in Paper II,the projection signi?-

cantly washes out the features in the real-space 2PCFs at

～2h ?1Mpc,and the projected 2PCFs better resemble a

power-law.The exception is the w p (r p )for the faintest sub-

sample of galaxies,where the ‘break’mentioned above is

clearly visible.To highlight the luminosity and type depen-

dence of w p (r p ),the lower panels of Fig.13plot the ratios of

w p (r p )to that of a reference sample de?ned as all (early-type

plus late-type)galaxies with ?19.5>M b J ?5log h >?20.5.For a given luminosity,the correlation amplitude is higher,

and the slope is steeper,for early-type galaxies than for late-

type galaxies.Signi?cant changes in the slope (and thus

deviations from a perfect power-law)occur at separations

r p ～2h ?1Mpc,which is at least qualitatively in agreement

with recent results from the SDSS (Zehavi et al.2003).

In order to facilitate a more direct comparison with

the 2dFGRS data,we ?t a single power-law relation of

the form (28)to these w p (r p )over the range 2h ?1Mpc <

r p <15h ?1Mpc.This range is also adopted by Norberg et

al.(2002a)when ?tting the projected 2PCFs obtained from

the 2dFGRS.Fig.14plots the real-space correlation lengths

r 0and the slopes γthus obtained as function of luminosity and galaxy type.The agreement between our MGRSs and the 2dFGRS is acceptable.The slight but systematic over-estimate of r 0is due to the e?ects discussed in Section 4.2.We now turn to a comparison of the projected correla-tion function for the entire,?ux limited surveys.The upper-left panel of Fig.15compares the w p (r p )obtained from our 8MSB and 4MB MGRSs with that of the 2dFGRS obtained by H03.The projected correlation functions from our MSBs and MBs agree well with each other (i.e.,the 1-σerrorbars overlap),and,at r p >～3h ?1Mpc,with the 2dFGRS results.Note that at r p >～10h ?1Mpc the w p (r p )obtained from the MB mocks is slightly larger than that obtained from the MSB mocks,again due to the e?ects discussed in Section 4.2.At large scales,w p (r p )is predominantly sensitive to the halo occupation numbers N (M ) and virtually independent of the second moment of P (N |M )or of details regarding the spatial distribution of satellite galaxies.The good agreement at large scales among di?erent MGRSs and with the observa-tions,therefore strongly supports our CLF and it shows that any ‘cosmic variance’among the di?erent MGRSs has only a relatively small impact on w p (r p ).On small scales,how-c 0000RAS,MNRAS 000,000–000

18Yang,Mo,Jing,van den Bosch &

Chu

Figure 15.The projected correlation function w p (r p )(top-left panel),the redshift-space correlation function ξ(s )(top-right),the quadrupole-to-monopole ratio q (s )(bottom-left),and the PVDs (bottom-right)for the samples of MSB (solid lines)and MB (dashed lines)surveys.Error bars,which are similar for MB and MSB results,are only shown for the MSB results for clarity.These errorbars are based on the variance of the 8MSB surveys.The open circles with errorbars correspond to the 2dFGRS results obtained by Hawkins et al.(2003),and are shown for comparison.Note that the MSBs and MBs give approximately the same results,but that there are marked di?erences between model predictions and observations.Note also that the model errorbars are in general larger than the di?erence in the mean between MB and MSB results,implying that these errorbars are statistical.

ever,the MGRSs reveal more correlation power (by about a

factor 2)than observed.On such scales,w p (r p )is sensitive

to our assumptions about the second moment of P (N |M )

and,to a lesser degree,the spatial distribution of satellite

galaxies.We shall return to this small-scale mismatch and

its implications in Section 6below.Rather than projecting ξ(r p ,π),one may also average ξ(r p ,π)along constant s =

Populating Dark Matter Haloes with galaxies

19

Figure 16.Same as Fig.15except that here we compare the results for the MSB sample with those of three alternative MGRSs in which we have modi?ed the CLF to yield cluster mass-to-light ratios of (M/L )cl =1000h (M /L)⊙(dotted lines),in which we adopt a velocity bias of b vel =0.6(dashed lines),and in which we adopt a cosmology with σ8=0.75(dot-dashed lines).All results correspond to the mean of the entire sample of 8MSB mock surveys.For clarity,no errorbars are plotted here,but they are similar to those shown in Fig.15.Note that both the (M/L )cl =1000h (M /L)⊙model and the σ8=0.75model are in good agreement with the observational data.

s >～6h ?1Mpc.At smaller redshift-space separations,how-

ever,the MGRSs slightly overpredict the correlation power.

Note that the MB samples predict higher ξ(s )on small scales

than the MSB samples.This di?erence comes from the fact

that the MB samples are incomplete for galaxies fainter than

M b J ?5log h =?18.4.To test this we construct a mock survey from the MSB sample,but only accepting galaxies

brighter than this.This yields a ξ(s )in excellent agreement with that of the MB samples over all scales.Thus,although the use of only large-box simulations can results in system-atic errors on small scale,the use of small-box simulations in the MSB samples does not cause any signicifant,systematic errors on large scale.

c 0000RAS,MNRAS 000,000–000

20Yang,Mo,Jing,van den Bosch&Chu 5.2Redshift Space Distortions

We now turn to a comparison of the detailed shape of ξ(r p,π).In particular,we focus on the distortions with re-spect to the real-space correlation functionξ(r)induced by the peculiar velocities of galaxies.

The two-dimensional correlation functionξ(r p,π)is of-ten modeled as a convolution of the real-space2PCFξ(r) and the conditional distribution function f(v12|r):

1+ξ(r p,π)=

∞?∞ 1+ξ(

1+(r/r0)2 (30) (Davis&Peebles1977)with y=|π?v12/H0|the separation in real-space along the line-of-sight.F=0corresponds to a Universe without any?ow other than the Hubble expansion, while F=1corresponds to stable clustering.Given the fairly ad hoc nature of this model,and the strong sensitivity to the uncertain value of F(Davis&Peebles1983),great care is required when interpreting any results based on this model.

A more robust model is based on linear theory and di-rectly modeling the infall velocities around density pertur-bations.Following Kaiser(1987)and Hamilton(1992)one can write the observed correlation function on linear scales as

ξlin(r p,π)=ξ0(s)P0(μ)+ξ2(s)P2(μ)+ξ4(s)P4(μ).(31) Here P l(μ)is the l th Legendre polynomial,andμis the cosine of the angle between the line-of-sight and the redshift-space separation s.According to linear perturbation theory the angular moments can be written as

ξ0(s)= 1+2β5 ξ(r),(32)ξ2(s)= 4β7 ξ(r)?

35 ξ(r)+5ξ(r)?7

ξ(r)=

3

r5 r0ξ(r′)r′4d r′(36) Given a value forβand the real-space correlation function, which can be obtained fromξ(r p,π)via the projected cor-relation function w p(r p),equation(31)yields a model for ξ(r p,π)on linear scales that takes proper account of the cou-pling between the density and velocity?elds.To model the non-linear virialized motions of galaxies within dark matter haloes,one convolves thisξlin(r p,π)with the distribution function of pairwise peculiar velocities f(v12|r).

1+ξ(r p,π)=

∞?∞[1+ξlin(r p,π?v12/H0)]f(v12|r)d v12(37)

Thus,by modelingξ(r p,π)one can hope to get both

an estimate ofβas well as information regarding the pair-wise peculiar velocity distribution.We follow H03,and as-sume that the real-space2PCF is a pure power-law,ξ(r)= (r/r0)?γ,and that f(v12|r)is an exponential that is inde-pendent of the real-space separation r:

f(v12|r)=f(v12)=12σ

12

exp ?√σ12 (38)

Using a simpleχ2minimization technique,we?t these mod-els,described by the four parametersβ,σ12,r0,andγ,to theξ(r p,π)in each of our8MSB and4MB MGRSs.The χ2is de?ned as

χ2= log[1+ξ]model?log[1+ξ]data

Populating Dark Matter Haloes with galaxies21 Table1.Best?t parameters.

The values ofβ,r0(in h?1Mpc),γ,andσ12(in km s?1)that

best?t theξ(r p,π)for8h?1Mpc

of di?erent MGRSs.Note that4MGRSs are used for MBs,while

8MGRSs are used for all other cases.The quoted values are the

mean and1σvariance of these MGRSs.The MGRSs denoted by

‘(M/L)cl’is similar to the MGRSs in the MSB set,except that

here the CLF is constrained to mass-to-light ratios for clusters of

(M/L)cl=1000h(M/L)⊙,rather than(M/L)cl=500h(M/L)⊙

as in MSB(see Section6.3).The MGRSs denoted by‘b vel’is

similar except for a velocity bias of b vel=σgal/σDM=0.6(see

Section6.2).The MGRSs denoted by‘σ8’is also similar except

that it adopts a?atΛCDM cosmology withσ8=0.75rather than

0.9(see Section6.4).The?nal line lists the best-?t parameters

obtained by Hawkins et al.(2003)by?tting theξ(r p,π)obtained

from the2dFGRS.Note that the errors in Hawkins et al.are

estimated from the spread of22Mock samples.

with those obtained by H03.First of all,we compute the

modi?ed quadrupole-to-monopole ratio

q(s)≡

ξ2(s)

s3 s0ξ0(s′)s′2d s′?ξ0(s).(40)

whereξl(s)is given by

ξl(s)=

2l+1

3

β?4

1+2

5β2

,(42)

this might indicate that the value ofβinherent to our MGRSs is too small compared to the real Universe.On the other hand,Cole,Fisher&Weinberg(1994)have shown that non-linear,small scale power can a?ect q(s)out to fairly large separations.Therefore the systematic overestimate of q at large s may simply be a re?ection of the random pecu-liar velocities being too large,rather than an inconsistency regarding the value ofβ.

The second statistic that we use to compare the redshift space distortions in our MGRSs with those of the2dFGRS are the PVDs,σ12(r p),as a function of projected radius, r p.Following H03,we keep r0,γandβ?xed at the‘global’values listed in Table1and determineσ12(r p)by minimiz-ingχ2in a number of independent r p bins .The results are shown in the lower-right panel of Fig.15.Whereas the 2dFGRS reveals aσ12(r p)that is almost constant with ra-dius at about500–600km s?1,our MGRSs reveal a strong increase fromσ12～600km s?1at r p=0.1h?1Mpc to σ12～900km s?1at r p=1.0h?1Mpc,followed by a decrease toσ12～500km s?1at r p=10h?1Mpc.Thus,at around 1h?1Mpc,our MGRSs dramatically overestimate the PVD. Although there is a non-negligible amount of scatter among the di?erent mock surveys,re?ecting the extreme sensitivity of the PVDs to the few richest systems in the survey,the variance among the8(4)MGRSs is small compared to the discrepancy.

As shown by Peacock et al.(2001)the best-?t values of σ12andβare highly degenerate.We have tested the impact of this degeneracy on ourσ12(r p)by repeating the same ex-ercise using a value forβthat is0.1larger(smaller)than the values listed in Table1.This leads to an increase(de-crease)ofσ12(r p)of the order of5percent(20percent)at projected radii of1h?1Mpc(10h?1Mpc).Given that our MGRSs overpredict the PVD at r p=1h?1Mpc by about70 percent,it is clear that this discrepancy is not a re?ection of theβ–σ12degeneracy.Thus,the standardΛCDM model seems to have a severe problem in matching the observed PVDs on intermediate scales.

6TOW ARDS A SELF-CONSISTENT MODEL FOR LARGE SCALE STRUCTURE

Our MGRSs,based on a?atΛCDM concordance cosmol-ogy with?m=0.3andσ8=0.9,and on a CLF that is required to yield cluster mass-to-light ratios of(M/L)cl= 500h(M/L)⊙,reveals clustering statistics that are overall in reasonable agreement with the data from the2dFGRS.Nev-ertheless,two discrepancies have come to light:the MGRSs predict too much power on small scales and PVDs that are too high.We now investigate possible ways to alleviate these discrepancies.

6.1Halo occupation models

The discrepancies between our MGRS and the2dFGRS re-sults might indicate a problem with our halo occupation models.Although the CLF is fairly well constrained by the observed luminosity function and the observed luminosity dependence of the correlation lengths(see Papers I and II), we have made additional assumptions regarding the second moments of the halo occupation number distributions and regarding the distribution of galaxies within individual dark matter haloes.

As we have shown in Section4.2the real space cor-relation function depends only very weakly on our method of distributing satellite galaxies within dark matter haloes (cf.Fig.8).We have veri?ed,using a number of tests,that modi?cations of the spatial distribution of satellite galaxies within dark matter haloes have no signi?cant in?uence on

Note that the PVDs thus obtained are a kind of average of the true PVD along the line-of-sight.Therefore,these PVDs should not be compared directly to the true PVD shown in Fig.9.

c 0000RAS,MNRAS000,000–000

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