DO DISTINCT COSMOLOGICAL MODELS PREDICT DEGENERATE …

DO DISTINCT COSMOLOGICAL MODELS PREDICT DEGENERATE HALO POPULATIONS?

Zheng Zheng,1Jeremy L. Tinker,

1David H.Weinberg,

1and Andreas A. Berlind

2

Received 2002 February 19; accepted 2002 April 25

ABSTRACT

Using cosmological N-body simulations, we investigate the influence of the matter density parameter �m

and the linear theory power spectrum PðkÞ on statistical properties of the dark matter halo population—themass function nðMÞ, two-point correlation function �ðrÞ, and pairwise velocity statistics v12ðrÞ and �12ðrÞ.For fixed linear theory PðkÞ, the effect of changing �m is simple: the halo mass scale M� shifts in proportionto �m, pairwise velocities (at fixed M=M�) are proportional to �0:6

m , and halo clustering at fixed M=M� isunchanged. If one simultaneously changes the power spectrum amplitude �8 to maintain the ‘‘ cluster nor-malization ’’ condition �8�

0:5m ¼ const, then nðMÞ stays approximately constant nearM � 5� 1014 h�1 M�,

and halo clustering and pairwise velocities are similar at fixedM. However, the shape of nðMÞ changes, witha decrease of �m from 0.3 to 0.2, producing a�30% drop in the number of low-mass halos. One can preservethe shape of nðMÞ over a large dynamic range by changing the spectral tilt ns or shape parameter C, but therequired changes are substantial—e.g., masking a decrease of �m from 0.3 to 0.2 requires Dns � 0:3 orD� � 0:15. These changes to PðkÞ significantly alter the halo clustering and halo velocities. The sensitivity ofthe dark halo population to cosmological model parameters has encouraging implications for efforts to con-strain cosmology and galaxy bias with observed galaxy clustering, since the predicted changes in the halopopulation cannot easily be masked by altering the way that galaxies occupy halos. A shift in�m alone wouldbe detected by any dynamically sensitive clustering statistic; a cluster normalized change to �8 and �m wouldrequire a change in galaxy occupation as a function of M=M�, which would alter galaxy clustering; and asimultaneous change to PðkÞ that preserves the halo mass function would change the clustering of the halosthemselves.

Subject headings: cosmology: theory — dark matter — galaxies: formation — galaxies: halos —large-scale structure of universe

1. INTRODUCTION

In cosmological models dominated by cold dark matter(CDM) and an unclustered energy component (such as acosmological constant), gravitational instability of primor-dial density fluctuations produces a population of dark mat-ter halos, each in approximate virial equilibrium.Depending on their mass, these halos may host individualgalaxies, galaxy groups, or rich galaxy clusters. Providedthat one focuses on systems of overdensity �=�� 200, andregards higher density structures as subsystems within theirparent halos, one finds that the halo population itself isinsensitive to the gas pressure forces that influence the sub-dominant baryon component. In this paper, we investigatewhether changes to cosmological parameters—specificallythe matter density parameter �m and parameters thatdescribe the shape and amplitude of the primordial powerspectrum—always produce measurable changes in the darkhalo population, or whether two models with different com-binations of these parameters can give rise to halo popula-tions that are effectively indistinguishable.

Our interest in this question is spurred by recent develop-ments in the theory of biased galaxy formation. The uncer-tain relation between galaxies and dark matter is theprimary limitation in testing cosmological models againstobservations of galaxy clustering. The ‘‘ halo occupation

distribution ’’ (HOD) characterizes this relation statisticallyin terms of the probability distribution PðNjMÞ that a haloof virial mass M contains N galaxies of a specified type,together with prescriptions that specify the relative spatialand velocity distributions of galaxies and dark matterwithin these halos. Numerous recent papers have shownthat the HOD framework is a powerful tool for analytic andnumerical calculations of clustering statistics, for modelingobserved clustering, and for characterizing the results ofsemianalytic or numerical studies of galaxy formation (e.g.,Kauffmann, Nusser, & Steinmetz 1997; Jing, Mo, & Borner1998; Kauffman et al. 1999; Benson et al. 2000; Ma & Fry2000; Peacock & Smith 2000; Seljak 2000; Berlind & Wein-berg 2002; Bullock, Wechsler, & Somerville 2002; Marinoni& Hudson 2002; Scoccimarro et al. 2001; Yoshikawa et al.2001; White, Hernquist, & Springel 2001). In particular,Peacock & Smith (2000), Marinoni & Hudson (2002), andBerlind & Weinberg (2002) have argued that the HOD canbe determined empirically from observed galaxy clustering,given an assumed cosmological model that determines themass function and spatial and velocity clustering of the darkhalo population. Empirical determinations of the HOD canprovide insight into the physics of galaxy formation, andthey may sharpen the ability of large-scale structure studiesto test cosmological models, since a model with an incorrectdark halo population may be unable to match the data forany choice of HOD (see Berlind & Weinberg 2002 for fur-ther discussion).

Suppose that we find a combination of cosmology andHOD that reproduces all aspects of observed galaxy cluster-ing. Can we infer that the cosmology and the derived HOD

1 Department of Astronomy, Ohio State University, Columbus, OH43210; [email protected],[email protected], [email protected].

2 Department of Astronomy and Astrophysics, The University ofChicago, Chicago, IL 60637; [email protected].

The Astrophysical Journal, 575:617–633, 2002 August 20

# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

617

are both correct? If there is another cosmological model thatpredicts an indistinguishable halo population, then theanswer is clearly no, since the combination of this alterna-tive cosmology with the same HOD would predict identicalgalaxy clustering. These cosmological models would bedegenerate in the sense that they could not be distinguishedby galaxy clustering data without relying on a predictivetheory of galaxy formation (which might yield differentHODs for the two cosmologies). Cosmological models thatpredict distinguishable halo populations could still bedegenerate in this sense, if changes to the HOD can maskthe differences in the halo populations; we will speculate onthis point in x 4, but we reserve a quantitative examinationof it to future work.

We focus our analyses on CDM cosmological modelswith Gaussian initial density fluctuation fields. Motivatedby cosmic microwave background measurements (Netter-field et al. 2002; Pryke et al. 2002), we restrict our attentionto spatially flat models with a cosmological constant or�m ¼ 1; however, we demonstrate in passing that galaxyclustering data at z ¼ 0 probably cannot distinguish a flatmodel from an open model with the same �m. The parame-ters that define our cosmological models are the matter den-sity parameter �m; the normalization �8 of the powerspectrum PðkÞ, which is the rms fluctuation of the lineardensity field filtered with a top-hat filter of radius8 h�1 Mpc; and the shape of PðkÞ, which we characterize bythe spectral index ns of the inflationary power spectrum andthe shape parameter C of the matter transfer function (see,e.g., Efstathiou, Bond, & White 1992). Although existingdata impose constraints on these parameters, individuallyor in combination, here we allow each to vary independentlyover a fairly broad range so that we can isolate the physicaleffects of the matter density, the amplitude of mass fluctua-tions, and the shape of the power spectrum on the resultinghalo population. We concentrate entirely on the halo popu-lations at z ¼ 0, since this is where galaxy clustering datawill be good enough to allow empirical HOD determina-tions in the near future.

Using N-body simulations described in x 2, we measure(in x 3) mass functions, two-point correlation functions,mean pairwise radial velocities, and pairwise velocity disper-sions of dark matter halos. First, we consider models thathave the same initial power spectrumPðkÞ but different mat-ter density parameter �m. We then move to cluster normal-ized models in which �8 is changed to compensate thechange in �m so that the amplitude of the halo mass func-tion is kept approximately fixed at a cluster scale. Finally,we investigate models in which both the amplitude and theshape of PðkÞ are changed in order to match the amplitudeand slope of the halo mass function at the cluster scale. Wereview our results and briefly discuss their implications forcosmological tests in x 4. Our results overlap previousnumerical studies of the halo mass function and halo clus-tering (recent examples include Jing 1998; Governato et al.1999; Colberg et al. 2000; Jenkins et al. 2001), but they differin the examination of controlled parameter sequences ratherthan specific cosmological models. Also, while some of thesestudies have focused on the large-scale correlations of clus-ter mass halos, we devote considerable attention to thelower masses and smaller spatial scales that are importantfor nonlinear galaxy clustering. Throughout the paper, weadopt a Hubble constant H0 ¼ 100 h km s�1 Mpc�1 ¼70 km s�1 Mpc�1 (Freedman et al. 2001).

2. NUMERICAL METHODS AND TESTS

In this work, statistics are mainly based on simulationsrun with a particle-mesh (PM) N-body code (Park 1990).We run PM simulations with different combinations of den-sity and power spectrum parameters. Each simulation fol-lows the evolution of 2003 particles in a periodic cube 200h�1 Mpc on a side, with a 4003 mesh to compute the gravita-tional force starting at z ¼ 19 and advancing to z ¼ 0 in 40equal steps of expansion factor a. The mass of each particleis 2:78� 1011�m h�1 M�. We adopt the parameterizationof Efstathiou et al. (1992) for the fluctuation power spec-trum PðkÞ. We divide our simulations into three categoriescorresponding to parameter changes in the matter densityand the normalization and the shape of the initial fluctua-tion power spectrum. For each category, four cosmologicalmodels are investigated with ð�m; ��Þ ¼ ð0:2; 0:8Þ, (0.3,0.7), (0.4, 0.6) and (1.0, 0.0). We generally regard the (0.3,0.7) model as the central one for comparison. In the centralmodel, the spectral index is ns ¼ 1:0, the spectral shapeparameter is � ¼ 0:20, and the rms mass fluctuation withinspheres of radius 8 h�1 Mpc is �8 ¼ 0:9, which is consistentwith the observed abundance of clusters (Eke, Cole, &Frenk 1996). For each model, we generate four independentrealizations, and we use the dispersion among these realiza-tions to estimate the statistical error bars on our clusteringmeasures associated with the finite simulation volume. Thetotal volume is comparable to that expected for the SloanDigital Sky Survey (York et al. 2000) out to its median red-shift zmed � 0:1. We use a friends-of-friends (FOF) algo-rithm (Davis et al. 1985) to identify the dark matter halos,and unless explicitly stated otherwise, the linking length is0.2 times the mean interparticle separation. With this link-ing length, the FOF algorithm picks out structures of typicalmass overdensity �=�� 200 (Davis et al. 1985).

The advantage of a PM code is that it is relatively cheapto run multiple simulations with a large dynamic range inmass. The limitation of a PM code is its moderate force reso-lution, but this limitation is not too serious for our pur-poses, since we only identify halos and do not attempt tomeasure their internal structure. We perform two tests tomake sure that the PM resolution is adequate for our pur-pose. First, we make a comparison between PM simulationsand simulations run with the GADGET tree code (Springel,Yoshida, & White 2001). Second, we perform a self-similarscaling test for the PM code.

GADGET is a publicly availableN-body code that calcu-lates particle accelerations by the hierarchical tree methodof Barnes & Hut (1986). Its advantages over the PM algo-rithm are its spatial and temporal adaptability. The gravita-tional softening length � can be set significantly smaller thanthe initial particle grid spacing to increase force resolution;however, if � is too small, two-body relaxation effects willbegin to influence the halo population. Particles have indi-vidual timesteps, which can vary continuously to ensureaccurate time integration given the adopted force resolu-tion. GADGET has fully periodic boundary conditions forcomoving integrations via the Ewald summation method(Hernquist, Bouchet, & Suto 1991).

We have compared the results of two simulations with1003 particles in a 100 h�1 Mpc box, one evolved with thePM code using a 2003 mesh (thus having equal resolution tothe larger simulations on which our results will be based),and one evolved with GADGET using � ¼ 0:3 h�1 Mpc.

618 ZHENG ET AL. Vol. 575

The initial conditions of the two simulations were identicalto each other and used the parameters of the central modeldescribed above. Slices through these simulations are plot-ted for comparison in Figure 1. The structures formed in thetwo simulations are very similar at the level of detail discern-ible in this plot. The halo mass functions of the two simula-tions, also shown in Figure 1, trace each other nearlyexactly. The PM simulation produces slightly (�10%) fewerhalos below 1013 h�1 M�, which is to be expected given itslower force resolution; the diameter of an overdensity 200sphere of 1013 h�1 M� is 1.04 h�1 Mpc, or about two PMgrid cells. When the mesh size on this PM simulation wasincreased to 3003, the resulting mass function was consistentwith the GADGET simulation even on these small-massscales.

In addition to the halo mass function, the quantitiesimportant to our analysis are the correlation functions forboth the mass distributions and the halo populations, whichare shown in Figure 2. As with the halo mass functions, thematter correlation functions for the PM and GADGETsimulations match very well. They begin to deviate at sepa-rations smaller than �0.5 h�1 Mpc, or under one mesh cell.

The halo correlation functions produced by the two meth-ods are also consistent. Larger simulations and multiplerealizations of each model are efficiently produced with thePM code, which takes up to an order of magnitude less com-puting time at the force resolution adopted here.

We next perform a self-similar scaling test for the PMcode. Assuming an Einstein–de Sitter universe (�m ¼ 1), weuse a pure power-law power spectrum PðkÞ / k�ns withns ¼ �1 (and no transfer function modulation) for the ini-tial density fluctuations of the scale-free model. A 4003 forcemesh is used to follow the gravitational evolution of 2003

particles. The normalization of the power spectrum is set sothat the nonlinear mass M� at the final output correspondsto 400 particles. We define M� by the condition�ðM�Þ ¼ �c ¼ 1:69, where �ðM�Þ is the rms linear massfluctuation on mass scale M�. The rms fluctuation is� ¼ 1:0 at about 1/26 of the box size. If we identify this scaleas 8 h�1 Mpc, the implied size of the periodic simulationcube is then 207.62 h�1 Mpc at z ¼ 0. Earlier outputs can beidentified with higher redshifts or with larger volumes atz ¼ 0. In the language of the redshift description, the simu-lation is evolved for 40 time steps from z ¼ 19, with four

Fig. 1.—Comparison of PM andGADGET simulations evolved from the same initial conditions. Top panels show slices through the particle distributions,100 h�1 Mpc on a side and 10 h�1 Mpc thick. Bottom panels show cumulative halo mass functions; in the right panel these are divided by the Jenkins et al.(2001) analytic mass function to allowmore detailed comparison. Theminimum groupmass plotted corresponds to 10 simulation particles.

No. 2, 2002 DEGENERATE HALO POPULATIONS? 619

outputs (at redshifts z ¼ 1:00, 0.60, 0.25, and 0.00), whenthe corresponding M� values are 50, 100, 200, and 400 par-ticles, respectively. The following results are averaged overfour independent realizations of the simulation, and theerror bars represent the dispersion among the four realiza-tions divided by

ffiffiffi3

pto yield the error on the mean.

Efstathiou et al. (1988) performed the first study on scale-free cosmological models using N-body simulations andshowed that many quantities in scale-free models have well-defined scaling behaviors. In an Einstein–de Sitter scale-freemodel, the cumulative halo mass functions at redshifts z1and z2 have the scaling behavior

NðM > M2; z2Þ ¼1þ z21þ z1

� �6=ð3þnsÞNðM > M1; z1Þ ;

M1 ¼1þ z21þ z1

� �6=ð3þnsÞM2 : ð1Þ

The left panel of Figure 3 shows the cumulative halo massfunction at different redshifts in our scale-free simulations

(solid curves). The dotted curves in Figure 3 are derived byscaling the mass function at z ¼ 0 according to the abovescaling rule. The scaling properties shown in the figure aregenerally very good. The high-mass end of the mass func-tion drops systematically below the self-similar scaling asM� increases, a likely result of the finite simulation volume,which suppresses the amplitude of fluctuations on the larg-est scales. Comparison of the first and last outputs suggeststhat this effect becomes noticeable above M � 2� 1015

h�1 M� at z ¼ 0 (for our adopted �8 ¼ 1:0 scaling), or amass scale �0.1% of the total simulation mass (10% inlength scale).

The N-point correlation functions �ðrÞ are expected tohave a similarity transformation with the variables ¼ rð1þ zÞ2=ð3þnsÞ for an Einstein–de Sitter universe (Efsta-thiou et al. 1988). The scaling rule for the correlation func-tions at redshifts z1 and z2 is then

�ðr2; z2Þ ¼ �ðr1; z1Þ; r1 ¼1þ z21þ z1

� �2=ð3þnsÞr2 : ð2Þ

Fig. 2.—Matter and halo correlation functions of the PM (dotted lines) and GADGET (solid lines) simulations shown in Fig. 1. The upper left panel showsthe matter correlation functions, and remaining panels show halo correlation functions for three different mass ranges, centered on M�=2, M�, and 2M�,whereM� ¼ 1:03� 1013 h�1 M� is the characteristic nonlinearmass (corresponding to 123 particles in these simulations). Results are averaged over four real-izations, and error bars show the run-to-run dispersion divided by

ffiffiffiffiffiffiffiffiffiffiffiffiffiN � 1

p¼

ffiffiffi3

pto yield the uncertainty in the mean.


From Figures 3 and 4 it is clear that mass correlation func-tions and halo correlation functions at different values ofM=M� obey the scaling rule quite well. There are somenoticeable departures from self-similar scaling in the M�=2panel at the two earliest output times (especially the ear-liest), when M�=2 corresponds to 25 and 50 particles,respectively.

The clustering of halos is biased relative to that of thematter. The halo bias factor bh can be defined via�hðrÞ ¼ b2h�mðrÞ, where �h and �m are the two-point correla-tion functions of halos and matter, respectively. The rela-tion between the clustering of halos and that of the matterhas been extensively studied based on analytical models andnumerical simulations (e.g., Cole &Kaiser 1989; Kashlinsky1991; Mo &White 1996; Jing 1998, 1999; Porciani, Catelan,& Lacey 1999; Sheth & Lemson 1999; Sheth & Tormen1999; Sheth, Mo, & Tormen 2001, and references therein).Mo&White (1996) give an analytic formula for the bias fac-tor of halos of a given mass based on an extended Press-Schechter analysis (Press & Schechter 1974). This formula isquite accurate for halos with M > M�. Jing (1998) empiri-cally modifies the formula to also fit the N-body simulationresults for halos of lower mass,

bhðM; zÞ ¼ 1

2�4þ 1

� �ð0:06�0:02nsÞ1þ �2 � 1

�c; 0

� �;

� ¼ �cðzÞ=�ðMÞ ; ð3Þ

where �ðMÞ is the rms fluctuation of the mass density at amass scaleM, �cðzÞ is the threshold density contrast for col-lapse of a homogeneous spherical overdense region at red-shift z, and �c; 0 � 1:69 for an Einstein–de Sitter universe.For ns ¼ �1 here, � ¼ ðM=M�Þ2=3. We derive the square ofthe bias factor numerically as a function ofM=M� at differ-ent redshifts by averaging �hðrÞ=�mðrÞ over comoving sepa-rations between 5 and 30 h�1 Mpc. In this range, the ratio isalmost constant. We compare the computed bias factor withthat given by Jing’s fitting formula in Figure 5. Two conclu-

sions can be drawn immediately from the comparison. First,since bias factors at different redshifts overlap with eachother when plotted as a function of M=M�, the simulationdisplays the correct scaling behavior. Second, the bias fac-tors agree well with Jing’s fitting formula. At higher mass,the computed bias is somewhat lower than that given byJing’s formula. In fact, the same trend can be found in Jing’splot (see his Fig. 2). We attribute this slight deviation in partto a sample volume effect and in part to systematic errors inthe approximate analytic formula given by Mo & White(1996; see Sheth et al. 2001).

Both the GADGET simulation test and the scale-freemodel test assure that the numerical artifacts in the PM sim-ulations have almost negligible effect on the statistics wecare about in this paper. For example, the mass scales thatwe are concerned with are generally higher than the levelwhere the PM and GADGET halo mass functions begin todeviate slightly. Moreover, the correlation functions aremeasured at scales e0.5 mesh cells, where the PM andGADGET results are consistent. The PM simulationsreproduce the expected scaling behaviors in the scale-freemodel, which also demonstrates that we canmeasure massesof halos and their spatial distribution correctly based onthese simulations. The scaling tests and code comparisonsuggest that there are some inaccuracies when the halo massis lower than 50 particles or more than 0.1% of the totalmass in the simulation, but even in these regimes thereshould be little impact on our conclusions because we com-pare different cosmologies evolved with the same code andnumerical parameters. The PM code is therefore an idealtool for our purpose in this paper.

3. COMPARISON OF HALO POPULATIONS

3.1. Changing�m with �8, ns, and C Fixed

We start with a simple case in which we fix the powerspectrum (as linearly evolved to z ¼ 0, with ns ¼ 1:0,

Fig. 3.—Test of self-similar scaling in PM simulations with scale-free initial conditions. Solid lines in the left panel show halo mass functions at four evolu-tionary stages, whenM� ¼ 50, 100, 200, and 400 particles ( from left to right). Dotted lines show the rightmost solid line scaled according to eq. (1). Points inthe right panel show two-point correlation functions of the mass at these four epochs. Dotted lines are derived by scaling the mass correlation function of thefinal output, whenM� ¼ 400Mp, according to eq. (2). Results are averaged over four simulations, and error bars on �ðrÞ show the dispersion among four inde-pendent runs, divided by

ffiffiffi3

p. For clarity, error bars are plotted only for the stage of M� ¼ 400Mp, and error bars for other stages have similar magnitude.

Physical scales are assigned by treating the final output as z ¼ 0 and adopting �8 ¼ 1:0, which implies a comoving size Lbox ¼ 207:62 h�1 M� of the simulationcube and a total massM ¼ 2:49� 1018 h�1 M� in the cube.


�8 ¼ 0:9, � ¼ 0:20) and change �m. In linear theory, theevolved fluctuations in these models are identical, and thisindependence of �m holds in the Zeldovich approximation(Zeldovich 1970) as well as its extension, the adhesionapproximation, as emphasized by Weinberg & Gunn(1990). In fact, it holds remarkably well into the fully non-linear regime, as shown in the top and middle panels of Fig-ure 6, where slices of particle distributions at z ¼ 0 formodels with different �m but identical initial conditions arecompared. These slices are taken from GADGET simula-tions that evolve the particle distribution from z ¼ 19. Par-ticle distributions of three spatially flat models withdifferent values of �m and one open model with �m ¼ 0:2are extremely similar to each other (although particle veloc-ities are higher for higher �m). Nusser & Colberg (1998)extend the Zeldovich/adhesion analytic explanation for thissimilarity of evolved structure by showing that the equa-tions of motion of a collisionless gravitating system of par-ticles in an expanding universe can be put in a form withalmost no dependence on �m and ��, if the linear perturba-tion growth factor is used as the time variable. The evolu-tionary histories of the four models in Figure 6 are different,

but this difference is captured almost entirely by the depend-ence of the linear growth factor on redshift. The bottom twopanels demonstrate this point, showing the particle distribu-tions of the flat �m ¼ 1:0 and �m ¼ 0:2 models at the red-shifts when the linear growth factor is 0.5, redshifts z ¼ 1:0and z ¼ 1:74, respectively (we define the growth factor to beunity at z ¼ 0). In most comparisons ofN-body simulationswith different �m, the amplitude and/or shape of PðkÞ isadjusted in concert with �m. Such adjustments are wellmotivated by observational and theoretical considerations,but they mask the fact that the influence of �m at fixed PðkÞis extremely simple. Earlier examples of matched compari-sons like those in Figure 6 include Figure 1 of Davis et al.(1985) and Figure 5 of Nusser & Colberg (1998).

Figure 6 suggests that halo mass functions for differentmodels in this case should be nearly the same when mea-sured as a function of particle number, i.e., mass divided by�m. This expectation is confirmed in Figure 7, which showsmass functions (from PM simulations) of four spatially flatmodels. In the left panel, all four (cumulative) mass func-tions have a similar shape and only shift horizontally rela-tive to each other. When the halo masses are divided by �m,

Fig. 4.—Test of self-similar scaling of the halo correlation functions, in mass ranges centered onM�=2,M�, 2M�, and 4M�, as indicated. Points show halocorrelation functions at the four evolutionary stages when M� ¼ 50, 100, 200, and 400 particles. Dotted lines are obtained by scaling the result of theM� ¼ 400 output (solid line) to earlier stages, according to eq. (2). Error bars show the dispersion among four independent runs, divided by

ffiffiffi3

p.


the four curves become nearly identical and overlap with the�m ¼ 1 curve. To allow a closer inspection, in the rightpanel we plot the ratio of the scaled mass functions to thatcalculated using Jenkins et al.’s (2001) fitting formula forthe central model. The maximum relative difference betweenthe four scaled mass functions is only about 10%.

The 10% residual differences reflect slight changes in thecharacteristic densities and profiles of halos in different cos-mologies. For Figure 7, we have identified halos with ourstandard FOF linking length parameter b ¼ 0:2, whichpicks out structures with mean overdensity �=�� 200(Davis et al. 1985; Lacey & Cole 1993), close to the value�=�� 178 predicted for the post-virialization overdensityin the spherical collapse model for an �m ¼ 1 universe(Gunn & Gott 1972; Peacock 1999). If we instead useb ¼ 0:16, roughly doubling the overdensity threshold(0:23=0:163 � 1:95), then halo masses drop by�20% as pre-viously linked particles are unlinked, but the level of agree-ment among different cosmological models stay the same.Another fairly common procedure is to scale b�3 in propor-tion to the virial overdensity predicted by the spherical col-lapse model, thus using a different b for each cosmologicalmodel. If we adopt this approach, taking the virial overden-sities from Eke et al. (1996), then the discrepancy of thescaled mass functions reverses sign and rises in amplitude toa maximum�20%. We conclude that the effect of a pure �m

change is described by the simple scaling of halo masses toan accuracy �10% for halos defined at fixed overdensity,that the discrepancies reflect the expected cosmologydependence of the characteristic virial overdensity, but thatthe magnitude of the cosmology correction predicted by thespherical collapse model is too large. Jenkins et al. (2001)also find better agreement among halo mass functions ofdifferent cosmological models for halos defined at constantoverdensity rather than overdensities scaled according to

the spherical collapse model. Henceforth we will use thefixed linking length b ¼ 0:2 for all models, but our resultswould not be substantially different if we used b ¼ 0:16 orscaled bwith cosmology.

The mass correlation functions �mðrÞ for these four mod-els are identical at large scales and gradually depart fromeach other at small scales (r < 1 h�1 Mpc), where the higheroverdensities for lower �m boost �mðrÞ. We calculate thetwo-point correlation function for halos of mass M bycounting pairs of halos with masses in the range (M=

ffiffiffi2

p,ffiffiffi

2p

M). Figure 8 shows halo correlation functions �hðrÞ forfour values of M=M�. In each case, the four �m models areindistinguishable from each other, as we would expect, sinceM� scales with �m, and comparisons at fixed M=M� there-fore remove any dependence on �m. The difference in �mðrÞarises from differences in spatial structure within the halosthemselves (which are difficult to discern in Fig. 6). We alsocalculate the halo bias factor from the mass and halo corre-lation functions (not shown). When plotted as a function ofM=M�, the bias factors of the four models are almost thesame, as expected, and they agree with Jing’s (1998) fittingformula to a similar degree as seen in Figure 5. The halo cor-relation function plummets at small r because two halo cen-ters cannot be separated by less than the sum of their virialradii. Halos for different �m have the same virial radius at agiven M=M� (in simulation terms, the scaling of M� with�m comes from the particle mass, not a change in halo size),so even this exclusion signature is the same in differentmodels.

Although mass functions and spatial clustering of halosare nearly identical as a function ofM=M�, different valuesof �m produce different velocity fields. Figure 9 showsthe mean pairwise radial (inward) velocity, v12 ��ðv1 � v2Þ x ðr1 � r2Þ=jr1 � r2j, and the pairwise velocity dis-persion, �12 � hv212i � hv12i2, at M ¼ M� and M ¼ 8M�.Here vi and ri (i ¼ 1; 2) are the velocities and positions of ahalo pair, and the average is over all halo pairs with separa-tions around r. Both v12ðrÞ and �12ðrÞ drop as�m drops. Themean pairwise radial velocity v12ðrÞ increases for massivehalos and increases much faster for pairs with small separa-tions, a sign of the nonlinear infall velocities induced byhigh-mass halos. The velocity dispersion �12ðrÞ continuesrising out to large separations, and its amplitude at largescales seems to be nearly independent of halo mass. Linearperturbation theory predicts a relation between the peculiarvelocity and matter density fields,

vðxÞ / f ð�mÞZ

�ðxÞ ðx0 � xÞ

jx0 � xj3d3x0 ; ð4Þ

where �ðxÞ is the mass density contrast and f ð�mÞ � �0:6m

(see, e.g., Peebles 1993). In the four cosmological modelsconsidered here, the mass density contrasts �ðxÞ should bethe same, so linear velocity fields simply scale with �0:6

m . Thedotted curves in Figure 9 show the effect of dividing v12ðrÞand �12ðrÞ by �0:6

m ; the agreement of these curves with eachother and with the �m ¼ 1 curve demonstrates that theimpact of a pure �m change on halo peculiar velocities iswell described by the linear theory scaling even on nonlinearscales.

Figure 6 suggests that the agreement of appropriatelyscaled halo mass functions, correlation functions, and pair-wise velocity statistics should hold at other epochs for whichthe linear growth factors are equal. Although we do not

Fig. 5.—Halo bias factor as a function ofM=M� in the scale-free simula-tions. Points show b2 at the four evolutionary stages when M� ¼ 50, 100,200, and 400 particles. The solid line is calculated using Jing’s (1998) for-mula.


Fig. 6.—Insensitivity of darkmatter clustering to the values of�m and��. The top andmiddle panels show slices of particle distributions at z ¼ 0 from sim-ulations that have identical linear theory PðkÞ and Fourier phases but different combinations of �m and ��, as indicated. The two bottom panels are slices formodels (1.0, 0.0) and (0.2, 0.8) at redshifts corresponding to a linear growth factor of 0.5. All slices have the size of 100� 100� 10 h�1 Mpc3. All simulationswere performed with GADGET using 1003 particles in a ð100 h�1 MpcÞ3 volume and a force resolution of � ¼ 0:3 h�1 Mpc.

624

Fig. 8.—Influence of �m on the halo correlation function, for halos in mass ranges centered on M�=2, M�, 4M�, and 8M� as indicated. Curves in eachpanel, nearly superposed, show �hðrÞ for the four indicated combinations of ð�m; ��Þ. To preserve visual clarity, we plot error bars only on the central (0.3,0.7) model, but error bars for other models have similar magnitude. In each panel, open circles show themass correlation function of the central model.

Fig. 7.—Influence of�m on the halo mass function. The left panel shows cumulative halo mass functions from simulations with four different combinationsof ð�m; ��Þ as indicated. A second curve for each low-�m model shows the result of dividing halo masses by �m before computing N(>M). These scaledcurves are almost perfectly superposed on the solid curve representing the �m ¼ 1:0 model, demonstrating that a change to �m alone simply shifts the massscale of the halo mass function. The right panel allows closer inspection of this result, suppressing dynamic range by dividing each mass function by the ana-lytic fitting formula of Jenkins et al. (2001) computed for the parameters of the central model ð�m ¼ 0:3; �� ¼ 0:7Þ.

625

show the results here, we have verified explicitly that thematching of scaled statistical measures in Figures 7, 8, and 9holds equally well at the redshifts when the four modelshave growth factors of 0.5 (z ¼ 1:0, 1.36, 1.50, and 1.74 for�m ¼ 1:0; 0.4, 0.3, and 0.2). Thus, for models with the samePðkÞ shape and the same �8 at z ¼ 0, the influence of�m and�� on the evolutionary history of the halo population isentirely encoded in the linear growth factor.

3.2. Cluster-normalizedModels with Fixed PðkÞ ShapeChanging �m on its own preserves the shape of the halo

mass function but shifts the mass scale in proportion to �m.This scaling implies that the threshold volume for collapse isindependent of �m, given the same power spectrum. If werequire that the four cosmological models that differ in �m

have halo mass functions that match in amplitude at somephysical mass scale (rather than M=M�), we need to adjustthe power spectrum in a way that builds more massive halosin low-�m models and fewer massive halos in high-�m mod-els. In other words, we need a larger threshold volume for

collapse in a low-�m model so that its nonlinear mass M�increases. Therefore, if the shape of the power spectrum iskept unchanged, a higher �8 is necessary in a lower �m

model in order to compensate for the lower mean massdensity.

Putting this idea in quantitative form, White, Efstathiou,& Frenk (1993) argued that the observed abundances ofmassive clusters of galaxies impose the constraint�8 � 0:57��0:56

m , with little dependence on the assumedshape of PðkÞ. Many authors have revisited, refined, andrecalibrated this ‘‘ cluster normalization ’’ constraint (seePierpaoli, Scott, & White 2001 and references therein). Onewidely used formulation is that of Eke et al. (1996), whofound �8 ¼ ð0:52� 0:04Þ��0:52þ0:13�m

m for a spatially flatuniverse. Inspired by these results, we now consider asequence of cluster normalized models in which we keep theshape of PðkÞ fixed but scale the amplitude as�8 ¼ 0:9ð�m=0:3Þ�0:5.

Figure 10 shows cumulative halo mass functions for thesecluster normalized models. Mass functions of models with�m ¼ 0:2, 0.3, and 0.4 match in amplitude at a cluster scale

Fig. 9.—Influence of �m on the mean pairwise velocity (top) and the pairwise velocity dispersion (bottom) of halos with massM � M� (left) andM � 8M�(right). Error bars are plotted only for the central model; they increase with �m and are about twice as large for �m ¼ 1 as for �m ¼ 0:2. Dotted curves areobtained by dividing the velocity by �0:6

m ; their agreement with the �m ¼ 1 curve demonstrates that the �m influence is captured almost entirely by the lineartheory velocity scaling.


(M � 5� 1014 h�1 M�). The mass function of the �m ¼ 1model crosses the others at a somewhat lower mass scale,M � 2� 1014 h�1 M�. The discrepancy in matched massscale just shows that the �8 / ��0:5

m scaling does not hold allthe way to �m ¼ 1, and the Eke et al. (1996) formula indeedimplies that we should adopt a somewhat higher �8 for�m ¼ 1. However, the crucial result for our purposes is thatthe mass functions of these cluster normalized models havesystematically different shapes, and there is no value of �8

we could choose that would make the �m ¼ 1 model matchthe others at more than a single mass scale. Over the smallerrange �m ¼ 0:2 0:4, the cluster normalization conditionkeeps the halo mass functions reasonably well aligned overabout a decade in mass, but by M � 1013 h�1 M� the halospace densities of the �m ¼ 0:2 and 0.4 models differ fromthose of the central model by�30%, and from each other bynearly a factor of 2.

The reason that cluster normalization does not preservethe shape of the halo mass function is that raising �8

increasesM� but simultaneously drives down the space den-sity of M� clusters by increasing the scale of nonlinearity.For models with the same PðkÞ shape and the same value ofM�, the space density of halos at fixed M=M� is propor-tional to �m (see eq. [7] below). The mass functions of clus-ter normalized models with different �m must thereforecross at different values of M=M�, and because the shapesremain the same as a function of M=M�, they cannot stayaligned as a function ofM.

In these cluster normalized models, the amplitude of�mðrÞ increases as �m decreases, as shown in the upper leftpanel of Figure 11. At large scales, where the mass densityfield is still linear, �mðrÞ is just proportional to the amplitudeof its Fourier transform, the linear power spectrum, andhence to �2

8. The roughly constant logarithmic offset in Fig-ure 11 shows that this multiplicative scaling holds approxi-mately even into the strongly nonlinear regime. However,the remaining panels of Figure 11 show that the halo corre-lation functions are very similar at a fixed physical mass.Qualitatively, this similarity makes sense, since a given massscale corresponds to a higher value of M=M� when �m ishigher, and the correspondingly higher halo bias tends to

compensate for the lower mass clustering amplitude. How-ever, it is impressive just how well this cancellation betweenmass clustering and halo bias works quantitatively. Thehalo exclusion signature does shift to a larger scale for lower�m because of the larger virial radii needed to compensatefor the lower mass density.

Figure 12 shows that the mean pairwise velocity and pair-wise dispersion increase with �m in this cluster normalizedsequence, but the dependence is weak, especially at largeseparations. As in Figure 9, more massive halos have higherpairwise velocities, especially at small separations. The weakdependence on �m at large scales can be understood withthe help of equation (4). In linear theory, velocity is propor-tional to f ð�mÞ�8, which, when combined with the clusternormalization condition, reduces to approximately �0:1

m .Nonlinear evolution at small separations amplifies the dif-ference between models, but only slightly.

3.3. MatchingMass Functions

By varying the amplitude of the power spectrum with �m,we can match the amplitudes of the cumulative halo massfunctions at a given mass scale. However, this combinationof parameter changes does not guarantee a match betweenthe shapes of the mass functions at that mass scale. In orderto match both the amplitude and the shape of the halo massfunctions, we must adjust the relative amplitudes of fluctua-tions at different scales, or, in other words, the shape ofPðkÞ. We therefore need to determine the appropriate com-binations of parameters (�8 and parameters related to theshape of the power spectrum) for each �m model so that allmodels produce halo mass functions that agree in bothamplitude and shape.

3.3.1. Analytic Solution to Parameter Combinations

Instead of searching the parameter space to obtain theright power spectrum combinations, we make use of theanalytic form of the halo mass function. There are threecommonly used analytic formulae for halo mass functions.Press & Schechter (1974) developed the first theoreticalframework based on the spherical collapse model (also see

Fig. 10.—Halo mass functions for cluster normalized models, in which the shape of PðkÞ is fixed but the amplitude is scaled by �8 / ��0:5m . The left panel

shows halo mass functions from simulations with four different ð�m; ��Þ, as indicated. In the right panel, these mass functions are divided by the Jenkins et al.(2001) fitting formula for the central (0.3, 0.7) model.


Bond et al. 1991) and provided the first theoretical formula(PS formula) for the halo mass function, which is still widelyused. The PS formula is known to underestimate the haloabundance in the high-mass tail (see Jenkins et al. 2001 andreferences therein). Sheth & Tormen (1999) give an analyticformula (ST formula) obtained by fitting to the results ofN-body simulations, which turns out to be consistent with theellipsoidal collapse model (Sheth et al. 2001). On the basisof more simulation results, Jenkins et al. (2001) provide themost accurate fitting formula for halo mass functions. How-ever, this formula includes an absolute value of a function,which makes it difficult to use for our present purpose:obtaining an analytic solution of parameter combinationsthat lead to a good match in halo mass functions. We there-fore use the slightly less accurate ST formula.

In the notation of Jenkins et al. (2001), the mass functionis defined in terms of the dimensionless function

f ð�Þ � M

�0

dnðMÞd ln��1

; ð5Þ

and the ST formula for the halo mass function can be

expressed as (Sheth & Tormen 1999)

f ð�Þ ¼ A

ffiffiffiffiffi2a

�

r1þ ða�2Þ�p� �

exp�a�2

2

� �; � ¼ �c

�ðMÞ ; ð6Þ

where nðMÞ is the abundance of halos with mass less thanM, �c � 1:69 is the threshold density contrast for collapse,�ðMÞ is the rms fluctuation of the mass density at a massscale M, �0 is the mean density of the current universe,A ¼ 0:3222, a ¼ 0:707, and p ¼ 0:3. The mass function canbe written as

dn

d lnM¼ dn

d ln��1

d ln��1

d lnM¼ �

�0M

f ð�Þ ; ð7Þ

where � ¼ d ln��1=d lnM ¼ ð3þ neffÞ=6 characterizes thelocal index neff of the power spectrum and is determined bythe shape of the power spectrum. The slope of the massfunction is

d

d lnM

dn

d lnM

� �¼ dn

d lnM�d ln f

d ln �� 1þ d ln�

d lnM

� �: ð8Þ

Fig. 11.—Mass and halo correlation functions for cluster normalizedmodels. The upper left panel shows the mass correlation function. The remaining threepanels show correlation functions of halos in mass ranges centered onMc, 4Mc, and 8Mc, whereMc ¼ 1:03� 1013 h�1 M� corresponds toM� of the central(0.3, 0.7) model. Open circles show themass correlation function for the central model. Error bars are plotted for the central model and have similar magnitudefor other models.


We use a power law to locally approximate the power spec-trum (so that the last term on the right-hand side of eq. [8]reduces to zero).

We require that the halo mass function of a given cosmo-logical model match as closely as possible that of the centralmodel at some mass scale. In other words, at this mass scale,both the amplitudes (eq. [7]) and the slopes (eq. [8]) of thetwo halo mass functions should be equal. We thus solvethese two equations for two unknowns (�, �) for each cos-mological model. All the information contained in thepower spectrum is fully embedded in (�, �). Therefore, bysolving for these two parameters at a given mass scale, wecan also determine the normalization �8 and shape of thepower spectrum.

The power spectrum can be expressed as PðkÞ /knsT2ðk; �Þ, where ns is the spectral index of the inflation-ary power spectrum and Tðk; �Þ is the transfer functionwith shape parameter C. A change in either ns or C leads to achange in the shape of the power spectrum. We first keep Cfixed and only tilt PðkÞ. Since � ¼ ð3þ neffÞ=6 and neff issimply the sum of ns and an index given by the transfer func-tion, it is straightforward to obtain the change in ns withrespect to that of the central model: Dns ¼ 6D�. Once the

shape of PðkÞ is determined, the normalization �8 can beobtained from �. Assuming that mass functions are matchedagainst that of the central model at a cluster mass scaleM ¼ 5� 1014 h�1 M�, we calculate ns and �8 as a functionof �m. The left panel of Figure 13 shows that both ns and �8

increase sharply toward low �m and drop slowly towardhigh �m. In order to match with the halo mass function ofthe central model, we need ns ¼ 1:33, 0.82, 0.42 and�8 ¼ 1:16, 0.78, 0.55 for models with �m ¼ 0:2, 0.4, and 1.0,respectively.

Alternatively, we can keep the spectral index ns fixed(ns ¼ 1 in our calculation) and change the shape parameterC. The right panel of Figure 13 shows C as a function of �m

when halo mass functions are matched at M ¼ 5�1014 h�1 M�. The shape parameter C also has a sharpincrease toward low �m and a slow drop toward high �m.For models with �m ¼ 0:2, 0.4, and 1.0, solutions to C are0.35, 0.14, and 0.06, respectively. In this case, the normaliza-tion �8 of each model is almost identical to that of the tiltedmodel, which indicates that the two ways of changing theshape of the power spectrum are equivalent at this massscale. Although pure-ns and pure-C changes alter the powerspectrum in different ways, they produce power spectra (and

Fig. 12.—Halo velocity statistics for cluster normalized models. Top panels show the mean pairwise radial velocities for halos with M � Mc (left) andM � 8Mc (right), where Mc ¼ 1:03� 1013 h�1 M� corresponds to M� of the central model. Bottom panels show pairwise radial velocity dispersions. Errorbars are plotted for the central model and have similar magnitude for other models.


thus mass functions) that match closely over a fairly largedynamic range, covering most of the regime in which � is onthe order of unity. The near degeneracy can be broken byexamining the low-mass end of the halo mass functions.More generally, we could allow both ns and C to change. Inthat case, for each value of �m, there would be a one-dimen-sional locus in the ns-� plane along which models would sat-isfy the requirement of matching the central model’s halomass function at some mass scale. However, we expect thatresults from this more general case would be similar to thoseof the two simple cases.

3.3.2. Simulation Results

Using the power spectrum parameters given by the aboveanalytic solutions, we run simulations and make compari-sons with the central model. Here we only show results for

different tilt models, since models with different values of Cgive similar results.

Figure 14 shows cumulative halo mass functions for thefour cosmological models. The mass functions agree witheach other quite well over a large mass range (from�1013 to�1015 h�1 M�), with only �10% relative differences. We donot expect the cumulative mass functions to match exactly,in amplitude and slope at M ¼ 5� 1014 h�1 M�, for threereasons. First, the ST formula is not a perfect fit to the halomass functions in the simulations. Second, we locallyapproximate the power spectrum as a power law in order toobtain analytic solutions for the power spectrum parame-ters. Third, we require a match for differential mass func-tions instead of cumulative mass functions. Nevertheless,the cumulative mass functions trace each other remarkablywell. The nonlinear mass scales are quite different for thesefour models: M� ¼ 3:10� 1013, 1:03� 1013, 3:66� 1012,

Fig. 13.—Changes in power spectrum parameters required to match the amplitude and slope of the halo mass function atM ¼ 5� 1014 h�1 M�. The leftpanel shows the values of �8 (dotted line) and ns (solid line) required to match the mass function of a model with �m ¼ 0:3, �8 ¼ 0:9, ns ¼ 1:0, and � ¼ 0:2,when C is fixed at 0.2. The values of �8 we adopt for cluster-normalized models with fixed PðkÞ shape are plotted as a dashed line, for comparison. The rightpanel shows the required value of C if ns is held fixed at 1.0; the corresponding values of �8 are nearly identical to those in the left panel. Curves are calculatedusing the Sheth-Tormen (1999) analytic mass function, as discussed in the text.

Fig. 14.—Halo mass functions for models that have been matched at M ¼ 5� 1014 h�1 M� by changing ns and �8. As expected, the change in ns yieldsmuch better agreement of the mass functions than found for cluster normalized models with no change to the shape of PðkÞ (see Fig. 10).


and 7:50� 108 h�1 M�, for �m ¼ 0:2, 0.3, 0.4, and 1.0,respectively.

Figure 15 shows two-point correlation functions of themass and halos. Mass is more strongly clustered in low �m

models, having a correlation function that is steeper andhigher in amplitude than in higher�m models. However, theclustering of halos exhibits the opposite trend. Halos in the�m ¼ 1 model are highly biased with respect to those ofother models, as one might guess from the low nonlinearmass of this model, and this bias more than compensates forthe weaker mass clustering. Conversely, at the mass scaleM ¼ 1:03� 1013 h�1 M�, shown in the upper right panel ofFigure 15, halos in the �m ¼ 0:2 model are actually anti-biased with respect to the mass. Figure 11 shows that forcluster normalized models with fixed PðkÞ shape, the effectsof bias almost exactly cancel the change in �8, leaving �hðrÞnearly independent of �m. Since our mass function match-ing approach imposes a redder PðkÞ shape (lower ns or C)for higher �m, and increased large-scale power amplifies theimportance of bias, it is not surprising that bias overcompen-sates mass clustering changes in these models, leaving �hðrÞsteadily dependent on�m.

Figure 16 shows the mean pairwise radial velocity andpairwise velocity dispersion for the four cosmological mod-els. As in the case of cluster normalized models, models withhigher �m have larger mean pairwise velocities and velocitydispersions than lower�m models. The dependence on�m isstronger and more systematic than that in Figure 12, again asign of the redder power spectrum of higher�m models.

4. SUMMARY AND DISCUSSION

We have examined how changes to the matter densityparameter �m and the shape and amplitude of the linearpower spectrum PðkÞ affect the halo mass function, the two-point halo correlation function, and the first and secondmoments of the halo pairwise velocity distribution. Achange in �m, at fixed PðkÞ, simply shifts the halo massscale. Therefore, if halo masses are scaled in proportion to�m, halo populations of different �m models have identicalmass functions and clustering properties. However, meanpairwise velocities and pairwise velocity dispersions, whichscale as �0:6

m on large scales (and fairly far into the nonlinearregime), break this degeneracy. A change of the power spec-

Fig. 15.—Mass and halo correlation functions for the four models of Fig. 14, in the same format as Fig. 11. Halo mass ranges are centered onMc, 4Mc, and8Mc, where Mc ¼ 1:03� 1013 h�1 M� corresponds to M� of the central model. Open circles in these three panels show the mass correlation function of thismodel. Error bars are plotted for the central model and have similar magnitude for other models. Changing ns to match the shapes of the halo mass functionsleads to substantial differences in halo clustering, in contrast to Fig. 11.


trum amplitude (�8) along with �m, maintaining a clusternormalization condition �8 / ��0:5

m , produces halo popula-tions that have nearly identical spatial clustering at a fixedphysical mass scale and systematic but small differences inthe mean pairwise velocities and pairwise velocity disper-sions. However, while these cluster-normalized models havethe same halo space density at M � several� 1014 M�, theshapes of their mass functions are systematically different. Ifwe further allow the shape of the power spectrum to change(either by tilting it or by changing its shape parameter C),we can produce halo populations whose mass functionsmatch very well in both amplitude and shape, over a largemass range. However, the changes to the power spectrumshape cause correlation functions, mean pairwise velocities,and velocity dispersions for halos in the same mass range todiffer. We conclude that the halo populations produced bydistinct cosmological models are not degenerate. If they areindistinguishable by one statistic, they can be told apartusing another statistic.

Our results imply that a perfect knowledge of the halopopulation and its properties would allow us to pin downthe underlying cosmological parameters—specifically, thevalue of �m and the shape and amplitude of PðkÞ—even

before bringing in constraints from other cosmologicalmeasurements. However, a galaxy redshift survey detectsgalaxies rather than halos. Could changes to the halo occu-pation distribution (HOD) mask the differences in the halopopulations of different cosmological models? Models thatdiffer only in �m lead to the same halo mass function andhalo clustering in terms of M=M�, but with M� / �m.Their halos could thus be populated the same way as a func-tion of M=M� to produce indistinguishable galaxy spatialclustering. However, any dynamically sensitive statistics—virial masses of clusters and groups, pairwise velocity dis-persions, the parameter combination �0:6

m =b inferred fromredshift-space distortions, the galaxy-mass correlation func-tion from galaxy-galaxy lensing—would distinguish modelswith different values of �m immediately. Velocity biaswithin halos could mask some of these changes, but not allof them (Berlind & Weinberg 2002). Cluster-normalizedmodels with different �m and the same PðkÞ shape have dif-ferent halo mass functions, and they would thereforerequire a different PðNjMÞ in order to keep the galaxy den-sity fixed. This change would likely cause differences in thegalaxy clustering even though the halo clustering is similar,since galaxy clustering is highly sensitive to PðNjMÞ (Ber-

Fig. 16.—Halo velocity statistics for the four models shown in Figs. 14 and 15, in the same format as Fig. 12. Error bars are plotted for the central model;they increase with�m and are about twice as large for�m ¼ 1 as for�m ¼ 0:2. Changing ns to match the halo mass function shapes leads to significant system-atic differences in the mean pairwise velocities and pairwise velocity dispersions as a function of�m, larger than those in Fig. 12.


lind &Weinberg 2002 and references therein). Changing theshape of PðkÞ by the amount required to match mass func-tion shapes makes substantial differences to the halo cluster-ing and velocities, which seem unlikely to be masked by achange in the HOD.

These speculations suggest that distinct cosmologicalmodels, which produce nondegenerate halo populations,cannot have galaxy populations that are indistinguishablein every spatial and velocity statistic, even if one allows com-plete freedom in the way that galaxies occupy these halos.We reserve a detailed investigation of this issue for futurework. If our optimistic speculation holds true, then high-

precision measurements of galaxy clustering and galaxy-galaxy lensing should impose strong constraints on cosmo-logical models without reliance on a priori models of galaxybias.

We thank Volker Springel for advice on GADGET andChangbom Park for the use of his PM code. D. W. thanksthe Institute for Advanced Study for its hospitality and theAmbrose Monell Foundation for its financial support dur-ing the final stages of this work. This work was supportedby NSF grant AST 00-98584.

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