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Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 29 November 2018 (MN LATEX style file v1.4)

Tracing the Nature of Dark Energy with Galaxy

Distribution

P. Solevi1,2 ⋆, R. Mainini1,2, S.A. Bonometto1,2, A.V. Maccio 3, A. Klypin4 &

S. Gottlober 5

1 Physics Department G. Occhialini, Universita degli Studi di Milano–Bicocca, Piazza della Scienza 3, I20126 Milano (Italy)2 I.N.F.N., Sezione di Milano (Italy)3 Institute for Theoretical Physics, University of Zurich, Winterturstrasse 190, CH–8057 Zurich (Switzerland)4 Astronomy Department, New Mexico State University, Box 30001, Department 4500, Las Cruces, NM 88003-00015 Astrophysikalisches Institut Potsdam, An der Sternwarte 16,14482 Potsdam (Germany)

29 November 2018

ABSTRACT

Dynamical Dark Energy (DE) is a viable alternative to the cosmological constant.Yet, constructing tests to discriminate between Λ and dynamical DE models is difficultbecause the differences are not large. In this paper we explore tests based on the galaxymass function, the void probability function (VPF), and the number of galaxy clusters.At high z the number density of clusters shows large differences between DE models,but geometrical factors reduce the differences substantially. We find that detecting amodel dependence in the cluster redshift distribution is a hard challenge. We show thatthe galaxy redshift distribution is potentially a more sensitive characteristics. We doso by populating dark matter halos in N−body simulations with galaxies using well-tested Halo Occupation Distribution (HOD). We also estimate the Void ProbabilityFunction and find that, in samples with the same angular surface density of galaxiesin different models, the VPF is almost model independent and cannot be used asa test for DE. Once again, geometry and cosmic evolution compensate each other.By comparing VPF’s for samples with fixed galaxy mass limits, we find measurabledifferences.

Key words: methods: analytical, numerical – galaxies: clusters – cosmology: theory– dark energy

1 INTRODUCTION

High redshift supernovae, anisotropies of the cosmic mi-crowave background (CMB), as well as data on the large-scale galactic distribution (Riess et al. 1988, Perlmutter et al1988, Tegmark et al. 2001, De Bernardis et al 2000, Hananyet al 2000, Halverson et al 2001, Spergel et al 2003, Percivalet al. 2002, Efstathiou et al 2002) indicate that ∼ 70% of theworld contents are due to a smooth component with largelynegative pressure. This component is dubbed dark energy(DE). Recently Maccio, Governato & Horellou (2005) pre-sented further arguments in favor of DE based on the lo-cal (∼ 5Mpc) Hubble flow of galaxies. The nature of DEis still open for debate. Candidates range from a positivecosmological constant Λ – yielding a ΛCDM cosmology – tomodels with a slowly evolving self–interacting scalar field φ(dynamical DE; Ratra & Peebles 1988; Wetterich 1988) to

⋆ E-mail: solevi@mib.infn.it

even more exotic physics of extra dimensions (e.g., Dvali &Turner 2003).

ΛCDM cosmologies are easy to study and fit most data.Unfortunately, to give a physical motivation to the value ofΛ, we need a fine–tuning of vacuum energy at the end of thelast phase transition. To rival the success of ΛCDM other DEmodels ought to predict observables hardly distinguishablefrom it, so that discriminatory tests on DE nature are noteasy to devise.

Up to now, most tests based on large scale structuredealt with the evolution of the cluster distribution. Usingthe Press-Schechter-type approximations (Press & Schechter1974, PS hereafter; Sheth & Tormen, 1999, 2002 ST; Jenk-ins 2001) the expected dependence of the cluster mass func-tion on DE nature was extensively studied (see, e.g., Wang& Steinhardt 1998, Haiman et al 2001, Majundar & Mohr2003, Mainini, Maccio & Bonometto 2003). The results wereused to predict the redshift dependence of various observ-ables, such as temperatures (T ) or photon counts (N). InSec. 2, we compare the (virial) mass functions for different

c© 2005 RAS

2 Solevi et al.

DE. An important – and often overlooked – factor is thedependence of halo concentration c on the DE equation ofstate. For given virial mass the concentration varies with DEnature up to 80%, as shown by simulations (Klypin, Maccio,Mainini & Bonometto 2003; Linder & Jenkins 2003; Kuhlenet al. 2005). The model dependence of c is so strong thatit can be used as a possible discriminatory test for stronglensing measurements (Dolag et al 2004, Maccio 2005).

In this paper we shall focus on galactic ∼ 1012h−1M⊙

scales which are also interesting for testing models of DE.In order to make a prediction we first need to know howto generate a distribution of galaxies, not just dark matterhalos. There are different ways for doing this. We decidedto use recent results on the Halo Occupation Distribution(HOD): a probability to find N galaxies in a halo of massM . The HOD properties have been studied in details (Sel-jak 2000; Benson 2001; Bullock, Wechsler & Somerville 2002;Zheng et al. 2002; Berlind & Weinberg 2002; Berlind et al.2003; Magliocchetti & Porciani 2003, Yang, van den Bosch& Mo 2003; van den Bosch, Yang & Mo 2003). Numeri-cal simulations including gas dynamics (White, Hernquist& Springel 2001; Yoshikawa et al., 2001; Pearce et al. 2001;Nagamine et al. 2001;Berlind et al. 2003, Yang et al. 2004)or semi-analytical models of galaxy formation (Kauffmann,Nusser & Steinmetz 1997; Governato et al. 1998; Kauffmannet al. 1999a,b; Benson et al. 2000a,b; Sheth & Diaferio 2001;Somerville et al. 2001; Wechsler et al. 2001; Benson et al.2003a; Berlind et al. 2003) were used, to find a law tellingus how halos split in sub–halos hosting individual galaxies.

In this way we formulate predictions on galaxy massfunctions and their z–dependence. In order to compare pre-dictions with data we then need to disentangle the evolutionof the mass function from the evolution of theM/L ratio, be-cause observations provide luminosities of galaxies and ourestimates give us halo masses. In this respect, one of the aimsof this work is to estimate how precisely the M/L evolutionshould be known, in order to use data on galactic scales totest DE nature. Potentially,M/L evolution can be predictedusing galactic evolution models (see, e.g., Bressan, Chiosi &Fagotto 1994, Portinari, Sommer-Larsen & Tantalo 2004,and references therein). Such predictions can be comparedwith weak lensing results or satellite dynamics (Prada et al2003). The latter methods will provide estimates of virialM/L for samples of galaxies at different z. This is exactlywhat we need in order to test different models for DE. Herewe find that, for some statistics, the expected signal, i.e.,the differences between models are rather large. So, there isa hope to detect DE effects in spite of uncertainties in M/Lratios. Certainly, if galactic evolution predictions and high–zM/L estimates are compared, one can hardly prescind fromtaking into account accurately the impact of DE nature.

We run a series of N−body simulations with differentequations of state. In these models the ratio w = pde/ρde ofthe DE pressure to the energy density varies with z accord-ing to field dynamics. The models considered here are theRatra-Peebles (RP, Ratra & Peebles 1988) and the mod-els with the supergravity (SUGRA, Brax & Martin 1999,2001; Brax, Martin & Riazuelo, 2000). Appendix A givesa short summary on these models. Each model is speci-fied by an additional parameter – the energy scale Λ of theself–interacting potential of the scalar field. Here we take

Λ = 103GeV for both models. In RP (SUGRA), w showsslow (fast) variations with z.

The void probability function is an obvious candidatefor discriminating models. Fluctuations grow differently inthe models. So, one may expect some differences in VPF.We use galaxy distributions to estimate the void probabilityfunction (VPF) at different redshifts. While measuring VPFin simulations is straightforward, mimicking observations isslightly more complicated because it requires corrections forgeometrical effects and because the answer depends on adefinition of galactic populations. At z = 0 no model de-pendence of the VPF is expected or found. Predictions athigher z depend on how galaxy samples are defined. In par-ticular, we show that, if equal angular density samples areconsidered, VPF results are independent of the DE nature.On the contrary, if we select samples above fixed galacticmass M , a significant signal is found, which can be usefulfor testing the DE nature.

The plan of the paper is as follows: In Sec. 2 we discusshow geometry and galactic evolution affect the redshift dis-tributions of galaxies and clusters. In Sec. 3 and 4 we discussthe simulations and prescriptions of populating halos withgalaxies. In Sec. 5 and 6 results on redshift distributions andthe VPF are given. Sec. 7 is finally devoted to discussing ourresults and future perspectives.

2 GEOMETRICAL AND EVOLUTIONARY

FACTORS

Let us consider a set of objects whose mass function is n(>M, z). In a spatially flat geometry, their number between zand z +∆z, in a unit solid angle, is given by

N(> M, z,∆z) =

∫ z+∆z

z

dz′D(z′) r2(z′)n(> M, z′) (1)

with D(z) = dr/dz. For flat models:

D(z) =c

Ho

√

Ωm(z)

Ωmo(1 + z)3, r(z) =

∫ z

0

dz′D(z′) (2)

Here Ωm(z) is the matter density parameter at the redshiftz. The Friedman equation can be written in the form

H2(z) =8π

3Gρmo(1 + z)3

Ωm(z)= H2

oΩmo(1 + z)3

Ωm(z). (3)

Along the past light cone, a dr = −c dt. So, by dividingthe two sides by dz = −da/a2, one finds that a dr/dz =a2c dt/da. Accordingly, D(z) = dr/dz = c/H(z) is derivedfrom eq. (3).

When w is a constant, a useful expression

D2(z) = (c/Ho)(1 + z)−3[Ωmo + (1− Ωmo)(1 + z)3w ]−1 (4)

can be obtained, which allows one to see that the geometricalfactors increase both when w decreases and Ωmo decreasesin models with w < 0.

An extension to dynamical DE can be performed eitherby using the interpolating expressions yielding Ωm(z), forRP and SUGRA models, provided by Mainini et al (2003b)or, equivalently, through direct numerical integration. Weobtain the geometrical factor r2(z)D(z) shown in the upperpanel of Fig. 1.

c© 2005 RAS, MNRAS 000, 000–000

Tracing the Nature of Dark Energy with Galaxy Distribution 3

Figure 1. Geometrical and evolutionary terms on the clustermass scale. In both panels ΛCDM with Ωmo = 0.2 is aboveΛCDM with Ωmo = 0.3. On the contrary, in the upper and lowerpanels RP and SUGRA lay on the opposite sides of ΛCDM. Inthe latter case, cancellation between geometrical and evolutionaryeffects is therefore expected.

Figure 2. The upper panel shows how the redshift distributionon the cluster mass scale depends on the models. The cancel-lation expected from Fig. 1 has occurred and the models withΩmo = 0.3 are quite close, while ΛCDM grantes large differenceswhen varying Ωmo. In the lower panel, similar curves are shownfor halos on galactic mass scales. In this case, evolution is almostmodel independent and the geometrical factor causes an apprecia-ble difference. On these scales a lower Ωmo yields a different halodensity, unless the spectrum is normalized to unphysical levels.Hence, only models with Ωmo = 0.3 are shown.

Here Ωmo = 0.3 or 0.2 for ΛCDM, while Ωmo = 0.3for SUGRA and RP models (for whom Λ = 103GeV). Ho

is 70 km/s/Mpc in all models (h = 0.7). In the absence ofnumber density evolution, the upper panel of Fig. 1 showsthe dependence of the angular number density on z.

In the PS formulation, the expected differential clusternumber density n(M), at a given time, is then given by theexpression

f(ν)νd log ν =M

ρmn(M)Md logM . (5)

Here ρm is the matter density, ν = δc/σM is the bias factor,M is the mass scale considered. σM is the r.m.s. densityfluctuation on the scale M and δc is the amplitude that, inthe linear theory, fluctuations have in order that, assumingspherical evolution, full recollapse is attained exactly at thetime considered (in a standard CDM model this value is∼1.68; the difference, in other model, ranges around a fewpercent). As usual, we took a Gaussian f(ν) distribution.

Together with eq. (5), we must take into account thevirialization condition, which yields significantly differentdensity contrasts ∆v in different DE models. Further detailscan be found in Mainini et al (2003a).

In the lower panel of Fig. 1 we show the evolution ofthe number of halos of mass > 1014h−1M⊙, in comoving vol-umes. All models are normalized to the same cluster numbertoday and the redshift dependence of n(> M, z) is clearlyunderstandable, on qualitative bases: When ΛCDM modelsare considered, the evolution is faster as we approach stan-dard CDM. On the contrary, RP and SUGRA yield a slowerevolution than ΛCDM.

The important issue is that, while both the geometricalfactor and the evolutionary factor of ΛCDM (with Ωmo =0.2) lay above ΛCDM (with Ωmo = 0.3), RP and SUGRAfactors lay on the opposite sides of ΛCDM.

When the two factors are put together this causes theeffect shown in the upper panel of Fig. 2, a strong signalon Ωmo and a widespread cancellation for DE models, com-pared to ΛCDM. Discriminating between different DE na-tures, from this starting point, is unavoidably a hard chal-lenge.

Geometrical factors do not depend on the mass scale.Instead, evolutionary factors are known to have a strongerdependence on the model for larger masses. As cancellationis almost complete on cluster scales, it is to be expected thatgeometrical factors yield a significant signal on lower massscales. This is shown on the lower panel of Fig. 2, wherehalos of 1012h−1M⊙ are considered. A halos of this mass isexpected to host a normal galaxy. More massive halos areexpected to host many galaxies. Hence, this plot cannot bedirectly compared with observations. Its main significance isthat such lower mass scales deserve to be inspected becauseDE signals are expected to be strong enough on these scales.

3 SIMULATIONS

The simulations run for this work are based on a ΛCDMmodel and two dynamical DE models, with the same mat-ter density and Hubble parameters (Ωmo = 0.3 and h = 0.7).The simulations are run using the Adaptive Refinement Treecode (ART; Kravtsov, Klypin & Khokhlov 1997). The ART

c© 2005 RAS, MNRAS 000, 000–000

4 Solevi et al.

code starts with a uniform grid, which covers the whole com-putational box. This grid defines the lowest (zeroth) level ofresolution of the simulation. The standard Particles-Meshalgorithms are used to compute density and gravitationalpotential on the zeroth-level mesh. The ART code reacheshigh force resolution by refining all high density regions us-ing an automated refinement algorithm. The refinements arerecursive: the refined regions can also be refined, each subse-quent refinement having half of the previous level’s cell size.This creates a hierarchy of refinement meshes of differentresolution, size, and geometry covering regions of interest.Because each individual cubic cell can be refined, the shapeof the refinement mesh can be arbitrary and match effec-tively the geometry of the region of interest.

The criterion for refinement is the local density of par-ticles: if the number of particles in a mesh cell (as estimatedby the Cloud-In-Cell method) exceeds the level nthresh, thecell is split (“refined”) into 8 cells of the next refinementlevel. The refinement threshold may depend on the refine-ment level. The code uses the expansion parameter a asthe time variable. During the integration, spatial refine-ment is accompanied by temporal refinement. Namely, eachlevel of refinement, l, is integrated with its own time step∆al = ∆a0/2

l, where ∆a0 is the global time step of thezeroth refinement level. This variable time stepping is veryimportant for accuracy of the results. As the force resolutionincreases, more steps are needed to integrate the trajectoriesaccurately. Extensive tests of the code and comparisons withother numerical N-body codes can be found in Kravtsov(1999) and Knebe et al. (2000).

The code was modified to handle DE of different types,according to the prescription of Mainini et al. (2003b). Mod-ifications include effects due to the change in the rate of theexpansion of the Universe and on initial conditions, keep-ing into account also spatial fluctuations of the scalar fieldbefore they enter the horizon.

In this paper we use 4 new simulations. The mod-els are normalized assuming σ8 = 0.9. They are run ina box of 100 h−1Mpc. We use 2563 particles with massmp = 4.971 · 109 h−1M⊙. The nominal force resolution is3h−1kpc. All models are spatially flat, while Ωmo = 0.3 andh = 0.7. The two ΛCDM models start from different ran-dom numbers and are indicated as ΛCDM1 and ΛCDM2.The two DE models, named RP3 and SUGRA3, are startedfrom the same random numbers of ΛCDM1.

4 GALAXIES IN HALOS

Halos made by more than 30 particles were found in sim-ulations by the same spherical overdensity (SO) algorithmused in Klypin et al (2003). The algorithm locates all non-overlapping largest spheres where the density contrast at-tains a given value ∆v. Density contrasts are assigned thevirialization values, which depend on the redshift z and onparameters of the DE. For instance, at z = 0, ∆v = 101.0for ΛCDM, 119.4 for SUGRA3 and 140.1 for RP3; ∆v valuesfor higher z can be found in Mainini, Maccio & Bonometto(2003; see also Mainini et al 2003b).

Figure 3 shows the halo mass function in the LCDM1,the SUGRA3, and the RP3 models at z = 0. Differences be-tween the models are only due to different w(z), while their

Figure 3. Halo mass function at z = 0. Results forΛCDM (dashed line), RP (solid line), and SUGRA (dotted line)overlap.

σ8 is identical. Accordingly, at z = 0 their mass functionsalmost overlap and are well fitted by a the Sheth-Tormen(ST, Sheth & Tormen, 1999) approximation.

However, when we consider galaxies, we need to useanother approach because individual halos may host manygalaxies of different mass and luminosity. In order to assigngalaxies to halos we use a HOD. This is a relatively novelapproach to locate galaxies in each halo. It can be used indifferent ways. Full-scale semi-analytical methods can pre-dict such quantities as the luminosity, colors, star-formationrates. Unfortunately, many important mechanisms are stillpoorly understood making the results less reliable. It seemstherefore advisable to minimize the physical input, keepingjust to gravitational dynamics.

In this paper we use a prescription consistent with theresults of Kravtsov et al (2004). We utilize an analyticalexpression recently proposed by Vale & Ostriker (2004), butparameters of the approximation are different form Vale &Ostriker (2004) and tuned to produce a good fit to resultsof Kravtsov et al (2004). The approximation is based onthe assumption that the probability Ps(Ns|M) for a halo ofmass M to host Ns subhalos is approximately universal.We take the Schechter approximation

N(m|M) dm = Adm

βM

(

m

βM

)−α

exp(

− m

βM

)

(6)

for the number of subhaloes with masses in the range m tom+ dm, for a parent halo of mass M . A must be such thatthe total mass in subhaloes,

∫

∞

0dmmN(m|M), is a fraction

γM of the parent halo mass. Therefore A = γ/βΓ(2 − α),so that the number of sub–halos of mass m is

nsh(m) =

∫

∞

0

dM N(m|M)nh(M) , (7)

(nh(M) is the halo mass function) independently from the

c© 2005 RAS, MNRAS 000, 000–000

Tracing the Nature of Dark Energy with Galaxy Distribution 5

Figure 4. Comparison of different halo occupation distributions.The solid line is the HOD given by Kravtsov et al (2004). Othercurves are obtained from eq.(8) for different parameters γ as in-dicated in the plot.

parent halo mass. The expression (6) yields the followingnumber of subhaloes with mass > m in a halo of mass M :

Nsh(> m,M) =γ

βΓ(2− α)

∫

∞

m/βM

dx x−α exp(−x) . (8)

Sub–haloes will be then identified with galaxies. Figure4 shows that the expression approximates the results ofKravtsov et al (2004) once we fit parameters α, β, γ and addto the expression (8) a unity, i.e. the halo as a sub–halo ofitself. For large haloes, to be interpreted as galaxy clusters,this sub–halo could be the central cD; but adding an extraobject, in such large galaxy sets, is just a marginal reset. Forsmall haloes, instead, it is important not to forget that theyrepresent a galaxy, as soon as they exceed the galaxy massthreshold.

In a different context, Vale & Ostriker (2004) use thevalue γ = 0.18, β = 0.39. Owing to the use we make ofeq. (8), γ = 0.7 appears more adequate.

If the (differential) halo mass function nh(M) is known,the sub–halo mass function is

Nsh(> m) =γ

βΓ(2− α)×

×∫

∞

m/γ

dM nh(M)

∫

∞

m/βM

dxx−α exp(−x) . (9)

If we perform nonlinear predictions, nh(M) is obtained fromthe expression (5) or from the corresponding expressions inthe ST case. When we deal with simulations, halo masseshave discrete values mν = νmp, appearing n

(ν)h times, up to

a top mass νMmp. Then

Nsh(> m) =γ

βΓ(2− α)×

Figure 5. Galaxy (cumulative) mass function at z = 0 fromsimulations (dots) vs. an integral Schechter function, with Φo, αand M∗ shown in the frame.

×νM∑

ν= mγmp

n(ν)h

∫

∞

νm/βM

dxx−α exp(−x) . (10)

In Figure 5 we plot the galaxy mass function obtainedwith eq. (10), using the mass function of halos in the simu-lations, and identifying sub–haloes with galaxies. At z = 0model differences are unappreciable and the plotted functionholds for all models. We also plot a Schechter function withthe parameters shown in the frame, selected to minimize theratios between differential values at all points. As expected,the two curves are close. In fact, there must be some rela-tion between masses and luminosities, but the Mg/Lg ratioshould not be a constant.

5 GALAXY ANGULAR DENSITY IN MODELS

WITH DIFFERENT DE

Let us now consider the galaxy mass function at higher z,for the different models. According to eq. (1), the numberof galaxies with mass > m, in a solid angle ∆θ2 (∆θ ≪ 1),between redshifts z and z +∆z, is

Ng(> m, z;∆z,∆θ)

∆z∆θ2≃ c

r2(z)

H(z)ng(> m, z) , (11)

if ng(> m,z) is the comoving number density of galaxieswith mass > m at a redshift z. The galaxy density canbe obtained from the subhalo mass functions (9) and (10).Accordingly, the average angular distance θgg is given by

θgg(> m, z)√∆z ≃ 1

r(z)

[

H(z)

ng(> m, z)

]1/2

. (12)

Thus, the l.h.s. expression is independent of the particularvolume considered.

c© 2005 RAS, MNRAS 000, 000–000

6 Solevi et al.

Such θgg(> m, z) therefore depends on geometry, halomass function and HOD. We however expect that the red-shift dependence mostly arises from geometry, while evolu-tion plays a significant role at higher z. In fact, the maindifference between this and the cluster case is that evolu-tion is mild and discrepancies between models, in comovingvolumes, up to z ∼ 2, are even weaker.

Let θΛ(z), θSU(z), θRP (z) be the mean angular dis-tances between galaxies at redshift z for the ΛCDM, SUGRAand RP models, respectively. Besides these functions, letus also consider the angular distance θgeo(z) obtained fromeq. (12), keeping the value of ng(> m, z = 0) at any redshiftand the ΛCDM geometry. Therefore, θgeo(z), although thesymbol has no reference to ΛCDM, describes the behaviorof the angular separation in a ΛCDM model, in the limit ofno halo evolution.

In Figure 6, we compare results obtained from simu-lations with the ST predictions in different models. Ratherthan presenting θmod(z), we plot the fractional difference(θmod − θgeo)/θgeo. In principle, error bars can be evaluatedin two ways: (i) by comparing ΛCDM1 with ΛCDM2 (cos-mic variance) and (ii) by comparing the differences betweenmodels. The latter can be done only at present because themodels have the same power spectrum only at z = 0. Thedifferences between models exist because the fluctuationsgrow differently in the past. At larger z, this evolutionary

variance should be smaller, but is not easy to evaluate. Weuse differences between models at z = 0 as a rough estimateof error bars at all redshifts. Judging by the differences be-tween ΛCDM1 and ΛCDM2, the cosmic variance seems tobe smaller by a factor of three than the evolutionary differ-ences. We find similar behaviour for different galaxy masses.In all cases, differences between models can be clearly seen.

The largest differences between models are attained atz ∼ 1. Let us remind that the plot shows the fractional dif-ferences between DE models and the z–dependence due tothe mere ΛCDM geometry. At z ≃ 1, this difference is just≃ 5% for ΛCDM while, for SUGRA, it is ∼ 20%, becauseof the different geometry and a still slower cosmological evo-lution. Even larger is the difference with RP. This compareswith an evolutionary variance hardly exceeding ∼ 2%, if theeffective comoving volume inspected is ∼ 106h−3Mpc3. For∆z ∼ 0.1, this corresponds to δθ ∼ 30–40o.

The discriminating power of this theoretical predictionis to be still compared with two possible sources of error: (i)Peculiar velocities, setting individual galaxies into an ap-parent redshift band different from the one to which theybelong. (ii) Luminosity evolution.

Overcoming the latter point is critical to the use ofgalaxies as indicators of DE nature and we shall devote thewhole next section to the impact of luminosity evolution.The conclusion of this discussion is that galaxies are indeeda possible indicator of DE nature, but more work is neededbefore they can be efficiently used to this aim.

In this section we shall report the result of a test per-formed to evaluate the impact of redshift displacements dueto peculiar velocities.

We divided the volume L3 of the box into cells of side10−1L, with volume 10−3L3. 50 such cells were selected atrandom at each redshift and replaced by the cells at theclosest redshift considered with their galaxy contents. Theredshift displacement between the original cell and its re-

Figure 6. The redshift dependence of the mean fractional angularseparation in different models. Points with error bars show resultsof simulations. Curves indicate ST predictions. The plot shows astrong dependence of the galaxy redshift distribution on the DEnature.

placements is ±0.5. For a change of 5% in galaxy contentsthe average shift of θgg is ∼ 1.2%.

This shift lays well below the error bars shown in Figure6. However, the average errors due to evolutionary and cos-mic variance plus peculiar velocities are ∼ 2.4% and thewhole error never exceeds 2.6–2.7 % . We therefore arguethat these sources of error do not affect the robustness ofresults.

6 DEPENDENCE ON MASS–LUMINOSITY

RELATION

Let us now consider galaxy evolution, which is expected tocause z–dependence of the average Mg/Lg ratio, but canalso yield a z–dependence of the Mg/Lg distribution aboutsuch average, in a way which may depend on the mass rangeconsidered. From an observational point of view, when weconsider galaxies of various luminosities Lg, we must thentake into account that their expected mass Mg could bedistributed with different laws at different z and Lg .

Let us therefore consider the galaxy distribution on theMg, Lg plane at a given z, yielding the galaxy number

dN = D(Mg , Lg , [z]) dMg dLg (13)

in the infinitesimal area dMg dLg about the point Mg , Lg .We put z in brackets to outline that, in respect to it, Dis not a distribution but a function . Obviously we expect astrong correlation between Mg and Lg , at any z, so that itmakes sense to consider an average Mg/Lg ratio.

Once the distribution D is assigned, the distributions

c© 2005 RAS, MNRAS 000, 000–000

Tracing the Nature of Dark Energy with Galaxy Distribution 7

on Mg ( at fixed Lg ) and on Lg (at fixed Mg ) read

φ(Mg, [z]) =

∫

dLg D(Mg , Lg , [z]) ,

ψ(Lg, [z]) =

∫

dMg D(Mg, Lg , [z]).

The number dN is also the product of φ(Mg, [z]) times thedistribution on luminosities at fixed mass Mg:

dN = φ(Mg, [z])Q(Lg; [Mg , z]) dMg dLg . (14)

Equating the r.h.s.’s of eqs. (13) and (14) yields

Q(Lg; [Mg , z]) =D(Mg, Lg , [z])∫

dlD(Mg , l, [z])(15)

and, similarly, the distribution on masses at given Lg , reads

P (Mg; [Lg , z]) =D(Mg , Lg , [z])

∫

dmD(m,Lg, [z]). (16)

We can now use P to work out the average Mg/Lg at fixedLg, and the distribution on Mg/Lg about such average.Clearly⟨

Mg

Lg

⟩

Lg ,z

=1

Lg

∫

dmmP (m; [Lg , z]) =

=1

Lg

∫

dmmD(m,Lg , [z])∫

dmD(m,Lg , [z]), (17)

while the distribution

D(Mg ; [Lg , z]) =Mg

Lg

D(Mg , Lg , [z])∫

dmD(m,Lg , [z])(18)

tells us how Mg/Lg is distributed around 〈Mg/Lg〉Lg ,z.

The impact of the evolution of stellar populations (orother mechanisms) on the Mg/Lg ratio can be fully ex-pressed through the distribution D(Mg , Lg ; [z]) in eq. (13).From it we can work out an average mass/luminosity ra-tio 〈Mg/Lg〉 and the distribution on masses D; they bothdepend on Lg and z.

Let us now try to inspect how such variable D distribu-tion affects our results, taking into account that we mostlyignore how such variations occur. Accordingly, we shall pro-ceed as follows: we define a “wild” distribution, that we ex-pect to spread the Mg/Lg ratio, at fixed Lg, farther fromaverage than any physical D, at any z, will do. The effectscaused by such wild distribution should then be an overes-timate of the effects of the actual distributions. Should theycause just a minor perturbation in estimates, all we haveto worry about is the redshift dependence of the average〈Mg/Lg〉Lg

.More in detail, we shall allow that a galaxy of given

mass Mg has a luminosity in an interval L1, L2 with L2 ∼20L1. We test this prescription without direct reference toluminosities: In the sample of galaxies obtained through theHOD, at each z, each galaxy mass (m) is replaced by a massm′ = m + ∆mR, R being a random number with normaldistribution and unit variance. We take ∆m = 0.8m, butreplace all m′ < 0.1m with 0.1m, as well as all m′ > 1.9mwith 1.9m. The shift is therefore symmetric on m (not onlogm). The operation causes a slight increase of the massfunction above M ∼ 2.8 · 1011h−1M⊙ (by a few percents),

as there are however more lighter galaxies coming upwardsthan heavier galaxies going downwards (below M , the low–mass cut–off of the mass function, set by the mass resolutionof our simulations – see section 3 – begins to cause shortageof transfers upwards). The operation is then completed byreducing all masses by a (small) constant factor, so that,summing up all masses of objects with mass Mg > M , wehave the same total mass as before the operation. This lowersthe limit below which the mass function preserves its initialshape, but we never use galaxy samples including massesbelow 3 · 1011h−1M⊙.

We re–estimated θmod

√∆z using the new masses, for

the same mass limits as before, and compare the changesobtained in this way with the Poisson uncertainty, due tothe finite number of galaxies in each sample.

We find that the error obtained from the above proce-dure ranges between 20 and 40% of the Poisson error.

This output tells us that the evolution of the physi-cal distribution can be expected to redistribute results wellinside Poisson uncertainty. If we expect this redistributionto be random, the top value of the whole expected error isthen 3%; if we attribute a systematic character to it and re-frain from performing a quadratic sum with the other errorsources considered, the overall possible error is still within3.8%. It should be outlined that here we pushed all errorsources to their maximum; thus, we believe that the aboveestimates are safely conservative.

Let us now discuss how the evolution of the averageMg/Lg ratio can affect the use of galaxies to detect DEnature. In principle, one could use the z dependence ofD(Mg, Lg), to obtain the z–dependence of the 〈Mg/Lg〉Lg

ratio. More realistically, suitable data sets can directly pro-vide the z–dependence of 〈Mg/Lg〉Lg , with some residualuncertainty. The basic issue is then: How well the z evolu-tion of 〈Mg/Lg〉Lg is to be known, in order that we can testDE nature ?

Figure 6 is however devised so to provide a direct replyto this question: If we double the values of δθgg/θgg pro-vided there, we have a fair estimate the difference betweenevolution rates of R = 〈Mg/Lg〉Lg (z)/〈Mg/Lg〉Lg (z = 0)needed just to cover the differences between geometry anddynamics for the ΛCDM model or to compensate the differ-ences between the ΛCDM geometry and whole evolution fordynamical DE models, (in the limit Mg ≪ M∗, where M∗

is the mass scale appearing in a Schechter–like expression).Figure 6 can just be interpreted in both senses, changing thename of the ordinate.

For instance, at z = 0.5, an uncertainty ∼ 20% (35%)on the evolution of Mg/Lg is needed to hide the differencebetween ΛCDM and SUGRA (RP).

Let us show this point. Owing to eq. (12), θgg ∝ n−1/2g ,

so that a shift δθgg in the observed angular distance arisesfrom a shift

δng

ng≃ 2

δθggθgg

(19)

in the galaxy number density

ng(> Mg) = ng

(

> LgMg

Lg

)

. (20)

The latter shift, as shown in eq. (20), can arise from a shifton Mg/Lg , being

c© 2005 RAS, MNRAS 000, 000–000

8 Solevi et al.

δng = ng(Mg)Lg δ

(

Mg

Lg

)

; (21)

here ng(Mg) is the differential mass function, obtained bydifferentiating the integral mass function ng(> Mg). There-fore,

δng(> Mg)

Mgng(Mg)≃ δ(Mg/Lg)

(Mg/Lg)(22)

and

δ(Mg/Lg)

(Mg/Lg)≃ δng(> Mg)

ng(> Mg)

ng(> Mg)

Mgng(Mg). (23)

If we approximate the integral mass function by a Schechterexpression, it is |mn(m)/n(> m)| = 1 +m/M∗, so that

δ(Mg/Lg)

(Mg/Lg)≃ δng(> Mg)

ng(> Mg)

(

1 +Mg

M∗

)−1

. (24)

Using this equation together with eqs. (19) and (23), wehave the relation

δ(Mg/Lg)

(Mg/Lg)≃ 2

(

1 +Mg

M∗

)−1 δθggθgg

(25)

telling us how to use Figure 6 to estimate the evolution ofMg/Lg needed to yield the same effects of a change in DEnature. This equation also tells us how to use Figure 6 formasses approaching M∗.

7 THE VOID PROBABILITY FUNCTION

Let us randomly throw spheres of radius R, in a space whereobjects of various masses M are set. The probability Po(R)of finding no object with M > Mtr in them, if the VPF forobjects of mass > Mtr.

We expect and find no model dependence in the galaxy

VPF’s at z = 0. At z > 0, a critical issue is how Mtr is set.One can simply plan to determine the galaxy masses Mg

from data (e.g., from Lg values), so to select galaxies withMg > Mtr. As widely outlined, this choice involves severalcomplications. Another option is taking the most luminous

galaxies up to an average angular distance θgg.Each threshold Mtr, for any z and ∆z, yields a value

of θgg. Fig. 6 shows how θgg depends on the model at fixedthreshold. Viceversa, if we keep, for that z and ∆z, a fixedθgg, the relative Mtr varies with models. We can comparemodels either at fixed Mtr or at fixed θgg. Dealing withobservations, the latter option is easier, but mixes up theintrinsical VPF dependence on the model and other features,which also depend on the model.

Besides of threshold setting, another issue bears a greatoperational relevance. In principle, VPF’s can be evaluatedin the comoving volumes where galaxies are set and com-pared there with VPF’s from data. This is however unade-quate to evaluate how discriminatory is the VPF statistics.To do so, we rather follow the following steps:(i) From cartesian coordinates xi (i = 1, 2, 3) in comovingvolumes we work out the redshift and the celestial coordi-nates z, θ, φ that an observer, set at z = 0, would measure.This is done by using the geometry of the model.(ii) Data also give z, θ, φ, for each galaxy. But, to estimatethe VPF, they must be translated into cartesian coordinatesxi. An observer can only perform such translation by using

the geometry of a fiducial model, e.g., ΛCDM. The secondstep, to forge predictions, therefore amounts to re–transformz, θ, φ into cartesian coordinates xi, but using now the fidu-cial ΛCDM geometry; xi’s coincide with the xi’s only for aΛCDM cosmology. Let us call fiducial space, the environ-ment where galaxies are now set.(iii) We then estimate the VPF’s, for all models, in the fidu-cial space; these VPF’s should be compared with observa-tional data, but can also be compared one another to assesshow discriminatory this statistics can be.

Although comparing predictions for VPF’s in comovingvolumes, therefore, bears little discriminatory meaning, ouroutputs are more easily explained if we start from comovingspace VPF’s. Let us recall that, on galaxy scales, evolution-ary differences, up to z ∼ 2, are modest. Accordingly, foran assigned Mtr, we find just marginal discrepancies, as isshown in Fig. 7 forMtr = 6 ·1011M⊙h

−1. Differences amongVPF’s arise, of course, if we do not fix Mtr, but θgg, as areflex of θgg differences. These VPF’s are shown in Fig. 8.

VPF’s in respect of comoving coordinates bear a strictanalogy with mass functions in comoving volumes. The factthat geometry erases almost any signal on the cluster massfunction is analogous to what happens for the VPF, whenwe pass from the comoving to the fiducial space. This fact isfar from trivial. Let us compare these VPF’s with the VPF’sin a Poisson sample with the same θgg (Fig. 10) and remindthat (White 1979)

Po(R) = exp

[

−NR +

∞∑

n=2

(−NR)n

n!ξ(n)(R)

]

. (26)

Here NR is the average number of points in a sphere of radiusR; ξ(n)(R) are the n–point functions averaged within thesame sphere. For the Poisson sample Po(R) = exp(−NR),as all ξ(n)(R) vanish. The difference between Poisson VPFand model VPF is to be fully ascribed to ξ(n), as NR is setequal. This difference is huge, in respect to the differencesbetween models, which should arise because of ξ(n) shifts.Their paucity indicates that density renormalization almosterases the shifts in correlation functions of all orders.

The cancellation between geometrical and θgg effects,shown in Fig. 9, indicates that the passage from comovingto fiducial coordinates bears a weight comparable with thedifferences shown in Fig. 8. It therefore comes as no surprisethat VPF’s, for fixed Mtr, almost absent in the comovingspace, are significant in the fiducial space. They are shownin Fig. 11 and, as is expected, the curves of different modelsappear in the opposite order in respect to Fig. 8.

If the redshift dependence of the Mg/Lg ratio is un-der control, Fig. 11 shows a discriminatory prediction whichcan be compared with data. Once again, the problem con-cerns both the evolution of the mean Mg/Lg ratio and sin-gle galaxy deviations from average. Supposing that the av-erage Mg/Lg evolution is under control, we can estimatethe impact of individual deviations by replacing the sharpthreshold on Mg with a soft threshold, substituting eachgalaxy mass Mg with Mg +∆Mg R, as in the previous sec-tion. This test was performed for samples as wide as thosein our 100 h−1Mpc box and the effects of such replacementare modest, amounting to ∼ 10% of the difference foundbetween ΛCDM and SUGRA.

Accordingly, the critical issue concerns the meanMg/Lg

c© 2005 RAS, MNRAS 000, 000–000

Tracing the Nature of Dark Energy with Galaxy Distribution 9

Figure 7. VPF’s in comoving volumes for Mtr = 6 ·1011M⊙h−1.The four panels refer to redshift 0.5, 1, 1.5, 2, as indicated in theframes. The galaxy numbers in the simulation box are reported,for each model and redshift. Solid, dashed and dotted lines as inFig. 6.

ratio. By comparing V PF ’s for different thresholds, we canhowever see that, in order that the shift of Mtr induces aVPF shift similar to differences between models, Mtr is tobe displaced by a factor 1.7–1.8. Uncertainties ∼ 10% onthe mean Mg/Lg ratio would therefore leave intact the dis-criminatory power of the VPF statistics.

Before concluding this section, let us finally commenton sample variance. All ΛCDM VPF’s were estimated onthe ΛCDM1 simulation. Differences between ΛCDM1 andΛCDM2 are however small and could not be appreciatedin the above plots. Accordingly, sample variance is not arelevant limit to the use of VPF’s.

8 CONCLUSIONS

Dark Energy modifies the rate of cosmic expansion in theepoch when a substantial fraction of fluctuations on clus-ter scales reach their turnaround. Therefore, it seems quitenatural to trace the redshift dependence of w(z) = pde/ρdeusing the cluster mass function at different redshifts.

Unfortunately, the situation is more complicated. Theevolution of w(z) affects observations in two ways. First, itcauses objects to form and evolve with different rate. Second,it results in a different mapping of comoving coordinates ofgalaxies to observed angular positions and redshifts. Then,on a cluster mass scale (∼ 1014h−1M⊙), the evolutionaryand geometrical effects tend to cancel, which makes clusterssomewhat problematic for testing the equation of state.

In this paper we discuss the use of scales where theevolutionary effects are minimized, so that the geometricaleffects leave a clearer imprint. In a sense, this is not a newprocedure: an analogous idea is utilized when the DE equa-

Figure 8. VPF’s in comoving volumes for fixed angular density.The four panels refer to redshift 0.5, 1, 1.5, 2, as indicated in theframes. Numbers in the frames as in Fig. 7. Solid, dashed anddotted lines as in Fig. 6.

Figure 9. VPF’s in fiducial volumes for fixed angular density.Numbers in the frames as for Fig. 7. Solid, dashed and dottedlines as in Fig. 6.

tion of state is tested by using a standard candle. Obviously,galaxies are not standard candles themselves, but they canprovide a “standard meter” through their (almost) model–independent abundance and evolution.

Previous analysis, which focused on clusters, tried toovercome the above difficulties by making recourse to vari-ous features. For example, the evolutionary dependence on wis preserved if one considers masses well above 1014h−1M⊙.Unfortunately, clusters with masses ∼ 1015h−1M⊙ or largerare rare today and surely are even more rare in the past.Some analysis (see, e.g., Haiman et al. 2000) stressed a pos-sible role of very massive clusters at large z. Yet, the numberof such clusters cannot be large and, thus, comparing suchpredictions with observations is a hard challenge. There isalso another problem with using cluster masses. Usually themass function is estimated with a PS–like approximations,

c© 2005 RAS, MNRAS 000, 000–000

10 Solevi et al.

Figure 10. Differences between VPF’s of various models andthe VPF for a Poisson sample, in fiducial volumes for fixed an-gular density. They arise from the sum of n–point correlationfunctions in eq. (26). Tiny residual differences between models,almost unappreciable in the previous plot, arise from differencesbetween their n–point functions, clearly almost erasen by geomet-rical renormalization.

Figure 11. VPF’s in fiducial volumes for fixed mass limit (seetext). Numbers in the frames as for Fig. 7. Solid, dashed anddotted lines as in Fig. 6.

which are well tested with simulations. The “virial” radiusRv is defined so that inside Rv the density contrast is ∆v.The value of ∆v depends on the redshift and on DE model.On the contrary, data are typically analized with a standarddensity contrast ∆c ≃ 180 (or 200). Increasing ∆c reducesthe amplitude of the mass function. If mass functions de-fined with variable ∆v (almost) overlap one another, massfunctions defined with constant ∆c can be different. In orderto account for these differences in the definitions, one needsto assume some shape for the density profile in the outskirtsof clusters. This is typically done by using a NFW profilewith concentration cs ≃ 5. As we deal with rather peripheral(virial) cluster regions, we can neglect the spread of actualvalues of concentration. However, when different w(z) are

considered, the mean cs changes substantially, up to 80%(Klypin et al 2003; Kuhlen et al. 2005). The differences be-tweenM200 andMv are not large – 10–15%. Approximatelythe same percent of the difference depends on w. This maybe still important. Neglecting these corrections may lead tosubstantial systematic errors.

As an alternative to using galaxy clusters, so avoidingthese and other problems, here we suggest to exploit the de-pendence on DE nature of the redshift distribution of galax-ies and argue that the difficulties of this approach can beovercome.

A first problem is that one needs to know how to treatsubhalos of more massive halos because a large fraction ofgalaxies is hosted by subhalos. To populate massive haloswith galaxies we use recent results on the halo occupationdistribution (HOD). Accordingly, we believe that this diffi-culty can be readily overcome.

Dealing with galaxies also requires knowledge of theirmasses Mg . The Mg −Lg relation is more complex than therelation between Mv, X-ray flux and T in clusters.

Galactic evolution studies and/or techniques aiming tocompare dynamical or lensing masses with luminosities canbe used to this aim (Bressan et al 1994, Portinari et al 2004,Prada et al 2003). It is also known that there is no one–to–one correspondence between Lg and Mg , as the luminositiesof two galaxies of the same mass can be quite different, up toone order of magnitude. We then considered separately twodifferent issues: (i) How well we must know the evolutionof 〈Mg/Lg〉Lg . (ii) Which can be the impact of fluctuationsabout such average value, taking into account that the dis-tribution about average can depend on z and mass.

How precisely 〈Mg/Lg〉Lg is to be known, in order thatdifferent cosmologies can be safely discriminated, is shownby Fig. 6, according to Sec. 6: If we double the values ofδθgg/θgg provided there, we have a fair estimate the evolu-tion rate R = 〈Mg/Lg〉Lg (z)/〈Mg/Lg〉Lg (z = 0) just cov-ering the differences between models. Typically, if the es-timated 〈Mg/Lg〉Lg evolution is reliable at a ∼ 10% level,different models could be discriminated.

To reach this goal, fresh observational material isneeded, but no conceptual difficulty is apparently involvedin its acquisition.

The spread of the Mg/Lg ratio, around its averagevalue, also involves a delicate issue, as it could cause sys-tematics. Here we reported the effect of a random spread ofLg in a luminosity interval L1, L2 with L2 ≃ 20L1. If thephysical distribution of luminosities, for any mass and at anyredshift, is within these limits, then possible systematics arewell within errors due to other effects.

Further tests on Lg spread were also performed, whichare not reported in detail in this paper. They apparentlyindicate that really wide and ad–hoc distributions are nec-essary, in order that possible systematics exceed Poisson un-certainty.

Although bearing these reserves in mind, we concludethat estimates of the redshift dependence of the averageMg/Lg , reliable within ∼ 10%, can enable us to obtain afair information on DE nature.

In this paper we also discuss various tests based on thevoid probability function. We find that the VPF is almostmodel independent when estimated for samples with con-stant angular number density of galaxies. This result ap-

c© 2005 RAS, MNRAS 000, 000–000

Tracing the Nature of Dark Energy with Galaxy Distribution 11

parently suggests that n–point functions are almost modelindependent, at any z, once distances are suitably rescaled.If the mean Mg/Lg evolution is under control, at the 10%level, we can also assess that VPF, for samples defined with agiven Mtr, puts in evidence geometrical differences betweenmodels and provides a discriminatory statistics.

Combining the simulated halo distribution with theHOD provides an effective tool for testing the equation ofstate of DE. More work is needed to define the Lg–Mg rela-tion and, possibly, to reduce systematic effects. It is justifiedto expect that the z–dependence of galaxy distribution, indeep galaxy samples, will allow us to constrain the DE na-ture even more reliably than the density of galaxy clustersin future complilations, sampling them up to large z’s.

Acknowledgments: We thank Loris Colombo and CesareChiosi for stimulating discussions. An anonimous referee isalso to be thanked for improving the arguments now inSec. 6. A.K. acknowledges financial support of NSF andNASA grants to NMSU. S.G. acknowledges financial supportby DAAD. Our simulations were done at the LRZ computercenter in Munich, Germany.

APPENDIX A: DYNAMICAL DARK ENERGY

MODELS

Dynamical DE is to be ascribed to a scalar field, φ, self–interacting through an effective potential V (φ), whose dy-namics is set by the Lagrangian density:

LDE = −1

2

√−g (∂µφ∂µφ+ V (φ)) . (A1)

Here g is the determinant of the metric tensor gµν =a2(τ )dxµdxν (τ is the conformal time). In this work weneed to consider just a spatially homogeneous φ (∂iφ ≪ φ;i = 1, 2, 3; dots denote differentiation with respect to τ ); theequation of motion is then:

φ+ 2a

aφ+ a2

dV

dφ= 0 . (A2)

Energy density and pressure, obtained from the energy–momentum tensor Tµν , are:

ρ = −T 00 =

φ2

2a+ V (φ) , p =

1

3T ii =

φ2

2a− V (φ) , (A3)

so that the state parameter

w ≡ p

ρ=φ2/2a− V (φ)

φ2/2a+ V (φ)(A4)

changes with time and is negative as soon as the potentialterm V (φ) takes large enough values.

The evolution of dynamical DE depends on details ofthe effective potential V (φ). Here we use the model proposedby Ratra & Peebles (1988), that yield a rather slow evolutionof w, and the model based on supergravity (Brax & Martin1999, 2001; Brax, Martin & Riazuelo 2000). The latter givesa much faster evolving w. The RP and SUGRA potentials

V (φ) =Λ4+α

φαRP, (A5)

V (φ) =Λ4+α

φαexp(4πGφ2) SUGRA (A6)

cover a large spectrum of evolving w. These potentials allowtracker solutions, yielding the same low–z behavior that isalmost independent of initial conditions. In eqs. (A5) and(A6), Λ is an energy scale in the range 102–1010 GeV, rel-evant for the physics of fundamental interactions. The po-tentials depend also on the exponent α. Fixing Λ and α, theDE density parameter Ωde,o is determined. Here we ratheruse Λ and Ωde,o as independent parameters. In particular,numerical results are given for Λ = 103GeV.

The RP model with such Λ value is in slight disagree-ment with low-l multipoles of the CMB anisotropy spectrumdata. Agreement may be recovered with smaller Λ’s, whichhowever loose significance in particle physics. The SUGRAmodel considered here, on the contrary, is in fair agreementwith all available data.

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