The impact of Early Dark Energy on non-linear structure formation
aa r X i v : . [ a s t r o - ph ] S e p Mon. Not. R. Astron. Soc. , 1–17 (2007) Printed 15 August 2018 (MN L A TEX style file v2.2)
The impact of Early Dark Energy on non-linear structure formation
Margherita Grossi , Volker Springel Max-Planck Institut fuer Astrophysik, Karl-Schwarzschild Strasse 1, D-85748 Garching, Germany([email protected]),([email protected])
Accepted ???. Received ???; in original form
ABSTRACT
We study non-linear structure formation in high-resolution simulations of Early Dark Energy(EDE) cosmologies and compare their evolution with the standard Λ CDM model. In EarlyDark Energy models, the impact on structure formation is expected to be particularly strongbecause of the presence of a non-negligible dark energy component even at very high red-shift, unlike in standard models that behave like matter-dominated universes at early times. Infact, extensions of the spherical top-hat collapse model predict that the virial overdensity andlinear threshold density for collapse should be modified in EDE model, yielding significantmodifications in the expected halo mass function. Here we present numerical simulations thatdirectly test these expectations. Interestingly, we find that the Sheth & Tormen formalism forestimating the abundance of dark matter halos continues to work very well in its standardform for the Early Dark Energy cosmologies, contrary to analytic predictions. The residualsare even slightly smaller than for Λ CDM. We also study the virial relationship between massand dark matter velocity dispersion in different dark energy cosmologies, finding excellentagreement with the normalization for Λ CDM as calibrated by Evrard et al. (2008). The earliergrowth of structure in EDE models relative to Λ CDM produces large differences in the massfunctions at high redshift. This could be measured directly by counting groups as a functionof the line-of-sight velocity dispersion, skirting the ambiguous problem of assigning a massto the halo. Using dark matter substructures as a proxy for member galaxies, we demonstratethat even with 3-5 members sufficiently accurate measurements of the halo velocity dispersionfunction are possible. Finally, we determine the concentration-mass relationship for our EDEcosmologies. Consistent with the earlier formation time, the EDE halos show higher concen-trations at a given halo mass. We find that the magnitude of the difference in concentration iswell described by the prescription of Eke et al. (2001) for estimating halo concentrations.
Key words: early universe – cosmology: theory – galaxies: formation
Arguably the most surprising result of modern cosmology is thatall matter (including both atoms and non-baryonic dark matter) ac-counts for only a quarter of the total energy density of the Uni-verse today, while the rest is contributed by a dark energy field. In1999, observations of type Ia supernovae by the Supernovae Cos-mology Project (Riess et al. 1999; Riess 2004) and the relative ac-curate measurements of the distances to this objects (Perlmutter1999; Kowalski 2008) demonstrated that the expansion of the Uni-verse is accelerated today; there hence exists a mysterious force thatacts against the pull of gravity. Nowadays, the inference that this iscaused by dark energy can be made with significant confidence, asthe observational evidence has further firmed up. In fact, we havegood reason to believe that we live in a flat universe with an upperlimit of Ω m . for the matter density today, based on cosmicmicrowave background measurements and a host of other observa-tional probes (Komatsu et al. 2008, e.g.). These observations yield a consistent picture, the so-called concordance cosmology, and arein agreement with predictions of the inflationary theory.The physical origin of dark energy is however unknown and amajor puzzle for theoretical physics. A nagging outstanding prob-lem is that most quantum field theories predict a huge cosmologicalconstant from the energy of the quantum vacuum, up to 120 ordersof magnitude too large. There is hence no simple natural explana-tion for dark energy, and one has to be content with phenomenolog-ical models at this point. Two proposed forms of dark energy are thecosmological constant, a constant energy density filling space ho-mogeneously, and scalar fields such as quintessence. In particular,‘tracking quintessence’ models attempt to alleviate the coincidenceproblem of the cosmological constant model. More exotic modelswhere the dark energy couples to matter fields or can cluster itselfhave also been proposed.In light of the many theoretical possibilities, the hope is thatfuture observational constraints on dark energy will enable progressin the understanding of this puzzling phenomenon. This requires c (cid:13) Grossi & Springel the exploitation of the subtle influence of dark energy on structureformation, both on linear and non-linear scales. As the expectedeffects are generally small for many of the viable dark energy sce-narios, it is crucial to be able to calculate structure formation indark energy cosmologies with sufficient precision to tell the dif-ferent models apart, and to be able to correctly interpret observa-tional data. For example, in order to use the abundance of clustersof galaxies at different epochs to measure the expansion historyof the universe, one needs to reliably know how the cluster massfunction evolves with time in different dark energy cosmologies.Numerical simulations are the most accurate tool available to ob-tain the needed theoretical predictions, and they are also crucial fortesting the results of more simplified analytic calculations.In this study, we carry out such non-linear simulations for aparticular class of dark energy cosmologies, so-called Early DarkEnergy (EDE) models where dark energy might constitute an ob-servable fraction of the total energy density of our Universe at thetime of matter radiation equality or even big-bang nucleosynthe-sis. While in the cosmological constant scenario, the fraction indark energy is negligible at high redshift, in such models the en-ergy fraction is a few per cent during recombination and structureformation, which introduces interesting effects due to dark energyalready at high redshift. In particular, for an equal amplitude ofclustering today, we expect structures to form earlier in such cos-mologies than in Λ CDM. This could be useful to alleviate the ten-sion between a low σ normalization suggested by current observa-tional constraints from the CMB on one hand, and the observationsof relatively early reionization and the existence of a population ofmassive halos present already at high redshift on the other hand.Recently, Bartelmann et al. (2006) studied two particular EDEmodels, evaluating the primary quantities relevant for structure for-mation, such as the linear growth factor of density perturbation, thecritical density for spherical collapse and the overdensity at virial-ization, and finally the halo mass function. In the two models an-alyzed, they found that the effect of EDE on the geometry of theUniverse is only moderate, for example, distance measures can bereduced by . Assuming the same expansion rate today, suchmodels are younger compared to Λ CDM. At early times, the age ofthe universe should differ by approximately − .However, when Bartelmann et al. (2006) repeated the calcu-lation of the spherical collapse model in the EDE cosmology, afew nontrivial modifications appeared. The evolution of a homoge-neous, spherical overdensity can be traced utilizing both the virialtheorem and the energy conservation between the collapse and theturn around time (see also Lacey & Cole 1993; Wang & Steinhardt1998). Bartelmann et al. (2006) obtained the value of the virialoverdensity as a function of the collapse redshift, translating theeffect of the early dark energy in an extra contribution to the po-tential energy at early times. They found that the virial overdensityshould be slightly enlarged by EDE, because a faster expansionof the universe means that, by the time a perturbation has turnedaround and collapsed to its final radius, a larger density contrast hasbeen produced. However, at the same time they found that the lin-early extrapolated density contrast corresponding to the collapsedobject should be significantly reduced.These two results based on analytic expectations have a pro-nounced influence on the predicted mass function of dark matterhalos. In EDE models, the cluster population expected from thePress-Schechter or Sheth-Tormen formalism grows considerablyrelative to Λ CDM, as a result of the lowered value of the critical lin-ear density contrast δ c for collapse. This effect can be compensatedfor by lowering the normalization parameter σ in order to obtain the same abundance of clusters today. In this case, one would how-ever still expect a higher cluster abundance in EDE at high redshift,due to the earlier growth of structure in this model.An open question is whether the EDE really participates inthe virialization process in the way assumed in the analytic model-ing. Similarly, it is not clear whether the excursion set formalism ofSheth & Tormen yields an equally accurate description of the non-linear mass function of halos in EDE cosmologies as in Λ CDM.Because accurate theoretical predictions for the halo mass functionare a critical ingredient for constraining cosmological parameters(in particular Ω m and Ω Λ ) as well as models of galaxy formation,it is important to test these predictions for the EDE cosmology indetail with numerical N-body simulations. In particular, we want toprobe whether the fraction f of matter ending up in objects largerthan a given mass M at some redshift z can be found by only look-ing at the properties of the linearly evolved density field at thisepoch, using the ordinary ST formalism, or whether there is somedependence on redshift, power spectrum or dark energy parameters,as suggested by Bartelmann et al. (2006).A further interest in EDE cosmologies stems from the fact thatfor a given σ , the EDE models predict a substantially slower evo-lution of the halo population than in the Λ CDM model. This couldexplain the higher normalization cosmology expected from clus-ter studies relative to analysis of the CMB. The value of σ , fora given cosmology, provides also a measure of the expected bias-ing parameter that relates the galaxy and the mass distribution. Theearly dark energy cosmologies could hence reduce the current mildtension between cluster data and the CMB observations. We notethat halos in cosmologies with EDE are also expected to be moreconcentrated than in Λ CDM; because the density of the Universewas greater at early times, objects that virialized at high redshift aremore compact than those that virialized more recently.Previous numerical simulations of a quintessence componentwith a changing equation of state (EOS) explored two particularpotentials: SUGRA and Ratra Peebles (RP), which differ becauseRP has a more smoothly decreasing w and consequently a verydifferent evolution in the past. Both Linder & Jenkins (2003) andKlypin et al. (2003) analyzed the influence of the dark energy onthe halo mass function in order to extrapolate the abundance ofstructure at different epochs and to compare it with existent theo-retical models. They used different numerical codes: the publiclyavailable code GADGET , in the first project, and the Adaptive Re-finement Tree code (Kravtsov et al. 1997), in the second. They con-cluded that the best way to understand which dark energy Universefit the observations best is to look at the growth history of halosand the evolution of their properties with time. Dolag et al. (2004)focused on the modification of the concentration parameter withmass and redshift, for the same cosmologies, based on high reso-lution simulations of a sample of massive halos. A limited num-ber of numerical studies also considered the possibility of a cou-pling of the dark energy field with dark matter (Mainini et al. 2003;Macci`o et al. 2004).In this paper, we carry out several high resolution simulationsof dark energy cosmologies in order to accurately measure thequantitative impact of early dark energy on abundance and struc-ture of dark matter halos. To this end, we in particular measure halomass functions and evaluate the agreement/disagreement with dif-ferent analytic fitting functions. We also test how well the growthof the mass function can be tracked with dynamical measure basedon the velocity dispersion of dark matter substructures, which canserve as a proxy for the directly measurable line-of-sight motion ofgalaxies or line widths in observations, and gets around the usual c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies ambiguities arising from different possible mass definitions for ha-los. Finally, we also present measurements of halo concentrations,and of the relation between dark matter velocity dispersion and halomass. While finalizing this paper, Francis et al. (2008) submitted apreprint which also studies numerical simulations of EDE cosmolo-gies. Their work provides a different analysis and is complementaryto our study, but it reaches similar basic conclusions about the halomass function.This paper is organized as follows. After a brief introductionto the Early Dark Energy models in Section 2, we present the simu-lations and also give details on our numerical methods in Section 3.In Section 4, we study the mass function of halos for the differentcosmologies, and as a function of redshift. Then, in Section 5 weinvestigate the properties of halos by studying the virial relationbetween mass and dark matter velocity dispersion, as well as themass–concentration relationship. In Section 6, we consider the ve-locity distribution function and prospects for measuring it in obser-vations. Finally, we discuss our results and present our conclusionsin Section 7. The influence of dark energy on the evolution of the Universe isgoverned by its equation of state, p = wρc . (1)A cosmological constant has w Λ = − at all redshift, while a dis-tinctive feature of the Early Dark Energy (hereafter EDE) models aswell as of other models such as quintessence is that their equationof state parameter, w de ( z ) , varies during cosmic history.Negative pressure at all times implies that the energy densityparameter will fall to zero very steeply for increasing redshift. If,however, we allow the equation of state parameter to rise abovezero, we can construct models in which Ω de ( z ) has a small posi-tive value at all epochs, depending on the cosmological backgroundmodel we adopt. While canonical dark energy models with nearconstant behaviour for w do not predict any substantial dark en-ergy effect at z > , in such EDE models the contribution of darkenergy to the cosmic density can be of order of a few percent evenat very high redshift.We are here investigating this interesting class of modelswhich are characterized by a low but non-vanishing dark energydensity at early times. Note that while the acceleration of the ex-pansion of the Universe is a quite recent phenomenon, the darkenergy responsible for this process could have an old origin. Infact, field theoretical models have been constructed that generi-cally cause such a dynamical behaviour (Ratra & Peebles 1988;Wetterich 1988; Ferreira & Joyce 1998; Liddle & Scherrer 1999).Wetterich (2004) proposed a useful parameterization of a fam-ily of cosmological models with EDE in terms of three parameters: • the amount of dark energy today, Ω de , (we assume a flat uni-verse, so Ω m, = 1 − Ω de , ), • the equation-of-state parameter w today, and • an average value Ω de , e of the energy density parameter at earlytimes (to which it asymptotes for z
7→ ∞ ).Figure 1 shows the redshift evolution of the equation-of-stateparameter in the four different cosmologies that we examine inthis study. As can be noticed, the EDE models approach the cos-mological constant scenario at very low redshift. We can compute
Figure 1.
Equation of state parameter w shown as a function of redshift forthe four different cosmological models considered in this work. In the twoearly dark energy models EDE1 and EDE2, shown with black solid and reddotted lines respectively, the value of w today is close to that of Λ CDM, butthe amount of dark energy at early times is non vanishing, as described bythe parameterization (2). the equation-of-state parameter for these early dark energy modelsfrom the fitting formula: w ( z ) = w (1 + by ) , (2)where b = − w ln (cid:16) / Ω de , e Ω de , e (cid:17) + ln (cid:16) − Ω m, Ω m, (cid:17) , (3)and y = ln (1 + z ) = − ln a . The parameter b characterizes thetime at which an approximately constant equation-of-state changesits behaviour.In Figure 2, we plot the evolution of the matter and energydensity parameters up to redshift z = 30 . The dark energy pa-rameter for EDE models evolves relatively slowly with respect to astandard Λ CDM cosmology. In fact, the critical feature of this pa-rameterization is a non-vanishing dark energy contribution duringrecombination and structure formation (see also Doran et al. 2001): ¯Ω de , sf = − ln a − Z a eq Ω de ( a ) d ln a. (4)For sufficiently low Ω de , e , the EDE models reproduce quite wellthe accelerated cosmic expansion in the present-day Universe andthey can be fine-tuned to agree both with low-redshift observationsand CMB temperature fluctuation results (Doran et al. 2005, 2007). We performed a series of cosmological N-body simulations fortwo early dark energy models ‘EDE1’ and ‘EDE2’, which have w = − . and w = − . , respectively, and a dark energydensity at early times of about − (see Tab. 1). For comparison,we have also calculated a model ‘DECDM’ with constant equationof state parameter equal to w = − . , and a conventional Λ CDM c (cid:13) , 1–17 Grossi & Springel den s i t y pa r a m e t e r s EDE2EDE1w=-0.6 Λ CDM
Figure 2.
Evolution of the density parameters Ω m ( z ) (dashed lines) and Ω de ( z ) (solid line) for the four cosmological models studied in this work.At redshift z = 30 , the dark energy contribution is orders of magnitudehigher for EDE models compared with a Λ CDM cosmology. Ω m, Ω de , h σ w Ω de ,e Λ CDM 0.25 0.75 0.7 0.8 -1. 0.DECDM 0.25 0.75 0.7 0.8 -0.6 0.EDE1 0.25 0.75 0.7 0.8 -0.93 × − EDE2 0.25 0.75 0.7 0.8 -0.99 × − Table 1.
Parameters of the N-Body simulations. The parameter Ω de , e de-scribes the amount of dark energy at early times, see equation (3). Thisvalue, together with w , the value of the equation state parameter today,and Ω de , , the amount of dark energy today, completely describes our EDEmodels. reference model. We shall refer with these labels to the differentmodels throughout the paper.In all our models, the matter density parameter today was cho-sen as Ω m = 0 . , and we consider a flat universe. The Hubbleparameter is h = H / (100 km s − Mpc − ) = 0 . and we as-sume Gaussian density fluctuations with a scale-invariant primor-dial power spectrum. The normalization of the linear power spec-trum extrapolated to z = 0 is σ = 0 . for all our simulationsin order to match the observed abundance of galaxy clusters to-day, irrespective of the cosmology. We also used the same spectralindex n = − throughout in order to focus our attention on possi-ble differences due to the dark energy contribution alone. For thesechoices, the models EDE1 and EDE2 are almost degenerate, buttheir proximity serves as a useful test for how well differences inthe results can be detected even for small variations in the EDE pa-rameters. This gives a useful illustration on how well one can hopeto be able to distinguish them in practice and provides realistic datafor testing the discriminative power of specific statistics.For our largest calculations we used particles in boxesof volume h − Mpc , resulting in a mass resolution of m p =5 . × h − M ⊙ and a gravitational softening length of ǫ =4 . h − kpc , kept fixed in comoving coordinates. All the simula-tions were started at redshift z init = 49 , and evolved to the present.For the simulations, we adapted the cosmological code GADGET-3 (based on Springel et al. 2001; Springel 2005) and the initial con-dition code
N-GENIC , in order to allow simulations with a time-variable equation of state. These simulations can be used to deter-mine the mass function also in the high-mass tail with reasonably H D E ( a ) / H Λ CD M ( a ) Λ CDMDECDMEDE1EDE2
Figure 3.
Hubble expansion rate for the models studied in this work. Allmodels are normalized with respect to the reference Λ CDM case. In themodels EDE1, EDE2, and in the model with constant w , the expansionrate of the universe is higher at early times. This has a strong effect on theevolution of the growth factor. small cosmic variance error, while at the same time probing downto interestingly small mass scales.In Figure 3, we plot the expansion function of the EDE mod-els relative to the Λ CDM case. We note that the only modificationrequired in the simulation code was to update the expression forcalculating the Hubble expansion rate, which needs to include thequintessence component. This term enters in both the kinematicsand the dynamics of the cosmological models.According to the Friedmann equation within a flat universe wehave H ( a ) = H (cid:20) Ω m, a + Ω de , exp (cid:18) − Z [1 + w ( a )] d ln a (cid:19)(cid:21) / . (5)The density of dark energy changes with the scale factor as: Ω de ( z ) = Ω de , exp (cid:18) − Z d ln a [1 + w ( a )] (cid:19) , (6)instead of simply being equal to Ω de , , as in the usual scenario. For w = − , the behaviour of a cosmological constant is recovered.If we interpret the modified expansion rate as being due to w ( z ) , as defined in equation (2), we find: H ( z ) /H = Ω de , (1 + z ) w h ( z ) + Ω m, (1 + z ) , (7)where ¯ w h ( z ) = w b ln (1 + z ) , (8)and b is given by the Eqn. (3).We can see that effectively the EDE models predict the ob-served effect of an acceleration in the expansion rate, and this hasconsequences on the global geometry of the Universe. We note thatthe dark energy term in Eqn. (6) just parametrizes our ignoranceconcerning the physical mechanism leading to an increase in ex-pansion rate. However, once the dependence of H on the scalefactor is fixed, the mathematical problem of calculating structuregrowth is then unambiguously defined.The evolution of Ω de , a affects not only the expansion rate ofthe background but also the formation of structures. The primaryinfluence of dark energy on the growth of matter density pertur-bations is however indirect and arises through the sensitive depen-dence of structure growth on the expansion rate of the universe. InFigure 4, we show the linear growth factor D divided by the scale c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies Figure 4.
Ratio of the growth factor of linear density perturbations andthe scale factor a , as a function of a . The four models are described inTable 1. The curves are normalized to unity at early times, i.e. we hereassume that the starting density contrast is the same in the four cosmologies.The models EDE1 and EDE2 show a significant difference in the growthfactor evolution even with small energy density at high redshift: structureshave to grow earlier to reach the same abundance as the Λ CDM modeltoday. factor
D/a as a function of time for all our models. All curves arenormalized so that they start from unity at early times.In order to rescale the power spectrum of matter fluctuationsto the redshift of the initial conditions ( z = 49 for all simula-tions), we introduced in our initial condition code the calculationof the growth factor for a general equation-of-state as given byLinder & Jenkins (2003): D ′′ + 32 (cid:20) − w ( a )1 + X ( a ) D ′ a + 32 X ( a )1 + X ( a ) Da (cid:21) = 0 , (9)where X ( a ) is the ratio of the matter density to the energy density: X ( a ) = Ω m, Ω de , exp (cid:20) − Z a d ln a ′ w (cid:0) a ′ (cid:1)(cid:21) , (10)and we allowed for a time-dependent equation of state, w ( a ) . Herewe define the growth factor as the ratio D = δ ( a ) /δ ( a i ) of theperturbation amplitude at scale factor a relative to the one at a i ,and we use the normalization condition D ( a eq ) = a eq .We can easily see that for very large redshift we recover thematter dominated behaviour in the Λ CDM case: D ( a ) ∝ a . On theother hand, as expected, the linear growth in the two EDE modelsfalls behind the green curve in Fig. 4, implying that they reach agiven amplitude at earlier times. In fact, the expansion rate in the Λ CDM cosmology is lower than in EDE models, which governsthe friction term ( ˙ a/a ) in the growth equation ¨ δ + 2 ˙ aa ˙ δ − πGρδ = 0 (11)of the perturbations.These formulae can be used to derive a suitable expressionfor the reduced linear overdensity δ c for collapse expected in EDEmodels (Bartelmann et al. 2006), which in turn suggests that thereare significant consequences for the process of non-linear structure [H] Figure 5.
Comparison of the non-linear power spectra of the four differentcosmological models studied here. The three panels give results for redshifts z = 0 , z = 3 and z = 5 , from top to bottom. The y -axes shows thedimensionless power ∆ = k P ( k ) as a function of k computed from thedark matter density field using a grid of points. All the simulationsare normalized to σ = 0 . for the linearly extrapolated density field today.The dashed lines indicate the expected linear power spectra. The predictionfrom Smith et al. (2003) for the Λ CDM cosmology is shown by the blackdot-dashed line. formation, an expectation that we will analyse later in detail. Insum, structures need to grow earlier in EDE models than in Λ CDMin order to reach the same amplitude at the present time. At anequal redshift, the initial conditions must hence be more evolvedin order to produce comparable results today. The DECDM showsa behaviour qualitatively similar to EDE1 and EDE2 (blue long-dashed line). c (cid:13) , 1–17 Grossi & Springel
In all our simulations, we have identified dark matter halos us-ing two methods: the friends-of-friends (FOF) algorithm with link-ing length b = 0 . , and the spherical overdensity (SO) group finder.Candidate groups with a minimum of 32 particles were retained bythe FOF group finder. In the SO algorithm, we first identify FOFgroups, and then select the particle with the minimum gravitationalpotential as their centres, around which spheres are grown that en-close a fixed prescribed mean density ∆ × ρ crit , where ρ crit is thecritical density. Different definitions of virial overdensity are in usein the literature, and we consider different values for ∆ where ap-propriate. The classical definition of NFW adopts ∆ = 200 in-dependent of cosmology, while sometimes also ∆ = 200 Ω m isused, corresponding to a fixed overdensity relative to the back-ground density. Finally, a value of ∆ ∼
178 Ω . m based on ageneralization of the spherical top-hat collapse model to low den-sity cosmologies can also be used. Note however that this may inprinciple depend on the dark energy cosmology (Bartelmann et al.2006), and is hence slightly ambiguous in these cosmologies.We have verified the correctness of our implementation ofearly dark energy in the simulation code by checking that it ac-curately reproduces the expected linear growth rate in these non-standard cosmologies. Recall that rather than normalizing the den-sity perturbations of the initial conditions to the same value at the(high) starting redshift, we determine them such that they shouldgrow to the same linear amplitude today in all of the models. Inpractice, we fix σ , the linearly extrapolated rms fluctuations intop-hat spheres of radius h − Mpc to the value . for the epoch z = 0 .In Figure 5, we show measurements of the power spectrum ofour different models at three different redshifts. While the modelsdiffer significantly at high redshift, the four different realizationsshow the same amplitude of the power spectrum, at least on largescales, at redshift z = 0 (left panel). The fluctuations on smallscales probably reflect the earlier structure formation time in theEDE models and the resulting differences in the non-linear halostructures. The good agreement of the power spectrum at the end,as well as a detailed comparison of the growth rate of the largestmodes in the box with the linear theory expectation (not shown),demonstrate explicitly that the EDE models are simulated accu-rately by the code, as intended.Note also that the power spectrum measurements show thatdue to the slower evolution of the linear growth factor in EDEmodels, the degeneracy between the models is lifted towards high-zsince this corresponds to more time for the different growth dynam-ics to take effect. Consequently, we expect a different evolution ofstructures back in time. Our main focus in the following will be tostudy the impact of a different equation of state for the dark energyupon the mass function of dark matter halos and its evolution withredshift. In this section we measure the halo abundance at different redshiftsand compare with analytic fitting functions proposed in the litera-ture. Our primary goal is to see to which extent dark energy modelscan still be described by these fitting formulae, and whether thereis any numerical evidence that supports the higher halo abundancepredicted for the EDE cosmologies (Bartelmann et al. 2006). Wewill mostly focus on halo mass functions determined with the FOFalgorithm with a linking length of 0.2, but we shall also considerSO mass functions later on. In Figure 7, we show our measured halo mass functions interms of the multiplicity function, which we define as f ( σ, z ) = Mρ d n ( M, z )d ln σ − (12)where ρ is the background density, n ( M, z ) is the abundanceof halos with mass less than M at redshift z , and σ is the massvariance of the power spectrum filtered with a top-hat mass scaleequal to M . We give results for the cosmological models Λ CDM,DECDM, EDE1 and EDE2, plotted as symbols, while the solidlines show various theoretical predictions.Note that we plot the mass function only in a limited massrange in order to avoid being dominated by counting statistics orresolution effects. To this end we only consider halos above a min-imum size of 200 particles. At the high mass end, individual ob-jects are resolved well, but the finite volume of the box limits thenumber of massive rare halos we can detect. We therefore plot themass function only up to the point where the Poisson error reaches ∼ (corresponding to minimum number of ∼ objects perbin). As is well known, the Press & Schechter mass function(Press & Schechter 1974), while qualitatively correct, disagreesin detail with the results of N-body simulations (Efstathiou et al.1988; White et al. 1993; Lacey & Cole 1994; Eke et al. 1996),specifically, the PS formula overestimates the abundance of halosnear the characteristic mass M ⋆ and underestimates the abundancein the high-mass tail. We therefore omit it in our comparison. Thediscrepancy is largely resolved by replacing the spherical collapsemodel of the standard Press & Schechter theory with the refined el-lipsoidal collapse model (Sheth & Tormen 1999; Sheth et al. 2001;Sheth & Tormen 2002). Indeed, in the top left panel of Figure 6 wecan see quite good agreement of the Sheth & Tormen mass function(ST) with our simulations at z = 0 . We stress that here the standardvalue of δ c = 1 . for the linear collapse threshold has been usedirrespective of the cosmological model. Two other well-known fit-ting formulae are that from Jenkins (central panel, Jenkins et al.2001, ‘J’) and that from Warren (right panel, Warren et al. 2006,‘W’), which differ only very slightly in the low-mass range. Wecompare our measurements with these models in the panels of themiddle and right columns. As we can see from the comparison be-tween the solid lines and the numerical data points, the differencesbetween the different theoretical models (which only rely on thelinearly evolved power spectrum at each epoch) and the simulationresults is very small.Figure 7 shows the redshift evolution of the mass function,in the form of separate comparison panels at redshifts z = 1 and z = 3 . While at z = 0 the different cosmologies agree rather wellwith each other, as expected based on the identical linear powerspectra, at redshift z = 1 we begin to see differences between themodels, and finally at z = 3 , we can observe a significantly highernumber density of groups and clusters in the non-standard dark en-ergy models. Notice that the model with constant w (blue line) be-haves qualitatively rather similar to the EDE models. In each of thepanels, we include a separate plot of the residuals with respect tothe analytic fitting functions. This shows that at z = 3 the agree-ment is clearly best for the ST formula.The differences between the models are most evident in theexponential tail of the mass function where it begins to fall off quitesteeply, in agreement with what is expected from the power spec-trum analysis. We can see that, at high- z , replacing the cosmolog-ical constant by an early dark energy scenario has a strong impacton the history of structure formation. In particular, non-linear struc- c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies M / ρ d n / d M Sheth & Tormen z=0. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Jenkins z=0. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Warren z=0. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] Figure 6.
Friends-of-friends multiplicity mass functions at z = 0 for the four dark energy models studied here. The solid lines in each panel represent themultiplicity function computed analytically either from the Sheth & Tormen formula (left panel), the Jenkins formula (central panel) or the Warren model(right panel). The symbols are the numerical simulation results for Λ CDM (green), DECDM (blue), EDE1 (orange) and EDE2 (red). We consider only haloswith more than particles and we apply an upper mass cut-off where the Poisson error reaches . In the lower plot of each figure we show the residualsbetween analytically expected and numerically determined mass functions for all models. The differences are typically below . The error bars showPoisson uncertainties due to counting statistics for all models. At z = 0 , the simulation results for all cosmologies are basically identical, which reflects thefact that we normalized the models such that they have the same linear power spectra today, with a normalization of σ = 0 . . tures form substantially earlier in such a model, such that a differ-ence in abundance of a factor of ∼ is reached already by z = 3 .This underlines the promise high redshift cluster surveys hold fordistinguishing different cosmological models, and in particular forconstraining the dynamical evolution of dark energy.We now want to assess in a more quantitative fashion the dif-ferences between our numerical halo mass functions and the ana-lytic fitting functions. In particular we are interested in the questionwhether we can objectively determine a preference for one of theanalytic models, and whether there is any evidence that the ordi-nary mass function formalism does work worse for the generalizeddark energy models than for Λ CDM. The latter would indicate thatthe critical linear overdensity threshold δ c needs to be revised forEDE models, as suggested by the analytic spherical collapse theory(Bartelmann et al. 2006).To this end we directly measure the goodness of the fit, whichwe define for the purposes of this analysis as: χ = X j /σ j ! − X i (MF i − MF TH ,i ) σ i MF ,i , (13)where MF TH ,i are the theoretical values, MF i are the simulationsresults, and we took into account a simple Poisson error in the def-inition of the goodness of fit. In Figure 8, we plot this value ex-pressed in percent for all simulations when compared with the the-oretical formulae of ST (solid line), Jenkins (dotted line) and War-ren (dashed line). We cannot identify a clearly superior behaviourof any the three fitting functions, at least at this level of resolu-tion; the models lie in a strip between approximately and error between z = 0 and z = 5 . There is some evidence that theST model does a bit better than the other fitting formulae for the Λ CDM cosmology at high redshift, but the opposite is true for thetwo EDE cosmologies and the Jenkins and Warren functions.Interestingly, the overall agreement between simulation re-sults and fitting functions is actually slightly worse for Λ CDM thanfor the non-standard dark energy cosmologies. There is hence notangible evidence that a revision of the mass function formalismis required to accurately describe EDE cosmologies. Our findingof a universal f ( σ ) is quantitatively different from the expecta-tion based on the analysis of the EDE models by Bartelmann et al. (2006). We find that only the different linear growth rate has tobe taken into account for describing the mass function in the earlydark energy cosmologies with the ST formalism, but there is noneed to modify the linear critical overdensity value. To make thispoint more explicit, we show in Figure 10 the mass function forthe EDE models and compare it to standard ST (solid lines), andto the expectations obtained taking into account a different den-sity contrast for EDE models (dashed lines). The predictions in thesecond case are based on the analytic study of Bartelmann et al.(2006) and the critical overdensity is proportional to ( a ) de , sf / (see Eqn. 4). Clearly, the proposed modification of δ c actuallyworsens the agreement, both for the halos selected according theFOF algorithm (top panel) or defined with respect to the virial over-density (bottom panel).In the plots we discussed above, we always employed the FOFhalo finder with standard linking length of b = 0 . to find the ha-los, and the masses were simply the FOF group masses, which ef-fectively correspond to the mass within an isodensity surface ofconstant overdensity relative to the background density. As the an-alytic mass function formulae have been calibrated with FOF halomass functions, we expect that they work best if the mass is definedin this way. However, we may alternatively also employ a differentmass definition based on the spherical overdensity (SO) approach,which allows one to take into account the time-dependent virialoverdensity ∆ predicted by generalizations of the spherical col-lapse model for dark energy cosmologies. In the bottom panel ofFigure 10 we can see that an even more marked disagreement re-sults when we take into account this arguably more consistent halodefinition.To stress this conclusion, in Figure 9, we show the residualsof our SO halo mass functions compared with the Sheth & Tormenprediction, as a function of redshift and for our different cosmolog-ical models, using the same procedure already applied to the FOFhalo finder results. In this case, the halos were defined as virializedregions that are overdense by a variable density threshold equal to In order to obtain the new values for the critical overdensity it is neces-sary to compute the virial overdensity by solving the equation of the gener-alized spherical collapse model.c (cid:13) , 1–17
Grossi & Springel M / ρ d n / d M Sheth & Tormen z=1. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Sheth & Tormen z=3. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Jenkins z=1. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Jenkins z=3. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Warren z=1. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] M / ρ d n / d M Warren z=3. Λ CDMDECDMEDE1EDE2 M [ h -1 M sun ] -0.50.00.5 ∆ [ M / ρ d n / d M ] Figure 7.
Friends-of-friends multiplicity mass functions for the four dark energy models studied here. The evolution towards high redshift is shown in termsof results at z = 1 (left column) and at z = 3 (right column). The solid lines in each plot represent the multiplicity function computed analytically from theSheth & Tormen formula (top row), the Jenkins formula (middle row) and the Warren formula (bottom row). The points are the numerical simulation resultsfor Λ CDM model (green), DECDM (blue), EDE1 (orange) and EDE2 (red). We consider only halos with more than particles and we apply an upper masscut-off where the Poisson error reaches . In the lower plot of each figure we show the residuals between analytically expected and numerically determinedmass functions for all models. The differences are typically below . The error bars show Poisson uncertainties due to counting statistics for all models.c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies E rr o r LCDMDECDMEDE1EDE2Sheth & Tormen Jenkins et al.Warren et al.
Figure 8.
Redshift dependence of our goodness of fit parameter χ (seeEqn. 13), expressed as percent, computed by comparing the theoretical ex-pectation for the multiplicity mass function with the simulation results. Allcosmological models are compared. The deviations are computed with re-spect to the Sheth & Tormen model (solid lines), the Jenkins et al. (dottedlines) and the Warren (dashed lines). ∆ c = 18 π + 82 x − x , (14)where x = Ω m ( z ) − , see Bryan & Norman (1998). This is thepredicted dependence of ∆ for Λ CDM, which we used for sim-plicity also for the other dark energy cosmologies. As expected,we see that the error increases relative to the FOF mass functions,with discrepancies of order at z = 0 . However, there is againno evidence that the non-standard dark energy cosmologies are de-scribed worse by the ST formalism than Λ CDM. Also, there is noimprovement in the accuracy of the fit when we introduce the mod-ified linear density contrast for the EDE models. On the contrary, asseen by the dotted lines, which represent the theoretical mass func-tion (based on Sheth & Tormen) modified according to the sphericaltop hat collapse theory proposed by Bartelmann et al. (2006).Our results thus suggest that the mass function depends pri-marily on the linear power spectrum and is only weakly, if at all,dependent on the details of the expansion history. This disagreeswith the expectations from the generalization of the top hat col-lapse theory, which are not confirmed by our numerical data. Infact, our simulations show that a description of the mass functionbased on the generalized TH calculation is incorrect at the accu-racy level reached here. While the dynamic range of our resultscould be improved by increasing the resolution and box-size of oursimulations, it appears unlikely that this could affect our basic con-clusions. Nevertheless, better resolution would be required if oneseeks to still further reduce the present residuals of order 5-15%between the fitting functions of ST, Jenkins or Warren.
Evrard et al. (2008) have shown that the dark matter velocity dis-persion of halos provides for accurate mass estimates once the re-lationship between mass and velocity dispersion is accurately cal-ibrated with the help of numerical simulations. They have demon- E rr o r LCDMDECDMEDE1EDE2Sheth & Tormen Sheth & Tormen dv
Figure 9.
Redshift dependence of our goodness of fit parameter χ (seeEqn. 13), expressed as percent, computed by comparing the theoretical ex-pectation for the multiplicity mass function with the simulation results. Allmodels are considered. Here we use the top hat halo mass definition to com-pute the mass function from the simulations, and the deviations are com-puted with respect to the standard Sheth & Tormen model (dotted-dashedlines), and the Sheth & Tormen formula computed from a generalization ofthe top-hat collapse theory (dashed lines). M / ρ d n / d M z=1.5 FOF
Sheth & Tormen δ c = 1.689Sheth & Tormen δ c from TH M [ h -1 M sun ] M / ρ d n / d M SO EDE1EDE2
Figure 10.
Multiplicity mass function at z = 1 . for the two EDE mod-els studied here. We want to highlight that the introduction of a mod-ified overdensity motivated by the generalized spherical collapse theory(Bartelmann et al. 2006) reduces the agreement between the simulation re-sults and the theoretical ST mass function. The measured points are in bet-ter agreement with the solid line (standard ST model) than with the dashedlines (ST modified model), the latter systematically overestimate the haloabundance. This holds true both if we consider the halos obtained fromthe FOF halo finder (top panel) or the one obtained by taking into accountthe theoretically motivated virial overdensity with the appropriate SO massdefinition (bottom panel).c (cid:13) , 1–17 Grossi & Springel
Figure 11.
The virial scaling relation at the present epoch for primary halos with mass larger than M ⊙ for the four models considered (from left to rightand top to bottom: Λ CDM, DECDM, EDE1, EDE2). The red solid line in each plot represents the Evrard et al. (2008) relation, while the blue dashed line isour best fit. The triangles are the simulation results: we employ a fixed critical threshold of ∆ = 200 to identify the dark matter halos. The insets show thedistributions of deviations in ln σ DM around the Evrard et al. (2008) fit. strated that there exists a quite tight power-law relation between themass of a halo and its one-dimensional velocity dispersion σ DM ,where σ = 13 N p N p X i =1 3 X j =1 | v i,j − ¯ v j | , (15)with v i,j being the j th component of the physical velocity of par-ticle i in the halo, N p is total number of halo particles within aradius that encloses a mean overdensity of ∆ = 200 with respectto the critical density, and ¯ v is the mean halo velocity. When virialequilibrium is satisfied, we expect that the specific thermal energyin a halo of mass M and of radius R will scale with its potentialenergy, GM/R , while the kinetic energy is proportional to M / .Since σ DM expresses the specific thermal energy in dark matter, wecan express the mean expected velocity dispersion as a function ofmass as σ DM ( M, z ) = σ DM , (cid:18) h ( z ) M M ⊙ (cid:19) α . (16)Here the fit parameters are the slope α of the relation, and the nor-malization σ DM , at a mass scale of h − M ⊙ . While the slope α just follows from the virial theorem if halos form a roughly self-similar family of objects (which they do to good approximation),the amplitude σ DM , of the relationship is a non-trivial outcomeof numerical simulations and reflects properties of the virialization process of the halos as well as their internal structure. Evrard et al.(2008) showed that a single fit is consistent with the numerical dataof a large set of N-body simulations of the Λ CDM cosmology, cov-ering a substantial dynamic range.However it is conceivable that the amplitude of the relation-ship will be slightly different in early dark energy cosmologies, asa result of the different virial overdensity that is predicted by thetop hat collapse in these cosmologies. If true, this would then alsohint at a different normalization of the relationship between totalSunyaev-Zeldovich decrement and mass, which would hence di-rectly affect observationally accessible probes of the cluster massfunction at high redshift.We here test whether we can find any difference in this re-lationship for our different dark energy cosmologies. In Figure 11,we plot the velocity dispersion of halos as a function of mass, in thefour different cosmologies we simulated. The halos were identifiedusing a spherical overdensity definition, where the virial radius r was determined as the radius that encloses a fixed multiple of times the critical density at the redshift z , and M being the cor-responding enclosed mass. We then determined the best-fit relationobtained from our numerical data (red solid lines). This fit is in verygood agreement with the results obtained by Evrard et al. (2008)(dotted blue lines), given by σ DM , = 1082 ± . − and α = 0 . ± . , a value consistent with the viral expectationof α = 1 / . The insets show the residuals about the fit at redshift c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies z = 0 . They have a log-normal distribution with a maximum of dispersion (for the DECDM model) around the power-law relation.The histograms are well fit by a log-normal with zero mean.We find that the halos closely follow a single virial relation,insensitive of the cosmological parameters, the epoch and also theresolution of the simulation. In particular, we do not find any sig-nificant differences for the EDE models, instead, the same form ofthe virial relation is preserved across the entire range of mass andredshift in the four simulations. The velocity dispersion-mass cor-relation hence appears to be global and very robust property of darkmatter halos which is not affected by different contributions of darkenergy to the total energy density of the universe.This is a reassuring result as it means that also in the case ofearly dark energy, clusters can be studied as a one parameter fam-ily and the calibration of dynamical mass estimates from internalcluster dynamics does not need to be changed. Differences in thenormalization should only reflect more or less frequent halo merg-ers and interactions, which can introduce an additional velocitycomponent (Espino-Briones et al. 2007; Faltenbacher & Mathews2007). As we have seen, for an equal normalization of the present-day lin-ear power spectrum, the dark matter halo mass function at z = 0 does not depend on the nature of dark energy. On one hand this isa welcome feature, as it simplifies using the evolution of the massfunction to probe the expansion history of the universe, but on theother hand it disappointingly does not provide for an easy handleto tell different evolutions apart based only on the present-day data.However, a discrimination between the models may still be made ifthe internal structure of halos is affected by the formation history,which would show up for example in their concentration distribu-tion. Cosmological simulations have consistently shown that thespherically averaged mass density profile of equilibrium dark mat-ter halos are approximately universal in shape. As a result, we candescribe the halo profiles by the NFW formula (Navarro et al. 1995,1996, 1997): ρ ( r ) ρ crit = δ c ( r/r s )(1 + r/r s ) , (17)where ρ crit = 3 H / πG is the critical density, δ c is the charac-teristic density contrast and r s is the scale radius of the halo. Theconcentration c is defined as the ratio between r and r s . Thequantities δ c and c are directly related by δ c = 2003 c [ln(1 + c ) − c/ (1 + c )] . (18)The concentration c is the only free parameter in Eqn. (18) ata given halo mass and these two quantities are known to be corre-lated. In fact, characteristic halo densities reflect the density of theuniverse at the time the halos formed; the later a halo is assembled,the lower is its average concentration.We have measured concentrations for our halos in the differentcosmologies using the same procedures as applied to the analysis ofthe Millennium simulation (Neto et al. 2007; Gao et al. 2008). Forour measurements, we take into account both relaxed and unrelaxedhalos. In the second case, the equilibrium state is assessed by meansof three criteria: (1) the fraction of mass in substructures with cen-ters inside the virial radius is small, f sub < . , (2) the normalized offset between the center-of-mass of the halo r cm and the poten-tial minimum r c is small, s = | r c − r cm | /r < . and (3)the virial ratio is sufficiently close to unity, T / | U | < . . Thesequantities provide a measure for the dynamical state of a halo, andconsidering these three conditions together guarantees in practicethat a halo is close to an equilibrium configuration, excluding theones with ongoing mergers, or with strong asymmetric configura-tions due to massive substructures.For all relaxed halos selected in this way, we computed aspherically averaged density profile by storing the halo mass inequally spaced bins in log ( r ) between the virial radius r and log ( r/r ) = − . . We used 32 bins for each halo and wechoose a uniform radial range in units of r for the fitting pro-cedure so that all halos are treated equally, regardless of the mass.We find that we obtain stable results when we use halos with morethan 3000 particles, consistent with the Power et al. (2003) criteria,while with fewer particles we notice resolution effects in the con-centration measurements, as both the gravitational softening anddiscreteness effects can artificially reduce the concentration. Thefinal mass range we explored is hence to h − M ⊙ .In Figure 12, we show our measured mass-concentration rela-tion for the different dark energy models at z = 0 . The four solidlines show the mean concentration as a function of mass. The boxesrepresent the 25 and 75 percentiles of the distribution, while thewhiskers indicate the 5 and 95 percentiles of the distributions. Wenote that the scatter of the concentration at a given mass is veryclose to a log-normal distribution. It is interesting to remark thatboth the mean and the dispersion decrease with mass. In fact, mas-sive halos form in some sense a more homogeneous population,because they have collapsed recently and so the formation redshiftis relatively close to the present epoch. On the other hand, less mas-sive halos have a wider distribution of assembly redshifts and thestructure of individual objects strongly depends on their particularaccretion histories. For them, the assumption that objects we ob-serve are just virialized is therefore inappropriate, especially forvery low mass halos. In Fig. 12 we take into account only the re-laxed halos, but we did an analogue measurement also for the wholesample, shown in Figure 13 at redshift (top panel) and at z = 1 (low panel).The correlation between mass and concentration approxi-mately follows a power law for the relaxed halos of the Λ CDMmodel. In the literature, the concentrations would be expected to besomewhat lower if a complete sample is considered that includesdisturbed halos. Comparing Figures 12 and 13 we notice that thisexpectation is confirmed, but the difference is not very pronounced,only about for the whole mass range. We also note that thenormalization σ = 0 . used for our simulations slightly lowersthe amplitude of the zero point of the relation (Macci`o et al. 2008)when compared to the WMAP-3 normalization, as halos tend toassemble later with lower σ and/or Ω m .When we compare our four simulated cosmologies we findthat, as expected, EDE halos of given mass have always higherconcentration at a given redshift than models with a cosmologicalconstant: they tend to form earlier and so they have a higher char-acteristic density. Nevertheless, the differences are not large, theydeviate by no more than ∼ at z = 0 over the entire mass rangewe studied for all halos and ∼ for the relaxed one. At higherredshift, the differences are only slightly bigger, of order of ∼ at z = 1 for the whole sample, and ∼ for halos in equilibriumconfiguration, suggesting that we anyway need reliable numericalcalibrations and highly accurate observational data to discriminatebetween the different cosmologies. Interestingly, the average con- c (cid:13) , 1–17 Grossi & Springel
11 12 13 14 15mass log M / h -1 M sun c on c en t r a t i on c relaxed halos z=0. LCDMDECDMEDE1EDE2
NFW modifiedEke et al 2001Bullock et al 2001NFW97
Figure 12.
Mass-concentration relation for relaxed halos in all our simulations. The boxes represent the 25 and 75 percentiles of the distribution with respectto the median value, while the whiskers show the 5 and 95 percentiles. We compare our results with the theoretical expectations from NFW, ENS, B01. Also,a modified NFW prescriptions with slightly modified parameters as updated by Gao et al. (2008) is shown (see Section 5 for details). centration is almost independent of mass when we consider z > ,as the average concentration of the more massive halos is similarat all redshifts (Gao et al. 2008) and we are then restricted to theexponential tail of the mass function.The change in concentration normalization relative to the cos-mological constant model is well represented by the ratio betweenthe linear growth factor of different models at very high redshift, c → c , ΛCDM D + ( ∞ ) D + , ΛCDM ( ∞ ) , (19)as suggested by Dolag et al. (2004). In Table 2, we compare theratio between the concentration at z = 0 both for the relaxed ha-los (second column) and for the whole sample (third column) withthe ratio between the asymptotic growth factor for the same cos-mologies (forth column). The order of magnitude of the two effectsis comparable, although the match is not perfect. Here the ratiosare computed for M ∼ × h − M ⊙ , where we have a largenumber density of halos.It is interesting to compare the concentrations we measurewith the various theoretical predictions that have been made forthis quantity. We investigate three popular descriptions for the con-centration: the classic Navarro, Frenk & White model (hereafterNFW), the model of Bullock et al. (2001, hereafter B01), and thatof Eke et al. (2001, hereafter ENS). Finally, we also plot the newmodified version of the original Navarro Frenk and White model,as recently proposed by Gao et al. (2008). Both the B01 model and Model c c , ΛCDM c ,ALL c ,ALL, ΛCDM D + (inf) D + , ΛCDM (inf) Λ CDM 1.000 1.000 1.000DECDM 1.256 1.275 1.228EDE1 1.218 1.232 1.229EDE2 1.255 1.273 1.252
Table 2.
Concentration and asymptotic growth factor in the four differentcosmologies studied here. The ratio between the concentration parametersat redshift z = 0 refers to a mass of M ∼ × h − M ⊙ that corre-sponds to the mass range that contains the majority of our halos. For eachmodel (first column) we give the c parameter relative to Λ CDM, takinginto account the relaxed halos (second column) or the whole sample (thirdcolumn). Finally, in the last column we show the linear growth factor atinfinity relative to the Λ CDM cosmology. the standard NFW have two free parameters that have been tuned toreproduce simulation results. In the original NFW prescription, thedefinition of the formation time of a halo is taken to be the redshiftat which half of its mass is first contained in a single progenitor: F = 0 . . The second parameter is the proportionality constant, C = 3000 , that relates the halo density scale to the mean cosmicdensity at the collapse redshift z coll . Recently, Gao et al. (2008) no-ticed that the evolution of the mass-concentration relation with red-shift can be approximated much better by setting F = 0 . . The c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies
11 12 13 14 15mass log M / h -1 M sun c on c en t r a t i on c all halos z=0. LCDMDECDMEDE1EDE2
NFW modifiedEke et al 2001Bullock et al 2001NFW97
11 12 13 14 15mass log M / h -1 M sun c on c en t r a t i on c all halos z=1. LCDMDECDMEDE1EDE2
NFW modifiedEke et al 2001Bullock et al 2001NFW97
Figure 13.
Mass-concentration relation for all halos in our simulations. Thetop panels refers to redshift z = 0 , while the bottom panel shows the resultsat z = 1 . The boxes represent the 25 and 75 percentiles of the distributionwith respect to the median value, while the whiskers show the 5 and 95percentiles. We compare our results with the theoretical expectations fromNFW, ENS, B01. Also, a modified NFW prescription with slightly modi-fied parameters as updated by Gao et al. (2008) is shown (see Section 5 fordetails). The concentration is lower with respect to the relaxed sampleat z = 0 for the Λ CDM model.
B01 model adopts as collapse redshift the epoch at which the typ-ical collapsing mass fulfills M ∗ ( a c ) = F M vir , with F = 0 . .They further assume that the concentration is a factor K = 3 . times the ratio between the scale factor at the time the halo is iden-tified and the collapse time. For K and F we use the values thatare indicated as the best parameters by Macci`o et al. (2007). Fi-nally, we compute the ENS prescriptions considering the effectiveamplitude of the power spectrum at the scale of the cluster mass.This quantity, rescaled for the linear growth factor of the simulatedcosmology, has to be constant. In this case, only one parameter, C σ = 28 , is needed. Bullock et al. (2001) and Eke et al. (2001)refer to the virial radius as the one including an overdensity givenby the generalized top hat collapse model. We have appropriatelyadapted these models such that the concentration of a halo is de-fined instead relative to radius r , as in the NFW model.Aside from B01, all three other model predictions yield con- centrations that agree reasonably well with the measured valuesat z = 0 . The B01 model underpredicts the relation at highmasses, where it gives has a sharp decline of the relation for M h − M ⊙ which is not seen in the simulations. In contrast,the NFW model is in reasonable agreement with the data at z = 0 for both halo samples. However, at z = 1 the evolution predicted bythe NFW model is less than what we find numerically, even whenwe consider the revised formulation proposed by Gao et al. (2008)(indicated as NFW modified). The NFW model with the new fit-ting parameters yields a reasonable fit at the high mass end, butperforms a bit worse than the original formulation at z = 0 , spe-cially at low masses. Unfortunately, for the NFW model the nor-malization is model dependent, so we cannot really capture all theeffects due to different cosmological parameters we use. Finally,the dashed black line in each plot shows the ENS model. This pre-scription gives the best match with our results and has been ableto reproduce the slope of the concentration-mass relation even athigher redshift.At a fixed mass, halos in the EDE cosmology are significantlyless concentrated than their counterparts in the Λ CDM cosmology.It is interesting to notice that the ENS model reproduces these dif-ferences quite well, without modifications of the original prescrip-tion. In Figure 14, we plot for each simulation the correspondingtheoretical expectation (dashed lines) for the sample of relaxed ha-los at z = 0 . For a low density universe the scaling of the lineargrowth factor with redshift leads to a greater difference betweenthe models. Dark halo concentrations depend both on the redshiftevolution of δ c and the amplitude of the power spectrum on massscales characteristic for the halo.These results for the concentration are particularly impor-tant since they demonstrate that quintessence cosmologies with thesame equation-of-state at present, but different redshift evolution,can produce measurable differences in the properties of the non-linear central regions of cluster-sized halos. However, the prospectsto observationally exploit these concentration differences to distin-guish different dark energy cosmologies are sobering. For one, thesystematic differences we measure for the concentrations are quitesmall compared to the statistical errors for the mean concentration,while at the same time the theoretical algorithms for predicting thehalo concentration perform quite differently already for the Λ CDMcosmology. Furthermore, directly measuring halo concentrations inobservations is not readily possible as it requires an accurate knowl-edge of the virial radius of a halo, a parameter which is poorly con-strained from observations. It therefore remains to be seen whetherthe effects of dark energy on the non-linear structure of dark haloscan be turned into a powerful tool to learn about the nature of darkenergy.
As we have seen, the different evolution of the halo mass functionis in principle a very sensitive probe of the expansion history ofthe universe, especially when the massive end of the mass functionis probed. Obtaining absolute mass estimates from observations ishowever problematic, and fraught with systematic biases and un-certainties. It is therefore important to look for new ways to counthalos which are more readily accessible by observations.One such approach lies in using the motion of galaxies ingroups or clusters of galaxies to measure the line-of sight veloc-ity dispersion, which in turn can be cast into an estimate of the totalvirial mass of the host halo. This relies on the assumption that the c (cid:13) , 1–17 Grossi & Springel
12 13 14 15mass log M / h -1 M sun c on c en t r a t i on c relaxed halos z=0. LCDMDECDMEDE1EDE2
Figure 14.
Mass-concentration relation for relaxed halos today. Here weshow the agreement between simulation results (symbols) and theoreticalpredictions from ENS (dashed lines), both for the Λ CDM and EDE cos-mologies. To this end we solve Eqn. (13) and (16) of Eke et al. (2001). Thedifferences between the four cosmologies are due mostly to the differencesin the growth factor evolution and consequently in the amplitude of thepower spectrum. The ENS formula works quite well also for EDE modelswithout modifications of the original prescription. σ -5 -4 -3 N ( > σ D M ) [ ( h - M p c ) - ] z=1.5 Λ CDM DECDM EDE1 EDE2
Figure 15.
The velocity function n ( σ ) as a function of halo mass, for allsatellites inside r . The shaded area indicates the differences between a Λ CDM model with σ = 0 . and the same model with σ = 0 . . It isinteresting to remark that this EDE models could justify a higher normal-ization cosmology. dynamics of the cluster or group galaxies is tracing out the darkmatter halo potential.Cluster and group galaxies can be identified with dark mat-ter sub-structures in N-body simulations (Springel et al. 2001;Vale & Ostriker 2004). Employing the bulk velocities of sub-halosas a simulation proxy for real galaxy velocities, we can hence builda velocity profile for any isolated halo, and estimate a line-of-sightvelocity dispersion, similarly as it is done for observed group cat-alogues of galaxies. This allows then to directly count halos (i.e.galaxy groups) as a function of line-of-sight velocity dispersion,bypassing the problematic point of assigning halo mass estimates.In Figure 15, we show our estimated cumulative velocity dispersion function for our four different cosmologies at redshift z = 1 . . This graph can be interpreted as being a different rep-resentation of the halo mass function, except that it is in principledirectly accessible by observations. For this measurement, we havederived the information on the velocities from the SUBFIND algo-rithm directly implemented in
GADGET-3 , which can find subhalosembedded in dark matter halos.An important aspect of this statistic is that it does not relyon the often ambiguous definition of a group mass. Instead, it canbe directly measured and is more readily accessed by observa-tions. In fact, studies based on the DEEP2 survey (Lin et al. 2004;Davis et al. 2005; Conroy et al. 2007) indicate that, if combinedwith both the velocity dispersion distribution of clusters from theSloan Digital Sky Survey and independent measurements of σ ,they will be able to constrain w to within approximately accu-racy. This method is almost independent of cosmological parame-ters, with the exception of σ , since a change in normalization canshift the space density of halos as a function of mass by a similaramount as done by the EDE models. This is illustrated in Figure 15by the shaded area, which represents the change of the velocity dis-persion function when σ is increased from . (green line) to . (upper limit of the shaded area). The velocity distribution functionof the EDE models then approaches the one that we would measurefor a Λ CDM model with higher σ .These kind of studies have strong motivations both from theobservational and theoretical point of view: there is little scatterbetween host galaxy luminosity and dark matter halo virial massand the velocity difference distribution of satellites and interlop-ers can be modeled as a Gaussian and a constant, respectively(Conroy et al. 2005; Faltenbacher & Diemand 2006).Figure 16 (left panel) shows the cumulative number of groupswith velocity dispersion above a given value, as a function in red-shift for the different models. We decided to count halos above avelocity dispersion of
300 km s − , where accurate measurementscan be expected also from observations. Note that there is alreadya very large difference between Λ CDM and EDE at redshift z = 1 .We find that there is almost no evolution in the cluster number inthe dark energy models, while Λ CDM drops by a factor of nearly10 up to redshift z = 3 . What is especially important here is therelative difference between the number counts of the two differentcosmologies. The fact that we do not need to introduce the mass inthis comparison give us the advantage of having no error derivedfrom the particular measurement procedure adopted for the mass.At a fixed velocity dispersion, we can directly probe the growthof the structure at each redshift, which depends on the equation ofstate parameter w . The slower evolution of the cluster population inEDE models is exactly what is expected to be observed also fromSunyaev-Zeldovich studies of large samples of clusters of galaxies.Combined with probes of the cluster internal velocity dispersionwe can hence hope to be able to derive stringent cosmological con-straints.We also remark that the relative difference between the num-ber of objects within these four simulations seems to be a quiterobust statistic which is invariant with respect to details of the mea-surement procedure. For example, in Figure 16 (right panel), wechange the number of considered subhalos in the halos to be a min-imum of 3, 4, and 5, but the velocity dispersion function relativeto the Λ CDM cosmology is essentially unchanged. In practice, thenumber of observable satellites per host halo suffers from limita-tions imposed by the magnitude limit of the survey. Our resultssuggest that the measured velocity dispersion should however berelatively insensitive to this selection effect. c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies -5 -4 N ( > σ S H = . e4 [ k m / s e c ] ) [ ( h - M p c ) - ] EDE2EDE1DECDMLCDM 0.0 0.5 1.0 1.5 2.0 2.5 3.0redshift-0.20.00.20.40.6 ∆ N ( > σ S H = . e4 ( k m / s e c ) ) EDE2EDE1DECDMLCDM N SH > 3N SH > 4N SH > 5 Figure 16.
Left panel: Comparison of the redshift evolution of the velocity dispersion function for all four cosmologies we simulated ( Λ CDM, DECDM,EDE1, and EDE2). Here the cumulative count of groups with velocity dispersion above σ = 300 km s − was used to measure the amplitude of the velocitydispersion function. Right panel: Differences in the number count when only halos with more than 3 (solid line), 4 (dotted line) or 5 (dashed line) substructuresare selected. Finally, we have also studied a few properties of the largestsubstructures in halos to see whether there is a difference in EDEcosmologies. In Figure 17, the small diamonds indicate the valuesof the ratio between M (the mass of the most massive subhalo)and M (the mass within a sphere of density 200 times the criticalvalue at redshift 0) for the first 200 most massive halos at redshift z = 3 . The filled circles represent the median of the distribution,computed in bins of 50 halos each, while the error bars mark the20-th and 80-th percentiles of the distribution. There is almost nodependence on parent halo mass, but we can notice a small, butsystematic tendency for the Λ CDM subhalos to be slightly moremassive. The dependence is quite weak, yet this behaviour is cleareven if the mass of the progenitor M tends to be lower on av-erage at this high redshift. This is symptomatic of the fact that the Λ CDM substructures are formed at lower redshift with respect towhat happens in the EDE models. This is also consistent with ex-pectations based on the observed dependence of substructure massfraction on halo mass (e.g. De Lucia et al. 2004). Once accretedonto a massive halo, subhalos suffer significant stripping, an effectthat is more important for substructures accreted at higher redshift,making the subhalos in the EDE models less massive on average.
In this study we have analyzed non-linear structure formation ina particular class of dark energy cosmologies, so called early darkenergy models where the contribution of dark energy to the total en-ergy density of the universe does not vanish even at high redshift,unlike in models with a cosmological constant and many other sim-ple quintessence scenarios. Our particular interest has been to testwhether analytic predictions for the halo mass function still reliablywork in such cosmologies. As the evolution of the mass function isone of the most sensitive probes available for dark energy, this isof crucial importance for the interpretation of future large galaxycluster surveys at high redshift. The mass function of EDE mod-els is also especially interesting because analytic theory based onextensions of the spherical collapse model predicts that the massfunction should be significantly modified (Bartelmann et al. 2006), ]-3.0-2.5-2.0-1.5-1.0-0.50.00.5
Log [ M / M ] z=3. EDE2EDE1DECDMLCDM
Figure 17.
Ratio of the mass of the two most massive substructures with re-spect to the mass of the parent halo. The small diamonds refer to individualhalos, while the filled circles are the median values. The error bars mark the20th and 80th percentiles of the distributions. and in particular be characterized by a different value of the lin-ear overdensity δ c for collapse, as well as a slightly modified virialoverdensity.We have carried out a set of high-resolution N-body simu-lations of two EDE models, and compared them with a standard Λ CDM cosmology, and a model with a constant equation of stateequal to w = − . . Interestingly, we find that the universality ofthe standard Sheth & Tormen formalism for estimating the halomass function also extends to the EDE models, at least at the < ∼ accuracy level that is reached also for the ordinary Λ CDMmodel. This means that we have found good agreement of the stan-dard ST estimate of the abundance of DM halos with our numer-ical results for the EDE cosmologies, without modification of theassumed virial overdensity and the linear density contrast thresh-old. This disagrees with the theoretical suggestions based on thegeneralized top-hat collapse. In fact, if we instead use the latter astheoretical prediction of the halo mass function, the deviations be- c (cid:13) , 1–17 Grossi & Springel tween the prediction and the numerical results become significantlylarger. We hence conclude that the constant standard value for thelinearly extrapolated density contrast can be used also for an analy-sis of early dark energy cosmologies. Very recently, similar resultswere also obtained by Francis et al. (2008), who studied the sameproblem in cosmological simulations with somewhat smaller massresolution.This results on the mass function appear to hold over thewhole redshift range we studied, from z = 0 to z = 3 . Since oursimulations were normalized to the same σ today, their mass func-tions and power spectra agree very well today, but towards higherredshift there are significant differences, as expected due to the dif-ferent histories of the linear growth factor in the different cosmolo-gies. In general, structure in the EDE cosmologies has to form sig-nificantly earlier than in Λ CDM to arrive at the same abundancetoday. For example, already by redshift z = 3 , the abundance ofgalaxy clusters of mass M = 5 × h − M ⊙ is higher in EDE1by a factor of ∼ . relative to Λ CDM.The earlier formation of halos in EDE models is also directlyreflected in the concentration of halos. While for a given σ we findthe same abundance of DM halos, the different formation historiesare still reflected in a subtle modification of the internal structure ofhalos, making EDE concentrations for all halo masses and redshiftsconsidered slightly higher. The difference is however quite small,but it would, for example, lead to a higher rate of dark matter anni-hilation in halos.Another relationship that appears to accurately hold equallywell in Λ CDM as in generalized dark energy cosmologies is thevirial scaling between mass and dark matter velocity dispersionthat Evrard et al. (2008) has found. In fact, we find that their nor-malization of this relation is accurately reproduced by all of oursimulations within the measurement uncertainties, independent ofcosmology. This also suggests that possible differences in the virialoverdensity of EDE halos must be very small, and that presumablythe relationship between total Sunyaev-Zeldovich decrement andhalo mass is unmodified as well.We show that counting the number of halos as a function ofthe line-of-sight velocity dispersion (of subhalos or galaxies), bothin simulations and observations, can probe the growth of struc-tures with redshift, and so put powerful constraints on the equa-tion of state parameters. This goal can be achieved by just identi-fying and counting groups in galaxy survey data such as DEEP2,and by comparing them with high-resolution N-body simulations.Precision measurements with this technique will still require accu-rate calibrations to deal with complications such as a possible ve-locity bias or selection effects in observational surveys. However,Davis et al. (2005) suggest that the DEEP2 survey alone has thepower to constrain w to an accuracy of 20% using velocity disper-sion data, which illustrates the promise of this technique. In combi-nation with other independent data, such as X-ray temperature andSZ decrement data, the constraints could be improved to an accu-racy of 5%, without the need to invoke a model for the ambiguoustotal mass of a halo.Distinguishing a time-varying dark energy component fromthe cosmological constant is a major quest of the present theoreticaland observational astronomy. One approach is to rely on classicalcosmological tests of the Hubble diagram, e.g. by pushing the su-pernova type Ia observations to much higher redshift. Another quitedirect geometrical probe is the observation of baryonic acoustic os-cillations in the matter distribution at different redshifts. Finally,the linear and non-linear evolution of cosmic structures providesanother opportunity to constrain dark energy. In this work we have used numerical N-body simulations to examine the difference instructure growth in early dark energy cosmologies. We have seenthat such simulations are essential to test the predictions of moresimplified analytic models, and to calibrate observational tests thattry to constrain the properties of dark energy with the abundanceand internal structure of dark matter halos. Our results show clearlythat the effects due to dynamical dark energy tend to be quite sub-tle, and can only be cleanly distinguished from ordinary Λ CDM inhigh accuracy simulations. This poses new challenges to improvethe precision of future generations of simulations, and at the sametime emphasizes the immense observational task to arrive at suffi-ciently precise data at high redshift to constrain the dark side of theuniverse with the required accuracy.
ACKNOWLEDGEMENTS
We acknowledge Matthias Bartelmann for providing us with hiscode LIBASTRO for the virial overdensity prediction in EDE mod-els. We also thank Klaus Dolag for useful discussions. Computa-tions were performed on the OPA machine at the computer cen-tre of the Max Planck Society. This research was supported by theDFG cluster of excellence Origin and Structure of the Universe.
REFERENCES
Bartelmann M., Doran M., Wetterich C., 2006, A&A, 454, 27Bryan G. L., Norman M. L., 1998, ApJ, 495, 80Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S., KravtsovA. V., Klypin A. A., Primack J. R., Dekel A., 2001, MNRAS,321, 559Conroy C., Newman J. A., Davis M., Coil A. L., Yan R., CooperM. C., Gerke B. F., Faber S. M., Koo D. C., 2005, ApJ, 635, 982Conroy C., Prada F., Newman J. A., Croton D., Coil A. L., Con-selice C. J., Cooper M. C., Davis M., Faber S. M., Gerke B. F.,Guhathakurta P., Klypin A., Koo D. C., Yan R., 2007, ApJ, 654,153Davis M., Gerke B. F., Newman J. A., the Deep2 Team 2005, inWolff S. C., Lauer T. R., eds, Observing Dark Energy Vol. 339of Astronomical Society of the Pacific Conference Series, Con-straining Dark Energy with the DEEP2 Redshift Survey. p. 128De Lucia G., Kauffmann G., Springel V., White S. D. M., LanzoniB., Stoehr F., Tormen G., Yoshida N., 2004, MNRAS, 348, 333Dolag K., Bartelmann M., Perrotta F., Baccigalupi C., MoscardiniL., Meneghetti M., Tormen G., 2004, A&A, 416, 853Doran M., Karwan K., Wetterich C., 2005, Journal of Cosmologyand Astro-Particle Physics, 11, 7Doran M., Robbers G., Wetterich C., 2007, Phys. Rev. D, 75,023003Doran M., Schwindt J.-M., Wetterich C., 2001, Phys. Rev. D, 64,123520Efstathiou G., Frenk C. S., White S. D. M., Davis M., 1988, MN-RAS, 235, 715Eke V. R., Cole S., Frenk C. S., 1996, MNRAS, 282, 263Eke V. R., Navarro J. F., Steinmetz M., 2001, ApJ, 554, 114Espino-Briones N., Plionis M., Ragone-Figueroa C., 2007, ApJ,666, L5Evrard A. E., Bialek J., Busha M., White M., Habib S., HeitmannK., Warren M., Rasia E., Tormen G., Moscardini L., Power C.,Jenkins A. R., Gao L., Frenk C. S., Springel V., White S. D. M.,Diemand J., 2008, ApJ, 672, 122 c (cid:13) , 1–17 ark matter structures in early dark energy cosmologies Faltenbacher A., Diemand J., 2006, MNRAS, 369, 1698Faltenbacher A., Mathews W. G., 2007, MNRAS, 375, 313Ferreira P. G., Joyce M., 1998, Phys. Rev. D, 58, 023503Francis M. J., Lewis G. F., Linder E. V., 2008, ArXiv e-prints,0808.2840Gao L., Navarro J. F., Cole S., Frenk C. S., White S. D. M.,Springel V., Jenkins A., Neto A. F., 2008, MNRAS, 387, 536Jenkins A., Frenk C. S., White S. D. M., Colberg J. M., Cole S.,Evrard A. E., Couchman H. M. P., Yoshida N., 2001, MNRAS,321, 372Klypin A., Macci`o A. V., Mainini R., Bonometto S. A., 2003, ApJ,599, 31Komatsu E., Dunkley J., Nolta M. R., Bennett C. L., Gold B.,Hinshaw G., Jarosik N., Larson D., Limon M., Page L., SpergelD. N., Halpern M., Hill R. S., Kogut A., Meyer S. S., TuckerG. S., Weiland J. L., Wollack E., Wright E. L., 2008, ArXiv e-prints, 0803.0547Kowalski M. e. a., 2008, ArXiv e-prints, 0804.4142Kravtsov A. V., Klypin A. A., Khokhlov A. M., 1997, ApJS, 111,73Lacey C., Cole S., 1993, MNRAS, 262, 627Lacey C., Cole S., 1994, MNRAS, 271, 676Liddle A. R., Scherrer R. J., 1999, Phys. Rev. D, 59, 023509Lin L., Koo D. C., Willmer C. N. A., Patton D. R., Conselice C. J.,Yan R., Coil A. L., Cooper M. C., Davis M., Faber S. M., GerkeB. F., Guhathakurta P., Newman J. A., 2004, ApJ, 617, L9Linder E. V., Jenkins A., 2003, MNRAS, 346, 573Macci`o A. V., Dutton A. A., van den Bosch F. C., 2008, ArXive-prints, 0805.1926Macci`o A. V., Dutton A. A., van den Bosch F. C., Moore B., PotterD., Stadel J., 2007, MNRAS, 378, 55Macci`o A. V., Quercellini C., Mainini R., Amendola L.,Bonometto S. A., 2004, Phys. Rev. D, 69, 123516Mainini R., Macci`o A. V., Bonometto S. A., 2003, New Astron-omy, 8, 173Navarro J. F., Frenk C. S., White S. D. M., 1995, MNRAS, 275,720Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493Neto A. F., Gao L., Bett P., Cole S., Navarro J. F., Frenk C. S.,White S. D. M., Springel V., Jenkins A., 2007, MNRAS, 381,1450Perlmutter S. e. a., 1999, ApJ, 517, 565Power C., Navarro J. F., Jenkins A., Frenk C. S., White S. D. M.,Springel V., Stadel J., Quinn T., 2003, MNRAS, 338, 14Press W. H., Schechter P., 1974, ApJ, 187, 425Ratra B., Peebles P. J. E., 1988, Phys. Rev. D, 37, 3406Riess A. G., Filippenko A. V., Li W., Treffers R. R., Schmidt B. P.,Qiu Y., Hu J., Armstrong M., Faranda C., Thouvenot E., Buil C.,1999, AJ, 118, 2675Riess A. G. e. a., 2004, ApJ, 607, 665Sheth R. K., Mo H. J., Tormen G., 2001, MNRAS, 323, 1Sheth R. K., Tormen G., 1999, MNRAS, 308, 119Sheth R. K., Tormen G., 2002, MNRAS, 329, 61Smith R. E., Peacock J. A., Jenkins A., White S. D. M., FrenkC. S., Pearce F. R., Thomas P. A., Efstathiou G., CouchmanH. M. P., 2003, MNRAS, 341, 1311Springel V., 2005, MNRAS, 364, 1105Springel V., White S. D. M., Tormen G., Kauffmann G., 2001,MNRAS, 328, 726Springel V., Yoshida N., White S. D. M., 2001, New Astronomy,6, 79 Vale A., Ostriker J. P., 2004, MNRAS, 353, 189Wang L., Steinhardt P. J., 1998, ApJ, 508, 483Warren M. S., Abazajian K., Holz D. E., Teodoro L., 2006, ApJ,646, 881Wetterich C., 1988, Nuclear Physics B, 302, 668Wetterich C., 2004, Physics Letters B, 594, 17White S. D. M., Efstathiou G., Frenk C. S., 1993, MNRAS, 262,1023 c (cid:13)000