Extended Press-Schechter Research Articles

The abundance of dark matter haloes down to Earth mass

ABSTRACT We use the voids-within-voids-within-voids (VVV) simulations, a suite of successive nested N-body simulations with extremely high resolution (denoted, from low to high resolution, by L0 to L7), to test the Press–Schechter (PS), Sheth–Tormen (ST), and extended Press–Schechter (EPS) formulae for the halo abundance over the entire mass range, from minihaloes of 10−6 M⊙, to cluster haloes of 1015 M⊙, at different redshifts, from z = 30 to the present. We find that at z = 0 and z = 2, ST best reproduces the results of L0, which has the mean cosmic density (overdensity δ = 0), at 1011−15 M⊙. The higher resolution levels (L1–L7) are biased underdense regions (δ < −0.6). The EPS formalism takes this into account since it gives the mass function of a region conditioned, in this case, on having a given underdensity. EPS provides good matches to these higher levels, with deviations ≲20 per cent, at 10−6−12.5 M⊙. At z ∼ 7−15, the ST predictions for L0 and the EPS for L1–L7 show somewhat larger deviations from the simulation results. However, at even higher redshifts, z ∼ 30, EPS fits the simulations well again. We confirm our results by picking more subvolumes from the L0 simulation, finding that our conclusions depend only weakly on the size and overdensity of the region. The good agreement of EPS with the higher level simulations implies that PS (or ST) gives an accurate description of the total halo mass function in representative regions of the universe.

Stochastic gravitational wave background from the collisions of dark matter halos

We investigate the effect of the dark matter (DM) halos collisions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on the stochastic gravitational wave background. We first calculate the gravitational wave signal of a single collision event, assuming point masses and linear perturbation theory. Then we proceed to the calculation of the energy spectrum of the collective effect of all dark matter collisions in the Universe. Concerning the DM halo collision rate, we show that it is given by the product of the number density of DM halos, which is calculated by the extended Press–Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by simulation results, with a function of the linear growth rate of matter density through cosmological evolution. Hence, integrating over all mass and distance ranges, we finally extract the spectrum of the stochastic gravitational wave background created by DM halos collisions. As we show, the resulting contribution to the stochastic gravitational wave background is of the order of hc≈10-29\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{c} \\approx 10^{-29}$$\\end{document} in the band of f≈10-15Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f \\approx 10^{-15} Hz$$\\end{document}. However, in very low frequency band, it is larger. With current observational sensitivity it cannot be detected.

The halo mass function and inner structure of ETHOS haloes at high redshift

ABSTRACT We study the halo mass function and inner halo structure at high redshifts (z ≥ 5) for a suite of simulations within the structure formation ETHOS framework. Scenarios such as cold dark matter (CDM), thermal warm dark matter (WDM), and dark acoustic oscillations (DAOs) of various strengths are contained in ETHOS with just two parameters hpeak and kpeak, the amplitude and scale of the first DAO peak. The extended Press–Schechter (EPS) formalism with a smooth-k filter is able to predict the cut-off in the halo mass function created by the suppression of small-scale power in ETHOS models (controlled by kpeak), as well as the slope at small masses that is dependent on hpeak. Interestingly, we find that DAOs introduce a localized feature in the mass distribution of haloes, resulting in a mass function that is distinct in shape compared to either CDM or WDM. We find that the halo density profiles of all ETHOS models are well described by the NFW profile, with a concentration that is lower than in the CDM case in a way that is regulated by kpeak. We show that the concentration–mass relation for DAO models can be well approximated by the mass assembly model based on the EPS theory, which has been proposed for CDM and WDM elsewhere. Our results can be used to perform inexpensive calculations of the halo mass function and concentration–mass relation within the ETHOS parametrization without the need of N-body simulations.

The halo mass function in alternative dark matter models

ABSTRACT The claimed detection of large amounts of substructure in lensing flux anomalies, and in Milky Way stellar stream gap statistics, has led to a step change in constraints on simple warm dark matter models. In this study, we compute predictions for the halo mass function both for these simple models and for comprehensive particle physics models of sterile neutrinos and dark acoustic oscillations. We show that the mass function fit of Lovell et al. underestimates the number of haloes less massive than the half-mode mass, $M_\mathrm {hm}$, by a factor of 2, relative to the extended Press–Schechter (EPS) method. The alternative approach of applying EPS to the Viel et al. matter power spectrum fit instead suggests good agreement at $M_\mathrm {hm}$ relative to the comprehensive model matter power spectrum results, although the number of haloes with mass $\rm{\lt} M_\mathrm {hm}$ is still suppressed due to the absence of small-scale power in the fitting function. Overall, we find that the number of dark matter haloes with masses $\rm{\lt} 10^{8}{\, \rm M_\odot }$ predicted by competitive particle physics models is underestimated by a factor of ∼2 when applying popular fitting functions, although careful studies that follow the stripping and destruction of subhaloes will be required in order to draw robust conclusions.

The accretion history of dark matter haloes – III. A physical model for the concentration–mass relation

We present a semi-analytic, physically motivated model for dark matter halo concentration as a function of halo mass and redshift. The semi-analytic model combines an analytic model for the halo mass accretion history (MAH), based on extended Press Schechter (EPS) theory, with an empirical relation between concentration and formation time obtained through fits to the results of numerical simulations. Because the semi-analytic model is based on EPS theory, it can be applied to wide ranges in mass, redshift and cosmology. The resulting concentration-mass (c-M) relations are found to agree with the simulations, and because the model applies only to relaxed halos, they do not exhibit the upturn at high masses or high redshifts found by some recent works. We predict a change of slope in the z=0 c-M relation at a mass scale of $10^{11}\rm{M}_{\odot}$. We find that this is due to the change in the functional form of the halo MAH, which goes from being dominated by an exponential (for high-mass halos) to a power-law (for low-mass halos). During the latter phase, the core radius remains approximately constant, and the concentration grows due to the drop of the background density. We also analyse how the c-M relation predicted by this work affects the power produced by dark matter annihilation, finding that at z = 0 the power is two orders of magnitude lower than that obtained from extrapolating best-fitting c-M relations. We provide fits to the c-M relations as well as numerical routines to compute concentrations and MAHs.

Generating merger trees for dark matter haloes: a comparison of methods

Halo merger trees describe the hierarchical mass assembly of dark matter haloes, and are the backbone for modeling galaxy formation and evolution. Merger trees constructed using Monte Carlo algorithms based on the extended Press-Schechter (EPS) formalism are complementary to those extracted from N-body simulations, and have the advantage that they are not trammeled by limited numerical resolution and uncertainties in identifying (sub)haloes and linking them between snapshots. This paper compares multiple EPS-based merger tree algorithms to simulation results using four diagnostics: progenitor mass function (PMF), mass assembly history (MAH), merger rate per descendant halo, and the unevolved subhalo mass function (USMF). In general, algorithms based on spherical collapse yield major-merger rates that are too high by a factor of two, resulting in MAHs that are systematically offset. Assuming ellipsoidal collapse solves most of these issues, but the particular algorithm investigated here that incorporates ellipsoidal collapse dramatically overpredicts the minor-merger rate for massive haloes. The only algorithm in our comparison that yields MAHs, merger rates, and USMFs in good agreement with simulations, is that by Parkinson et al. (2008). However, this is not a true EPS-based algorithm as it draws its progenitor masses from a PMF calibrated against simulations, rather than `predicted' by EPS. Finally we emphasize that the benchmarks used to test the EPS algorithms are obtained from simulations and are hampered by significant uncertainties themselves. In particular, MAHs and halo merger rates obtained from simulations by different authors reveal discrepancies that easily exceed 50 percent, even when based on the same simulation. Given this status quo, merger trees constructed using the Parkinson et al. algorithm are as accurate as those extracted from N-body simulations.

Halo mass function and the free streaming scale

The nature of structure formation around the particle free streaming scale is still far from understood. Many attempts to simulate hot, warm, and cold dark matter cosmologies with a free streaming cutoff have been performed with cosmological particle-based simulations, but they all suffer from spurious structure formation at scales below their respective free streaming scales -- i.e. where the physics of halo formation is most affected by free streaming. We perform a series of high resolution numerical simulations of different WDM models, and develop an approximate method to subtract artificial structures in the measured halo mass function. The corrected measurements are then used to construct and calibrate an extended Press-Schechter (EPS) model with sharp-$k$ window function and adequate mass assignment. The EPS model gives accurate predictions for the low redshift halo mass function of CDM and WDM models, but it significantly under-predicts the halo abundance at high redshifts. By taking into account the ellipticity of the initial patches and connecting the characteristic filter scale to the smallest ellipsoidal axis, we are able to eliminate this inconsistency and obtain an accurate mass function over all redshifts and all dark matter particle masses covered by the simulations. As an additional application we use our model to predict the microhalo abundance of the standard neutralino-CDM scenario and we give the first quantitative prediction of the mass function over the full range of scales of CDM structure formation.

Dark matter halo merger histories beyond cold dark matter – I. Methods and application to warm dark matter

We describe a methodology to accurately compute halo mass functions, progenitor mass functions, merger rates and merger trees in non-cold dark matter universes using a self-consistent treatment of the generalized extended Press-Schechter formalism. Our approach permits rapid exploration of the subhalo population of galactic halos in dark matter models with a variety of different particle properties or universes with rolling, truncated, or more complicated power spectra. We make detailed comparisons of analytically derived mass functions and merger histories with recent warm dark matter cosmological N-body simulations, and find excellent agreement. We show that, once the accretion of smoothly distributed matter is accounted for, coarse-grained statistics such as the mass accretion history of halos can be almost indistinguishable between cold and warm dark matter cases. However, the halo mass function and progenitor mass functions differ significantly, with the warm dark matter cases being strongly suppressed below the free-streaming scale of the dark matter. We demonstrate the importance of using the correct solution for the excursion set barrier first-crossing distribution in warm dark matter - if the solution for a flat barrier is used instead the truncation of the halo mass function is much slower, leading to an overestimate of the number of low mass halos.

An improved model for the dynamical evolution of dark matter subhaloes

Using an analytical model, we study the evolution of subhalo, including its mass, angular momentum and merging time-scale. This model considers the dominant processes governing subhalo evolution, such as dynamical friction, tidal stripping and tidal heating. We find that in order to best match the evolution of angular momentum measured from N-body simulation, mass stripping by tidal force should become inefficient after subhalo has experienced a few passages of pericentre. It is also found that the often used Coulomb logarithm ln M/m has to be revised to best fit the merging time-scales from simulation. Combining the analytical model with the extended Press–Schechter (EPS) based merger trees, we study the subhalo mass function, and their spatial distribution in a Milky Way (MW) type halo. By tuning the tidal stripping efficiency, we can gain a better match to the subhalo mass function from simulation. The predicted distribution of subhaloes is found to agree with the distribution of MW satellites, but is more concentrated than the simulation results. The radial distribution of subhaloes depends weakly on subhaloes mass at both present day and the time of accretion, but strongly on the accretion time. Using the improved model, we measure the second moment of the subhalo occupation distribution, and it agrees well with the results of Kravtsov et al. and Zheng et al.

The birth and growth of neutralino haloes

We use the Extended-Press-Schechter (EPS) formalism to study halo assembly histories in a standard $\Lambda$CDM cosmology. A large ensemble of Monte Carlo random walks provides the {\it entire} halo membership histories of a representative set of dark matter particles, which we assume to be neutralinos. The first generation halos of most particles do not have a mass similar to the free-streaming cut-off $M_{f.s.}$ of the neutralino power spectrum, nor do they form at high redshift. Median values are $M_1 = 10^5$ to $10^7M_{f.s.}$ and $z_1 = 13$ to 8 depending on the form of the collapse barrier assumed in the EPS model. For almost a third of all particles the first generation halo has $M_1>10^9M_{f.s.}$. At redshifts beyond 20, most neutralinos are not yet part of any halo but are still diffuse. These numbers apply with little modification to the neutralinos which are today part of halos similar to that of the Milky Way. Up to 10% of the particles in such halos were never part of a smaller object; the typical particle has undergone $\sim 5$ "accretion events' where the halo it was part of falls into a more massive object. Available N-body simulations agree well with the EPS predictions for an "ellipsoidal" collapse barrier, so these may provide a reliable extension of simulation results to smaller scales. The late formation times and large masses of the first generation halos of most neutralinos imply that they will be disrupted with high efficiency during halo assembly.

How to grow a healthy merger tree

We investigate seven Monte Carlo algorithms -- four old and three new -- for constructing merger histories of dark matter halos using the extended Press-Schechter (EPS) formalism based on both the spherical and ellipsoidal collapse models. We compare, side-by-side, the algorithms' abilities at reproducing the analytic EPS conditional (or progenitor) mass function over a broad range of mass and redshift (z=0 to 15). Among the four old algorithms (Lacey & Cole 1993, Kauffmann & White 1993, Somerville & Kolatt 1999, Cole et al 2000), we find that only KW93 produces a progenitor mass function that is consistent with the EPS prediction for all look-back redshifts. The origins of the discrepancies in the other three algorithms are discussed. Our three new algorithms are designed to generate the correct progenitor mass function at each timestep. We show that this is a necessary and sufficient condition for consistency with EPS at any look-back time. We illustrate the differences among the three new algorithms and KW93 by investigating two other conditional statistics: the mass function of the i_{th} most massive progenitors and the mass function for descendants with N_p progenitors.

Merger rates of dark matter haloes

We derive analytic merger rates for dark matter haloes within the framework of the extended Press–Schechter (EPS) formalism. These rates become self-consistent within EPS once we realize that the typical merger in the limit of a small time-step involves more than two progenitors, contrary to the assumption of binary mergers adopted in earlier studies. We present a general method for computing merger rates that span the range of solutions permitted by the EPS conditional mass function, and focus on a specific solution that attempts to match the merger rates in N-body simulations. The corrected EPS merger rates are more accurate than the earlier estimates of Lacey & Cole by ∼20 per cent for major mergers and by up to a factor of ∼3 for minor mergers of mass ratio 1:104. Based on the revised merger rates, we provide a new algorithm for constructing Monte Carlo EPS merger trees, which could be useful in semi-analytic modelling. We provide analytic expressions and plot numerical results for several quantities that are very useful in studies of galaxy formation. This includes (i) the rate of mergers of a given mass ratio per given final halo, (ii) the fraction of mass added by mergers to a halo and (iii) the rate of mergers per given main progenitor. The creation and destruction rates of haloes serve for a self-consistency check. Our method for computing merger rates can be applied to conditional mass functions beyond EPS, such as those obtained by the ellipsoidal collapse model or extracted from N-body simulations.

The nearly universal merger rate of dark matter haloes in ΛCDM cosmology

We construct merger trees from the largest data base of dark matter haloes to date provided by the Millennium Simulation to quantify the merger rates of haloes over a broad range of descendant halo mass (10 12 ≤ M 0 ≤ 1015 M ⊙ ), progenitor mass ratio (10 -3 ≤ ξ ≤ 1), and redshift (0 ≤ z ≤ 6). We find the mean merger rate per halo, Bin, to have very simple dependence on M 0 , ξ, and z, and propose a universal fitting form for B/n that is accurate to 10-20 per cent. Overall, B/n depends very weakly on the halo mass (oc M 0 0 .08) and scales as a power law in the progenitor mass ratio (oc ξ -2 ) for minor mergers (ξ ≤ 0.1) with a mild upturn for major mergers. As a function of time, we find the merger rate per Gyr to evolve roughly as (1 + z) nm with r m = 2-2.3, while the rate per unit redshift is nearly independent of z. Several tests are performed to assess how our merger rates are affected by e.g. the time interval between Millennium outputs, binary versus multiple progenitor mergers, and mass conservation and diffuse accretion during mergers. In particular, we find halo fragmentations to be a general issue in merger tree construction from N-body simulations and compare two methods for handling these events. We compare our results with predictions of two analytical models for halo mergers based on the extended Press-Schechter (EPS) model and the coagulation theory. We find that the EPS model overpredicts the major merger rates and underpredicts the minor merger rates by up to a factor of a few.

The statistical properties of Λ cold dark matter halo formation

We present a comparison of the statistical properties of dark matter halo merger trees extracted from the Millennium Simulation with Extended Press–Schechter (EPS) formalism and the related galform Monte Carlo method for generating ensembles of merger trees. The volume, mass resolution and output frequency make the Millennium Simulation a unique resource for the study of the hierarchical growth of structure. We construct the merger trees of present-day friends-of-friends groups and calculate a variety of statistics that quantify the masses of their progenitors as a function of redshift, accretion rates, and the redshift distribution of their most recent major merger. We also look in the forward direction and quantify the present-day mass distribution of haloes into which high-redshift progenitors of a specific mass become incorporated. We find that the EPS formalism and its Monte Carlo extension capture the qualitative behaviour of all these statistics, but as redshift increases they systematically underestimate the masses of the most massive progenitors. This shortcoming is worst for the Monte Carlo algorithm. We present a fitting function to a scaled version of the progenitor mass distribution and show how it can be used to make more accurate predictions of both progenitor and final halo mass distributions.

On the assembly history of dark matter haloes

We study the mass assembly history (MAH) of dark matter haloes. We compare MAHs obtained using (i) merger trees constructed with the extended Press-Schechter (EPS) formalism, (ii) numerical simulations and (iii) the Lagrangian perturbation code PINOCCHIO. We show that the PINOCCHIO MAHs are in excellent agreement with those obtained using numerical simulations, while the EPS formalism predicts MAHs that occur too late. PINOCCHIO, which is much less CPU intensive than N-body simulation, can be run on a simple personal computer, and does not require any labour intensive post-simulation analysis, therefore provides a unique and powerful tool to investigate the growth history of dark matter haloes. Using a suite of 55 PINOCCHIO simulations, with 256(3) particles each, we study the MAHs of 12 924 cold dark matter (CDM) haloes in a Lambda CDM concordance cosmology. This is by far the largest set of haloes used for any such analysis. For each MAH we derive four different formation redshifts, which characterize different epochs during the assembly history of a dark matter halo. We show that haloes less massive than the characteristic non-linear mass scale establish their potential wells much before they acquire most of their mass. The time when a halo reaches its maximum virial velocity roughly divides its mass assembly into two phases, a fast-accretion phase which is dominated by major mergers, and a slow-accretion phase dominated by minor mergers. Each halo experiences about 3 +/- 2 major mergers since its main progenitor had a mass equal to 1 per cent of the final halo mass. This major merger statistic is found to be virtually independent of halo mass. However, the average redshift at which these major mergers occur is strongly mass dependent, with more massive haloes experiencing their major mergers later.

Natural downsizing in hierarchical galaxy formation

Stellar-population analyses of today's galaxies show ‘downsizing’, where the stars in more massive galaxies tend to have formed earlier and over a shorter time-span. We show that this phenomenon is not necessarily ‘antihierarchical’ but rather has its natural roots in the bottom-up clustering process of dark-matter haloes. While the main progenitor does indeed show an opposite effect, the integrated mass in all the progenitors down to a given minimum mass shows a robust downsizing that is qualitatively similar to what has been observed. These results are derived analytically from the standard extended Press–Schechter (EPS) theory, and are confirmed by merger trees based on EPS or drawn from N-body simulations. The downsizing is valid for any minimum mass, as long as it is the same for all haloes at any given time, but the effect is weaker for smaller minimum mass. If efficient star formation is triggered by atomic cooling, then a minimum halo mass arises naturally from the minimum virial temperature for cooling, T≃ 104 K, though for such a small minimum mass the effect is weaker than observed. Baryonic feedback effects, which are expected to stretch the duration of star formation in small galaxies and shut it down in massive haloes at late epochs, are likely to play a subsequent role in shaping up the final downsizing behaviour. Other appearances of downsizing, such as the decline with time of the typical mass of star-forming galaxies, may not be attributed to the gravitational clustering process but rather arise from the gas processes.

Supermassive black hole merger rates: uncertainties from halo merger theory

The merger of two supermassive black holes is expected to produce a gravitational-wave signal detectable by the Laser Interferometer Space Antenna(LISA). The rate of supermassive-black-hole mergers is intimately connected to the halo merger rate, and the extended Press-Schechter (EPS) formalism is often employed when calculating the rate at which these events will be observed by LISA. This merger theory is flawed and provides two rates for the merging of the same pair of haloes. We show that the two predictions for the LISA supermassive-black-hole-merger event rate from EPS merger theory are nearly equal because mergers between haloes of similar masses dominate the event rate. An alternative merger rate may be obtained by inverting the Smoluchowski coagulation equation to find the merger rate that preserves the Press–Schechter halo abundance, but these rates are only available for power-law power spectra. We compare the LISA event rates derived from the EPS merger formalism to those derived from the merger rates obtained from the coagulation equation and find that the EPS LISA event rates are 30 per cent higher for a power spectrum spectral index that approximates the full Λ cold dark matter result of the EPS theory.

Masses of high-z galaxy hosting haloes from angular clustering and their evolution in the cold dark matter model★

We examine masses of hosting haloes of two photometrically selected high-z galaxy samples: the old passively evolving galaxies (OPEGs) at z ∼ 1 and Lyman break galaxies (LBGs) at z ∼ 4 both taken from the Subaru/XMM-Newton Deep Survey (SXDS). The large survey area of the SXDS (1 deg 2 ) allows us to measure the angular two-point correlation functions to a wide separation of >10 arcmin with a good statistical quality. We utilize the halo model prescription for estimating characteristic masses of hosting haloes from the measured large-scale clustering amplitudes. It is found that the hosting halo mass positively correlates with the luminosity of galaxies. Then, adopting the extended Press-Schechter (EPS) model, we compute the predictions for the mass evolution of the hosting haloes in the framework of the cold dark matter (CDM) cosmology in order to make an evolutionary link between the two galaxy samples at different redshifts and to identify their present-day descendants by letting their haloes evolve forward in time. It is found that, in the view of the mass evolution of hosting haloes in the CDM model, bright (i'? i'* + 1) LBGs are consistent with being the progenitor of the OPEGs, whereas it is less likely that the LBG population, as a whole, has evolved into the OPEG population. It is also found that the present-day descendants of both the bright LBGs and OPEGs are likely to be located in massive systems such as groups of galaxies or clusters of galaxies. Finally, we estimate the hosting halo mass of local early-type galaxy samples from the 2dF and Sloan Digital Sky Survey (SDSS) based on the halo model, and it turns out that their expected characteristic mass of hosting haloes is in good agreement with the EPS predictions for the descendant's mass of both the bright LBGs and OPEGs.

Early structure in CDM

We use a novel technique to simulate the growth of one of the most massive progenitors of a supercluster region from redshift z 80, when its mass was about 10 M, until the present day. Our nested sequence of N-body resimulations allows us to study in detail the structure both of the dark matter object itself and of its environment. Our effective resolution is optimal at redshifts of 49, 29, 12, 5 and 0 when the dominant object has mass 1.2 × 105, 5 × 107, 2 × 1010, 3 × 1012 and 8 × 1014 h1 M, respectively, and contains 106 simulation particles within its virial radius. Extended Press–Schechter (EPS) theory correctly predicts both this rapid growth and the substantial overabundance of massive haloes we find at early times in regions surrounding the dominant object. Although the large-scale structure in these regions differs dramatically from a scaled version of its present-day counterpart, the internal structure of the dominant object is remarkably similar. Molecular hydrogen cooling could start as early as z 49 in this object, while cooling by atomic hydrogen becomes effective at z 39. If the first stars formed in haloes with virial temperature 2000 K, their comoving abundance at z= 49 should be similar to that of dwarf galaxies today, while their comoving correlation length should be 2.5 h1 Mpc.

Formation time-distribution of dark matter haloes: theories versus N-body simulations

In this paper we use numerical simulations to test the formation time distribution of dark matter haloes predicted by the analytical excursion set approaches. The formation time distribution is closely linked to the conditional mass function and this test is therefore an indirect probe of this distribution. The excursion set models tested are the extended Press-Schechter (EPS) model, the ellipsoidal collapse model, and the non-spherical collapse boundary model. Three sets of simulations (six realizations) have been used to investigate the halo formation time distribution for halo masses ranging from dwarf-galaxy-like haloes (M = 10 - 3 M * , where M * is the characteristic non-linear mass scale) to massive haloes of M = 8.7M * . None of the models can match the simulation results at both high and low redshift. In particular, dark matter haloes formed generally earlier in our simulations than predicted by the EPS model. This discrepancy might help explain why semi-analytical models of galaxy formation, based on EPS merger trees, underpredict the number of high redshift galaxies compared with recent observations.