Evaluating approximations for halo merging histories

We study the merging history of dark matter haloes in N -body simulations and semi-analytical 'merger trees' based on the extended Press–Schechter (EPS) formalism. The main focus of our study is the joint distribution of progenitor number and mass as a function of redshift and parent halo mass. We begin by investigating the mean quantities predicted directly by the Press–Schechter (PS) and EPS formalism, such as the halo mass and conditional mass functions, and compare these predictions with the results of the simulations. The higher moments of this distribution are not predicted by the EPS formalism alone and must be obtained from the merger trees. We find that the Press–Schechter model deviates from the simulations at the level of 30–50 per cent on certain mass scales, and that the sense of the discrepancy changes as a function of redshift. We show that this discrepancy is reflected in the higher moments of the distribution of progenitor mass and number. We investigate some related statistics such as the accretion rate and the mass ratio of the largest two progenitors. For galaxy sized haloes ( M ∼10¹² M_⊙), we find that the merging history of haloes, as represented by these statistics, is well reproduced in the merger trees compared with the simulations. The agreement deteriorates for larger mass haloes. We conclude that merger trees based on the extended Press–Schechter formalism provide a reasonably reliable framework for semi-analytical models of galaxy formation.     

Cosmological evolution and hierarchical galaxy formation

We calculate the rate at which dark matter haloes merge to form higher mass systems. Two complementary derivations using Press–Schechter theory are given, both of which result in the same equation for the formation rate. First, a derivation using the properties of the Brownian random walks within the framework of Press–Schechter theory is presented. We then use Bayes' theorem to obtain the same result from the standard Press–Schechter mass function. The rate obtained is shown to be in good agreement with results from Monte Carlo and N -body simulations. We illustrate the usefulness of this formula by calculating the expected cosmological evolution in the rate of star formation that is due to short-lived, merger-induced starbursts. The calculated evolution is well-matched to the observed evolution in ultraviolet luminosity density, in contrast to the lower rates of evolution that are derived from semi-analytic models that do not include a dominant contribution from starbursts. Hence we suggest that the bulk of the observed ultraviolet starlight at z >1 arises from merger-induced starbursts. Finally, we show that a simple merging-halo model can also account for the bulk of the observed evolution in the comoving quasar space density.     

Testing the modified Press–Schechter model against N-body simulations

A modified version of the extended Press–Schechter model for the growth of dark-matter haloes was introduced in two previous papers, with the aim of explaining the mass–density relation shown by haloes in high-resolution cosmological simulations. In this model, major mergers are well separated from accretion, thereby allowing a natural definition of halo formation and destruction. This makes it possible to derive analytic expressions for halo formation and destruction rates, the mass accretion rate and the probability distribution functions of halo formation times and progenitor masses. The stochastic merger histories of haloes can be readily derived and easily incorporated into semi-analytical models of galaxy formation, thus avoiding the usual problems encountered in the construction of Monte Carlo merger trees from the original extended Press–Schechter formalism. Here we show that the predictions of the modified Press–Schechter model are in good agreement with the results of N -body simulations for several scale-free cosmologies.     

Extended Press–Schechter theory and the density profiles of dark matter haloes

An inside–out model for the formation of haloes in a hierarchical clustering scenario is studied. The method combines the picture of the spherical infall model and a modification of the extended Press–Schechter theory. The mass accretion rate of a halo is defined to be the rate of its mass increase due to minor mergers. The accreted mass is deposited at the outer shells without changing the density profile of the halo inside its current virial radius. We applied the method to a flat Λ-cold dark matter universe. The resulting density profiles are compared with analytical models proposed in the literature, and a very good agreement is found. A trend is found of the inner density profile to become steeper for larger halo mass, which also results from recent N -body simulations. Additionally, present-day concentrations as well as their time evolution are derived and it is shown that they reproduce the results of large cosmological N -body simulations.     

The universal mass accretion history of cold dark matter haloes

We use the extended Press–Schechter formalism to investigate the rate at which cold dark matter haloes accrete mass. We discuss the shortcomings of previous methods that have been used to compute the mass accretion histories of dark matter haloes, and present an improved method based on the N -branch merger tree algorithm of Somerville & Kolatt. We show that this method no longer suffers from inconsistencies in halo formation times, and compare its predictions with high-resolution N -body simulations. Although the overall agreement is reasonable, there are slight inconsistencies which are most easily interpreted as a reflection of ellipsoidal collapse (as opposed to spherical collapse assumed in the Press–Schechter formalism). We show that the average mass accretion histories follow a simple, universal profile, and we present a simple recipe for computing the two scale-parameters which is applicable to a wide range of halo masses and cosmologies. Together with the universal profiles for the density and angular momentum distributions of cold dark matter haloes, these universal mass accretion histories provide a simple but accurate framework for modelling the structure and formation of dark matter haloes. In particular, they can be used as a backbone for modelling various aspects of galaxy formation where one is not interested in the detailed effects of merging. As an example we use the universal mass accretion history to compute the rate at which dark matter haloes accrete mass, which we compare with the cosmic star formation history of the Universe.     

Surveying the sky with the Arcminute MicroKelvin Imager: expected constraints on galaxy cluster evolution and cosmology

We discuss prospects for cluster detection via the Sunyaev–Zel'dovich (SZ) effect in a blank field survey with the interferometer array, the Arcminute MicroKelvin Imager (AMI). Clusters of galaxies selected in the SZ effect probe cosmology and structure formation with little observational bias, because the effect measures integrated gas pressure directly, and does so independently of cluster redshift.
We use hydrodynamical simulations in combination with the Press–Schechter expression to simulate SZ cluster sky maps. These are used with simulations of the observation process to gauge the expected SZ cluster counts. Even with a very conservative choice of parameters we find that AMI will discover at least several tens of clusters every year with     the numbers depend on factors such as the mean matter density, the density fluctuation power spectrum and cluster gas evolution. The AMI survey itself can distinguish between these to some degree, and parameter degeneracies are largely eliminated given optical and X-ray follow-up of these clusters; this will also permit direct investigation of cluster physics and what drives the evolution.     

Dark energy and the evolution of spherical overdensities

We use the non-linear spherical model in cold dark matter (CDM) cosmologies with dark energy to investigate the effects of dark energy on the growth of structure and the formation of virialized structures. We consider dark energy models with a constant equation-of-state parameter w . For  −1 < w < −1/3  , clusters form earlier and are more concentrated in quintessence than in ΛCDM models, but they form later and are less concentrated than in the corresponding open model with the same matter density and no dark energy. We point out some confusion in the literature around the expression of the collapse factor (ratio of the radius of the sphere at virialization to that at turnaround) derived from the virial theorem. We use the Sheth & Tormen extension of the Press–Schechter framework to calculate the evolution of the cluster abundance in different models and show the sensitivity of the cluster abundance to both the amplitude of the mass fluctuations, σ₈, and the σ₈– w normalization, selected to match either the cosmic microwave background observations or the abundance of X-ray clusters.     

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.     

An application of the Kerr black hole fly-wheel model to statistical properties of QSOs/AGNs

The aim of this work is to demonstrate the properties of the magnetospheric model around Kerr black holes (BHs), the so-called fly-wheel (rotation driven) model. The fly-wheel engine of the BH–accretion disc system is applied to the statistics of QSOs/AGNs. In the model, the central BH is assumed to be formed at z ∼10² and obtains nearly maximum but finite rotation energy (∼extreme Kerr BH) at the formation stage. The inherently obtained rotation energy of the Kerr BH is released through a magnetohydrodynamic process. This model naturally leads to a finite lifetime of AGN activity.
Nitta, Takahashi & Tomimatsu clarified the individual evolution of the Kerr BH fly-wheel engine, which is parametrized by BH mass, initial Kerr parameter, magnetic field near the horizon and a dimensionless small parameter. We impose a statistical model for the initial mass function (IMF) of an ensemble of BHs using the Press–Schechter formalism. With the help of additional assumptions, we can discuss the evolution of the luminosity function and the spatial number density of QSOs/AGNs.
By comparing with observations , it is found that a somewhat flat IMF and weak dependence of the magnetic field on the BH mass are preferred. The result explains well the decrease of very bright QSOs and decrease of population after z ∼2.     

The build-up of haloes within Press–Schechter theory

Modelling the build-up of haloes is important for linking the formation of galaxies with cosmological models. A simple model of halo growth is provided by Press–Schechter (PS) theory, where the initial field of density fluctuations is smoothed using spherically symmetric filters centred on a given position to obtain information about the likelihood of later collapse on varying scales. In this paper the predicted halo mass growth is compared for three filter shapes: Gaussian, top-hat and sharp k -space. Preliminary work is also presented analysing the build-up of haloes within numerical simulations using a friends-of-friends group finder. The best-fit to the simulation mass function was obtained using PS theory with a top-hat filter. By comparing both the backwards conditional mass function, which gives the distribution of halo progenitors, and the distribution of halo mergers in time, the build-up of haloes in the simulations is shown to be better fitted by PS theory with a sharp k -space filter. This strengthens previous work, which also found the build-up of haloes in simulations to be well matched to PS theory with a sharp k -space filter by providing a direct comparison of different filters and by extending the statistical tools used to analyse halo mass growth. The usefulness of this work is illustrated by showing that the cosmological evolution in the proportion of haloes that have undergone recent merger is predicted to be independent of mass and power spectrum and to only depend upon cosmology. Recent results from observations of field galaxies are shown to match the evolution expected, but are not sufficiently accurate to distinguish usefully between cosmological parameters.     

The bias field of dark matter haloes

Accepted 1998 January 26. Received 1998 January 26; in original form 1997 August 13This paper presents a stochastic approach to the clustering evolution of dark matter haloes in the Universe. Haloes, identified by a Press–Schechter-type algorithm in Lagrangian space, are described in terms of 'counting fields', acting as non-linear operators on the underlying Gaussian density fluctuations. By ensemble-averaging these counting fields, the standard Press–Schechter mass function as well as analytic expressions for the halo correlation function and corresponding bias factors of linear theory are obtained, extending the recent results by Mo & White. The non-linear evolution of our halo population is then followed by solving the continuity equation, under the sole hypothesis that haloes move by the action of gravity. This leads to an exact and general formula for the bias field of dark matter haloes, defined as the local ratio between their number density contrast and the mass density fluctuation. Besides being a function of position and 'observation' redshift, this random field depends upon the mass and formation epoch of the objects and is both non-linear and non-local. The latter features are expected to leave a detectable imprint on the spatial clustering of galaxies, as described, for instance, by statistics like the bispectrum and the skewness. Our algorithm may have several interesting applications, among which is the possibility of generating mock halo catalogues from low-resolution N -body simulations.     

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.     

Constraining our Universe with X-ray and optical cluster data

We have used recent X-ray and optical data in order to impose some constraints on the cosmology and cluster scaling relations.
Generically, two kinds of hypotheses define our model. First, we consider that the cluster population is well described by the standard Press–Schechter (PS) formalism, and secondly, these clusters are assumed to follow scaling relations with mass: temperature–mass ( T – M ) and X-ray luminosity–mass ( L _x– M ) .
In contrast with many other authors we do not assume specific scaling relations to model cluster properties such as the usual T – M virial relation or an observational relation or an observational determination of the L _x– T relation. Instead we consider general unconstrained parameter scaling relations.
With the previous model (PS plus scalings) we fit our free parameters to several X-ray and optical data sets with the advantage over preceding works that we consider all the data sets at the same time. This prevents us from being inconsistent with some of the available observations. Among other interesting conclusions, we find that only low-density universes are compatible with all the data considered and that the degeneracy between Ω_m and σ ₈ is broken. Also we obtain interesting limits on the parameters characterizing the scaling relations.     

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 256³ particles each, we study the MAHs of 12 924 cold dark matter (CDM) haloes in a ΛCDM 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.     

On the distribution of haloes, galaxies and mass

The stochasticity in the distribution of dark haloes in the cosmic density field is reflected in the distribution function   P _V ( N _h| δ _m)  , which gives the probability of finding N _h haloes in a volume V with mass density contrast δ _m. We study the properties of this function using high-resolution N -body simulations, and find that   P _V ( N _h| δ _m)  is significantly non-Poisson. The ratio between the variance and the mean goes from ∼1 (Poisson) at  1+ δ _m≪1  to <1 (sub-Poisson) at  1+ δ _m∼1  to >1 (super-Poisson) at  1+ δ _m≫1  . The mean bias relation is found to be well described by halo bias models based on the Press–Schechter formalism. The sub-Poisson variance can be explained as a result of halo exclusion, while the super-Poisson variance at high δ _m may be explained as a result of halo clustering. A simple phenomenological model is proposed to describe the behaviour of the variance as a function of δ _m. Galaxy distribution in the cosmic density field predicted by semi-analytic models of galaxy formation shows similar stochastic behaviour. We discuss the implications of the stochasticity in halo bias to the modelling of higher order moments of dark haloes and of galaxies.     

A comparison of simulated and analytic major merger counts

We use large volume, high resolution, N -body simulations of three different ΛCDM models, with different clustering strengths, to generate dark-matter halo merging histories. Over the reliable range of halo masses, roughly galaxy groups to rich clusters of galaxies, we quantify the number density of major mergers for two different time intervals and compare them with analytic predictions based on the extended Press–Schechter theory.     

The non-Gaussian tail of cosmic-shear statistics

Owing to gravitational instability, an initially Gaussian density field develops non-Gaussian features as the Universe evolves. The most prominent non-Gaussian features are massive haloes, visible as clusters of galaxies. The distortion of high-redshift galaxy images because of the tidal gravitational field of the large-scale matter distribution, called cosmic shear, can be used to investigate the statistical properties of the large‐scale structure (LSS) . In particular, non-Gaussian properties of the LSS will lead to a non-Gaussian distribution of cosmic-shear statistic. The aperture mass ( M _ap) statistics, recently introduced as a measure for cosmic shear, is particularly well suited for measuring these non-Gaussian properties. In this paper we calculate the highly non-Gaussian tail of the aperture mass probability distribution, assuming Press–Schechter theory for the halo abundance and the 'universal' density profile of haloes as obtained from numerical simulations. We find that for values of M _ap much larger than its dispersion, this probability distribution is closely approximated by an exponential, rather than a Gaussian. We determine the amplitude and shape of this exponential for various cosmological models and aperture sizes, and show that wide-field imaging surveys can be used to distinguish between some of the currently most popular cosmogonies. Our study here is complementary to earlier cosmic-shear investigations, which focused more on two- and three-point statistical properties.     

Understanding the redshift evolution of the luminosity functions of Lyman α emitters

We present a semi-analytical model of star formation which explains simultaneously the observed ultraviolet (UV) luminosity function (LF) of high-redshift Lyman break galaxies (LBGs) and LFs of Lyman α emitters. We consider both models that use the Press–Schechter (PS) and Sheth–Tormen (ST) halo mass functions to calculate the abundances of dark matter haloes. The Lyman α LFs at   z ≲ 4  are well reproduced with only ≲10 per cent of the LBGs emitting Lyman α lines with rest equivalent width greater than the limiting equivalent width of the narrow band surveys. However, the observed LF at   z > 5  can be reproduced only when we assume that nearly all LBGs are Lyman α emitters. Thus, it appears that  4 < z < 5  marks the epoch when a clear change occurs in the physical properties of the high-redshift galaxies. As Lyman α escape depends on dust and gas kinematics of the interstellar medium (ISM), this could mean that on an average the ISM at   z > 5  could be less dusty, more clumpy and having more complex velocity field. All of these will enable easier escape of the Lyman α photons. At   z > 5  , the observed Lyman α LF are well reproduced with the evolution in the halo mass function along with very minor evolution in the physical properties of high-redshift galaxies. In particular, up to   z = 6.5  , we do not see the effect of evolving intergalactic medium opacity on the Lyman α escape from these galaxies.