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.     

Tidal effects and the environment dependence of halo assembly

We explore a possible origin for the puzzling anti-correlation between the formation epoch of galactic dark-matter haloes and their environment density. This correlation has been revealed from cosmological N -body simulations and is in conflict with the extended Press–Schechter model of halo clustering. Using similar simulations, we first quantify the straightforward association of an early formation epoch with a reduced mass-growth rate at late times. We then find that a primary driver of suppressed growth, by accretion and mergers, is tidal effects dominated by a neighbouring massive halo. The tidal effects range from a slowdown of the assembly of haloes due to the shear along the large-scale filaments that feed the massive halo to actual mass loss in haloes that pass through the massive halo. Using the restricted three-body problem, we show that haloes are prone to tidal mass loss within 1.5 virial radii of a larger halo. Our results suggest that the dependence of the formation epoch on environment density is a secondary effect induced by the enhanced density of haloes in filaments near massive haloes where the tides are strong. Our measures of assembly rate are particularly correlated with the tidal field at high redshifts   z ∼ 1  .     

Analytical approach to subhalo population in dark matter haloes

In the standard model of cosmic structure formation, dark matter haloes form by gravitational instability. The process is hierarchical: smaller systems collapse earlier, and later merge to form larger haloes. The galaxy clusters, hosted by the largest dark matter haloes, are at the top of this hierarchy and representing the largest as well as the last structures formed in the Universe, while the smaller and first haloes are those Earth-sized dark subhaloes that have been both predicted by theoretical considerations and found in numerical simulations, though there do not exist any observational hints of their existence. The probability that a halo of mass m at redshift z will be part of a larger halo of mass M at the present time can be described in the frame of the extended Press & Schecter theory making use of the progenitor (conditional) mass function. Using the progenitor mass function, we calculate analytically, at redshift zero, the distribution of subhaloes in mass, formation epoch and rarity of the peak of the density field at the formation epoch. That is done for a Milky Way size system, assuming both a spherical and an ellipsoidal collapse model. Our calculation assumes that small progenitors do not lose mass due to dynamical processes after entering the parent halo, and that they do not interact with other subhaloes. For a Λ cold dark matter power spectrum, we obtain a subhalo mass function  d n /d m   proportional to   m ^−α  with a model-independent  α∼ 2  . Assuming that the dark matter is a weakly interacting massive particle, the inferred distributions are used to test the feasibility of an indirect detection in the γ-ray energy band of such a population of subhaloes with a Gamma-ray Large Area Space Telescope like satellite.     

Modelling angular-momentum history in dark-matter haloes

We model the acquisition of spin by dark-matter haloes in semi-analytic merger trees. We explore two different algorithms: one in which halo spin is acquired from the orbital angular momentum of merging satellites, and another in which halo spin is gained via tidal torquing on shells of material while still in the linear regime. We find that both scenarios produce the characteristic spin distribution of haloes found in N -body simulations, namely, a log-normal distribution with mean ≈ 0.04 and standard deviation ≈ 0.5 in the log. A perfect match requires fine-tuning of two free parameters. Both algorithms also reproduce the general insensitivity of the spin distribution to halo mass, redshift and cosmology seen in N -body simulations. The spin distribution can be made strictly constant by physically motivated scalings of the free parameters. In addition, both schemes predict that haloes that have had recent major mergers have systematically larger spin values. These algorithms can be implemented within semi-analytic models of galaxy formation based on merger trees. They yield detailed predictions of galaxy properties that strongly depend on angular momentum (such as size and surface brightness) as a function of merger history and environment.     

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.     

Environmental dependence of dark matter halo growth – I. Halo merger rates

In an earlier paper, we quantified the mean merger rate of dark matter haloes as a function of redshift z , descendant halo mass M ₀, and progenitor halo mass ratio ξ using the Millennium simulation of the Λ cold dark matter cosmology. Here, we broaden that study and investigate the dependence of the merger rate of haloes on their surrounding environment. A number of local mass overdensity variables, both including and excluding the halo mass itself, are tested as measures of a halo's environment. The simple functional dependence on   z , M ₀  , and ξ of the merger rate found in our earlier work, is largely preserved in different environments, but we find that the overall amplitude of the merger rate has a strong positive correlation with the environmental densities. For galaxy-mass haloes, we find mergers to occur ∼2.5 times more frequently in the densest regions than in voids at both   z = 0  and higher redshifts. Higher mass haloes show similar trends. We present a fitting form for this environmental dependence that is a function of both mass and local density and valid out to   z = 2  . The amplitude of the progenitor (or conditional) mass function shows a similar correlation with local overdensity, suggesting that the extended Press–Schechter model for halo growth needs to be modified to incorporate environmental effects.     

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.     

Dark matter halo creation in moving barrier models

In hierarchical models of structure formation, the time derivative of the halo mass function may be thought of as the difference of two terms – a creation term, which describes the increase in the number of haloes of mass m from mergers of less massive objects, and a destruction term, which describes the decrease in the number of m -haloes as these merge with other haloes, creating more massive haloes as a result. The first part of this paper focuses on estimating the distribution of times when these creation events take place. In models where haloes form from a spherical collapse, this distribution can be estimated from the same formalism which is used to estimate halo abundances: the constant-barrier excursion-set approach. In the excursion-set approach, moving rather than constant barriers are necessary for estimating halo abundances when the collapse is triaxial. First, we generalize the excursion-set estimate of the creation time distribution by incorporating ellipsoidal collapse. Then, we show that these moving barrier based predictions are in better agreement with measurements in numerical simulations than are the corresponding predictions of the spherical collapse model. In the second part of the paper, we link the creation time distribution to the creation term mentioned above. For this quantity, the improvement provided by the ellipsoidal collapse model is more evident. These results should be useful for studies of merger-driven star formation rates and active galactic nucleus activity. We also present a similar study of the creation of haloes conditioned on belonging to an object of a certain mass today, and reach similar conclusions – the moving barrier based estimates are in substantially better agreement with the simulations. This part of the study may be useful for understanding the tendency for the oldest stars to exist in the most massive objects, and for star formation to only occur in lower mass objects at late times.     

An analytic model for the epoch of halo creation

In this paper we describe the Bayesian link between the cosmological mass function and the distribution of times at which isolated haloes of a given mass exist. By assuming that clumps of dark matter undergo monotonic growth on the time-scales of interest, this distribution of times is also the distribution of 'creation' times of the haloes. This monotonic growth is an inevitable aspect of gravitational instability. The spherical top-hat collapse model is used to estimate the rate at which clumps of dark matter collapse. This gives the prior for the creation time given no information about halo mass. Applying Bayes' theorem then allows any mass function to be converted into a distribution of times at which haloes of a given mass are created. This general result covers both Gaussian and non-Gaussian models. We also demonstrate how the mass function and the creation time distribution can be combined to give a joint density function, and discuss the relation between the time distribution of major merger events and the formula calculated. Finally, we determine the creation time of haloes within three N -body simulations, and compare the link between the mass function and creation rate with the analytic theory.     

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.     

A marked correlation function analysis of halo formation times in the Millennium Simulation

We study the environmental dependence of the formation epoch of dark matter haloes in the Millennium Simulation: a ten billion particle N -body simulation of standard Lambda cold dark matter cosmology. A sensitive test of this dependence – the marked correlation function – reveals highly significant evidence that haloes of a given mass form earlier in denser regions. We define a marked cross-correlation function, which helps quantify how this effect depends upon the choice of the halo population used to define the environment. The mean halo formation redshift as a function of the local overdensity in dark matter is also well determined, and we see an especially clear dependence for galaxy-sized haloes. This contradicts one of the basic predictions of the excursion set model of structure formation, even though we see that this theory predicts other features of the distribution of halo formation epochs rather well. It also invalidates an assumption usually employed in the popular halo, or halo occupation distribution, models of galaxy clustering, namely that the distribution of halo properties is a function of halo mass but not of halo environment.     

On the Reliability of Merger-Trees and the Mass-Growth Histories of Dark Matter Haloes

We have used merger-trees realizations to study the formation of dark matter haloes. The construction of merger-trees is based on three different pictures about the formation of structures in the Universe. These pictures include the spherical collapse (SC), the ellipsoidal collapse (EC) and the non-radial collapse (NR). The reliability of merger-trees has been examined comparing their predictions related to the distribution of the number of progenitors, as well as the distribution of formation times, with the predictions of analytical relations. The comparison yields a very satisfactory agreement. Subsequently, the mass-growth histories (MGH) of haloes have been studied and their formation scale factors have been derived. This derivation has been based on two different definitions that are (a) the scale factor when the halo reaches half its present day mass and (b) the scale factor when the mass-growth rate falls below some specific value. Formation scale factors follow approximately power laws of mass. It has also been shown that MGHs are in good agreement with models proposed in the literature that are based on the results of N-body simulations. The agreement is found to be excellent for small haloes but, at the early epochs of the formation of large haloes, MGHs seem to be steeper than those predicted by the models based on N-body simulations. This rapid growth of mass of heavy haloes is likely to be related to a steeper central density profile indicated by the results of some N-body simulations.     

How big were the first cosmological objects?

We calculate the cooling times at constant density for haloes with virial temperatures from 100 K to  1×10⁵ K  that originate from a 3 σ fluctuation of a CDM power spectrum in three different cosmologies. Our intention is to determine the first objects that can cool to low temperatures, but not to follow their dynamical evolution. We identify two generations of haloes: those with low virial temperatures,   T _vir≲9000 K  that remain largely neutral, and those with larger virial temperatures that become ionized. The lower temperature, lower mass haloes are the first to cool to 75 per cent of their virial temperature. The precise temperature and mass of the first objects are dependent upon the molecular hydrogen (H₂) cooling function and the cosmological model. The higher mass haloes collapse later but, in this paradigm, cool much more efficiently once they have done so, first via electronic transitions and then via molecular cooling: in fact, a greater residual ionization once the haloes cool below 9000 K results in an enhanced H₂ production and hence a higher cooling rate at low temperatures than for the lower mass haloes, so that within our constant-density model it is the former that are the first to cool to really low temperatures. We discuss the possible significance of this result in the context of CDM models in which the shallow slope of the initial fluctuation spectrum on small scales leads to a wide range of halo masses (of differing overdensities) collapsing over a small redshift interval. This 'crosstalk' is sufficiently important that both high- and low-mass haloes collapse during the lifetimes of the massive stars which may be formed at these epochs. Further investigation is thus required to determine which generation of haloes plays the dominant role in early structure formation.     

Survival of substructure within dark matter haloes

Using high-resolution cosmological N -body simulations, we investigate the survival of dark matter satellites falling into larger haloes. Satellites preserve their identity for some time after merging. We compute their loss of mass, energy and angular momentum as they are dissolved by dynamical friction, tidal forces and collisions with other satellites. We also analyse the evolution of their internal structure. Satellites with less than a few per cent of the mass of the main halo may survive for several billion years, whereas larger satellites rapidly sink into the centre of the main halo potential well and lose their identity. Penetrating encounters between satellites are frequent and may lead to significant mass loss and disruption. Only a minor fraction of cluster mass (10–15 per cent on average) is bound to substructure at most redshifts of interest. We discuss the application of these results to the survival and extent of dark matter haloes associated with galaxies in clusters, and to their interactions. We find that a minor fraction of galaxy-size dark matter haloes are disrupted by redshift z  = 0. The fraction of satellites undergoing close encounters is similar to the observed fraction of interacting or merging galaxies in clusters at moderate redshift.     

On the origin of cold dark matter halo density profiles

N -body simulations predict that cold dark matter (CDM) halo-assembly occurs in two phases: (i) a fast-accretion phase with a rapidly deepening potential well; and (ii) a slow-accretion phase characterized by a gentle addition of mass to the outer halo with little change in the inner potential well. We demonstrate, using one-dimensional simulations, that this two-phase accretion leads to CDM haloes of the Navarro, Frenk & White (NFW) form and provides physical insight into the properties of the mass-accretion history that influence the final profile. Assuming that the velocities of CDM particles are effectively isotropized by fluctuations in the gravitational potential during the fast-accretion phase, we show that gravitational collapse in this phase leads to an inner profile  ρ( r ) ∝ r ⁻¹  . Slow accretion on to an established potential well leads to an outer profile with  ρ( r ) ∝ r ⁻³  . The concentration of a halo is determined by the fraction of mass that is accreted during the fast-accretion phase. Using an ensemble of realistic mass-accretion histories, we show that the model predictions of the dependence of halo concentration on halo formation time and, hence, the dependence of halo concentration on halo mass, and the distribution of halo concentrations all match those found in cosmological N -body simulations. Using a simple analytic model that captures much of the important physics, we show that the inner   r ⁻¹  profile of CDM haloes is a natural result of hierarchical mass assembly with an initial phase of rapid accretion.     

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.