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.     

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.     

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.     

Effects of correlation between merging steps on the global halo formation

The excursion set theory of halo formation is modified by adopting the fractional Brownian motion, to account for possible correlation between merging steps. We worked out analytically the conditional mass function, halo merging rate and formation time distribution in the spherical collapse model. We also developed an approximation for the ellipsoidal collapse model and applied it to the calculation of the conditional mass function and the halo formation time distribution. For models in which the steps are positively correlated, the halo merger rate is enhanced when the accreted mass is less than  ∼25 M *  , while for the negatively correlated case this rate is reduced. Compared with the standard model in which the steps are uncorrelated, the models with positively correlated steps produce more aged population in small mass haloes and more younger population in large mass haloes, while for the models with negatively correlated steps the opposite is true. An examination of simulation results shows that a weakly positive correlation between successive merging steps appears to fit best. We have also found a systematic effect in the measured mass function due to the finite volume of simulations. In future work, this will be included in the halo model to accurately predict the three-point correlation function estimated from simulations.     

Merging history trees for dark matter haloes: tests of the Merging Cell Model in a CDM cosmology

The merging history of dark matter haloes is computed with the Merging Cell Model proposed by Rodrigues & Thomas. While originally discussed in the case of scale-free power spectra, it is developed and tested here in the framework of the cold dark matter cosmology. The halo mass function, the mass distribution of progenitors and child haloes, as well as the probability distribution of formation times, have been computed and compared with the available analytic predictions. The halo autocorrelation function has also been obtained (a first for a semi-analytic merging tree), and tested against analytic formulae. An overall good agreement is found between results of the model, and the predictions derived from the Press & Schechter theory and its extensions. More severe discrepancies appear when formulae that better describe N -body simulations are used for comparison. In many instances, the model can be a useful tool for following the hierarchical growth of structures. In particular, it is suitable for addressing the issue of the formation and evolution of galaxy clusters, as well as the population of Lyman-break galaxies at high redshift, and their clustering properties.     

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.     

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.     

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.     

Gravitational collapse in an expanding background and the role of substructure – I. Planar collapse

We study the interplay of clumping at small scales with the collapse and relaxation of perturbations at much larger scales. We present results of our analysis when the large-scale perturbation is modelled as a plane wave. We find that in the absence of substructure, collapse leads to formation of a pancake with multistream regions. Dynamical relaxation of the plane wave is faster in the presence of substructure. Scattering of substructures and the resulting enhancement of transverse motions of haloes in the multistream region lead to a thinner pancake. In turn, collapse of the plane wave leads to formation of more massive collapsed haloes as compared to the collapse of substructure in the absence of the plane wave. The formation of more massive haloes happens without any increase in the total mass in collapsed haloes. A comparison with the Burgers equation approach in the absence of any substructure suggests that the preferred value of effective viscosity depends primarily on the number of streams in a region.     

An excursion set model for the distribution of dark matter and dark matter haloes

A model of the gravitationally evolved dark matter distribution, in the Eulerian space, is developed. It is a simple extension of the excursion set model that is commonly used to estimate the mass function of collapsed dark matter haloes. In addition to describing the evolution of the Eulerian space distribution of the haloes, the model allows one to describe the evolution of the dark matter itself. It can also be used to describe density profiles, on scales larger than the virial radius of these haloes, and to quantify the way in which matter flows in and out of Eulerian cells. When the initial Lagrangian space distribution is white noise Gaussian, the model suggests that the Inverse Gaussian distribution should provide a reasonably good approximation to the evolved Eulerian density field, in agreement with numerical simulations. Application of this model to clustering from more general Gaussian initial conditions is discussed at the end.     

Dynamical friction in dwarf galaxies

We present a simplified analytic approach to the problem of the spiralling of a massive body orbiting within the dark halo of a dwarf galaxy. This dark halo is treated as the core region of a King distribution of dark matter particles, in consistency with the observational result of dwarf galaxies having solid-body rotation curves. Thus we derive a simple formula which provides a reliable and general first-order solution to the problem, totally analogous to the one corresponding to the dynamical friction problem in an isothermal halo. This analytic approach allows a clear handling and a transparent understanding of the physics and the scaling of the problem. A comparison with the isothermal case shows that in the core regions of a King sphere, dynamical friction proceeds at a different rate, and is sensitive to the total core radius. Thus, in principle, observable consequences may result. In order to illustrate the possible effects, we apply this formula to the spiralling of globular cluster orbits in dwarf galaxies, and show how present-day globular cluster systems could, in principle, be used to derive better limits on the structure of dark haloes around dwarf galaxies, when the observational situation improves. As a second application, we study the way a massive black hole population forming a fraction of these dark haloes would gradually concentrate towards the centre, with a consequent deformation of an originally solid-body rotation curve. This effect allows us to set limits on the fraction/mass of any massive black hole minority component of the dark haloes of dwarf galaxies. In essence, we take advantage of the way the global matter distribution fixes the local distribution function for the dark matter particles, which in turn determines the dynamical friction problem.     

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 assembly bias of dark matter haloes to higher orders

We use an extremely large volume  (2.4  h ⁻³ Gpc³)  , high-resolution N -body simulation to measure the higher order clustering of dark matter haloes as a function of mass and internal structure. As a result of the large simulation volume and the use of a novel 'cross-moment' counts-in-cells technique which suppresses discreteness noise, we are able to measure the clustering of haloes corresponding to rarer peaks than was possible in previous studies; the rarest haloes for which we measure the variance are 100 times more clustered than the dark matter. We are able to extract, for the first time, halo bias parameters from linear up to fourth order. For all orders measured, we find that the bias parameters are a strong function of mass for haloes more massive than the characteristic mass   M _*  . Currently, no theoretical model is able to reproduce this mass dependence closely. We find that the bias parameters also depend on the internal structure of the halo up to fourth order. For haloes more massive than   M _*  , we find that the more concentrated haloes are more weakly clustered than the less concentrated ones. We see no dependence of clustering on concentration for haloes with masses   M < M _*  ; this is contrary to the trend reported in the literature when segregating haloes by their formation time. Our results are insensitive to whether haloes are labelled by the total mass returned by the friends-of-friends group finder or by the mass of the most massive substructure. This implies that our conclusions are not an artefact of the particular choice of group finding algorithm. Our results will provide important input to theoretical models of galaxy clustering.     

Evolution of galaxy clustering

We study the evolution of the correlation function of dark matter haloes in the CDM class of models. We show that the halo correlation function does not evolve in proportion with the correlation function of the underlying mass distribution. The earliest haloes to collapse, which correspond to rare peaks in the density field, cluster very strongly. The amplitude of the halo correlation function decreases from its initial, large, value. This decrease continues until the average peaks have collapsed, after which the amplitude grows slowly. This behaviour is shown to be generic and the epoch of minimum amplitude depends only on the rms  fluctuations in mass at the relevant scale and, to a much smaller extent, on the slope of the power spectrum at that scale. We discuss the relevance of this result for interpretation of observations of galaxy and quasar clustering.     

The distribution of ejected subhaloes and its implication for halo assembly bias

Using a high-resolution cosmological N -body simulation, we identify the ejected population of subhaloes, which are haloes at redshift   z = 0  but were once contained in more massive 'host' haloes at high redshifts. The fraction of the ejected subhaloes in the total halo population of the same mass ranges from 9 to 4 per cent for halo masses from  ∼10¹¹  to  ∼10¹²  h ⁻¹ M_⊙  . Most of the ejected subhaloes are distributed within four times the virial radius of their hosts. These ejected subhaloes have distinct velocity distribution around their hosts in comparison to normal haloes. The number of subhaloes ejected from a host of given mass increases with the assembly redshift of the host. Ejected subhaloes in general reside in high-density regions, and have a much higher bias parameter than normal haloes of the same mass. They also have earlier assembly times, so that they contribute to the assembly bias of dark matter haloes seen in cosmological simulations. However, the assembly bias is not dominated by the ejected population, indicating that large-scale environmental effects on normal haloes are the main source for the assembly bias.     

The survival and disruption of cold dark matter microhaloes: implications for direct and indirect detection experiments

If the dark matter particle is a neutralino, then the first structures to form are cuspy cold dark matter (CDM) haloes collapsing after redshifts   z ≈ 100  in the mass range  10⁻⁶–10⁻³ M_⊙  . We carry out a detailed study of the survival of these microhaloes in the Galaxy as they experience tidal encounters with stars, molecular clouds, and other dark matter substructures. We test the validity of analytic impulsive heating calculations using high-resolution N -body simulations. A major limitation of analytic estimates is that mean energy inputs are compared to mean binding energies, instead of the actual mass lost from the system. This energy criterion leads to an overestimate of the stripped mass and an underestimate of the disruption time-scale, since CDM haloes are strongly bound in their inner parts. We show that a significant fraction of material from CDM microhaloes can be unbound by encounters with Galactic substructure and stars; however, the cuspy central regions remain relatively intact. Furthermore, the microhaloes near the solar radius are those which collapse significantly earlier than average and will suffer very little mass-loss. Thus, we expect a fraction of surviving bound microhaloes, a smooth component with narrow features in phase space, which may be uncovered by direct detection experiments, as well as numerous surviving cuspy cores with proper motions of arcminutes per year, which can be detected indirectly via their annihilation into gamma-rays.