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Register a new userDark matter halo merger and accretion probabilities in the excursion set formalism
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The merger and accretion probabilities of dark matter halos have so far only been calculated for an infinitesimal time interval. This means that a Monte-Carlo simulation with very small time steps is necessary to find the merger history of a parent halo. In this paper we use the random walk formalism to find the merger and accretion probabilities of halos for a finite time interval. Specifically, we find the number density of halos at an early redshift that will become part of a halo with a specified final mass at a later redshift, given that they underwent $n$ major mergers, $n=0,1,2,...$ . We reduce the problem into an integral equation which we then solve numerically. To ensure the consistency of our formalism we compare the results with Monte-Carlo simulations and find very good agreement. Though we have done our calculation assuming a flat barrier, the more general case can easily be handled using our method. This derivation of finite time merger and accretion probabilities can be used to make more efficient merger trees or implemented directly into analytical models of structure formation and evolution.
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The excursion set model provides a convenient theoretical framework to derive dark matter halo abundances. This paper generalizes the model by introducing a more realistic merging and collapse process. A new parameter regulates the influence of the environment and thus the coherence (non-Markovianity) of the merging and the collapse of individual mass shells. The model mass function also includes the effects of an ellipsoidal collapse. Analytic approximations of the halo mass function are derived for scale-invariant power spectra with the slopes $n=0,-1,-2$. The $n=-2$ mass function can be compared with the results obtained from the `Hubble volume simulations. A significant detection of non-Markovian effects is found for an assumed accuracy of the simulated mass function of 10%.
The simplest stochastic halo formation models assume that the traceless part of the shear field acts to increase the initial overdensity (or decrease the underdensity) that a protohalo (or protovoid) must have if it is to form by the present time. Equivalently, it is the difference between the overdensity and (the square root of the) shear that must be larger than a threshold value. To estimate the effect this has on halo abundances using the excursion set approach, we must solve for the first crossing distribution of a barrier of constant height by the random walks associated with the difference, which is now (even for Gaussian initial conditions) a non-Gaussian variate. The correlation properties of such non-Gaussian walks are inherited from those of the density and the shear, and, since they are independent processes, the solution is in fact remarkably simple. We show that this provides an easy way to understand why earlier heuristic arguments about the nature of the solution worked so well. In addition to modelling halos and voids, this potentially simplifies models of the abundance and spatial distribution of filaments and sheets in the cosmic web.
We derive approximated, yet very accurate analytical expressions for the abundance and clustering properties of dark matter halos in the excursion set peak framework; the latter relies on the standard excursion set approach, but also includes the effects of a realistic filtering of the density field, a mass-dependent threshold for collapse, and the prescription from peak theory that halos tend to form around density maxima. We find that our approximations work excellently for diverse power spectra, collapse thresholds and density filters. Moreover, when adopting a cold dark matter power spectra, a tophat filtering and a mass-dependent collapse threshold (supplemented with conceivable scatter), our approximated halo mass function and halo bias represent very well the outcomes of cosmological $N-$body simulations.
We present a new Monte-Carlo algorithm to generate merger trees describing the formation history of dark matter halos. The algorithm is a modification of the algorithm of Cole et al (2000) used in the GALFORM semi-analytic galaxy formation model. As such, it is based on the Extended Press-Schechter theory and so should be applicable to hierarchical models with a wide range of power spectra and cosmological models. It is tuned to be in accurate agreement with the conditional mass functions found in the analysis of merger trees extracted from the LCDM Millennium N-body simulation. We present a comparison of its predictions not only with these conditional mass functions, but also with additional statistics of the Millennium Simulation halo merger histories. In all cases we find it to be in good agreement with the Millennium Simulation and thus it should prove to be a very useful tool for semi-analytic models of galaxy formation and for modelling hierarchical structure formation in general. We have made our merger tree generation code and code to navigate the trees available at http://star-www.dur.ac.uk/~cole/merger_trees .
Our heuristic understanding of the abundance of dark matter halos centers around the concept of a density threshold, or barrier, for gravitational collapse. If one adopts the ansatz that regions of the linearly evolved density field smoothed on mass scale M with an overdensity that exceeds the barrier will undergo gravitational collapse into halos of mass M, the corresponding abundance of such halos can be estimated simply as a fraction of the mass density satisfying the collapse criterion divided by the mass M. The key ingredient of this ansatz is therefore the functional form of the collapse barrier as a function of mass M or, equivalently, of the variance sigma^2(M). Several such barriers based on the spherical, Zeldovich, and ellipsoidal collapse models have been extensively discussed. Using large scale cosmological simulations, we show that the relation between the linear overdensity and the mass variance for regions that collapse to form halos by the present epoch resembles expectations from dynamical models of ellipsoidal collapse. However, we also show that using such a collapse barrier with the excursion set ansatz predicts a halo mass function inconsistent with that measured directly in cosmological simulations. This inconsistency demonstrates a failure of the excursion set ansatz as a physical model for halo collapse. We discuss implications of our results for understanding the collapse epoch for halos as a function of mass, and avenues for improving consistency between analytical models for the collapse epoch and the results of cosmological simulations.
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