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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 .
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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.
We present TreeFrog, a massively parallel halo merger tree builder that is capable comparing different halo catalogues and producing halo merger trees. The code is written in c++11, use the MPI and OpenMP APIs for parallelisation, and includes python tools to read/manipulate the data products produced. The code correlates binding energy sorted particle ID lists between halo catalogues, determining optimal descendant/progenitor matches using multiple snapshots, a merit function that maximises the number of shared particles using pseudo-radial moments, and a scheme for correcting halo merger tree pathologies. Focusing on VELOCIraptor catalogues for this work, we demonstrate how searching multiple snapshots spanning a dynamical time significantly reduces the number of stranded halos, those lacking a descendant or a progenitor, critically correcting poorly resolved halos. We present a new merit function that improves the distinction between primary and secondary progenitors, reducing tree pathologies. We find FOF accretion rates and merger rates show similar mass ratio dependence. The model merger rates from Poole et al, (2017) agree with the measured net growth of halos through mergers.
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.
Merger tree codes are routinely used to follow the growth and merger of dark matter haloes in simulations of cosmic structure formation. Whereas in Srisawat et. al. we compared the trees built using a wide variety of such codes here we study the influence of the underlying halo catalogue upon the resulting trees. We observe that the specifics of halo finding itself greatly influences the constructed merger trees. We find that the choices made to define the halo mass are of prime importance. For instance, amongst many potential options different finders select self-bound objects or spherical regions of defined overdensity, decide whether or not to include substructures within the mass returned and vary in their initial particle selection. The impact of these decisions is seen in tree length (the period of time a particularly halo can be traced back through the simulation), branching ratio (essentially the merger rate of subhalos) and mass evolution. We therefore conclude that the choice of the underlying halo finder is more relevant to the process of building merger trees than the tree builder itself. We also report on some built-in features of specific merger tree codes that (sometimes) help to improve the quality of the merger trees produced.
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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