How to plant a merger tree (original) (raw)

Merger rates of dark matter haloes from merger trees in the extended Press-Schechter theory

Eprint Arxiv 0905 0793, 2009

We construct merger trees based on the extended Press-Schechter theory (EPS) in order to study the merger rates of dark matter haloes over a range of present day mass ($10^{10}M_{\sun}\leq M_0 \leq10^{15}M_{\sun}$), progenitor mass (5times10−3leqxileq1(5\times10^{-3}\leq \xi \leq1(5times10−3leqxileq1) and redshift ($0\leq z\leq 3$). We used the first crossing distribution of a moving barrier of the form B(S,z)=p(z)+q(z)SgammaB(S,z)=p(z)+q(z)S^{\gamma}B(S,z)=p(z)+q(z)Sgamma, proposed by Sheth & Tormen, to take into account the ellipsoidal nature of collapse. We find that the mean merger rate per halo Bm/nB_m/nBm/n depends on the halo mass MMM as M0.2M^{0.2}M0.2 and on the redshift as (mathrmddeltac(z)/mathrmdz)1.1(\mathrm{d}\delta_c(z)/\mathrm{d}z)^{1.1}(mathrmddeltac(z)/mathrmdz)1.1. Our results are in agreement with the predictions of N-body simulations and this shows the ability of merger-trees based on EPS theory to follow with a satisfactory agreement the results of N-body simulations and the evolution of structures in a hierarchical Universe.

Generating merger trees for dark matter haloes: a comparison of methods

Halo merger trees describe the hierarchical assembly of dark matter haloes, and are the backbone for modelling galaxy formation and evolution. Merger trees constructed using Monte Carlo algorithms based on the extended Press–Schechter (EPS) formalism are complementary to using N-body simulations and have the advantage that they are not trammelled by limited numerical resolution and uncertainties in identifying and linking (sub)haloes. This paper compares multiple EPS-based merger tree algorithms to simulation results using four diagnostics: progenitor mass function, mass assembly history (MAH), merger rate per descendant halo and the unevolved subhalo mass function. Spherical collapse-based methods typically overpredict major-merger rates, whereas ellipsoidal collapse dramatically overpredicts the minor-merger rate for massive haloes. The only algorithm in our comparison that yields results in good agreement with simulations is that by Parkinson et al. (P08). We emphasize, though, that the simulation results used as benchmarks in testing the merger trees are hampered by significant uncertainties themselves: MAHs and merger rates from different studies easily disagree by 50 per cent, even when based on the same simulation. Given this status quo, the P08 merger trees can be considered as accurate as those extracted from simulations.

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.

Evaluating approximations for halo merging histories

Monthly Notices of the Royal Astronomical Society, 2000

We study the merging history of dark matter haloes in N-body simulations and semianalytical`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 , 10 12 M (Y 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.

The merger rates and mass assembly histories of dark matter haloes in the two Millennium simulations

Monthly Notices of The Royal Astronomical Society, 2010

We construct merger trees of dark matter haloes and quantify their merger rates and mass growth rates using the joint dataset from the Millennium and Millennium-II simulations. The finer resolution of the Millennium-II Simulation has allowed us to extend our earlier analysis of halo merger statistics to an unprecedentedly wide range of descendant halo mass (10^10 < M0 < 10^15 Msun), progenitor mass ratio (10^-5 < xi < 1), and redshift (0 < z < 15). We update our earlier fitting form for the mean merger rate per halo as a function of M_0, xi, and z. The overall behavior of this quantity is unchanged: the rate per unit redshift is nearly independent of z out to z~15; the dependence on halo mass is weak (M0^0.13); and it is nearly a power law in the progenitor mass ratio (xi^-2). We also present a simple and accurate fitting formula for the mean mass growth rate of haloes as a function of mass and redshift. This mean rate is 46 Msun/yr for 10^12 Msun haloes at z=0, and it increases with mass as M^{1.1} and with redshift as (1+z)^2.5 (for z > 1). When the fit for the mean mass growth rate is integrated over a halo's history, we find excellent match to the mean mass assembly histories of the simulated haloes. By combining merger rates and mass assembly histories, we present results for the number of mergers over a halo's history and the statistics of the redshift of the last major merger.

Merger rates of dark matter haloes: a comparison between EPS and N-body results

Astrophysics and Space Science, 2011

We calculate merger rates of dark matter haloes using the Extended Press-Schechter approximation (EPS) for the Spherical Collapse (SC) and the Ellipsoidal Collapse (EC) models. Merger rates have been calculated for masses in the range 1010 M ⊙ h-1 to 1014 M ⊙ h-1 and for redshifts z in the range 0 to 3 and they have been compared with merger rates that have been proposed by other authors as fits to the results of N-body simulations. The detailed comparison presented here shows that the agreement between the analytical models and N-body simulations depends crucially on the mass of the descendant halo. For some range of masses and redshifts either SC or EC models approximate satisfactory the results of N-body simulations but for other cases both models are less satisfactory or even bad approximations. We showed, by studying the parameters of the problem that a disagreement—if it appears—does not depend on the values of the parameters but on the kind of the particular solution used for the distribution of progenitors or on the nature of EPS methods. Further studies could help to improve our understanding about the physical processes during the formation of dark matter haloes.

Predicting the Number, Spatial Distribution, and Merging History of Dark Matter Halos

The Astrophysical Journal, 2002

We present a new algorithm (PINOCCHIO, PINpointing Orbit-Crossing Collapsed HIerarchical objects) to predict accurately the formation and evolution of individual dark matter haloes in a given realization of an initial linear density field. Compared with the halo population formed in a large (360 3 particles) collisionless simulation of a CDM universe, our method is able to predict to better than 10 per cent statistical quantities such as the mass function, two-point correlation function and progenitor mass function of the haloes. Masses of individual haloes are estimated accurately as well, with errors typically of order 30 per cent in the mass range well resolved by the numerical simulation. These results show that the hierarchical formation of dark matter haloes can be accurately predicted using local approximations to the dynamics when the correlations in the initial density field are properly taken into account. The approach allows one to automatically generate a large ensemble of accurate merging histories of haloes with complete knowledge of their spatial distribution. The construction of the full merger tree for a 256 3 realisation requires a few hours of CPU-time on a personal computer, orders of magnitude faster than the corresponding N -body simulation would take, and without needing any extensive post-processing. The technique can be efficiently used, for instance, for generating the input for galaxy formation modeling.

Merger history trees of dark matter haloes in moving barrier models

Monthly Notices of the Royal Astronomical Society, 2008

We present an algorithm for generating merger histories of dark matter haloes. The algorithm is based on the excursion set approach with moving barriers whose shape is motivated by the ellipsoidal collapse model of halo formation. In contrast to most other merger-tree algorithms, ours takes discrete steps in mass rather than time. This allows us to quantify effects which arise from the fact that outputs from numerical simulations are usually in discrete time bins. In addition, it suggests a natural set of scaling variables for describing the abundance of halo progenitors; this scaling is not as general as that associated with a spherical collapse. We test our algorithm by comparing its predictions with measurements in numerical simulations. The progenitor mass fractions and mass functions are in good agreement, as is the predicted scaling law. We also test the formation-redshift distribution, the mass distribution at formation, and the redshift distribution of the most recent major merger; all are in reasonable agreement with N-body simulation data, over a broad range of masses and redshifts. Finally, we study the effects of sampling in discrete time snapshots. In all cases, the improvement over algorithms based on the spherical collapse assumption is significant.

Evaluating Semi-Analytic Halo Merging Histories

We evaluate the accuracy of semi-analytic merger-trees by comparing them with the merging histories of dark-matter halos in N-body simulations, focusing on the joint distribution of the number of progenitors and their masses. We first confirm that the halo mass function as predicted directly by the Press-Schechter (PS) model deviates from the simulations by up to 50% depending on the mass scale and redshift, while the means of the projected distributions of progenitor number and mass for a halo of a given mass are more accurately predicted by the Extended PS model. We then use the full merger trees to study the joint distribution as a function of redshift and parent-halo mass. We find that while the deviation of the mean quantities due to the inaccuracy of the Extended PS model partly propagates into the higher moments of the distribution, the merger-tree procedure does not introduce a significant additional source of error. In particular, certain properties of the merging history such as the mass ratio of the progenitors and the total accretion rate are reproduced quite accurately for galaxy sized halos (∼ 10 12 M ⊙), and less so for larger masses. We conclude that although there could be ∼ 50% deviations in the absolute numbers and masses of progenitors and in the higher order moment of these distributions, the relative properties of progenitors for a given halo are reproduced fairly well by the merger trees. They can thus provide a useful framework for modelling galaxy formation once the above-mentioned limitations are taken into account.

Galaxy formation using halo merger histories taken from N-body simulations

Monthly Notices of the Royal Astronomical Society, 2003

We develop a hybrid galaxy formation model which uses outputs from an N-body simulation to follow the merger histories (or "merger trees") of dark matter halos and treats baryonic processes, such as the cooling of gas within halos and subsequent star formation, using the semi-analytic model of Cole et al. We compare this hybrid model to an otherwise identical model which utilises merger tree realisations generated by a Monte-Carlo algorithm and find that, apart from the limited mass resolution imposed by the N-body particle mass, the only significant differences between the models are due to the known discrepancy between the distribution of halo progenitor masses predicted by extended Press Schechter theory and that found in N-body simulations. We investigate the effect of limited mass resolution on the hybrid model by comparing to a purely semi-analytic model with greatly improved mass resolution. We find that the mass resolution of the simulation we use, which has a particle mass of 1.4 × 10 10 h −1 M ⊙ , is insufficient to produce a reasonable luminosity function for galaxies with magnitudes in the b J band fainter than -17.