We study merger histories of dark-matter haloes in a suite of N-body simulations that span different cosmological models. The simulated cases include the up-to-date WMAP5 cosmology and other test cases based on the Einstein-deSitter cosmology with different power spectra. We provide a robust fitting function for the conditional mass function (CMF) of progenitor haloes of a given halo. This fit is valid for the different cosmological models and for different halo masses and redshifts, and it is a significant improvement over earlier estimates. Based on this fit, we develop a simple and accurate technique for transforming the merger history of a given simulated halo into haloes of different mass, redshift and cosmology. Other statistics such as main-progenitor history and merger rates are accurately transformed as well. This method can serve as a useful tool for studying galaxy formation. It is less sensitive to the low accuracy of the fit at small time-steps, and it can thus replace the more elaborate task of construction Monte-Carlo realizations. As an alternative approach, we confirm the earlier finding by Neistein & Dekel that the main-progenitor follows a log-normal distribution. This property of merger trees allows us to better capture their behaviour as a function of time and descendant mass, but a broader suite of simulations is required for evaluating the dependence of the log-normal parameters on the cosmological model.
Deep Dive into Universal Merger Histories of Dark-Matter Haloes.
We study merger histories of dark-matter haloes in a suite of N-body simulations that span different cosmological models. The simulated cases include the up-to-date WMAP5 cosmology and other test cases based on the Einstein-deSitter cosmology with different power spectra. We provide a robust fitting function for the conditional mass function (CMF) of progenitor haloes of a given halo. This fit is valid for the different cosmological models and for different halo masses and redshifts, and it is a significant improvement over earlier estimates. Based on this fit, we develop a simple and accurate technique for transforming the merger history of a given simulated halo into haloes of different mass, redshift and cosmology. Other statistics such as main-progenitor history and merger rates are accurately transformed as well. This method can serve as a useful tool for studying galaxy formation. It is less sensitive to the low accuracy of the fit at small time-steps, and it can thus replace the
The growth of dark-matter haloes through merging and accretion is driving the formation and evolution of galaxies. An accurate theoretical prediction for the way haloes grow is thus a key element in the effort to develop a theoretical understanding of the processes associated with galaxy formation, including star formation, the growth of black holes in galaxy centers and their appearance as quasars.
The conditional mass function (hereafter CMF) has been an important tool in quantifying the growth of haloes. It is defined at a given time as the average number of progenitors that will merge into a descendant halo at a later time. The CMF was introduced in the context of the Extended Press-Schechter formalism based on excursion-set theory (hereafter EPS, Bond et al. 1991;Bower 1991;Lacey & Cole 1993). Recent theoretical predictions use a variant of the EPS formalism in which the spherical collapse model is replaced by an ellipsoidal collapse model (Sheth & Tormen 2002;Moreno et al. 2008;Zhang et al. 2008). An empirical fit to the CMF based on N -body simulation was presented by Cole et al. (2008).
Although the excursion-set models that employ ellipsoidal collapse are successful in reproducing the overall (unconditional)
⋆ E-mail: eyal@mpa-garching.mpg.de halo mass function as derived from N -body simulations these models do not provide accurate CMF predictions. Most of the current analytic predictions of the CMF deviate from the results of Nbody simulations by a multiplicative factor of a few, especially for the number of massive progenitors. The empirical fit of Cole et al. (2008) does better, but its inaccuracies may reach the level of 50%. As shown by Cole et al. (2008), these inaccuracies are valid whenever the standard variables are used, and they cannot be reduced by a different functional fit. In addition, this fit was calibrated against the Millennium simulation (Springel et al. 2005) that was based on the cosmological parameters from the first-year data of the Wilkinson Microwave Anisotropy Probe (WMAP), it should be interesting to evaluate the CMF for the more up-to-date cosmological parameters that emerge from the fifth-year WMAP data (e.g. Komatsu et al. 2009).
The main goal of this paper is to provide a more accurate and more robust empirical description of the CMF as measured from N -body simulations. We work out the possible scaling laws which can be applied to the CMF in order to capture its detailed properties over a large range of cosmological models, halo masses and redshifts. The result is a fitting function that offers a significant improvement in accuracy over previous studies. The empirical fit can help us distinguish between different analytical models of structure formation, and can guide us to new improved versions of them. Such a fit can be used for generating Monte-Carlo merger trees, which will accurately reproduce the results of N -body simulations. The effect of cosmology, environment density and different dark energy models can be studied, relating these factors to haloes and galaxies (e.g. Macciò et al. 2008). These issues and additional applications are discussed below in more detail.
Currently, there is still some freedom in the definitions of halo mass and merger trees. Accurate CMF fit may help us to test different algorithms for constructing merger trees, including assigning masses to haloes and relating progenitors to their descendants. It will allow us to quantify the effects of different construction schemes for merger trees, and especially to identify the scheme that results in the simplest and most universal CMF.
A significant progress has been made in quantifying the overall halo mass function, namely the abundance of haloes of a given mass at a given redshift. The analytical model of Sheth & Tormen (1999) offers a significant improvement over the classic estimate of Press & Schechter (1974). Furthermore, the growing volume and dynamical range of N -body simulations allow an accurate measurement of the halo mass function with small sampling scatter and cosmic variance (e.g. Jenkins et al. 2001;Warren et al. 2006;Reed et al. 2007). Still, there are significant deviations between the analytic approximations and the simulations, which can increase as a function of redshift and reach a few tens of percents at z = 2.5 (Tinker et al. 2008). For one thing, an accurate knowledge of the CMF can provide a better estimate of the halo mass function and its evolution at high redshift. However, the more interesting feature of the CMF is that it involves much more information than the halo mass function, and it can still be derived with a manageable statistical uncertainty.
Using the CMF, we present a new method for scaling a given set of merger trees into a different cosmological model, mass resolution or redshift. Although this technique requires an existing database of merger trees extracted from a given N -body simulation, it offers an easy transformation to the desired
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