Mass function of dark matter halos

TL;DR

This work tackles the challenge of predicting the dark matter halo mass function across more than four orders of magnitude in mass for CDM cosmologies. By combining extensive N-body simulations (including the large Hubble Volume runs) with two halo-finding methods, the authors demonstrate a near-universal mass function when expressed in terms of $f(\sigma)$, largely independent of redshift, cosmology, and initial power spectrum, provided halos are defined at fixed overdensity. They show that the Press-Schechter prediction deviates in detail, while the Sheth-Tormen form provides an excellent fit across a broad range of models, yielding a practical, physically motivated fitting formula that remains accurate across redshifts and parameter space. This universal description enables precise predictions for cluster abundances and supports robust cosmological inferences from large-scale structure surveys, with quantified systematic uncertainties and accessible data/software for broader use.

Abstract

We combine data from a number of N-body simulations to predict the abundance of dark halos in Cold Dark Matter universes over more than 4 orders of magnitude in mass. A comparison of different simulations suggests that the dominant uncertainty in our results is systematic and is smaller than 10--30% at all masses, depending on the halo definition used. In particular, our ``Hubble Volume'' simulations of \tcdm and \lcdm cosmologies allow the abundance of massive clusters to be predicted with uncertainties well below those expected in all currently planned observational surveys. We show that for a range of CDM cosmologies and for a suitable halo definition, the simulated mass function is almost independent of epoch, of cosmological parameters, and of initial power spectrum when expressed in appropriate variables. This universality is of exactly the kind predicted by the familiar Press-Schechter model, although this model predicts a mass function shape which differs from our numerical results, overestimating the abundance of ``typical'' halos and underestimating that of massive systems.

Figures (12)

• Figure 1: Friends-of-friends differential mass functions for dark matter halos in the $\tau$CDM and $\Lambda$CDM simulations. Halos were identified using linking lengths of 0.2 and 0.164 respectively. The different curves correspond to the various simulations detailed in Table 1. The mass resolution in the simulations varies by more than two orders of magnitude and the volume surveyed by more than four orders of magnitude. In all cases, the mass functions are truncated at the low mass end at the mass corresponding to 20 particles, and at the high mass end at the point where the predicted Poisson abundance errors reach 10%. The simulations match up very well.
• Figure 2: Spherical overdensity differential mass function for dark halos in the $\tau$CDM and $\Lambda$CDM simulations. Halo masses were defined at mean interior overdensities of 180 and 324 respectively. The different curves correspond to the various simulations detailed in Table 1 and are truncated as in Fig.1. The simulations do not match up as well here as in Fig.1. Simulations with coarse mass resolution seem to underestimate the halo abundance near their lower mass limit. See the text for further details.
• Figure 3: A comparison of analytic models with the halo mass function at $z=0$ in our N-body simulations of the $\tau$CDM cosmology. Halos were found using the FOF algorithm with $b=0.2$. The short dashed lines show results from the individual $\tau$CDM simulations used in this paper, the solid curve is the fit of eqn. (\ref{['tcdm_fit']}) to the combined results of the simulations, while the dashed line shows the P-S prediction and the dotted line, the S-T prediction, both using $\delta_c=1.686$. The arrow marks the characteristic mass scale, $M_*$, where $\sigma(M_*)=\delta_c$, and corresponds to the position of the peak in the Press-Schechter mass multiplicity function. Note that we use natural logarithms in this plot.
• Figure 4: A comparison of analytic models with the halo mass function at $z=0$ in our N-body simulations of the $\Lambda$CDM cosmology. Halos were found using the FOF algorithm with $b=0.164$. The short dashed lines show results from the individual $\Lambda$CDM simulations used in this paper, the solid curve is the fit of eqn. (\ref{['lcdm_fit']}) to the combined results of the simulations, while the long dashed line shows the P-S prediction and the dotted line, the S-T prediction, both using $\delta_c=1.675$. The arrow marks the characteristic mass scale, $M_*$, where $\sigma(M_*)=\delta_c$, and corresponds to the position of the peak in the Press-Schechter mass function. Note that we use natural logarithms in this plot.
• Figure 5: A comparison of mass functions in different simulations at different epochs. The full curves show the FOF(0.2) mass functions for the $\tau$CDM-hub simulation at redshifts $z=0, 0.18, 0.44$ and 0.78. The dashed curves show the corresponding functions for the SCDM simulations of Governato et al (1999) at four epochs for which $\sigma_8=$1.0,0.7,0.47 and 0.35 respectively. The heavy dashed curve shows the P-S model function. Both simulation datasets show a weak trend with $\sigma_8$, but the trends are opposite! See the text for discussion. Note that we use natural logarithms in this plot.