The Halo Mass Function: High-Redshift Evolution and Universality

TL;DR

This work presents a comprehensive assessment of the halo mass function in ΛCDM from $z=20$ to $z=0$ using 60 nested-volume N-body simulations, achieving percent-level accuracy across a wide mass range ($M \sim 10^7$ to $10^{13.5}\,h^{-1}M_\odot$). It rigorously analyzes systematic effects—starting redshift, force/mass resolution, time stepping, and finite-volume corrections—and adopts FOF halos with $b=0.2$ along with the Warren sampling correction. The results show that the Press-Schechter form severely underpredicts halos at high redshift, while modern fits (Jenkins, Warren, Reed) generally agree within ~20% after volume corrections; universality is approximately recovered when finite-volume effects are accounted for. The findings provide practical guidance for high-redshift structure predictions and reionization modeling, confirming near-universal behavior of the mass function under careful error control and offering a robust benchmark for future simulations and analytic fits.

Abstract

We study the formation of dark matter halos in the concordance LCDM model over a wide range of redshifts, from z=20 to the present. Our primary focus is the halo mass function, a key probe of cosmology. By performing a large suite of nested-box N-body simulations with careful convergence and error controls (60 simulations with box sizes from 4 to 256 Mpc/h, we determine the mass function and its evolution with excellent statistical and systematic errors, reaching a few percent over most of the considered redshift and mass range. Across the studied redshifts, the halo mass is probed over 6 orders of magnitude (10^7 - 10^13.5 M_sun/h). Historically, there has been considerable variation in the high redshift mass function as obtained by different groups. We have made a concerted effort to identify and correct possible systematic errors in computing the mass function at high redshift and to explain the discrepancies between some of the previous results. We discuss convergence criteria for the required force resolution, simulation box size, halo mass range, initial and final redshift, and time stepping. Because of conservative cuts on the mass range probed by individual boxes, our results are relatively insensitive to simulation volume, the remaining sensitivity being consistent with extended Press-Schechter theory. Previously obtained mass function fits near z=0, when scaled by linear theory, are in good agreement with our results at all redshifts, although a mild redshift dependence consistent with that found by Reed and collaborators exists at low redshifts.

Figures (15)

• Figure 1: Ratio of the Jenkins, PS, and ST mass function fits with respect to the Warren fit for five different redshifts over a range of halo masses. Top to bottom: Redshifts $z=0$, 5, 10, 15, and 20. Note that the ranges of the axes are different in the different panels. We do not show the Jenkins fit below masses of 10$^{11} h^{-1} M_\odot$ at $z=0$, since it is not valid for such low masses at that redshift.
• Figure 2: Halo growth function based on the Warren mass function fit for different mass bins. The curves for the lower mass bins have a maximum at $z>0$ which reflects a crossover of the mass functions at different redshifts.
• Figure 3: Summary of recent work on the mass function at high redshift. The mass function fits are shown at $z=10$ (top) and $z=20$ (bottom) for the cosmology used throughout this paper (the other groups used slightly different parameters). At $z=10$, Jang-Condell & Hernquist (2001) (gray shaded region) cover the very low mass range using a very small box, as do Cen et al. (2004) (green shaded region). The larger boxes of Reed07 and Springel et al. (2005) (red shaded region) lead to results at higher halo masses. Note that in this regime the PS fit deviates substantially from the other fits, while at the very low mass end all fits tend to merge. Our suite of variable box sizes covers a mass range of 10$^7$ to 10$^{13.5}\,h^{-1}M_\odot$ between $z=0$ and 20, a much larger range than previously covered by any group with a uniform set of simulations. At $z=20$ Yoshida et al. (2003a, 2003b, 2003c, 2003d) cover the very low mass end of the mass function, while Zahn et al. (2007) investigate larger mass halos. Our simulations overlap with both of them at the edges. By combining a heterogeneous set of simulations, Reed07 cover a wide range in mass and redshift. Figure quality reduced for the arXiv version of the paper.
• Figure 4: Probability distribution of $|\nabla{\bf }|$ in units of the interparticle spacing $\Delta_{\rm p}$. All curves shown are drawn from $256^3$ particle simulations from an initial density grid of $256^3$ zones. The physical box sizes are $126\,h^{-1}$Mpc (black line), $32\,h^{-1}$Mpc (red line), and $8\,h^{-1}$Mpc (green line). As expected, $\langle|\nabla{\bf }|\rangle$ increases with decreasing box size (which is equivalent to increasing force resolution). Therefore, $z_{\rm in}$ and $z_{\rm cross}$ are higher for the smaller boxes.
• Figure 5: Average redshift of first crossing (top) and highest redshift of first crossing (bottom) as a function of box size. The initial conditions (five different realizations) are shown for boxes between 1 and 512$\,h^{-1}$Mpc with 128$^3$ and 256$^3$ particles. For each initial condition, $z_{\rm cross}^{\rm first}$ and $z_{\rm cross}^{\rm rms}$ are shown by the crosses. The solid lines show the average from the five realizations. As expected, scatter from the different realizations is larger for smaller boxes. These plots provide estimates of the required initial redshift for a simulation since $|\nabla |/\Delta_{\rm p}$ is $z$-independent in the Zel'dovich approximation (see text).