Toward a halo mass function for precision cosmology: the limits of universality

TL;DR

This work addresses precision cosmology needs by recalibrating the halo mass function using a large suite of collisionless $\Lambda$CDM simulations with spherical overdensity masses defined at overdensities $\Delta$. The authors introduce a fitting framework where $\frac{dn}{dM}$ is expressed via $f(\nu)$ with $\nu \equiv \delta_c/\sigma(M)$ and $f(\nu)=A[(\nu/b)^{-a}+1]\exp(-c/\nu^2)$, calibrating parameters for multiple $\Delta$ and monitoring redshift evolution up to $z\approx2.5$. They demonstrate that the mass function is not universal at percent-level accuracy: the amplitude and even the shape evolve with redshift and depend on the mass definition, challenging prior universality assumptions, and showing that observable-linked SO masses are preferable to FOF masses for abundance forecasts. The findings have significant implications for cluster counts as cosmological probes and highlight the need for meticulous treatment of numerical resolution, initial conditions, and potential baryonic effects in future precision work. The paper provides a practical, Δ-dependent parameter set and interpolation strategy to predict halo abundances across a wide range of overdensities and redshifts.

Abstract

We measure the mass function of dark matter halos in a large set of collisionless cosmological simulations of flat LCDM cosmology and investigate its evolution at z<~2. Halos are identified as isolated density peaks, and their masses are measured within a series of radii enclosing specific overdensities. We argue that these spherical overdensity masses are more directly linked to cluster observables than masses measured using the friends-of-friends algorithm (FOF), and are therefore preferable for accurate forecasts of halo abundances. Our simulation set allows us to calibrate the mass function at z=0 for virial masses in the range 10^{11} Msol/h < M < 10^{15} Msol/h, to <~ 5%. We derive fitting functions for the halo mass function in this mass range for a wide range of overdensities, both at z=0 and earlier epochs. In addition to these formulae, which improve on previous approximations by 10-20%, our main finding is that the mass function cannot be represented by a universal fitting function at this level of accuracy. The amplitude of the "universal" function decreases monotonically by ~20-50%, depending on the mass definition, from z=0 to 2.5. We also find evidence for redshift evolution in the overall shape of the mass function.

Figures (15)

• Figure 1: A graphical key for the list of simulations in Table 1. The upper panel shows point styles for all the WMAP1 simulations ordered by the box size. Each simulation is represented with a different color, while different point types represent different numerical codes: circles=HOT, squares=ART, triangles=GADGET2. The lower panel plots all WMAP3 simulations, as well as H384$\Omega$, the low-$\Omega_m$ simulation. See Table 1 for the details of each simulation.
• Figure 2: Comparison between spherical overdensity masses and friends-of-friends masses for the same sample of objects from H384, L250, and L1000W. Panel (a) compares the masses of $\Delta=200$ halos to FOF halos with $l=0.2$. The symbols represent the median mass ratio, for objects binned by $M_{200}$. The curves show the upper 90% and lower 10% bounds of the distribution of mass ratios in each $M_{200}$ bin: solid for H384, dashed for L250, and dotted for L1000W. The asymmetry in the mass ratio distribution reflects the tendency of FOF to link objects together. Panel (b) compares $\Delta=1600$ halos to FOF objects with $l=0.1$. Panel (c) shows the distribution of mass ratios, $r_M = M_{200}/M_{\rm FOF.2}$, for halos $13\le \log\,M_{200}\le 14$ ( solid line). The long tail of the distribution at $r_M<0.5$ indicates SO halos that are linked with other virialized objects in the FOF halo-finding process. The dotted line is the same distribution at $z=1.25$. Panel (d) shows the distribution of $r_M$ for the same mass range, for the $\Delta=1600$ and and FOF linking length $l=0.1$. Solid and dotted lines are $z=0$ and $z=1.25$, respectively. Both panels (c) and (d) show results for the L250 run.
• Figure 3: The halo density profiles are compared to analytic predictions for three different simulations. In each panel, the dotted curve represents the mean interior density given by an NFW profile with $c(M)$ from dolag_etal:04. The shaded region is the expected scatter assuming $_{\log c}=0.12$. The solid curves with errorbars represent the numerical results. The left panel shows results from H384 for all halos $M>10^{14.5}$$h^{-1}\,$M$_\odot$. The center and right panels show results for halos $M>10^{15}$$h^{-1}\,$M$_\odot$. The center and left panel demonstrate that halo profiles are well resolved in these simulations. The right panel, shows significant deviations from the expected NFW profile at $r<0.1R_{200}$ in the lower-resolution L1280 simulation.
• Figure 4: Test of the resolution of the large-volume simulations, L500, L1000W, and one realization of L1280. In each panel, the mass functions are plotted as residuals with respect to the best-fit $f( )$ function from Table 2. The symbols represent the mass functions measured directly from the simulations at each $\Delta$. The curves are mass functions inferred from the $\Delta=200$ halo catalog of each simulation, where the mass of each $\Delta=200$ halo is scaled to higher overdensities assuming an analytic NFW halo (including scatter in concentrations at fixed mass). For the two higher resolution simulations, the scaled and true mass function are in agreement. Due to insufficient resolution, the L1280 mass function falls below the scaled mass function at high $\Delta$.
• Figure 5: The measured mass functions for all WMAP1 simulations, plotted as $(M^2/\bar{ }_m)\,dn/dM$ against $\log\,M$. The solid curves are the best-fit functions from Table 2. The three sets of points show results for $\Delta=200$, 800, and 3200 (from top to bottom). To provide a rough scaling between $M$ and $^{-1}$, the top axis of the plot shows $^{-1}$ for this mass range for the WMAP1 cosmology. The slight offset between the L1280 results and the solid curves is due to the slightly lower value of $\Omega_m=0.27$.