Precision Determination of the Mass Function of Dark Matter Halos

TL;DR

This work precisely determines the dark matter halo mass function in flat $Λ$CDM by combining sixteen nested-volume $1024^3$ simulations with a corrected friends-of-friends halo finder and an extended maximum-likelihood fit to the multiplicity function $f(σ)=A(σ^{-a}+b) e^{-c/σ^2}$. The authors report calibrated parameters for $f(σ)$ and demonstrate that common forms such as the Press-Schechter, Jenkins, and Sheth–Tormen fits do not fully describe the simulated mass function, as well as evidence for potential non-universality of $f(σ)$ under cosmology variations. The mass function’s statistical and systematic uncertainties are quantified and shown to have negligible impact on future cluster-survey parameter forecasts, though cosmology dependence and halo-finder systematics warrant further study. Overall, this calibration provides a robust basis for interpreting cluster counts and galaxy clustering within the $Λ$CDM framework, while underscoring the need for improved halo definitions and bias analyses in precision cosmology.

Abstract

The predicted mass function of dark matter halos is essential in connecting observed galaxy cluster counts and models of galaxy clustering to the properties of the primordial density field. We determine the mass function in the concordance $Λ$CDM cosmology, as well as its uncertainty, using sixteen $1024^3$-particle nested-volume dark-matter simulations, spanning a mass range of over five orders of magnitude. Using the nested volumes and single-halo tests, we find and correct for a systematic error in the friends-of-friends halo-finding algorithm. We find a fitting form and full error covariance for the mass function that successfully describes the simulations' mass function and is well-behaved outside the simulations' resolutions. Estimated forecasts of uncertainty in cosmological parameters from future cluster count surveys have negligible contribution from remaining statistical uncertainties in the central cosmology multiplicity function. There exists a potentially non-negligible cosmological dependence (non-universality) of the halo multiplicity function.

Figures (3)

• Figure 1: Shown are the central values of the binned mass functions from sixteen $1024^3$ simulations of the $\Lambda$CDM universe as crosses, with simulations in different colors. The best-fit form for the mass function we calculate is in solid (red), the Jenkins fit dashed (purple), and the Sheth-Tormen fit in dot-dashed (dark grey). Goodness-of-fit is poorly judged on this extreme log scale; it is more clearly resolved in the linear residuals of Fig. \ref{['residuals']}.
• Figure 2: Shown are the residuals from the binned simulation data to the fit presented in this work as square data points of different colors per simulation. The Jenkins fit is the solid (purple) line, ST original fit the dashed (dark gray) line, the ST fit with parameters $A,a,p$ free with dot-dashed line (red), and the ST fit with $a,p$ free and amplitude $A$ set to require all dark matter in halos as a triple-dot-dashed line (light gray). The binned mass function from the Virgo Hubble Volume simulation are the asterisk points with errors (pink).
• Figure 3: Residuals are shown for simulations with varied cosmological parameters with that cosmologies' mass function using the multiplicity function predicted from the central cosmology. Cosmological parameter are held constant, except triangles are for $\sigma_8 = 0.8$; squares are for $\Omega_m =0.2$; circles for $h=0.6$ and $\sigma_8 = 0.8$; diamonds are for $\Omega_m =0.2$ and $\sigma_8 = 1.2$.