Cluster Cosmology Redux: A Compact Model of the Halo Mass Function

Massive halos hosting groups and clusters of galaxies imprint coherent, arcminute-scale features across the spectrophotometric sky, especially optical-IR clusters of galaxies, distortions in the sub-mm CMB, and extended sources of X-ray emission. Statistical modeling of such features often rely upon the evolving space-time density of dark matter halos -- the halo mass function (HMF) -- as a common theoretical ground for cosmological, astrophysical and fundamental physics studies. We propose a compact (eight parameter) representation of the HMF with readily interpretable parameters that stem from polynomial expansions, first in terms of log-mass, then expanding those coefficients similarly in redshift. We demonstrate good ($\sim \! 5\%$) agreement of this form, referred to as the dual-quadratic (DQ-HMF), with Mira-Titan N-body emulator estimates for halo masses above $10^{13.7} h^{-1} {\rm M}_\odot$ over the redshift range $0.1 < z < 1.5$, present best-fit parameters for a Planck 2018 cosmology, and present parameter variation in the $\sigma_8 - \Omega_{\rm m}$ plane. Convolving with a minimal mass-observable relation (MOR) yields closed-form expressions for counts, mean mass, and mass variance of cluster samples characterized by some observable property. Performing information-matrix forecasts of potential parameter constraints from existing and future surveys under different levels of systematic uncertainties, we demonstrate the potential for percent-level constraints on model parameters by an LSST-like optical cluster survey of 300,000 clusters and a richness-mass variance of $0.3^2$. Even better constraints could potentially be achieved by a survey with one-tenth the sample size but with a reduced selection property variance of $0.1^2$. Potential benefits and extensions to the basic MOR parameterization are discussed.