Cluster cosmology redux: A compact representation for the halo mass function

Groups and clusters of galaxies imprint coherent, arcminute-scale features across the spectrophotometric sky, especially optical-IR clusters of galaxies, spectral distortions in the cosmic microwave background, and extended sources of X-ray emission. The space–time density of the host dark matter halo population – the halo mass function (HMF) – is a common theoretical basis for modelling such observable features. We explore a compact representation – a dual-quadratic (DQ-HMF) form – that features readily interpretable parameters representing polynomial expansions of the space–time number density surface, first in terms of log-mass, then in redshift. The DQ-HMF form fits Mira-Titan N-body emulator expectations for halo masses 1013.7-14.5 \, h⁻¹\, M_ over redshifts, 0.1 < z < 1.5 to within ~ \! 5~\ per\ cent. We provide best-fitting parameters for a Planck 2018 cosmology and demonstrate model self-similarity in the Ωm−\, σ ₈ 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 an observable property. Performing information-matrix forecasts of potential parameter constraints from existing and future surveys, 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 0.32. Even better constraints could potentially be achieved by a survey with one-tenth the sample size but with a reduced selection property variance of 0.12. Potential benefits and extensions to the basic MOR parametrization are discussed.