Cell Count Moments in the Halo Model

TL;DR

This paper analyzes cell count moments in the halo model, comparing theoretical predictions with simulations, and explores how clustering and halo structure influence statistical measures of halo and galaxy distributions.

Contribution

It introduces a simple point cluster model for large-scale moments and examines the impact of halo structure and bias on small-scale clustering predictions.

Findings

Large-scale moments approach halo distribution values

Hierarchical clustering with shape-dependent amplitudes

Differences in galaxy and mass statistics on small scales

Abstract

We study cell count moments up to fifth order of the distributions of haloes, of halo substructures as a proxy for galaxies, and of mass in the context of the halo model and compare theoretical predictions to the results of numerical simulations. On scales larger than the size of the largest cluster, we present a simple point cluster model in which results depend only on cluster-cluster correlations and on the distribution of the number of objects within a cluster, or cluster occupancy. The point cluster model leads to expressions for moments of galaxy counts in which the volume-averaged moments on large scales approach those of the halo distribution and on smaller scales exhibit hierarchical clustering with amplitudes $S_k$ determined by moments of the occupancy distribution. In this limit, the halo model predictions are purely combinatoric, and have no dependence on halo profile, concentration parameter, or potential asphericity. The full halo model introduces only two additional effects: on large scales, haloes of different mass have different clustering strengths, introducing relative bias parameters; and on the smallest scales, halo structure is resolved and details of the halo profile become important, introducing shape-dependent form factors. Because of differences between discrete and continuous statistics, the hierarchical amplitudes for galaxies and for mass behave differently on small scales even if galaxy number is exactly proportional to mass, a difference that is not necessarily well described in terms of bias.