The Halo Occupation Distribution and the Physics of Galaxy Formation

Abstract

The halo occupation distribution (HOD) describes the bias between galaxies and dark matter by specifying (1) the probability P ( N | M ) that a halo of virial mass M contains N galaxies of a particular class and (2) the relative spatial and velocity distributions of galaxies and dark matter within halos. We calculate and compare the HODs predicted by a smoothed particle hydrodynamics (SPH) simulation of a ΛCDM cosmological model (cold dark matter with a cosmological constant) and by a semianalytic galaxy formation model applied to the same cosmology. Although the two methods predict different galaxy mass functions, their HOD predictions for samples of the same space density agree remarkably well. In a sample defined by a baryonic mass threshold, the mean occupation function ⟨ N ⟩ M exhibits a sharp cutoff at low halo masses, a slowly rising plateau in which ⟨ N ⟩ climbs from 1 to 2 over nearly a decade in halo mass, and a more steeply rising high-occupancy regime at high halo mass. In the low-occupancy regime, the factorial moments ⟨ N ( N - 1)⟩ and ⟨ N ( N - 1)( N - 2)⟩ are well below the values of ⟨ N ⟩ 2 and ⟨ N ⟩ 3 expected for Poisson statistics, with important consequences for the small-scale behavior of the two- and three-point correlation functions. The HOD depends strongly on galaxy age, with high-mass halos populated mainly by old galaxies and low-mass halos by young galaxies. The distribution of galaxies within SPH halos supports the assumptions usually made in semianalytic calculations: the most massive galaxy lies close to the halo center and moves near the halo's mean velocity, while the remaining, satellite galaxies have the same radial profile and velocity dispersion as the dark matter. The mean occupation at fixed halo mass in the SPH simulation is independent of the halo's larger scale environment, supporting both the merger tree approach of the semianalytic method and the claim that the HOD provides a complete statistical characterization of galaxy bias. We discuss the connections between the predicted HODs and the galaxy formation physics incorporated in the SPH and semianalytic approaches. These predictions offer useful guidance to theoretical models of galaxy clustering, and they will be tested empirically by ongoing analyses of galaxy redshift surveys. By applying the HODs to a large-volume N -body simulation, we show that both methods predict slight departures from a power-law galaxy correlation function, similar to features detected in recent observational analyses.