Halo occupation numbers and galaxy bias

We propose a heuristic model that displays the main features of realistic theories for galaxy bias. We first show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes. The quasi-linear mass correlations are in fact reproduced well by a model of independent randomly-placed haloes. The distribution of galaxies within the halo density field depends on: (i) the efficiency of galaxy formation, as manifested by the halo occupation number– the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo. The first factor is constrained by the empirical luminosity function of groups. For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass. This second assumption is essential if small-scale galaxy correlations are to remain close to a single power law, rather than flattening in the same way as the correlations of the overall density field. These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density. We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat Ω=0.3 model (ΛCDM), but not for an Ω=1 model with the same power spectrum (τCDM). This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high Ω, and hence display enhanced small-scale clustering. Finally, the pairwise velocity dispersion of galaxies in the ΛCDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field. This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.

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