The Dark Side of the Halo Occupation Distribution

We analyze the halo occupation distribution (HOD) and two-point correlation function of galaxy-size dark matter halos using high-resolution dissipationless simulations of the concordance flat ΛCDM model. The halo samples include both the host halos and the ``subhalos,'' distinct gravitationally bound halos within the virialized regions of larger host systems. We find that the HOD, the probability distribution for a halo of mass M to host a number of subhalos N, is similar to that found in semianalytic and N-body+gasdynamics studies. Its first moment, <N>_M, has a complicated shape consisting of a step, a shoulder, and a power-law high-mass tail. The HOD can be described by Poisson statistics at high halo masses but becomes sub-Poisson for <N>_M<~4. We show that the HOD can be understood as a combination of the probability for a halo of mass M to host a central galaxy and the probability to host a given number N_s of satellite galaxies. The former can be approximated by a steplike function, while the latter can be well approximated by a Poisson distribution, fully specified by its first moment. The first moment of the satellite HOD can be well described by a simple power law <N_s>~M^β with β~1 for a wide range of number densities, redshifts, and different power spectrum normalizations. This formulation provides a simple but accurate model for the halo occupation distribution found in simulations. At z=0, the two-point correlation function (CF) of galactic halos can be well fitted by a power law down to ~100 h^-1 kpc with an amplitude and slope similar to those of observed galaxies. The dependence of correlation amplitude on the number density of objects is in general agreement with results from the Sloan Digital Sky Survey. At redshifts z>~1, we find significant departures from the power-law shape of the CF at small scales, where the CF steepens because of a more pronounced one-halo component. The departures from the power law may thus be easier to detect in high-redshift galaxy surveys than at the present-day epoch. They can be used to put useful constraints on the environments and formation of galaxies. If the deviations are as strong as indicated by our results, the assumption of the single power law often used in observational analyses of high-redshift clustering is dangerous and is likely to bias the estimates of the correlation length and slope of the correlation function.