Biasing and the distribution of dark matter haloes

In hierarchical models of gravitational clustering, virialized haloes are biased tracers of the matter distribution. As discussed by Mo & White (1996), this bias is nonlinear and stochastic. They developed a model which allows one to write down analytic expressions for the mean of the bias relation, in the initial Lagrangian, and the evolved, Eulerian, spaces. We provide analytic expressions for the higher order moments as well. In the initial Lagrangian space, each halo occupies a volume that is proportional to its mass. Haloes cannot overlap initially, so this gives rise to volume exclusion effects which can have important consequences for the halo distribution, particularly on scales smaller than that of a typical halo. Our model allows one to include these volume exclusion effects explicitly when computing the mean and higher order statistics of the Lagrangian space halo distribution. As a result of dynamical evolution, the spatial distribution of haloes in the evolved Eulerian space is likely to be different from that in the initial Lagrangian space. When combined with the Mo & White spherical collapse model, the model developed here allows one to quantify the evolution of the mean and scatter of the bias relation. We also show how their approach can be extended to compute the evolution, not just of the haloes, but of the dark matter distribution itself. Biasing and its evolution depend on the initial power spectrum. Clustering from Poisson and white noise Gaussian initial conditions is treated in detail, since, in these cases, exact analytical results are available. We conjecture that these results can be easily extended to provide an approximate but accurate model for the biasing associated with clustering from more general Gaussian initial conditions. For all initial power spectra studied here, the model predictions for the Eulerian bias relation are in reasonable agreement with numerical simulations of hierarchical gravitational clustering for haloes of a wide range of masses, whereas the predictions for the corresponding Lagrangian space quantities are accurate only for massive haloes.