On the distribution of haloes, galaxies and mass

The stochasticity in the distribution of dark haloes in the cosmic density field is reflected in the distribution function P-V (N (h) \delta (m) ), which gives the probability of finding N (h) haloes in a volume V with mass density contrast delta (m) . We study the properties of this function using high-resolution N -body simulations, and find that P-V (N (h) \delta (m) ) is significantly non-Poisson. The ratio between the variance and the mean goes from similar to1 (Poisson) at 1+delta (m) <<1 to <1 (sub-Poisson) at 1+delta (m) similar to1< to > 1 (super-Poisson) at 1+delta (m) >>1. The mean bias relation is found to be well described by halo bias models based on the Press-Schechter formalism. The sub-Poisson variance can be explained as a result of halo exclusion, while the super-Poisson variance at high delta (m) may be explained as a result of halo clustering. A simple phenomenological model is proposed to describe the behaviour of the variance as a function of delta (m) . Galaxy distribution in the cosmic density field predicted by semi-analytic models of galaxy formation shows similar stochastic behaviour. We discuss the implications of the stochasticity in halo bias to the modelling of higher order moments of dark haloes and of galaxies.