Stochasticity in Halo Bias

The stochasticity in the distribution of dark haloes in the cosmic density field is reflected in the distribution function $P_V(N|\delta_m)$ which gives the probability of finding N 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 $P_V(N|\delta_m)$ is significantly non-Poisson. The ratio between the variance and the mean goes from $\sim 1$ (Poisson) at $1+\delta_m\ll 1$ to $1$ (super-Poisson) at $1+\delta_m\gg 1$. 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 behavior 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 stochasticity. We discuss the implications of the stochasticity in halo bias to the modelling of high-order moments of dark haloes and of galaxies that form within them.