Nonlinear stochastic biasing from the formation epoch distribution of dark halos

We propose a physical model for nonlinear stochastic biasing of one-point statistics resulting from the formation epoch distribution of dark halos. In contrast to previous works based on extensive numerical simulations, our model provides for the first time an analytic expression for the joint probability function. Specifically, we derive the joint probability function of halo and mass density contrasts from the extended Press-Schechter theory. Since this function is derived in the framework of the standard gravitational instability theory assuming the random Gaussianity of the primordial density field alone, we expect that the basic features of the nonlinear and stochastic biasing predicted from our model are fairly generic. As representative examples, we compute the various biasing parameters in cold dark matter models as a function of a redshift and a smoothing length. Our major findings are that (1) the biasing of the variance evolves strongly as redshift, while its scale dependence is generally weak, and a simple linear biasing model provides a reasonable approximation at roughly R greater than or similar to 2(1 + z) h(-1) Mpc; and (2) the stochasticity exhibits moderate scale dependence, especially on R less than or similar to 20 h(-1) Mpc, but is almost independent of z. Comparison with the previous numerical simulations shows good agreement with the above behavior, indicating that the nonlinear and stochastic nature of the halo biasing can be essentially understood by taking into account the distribution of the halo mass and the formation epoch.