Weakly Non-linear Galaxy-Mass Density Relation in Stochastic Bias

The enormous data of the galaxies from the future sky survey might allow us to explore the time evolution of the galaxy distribution. It is also believed that the bias of the galaxies plays an important role for understanding the large-scale structure of the universe. In this paper, we investigate the time evolution of the bias in the stochastic formalism. Using the Egdeworth expansion, we derive a general formula for the weakly non-linear galaxy-mass density relation as a conditional mean. The resulting expression of the galaxy-mass density relation contains the joint moments of the total mass and galaxy distributions. The time dependence of the joint moments induces the time evolution of the galaxy-mass relation. The influence of the initial stochasticity on the evolution of the galaxy-mass relation is examined. We also find that the linear coefficient of the galaxy-mass relation is different from the one determined by using the two-point correlation functions of galaxies and mass density. Comparing with the skewness of the galaxy distribution, it turns out that the galaxy-mass density relation cannot be used as a tool for calculation of the skewness shown by Fry & Gaztañaga (1993), which is often used in the deterministic non-linear bias case. Subject headings: cosmology:theory−large scale structure of universe, biasing