Stochastic biasing and the galaxy-mass density relation in the weakly nonlinear regime

It is believed that the biasing of galaxies plays an important role in unde rstanding the large-scale structure of the universe. In general, the biasin g of galaxy formation could be stochastic. Furthermore, future galaxy surve ys might allow us to explore the time evolution of the galaxy distribution. In this paper the analytic study of the galaxy-mass density relation and i ts time evolution is presented within the framework of stochastic biasing. In the weakly nonlinear regime, we derive a general formula for the galaxy- mass density relation as a conditional mean using the Edgeworth expansion. The resulting expression contains the joint moments of the total mass and g alaxy distributions. Using the perturbation theory, we investigate the time evolution of the joint moments and examine the influence of the initial st ochasticity on the galaxy-mass density relation. The analysis shows that th e galaxy-mass density relation could be well approximated by the linear rel ation. Compared with the skewness of the galaxy distribution, we find that the estimation of the higher order moments using the conditional mean could be affected by the stochasticity. Therefore, the galaxy-mass density relat ion as a conditional mean should be used with caution as a tool for estimat ing the skewness and the kurtosis.