Stochastic biasing and weakly nonlinear evolution of power spectrum

Distribution of galaxies may be a biased tracer of the dark matter distribution and the relation between the galaxies and the total mass may be stochastic, nonlinear and time dependent. Since many observations of galaxy clustering will be done at high redshift, the time evolution of nonlinear stochastic biasing would play a crucial role for the data analysis of the future sky surveys. Recently, analytic study of the time evolution induced by gravity has been reported in the mildly nonlinear regime. Here, we further develop the nonlinear analysis including the next-to-leading order and attempt to clarify the nonlinear feature of the stochastic biasing. Employing the perturbative approach, we compute the one-loop correction of the power spectrum for the total mass, galaxies, and their cross correlation. Assuming that the initial distribution of galaxies is given by the local function, we specifically investigate the time evolution of the biasing parameter and the correlation coefficient deduced from the power spectra. On large scales, we find that the time evolution of the biasing parameter could deviate from the linear theory prediction in presence of the initial skewness, even though the scale dependence of the biasing is very weak. On the other hand, the deviation can be reduced if the stochasticity between the galaxies and the total mass exists. To explore the influence of nonlinear gravity, we focus on the quasi-linear scales, where the nonlinear growth of the total mass becomes important. It is recognized that the scale dependence of the biasing dynamically appears and the initial stochasticity could affect the time evolution of the scale dependence. The result is compared with the recent N-body simulation that the scale dependence of the halo biasing can appear on relatively large scales and the biasing parameter takes the lower value on smaller scales. Qualitatively, our weakly nonlinear results can explain this trend if the halo-mass biasing relation has the large scatter at high redshift.