Genus statistics for galaxy clusters and nonlinear biasing of dark matter halos

We derive an analytic formula of genus for dark matter halos assuming that the primordial mass density field obeys the random-Gaussian statistics. In particular, we for the first time take account of the nonlinear nature of t he halo biasing, in addition to the nonlinear gravitational evolution of th e underlying mass distribution. We examine in detail the extent to which th e predicted genus for halos depends on the redshift, smoothing scale of the density field, and the mass of those halos in representative cold dark mat ter models. In addition to the full model predictions, we derive an explici t perturbation formula for the halo genus which can be applied to a wider c lass of biasing model for dark matter halos. With the empirical relation be tween the halo mass and the luminosity/temperature of galaxy clusters, our prediction can be compared with the upcoming cluster redshift catalogs in t he optical and X-ray bands. Furthermore, we discuss the detectability of th e biasing and cosmological model dependence from the future data.