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 the halo biasing, in addition to the nonlinear gravitational evolution of the underlying mass distribution. We examine in detail the extent to which the predicted genus for halos depends on the redshift, smoothing scale of the density field, and the mass of those halos in representative cold dark matter models. In addition to the full model predictions, we derive an explicit perturbation formula for the halo genus which can be applied to a wider class of biasing model for dark matter halos. With the empirical relation between the halo mass and the luminosity/temperature of galaxy clusters, our prediction can be compared with the upcoming cluster redshift catalogs in the optical and X-ray bands. Furthermore, we discuss the detectability of the biasing and cosmological model dependence from the future data.