Statistics of Smoothed Cosmic Fields in Perturbation Theory. I. Formulation and Useful Formulae in Second‐Order Perturbation Theory

We formulate a general method for perturbative evaluations of statistics ofsmoothed cosmic fields, and provide useful formulas in application of theperturbation theory to various statistics. This formalism is an extensivegeneralization of the method used by Matsubara (1994) who derived a weaklynonlinear formula of the genus statistic in a 3D density field. Afterdescribing the general method, we apply the formalism to a series ofstatistics, including genus statistics, level-crossing statistics, Minkowskifunctionals, and a density extrema statistic, regardless of the dimensions inwhich each statistic is defined. The relation between the Minkowski functionalsand other geometrical statistics is clarified. These statistics can be appliedto several cosmic fields, including 3D density field, 3D velocity field, 2Dprojected density field, and so forth. The results are detailed for secondorder theory of the formalism. The effect of the bias is discussed. Thestatistics of smoothed cosmic fields as functions of rescaled threshold byvolume-fraction are discussed in the framework of second-order perturbationtheory. In CDM-like models, their functional deviations from linear predictionsplotted against the rescaled threshold are generally much smaller than thatplotted against the direct threshold. There is still slight meat-ball shiftagainst rescaled threshold, which is characterized by asymmetry in depths oftroughs in the genus curve. A theory-motivated asymmetry factor in genus curveis proposed.