Perturbative analysis of adaptive smoothing methods in quantifying large-scale structure

Smoothing operation to make a continuous density held from the observed poi ntlike distribution of galaxies is crucially important for topological or m orphological analysis of the large-scale structure, such as the genus stati stics or the area statistics (or, equivalently, the level-crossing statisti cs). It has been pointed out that the adaptive smoothing filters are more e fficient tools to resolve cosmic structures than the traditional spatially fixed filters. We study weakly nonlinear effects caused by two representati ve adaptive methods often used in smoothed hydrodynamical particle (SPH) si mulations. Using the framework of second-order perturbation theory, we calc ulate the generalized skewness parameters for the adaptive methods in the c ase of fluctuations that are initially power-law fluctuations. Then we appl y the multidimensional Edgeworth expansion method and investigate the weakl y nonlinear evolution of the genus statistics and the area statistics. Isod ensity contour surfaces are often parameterized by the volume fraction of t he regions above a given density threshold. We also discuss this parameteri zation method in perturbative manner.