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

Smoothing operation to make a continuous density held from the observed pointlike distribution of galaxies is crucially important for topological or morphological analysis of the large-scale structure, such as the genus statistics or the area statistics (or, equivalently, the level-crossing statistics). It has been pointed out that the adaptive smoothing filters are more efficient tools to resolve cosmic structures than the traditional spatially fixed filters. We study weakly nonlinear effects caused by two representative adaptive methods often used in smoothed hydrodynamical particle (SPH) simulations. Using the framework of second-order perturbation theory, we calculate the generalized skewness parameters for the adaptive methods in the case of fluctuations that are initially power-law fluctuations. Then we apply the multidimensional Edgeworth expansion method and investigate the weakly nonlinear evolution of the genus statistics and the area statistics. Isodensity contour surfaces are often parameterized by the volume fraction of the regions above a given density threshold. We also discuss this parameterization method in perturbative manner.