Perturbative Analysis of Adaptive Smoothing Methods in Quantifying Large‐Scale Structure

Abstract

Smoothing operation to make a continuous density Ðeld 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 Ðlters are more efficient tools to resolve cosmic structures than the traditional spatially Ðxed Ðlters. We study weakly nonlinear e †ects 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 Ñuctuations that are initially power-law Ñuctuations. 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.

Non-Gaussian Minkowski functionals and extrema counts in redshift space

“…Indeed, the fraction of volume occupied by a data set is readily available from the data, whereas specifying ν requires prior knowledge of the variance σ, a quantity which is typically the main unknown for such investigations. Following Gott et al (1987); Gott (1988); Gott et al (1989), Seto (2000) and Matsubara (2003), let us therefore introduce the threshold ν f = √ 2 erfc −1 (2fV ) to be used as an observable alternative to ν. The mapping between ν and ν f is monotonic and is implicitly given by the identity…”