Analytical First Crossing Distributions for Curved Barriers in Gaussian Density Fields

We study first crossing distributions of curved boundaries in Gaussian density fields. We construct the integral equation for the k-sharp filter, which is a Volterra integral equation of the second kind, and we examine both exact numerical solutions and analytical approximations. We study the case of a k-sharp filter and we have an initial-boundary value problem which admits simple analytical solutions, at least for the widely used in cosmology ellipsoidal barrier, that approximate the exact solution better than any of the models that have been proposed in the literature. Finally, we present multiplicity functions for dark matter structures and we show that for the proper choice of the parameters of the ellipsoidal barrier these are in a very satisfactory agreement with the results of N-body simulations.