Publication : t95/076

Abstract:
We introduce a new method for estimating the value of the expansion scalar $\theta \equiv H^{-1} {\bf \nabla \cdot v}$ -- the divergence of the peculiar velocity, expressed in units of the Hubble constant, $H\equiv 100h\,\rm{km s^{-1} Mpc^{-1}}$ -- in a general density and velocity fluctuation field sampled by discrete points. The method is based on the use of the Voronoi and Delaunay tessellations of these point distributions. The principal advantage of using these tools is that it provides us with an optimal estimator for determining the probability distribution function (PDF) of volume-averaged quantities; the latter are precisely the quantities on which the major share of the related analytical work has focussed. The PDFs in structure formation scenarios starting from Gaussian density and velocity fluctuation fields have been calculated analytically. As a test we here apply our estimator on an N-body simulation of such structure formation scenarios. The PDFs determined from the simulations agree very well with the analytical predictions. An important benefit of the present method is that, unlike previous methods, it is capable of probing accurately the velocity field statistics in regions of very low density, which in N-body simulations are typically sparsely sampled. In addition, we extend our method to estimating the shear and rotation of the velocity fields, and present the resulting PDFs of these quantities for the test simulation. In a forthcoming paper we will apply the newly developed tool to a plethora of structure formation scenarios, both of Gaussian and non-Gaussian initial conditions, in order to see in how far the velocity field PDFs are sensitive discriminators, highlighting fundamental physical differences between the scenarios.

publi.pdf