A new method for accurate estimation of velocity field statistics

We introduce two new methods to obtain reliable velocity field statistics from N-body simulations, or indeed from any general density and velocity fluctuation field sampled by discrete points, These methods, the Voronoi tessellation method and Delaunay tessellation method, are based on the use of the Voronoi and Delaunay tessellations of the point distribution defined by the locations at which the velocity held is sampled. In the Voronoi method the velocity is supposed to be uniform within the Voronoi polyhedra, whereas the Delaunay method constructs a velocity field by linear interpolation between the four velocities at the locations defining each Delaunay tetrahedron. The most important advantage of these methods is that they provide an optimal estimator for determining the statistics of volume-averaged quantities, as opposed to the available numerical methods that mainly concern mass-averaged quantities. As the major share of the related analytical work on velocity field statistics has focused on volume-averaged quantities, the availability of appropriate numerical estimators is of crucial importance for checking the validity of the analytical perturbation calculations. In addition, it allows us to study the statistics of the velocity field in the highly non-linear clustering regime. Specifically we describe in this paper how to estimate, in both the Voronoi and the Delaunay methods, the value of the volume-averaged expansion scalar theta = H(-1)del .upsilon (the divergence of the peculiar velocity, expressed in units of the Hubble constant H), as well as the value of the shear and the vorticity of the velocity field, at an arbitrary position. The evaluation of these quantities on a regular grid leads to an optimal estimator for determining the probability distribution function (PDF) of the volume-averaged expansion scalar, shear and vorticity. Although in most cases both the Voronoi and the Delaunay methods lead to equally good velocity field estimates, the Delaunay method may be slightly preferable. In particular it performs considerably better at small radii. Note that it is more CPU-time intensive while its requirement for memory space is almost a factor 8 lower than the Voronoi method. As a test we here apply our estimator to that of an N-body simulation of such structure formation scenarios. The PDF:; 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 a forthcoming paper we will apply the newly developed tool to a plethora of structure formation scenarios, of both Gaussian and non-Gaussian initial conditions, in order to see to what extent the velocity field PDFs are sensitive discriminators, highlighting fundamental physical differences between the scenarios.