A NEW METHOD FOR ACCURATE ESTIMATION OF VELOCITY-FIELD STATISTICS

We introduce two new methods to obtain reliable velocity field statist ics from N-body simulations, or indeed from any general density and ve locity fluctuation field sampled by discrete points, These methods, th e Voronoi tessellation method and Delaunay tessellation method, are ba sed on the use of the Voronoi and Delaunay tessellations of the point distribution defined by the locations at which the velocity held is sa mpled. In the Voronoi method the velocity is supposed to be uniform wi thin the Voronoi polyhedra, whereas the Delaunay method constructs a v elocity field by linear interpolation between the four velocities at t he locations defining each Delaunay tetrahedron. The most important ad vantage 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 q uantities. As the major share of the related analytical work on veloci ty field statistics has focused on volume-averaged quantities, the ava ilability 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 fi eld in the highly non-linear clustering regime. Specifically we descri be 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 an d the vorticity of the velocity field, at an arbitrary position. The e valuation of these quantities on a regular grid leads to an optimal es timator for determining the probability distribution function (PDF) of the volume-averaged expansion scalar, shear and vorticity. Although i n most cases both the Voronoi and the Delaunay methods lead to equally good velocity field estimates, the Delaunay method may be slightly pr eferable. In particular it performs considerably better at small radii . Note that it is more CPU-time intensive while its requirement for me mory space is almost a factor 8 lower than the Voronoi method. As a te st we here apply our estimator to that of an N-body simulation of such structure formation scenarios. The PDF:; determined from the simulati ons agree very well with the analytical predictions. An important bene fit of the present method is that, unlike previous methods, it is capa ble of probing accurately the velocity field statistics in regions of very low density, which in N-body simulations are typically sparsely s ampled, In a forthcoming paper we will apply the newly developed tool to a plethora of structure formation scenarios, of both Gaussian and n on-Gaussian initial conditions, in order to see to what extent the vel ocity field PDFs are sensitive discriminators, highlighting fundamenta l physical differences between the scenarios.