Non-linear Density-velocity relation from phase space dynamics.

 The fractional overdensity and the peculiar velocity are the two main variables that characterize the large scale inhomogeneities of our Universe. In linear theory the density and the velocity divergence are proportional but this relation breaks down in the non-linear regime. We obtain a non-linear extension of this relation by examining the dynamics of the perturbations in a joint density-velocity divergence phase space. The method uses spherical
collapse as a proxy to model the non-linear regime; however it does not rely on the knowledge of exact solutions and is therefore generalizable. We show that this relation acts like an attractor in the phase space of perturbations and we demonstrate how it can be exploited to remove degeneracies in cosmological parameters. Finally, we will discuss and extension of these ideas to the ellipsoidal collapse model and obtain the marginal pdfs of
density and velocity fields.