Recovery of the Cosmological Peculiar Velocity from the Density Field in   the Weakly Nonlinear Regime
  Using third-order perturbation theory, we derive a relation between the mean divergence of the peculiar velocity given density and the density itself. Our calculations assume Gaussian initial conditions and are valid for Gaussian filtering of the evolved density and velocity fields. The mean velocity divergence turns out to be a third-order polynomial in the density contrast. We test the power spectrum dependence of the coefficients of the polynomial for scale-free and standard CDM spectra and find it rather weak. Over scales larger than about 5 megaparsecs, the scatter in the relation is small compared to that introduced by random errors in the observed density and velocity fields. The relation can be useful for recovering the peculiar velocity from the associated density field, and also for non-linear analyses of the anisotropies of structure in redshift surveys. 
Michal Chodorowski (1), Ewa Lokas (1), Agnieszka Pollo (1) and Adi   Nusser (2) ((1) Copernicus Center, (2) Max Planck)