Warm dark matter at small scales: peculiar velocities and phase space density

We study the scale and redshift dependence of the power spectra for density perturbations and peculiar velocities, and the evolution of a coarse grained phase space density for (WDM) particles that decoupled during the radiation dominated stage. The (WDM) corrections are obtained in a perturbative expansion valid in the range of redshifts at which N-body simulations set up initial conditions, and for a wide range of scales. The redshift dependence is determined by the kurtosis $\beta_2$ of the distribution function at decoupling. At large redshift there is an enhancement of peculiar velocities for $\beta_2 > 1$ that contributes to free streaming and leads to further suppression of the matter power spectrum and an enhancement of the peculiar velocity autocorrelation function at scales smaller than the free streaming scale. Statistical fluctuations of peculiar velocities are also suppressed on these scales by the same effect. In the linearized approximation, the coarse grained phase space density features redshift dependent (WDM) corrections from gravitational perturbations determined by the power spectrum of density perturbations and $\beta_2$. For $\beta_2 > 25/21$ it \emph{grows logarithmically} with the scale factor as a consequence of the suppression of statistical fluctuations. Two specific models for WDM are studied in detail. The (WDM) corrections relax the bounds on the mass.