Coarse-grained cosmological perturbation theory

Semi-analytical methods, based on Eulerian perturbation theory, are a promising tool to follow the time evolution of cosmological perturbations at small redshifts and at mildly nonlinear scales. All these schemes are based on two approximations: the existence of a smoothing scale and the single-stream approximation, where velocity dispersion of the dark matter fluid, as well as higher moments of the particle distributions, are neglected. Despite being widely recognized, these two assumptions are, in principle, incompatible, since any finite smoothing scale gives rise to velocity dispersion and higher moments at larger scales. We describe a new approach to perturbation theory, where the Vlasov and fluid equations are derived in presence of a finite coarse-graining scale: this allows a clear separation between long and short distance modes and leads to a hybrid approach where the former are treated perturbatively and the effect of the latter is encoded in external source terms for velocity, velocity dispersion, and all the higher order moments, which can be computed from N-body simulations. We apply the coarse-grained perturbation theory to the computation of the power spectrum and the cross-spectrum between density and velocity dispersion, and compare the results with N-body simulations, finding good agreement.