Perturbation theory with dispersion and higher cumulants: Nonlinear regime

We present nonlinear solutions of Vlasov perturbation theory (VPT), describing gravitational clustering of collisionless dark matter with dispersion and higher cumulants induced by orbit crossing. We show that VPT can be cast into a form that is formally analogous to standard perturbation theory (SPT), but including additional perturbation variables, nonlinear interactions, and a more complex propagation. VPT nonlinear kernels have a crucial decoupling property: for fixed total momentum, the kernels become strongly suppressed when any of the individual momenta cross the dispersion scale into the nonlinear regime. This screening of UV modes allows us to compute nonlinear corrections to power spectra even for cosmologies with very blue power-law input spectra, for which SPT diverges. We compare predictions for the density and velocity divergence power spectra as well as the bispectrum at one-loop order to N-body results in a scaling universe with spectral indices -1 <= ns <= thorn 2. We find a good agreement up to the nonlinear scale for all cases, with a reach that increases with the spectral index ns. We discuss the generation of vorticity as well as vector and tensor modes of the velocity dispersion, showing that neglecting vorticity when including dispersion would lead to a violation of momentum conservation. We verify momentum conservation when including vorticity, and compute the vorticity power spectrum at two-loop order, necessary to recover the correct large-scale limit with slope nw 1/4 2. Comparing to our N-body measurements confirms the cross-over from k4 to k2 scaling at large scales. Our results provide a proof-of-principle that perturbative techniques for dark matter clustering can be systematically improved based on the known underlying collisionless dynamics.