  Standard cosmological perturbation theory (SPT) for the Large Scale Structure
(LSS) of the Universe fails at small scales (UV) due to strong nonlinearities
and to multistreaming effects. In Pietroni et al. 2011 a new framework was
proposed in which the large scales (IR) are treated perturbatively while the
information on the UV, mainly small scale velocity dispersion, is obtained by
nonlinear methods like N-body simulations. Here we develop this approach,
showing that it is possible to reproduce the fully nonlinear power spectrum
(PS) by combining a simple (and fast) 1-loop computation for the IR scales and
the measurement of a single, dominant, correlator from N-body simulations for
the UV ones. We measure this correlator for a suite of seven different
cosmologies, and we show that its inclusion in our perturbation scheme
reproduces the fully non-linear PS with percent level accuracy, for wave
numbers up to $k\sim 0.4\, h~{\rm Mpc^{-1}}$ down to $z=0$. We then show that,
once this correlator has been measured in a given cosmology, there is no need
to run a new simulation for a different cosmology in the suite. Indeed, by
rescaling this correlator by a proper function computable in SPT, the
reconstruction procedure works also for the other cosmologies and for all
redshifts, with comparable accuracy. Finally, we clarify the relation of this
approach to the Effective Field Theory methods recently proposed in the LSS
context.
