Impact of shell crossing and scope of perturbative approaches, in real and redshift space

Aims. We study the effect of nonperturbative corrections associated with the behavior of particles after shell crossing on the matter power spectrum. We compare their amplitude with the perturbative terms that can be obtained within the fluid description of the system, to estimate the range of scales where such perturbative approaches are relevant. Methods. We use the simple Zeldovich dynamics as a benchmark, as it allows the exact computation of the full nonlinear power spectrum and of perturbative terms at all orders. Then, we introduce a sticky model that coincides with the Zeldovich dynamics before shell crossing but shows a different behavior afterwards. Thus, their power spectra only differ in their nonperturbative terms. We consider both the real-space and redshift-space power spectra. Results. We find that the potential of perturbative schemes is greater at higher redshift for a ACDM cosmology. For the real-space power spectrum, one can go up to order 66 of perturbation theory at z = 3, and to order 9 at z = 0, before the nonperturbative correction surpasses the perturbative correction of that order. This allows us to increase the upper bound on k where systematic theoretical predictions may be obtained by perturbative schemes, beyond the linear regime, by a factor similar to 26 at z = 3 and similar to 6.5 at z = 0. This provides a strong motivation to study perturbative resummation schemes, especially at high redshifts z >= 1. In the context of cosmological reconstruction methods, the Monge-Ampere-Kantorovich scheme appears to be close to optimal at z = 0. There also seems to be little room for improvement over current reconstruction methods of the baryon acoustic oscillations at z = 0. This can be understood from the small number of perturbative terms that are relevant at z = 0, before nonperturbative corrections dominate. We also point out that the rise of the power spectrum on the transition scale to the nonlinear regime strongly depends on the behavior of the system after shell crossing. We find similar results for the redshift-space power spectrum, with characteristic wavenumbers that are shifted to lower values as redshift-space distortions amplify higher order terms of the perturbative expansions while decreasing the resummed nonlinear power at high k.