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

Abstract

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 ΛCDM 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 ∼26 at z = 3 and ∼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-Ampère-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.

Combining perturbation theories with halo models

“…More precisely, the range of k where there is a noticeable mismatch before the 1-halo term becomes dominant (larger than P L ) is somewhat more extended than at z = 0.35. This agrees with the results of Valageas (2010a), where it was found (within the Zeldovich framework) that the scope of perturbation theory is somewhat greater at higher redshift for CDM power spectra, in the sense that the range where higher order perturbative terms are important (i.e. larger than the non-perturbative correction associated with shell crossing effects) is wider and that the perturbative expansion makes sense up to higher orders.…”

“…This means that on these scales the departure from the linear power is not yet due to the non-perturbative contribution associated with the 1-halo term, but to higher order perturbative terms. This agrees with the results of Valageas (2010a), based on the Zeldovich dynamics, which show that many orders of perturbation theory are relevant before the power spectrum is dominated by the non-perturbative contribution associated with shell crossing or halo formation (typically one can go up to order P 9 L at z = 0 and P 66 L at z = 3 for a ΛCDM cosmology, at least for this simpler case). This motivates the use of perturbative approaches that include higher order contributions.…”

“…In agreement with the discussion in Sect. 5 and with Valageas (2010a), this expresses the fact that for CDM power spectra the scope of perturbative expansions is somewhat broader at higher z, in the sense that the range of k where higher order terms play a role is wider. This corresponds to the interval [k 1loop , k s.c. ] where terms beyond linear order are already significant, as compared with the linear power P L , but still larger than the non-perturbative correction associated for instance with shell crossing effects.…”

“…There, the mapping x(q) is given by the linear displacement field Ψ L , as x(q) = q + Ψ L (q), so that for Gaussian initial conditions (i.e. Ψ L (q) is Gaussian) one can easily compute the average (6) and derive an explicit analytic expression for the density power spectrum (Schneider & Bartelmann 1995;Taylor & Hamilton 1996;Valageas 2010a). The second term in Eq.…”

“…In Valageas (2010a) such a decomposition between perturbative and non-perturbative contributions to the density power spectrum was obtained within the framework of the Zeldovich dynamics. More precisely, a "sticky model" that only differs from the Zeldovich dynamics beyond shell crossing was introduced.…”