Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations

TL;DR

The paper addresses nonlinear cosmological perturbations and the accuracy of the matter power spectrum for future surveys. It develops a BBGKY-like differential equation framework that evolves the two- and three-point correlators, with the four-point connected function (trispectrum) set to zero, thereby resumming an infinite class of perturbative corrections. Cosmology enters through a time-dependent $\Omega$ matrix, allowing straightforward extension to general models (e.g., modified gravity, massive neutrinos) without relying on propagator-based RG formalisms. Numerical BAO results for $\Lambda$CDM and $w=-0.8$ show good agreement with $N$-body simulations up to $k \sim 0.3-0.35\,h/\mathrm{Mpc}$, and velocity spectra are less nonlinear, indicating the method's practical utility and potential for further improvements by including the trispectrum.

Abstract

Nonlinear effects are crucial in order to compute the cosmological matter power spectrum to the accuracy required by future generation surveys. Here, a new approach is presented, in which the power spectrum, the bispectrum and higher order correlations, are obtained -- at any redshift and for any momentum scale -- by integrating a system of differential equations. The method is similar to the familiar BBGKY hierarchy. Truncating at the level of the trispectrum, the solution of the equations corresponds to the summation of an infinite class of perturbative corrections. Compared to other resummation frameworks, the scheme discussed here is particularly suited to cosmologies other than LambdaCDM, such as those based on modifications of gravity and those containing massive neutrinos. As a first application, we compute the Baryonic Acoustic Oscillation feature of the power spectrum, and compare the results with perturbation theory, the halo model, and N-body simulations. The density-velocity and velocity-velocity power spectra are also computed, showing that they are much less contaminated by nonlinearities than the density-density one. The approach can be seen as a particular formulation of the renormalization group, in which time is the flow parameter.

Figures (7)

• Figure 1: The $O(\gamma^0)$ contributions to the power spectrum and the bispectrum.
• Figure 2: The $O(\gamma)$ contributions to the bispectrum and the $O(\gamma^2)$ (first line) and $O(\gamma)$ (second line) ones to power spectrum.
• Figure 3: $O(\gamma^3)$ corrections to the BS included (left) and not included (right) in the approximation scheme considered in this paper, namely $Q_{abcd}=0$.
• Figure 4: Power spectra at redshift $z=1$ (divided by a smooth one). The continuous line is the result of the present paper, compared with linear theory (dotted), 1-loop PT (dash-dotted), the halo approach of ref. S03 (dashed). The dots with error bars are taken from the $N$-body simulationd of ref. JK06. The background cosmology is a spatially flat $\Lambda$CDM model with $\Omega_\Lambda^0=0.73$, $\Omega_b^0=0.043$, $h=0.7$, $n_s=1$, $\sigma_8 = 0.8$.
• Figure 5: The density-density ($P_{\delta\delta}$), density-velocity ($P_{\delta\theta}$), and velocity-velocity ($P_{\theta\theta}$) power spectra at $z=1$. The dotted line is linear theory, and the dash-dotted one the linear theory multiplied by $\exp(-k^2 \sigma_v^2(e^\eta-1)^2)$.