Evolution Equation for Non-linear Cosmological Perturbations

TL;DR

This paper introduces a new gravitational potential-based equation to analyze non-linear cosmological matter perturbations, successfully reproducing key features of existing advanced theories and simplifying the study of small-scale suppression effects.

Contribution

A novel, potential-based evolution equation for non-linear cosmological perturbations that aligns with and simplifies existing renormalized perturbation theories.

Findings

Perturbative solutions match standard results.

Mean field approximation reproduces exponential suppression.

Equation captures key non-linear features.

Abstract

We present a novel approach, based entirely on the gravitational potential, for studying the evolution of non-linear cosmological matter perturbations. Starting from the perturbed Einstein equations, we integrate out the non-relativistic degrees of freedom of the cosmic fluid and obtain a single closed equation for the gravitational potential. We then verify the validity of the new equation by comparing its approximate solutions to known results in the theory of non-linear cosmological perturbations. First, we show explicitly that the perturbative solution of our equation matches the standard perturbative solutions. Next, using the mean field approximation to the equation, we show that its solution reproduces in a simple way the exponential suppression of the non-linear propagator on small scales due to the velocity dispersion. Our approach can therefore reproduce the main features of the renormalized perturbation theory and (time)-renormalization group approaches to the study of non-linear cosmological perturbations. We conclude by a preliminary discussion of the nature of the full solutions of the equation and their significance.