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, with some possibly important differences. We conclude by a preliminary discussion of the nature of the full solutions of the equation and their significance.
Highlights
The Large-Scale Structure (LSS) of the universe grows from an initial nearly scale-invariant spectrum of Gaussian fluctuations due to a gravitational instability
The Lagrangian resummation theory (LRT) [10] reproduces the standard perturbation theory (SPT) power spectrum at the lowest nontrivial order and yields a non-perturbative prediction for the power spectrum that corresponds to resumming an infinite set of terms in SPT
The goal of this paper is to present a novel approach to study the non-linear evolution of matter perturbations which is entirely based on the gravitational potential
Summary
The Large-Scale Structure (LSS) of the universe grows from an initial nearly scale-invariant spectrum of Gaussian fluctuations due to a gravitational instability. The matter distribution of our universe today is well described on large scales by linear perturbation theory about a homogeneous and isotropic background. On length scales below about 10 Mpc the dynamics of matter is highly non-linear, so to describe its evolution in a qualitative way one has to resort numerical N-body simulations. At intermediate (quasi-linear) scales, the evolution of matter may be described analytically by extending standard perturbation theory (SPT) [1]. The n-th order term of this series grows as the n-th power of the scale factor a (for a pressureless fluid) which affects its convergence properties and limits its applicability. Qualitative comparisons to simulations have shown that the domain of applicability of second-order perturbation theory (in the linear power spectrum) is limited, at redshift z = 0, to wavenumbers of about 0.05 h/Mpc
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