The Energy-Momentum Tensor for Cosmological Perturbations

TL;DR

This work introduces a gauge-invariant effective energy-momentum tensor (EMT) for cosmological perturbations to quantify back-reaction on a FRW background. It develops a gauge-invariant formulation using $Q=e^{{\\cal L}_X}q$, yielding $G_{\\ u\\mu}(Q_0)=8\\pi G[T_{\\nu\\mu}(Q_0)+\\tau_{\\nu\\mu}(\\delta Q)]$, with explicit EMTs for scalar and tensor perturbations. In both long- and short-wavelength limits, long-wavelength scalar and tensor perturbations produce negative energy densities, and for inflationary scalar perturbations the EOS is de Sitter-like ($p\\approx -\\rho$). Applied to chaotic inflation with $V(\\varphi)=\\tfrac{1}{2}m^2\\varphi^2$, back-reaction can become dynamically important before the end of inflation, potentially shortening the inflationary period, and the results underscore the significance of gauge-invariant back-reaction in early-Universe cosmology.

Abstract

We study the effective energy-momentum tensor (EMT) for cosmological perturbations and formulate the gravitational back-reaction problem in a gauge invariant manner. We analyze the explicit expressions for the EMT in the cases of scalar metric fluctuations and of gravitational waves and derive the resulting equations of state. The formalism is applied to investigate the back-reaction effects in chaotic inflation. We find that for long wavelength scalar and tensor perturbations, the effective energy density is negative and thus counteracts any pre-existing cosmological constant. For scalar perturbations during an epoch of inflation, the equation of state is de Sitter-like.