Back Reaction of Cosmological Perturbations

TL;DR

The paper develops a gauge-invariant framework to quantify how cosmological perturbations back-react on the background through an effective energy-momentum tensor $\tau_{\mu\nu}$. It shows that in inflationary spacetimes the dominant infrared scalar modes yield a negative cosmological-constant–like contribution with $p_{br} \approx -\rho_{br}$ and $\rho_{br} < 0$ that grows in time. Applying the formalism to chaotic inflation with $V(\varphi) = \tfrac{1}{2} m^2 \varphi^2$, the authors derive a condition under which back-reaction can shorten inflation and provide a concrete estimate for $\rho_{br}$ relative to the background density, highlighting a potential connection to stochastic, self-reproducing scenarios. They speculate that the accumulated back-reaction could dynamically relax the cosmological constant, potentially driving $\Omega_\Lambda$ toward order unity, while acknowledging the speculative nature and the need to extend the analysis beyond the homogeneous background and to higher-order gauge invariance.

Abstract

The presence of cosmological perturbations affects the background metric and matter configuration in which the perturbations propagate. This effect, studied a long time ago for gravitational waves, also is operational for scalar gravitational fluctuations, inhomogeneities which are believed to be more important in inflationary cosmology. The back-reaction of fluctuations can be described by an effective energy-momentum tensor. The issue of coordinate invariance makes the analysis more complicated for scalar fluctuations than for gravitational waves. We show that the back-reaction of fluctuations can be described in a diffeomorphism-invariant way. In an inflationary cosmology, the back-reaction is dominated by infrared modes. We show that these modes give a contribution to the effective energy-momentum tensor of the form of a negative cosmological constant whose absolute value grows in time. We speculate that this may lead to a self-regulating dynamical relaxation mechanism for the cosmological constant. This scenario would naturally lead to a finite remnant cosmological constant with a magnitude corresponding to $Ω_Λ \sim 1$.