Back Reaction And Local Cosmological Expansion Rate

TL;DR

The paper investigates whether cosmological perturbations back-react on the local expansion rate in general relativity by defining a local observable through $\Theta = u^{\mu}{}_{;\mu}$ with $\dot{S}/S = \frac{1}{3}\Theta$ and computing to second order in perturbations. It analyzes both a matter-dominated background and a single-field inflationary scenario, showing that the leading infrared back-reaction terms cancel when the observable is evaluated on surfaces of fixed scalar field (a physical clock) but not when evaluated at fixed background time, highlighting gauge-invariance concerns. The results support the view that back-reaction depends on the chosen observable and clock, aligning with prior work and suggesting that two-field models could exhibit nonzero IR back-reaction. The work provides a framework to assess infrared effects in cosmology and points to future studies of multi-field scenarios and higher-order IR terms.

Abstract

We calculate the back reaction of cosmological perturbations on a general relativistic variable which measures the local expansion rate of the Universe. Specifically, we consider a cosmological model in which matter is described by a single field. We analyze back reaction both in a matter dominated Universe and in a phase of scalar field-driven chaotic inflation. In both cases, we find that the leading infrared terms contributing to the back reaction vanish when the local expansion rate is measured at a fixed value of the matter field which is used as a clock, whereas they do not appear to vanish if the expansion rate is evaluated at a fixed value of the background time. We discuss possible implications for more realistic models with a more complicated matter sector.