On energy-momentum tensor for gravitational waves in $f(R)$ gravity

TL;DR

This work extends the Isaacson high-frequency averaging formalism to vacuum $f(R)$ gravity on a de Sitter background by reformulating the theory in scalar-tensor form and deriving linear perturbations. It decouples tensor and scalar perturbations on dS, derives the second-order source $J^{(2)}_{ik}$, and computes the Brill-Hartle averaged energy-momentum tensor for gravitational waves on the dS background. The authors then implement the Isaacson back-reaction framework within $f(R)$, analyzing regimes of negligible and significant back-reaction and demonstrating that leading-order energy-momentum expressions closely mirror GR-like results with a modified prefactor $f'_0$. The results generalize prior flat-background analyses and provide a consistent methodology for back-reaction studies in cosmologically viable $f(R)$ theories.

Abstract

The classical Isaacson's procedure for describing back-reaction of the averaged energy-momentum for high frequency gravitational waves is generalized to the $f(R)$ gravity case. From the beginning it is assumed that an initial background could be arbitrary one. Then, we restrict the background to be de Sitter, which is a novelty regarding the study of a back-reaction in $f(R)$ gravity. Consideration of the de Sitter space as a background spacetime allows us to provide the averaging procedure completely. Using the results on the de Sitter space and generalizing the Isaacson procedure, we construct the averaged energy-momentum on an additionally curved (averaged) background. Consistency tests for de Sitter spacetime are performed both at the background and perturbative regimes. Our results generalize previous studies in which the authors consider the flat (Minkowski) spacetime as the initial background.