On Isaacson's Procedure in $f(R)$-Gravity with Second Order Metric Perturbations

Abstract

In the recent paper arXiv:2504.02253 we generalized classical Isaacson's procedure for describing back-reaction of the averaged energy-momentum for high frequency gravitational waves to $f(R)$ gravity case with a choice of the de Sitter background. To describe this second order effect we used standard first order decomposition of metric tensor. In this paper, we take into account possible contribution to this process from second order term in metric tensor decomposition. We find that on the de Sitter background all possible contributions from this term are strongly suppressed and therefore results obtained in the paper arXiv:2504.02253 are left unchanged.