Cosmological scalar and tensor perturbations with a scalar field: quadratic-order effective energy-momentum tensor

TL;DR

This work analyzes the back-reaction of inflationary perturbations by constructing a second-order effective energy-momentum tensor $\hat{\tau}_{\mu\nu}$ from quadratic combinations of scalar and tensor metric perturbations and their coupling to the inflaton. It decomposes 2EMT into scalar-only $\mathcal{S}$, tensor-only $\mathcal{T}$, and scalar-tensor coupled $\mathcal{ST}$ parts, and evaluates their behavior in slow-roll inflation under three gauges (longitudinal, spatially flat, and comoving) in both long- and short-wavelength limits. The results show a wavelength-dependent pattern: in the long-wavelength regime, tensor and ST contributions can dominate, while in the short-wavelength regime the scalar part is dominant; the ST terms can contribute nontrivially, including off-diagonal shear-like components, with strong gauge dependence. The findings emphasize that back-reaction from perturbations is non-negligible and gauge-sensitive, motivating further work on additional gauges, potential observational signatures, and extensions to post-inflationary dynamics or multi-field scenarios.

Abstract

We introduce the scalar and tensor modes of the gravitational perturbation in the presence of a scalar field which describes inflation. We investigate the back-reaction of the perturbations to the background by studying the effective energy-momentum tensor (2EMT) which is the second order constructed by the quadratic terms of the linear perturbations. 2EMT is gauge dependent due to the scalar mode. We obtain 2EMT in the slow-roll stage of inflation, and get its cosmological expressions in three (longitudinal, spatially flat, and comoving) gauge conditions. We find that the pure scalar-mode part in 2EMT is stronger in the short-wavelength limit, while the parts involved with the tensor mode (the pure tensor-mode part and the scalar-tensor coupled part) are stronger in the long-wavelength limit.