Tensor mode backreaction during slow-roll inflation

We consider the backreaction of the long wavelength tensor modes produced during a slow-roll inflationary regime driven by a single scalar field in a spatially flat Friedmann-Lemaître-Robertson-Walker background geometry. We investigate the effects on nonlocal observables such as the effective (averaged) expansion rate and equation of state at second order in cosmological perturbation theory. The coupling between scalar and tensor perturbations induces at second-order new tensor backreaction terms beyond the one already present in a de Sitter background. We analyze in detail the effects seen by the class of observers comoving with the inflaton field (taken as a clock) and the class of free-falling observers. In both cases the quantum backreaction is at least $1/\epsilon$ (with $\epsilon$ the slow-roll parameter) larger than the one which can be naively inferred from a de Sitter background. In particular, we compute the effect for a free massive inflaton model and obtain in both cases a quantum correction on the background expansion rate of the order of $H^4/(m^2{M}_{\text{Pl}}^2)$. A short discussion on the issue of the breakdown of perturbation theory is given.