No One Loop Back_Reaction In Chaotic Inflation

TL;DR

This work probes whether quantum gravitational back-reaction from scalar perturbations can slow chaotic inflation at one loop by employing an invariant observable ${\cal A} \equiv _c^{-1}$ to measure local expansion. Through a careful slow-roll and long-wavelength analysis, expanded to quadratic order in initial data and expressed on invariant 3-surfaces, the authors find complete cancellation of all potentially secular terms in ${\cal A}_{\rm inv}$, yielding ${\cal A}_{\rm inv} = -\frac{1}{2H^2} + O(\Phi^3)$ with $\Phi$ the accumulated Newtonian potential. They also develop a stochastic-sampling framework to study back-reaction beyond lw/sr, concluding that fluctuations either have definite sign (local case) or are negligible (nonlocal case) at this order. The results indicate that, at one loop and within these approximations, back-reaction does not modify the background expansion, though higher-order effects or relaxing the lw/sr limits could still produce nontrivial dynamics. Overall, the paper strengthens the view that back-reaction must manifest beyond leading approximations or in nonlocal, higher-loop regimes, while providing a robust invariant formalism and a stochastic framework for future exploration.

Abstract

We use an invariant operator to study the quantum gravitational back-reaction to scalar perturbations during chaotic inflation. Our operator is the inverse covariant d'Alembertian expressed as a function of the local value of the inflaton. In the slow roll approximation this observable gives $-1/(2 H^2)$ for an arbitrary homogeneous and isotropic geometry, hence it is a good candidate for measuring the local expansion rate even when the spacetime is not perfectly homogeneous and isotropic. Corrections quadratic in the scalar creation and annihilation operators of the initial value surface are included using the slow-roll and long wavelength approximations. The result is that all terms which could produce a significant secular back-reaction cancel from the operator, before one even takes its expectation value. Although it is not relevant to the current study, we also develop a formalism for using stochastic samples to study back-reaction.