IR divergence does not affect the gauge-invariant curvature perturbation

TL;DR

This work tackles the problem of infrared divergences in inflationary loop corrections by arguing that such divergences can be gauge artifacts in single-field models. The authors construct a genuine gauge-invariant observable—a two-point function of the spatial curvature evaluated at geodesic-defined coordinates—and show that, at leading order in slow-roll, the infrared divergence disappears in a scale-invariant Bunch-Davies vacuum. They demonstrate how residual gauge freedom, encoded in boundary data and a vector $G_i(x)$, can contaminate naive curvature variables, and argue that a proper gauge-invariant construction avoids these issues. The study highlights the importance of gauge-invariant definitions and the subtleties of initial vacuum specification for reliable predictions, with extensions to higher-order corrections and transverse-traceless modes left for future work.

Abstract

We address the infrared(IR) divergence problem during inflation that appears in the loop corrections to the primordial perturbations. In our previous paper, we claimed that, at least in single field models, the IR divergence is originating from the gauge artifact. Namely, diverging IR corrections should not appear in genuine gauge-invariant observables. We propose here one simple but explicit example of such gauge-invariant quantities. Then, we explicitly calculate such a quantity to find that the IR divergence is absent at the leading order in the slow-roll approximation for the usual scale invariant vacuum state. At the same time we notice that there is a subtle issue on the gauge-invariance in how to specify the initial vacuum state.