Power-counting during single-field slow-roll inflation

TL;DR

The paper develops a systematic power-counting framework for inflationary perturbation theory within an effective field theory of gravity, clarifying the domain of validity for semiclassical methods by expressing correlators in terms of the slow-roll parameter $ε$ and the ratio $H/M_p$. By extending GREFT techniques to scalar–metric inflation and introducing a detailed vertex-counting scheme, it shows how observables scale with $H$, $M_p$, $M$, and $ε$, and identifies when higher-derivative operators or loop corrections become relevant. Specialization to single-field slow-roll recovers familiar results for two-point and higher-point correlators, illustrating the hierarchy $B_{ζζ} \,\sim\, H^2/(ε M_p^2)$ and $B_{TT} \,\sim\, H^2/M_p^2$ and outlining the leading bispectrum and trispectrum scalings. The analysis then explores boundary cases, including small sound speed and eternal inflation regimes, highlighting when the standard perturbative expansion may fail and when extended counting or quantum corrections must be incorporated, with implications for the reliability of inflationary predictions in these scenarios.

Abstract

We elucidate the counting of the relevant small parameters in inflationary perturbation theory. Doing this allows for an explicit delineation of the domain of validity of the semi-classical approximation to gravity used in the calculation of inflationary correlation functions. We derive an expression for the dependence of correlation functions of inflationary perturbations on the slow-roll parameter $ε= -\dot{H}/H^2$, as well as on $H/M_p$, where $H$ is the Hubble parameter during inflation. Our analysis is valid for single-field models in which the inflaton can traverse a Planck-sized range in field values and where all slow-roll parameters have approximately the same magnitude. As an application, we use our expression to seek the boundaries of the domain of validity of inflationary perturbation theory for regimes where this is potentially problematic: models with small speed of sound and models allowing eternal inflation.

Figures (1)

• Figure 1: Leading Feynman graphs corresponding to the left hand sides of eq. \ref{['eqn:bispecest1']} and eq. \ref{['eqn:bispecest2']}