Inflationary Infrared Divergences: Geometry of the Reheating Surface vs. delta N Formalism

TL;DR

This work tackles infrared divergences in inflationary curvature perturbations by proposing a simple modification of the δN formalism that accounts for Hubble-scale fluctuations at horizon exit. By treating the Hubble parameter as a function of the perturbed inflaton value, the authors derive a log-enhanced correction to the curvature power spectrum whose coefficient is the second derivative of the leading spectrum with respect to the inflaton, i.e. (1/2) d^2/dphi^2 (N_phi^2 P_chi). The same log term emerges from a purely geometric calculation on the reheating surface, providing a nontrivial cross-check and supporting the modified δN approach. The results clarify the physical origin of IR logs (coordinate vs invariant definitions) and suggest that multi-field models could amplify these effects, motivating further study of higher-point functions and extensions beyond single-field slow-roll.

Abstract

We describe a simple way of incorporating fluctuations of the Hubble scale during the horizon exit of scalar perturbations into the delta N formalism. The dominant effect comes from the dependence of the Hubble scale on low-frequency modes of the inflaton. This modifies the coefficient of the log-enhanced term appearing in the curvature spectrum at second order in field fluctuations. With this modification, the relevant coefficient turns out to be proportional to the second derivative of the tree-level spectrum with respect to the inflaton phi at horizon exit. A logarithm with precisely the same coefficient appears in a calculation of the log-enhancement of the curvature spectrum based purely on the geometry of the reheating surface. We take this agreement as strong support for the proposed implementation of the delta N formalism. Moreover, our analysis makes it apparent that the log-enhancement of the inflationary power-spectrum is indeed physical if this quantity is defined using a global coordinate system on the reheating surface (or any other post-inflationary surface of constant energy density). However, it can be avoided by defining the spectrum using invariant distances on this surface.