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

TL;DR

This paper introduces a modification to the delta N formalism to incorporate Hubble scale fluctuations during horizon exit, revealing a geometric origin for infrared divergences in inflationary spectra and emphasizing the importance of coordinate choices.

Contribution

It proposes a simple method to include Hubble scale fluctuations in the delta N formalism, linking the log-enhancement to the geometry of the reheating surface and clarifying the physicality of infrared divergences.

Findings

The modified delta N formalism accounts for Hubble scale fluctuations.

The log-enhanced term coefficient relates to the second derivative of the spectrum.

The geometric calculation confirms the formalism's predictions.

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.