On the divergences of inflationary superhorizon perturbations

TL;DR

This work analyzes infrared divergences in inflationary perturbation theory using stochastic inflation and the ΔN formalism. Eternal inflation regulates IR fluctuations, yielding finite one-point correlators for the inflaton and curvature perturbation, while a renormalization-group–like running describes how correlator coefficients transform when changing the infrared cutoff. At one loop, the authors show that infrared effects can be absorbed into background redefinitions, making observable two-point differences RG-invariant and enabling consistent predictions across scales. A concrete free-field example demonstrates an infrared-finite, renormalization-point–independent relation between two- and three-point functions and reveals a logarithmic separation dependence that remains cutoff-independent. The results provide a principled framework for finite, scale-consistent predictions of curvature perturbations and clarify the role of the IR cutoff in inflationary cosmology.

Abstract

We discuss the infrared divergences that appear to plague cosmological perturbation theory. We show that within the stochastic framework they are regulated by eternal inflation so that the theory predicts finite fluctuations. Using the $ΔN$ formalism to one loop, we demonstrate that the infrared modes can be absorbed into additive constants and the coefficients of the diagrammatic expansion for the connected parts of two and three-point functions of the curvature perturbation. As a result, the use of any infrared cutoff below the scale of eternal inflation is permitted, provided that the background fields are appropriately redefined. The natural choice for the infrared cutoff would of course be the present horizon; other choices manifest themselves in the running of the correlators. We also demonstrate that it is possible to define observables that are renormalization group invariant. As an example, we derive a non-perturbative, infrared finite and renormalization point independent relation between the two-point correlators of the curvature perturbation for the case of the free single field.