A Leading Logarithm Approximation for Inflationary Quantum Field Theory

TL;DR

This work develops a stochastic-inflation framework to sum leading infrared logarithms in inflationary quantum field theory, generalizing Starobinskiĭ's approach to theories with derivative interactions and constrained fields. Using two simple scalar models, it derives practical rules for constructing Langevin-type equations that reproduce $\\ln(a)$- and $\\ln^2(a)$-level growth of observables, and it demonstrates how to obtain closed-form stochastic equations and even operator solutions that reveal a running field-strength renormalization. The results show that derivative couplings and constraints can be integrated into the leading-log formalism, providing a nonperturbative handle on infrared effects during inflation and pointing toward applications to quantum gravity. Together, these insights advance the program of understanding backreaction and long-time dynamics in inflationary backgrounds via nonperturbative stochastic methods.

Abstract

During inflation explicit perturbative computations of quantum field theories which contain massless, non-conformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskiĭ's technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features.