Generation of fluctuations during inflation: comparison of stochastic and field-theoretic approaches

TL;DR

This work proves that stochastic inflation and standard perturbative QFT in curved space-time yield identical leading-order predictions for light scalar fluctuations during inflation, for both test fields and inflaton-driven cases, within slow-roll. It clarifies that using the number of e-folds $N$ as the time variable is essential when computing gauge-invariant inflaton and metric fluctuations, establishing exact equivalence with QFT results including higher-loop structures (up to four loops in de Sitter). The authors develop a renormalization framework via adiabatic subtraction, show IR dynamics dominate over UV, and explore consequences of particle production such as possible second inflationary stages and exacerbation of the moduli problem. Together, these results validate the stochastic approach as a reliable, nonperturbative tool for inflationary fluctuations and illuminate the interplay between stochastic and field-theoretic descriptions in realistic, slowly evolving backgrounds.

Abstract

We prove that the stochastic and standard field-theoretical approaches produce exactly the same results for the amount of light massive scalar field fluctuations generated during inflation in the leading order of the slow-roll approximation. This is true both in the case for which this field is a test one and inflation is driven by another field, and the case for which the field plays the role of inflaton itself. In the latter case, in order to calculate the average of the mean square of the gauge-invariant inflaton fluctuation, the logarithm of the scale factor $a$ has to be used as the time variable in the Fokker-Planck equation in the stochastic approach. The implications of particle production during inflation for the second stage of inflation and for the moduli problem are also discussed. The case of a massless self-interacting test scalar field in a de Sitter background with a zero initial renormalized mean square is also considered in order to show how the stochastic approach can easily produce results corresponding to diagrams with an arbitrary number of scalar field loops in the field-theoretical approach (explicit results up to 4 loops inclusive are presented).