Stochastic growth of quantum fluctuations during slow-roll inflation

TL;DR

This paper analyzes the growth of quantum fluctuations during slow-roll inflation, comparing test fields with inflaton fluctuations, and introduces a stochastic approach using e-folds as the time variable to derive bounds in multi-field models.

Contribution

It provides a detailed comparison of quantum fluctuation growth for various test fields and inflaton, and develops a stochastic formalism with e-folds as time for inflationary models.

Findings

Inflaton fluctuations grow faster than test fields with non-negative mass in most models.

Hybrid inflation can allow test fields to dominate inflaton fluctuations.

The stochastic formalism with e-folds as time yields bounds in two-field inflation models.

Abstract

We compute the growth of the mean square of quantum fluctuations of test fields with small effective mass during a slowly changing, nearly de Sitter stage which took place in different inflationary models. We consider a minimally coupled scalar with a small mass, a modulus with an effective mass $ \propto H^2$ (with $H$ as the Hubble parameter) and a massless non-minimally coupled scalar in the test field approximation and compare the growth of their relative mean square with the one of gauge-invariant inflaton fluctuations. We find that in most of the single field inflationary models the mean square gauge invariant inflaton fluctuation grows {\em faster} than any test field with a non-negative effective mass. Hybrid inflationary models can be an exception: the mean square of a test field can dominate over the gauge invariant inflaton fluctuation one on suitably choosing parameters. We also compute the stochastic growth of quantum fluctuation of a second field, relaxing the assumption of its zero homogeneous value, in a generic inflationary model; as a main result, we obtain that the equation of motion of a gauge invariant variable associated, order by order, with a generic quantum scalar fluctuation during inflation can be obtained only if we use the number of e-folds as the time variable in the corresponding Langevin and Fokker-Planck equations for the stochastic approach. We employ this approach to derive some bounds in the case of a model with two massive fields.