Stochastic growth of quantum fluctuations during slow-roll inflation

TL;DR

This work analyzes the stochastic growth of quantum fluctuations during slow-roll inflation, contrasting test fields with small or modulated masses against gauge-invariant inflaton fluctuations. By formulating diffusion in terms of the number of e-folds $N$ and studying four representative inflationary backgrounds, the authors quantify how test fields with $m^2$, $m^2 \propto H^2$, or non-minimal coupling accumulate fluctuations, and they compare this growth to the inflaton's gauge-invariant fluctuations. A key result is that inflaton fluctuations typically dominate, except in certain hybrid inflation scenarios where test-field fluctuations can prevail for suitable parameter choices. The paper also extends the stochastic framework to two-field models, showing that $N$ is the natural time variable for the corresponding Langevin/Fokker-Planck equations, and provides concrete bounds on the homogeneous mode and masses in a two-field quadratic system. Overall, the findings refine our understanding of quantum fluctuations during realistic inflation and their potential backreaction, with implications for the moduli problem and early-universe dynamics.

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.

Figures (4)

• Figure 1: Evolution of the mean square quantum fluctuations (in units of $m_{\rm pl}^2$) versus the number of e-folds $N$ for the quadratic chaotic model. For the inflationary background we have chosen the inflationary trajectory in Eq. (\ref{['htraj_quadratic']}) with $m=10^{-6} \, m_{\rm pl}$ and $H_i = 10 \, m$. The mean square gauge invariant inflaton fluctuation (thick line) dominates over those of test fields ($m_\chi \simeq 0.3 m$ is the solid line, $c=0.02$ is the dashed line, $\xi=0.001$ is the dotted line).
• Figure 2: Evolution of the mean square quantum fluctuations (in units of $m_{\rm pl}^2$) versus the number of e-folds $N$ for the small field inflationary model in Eq. (\ref{['hybridorsmall']}). For the inflationary background we have chosen $V_0=2.6 \times 10^{-12} m_{\rm pl}^4$, $M = 0.85 \times 10^{-6} m_{\rm pl}$ and $\phi_i = 0.3 \, m_{\rm pl}$ as parameters. The mean square gauge invariant inflaton fluctuation (thick line) dominates over those of test fields ($m_\chi = 10^{-2} H_0$ is the solid line, $c=0.1$ is the dashed line, $\xi=0.05$ is the dotted line).
• Figure 3: Evolution of the mean square quantum fluctuations (in units of $m_{\rm pl}^2$) versus the number of e-folds $N$ for the hybrid model in Eq. (\ref{['hybridorsmall']}). For the inflationary background we have chosen $V_0=2.6 \times 10^{-12} m_{\rm pl}^4$, $M = 1.8 \times 10^{-6} m_{\rm pl}$ and $\phi_i = 0.3 \, m_{\rm pl}$ as parameters. In this case the mean square of moduli can dominate over the mean square of gauge invariant inflaton fluctuation (thick line): the parameters chosen are $m_\chi = 10^{-2} H_0$ (solid line), $c=0.002$ (dashed line), $\xi=0.05$ (dotted line).
• Figure 4: Evolution of the mean square quantum fluctuations (in units of $m_{\rm pl}^2$) versus the number of e-folds $N$ for the exponential potential. For the inflationary background we have chosen the inflationary trajectory in Eq. (\ref{['htraj_exp']}) with $p=100$ and $t_i = 10^7 \, m_{\rm pl}^{-1}$. The mean square gauge invariant inflaton fluctuation (thick line) dominates over those of test fields ($m_\chi = 10^{-6} \, m_{\rm pl}$ is the solid line, $c=0.1$ is the dashed line, $\xi=0.05$ is the dotted line).