Stabilization of Starobinsky-Vilenkin stochastic inflation by an environmental noise

TL;DR

This paper explores how environmental noise influences the stability of Starobinsky-Vilenkin stochastic inflation, showing that noise can regulate inflation dynamics and eliminate the need for boundary conditions to prevent infinite inflation.

Contribution

It introduces a model combining environmental noise with quantum fluctuations in inflation, providing explicit stationary solutions and analyzing the transition from cold to warm inflation.

Findings

Environmental noise can ensure the integrability of the stationary probability distribution.

Quantum noise influences the asymptotic behavior of inflationary models.

Transition from cold to warm inflation can be signaled by changes in the scale factor and Hubble parameter dependence.

Abstract

We discuss the inflaton $\phi$ in an interaction with an infinite number of fields treated as an environment (noise) with a friction $\gamma^{2}>0$. In a Markovian approximation we obtain a stochastic wave equation (appearing also in the warm inflation models). After the replacement of the environment by the white noise, the stochastic wave equation violates the energy conservation if $\gamma\neq 0$. We introduce a dark energy as a compensation of the inflaton energy-momentum. We add to the classical wave equation the Starobinsky-Vilenkin noise which in the slow-roll approximation describes the quantum fluctuations in an expanding metric. We investigate the resulting consistent Einstein-Klein-Gordon system in the slow-roll regime. We obtain Fokker-Planck equation for the probability distribution of the inflaton assuming that the dependence of the scale factor $a$ and the Hubble variable $ H$ on the field $\phi$ is known. We obtain explicit stationary solutions of the Fokker-Planck equation assuming that $a(\phi)$ and $H(\phi)$ can approximately be determined in a slow-roll regime with the neglect of noise. We extend the results to the multifield D-dimensional configuration space. We show that in the regime $a(\phi)^{3}H(\phi)^{5}\rightarrow \infty$ the quantum noise determines the asymptotic behaviour of the stationary distribution. If $a(\phi)^{3}H(\phi)^{5}$ stays finite then the environmental noise ensures the integrability of the stationary probability. In such a case there is no need to introduce boundary conditions with the purpose to eliminate infinite inflation. The variation of $a(\phi)^{3}H(\phi)^{5}$ could be interpreted as a sign of a transition from cold inflation to warm inflation.