Stochastic Effects in Anisotropic Inflation

TL;DR

This paper analyzes stochastic effects in anisotropic inflation with a U(1) gauge field, deriving Langevin equations, showing equilibrium behavior of the electric field, and assessing implications for CMB anisotropy.

Contribution

It provides an analytical solution to the stochastic dynamics of gauge fields in anisotropic inflation, revealing equilibrium states and their impact on observable anisotropy.

Findings

Electric field reaches a stationary regime due to stochastic effects.

Stationary electric field value differs from classical attractor.

Potential for significant quadrupolar anisotropy consistent with CMB constraints.

Abstract

We revisit the stochastic effects in the model of anisotropic inflation containing a $U(1)$ gauge field. We obtain the Langevin equations for the inflaton and gauge fields perturbations and solve them analytically. We show that if the initial value of the electric field is larger than its classical attractor value, then the random stochastic forces associated with the gauge field is balanced by the frictional damping (classical) force and the electric field falls into an equilibrium (stationary) regime. As a result, the classical attractor value of the electric field is replaced by its stationary value. We show that the probability of generating quadrupolar statistical anisotropy consistent with CMB constraints can be significant.