Stochastic formalism for U(1) gauge fields in axion inflation

TL;DR

This paper develops a stochastic formalism for U(1) gauge fields coupled to a pseudo-scalar during inflation, deriving Langevin equations and analyzing fluctuations and isotropy at the Hubble scale.

Contribution

It introduces a novel stochastic approach to describe gauge fields with Chern-Simons coupling during inflation, including analytic solutions and numerical validation.

Findings

Electromagnetic fields fluctuate around analytically predicted values.

Hubble scale fields change direction and amplitude over time.

Global isotropy is maintained despite local anisotropy.

Abstract

We develop the stochastic formalism for $\mathrm{U}(1)$ gauge fields that has the Chern-Simons coupling to a rolling pseudo-scalar field during inflation. The Langevin equations for the physical electromagnetic fields are derived and the analytic solutions are studied. Using numerical simulation we demonstrate that the electromagnetic fields averaged over the Hubble scale continuously change their direction and their amplitudes fluctuate around the analytically obtained expectation values. Though the isotropy is spontaneously broken by picking up a particular local Hubble patch, each Hubble patch is understood independent and the isotropy is conserved globally by averaging all the Hubble patches.