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

We develop the stochastic formalism for 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.

Automated summary

In this study, we develop a stochastic formalism for U(1) gauge fields coupled with a rolling pseudo-scalar field during inflation. We derive the Langevin equations for the physical electromagnetic fields and study their analytic solutions. Through numerical simulation, we demonstrate that the averaged electromagnetic fields over the Hubble scale continuously change their direction and fluctuate around the analytically obtained expectation values. While the isotropy is spontaneously broken in each local Hubble patch, it is globally preserved by averaging all the Hubble patches.