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

Understand

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.