Gradient expansion formalism for a generic model of inflationary magnetogenesis

Abstract

We study the generation of electromagnetic fields during inflation in a model with kinetic and axial couplings to the inflaton field using the gradient expansion formalism. This formalism allows us to simultaneously take into account the possible presence of two nonlinear phenomena: (i) the backreaction of the generated electromagnetic fields on the evolution of the inflaton and (ii) the creation of pairs of charged fermions from the physical vacuum (the Schwinger effect). We model the latter phenomenon by using the generalized Ohmic form of the induced current, \(\vec {J}=\sigma _{E}\vec {E}+\sigma _{B}\vec {B}\), with \(\sigma _E\) and \(\sigma _B\) being the electric and magnetic conductivities. We derive the system of equtions of the gradient expansion formalism for generic kinetic and axial coupling functions as well as Schwinger conductivities. Further, in order to test our system of equations, we apply it to a specific case of the kinetic coupling in the exponential Ratra form and the linear axial coupling function for a few benchmark points in the parameter space. To estimate the accuracy of the obtained numerical results, we perform a comparison with the results of mode-by-mode solution in the Fourier space. We show that the backreaction causes a noticeable increase in the duration of the inflationary epoch while the Schwinger effect strongly suppresses the produced electromagnetic fields and lifts their backreaction.