Influence of backreaction of electric fields and Schwinger effect on
  inflationary magnetogenesis
  We study the generation of electromagnetic fields during inflation when the
conformal invariance of Maxwell's action is broken by the kinetic coupling
$f^{2}(\phi)F_{\mu\nu}F^{\mu\nu}$ of the electromagnetic field to the inflaton
field $\phi$. We consider the case where the coupling function $f(\phi)$
decreases in time during inflation and, as a result, the electric component of
the energy density dominates over the magnetic one. The system of equations
which governs the joint evolution of the scale factor, inflaton field, and
electric energy density is derived. The backreaction occurs when the electric
energy density becomes as large as the product of the slow-roll parameter
$\epsilon$ and inflaton energy density, $\rho_{E}\sim \epsilon \rho_{\rm inf}$.
It affects the inflaton field evolution and leads to the scale-invariant
electric power spectrum and the magnetic one which is blue with the spectral
index $n_{B}=2$ for any decreasing coupling function. This gives an upper limit
on the present-day value of observed magnetic fields below $10^{-22}\,{\rm G}$.
It is worth emphasizing that since the effective electric charge of particles
$e_{\rm eff}=e/f$ is suppressed by the coupling function, the Schwinger effect
becomes important only at the late stages of inflation when the inflaton field
is close to the minimum of its potential. The Schwinger effect abruptly
decreases the value of the electric field, helping to finish the inflation
stage and enter the stage of preheating. It effectively produces the charged
particles, implementing the Schwinger reheating scenario even before the fast
oscillations of the inflaton. The numerical analysis is carried out in the
Starobinsky model of inflation for the powerlike $f\propto a^{\alpha}$ and
Ratra-type $f=\exp(\beta\phi/M_{p})$ coupling functions.
