Backreaction of electromagnetic fields and the Schwinger effect in pseudoscalar inflation magnetogenesis

We study magnetogenesis in axionlike inflation driven by a pseudoscalar field $\phi$ coupled axially to the electromagnetic (EM) field $(\beta/M_{p})\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$ with dimensionless coupling constant $\beta$. A set of equations for the inflaton field, scale factor, and expectation values of quadratic functions of the EM field is derived. These equations take into account the Schwinger effect and the backreaction of generated EM fields on the Universe expansion. It is found that the backreaction becomes important when the EM energy density reaches the value $\rho_{\rm EM}\sim (\sqrt{2\epsilon}/\beta)\rho_{\rm inf}$ ($\epsilon$ is the slow-roll parameter and $\rho_{\rm inf}$ is the energy density of the inflaton) slowing down the inflaton rolling and terminating magnetogenesis. The Schwinger effect becomes relevant when the electric energy density exceeds the value $\rho_{E}\sim \alpha_{\rm EM}^{-3} (\rho_{\rm tot}^{2}/M_{p}^{4})$, where $\rho_{\rm tot}=3H^{2}M_{p}^{2}$ is the total energy density and $\alpha_{\rm EM}$ is the EM coupling constant. For large $\beta$, produced charged particles could constitute a significant part of the Universe energy density even before the preheating stage. Numerically studying magnetogenesis in the $\alpha$-attractor model of inflation, we find that it is possible to generate helical magnetic fields with the maximal strength $10^{-15}\,{\rm G}$, however, only with the correlation length of order $1\,{\rm pc}$ at present.