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We study magnetogenesis in axionlike inflation driven by a pseudoscalar field φ coupled axially to the electromagnetic (EM) field (β/Mp)φFµνF µν with dimensionless coupling constant β. 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 ρEM ∼ ( √ 2 /β)ρ inf ( is the slow-roll parameter and ρ 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 ρE ∼ α −3 EM (ρ 2 tot /M 4 p ), where ρtot = 3H 2 M 2 p is the total energy density and αEM is the EM coupling constant. For large β, produced charged particles could constitute a significant part of the Universe energy density even before the preheating stage. Numerically studying magnetogenesis in the α-attractor model of inflation, we find that it is possible to generate helical magnetic fields with the maximal strength 10 −15 G, however, only with the correlation length of order 1 pc at present.

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 (φ)FµνF µν of the electromagnetic field to the inflaton field φ. We consider the case where the coupling function f (φ) 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 ǫ and inflaton energy density, ρE ∼ ǫρ 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 nB = 2 for any decreasing coupling function. This gives an upper limit on the present-day value of observed magnetic fields below 10 −22 G. It is worth emphasizing that since the effective electric charge of particles e 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 ∝ a α and Ratra-type f = exp(βφ/Mp) coupling functions.