We use non-linear gauge-invariant perturbation theory to study the
interaction of an inflation produced seed magnetic field with density and
gravitational wave perturbations in an almost
Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime. We compare the effects
of this coupling under the assumptions of poor conductivity, infinite
conductivity and the case where the electric field is sourced via the coupling
of velocity perturbations to the seed field in the ideal magnetohydrodynamic
(MHD) regime, thus generalizing, improving on and correcting previous results.
We solve our equations for long wavelength limits and numerically integrate the
resulting equations to generate power spectra for the electromagnetic field
variables, showing where the modes cross the horizon. We find that the rotation
of the electric field dominates the power spectrum on small scales, in
agreement with previous arguments.Comment: 16 pages, 3 figures, published in PR
