Superhorizon magnetic fields

[Abridged] We analyze the evolution of superhorizon-scale magnetic fields from the end of inflation till today. Whatever is the mechanism responsible for their generation during inflation, we find that a given magnetic mode with wavenumber $k$ evolves, after inflation, according to the values of $kη_e$, $n_{\mathbf{k}}$, and $Ω_k$, where $η_e$ is the conformal time at the end of inflation, $n_{\mathbf{k}}$ is the number density spectrum of inflation-produced photons, and $Ω_k$ is the phase difference between the two Bogolubov coefficients which characterize the state of that mode at the end of inflation. For any realistic inflationary magnetogenesis scenario, we find that $n_{\mathbf{k}}^{-1} \ll |kη_e| \ll 1$, and three evolutionary scenarios are possible: ($i$) $|Ω_k \mp π| = \mathcal{O}(1)$, in which case the evolution of the magnetic spectrum $B_k(η)$ is adiabatic, $a^2B_k(η) = \mbox{const}$, with $a$ being the expansion parameter; ($ii$) $|Ω_k \mp π| \ll |kη_e|$, in which case the evolution is superadiabatic, $a^2B_k(η) \propto η$; ($iii$) $|kη_e| \ll |Ω_k \mp π| \ll 1$ or $|kη_e| \sim |Ω_k \mp π| \ll 1$, in which case an early phase of adiabatic evolution is followed, after a time $η_\star \sim |Ω_k \mp π|/k$, by a superadiabatic evolution. Once a given mode reenters the horizon, it remains frozen into the plasma and then evolves adiabatically till today. As a corollary of our results, we find that inflation-generated magnetic fields evolve adiabatically on all scales and for all times in conformal-invariant free Maxwell theory, while they evolve superadiabatically after inflation on superhorizon scales in the non-conformal-invariant Ratra model. The latter result supports our recent claim that the Ratra model can account for the presence of cosmic magnetic fields.

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