Superhorizon magnetic fields

TL;DR

This paper investigates the evolution of superhorizon magnetic fields from inflation to present, identifying different evolutionary scenarios based on initial conditions and supporting the Ratra model for cosmic magnetic field generation.

Contribution

It provides a detailed analysis of magnetic field evolution post-inflation, clarifying conditions for adiabatic and superadiabatic growth, and supports the Ratra model as a viable mechanism.

Findings

Magnetic fields evolve adiabatically in conformal Maxwell theory.

In the Ratra model, magnetic fields evolve superadiabatically after inflation.

The evolution depends on initial phase differences and mode parameters.

Abstract

[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 $k\eta_e$, $n_{\mathbf{k}}$, and $\Omega_k$, where $\eta_e$ is the conformal time at the end of inflation, $n_{\mathbf{k}}$ is the number density spectrum of inflation-produced photons, and $\Omega_k$ 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 $n_{\mathbf{k}}^{-1} \ll |k\eta_e| \ll 1$, and three evolutionary scenarios are possible: ($i$) $|\Omega_k \mp \pi| = \mathcal{O}(1)$, in which case the evolution of the magnetic spectrum $B_k(\eta)$ is adiabatic, $a^2B_k(\eta) = \mbox{const}$, with $a$ being the expansion parameter; ($ii$) $|\Omega_k \mp \pi| \ll |k\eta_e|$, in which case the evolution is superadiabatic, $a^2B_k(\eta) \propto \eta$; ($iii$) $|k\eta_e| \ll |\Omega_k \mp \pi| \ll 1$ or $|k\eta_e| \sim |\Omega_k \mp \pi| \ll 1$, in which case an early phase of adiabatic evolution is followed, after a time $\eta_\star \sim |\Omega_k \mp \pi|/k$, 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.