Primordial magnetic seed field amplification by gravitational waves

TL;DR

This paper develops a gauge-invariant perturbation theory framework to show how gravitational waves can amplify primordial magnetic seed fields, potentially explaining the magnetic fields observed in galaxies today.

Contribution

It introduces a self-consistent second-order gauge-invariant method to analyze the non-linear coupling between gravitational waves and magnetic fields, revealing how gravitational waves can significantly amplify seed magnetic fields during inflation.

Findings

Magnetic field amplification is proportional to shear anisotropy and initial scale.

A seed field of 10^{-34} G can be amplified if shear anisotropy exceeds 10^{-40}.

Weak field and gauge-invariant approaches agree under specific conditions.

Abstract

Using second-order gauge-invariant perturbation theory, a self-consistent framework describing the non-linear coupling between gravitational waves and a large-scale homogeneous magnetic field is presented. It is shown how this coupling may be used to amplify seed magnetic fields to strengths needed to support the galactic dynamo. In situations where the gravitational wave background is described by an `almost' Friedmann-Lema{\^i}tre-Robertson-Walker (FLRW) cosmology we find that the magnitude of the original magnetic field is amplified by an amount proportional to the magnitude of the gravitational wave induced shear anisotropy and the square of the field's initial co-moving scale. We apply this mechanism to the case where the seed field and gravitational wave background are produced during inflation and find that the magnitude of the gravitational boost depends significantly on the manner in which the estimate of the shear anisotropy at the end of inflation is calculated. Assuming a seed field of $10^{-34}$ $\rm{G}$ spanning a comoving scale of about $10 \rm{kpc}$ today, the shear anisotropy at the end of inflation must be at least as large as $10^{-40}$ in order to obtain a generated magnetic field of the same order of magnitude as the original seed. Moreover, contrasting the weak field approximation to our gauge-invariant approach, we find that while both methods agree in the limit of high conductivity, their corresponding solutions are otherwise only compatible in the limit of infinitely long-wavelength gravitational waves.