On inhomogeneous magnetic seed fields and gravitational waves within the MHD limit

TL;DR

This study uses second-order gauge-invariant perturbation theory within the MHD framework to analyze how gravitational waves interact with inhomogeneous magnetic fields in the early universe, finding no significant amplification at large scales.

Contribution

It extends previous work by incorporating spatial inhomogeneities in magnetic fields using MHD, showing that inhomogeneity does not enhance magnetic field amplification on super-horizon scales.

Findings

No amplification of magnetic fields at super-horizon scales.

MHD effects dominate at sub-horizon scales.

Inhomogeneities do not significantly boost magnetic field generation.

Abstract

In this paper we apply second-order gauge-invariant perturbation theory to investigate the possibility that the non-linear coupling between gravitational waves (GW) and a large scale inhomogeneous magnetic field acts as an amplification mechanism in an `almost' Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe. The spatial inhomogeneities in the magnetic field are consistently implemented using the magnetohydrodynamic (MHD) approximation, which yields an additional source term due to the interaction of the magnetic field with velocity perturbations in the plasma. Comparing the solutions with the corresponding results in our previous work indicates that, on super-horizon scales, the interaction with the spatially inhomogeneous field in the dust regime induces the same boost as the case of a homogeneous field, at least in the ideal MHD approximation. This is attributed to the observation that the MHD induced part of the generated field effectively only contributes on scales where the coherence length of the initial field is less than the Hubble scale. At sub-horizon scales, the GW induced magnetic field is completely negligible in relation to the MHD induced field. Moreover, there is no amplification found in the long-wavelength limit.