Gravitational wave production: A strong constraint on primordial magnetic fields

We compute the gravity waves induced by anisotropic stresses of stochastic primordial magnetic fields. The nucleosynthesis bound on gravity waves is then used to derive a limit on the magnetic field amplitude as a function of the spectral index. The obtained limits are extraordinarily strong: If the primordial magnetic field is produced by a causal process, leading to a spectral index $n>~2$ on superhorizon scales, galactic magnetic fields produced at the electroweak phase transition or earlier have to be weaker than $B_{\lambda }<~{10}^{-27}\hspace{0.5em}\mathrm{G}.$ If they are induced during an inflationary phase (reheating temperature $T\sim {10}^{15}\hspace{0.5em}\mathrm{GeV})$ with a spectral index $n\sim 0,$ the magnetic field has to be weaker than $B_{\lambda }<~{10}^{-39}\hspace{0.5em}\mathrm{G}.$ Only very red magnetic field spectra, $n\sim -3,$ are not strongly constrained. We also find that a considerable amount of the magnetic field energy is converted into gravity waves. The gravity wave limit derived in this work rules out most of the proposed processes for primordial seeds for the large scale magnetic fields observed in galaxies and clusters.