Gravitational wave production: A strong constraint on primordial magnetic fields

We compute the gravity waves induced by anisotropic stresses of stochasticprimordial magnetic fields. The nucleosynthesis bound on gravity waves is thenused to derive a limit on the magnetic field amplitude as function of thespectral index. The obtained limits are extraordinarily strong: If theprimordial magnetic field is produced by a causal process, leading to aspectral index $n\ge 2$ on super horizon scales, galactic magnetic fieldsproduced at the electroweak phase transition or earlier have to be weaker than$B_\la \le 10^{-27}$Gauss! If they are induced during an inflationary phase(reheating temperature $T\sim 10^{15}$GeV) with a spectral index $n\sim 0$, themagnetic field has to be weaker than $B_\la \le 10^{-39}$Gauss! Only very redmagnetic field spectra, $n\sim -3$ are not strongly constrained. We also findthat a considerable amount of the magnetic field energy is converted intogravity waves. The gravity wave limit derived in this work rules out most ofthe proposed processes for primordial seeds for the large scale coherentmagnetic fields observed in galaxies and clusters.Comment: 11pages 2 figures Replaced with the significantly expanded version accepted for publication in PRD. More detailed explanations, same conclusion