A Possible Origin of Magnetic Fields in Galaxies and Clusters: Strong   Magnetic fields at z~10?
  We propose that strong magnetic fields should be generated at shock waves associated with formation of galaxies or clusters of galaxies by the Weibel instability, an instability in collisionless plasmas. The strength of the magnetic fields generated through this mechanism is close to the order of those observed in galaxies or clusters of galaxies at present. If the generated fields do not decay rapidly, this indicates that strong amplification of magnetic fields after formation of galaxies or clusters of galaxies is not required. This mechanism could have worked even at a redshift of ~10, and therefore the generated magnetic fields may have affected the formation of stars in protogalaxies. This model will partially be confirmed by future observations of nearby clusters of galaxies. Mechanisms that preserve the magnetic fields for a long time without considerable decay are discussed. 
Yutaka Fujita, Tsunehiko N. Kato (NAOJ)