A Note on the Cosmic Evolution of the Axion in a Strong Magnetic Field
  It has been pointed out in the literature that in the presence of an external magnetic field the axion mass receives an electromagnetic contribution. We show that if a magnetic field with energy density larger than ~10^{-8} times the energy density of the Universe existed at temperatures of a few GeV, that contribution would be dominant and consequently the cosmic evolution of the axion field would change substantially. In particular, the expected axion relic abundance would be lowered, allowing a small relaxation of the present cosmological bound on the Peccei-Quinn constant. 
L. Campanelli and M. Giannotti