Evolution and thermalization of dark matter axions in the condensed   regime
  We discuss the possibility that dark matter axions form a Bose-Einstein condensate (BEC) due to the gravitational self-interactions. The formation of BEC occurs in the condensed regime, where the transition rate between different momentum states is large compared to the energy exchanged in the transition. The time evolution of the quantum state occupation number of axions in the condensed regime is derived based on the in-in formalism. We recover the expression for the thermalization rate due to self interaction of the axion field, which was obtained in the other literature. It is also found that the leading order contributions for interactions between axions and other species vanish, which implies that the axion BEC does not give any significant modifications on standard cosmological parameters. 
Ken'ichi Saikawa and Masahide Yamaguchi
hep-ph astro-ph.CO hep-th