Analytic approach to the motion of cosmological phase transition fronts
  We consider the motion of planar phase-transition fronts in first-order phase transitions of the Universe. We find the steady state wall velocity as a function of a friction coefficient and thermodynamical parameters, taking into account the different hydrodynamic modes of propagation. We obtain analytical approximations for the velocity by using the thin wall approximation and the bag equation of state. We compare our results to those of numerical calculations and discuss the range of validity of the approximations. We analyze the structure of the stationary solutions. Multiple solutions may exist for a given set of parameters, even after discarding non-physical ones. We discuss which of these will be realized in the phase transition as the stationary wall velocity. Finally, we discuss on the saturation of the friction at ultra-relativistic velocities and the existence of runaway solutions. 
Ariel Megevand and Alejandro D. Sanchez
astro-ph.CO hep-ph hep-th