Kinetics of Sub-Critical Bubbles and the Electroweak Transition

We investigate the role of large amplitude sub-critical thermal fluctuations in the dynamics of first order phase transitions. In particular, we obtain a kinetic equation for the number density of sub-critical fluctuations of the broken-symmetric phase within the symmetric phase, modeled as spherical bubbles, and solve it analytically for temperatures above the critical temperature. We study the approach to equilibrium and obtain the equilibrium distribution of sub-critical bubbles of the unstable phase by examining three possible mechanisms responsible for their removal; their shrinking, their coupling to thermal noise, and by thermal fluctuations of the true vacuum inside them. We show that for sufficiently strong transitions, either the shrinking or the coupling to thermal noise dominate the dynamics. As the strength of the transition weakens we show that sub-critical fluctuations become progressively more important, as a larger fraction of the total volume is occupied by the broken-symmetric phase, until the point where our analytical approach breaks down. Our investigation suggests that pre-transitional phenomena may considerably change the dynamics of sufficiently weak first-order transitions. We apply our results to the standard electroweak transition.