Defect formation rates in cosmological first-order phase transitions

In cosmological first-order phase transitions, the progress of true-vacuum bubbles is expected to be significantly retarded by the interaction between the bubble wall and the hot plasma. It has been claimed that this leads to a significant reduction in the number of topological defects formed per bubble, as a result of phase equilibration between bubbles. This claim has been verified for spontaneously broken global symmetries. We perform a series of simulations of complete phase transitions in the $\mathrm{}(2+1)\mathrm{}$-dimensional $U\left(1\right)\mathrm{}$-Abelian Higgs model, for a range of bubble wall velocities, in order to obtain a quantitative measure of the effect of bubble wall speed on the number density of topological defects. We find that the number of defects formed is (i) significantly lower in the local than the global case and (ii) decreases exponentially as a function of wall velocity. Slow-moving bubbles also lead, however, to the nucleation of more bubbles before the phase transition is complete. Our simulations show that this is in fact the dominant effect, and so we predict more defects per unit volume as a result of the subluminal bubble wall terminal velocity.