  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 2+1-dimensional U(1)-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 {\em more} defects per unit volume as a result of the sub-luminal
bubble wall terminal velocity.
