Defect formation rates in cosmological first-order phase transitions

Abstract

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 more defects per unit volume as a result of the sub-luminal bubble wall terminal velocity.

Bubble nucleation and growth in very strong cosmological phase transitions

“…The steady state wall velocity is given by the equation F dr = F fr , and can be computed using Eqs. (22)(23)(24)(25)(26). Like for S(T ), the result does not depend on the scale.…”

“…This results in the formation of a smaller number of defects for smaller velocities. On the other hand, for very strong phase transitions we generally have v w ≃ 1, in which case the dynamics of phase equilibration can be neglected (see, e.g., [24]).…”

“…where α T = ∆ε/(a + T 4 + ). For 0 < v ± < 1, the relation (24) has four branches. In the first place, the signs in front of the square root give two values of v − , either supersonic or subsonic, for a given v + .…”

Bubble nucleation and growth in slow cosmological phase transitions

“…Different kinds of simulations have been performed (mainly in 2+1 dimensions; see e.g. [42,43,44]) to study the dependence of defect formation on the dynamics of the phase transition. In these simulations, a constant wall velocity as well as a constant nucleation rate were assumed.…”

“…1. Then, to solve the condition (44) we may use the exponential-rate approximation. Assuming small I, we have ḟ− ≃ İ, and using the result (40) we have 8πv 3 w Γ(t m )/β 3 * = 3H c /r.…”