Bubble wall correlations in cosmological phase transitions

Abstract

We study statistical relationships between bubble walls in cosmological first-order phase transitions. We consider the conditional and joint probabilities for different points on the walls to remain uncollided at given times. We use these results to discuss space and time correlations of bubble walls and their relevance for the consequences of the transition. In our statistical treatment, the kinematics of bubble nucleation and growth is characterized by the nucleation rate and the wall velocity as functions of time. We obtain general expressions in terms of these two quantities, and we consider several specific examples and applications.

Model-independent features of gravitational waves from bubble collisions

“…A convenient choice for t * is the time t e for which I(t e ) = 1, i.e., when f + has decreased to e −1 . Indeed, at t = t e the average nucleation rate Γ(t)f + (t), as well as the total uncollided wall area S tot (t) , take their maximum [135]. By definition of t e we have I(t) = e β(t−te) , and…”

“…The value Γ(0) = 1/8π is not exactly of order 1 in this case. This is because a different time parameter t b > β −1 would be more representative of the duration of the phase transition (see, e.g., [135]). However, we shall use β since it is the standard parameter.…”

“…This probability was studied in Ref. [135]. It is proportional to e −I(t) e −I(t ) e I∩ , where the last factor takes into account the fact that the probabilities for each point are not independent 3 .…”

“…Here, Θ is the Heaviside step function, and we have used the notation r = R(t , t), r = R(t , t ) (for more details and interpretation, see [135] or [125]). The final expressions from Ref.…”

“…For instance, we may compare an exponential nucleation rate and a simultaneous nucleation with the same duration of the phase transition. We may take, as an estimation, the time ∆t = t 2 − t 1 between the moment t 1 at which f + = 0.99 and t 2 at which f + = 0.01 [135].…”