Bubble nucleation in first-order inflation and other cosmological phase transitions

We address in some detail the kinematics of bubble nucleation and percolation in first-order cosmological phase transitions, with the primary focus on first-order inflation. We study how a first-order phase transition completes, describe measures of its progress, and compute the distribution of bubble sizes. For example, we find that the typical bubble size in a successful transition is of order 1% to 100% of the Hubble radius, and depends very weakly on the energy scale of the transition. We derive very general conditions that must be satisfied by Γ/${\mathit{H}}^4$ to complete the phase transition (Γ=bubble nucleation rate per unit volume; H=expansion rate; physically, Γ/${\mathit{H}}^4$ corresponds to the volume fraction of space occupied by bubbles nucleated over a Hubble time). In particular, Γ/${\mathit{H}}^4$ must exceed 9/4π to successfully end inflation. To avoid the deleterious effects of bubbles nucleated early during inflation on primordial nucleosynthesis and on the isotropy and spectrum of the cosmic microwave background radiation, during most of inflation Γ/${\mathit{H}}^4$ must be less than order ${10}^{\mathrm{-}4}$–${10}^{\mathrm{-}3}$. Our constraints imply that in a successful model of first-order inflation the phase transition must complete over a period of at most a few Hubble times and all but preclude individual bubbles from providing an interesting source of density perturbation. We note, though, that it is just possible for Poisson fluctuations in the number of moderately large-size bubbles to lead to interesting isocurvature perturbations, whose spectrum is not scale invariant. Finally, we analyze in detail several recently proposed models of first-order inflation.