Bubble nucleation and growth in very strong cosmological phase transitions

TL;DR

The paper analyzes bubble nucleation and growth in very strong cosmological phase transitions, where substantial supercooling can invalidate the standard exponential time dependence of the nucleation rate. It introduces a Gaussian approximation for the nucleation rate near the minimum of the action $S(T)$ and derives analytic solutions for the evolution of the transition, including bubble growth, number density, and size distribution. By comparing Gaussian and exponential forms to full numerical results, it establishes the regimes of validity for each and explores implications for cosmic remnants such as gravitational waves. The work provides practical guidance for incorporating Gaussian-rate nucleation into simulations and for interpreting GW signals from strong transitions.

Abstract

Strongly first-order phase transitions, i.e., those with a large order parameter, are characterized by a considerable supercooling and high velocities of phase transition fronts. A very strong phase transition may have important cosmological consequences due to the departures from equilibrium caused in the plasma. In general, there is a limit to the strength, since the metastability of the old phase may prevent the transition to complete. Near this limit, the bubble nucleation rate achieves a maximum and thus departs from the widely assumed behavior in which it grows exponentially with time. We study the dynamics of this kind of phase transitions. We show that in some cases a gaussian approximation for the nucleation rate is more suitable, and in such a case we solve analytically the evolution of the phase transition. We compare the gaussian and exponential approximations with realistic cases and we determine their ranges of validity. We also discuss the implications for cosmic remnants such as gravitational waves.

Figures (11)

• Figure 1: The nucleation action as a function of temperature, for different values of $A/v$. The blue curve corresponds to $S=S_4$ and the rest to $S=S_3/T$. The horizontal lines indicate the value $4\log (M_{P}/v)$ for $v=1\mathrm{MeV }$ (upper line), $100\mathrm{GeV}$ (central line), and $10^{8}\mathrm{GeV}$ (lower line).
• Figure 2: The wall velocity and order parameter at different temperatures.
• Figure 3: The curves of $S(T)$ for (from bottom to top) $A/v=0.1$, $0.11$, $0.12$, $0.124$, $0.126$, $0.128$, $0.129$, and $0.131$. The dots correspond to the reference points $H$ (black), $N$ (red), $I$ (blue), $P$ (green), $E$ (orange), and $F$ (purple), for $v=250 \mathrm{GeV}$.
• Figure 4: The evolution of several quantities, for the cases $A/v=0.11$ (top), $A/v=0.126$ (middle) and $A/v=0.128$ (bottom). The left panels show the fraction of volume $f_{+}$, the nucleation rate $\Gamma$ and the average nucleation rate $\bar{\Gamma}$. The right panels show the average radius $\bar{R}$ and the bubble separation $d$.
• Figure 5: The bubble size distribution $dn/dR$ at several times, for $A/v=0.11$ (left), $A/v=0.126$ (center), and $A/v=0.128$ (right). The long ticks indicate the radius of the bubbles nucleated at $t=t_{N}$.