Energy Budget of Cosmological First-order Phase Transitions

TL;DR

The paper develops a comprehensive, model-independent hydrodynamic framework for cosmological first-order phase transitions, unifying detonations, deflagrations, hybrids, and runaway walls. It derives general relations for bubble-wall and plasma velocities, computes the energy-division efficiency κ, and analyzes how friction and vacuum energy determine whether walls reach a steady state or runaway, with explicit connections to gravitational wave production. By introducing α_N and α_+ as measures of transition strength and detailing microscopic versus phenomenological friction, it provides contour plots and analytic fits that map regimes and predict GW-relevant quantities. The results challenge simplistic CJ-based assumptions, showing that in strong transitions much of the energy can reside in the wall, suppressing turbulent plasma motions and affecting GW spectra. The work thus offers essential, model-agnostic inputs for baryogenesis and GW forecasts across a wide class of theories.

Abstract

The study of the hydrodynamics of bubble growth in first-order phase transitions is very relevant for electroweak baryogenesis, as the baryon asymmetry depends sensitively on the bubble wall velocity, and also for predicting the size of the gravity wave signal resulting from bubble collisions, which depends on both the bubble wall velocity and the plasma fluid velocity. We perform such study in different bubble expansion regimes, namely deflagrations, detonations, hybrids (steady states) and runaway solutions (accelerating wall), without relying on a specific particle physics model. We compute the efficiency of the transfer of vacuum energy to the bubble wall and the plasma in all regimes. We clarify the condition determining the runaway regime and stress that in most models of strong first-order phase transitions this will modify expectations for the gravity wave signal. Indeed, in this case, most of the kinetic energy is concentrated in the wall and almost no turbulent fluid motions are expected since the surrounding fluid is kept mostly at rest.

Figures (14)

• Figure 1: Contours of the fluid velocities $v_+$ and $v_-$ in the wall frame for fixed $\alpha_+$. In the shaded region in the top-left no consistent solutions to hydrodynamic equations exist. Flow profiles in the shaded region in the bottom-right decay into hybrid solutions, with $v_-=c_s^-$ (see sect. \ref{['subsec:hybrids']}).
• Figure 2: General profiles of the fluid velocity $v(\xi)$ in the frame of the bubble center (with $c_s^2=1/3$). Detonation curves start below $\mu(\xi,v)=c_s$ (dashed-doted curve) and end at $(\xi,v)=(c_s,0)$. Deflagration curves start below $v=\xi$ and end at $\mu(\xi,v) \xi = c_s^2$ (dashed curve) corresponding to the shock front, as explained in the text. There are no consistent solutions in the shaded regions.
• Figure 3: Pictorial representation of expanding bubbles of different types. The black circle is the phase interface (bubble wall). In green we show the region of non-zero fluid velocity.
• Figure 4: Examples of the fluid velocity (in the plasma rest frame), enthalpy and entropy profiles for a subsonic deflagration, a deflagration with rarefaction wave (hybrid) and a detonation, for $a_-/a_+=0.85$. The bubble of broken phase is in gray. For detonations, the fluid kinetic energy and thermal energy are concentrated near the wall but behind it i.e. inside the bubble, while they are located outside (mostly outside) of the bubble for deflagrations (hybrids).
• Figure 5: Fluid velocity profiles (in the plasma rest frame) for deflagrations, hybrids and detonations, for different wall velocities and $\alpha_N=0.3$.