Spherical and non-spherical bubbles in cosmological phase transitions

TL;DR

The paper analyzes how the geometry of expanding bubble walls in a first-order cosmological phase transition affects the hydrodynamics and energy transfer to the surrounding plasma, with a focus on the efficiency factor $\kappa$ that quantifies kinetic energy in bulk motions. Using a bag equation of state, it derives and compares fluid profiles for spherical, cylindrical, and planar walls across detonations, subsonic deflagrations, and supersonic deflagrations, showing that $\kappa$ is remarkably similar across geometries except at small wall speeds. A key result is the analytic solvability for planar walls, providing closed-form expressions for $\kappa$ in all hydrodynamic modes and revealing that Jouguet branches maximize energy injection. Section 5 translates these hydrodynamic insights into gravitational-wave predictions, highlighting turbulence as the dominant GW source and showing that supersonic deflagrations can yield strong signals within the electroweak-scale regime. Overall, the work supports planar-wall results as robust benchmarks for GW estimates and clarifies the limited role of wall geometry in determining bulk-fluid perturbations during phase transitions.

Abstract

The cosmological remnants of a first-order phase transition generally depend on the perturbations that the walls of expanding bubbles originate in the plasma. Several of the formation mechanisms occur when bubbles collide and lose their spherical symmetry. However, spherical bubbles are often considered in the literature, in particular for the calculation of gravitational waves. We study the steady state motion of bubble walls for different bubble symmetries. Using the bag equation of state, we discuss the propagation of phase transition fronts as detonations and subsonic or supersonic deflagrations. We consider the cases of spherical, cylindrical and planar walls, and compare the energy transferred to bulk motions of the relativistic fluid. We find that the different wall geometries give similar perturbations of the plasma. For the case of planar walls, we obtain analytical expressions for the kinetic energy in the bulk motions. As an application, we discuss the generation of gravitational waves.

Figures (8)

• Figure 1: Left: the fluid velocity profile of a detonation with $v_{w}=0.8$ and $\alpha _{N}=0.1$ for a spherical wall (solid), a cylindrical wall (dashed), and a planar wall (dotted). Right: the corresponding kinetic energy density profiles.
• Figure 2: Left: the fluid velocity profile of a deflagration with $v_{w}=0.4$ and $\alpha _{N}=0.1$ for a spherical wall (solid), a cylindrical wall (dashed), and a planar wall (dotted). Right: the corresponding kinetic energy density profiles.
• Figure 3: Left: the fluid velocity profile of a deflagration with $v_{w}=0.7$ and $\alpha _{N}=0.1$ for a spherical wall (solid), a cylindrical wall (dashed), and a planar wall (dotted). Right: the corresponding kinetic energy density profiles.
• Figure 4: The thickness $\delta \xi$ of the region around the wall inside which the kinetic energy density decreases to a half of its maximum value, for $\alpha _{N}=0.1$ as a function of $v_{w}$. Solid lines correspond to spherical bubbles, dashed lines correspond to cylindrical bubbles, and dotted lines correspond to planar walls. Subsonic deflagrations are plotted in blue, supersonic deflagrations are in black, and detonations in red.
• Figure 5: The efficiency factor $\kappa$ as a function of $v_{w}$ for $\alpha _{N}=0.01,0.03,0.1,0.3$. Solid lines correspond to the spherical case, dashed lines correspond to the cylindrical case, and dotted lines correspond to the planar case. Subsonic deflagrations are plotted in blue, supersonic deflagrations are in black, and detonations in red.