Supersonic Deflagrations in Cosmological Phase Transitions

TL;DR

The paper investigates hydrodynamics of bubble growth in cosmological first-order phase transitions, focusing on deflagration and detonation modes. It shows that strong deflagrations are forbidden and that supersonic deflagrations can exist as a Jouguet deflagration followed by a rarefaction wave, with front velocity lying between $c_s$ and the speed of light depending on supercooling. Using a bag EOS and similarity solutions, along with a dynamical order-parameter description, the authors map stationary final states and demonstrate that supersonic deflagrations fill the velocity gap between weak deflagrations and weak detonations, while dynamical evolution often selects the Jouguet deflagration plus rarefaction over near-Jouguet detonations. The results imply that, for sufficiently large supercooling, these modes can influence bubble growth and potential gravitational-wave signatures in QCD and electroweak phase transitions.

Abstract

The classification of the hydrodynamical growth mechanisms for the spherical bubbles of the low-temperature phase in cosmological phase transitions is completed by showing that the bubbles can grow as supersonic deflagrations. Such deflagrations consist of a Jouguet deflagration, followed by a rarefaction wave. Depending on the amount of supercooling, the maximal velocity of supersonic deflagrations varies between the sound and the light velocities. The solutions faster than supersonic deflagrations are weak detonations.

Figures (7)

• Figure 1: The solutions of the model of Ref. IKKL, in the case of planar symmetry. In 1+3 dimensions, the solid lines remain the same, but the exact functional form of the dotted lines changes a little. For detonations, the $x$-axis is the temperature just behind the phase transition surface; for deflagrations, it is the temperature in the center of the expanding bubble. The $y$-axis is the temperature just in front of the phase transition surface. The region forbidden kinematically, by the non-negativity of entropy production, or by boundary conditions, has been covered with the darker shade, assuming the types of solutions discussed in Sec. \ref{['defldet']}. The two allowed regions represent the deflagrations and the weak detonations. Dashed lines indicate the Jouguet processes, and strong deflagrations are covered with the lighter shade. The meaning of the different curves has been explained in the text. The nucleation temperatures $T_f/T_c$ corresponding to $\alpha$, $\beta$, $\gamma$, $\delta$ and $\epsilon$ are $0.86$, $0.89$, $0.92$, $0.95$ and $0.98$. The (QCD-type) parameters used in this figure are $L=0.1T_c^4$, $\sigma=0.1T_c^3$ and $l_c=6T_c^{-1}$. Given the expansion rate of the Universe, the actual nucleation temperature $T_f/T_c$ is then about $0.86$ (Fig. 1 in Ref. IKKL2).
• Figure 2: Temperature profiles at $t=14400T_c^{-1}$ for different values of $\eta$. All the profiles are moving to the right. The arrows indicate the location of the phase transition front. For $\eta=0.20T_c$, the solution is a weak deflagration; for $\eta=0.17T_c$, it is a Jouguet deflagration followed by a rarefaction wave; for $\eta=0.16T_c$ and $\eta=0.12T_c$, the solution is a weak detonation. The velocity of the phase transition front changes from $0.56$ at $\eta=0.20T_c$ to $0.74$ at $\eta=0.12T_c$.
• Figure 3: Energy-density (upper set) and flow velocity (lower set) profiles for spherical phase transition bubbles. We show 13 profiles, all for the same initial energy density $\epsilon_f = 100 B$, but with different phase transition front velocities. The six slowest (on the left) are weak deflagrations. The seventh is a Jouguet deflagration bubble, with $\xi_{\rm defl} = c_s$. The eighth is a supersonic deflagration bubble, which has a Jouguet deflagration front preceded by a shock and followed by a rarefaction wave. The ninth is a supersonic deflagration bubble almost at the limit where it becomes a Jouguet detonation. The shocked region is now so thin that it appears as a mere vertical line in the figure. The last four (on the right) are weak detonations.
• Figure 4: A supersonic deflagration bubble. The upper curve is the energy density profile and the lower curve the flow velocity profile. This figure is for a supersonic spherical deflagration bubble with an initial energy density $\epsilon_f = 100 B$ and shock velocity $\xi_{\rm sh} = 0.625$. The velocity of the phase transition front is $\xi_{\rm defl} = 0.6163$.
• Figure 5: The energy densities (a) and flow velocities (b) at special points of the bubble profile (see Fig. 4). These quantities are shown as a function of the phase transition velocity $\xi_{\rm p.t.}$ for a fixed initial energy density $\epsilon_f = 100 B$. The solution switches from a weak deflagration bubble to a supersonic deflagration bubble at $\xi_{\rm p.t.} = c_s = 0.57735$ and to a weak detonation bubble at $\xi_{\rm p.t.} = \xi_{\rm det,J} = 0.6534$.