Analytic approach to the motion of cosmological phase transition fronts

TL;DR

The paper develops analytic expressions for the steady-state velocity of planar cosmological phase-transition fronts by combining the thin-wall approximation with a bag EOS. By solving algebraic relations that couple hydrodynamics, thermodynamics, and friction, it maps how the wall velocity $v_w$ depends on the friction coefficient and thermodynamic parameters, and identifies multiple potential stationary states (detonations, deflagrations, Jouguet solutions). It analyzes the structure and stability of these solutions, discusses the validity of the approximations against numerical results, and explores how friction saturation can lead to runaway behavior. This framework enables rapid exploration of phase-transition dynamics and informs predictions for gravitational-wave signals and baryogenesis scenarios.

Abstract

We consider the motion of planar phase-transition fronts in first-order phase transitions of the Universe. We find the steady state wall velocity as a function of a friction coefficient and thermodynamical parameters, taking into account the different hydrodynamic modes of propagation. We obtain analytical approximations for the velocity by using the thin wall approximation and the bag equation of state. We compare our results to those of numerical calculations and discuss the range of validity of the approximations. We analyze the structure of the stationary solutions. Multiple solutions may exist for a given set of parameters, even after discarding non-physical ones. We discuss which of these will be realized in the phase transition as the stationary wall velocity. Finally, we discuss on the saturation of the friction at ultra-relativistic velocities and the existence of runaway solutions.

Figures (9)

• Figure 1: $v_{+}$ vs $v_{-}$ for the bag EOS ($c_+=c_-=1/\sqrt{3}$), for $\alpha_+=0.1$. The upper branch corresponds to detonations and the lower to deflagrations.
• Figure 2: Some fluid profiles for the bag EOS with $\alpha _{n}=0.1$ and $\alpha_c=0.06$ ($T_n/T_c\simeq 0.88$). Solid lines correspond to a weak deflagration with $v_w=0.3$, dashed lines to a Jouguet deflagration with $v_w=0.65$, and dashed-dotted lines to a weak detonation with $v_{w}=0.9$. Left: the fluid velocity. Right: the fluid temperature. The dotted line indicates the critical temperature.
• Figure 3: The solutions in the $T_{+}T_{-}$-plane for $a_{+}=(\pi ^{2}/90)g_{\ast }$ with $g_{\ast }=51.25$, $L=0.1T_{c}^{4}$, and $\eta=\tilde{\eta}T\sigma$, with $\sigma =0.1T_{c}^{3}$. The region with the dark shade is forbidden by kinematics or by the non-negativity of entropy production. The upper allowed region corresponds to deflagrations, and the lower one to detonations. The Jouguet processes are indicated by dashed lines, and the region with a lighter shade corresponds to strong deflagrations. Dotted lines correspond to fixed values of $T_{n}/T_{c}=0.86,0.89,0.92,0.95$ and 0.98 as indicated, and solid lines to $\tilde{\eta}$ fixed (see explanation in the text). The red line corresponds to $\tilde{\eta}=0$.
• Figure 4: The wall velocity for $g_{\ast }=51.25$, $L=0.1T_{c}^{4}$, and $\eta =\tilde{\eta}T\sigma$, with $\sigma =0.1T_{c}^{3}$. Detonations are plotted in blue, Jouguet deflagrations are plotted in black, and traditional deflagrations are in red. The dotted lines correspond to the sound and Jouguet velocities. Left: the wall velocity as a function of $\alpha _{n}$, for $\tilde{\eta} =0.05$. The range of $\alpha _{n}$ correspond to values of the temperature between $T_{n}=T_{c}$ and $T_{n}\simeq 0.8T_{c}$. Thus, the $\alpha_n$-axis begins at the value $\alpha_n=\alpha _{c}\simeq 4.45\times 10^{-3}$. Right: the wall velocity as a function of the friction, for $T_{n}=0.95T_{c}$ ($\alpha _{n}\simeq 5.46\times 10^{-3}$).
• Figure 5: The same as in Fig. \ref{['figdetodefla']} (right), but for $T_{n}=0.891T_{c}$. In the right panel only one of the multiple solutions has been chosen. The dashed line corresponds to the velocity of the shock front.