Detonations and deflagrations in cosmological phase transitions

TL;DR

The paper develops a self-consistent framework for bubble-wall propagation in cosmological first-order phase transitions by coupling hydrodynamic boundary conditions to a damping/friction term in the order-parameter equation, all within a bag equation of state. It computes the wall velocity $v_w$ as a function of the supercooling parameters $\alpha_c$ and $\alpha_N$ and the friction $\eta$, mapping regions where detonations or deflagrations are allowed and identifying parameter regimes with no stationary solution. The analysis is applied to a Standard Model extension with singlet scalars to derive the latent heat, nucleation temperature, and friction from microphysics, and to obtain both numerical results and analytic approximations for the two propagation modes. The results show that supersonic propagation generally requires strong phase transitions and sufficiently small friction, while CJ-type detonation predictions are not universally valid; the paper also provides ultrarelativistic and nonrelativistic approximations that are useful for gravitational wave predictions and baryogenesis scenarios. Overall, the work offers practical criteria and formulas to predict bubble-wall velocities across a broad class of cosmological phase transitions and guides future studies of their cosmological consequences.

Abstract

We study the steady state motion of bubble walls in cosmological phase transitions. Taking into account the boundary and continuity conditions for the fluid variables, we calculate numerically the wall velocity as a function of the nucleation temperature, the latent heat, and a friction parameter. We determine regions in the space of these parameters in which detonations and/or deflagrations are allowed. In order to apply the results to a physical case, we calculate these quantities in a specific model, which consists of an extension of the Standard Model with singlet scalar fields. We also obtain analytic approximations for the wall velocity, both in the case of deflagrations and of detonations.

Figures (14)

• Figure 1: The velocities on each side of the wall, in the reference frame of the wall.
• Figure 2: $v_{+}$ vs $v_{-}$ according to Eq. (\ref{['steinhardt']}), for $\alpha=0.1$
• Figure 3: Schematically, the profile of the density or the velocity of the fluid, induced by the moving wall.
• Figure 4: Detonation solutions for $\alpha _{c}\approx 1.27\times 10^{-2}$. Left: the velocities as functions of $\alpha_N$ for $\eta /L\approx 4.03\times 10^{-2}$. Right: the velocities as functions of $\eta/L$ for $\alpha _{N}\approx 1.48\times 10^{-2}$. Solid lines indicate the velocity $v_w$ of the wall, dashed lines the velocity $v_-$ of the outgoing fluid, and dotted lines the speed of sound (lower curves) and the Jouguet velocity (upper curves).
• Figure 5: Deflagration solutions for the same case as Fig. \ref{['figdeto']}. Solid lines indicate the velocity $v_w$ of the wall, dashed lines the velocity $v_+$ of the incoming fluid, and dotted lines the speed of sound.