Stability of cosmological deflagration fronts

TL;DR

This work analyzes the linear hydrodynamic stability of deflagration fronts in cosmological first-order phase transitions, incorporating separate temperature and velocity dependence on each side of the wall and reheating effects. By deriving perturbation equations from the field and fluid equations and solving for Fourier modes, it shows that large supercooling destabilizes any subsonic wall and that perturbations become unstable near the sound speed, with a newly identified range of marginally unstable wavelengths. Numerical results with the bag equation of state corroborate the analytic trends and reveal how reheating can stabilize otherwise unstable deflagrations, though strong supercooling may still drive instability. The findings have significant implications for electroweak baryogenesis and gravitational wave production, suggesting that stability is highly sensitive to model parameters and that instability may alter the cosmological consequences of the phase transition.

Abstract

In a cosmological first-order phase transition, bubbles of the stable phase nucleate and expand in the supercooled metastable phase. In many cases, the growth of bubbles reaches a stationary state, with bubble walls propagating as detonations or deflagrations. However, these hydrodynamical solutions may be unstable under corrugation of the interface. Such instability may drastically alter some of the cosmological consequences of the phase transition. Here, we study the hydrodynamical stability of deflagration fronts. We improve upon previous studies by making a more careful and detailed analysis. In particular, we take into account the fact that the equation of motion for the phase interface depends separately on the temperature and fluid velocity on each side of the wall. Fluid variables on each side of the wall are similar for weakly first-order phase transitions, but differ significantly for stronger phase transitions. As a consequence, we find that, for large enough supercooling, any subsonic wall velocity becomes unstable. Moreover, as the velocity approaches the speed of sound, perturbations become unstable on all wavelengths. For smaller supercooling and small wall velocities, our results agree with those of previous works. Essentially, perturbations on large wavelengths are unstable, unless the wall velocity is higher than a critical value. We also find a previously unobserved range of marginally unstable wavelengths. We analyze the dynamical relevance of the instabilities, and we estimate the characteristic time and length scales associated to their growth. We discuss the implications for the electroweak phase transition and its cosmological consequences.

Figures (13)

• Figure 1: Sketch of a deflagration in the wall frame.
• Figure 2: Sketch of fluid velocity and temperature profiles for a deflagration, in the reference frame of the bubble center.
• Figure 3: Sketch of fluid velocity profile for strong and Jouguet deflagrations.
• Figure 4: The real part of the dispersion relations $q_2(\Omega)$ (in black) and $q_3(\Omega)$ (in blue) for different values of the imaginary part $\mathrm{Im}(\Omega)$. Gray dashed lines indicate the asymptotes of these solutions. We also show the special solution $q_1(\Omega)$ in a dotted red line.
• Figure 5: $\hat{\Omega}$ vs $kd$ in the small $v_w$, small $L$ and small $\hat{\Omega}/v_w$ approximation, for $\bar{L}=0.01$, $T_+/T_c=0.995$ ($v_c\simeq 0.07$), and $v_w=0.05$ (left), $0.007$ (center) and $0.003$ (right).