Bubble wall velocity with out-of-equilibrium corrections

TL;DR

The paper develops a first-principles treatment of bubble-wall propagation during a cosmological first-order electroweak phase transition by solving the coupled scalar-field, hydrodynamic, and Boltzmann equations beyond local thermal equilibrium using a spectral algorithm for the collision integral. Focusing on the minimal $ ext{Z}_2$-symmetric singlet extension of the SM, the study finds that out-of-equilibrium corrections significantly increase friction on the wall, slowing $v_w$ and broadening the wall, with substantial qualitative and quantitative changes to the phase-transition dynamics. These OOE effects enhance electroweak baryogenesis efficiency and modify gravitational wave predictions, potentially improving the viability of baryogenesis in parameter regions that could be probed by future detectors, though GW signals may be reduced relative to LTE expectations. Overall, the work demonstrates that incorporating full non-equilibrium dynamics is crucial for accurate predictions of cosmological relics from first-order phase transitions and sets the stage for incorporating additional species and more complex models in future analyses.

Abstract

We study how out-of-equilibrium effects modify the steady-state propagation of bubble walls during a cosmological first-order electroweak phase transition. Going beyond the local thermal equilibrium approximation, we numerically solve the coupled system of scalar field, hydrodynamic and Boltzmann equations using a spectral algorithm that allows a first-principle treatment of the collision integral. This approach enables a quantitative assessment of non-equilibrium perturbations in the plasma and their backreaction on the wall motion. Focusing on the singlet extension of the Standard Model as a minimal benchmark scenario, we find that out-of-equilibrium corrections substantially enhance the effective friction on the expanding front, leading to slower wall velocities and broader wall profiles compared to the equilibrium case. These modifications have significant implications for cosmological observables. For instance, they enhance the efficiency of electroweak baryogenesis, thus improving the viability of baryon asymmetry generation within realistic parameter regions that can also be probed by future gravitational wave interferometers.

Figures (16)

• Figure 1: Total pressure $P_{tot}$ in terms of the wall velocity $v_w$ for two benchmark points, BP1 with $m_s=80$ GeV and $\lambda_{hs}=0.36$ and BP2 with $m_s=122$ GeV and $\lambda_{hs}=0.47$. Dashed lines are for the pressure evaluated in LTE, while solid lines are for the pressure evaluated including out-of-equilibrium contributions. The other three parameters $\delta_s$, $L_h$ and $L_s$ are fixed to the corresponding solution.
• Figure 2: Left panel: Contour plot of the (out-of-equilibrium) wall velocity $v_w$ in the parameter space. The colour gradient shows the variation of $v_w$, with the colour bar serving as a legend and $v_w$ being constant on black lines. In the light grey region no deflagration/hybrid solution is found. Right panel: Relative correction $\delta v_w$ to the LTE wall velocity. The coloured region is restricted to the subspace where a solution exists in LTE. The dark grey band is where a deflagration/hybrid solution emerges when out-of-equilibrium contributions are included, while the light grey band is as in the left panel.
• Figure 3: Histogram of the wall velocity $v_w$ in LTE (yellow bins) and with out-of-equilibrium contributions included (blue bins).
• Figure 4: Left panel: Contour plot of the (out-of-equilibrium) wall width $L_h$ in the parameter space. In the light grey region no deflagration/hybrid solution is found. Right panel: Relative correction $\delta L_h$ to the LTE wall width. The coloured region is restricted to the subspace where a solution exists. The dark grey band is where a deflagration/hybrid solution emerges when out-of-equilibrium contributions are included, while the light grey band is as in the left panel.
• Figure 5: Left panel. Scatter plot of the relative corrections $\delta v_w\%$, $\delta L_h\%$ and$\delta L_s\%$ in terms of the LTE wall velocity $v_w^{(\rm LTE)}$. Right panel. Histogram of the relative corrections $\delta v_w\%$ and $|\delta L_h|\%$ normalised to one. The inside figure is a histogram of the value we find for $v_w$ in LTE and including out-of-equilibrium contributions.