From friction scaling to an efficient method for estimating bubble wall velocity

Abstract

We present a unified description of first-order cosmological phase transition dynamics that links the phenomenological friction model employed in hydrodynamic simulations to the microscopic treatment based on Boltzmann equations. We derive an approximate analytical expression for the chemical potential and demonstrate that the resulting friction parameter $\tildeη$ follows a simple power-law dependence on the transition strength ($\propto v_n^4/T_n^4$). Incorporating this scaling into a phenomenological framework accurately reproduces the terminal wall velocities obtained from the full microscopic analysis performed using \texttt{WallGo}. This approach offers an efficient method to quantify out-of-equilibrium contributions to friction and reliably estimate bubble-wall velocities.

Figures (4)

• Figure 1: Left panel: Comparison of entropy production sources for the xSM benchmark obtained from the numerical solution of the Boltzmann equations, using WallGoEkstedt:2024fyq, with two approximate descriptions: the phenomenological ansatz \ref{['eq:s_ansatz']} and the CE expansion \ref{['eq:s_Ch_En']}. In both approximate cases, the overall normalization of the entropy source is fitted to the numerical Boltzmann result. This illustrates that both approaches accurately reproduce the shape of the microscopic entropy source. Right panel: Terminal bubble-wall velocities computed using the hydrodynamic matching conditions with entropy production given with eq. \ref{['eq:match_DSb']}, where the friction parameters are fixed by the fits shown in the left panel, compared to the wall velocities obtained directly from WallGo.
• Figure 2: The rescaled friction parameter $\tilde{\eta}$, extracted from the WallGoEkstedt:2024fyq friction, shown as a function of the ratio $\tfrac{v_n}{T_n}$. The red dashed line indicates the power-law relation given in eq. \ref{['eq:power_low']}, with $\zeta^{\rm (WG)} = 2.1\times10^{-4}$ obtained from fitting to the data.
• Figure 3: Left panel: terminal wall velocity obtained by solving numerically the equation for the chemical potential \ref{['eq:mu']} and using $v^{\rm (\mu)}_w$, versus the velocity predicted by WallGo code Ekstedt:2024fyq$v^{\rm (WG)}_w$. The only out-of-equilibrium process considered was the top annihilation $t\bar{t}\to g \bar{g}$. Right panel: The wall velocity computed based on the extrapolation of the phenomenological power-low rule \ref{['eq:power_low']}$v^{\rm (scal.)}_w$ against WallGo results. The free parameter in the power-low relation was inferred from fitting to the solutions obtained from the chemical potential, and reads $\zeta^{(\mu)}\approx2.5\times 10^{-4}$. In both panels the error bars represent numerical uncertainty of the WallGo solutions. Benchmarks which do not satisfy the perturbativity condition $v_n/T_n>g$ were plotted with lighter shade.
• Figure 4: Terminal wall velocity in the xSM model versus $\lambda_{HS}$ coupling for $m_s=90$ GeV and $\lambda_s=1$ (c.f. with ref. vandeVis:2025plm). Dashed, purple line shows the LTE limit, where out-of-equilibrium effects are neglected. Dashed, red line corresponds to the results obtained with near equilibrium approximation for the entropy source \ref{['eq:s_Ch_En']} with theoretically derived amplitude $\kappa=2.3/T_n$ as in Ekstedt:2025awx. The solid blue line shows the approximate method based on the estimation of the chemical potential \ref{['eq:frict_approx']}, assuming the only particle out of equilibrium is the top. The dotted blue line was obtained by extrapolating the phenomenological scaling of $\tilde{\eta}$\ref{['eq:power_low']}, with the coefficient inferred from chemical potential method (see discussion in the text). The orange and green lines depict full numerical solutions to the Boltzmann equations obtained with WallGoEkstedt:2024fyq, accounting only for the out-of-equilibrium top (green line) and for out-of-equilibrium top together with massive gauge bosons (orange line). The error bars on these lines show the uncertainties of the numerical solutions.