Fluctuation-Induced Friction in Bubble-Wall Dynamics of Cosmological First-Order Phase Transitions

Abstract

We study bubble-wall dynamics in cosmological first-order phase transitions in a two-scalar-field model, where the wall is formed by $φ$ and an additional real scalar $s$ couples through a portal interaction. We evolve the coupled classical field equations on the lattice and demonstrate that for an initial Bose--Einstein distribution of $s$ fluctuations at the nucleation temperature $T_n$, the resulting patchy background intermittently modulates the local driving pressure on the wall. The wall therefore undergoes alternating episodes of acceleration and deceleration and approaches a quasi-stationary propagation regime with a smaller time-averaged speed than in the decoupled limit. We further identify three familiar propagation profiles -- deflagration, detonation, and hybrid -- distinguished by where the dynamical $s$-sector energy density is concentrated relative to the wall. These effects can impact gravitational wave and baryogenesis predictions.

Figures (5)

• Figure 1: Potential slices $V(\phi,s,T_n)$ as functions of $\phi/T_c$ at the nucleation temperature $T_n$, evaluated for five fixed values of $s$ (see legend). Model parameters are the same as in Table \ref{['tab:benchmark_M2']}, and $\lambda_{\phi s}=1$.
• Figure 2: Spacetime evolution of the field $\phi$ along the $z$ axis. We show the line-out $\phi(t,z)$ taken on the central slice ($x=y=0$), i.e., along the $z$ axis through the bubble center. Top: vanishing portal coupling, $\lambda_{\phi s}=0$. Bottom: finite coupling, $\lambda_{\phi s}=7.5$.
• Figure 3: Bubble-wall velocity evolution extracted from the isosurface condition $\phi_\ast\equiv(\phi_{\rm false}+\phi_{\rm true})/2$. Black solid: $\lambda_{\phi s}=0$. Blue solid: $\lambda_{\phi s}=7.5$. The horizontal red dashed lines indicate the corresponding terminal velocities estimated at $tT_c=200$.
• Figure 4: Time evolution of a vacuum bubble in the two-scalar-field system. The simulation volume and discretization are $L T_c=500$, $dx\,T_c=0.1$, and $dt=0.2\,dx$. The parameters entering $V(\phi,s,T)$ are chosen to match the benchmark point used in the $3+1$ dimensions simulations, while the portal coupling is varied as $\lambda_{\phi s}=0.65$, $1.1$, and $1.8$ from top to bottom.
• Figure 5: Profiles of the $s$ dynamical energy density (left axis, $10^{-1}\rho_s/T_c^4$) and the field $\phi/T_c$ at the fixed time $tT_c=180$, shown as functions of the coordinate $\xi\equiv x/t$. In this representation $\xi$ plays the role of a velocity coordinate (in units of $c$), so the wall position directly visualizes the propagation speed. From top to bottom the portal coupling is $\lambda_{\phi s}=0.65$, $1.1$, and $1.8$ (other potential parameters are kept fixed).