Field-theoretic derivation of bubble-wall force

TL;DR

This work provides a covariant, first-principles derivation of the force on expanding bubbles during a first-order cosmological phase transition, casting the friction as the change in plasma entropy across the bubble with a γ^2 scaling in local thermal equilibrium. It develops a general relation F/V = -ΔP = (γ^2−1) T Δs from the renormalized energy-momentum tensor and applies it to a real scalar toy model and to the Standard Model with simple portal extensions. In LTE, bubbles rapidly reach a terminal velocity, whereas in the ballistic limit the friction saturates, potentially allowing runaway expansion; heavy degrees of freedom can dominate the friction in strong transitions. The framework unifies and extends previous semiclassical analyses, providing a principled basis for predicting bubble dynamics relevant to gravitational-wave spectra and baryogenesis.

Abstract

We derive a general quantum field theoretic formula for the force acting on expanding bubbles of a first order phase transition in the early Universe setting. In the thermodynamic limit the force is proportional to the entropy increase across the bubble of active species that exert a force on the bubble interface. When local thermal equilibrium is attained, we find a strong friction force which grows as the Lorentz factor squared, such that the bubbles quickly reach stationary state and cannot run away. We also study an opposite case when scatterings are negligible across the wall (ballistic limit), finding that the force saturates for moderate Lorentz factors thus allowing for a runaway behavior. We apply our formalism to a massive real scalar field, the standard model and its simple portal extension. For completeness, we also present a derivation of the renormalized, one-loop, thermal energy-momentum tensor for the standard model and demonstrate its gauge independence.

Figures (15)

• Figure 1: The propagation of a particle (horizontal solid black line) in presence of field dependent mass insertions, $m(x)=\sqrt{\lambda/2}\phi_0(x)$ (vertical dashed lines).
• Figure 2: The Feynman diagram that contributes at one-loop level to the energy-momentum tensor of a real scalar field. The dashed circle denotes the scalar propagator and the cross denotes the energy-momentum tensor insertion.
• Figure 3: The dimensionless change in the entropy density $\Delta s/T^3$ (solid blue) of a real scalar field thermal plasma across the nucleated bubble as a function of the scalar mass $m/T$. Since the scalar mass $m=\sqrt{\lambda/2}\phi_0(x)$ increases across the bubble as the scalar condensate increases, the entropy in the broken phase decreases. The maximum amount by which the entropy density can change is $2\pi^2T^3/45$, which is formally reached when $m\rightarrow \infty$, when the entropy density inside the bubble tends to zero (horizontal dashed).
• Figure 4: The bubble Lorentz factor $\gamma(v)$ as a function of the scalar mass $m/T$ for $\Delta {\cal P}=-0.01/\beta^4$ (green dashed), $\Delta {\cal P}=-0.1/\beta^4$ (solid black) and $\Delta {\cal P}=-0.5/\beta^4$ (solid orange).
• Figure 5: The bubble speed $v/c$ as a function of the scalar mass $m/T$ for the same choice of the parameters as in figure \ref{['bubble speed gamma']}: $\Delta {\cal P}=-0.01/\beta^4$ (green dashed), $\Delta {\cal P}=-0.1/\beta^4$ (solid black) and $\Delta {\cal P}=-0.5/\beta^4$ (solid orange).