Hydrodynamic backreaction force of cosmological bubble expansion

As a promising probe for the new physics beyond the standard model of particle physics in the early Universe, the predictions for the stochastic gravitational wave background from a cosmological first-order phase transition heavily rely on the bubble wall velocity determined by the bubble expansion dynamics. The bubble expansion dynamics is governed by the competition between the driving force from the effective potential difference and the backreaction force from a sum of the thermal force and friction force induced by the temperature jumping and out-of-equilibrium effects across the bubble wall, respectively. In this paper, we propose a hydrodynamic evaluation on this total backreaction force for a non-runaway steady-state bubble expansion, which, after evaluated at the wall interface, exactly reproduces the pressure difference $\Delta_\mathrm{wall}[(\bar{\gamma}^2-1)w]$ obtained previously from the junction condition of the total energy-momentum tensor at the wall interface, where $w$ is the enthalpy and $\bar{\gamma}\equiv(1-\bar{v}^2)^{-1/2}$ is the Lorentz factor of the wall-frame fluid velocity $\bar{v}$.