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 nonrunaway steady-state bubble expansion, which, after evaluated at the wall interface, exactly reproduces the pressure difference ${\mathrm{\Delta }}_{\text{wall}}[({\overline{\gamma }}^2-1)w]$ obtained previously from the junction condition of the total energy-momentum tensor at the wall interface, where $w$ is the enthalpy and $\overline{\gamma }\equiv (1-{\overline{v}}^2{)}^{-1/2}$ is the Lorentz factor of the wall-frame fluid velocity $\overline{v}$.

Figure 1

In the first panel, our hydrodynamic backreaction force exactly matches the driving force. In the second panel, the wall contribution to our hydrodynamic backreaction force exactly matches the previous estimation ${\mathrm{\Delta }}_{\text{wall}}(w{\overline{\gamma }}^2{\overline{v}}^2)$, both of which are balanced by the driving force acting on the wall. In the last two panel, $\mathrm{\Delta }(w{\overline{\gamma }}^2{\overline{v}}^2)=({\gamma }_w^2-1)\mathrm{\Delta }w$ are shown with respect to $v_w$ (left one) and ${\gamma }_w$ (right one). The asymptotic behaviors are indicated by the gray dashed lines.

Figure 2

The profiles of fluid velocity (first row), enthalpy $w/w_N$ (second row), temperature $T/T_N$ (third row), and $(w{\overline{v}}^2{\overline{\gamma }}^2)/(w_N{v}_w^2{\gamma }_w^2)$ (fourth row) for expansion modes of detonation (right column), deflagration (left column), and hybrid (middle column) types.