General backreaction force of cosmological bubble expansion

The gravitational-wave energy-density spectra from cosmological first-order phase transitions crucially depend on the terminal wall velocity of asymptotic bubble expansion when the driving force from the effective potential difference is gradually balanced by the backreaction force from the thermal plasma. Much attention has previously focused on the backreaction force acting on the bubble wall alone but overlooked the backreaction forces on the sound shell and shock-wave front, if any, which have been both numerically and analytically accomplished in our previous studies but only for a bag equation of state. In this paper, we will generalize the backreaction force on bubble expansion beyond the simple bag model.