Effective picture of bubble expansion
  Recently the thermal friction on an expanding bubble from the cosmic
first-order phase transition has been calculated to all orders of the
interactions between the bubble wall and thermal plasma, leading to a
$\gamma^2$-scaling instead of the previously estimated $\gamma^1$-scaling for
the thermal friction exerted on a fast-moving bubble wall with a Lorentz factor
$\gamma$. We propose for the first time the effective equation of motion (EOM)
for an expanding bubble wall in the presence of an arbitrary $\gamma$-scaling
friction to compute the efficiency factor from bubble collisions, which, in the
case of $\gamma^2$-scaling friction, is found to be larger than the recently
updated estimation when the bubble walls collide after starting to approach a
constant terminal velocity, leading to a slightly larger signal of the
gravitational waves background from bubble collisions due to its quadratic
dependence on the bubble collision efficiency factor, although the
$\gamma^2$-scaling friction itself has already suppressed the contribution from
bubble collisions compared to that with $\gamma^1$-scaling friction. We also
suggest a phenomenological parameterization for the out-of-equilibrium term in
the Boltzmann equation that could reproduce the recently found
$(\gamma^2-1)$-scaling of the friction term in the effective EOM of an
expanding bubble wall, which merits further study in future numerical
simulations of bubble expansion and collisions.
