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 γ 2 -scaling instead of the previously estimated γ 1 -scaling for the thermal friction exerted on a fast-moving bubble wall with a Lorentz factor γ. We propose for the first time the effective equation of motion (EOM) for an expanding bubble wall in the presence of an arbitrary γ-scaling friction to compute the efficiency factor from bubble collisions, which, in the case of γ 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 γ 2 -scaling friction itself has already suppressed the contribution from bubble collisions compared to that with γ 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 (γ 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.