Analytic derivation of GW spectrum from bubble collisions in FLRW Universe

We generalize the analytic formula for the gravitational-wave spectrum from bubble collisions during a cosmological first-order phase transition, under the thin-wall and envelope approximations, by incorporating the effect of cosmic expansion in the FLRW metric. Along with presenting the complete analytic expression and corresponding numerical results, we also derive simplified formulas valid in the large- and small-$k$ limits, as well as in the Minkovski limit. The latter expansion reveals that the Minkovski approximation breaks down for $\beta / H_* \lesssim 10$, where $\beta$ denotes the inverse duration of the phase transition and $H_*$ the Hubble parameter at its completion. Furthermore, the next-to-leading-order term contributes about a $10\%$ correction for $\beta / H_* \sim 140$, a typical value for the electroweak phase transition.