Maximal GW amplitude from bubble collisions in supercooled phase transitions

We extend analytic formulas for the gravitational-wave (GW) spectrum from first-order phase transitions to include cosmic expansion under the thin-wall and envelope approximations. We demonstrate that even for strongly supercooled transitions the GW amplitude is bounded from above. This conclusion is explicitly verified for several representative nucleation histories, including delta-function, power-law, and power-exponential types. Moreover, the spectral shape, amplitude, and peak frequency remain largely unaffected by the details of the nucleation rate once expressed in terms of the conformal variables evaluated at an appropriately defined characteristic collision time.