Impact of cosmic expansion on gravitational wave spectra from strongly supercooled first-order phase transitions

We compute the gravitational wave spectra from strongly supercooled first-order phase transitions, explicitly incorporating the evolution of the background metric across the transition from thermal inflation to radiation domination. We find that the spectral shape remains largely unchanged apart from a causality-induced super-horizon tail. However, in contrast to standard expectations, for slow transitions we show that the peak amplitude and frequency exhibit a weaker dependence on the transition rate $\beta$ than the usual scaling of $\propto {\beta }^{-2}$ and $\propto \beta$, respectively.