Gravitational waves from high-temperature vacuum decay in scale-invariant models: nanohertz vs. millihertz regimes

We present a comprehensive analysis of high-temperature vacuum decay and the resulting stochastic gravitational wave (GW) background within the framework of general scale-invariant models. The effective potential is constructed to include tree-level contributions, the Coleman-Weinberg correction, finite-temperature effects, and the Daisy resummation technique, culminating in a high-temperature form. We investigate the dynamics of the first-order phase transition, calculating the critical and nucleation temperatures, the supercooling parameter, and the key transition parameters $α$ (transition strength) and $β$ (inverse duration). The vacuum decay is found to be dominated by sphaleron transitions rather than quantum tunneling. We compute the full GW spectrum arising from bubble collisions, sound waves, and turbulence. An extensive numerical scan reveals two distinct phenomenological regimes: one produces nanohertz-frequency GW signals potentially detectable by Pulsar Timing Arrays (PTA), while the other yields millihertz-frequency signals that are prime targets for future space-based interferometers like the Laser Interferometer Space Antenna (LISA).