Phase Structure and Gravitational-Wave Phenomenology of a Thermal First-Order Phase Transition

Abstract

We investigate the phase structure and gravitational-wave (GW) phenomenology of a cosmological first-order phase transition described by the finite-temperature effective potential $V(\varphi ,T)=D(T^2-T_0^2){\varphi }^2-ET{\varphi }^3+\lambda {\varphi }^4/4$. We derive the critical temperature and the broken-phase order parameter and identify the dimensionless combination $E^2/(D\lambda )$ that controls the critical-temperature shift. We then construct a dense numerical atlas containing $30000$ parameter points and map the resulting transition parameters onto the characteristic GW frequency and peak amplitude. The scan resolves the multidimensional correlations among $T_{\ast }$, $\alpha$, $\beta /H_{\ast }$, $v_w$, $f_{\mathrm{peak}}$ and ${\Omega }_{\mathrm{GW}}^{\mathrm{peak}}{h}^2$. The present analysis is phenomenological: $T_{\ast }$ is defined by the prescription $T_{\ast }=0.95T_c$, while $\beta /H_{\ast }$ and $v_w$ are treated as scan inputs. Consequently, the resulting GW signals are not interpreted as first-principles predictions. We identify the additional ingredients required for a predictive calculation, including the thermal bounce action, nucleation and percolation temperatures, the transition duration and a microscopic treatment of bubble-wall friction. The resulting framework provides a systematic numerical characterization of the connection between the phase structure of the finite-temperature potential and the corresponding phenomenological GW parameter space.

24 pages, 8 Figures, 3 Tables