Probing thermal leptogenesis and dark matter through primordial gravitational waves from a supercooled universe

We explore the cosmological dynamics of a supercooled first-order phase transition in the classically conformal $U(1)_{B-L}$ extension of the Standard Model, where radiative symmetry breaking simultaneously generates the right-handed neutrino (RHN) masses, and a strong stochastic gravitational-wave (GW) background. The slow decay of the scalar field into RHNs can induce an early matter-dominated (EMD) era whose duration is sensitive to the RHN mass and gauge coupling $g^\prime$. This non-standard cosmological phase reshapes the GW spectrum and leaves a distinctive RHN-mass-dependent spectral distortion that correlates with the flavour regime of thermal leptogenesis. Within this framework, one RHN can serve as a dark matter candidate produced nonthermally from scalar decays, while the remaining states generate the baryon asymmetry via thermal leptogenesis. For $g^\prime=0.5$, we identify such a parameter region, and show that with singlet extensions, even with a smaller gauge coupling, one can realise this mechanism for the three-flavour regime. The resulting GW signals, amplified by supercooling and modified by EMD, provide a unique window to probe the scale and flavour structure of leptogenesis in future high-frequency GW observations.