Strong First-Order Electroweak Phase Transition and Gravitational Waves in a $\mathbb{Z}_4$ Fermion-Scalar Dark Matter Model
We investigate whether a minimal $\mathbb{Z}_4$-symmetric fermion-scalar extension of the Standard Model can simultaneously realise viable dark matter, a strong electroweak phase transition, and a stochastic gravitational-wave signal. The model contains a real scalar singlet and a Dirac fermion, allowing thermal two-component dark matter, mixed WIMP-FIMP histories, and an effectively fermionic relic abundance generated by scalar decays. We impose theoretical consistency, the correct electroweak vacuum, and dark-matter constraints from relic density, direct detection, and invisible Higgs decays before using the surviving points as input for the finite-temperature analysis. This reveals that the compatibility between dark matter and a strong first-order electroweak phase transition is highly selective. After current dark-matter constraints are imposed, the strong-transition criterion along the Higgs direction is satisfied only in two viable regimes: the thermal two-component case with $M_ψ 2M_ψ$. By contrast, the thermal regime with $M_S<M_ψ$ and the stable mixed WIMP-FIMP scenario with $M_S<2M_ψ$ are largely concentrated at small portal couplings or near the Higgs-resonance region, and do not yield a strong transition in the parameter space considered. The successful transitions typically proceed through an intermediate singlet-like phase. For representative nucleating benchmark points in the viable strong-transition regions, we compute the gravitational-wave spectra from sound waves and turbulence. Some spectra enter the projected reach of future space-based interferometers, showing that detectable signals arise only in selected dark-matter-compatible regions where a sufficiently active Higgs portal appears in correlated combination with the scalar mass and the remaining dark sector parameters.