Two-Component Dark Matter and Strongly First-Order Electroweak Phase Transition via a Two-Step Mechanism in the Inert Doublet plus Real Singlet Model

We study a two-component dark matter model in which a real singlet scalar and an inert doublet scalar are simultaneously present alongside the Standard Model, each stabilized by an independent discrete symmetry. The real singlet undergoes a strong first-order transition at high temperature, establishing an intermediate phase with a non-zero singlet vacuum expectation value, which in turn generates the barrier required for the Higgs field to complete a strongly first-order electroweak transition at a lower temperature. This two-step mechanism naturally enforces the condition $v_c\mathrm{/}{T}_c\ge 1$, the sphaleron washout-avoidance criterion necessary for electroweak baryogenesis. The gravitational wave spectrum from both steps is computed; the second step produces a decihertz signal, peaking near $0.08$–$0.34\mathrm{H}\mathrm{z}$ across the viable parameter space and $0.14$–$0.22\mathrm{H}\mathrm{z}$ at the benchmark points, that reaches the sensitivity of DECIGO for the strongest transitions and of BBO more widely, while the first remains sub-threshold for all planned detectors. At the coupling values required to drive the phase transition, the singlet is generically underabundant as a thermal relic, so the inert doublet carries the dominant fraction of the observed relic density. Confronting the model with current constraints, we find that the LUX-ZEPLIN 2024 result excludes the entire electroweak-phase-transition-viable parameter space of the singlet sector across the full mass range, even after rescaling by the singlet relic fraction, establishing a complete no-go for the singlet as a viable thermal dark matter candidate in this framework.