Dark Matter and Electroweak Baryogenesis with Spontaneous $CP$ Violation in the Early Universe

Dark matter (DM) and the baryon asymmetry of the universe (BAU) are among the most compelling indications of physics beyond the Standard Model. We revisit the inelastic Higgs-portal complex singlet, a minimal framework in which a complex scalar splits into two nearly degenerate real states, with an off-diagonal Higgs-portal interaction that drives coannihilation to set the relic density, while the elastic DM-Higgs coupling can be tuned small enough to evade direct-detection limits. This setup naturally supports a strong first-order electroweak phase transition (SFOEWPT) and can account for the long-standing Galactic Center gamma-ray excess (GCE) via present-day DM annihilation into Higgs pairs. In this work, we show that the same framework, extended by a $Z_2$-symmetric dimension-6 $CP$-violating top Yukawa operator, can also generate the BAU via the electroweak baryogenesis (EWBG) mechanism. The cosmological history involves a two-step electroweak phase transition: first, the singlet fields acquire nonzero vacuum expectation values (vevs); then a strongly first-order transition occurs in which the Higgs develops its nonzero vev while the singlet vevs vanish. During this second step, both fields remain nonzero only within the advancing bubble wall, generating wall-localized $CP$ violation that biases sphaleron transitions and enables EWBG. After the phase transition, $CP$ and $Z_2$ symmetries are restored: the lightest singlet state becomes a stable DM candidate, while the vanishing singlet vevs allow the model to naturally satisfy the stringent constraints on $CP$ violation. We delineate the SFOEWPT-favored parameter space, identifying the criteria for the two-step phase transition region that simultaneously yields the observed BAU and relic density, explains the GCE, and predicts gravitational wave spectra accessible to next-generation space-based detectors.