Spontaneous CP violation electroweak baryogenesis and gravitational wave through multi-step phase transitions

In a singlet pseudoscalar extension of the two-Higgs-doublet model we discuss spontaneous CP violation electroweak baryogenesis via two different patterns of phase transitions (PTs): (i) two-step PTs whose first-step and second-step are strongly first-order; (ii) three-step PTs whose first-step is second-order and the second-step and third-step are strongly first-order. For the case of the two-step pattern, the first-step PT takes place at a high temperature, converting the origin phase into an electroweak symmetry broken phase and breaking the CP symmetry spontaneously. Thus, the baryon number is produced during the first-step PT. At the second-step PT, the phase is converted into the observed vacuum at zero temperature, and the CP-symmetry is restored. In both phases the sphaleron processes are sufficiently suppressed, which keep the baryon number unchanged. For the case of the three-step PTs, the pseudoscalar field firstly acquires a nonzero VEV, and VEVs of other fields still remain zero during the first-step PT. The following PTs and electroweak baryogenesis are similar to the case of the two-step PTs. In addition, the gravitational wave spectra can have one or two peaks through the two-step and the three-step PTs, and we discuss the detectability at the future gravitational wave detectors.