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CP non-invariance is strongly limited by present experiments, while extrasources of CP-violation are needed for a successful baryogenesis. Motivated bythose observations we consider a model which predicts spontaneous violation ofCP at high temperature and restoration of CP at present temperature of theUniverse. In addition we propose a dark matter (DM) candidate that meets allknown properties of DM. Looking for a minimal model that satisfies the aboveconditions leads us to extending the Standard Model (SM) of fundamentalinteractions by adding a complex singlet scalar $S$. We impose the $CP$ and$Z_2$ symmetries on the scalar potential. With the complex vacuum expectationvalue of $S$ at the temperature higher than the EW phase transition, the $CP$symmetry is spontaneously broken and a strong first-order electro-weak phasetransition is easily realized. Introducing a dimension-6 effective operatorthat gives new complex contributions to the top quark mass, we show that it iseasy to yield the observed baryon asymmetry in our Universe. On the other hand,the $CP$ and $Z_2$ symmetries are recovered after the EW phase transition sothat the present strong constraints on $CP$ violation can be satisfied and thelighter of $\Re S$ or $\Im S$ can be the dark matter candidate. By scanning theparameter space, we find regions where the model can explain the dark matterrelic abundance and the baryon asymmetry simultaneously while satisfying allother experimental constraints. Finally, we discuss the explicit $CP$ symmetrybreaking in the scalar potential that can help dynamically eliminate thedomains producing the negative baryon asymmetry. It is found that this can beachieved by a tiny explicit $CP$-violating phase of ${\cal O}(10^{-15})$.

Spontaneous $CP$-Violating Electroweak Baryogenesis and Dark Matter from a Complex Singlet Scalar