Circularly polarized gravitational waves from parity-violating scalar-tensor theory

We study both primordial gravitational waves (GWs) and scalar-induced gravitational waves (SIGWs) in a class of the parity-violating scalar-tensor (PVST) theory, of which the Lagrangian is the linear combination of seven ghost-free parity-violating scalar-tensor monomials dubbed the "Qi-Xiu" Lagrangians. At linear order, we obtain the quadratic action for tensor perturbations and show that parity-violating terms associated with L_1, L_2, L_5, L_6, and L_7 render the tensor propagation polarization dependent, leading to chiral primordial spectra and a nonvanishing degree of circular polarization. At second order, we derive the equation of motion for SIGWs and identify the explicit parity-violating source terms. In particular, L_3 and L_4 enter exclusively through the source term for SIGWs, allowing parity violation to arise even when the linear GWs' propagation remains effectively general-relativity-like. During the radiation-dominated era, we compute the fractional energy density of SIGWs for both monochromatic and log-normal curvature power spectra. We find that, around the peak frequency, SIGWs in PVST gravity exhibit characteristic deviations from those in general relativity, resulting in a nonzero degree of circular polarization.