Gravitational mode mixing around black holes in scalar-tensor theories with parity-violating terms

We investigate black holes and gravitational perturbations when both the scalar Gauss-Bonnet and dynamical Chern-Simons gravity sectors coexist in addition to the Einstein-Hilbert term, and both sectors are coupled to a single canonically normalized scalar field. The presence of the scalar Gauss-Bonnet gravity sector allows the scalar field to possess a non-vanishing background solution, resulting in additional couplings between odd and even-type gravitational perturbations arising from the dynamical Chern-Simons gravity sector. We illustrate the impact of these even-odd gravitational couplings in gravitational perturbations around a static spherically symmetric black hole. Although the couplings between the odd and even-type gravitational perturbations are known to appear in purely tensorial gravity theories with higher curvature corrections, we demonstrate it in scalar-tensor theories.