Perturbations of spinning black holes in dynamical Chern Simons gravity I: Parity breaking

The extension of Teukolsky formalism from General Relativity (GR) to modified gravity allows the study of gravitational perturbations of black holes with general spin in a wide class of beyond-GR (bGR) theories. Modified Teukolsky equations were found for bGR Petrov type I spacetimes which deviate not significantly from Petrov type D spacetimes in GR. This modified Teukolsky formalism can potentially be used to study these parity-breaking bGR theories, where the quasinormal modes (QNMs) of even and odd parity modes are shifted differently, such as dynamical Chern-Simons (dCS) and Einstein-dilaton Gauss-Bonnet (EdGB) theories. However, unlike metric perturbations, the (modified) Teukolsky equations do not naturally have definite parity. In this work, we first introduce how the definite-parity modes of Teukolsky equations are defined in GR and then show that this definition can be extended to modified gravity. We then derive the equations of motion of these definite-parity modes from the modified Teukolsky equations. We observe that parity is generally broken since the source terms in the modified Teukolsky equations usually mix modes with different frequencies. As a demonstration, we compare the parity-breaking structure of several simple examples in dCS and EdGB to the results using metric perturbations. We also briefly illustrate how to evaluate the shift of QNMs of these definite-parity modes within our modified Teukolsky formalism.