  General relativity, though the most successful theory of gravity, has been
continuously modified to resolve its incompatibility with quantum mechanics and
explain the origin of dark energy or dark matter. One way to test these
modified gravity theories is to study the gravitational waves emitted during
the ringdown of binary mergers, which consist of quasinormal modes. In several
modified gravity theories, the even- and odd-parity gravitational perturbations
of non-rotating and slowly rotating black holes have different quasinormal mode
frequencies, breaking the isospectrality of general relativity. For black holes
with arbitrary spin in modified gravity, there were no avenues to compute
quasinormal modes except numerical relativity, until recent extensions of the
Teukolsky formalism. In this work, we describe how to use the modified
Teukolsky formalism to study isospectrality breaking in modified gravity. We
first introduce how definite-parity modes are defined through combinations of
Weyl scalars in general relativity, and then, we extend this definition to
modified gravity. We then use the eigenvalue perturbation method to show how
the degeneracy in quasinormal mode frequencies of different parity is broken in
modified gravity. To demonstrate our analysis, we also apply it to some
specific modified gravity theories. Our work lays the foundation for studying
isospectrality breaking of quasinormal modes in modified gravity for black
holes with arbitrary spin.
