  Dynamical Chern-Simons (DCS) modified gravity is an attractive, yet
relatively unexplored, candidate to an alternative theory of gravity. The DCS
correction couples a dynamical scalar field to the gravitational field. In this
framework, we analyze the perturbation formalism and stability properties of
spherically symmetric black holes. Assuming that no background scalar field is
present, gravitational perturbations with polar and axial parities decouple. We
find no effect of the Chern-Simons coupling on the polar sector, while axial
perturbations couple to the Chern-Simons scalar field. The axial sector can
develop strong instabilities if the coupling parameter beta, associated to the
dynamical coupling of the scalar field, is small enough; this yields a
constraint on beta which is much stronger than the constraints previously known
in the literature.
