Hair loss in parity violating gravity

The recent detection of gravitational waves by the LIGO/VIRGO collaboration has allowed for the first tests of Einstein's theory in the extreme gravity regime, where the gravitational interaction is simultaneously strong, non-linear and dynamical. One such test concerns the rate at which the binaries inspiral, or equivalently the rate at which the gravitational wave frequency increases, which can constrain the existence of hairy black holes. This is because black holes with scalar hair typically excite dipole radiation, which in turn leads to a faster decay rate and frequency chirping. In this paper, we present mathematical proofs that scalar hair is not sourced in stationary, asymptotically flat, and axisymmetric spacetimes, including those appropriate to stars and black holes, when considering extensions of Einstein's theory that break parity in gravity, focusing on dynamical Chern-Simons theory as a particular toy model. This result implies that current electromagnetic observations of accreting black hole systems or of binary pulsars cannot constrain parity violation in gravity today.