Gravitational odd-parity perturbation of a Horndeski hairy black hole: quasinormal mode and parameter constraint

During the binary black hole coalescence, gravitational waves emitted at the ringdown stage can be well described by black hole perturbation theory, where the quasinormal modes (QNMs) become the important ingredient in modeling the pattern waveform. In general relativity (GR), the QNMs can be obtained from solving the Regge–Wheeler (RW) equation of a non-rotating black hole. While in Horndeski gravity, the isospectrality between the odd and even parity perturbations is broken due to the scalar field, the odd perturbation equation can be simplified into a modified RW equation from the perturbed action. In this paper, we propose a new auxiliary field and tortoise coordinate to refine the modified RW equation in Horndeski gravity, and calculate the QNM frequencies of the odd perturbation of a specific hairy black hole. We find that this proposal not only cures the superluminal propagation addressed in the previous literature, but also holds the original QNM spectrum of the odd perturbation. Moreover, our results indicate that such a Horndeski hairy black hole is stable under the odd perturbation, which is also verified by the time evolution of the perturbation. In particular, in contrast to GR, the modes with \(\ell =2\) can decay faster than modes with \(\ell >2\) for a certain range of the Horndeski hair, and the link between the null geodesics and QNM for the odd perturbation in the current theory is violated. We then use the ringdown QNMs to preliminarily investigate the signal-to-noise ratio (SNR) rescaled measurement error of the Horndeski hair. We obtain significant effects of the angular momentum and overtone on the error bound of the hair parameter. We hope that our findings will inspire further theoretical and phenomenological work on the testing of the no-hair theorem of black holes using gravitational wave physics.