Spectroscopy of bumpy BHs: non-rotating case

Recent detections of gravitational waves have made black hole quasinormal modes a powerful tool in testing predictions of general relativity. Understanding the spectrum of these quasinormal modes in a broad class of theories beyond general relativity and a variety of astrophysical environments around black holes remains vital. In this work, we study the quasinormal mode spectrum of parametrized deformations of a non-rotating black hole in the vacuum. Following Vigeland and Hughes, we model these parametrized deformations as axisymmetric multipole moments in the Weyl coordinates with amplitudes much less than the amplitude of the Schwarzschild potential. These tiny bumps in the black hole geometry satisfy the linearized vacuum Einstein equations and are asymptotically flat. We use the recently developed modified Teukolsky formalism to derive one decoupled differential equation for the radiative Weyl scalar $\Psi_0$. We then use the eigenvalue perturbation method to compute the quasinormal mode frequency shifts of both even- and odd-parity modes with $\ell=2,3$ and up to the overtone number $n=2$ for the Weyl multipoles with $\ell_W=2,3$. Our calculation provides an avenue to directly connect the multipole moments of a modified black hole spacetime to the QNM frequency shifts in a parametric way.