Constraining modifications of black hole perturbation potentials near the light ring with quasinormal modes

In modified theories of gravity, the potentials appearing in theSchr\"odinger-like equations that describe perturbations of non-rotating blackholes are also modified. In this paper we ask: can these modifications beconstrained with high-precision gravitational-wave measurements of the blackhole's quasinormal mode frequencies? We expand the modifications in a smallperturbative parameter regulating the deviation from the general-relativisticpotential, and in powers of $M/r$. We compute the quasinormal modes of themodified potential up to quadratic order in the perturbative parameter. Then weuse Markov-chain-Monte-Carlo (MCMC) methods to recover the coefficients in the$M/r$ expansion in an ``optimistic'' scenario where we vary them one at a time,and in a ``pessimistic'' scenario where we vary them all simultaneously. Inboth cases, we find that the bounds on the individual parameters are notrobust. Because quasinormal mode frequencies are related to the behavior of theperturbation potential near the light ring, we propose a different strategy.Inspired by Wentzel-Kramers-Brillouin (WKB) theory, we demonstrate that thevalue of the potential and of its second derivative at the light ring can berobustly constrained. These constraints allow for a more direct comparisonbetween tests based on black hole spectroscopy and observations of black hole`shadows'' by the Event Horizon Telescope and future instruments.