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

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In modified theories of gravity, the potentials appearing in the Schrodinger-like equations that describe perturbations of nonrotating black holes are also modified. Can these modifications be constrained with high-precision gravitational-wave measurements of the black hole’s quasinormal mode frequencies? By assuming the modifications to be proportional to a small perturbation parameter, and parametrized by a Taylor expansion in M/r, we compute the quasinormal modes of the modified potential up to quadratic order in the perturbative parameter. Either through a principal component analysis or via Markov-chain-Monte-Carlo methods we try to recover the Taylor coefficients in the M/r. In both cases, even if the overall reconstruction is good, we find that the bounds on the individual parameters are not robust. Because quasinormal mode frequencies are related to the behaviour of the perturbation potential near the light ring, we propose a different strategy. We map the Taylor expansion to the value of the potential and its derivatives at the peak by using Wentzelâ€“Kramersâ€“Brillouin theory, and we demonstrate that the value of the potential and its second derivative at the light ring can be robustly constrained. These constraints allow for a more direct comparison between tests based on black hole spectroscopy and observations of black hole “shadows” by the Event Horizon Telescope and future instruments.