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

TL;DR

This paper investigates how modifications to black hole perturbation potentials in alternative gravity theories can be constrained using gravitational-wave data, proposing a focus on the potential's behavior at the light ring for more robust tests.

Contribution

It introduces a new approach to constrain black hole potential modifications by focusing on the potential's value and curvature at the light ring, improving robustness over traditional quasinormal mode analysis.

Findings

Bounds on individual potential parameters are not robust.

Potential and its second derivative at the light ring can be constrained reliably.

This method facilitates comparison between gravitational-wave and black hole shadow observations.

Abstract

In modified theories of gravity, the potentials appearing in the Schr\"odinger-like equations that describe perturbations of non-rotating black holes are also modified. In this paper we ask: can these modifications be constrained with high-precision gravitational-wave measurements of the black hole's quasinormal mode frequencies? We expand the modifications in a small perturbative parameter regulating the deviation from the general-relativistic potential, and in powers of $M/r$. We compute the quasinormal modes of the modified potential up to quadratic order in the perturbative parameter. Then we use 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. In both cases, we find that the bounds on the individual parameters are not robust. Because quasinormal mode frequencies are related to the behavior of the perturbation potential near the light ring, we propose a different strategy. Inspired by Wentzel-Kramers-Brillouin (WKB) theory, we demonstrate that the value of the potential and of 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.