Constraining the geometry of rotating black holes with eikonal QNMs
Quasinormal modes of black holes provide a central avenue for confronting general relativity with gravitational-wave observations from the ringdown stage of compact binary coalescence. However, inferring black-hole properties or the background spacetime requires theory-dependent input. In this work, we connect a calibrated eikonal method, which relates quasinormal mode shifts to local metric properties near the light ring, with the underlying rotating black hole metric. Using Bayesian inference, we compare a metric-specific model based on the global Konoplya-Rezzolla-Zhidenko parametrization with a metric-agnostic approach based on local values and radial derivatives of the metric functions at the light ring. For each model, we perform both one-parameter and simultaneous multi-parameter analyses and quantify how their priors map onto the same local quantities. We find that uniform priors on the metric-specific coefficients can induce strongly nonuniform and skewed priors on these local metric quantities. For an injection compatible with general relativity, both models remain consistent with Kerr and yield similar posteriors for the orbital frequency and Lyapunov exponent despite their different induced priors. For injections representing deviations from general relativity, one-parameter analyses can fail to recover the injected local deviations and, in the metric-specific case, can bias the global metric reconstruction. The multi-parameter analyses of both models yield consistent constraints, suggesting sufficient robustness. However, they introduce substantial degeneracies and differences in marginalized parameters, demonstrating practical limitations when considering more flexible models. Finally, we apply the framework to approximate posterior information from the ringdown analysis of GW250114 and find that both models are consistent with the Kerr metric in general relativity.