Confronting eikonal and post-Kerr methods with numerical evolution of scalar field perturbations in spacetimes beyond Kerr

The accurate computation of quasinormal modes from rotating black holes beyond general relativity is crucial for testing fundamental physics with gravitational waves. In this study, we assess the accuracy of the eikonal and post-Kerr approximations in predicting the quasinormal mode spectrum of a scalar field on a deformed Kerr spacetime. To obtain benchmark results and to analyze the ringdown dynamics from generic perturbations, we further employ a 2+1-dimensional numerical time-evolution framework. This approach enables a systematic quantification of theoretical uncertainties across multiple angular harmonics, a broad range of spin parameters, and progressively stronger deviations from the Kerr geometry. We then confront these modeling errors with simple projections of statistical uncertainties in quasinormal mode frequencies as a function of the signal-to-noise ratio, thereby exploring the domain of validity of approximate methods for prospective high-precision black-hole spectroscopy. We also report that near-horizon deformations can affect prograde and retrograde modes differently and provide a geometrical explanation.