Challenges in Quasinormal Mode Extraction: Perspectives from Numerical solutions to the Teukolsky Equation

The intricacies of black hole ringdown analysis are amplified by the absence of a complete set of orthogonal basis functions for quasinormal modes. Although damped sinusoids effectively fit the ringdown signals from binary black hole mergers, the risk of overfitting remains, due to initial transients and nonlinear effects. In light of this challenge, we introduce two methods for extracting quasinormal modes in numerical simulations and qualitatively study how the transient might affect quasinormal mode fitting. In one method, we accurately fit quasinormal modes by using their spatial functional form at constant time hypersurfaces, while in the other method, we exploit both spatial and temporal aspects of the quasinormal modes. Both fitting methods leverage the spatial behavior of quasinormal eigenfunctions to enhance accuracy, outperforming conventional time-only fitting techniques at null infinity. We also show that we can construct an inner product for which the quasinormal eigenfunctions form an orthonormal (but not complete) set. We then conduct numerical experiments involving linearly perturbed Kerr black holes in horizon penetrating, hyperboloidally compactified coordinates, as this setup enables a more precise isolation and examination of the ringdown phenomenon. From solutions to the Teukolsky equation, describing scattering of an ingoing gravitational wave pulse, we find that the contributions from early-time transients can lead to large uncertainties in the fit to the amplitudes of higher overtones ($n\geq 3$). While the methods we discuss here cannot be applied directly to data from merger observations, our findings underscore the persistence of ambiguities in interpreting ringdown signals, even with access to both temporal and spatial information.