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

TL;DR

This work benchmarks the accuracy of eikonal and post-Kerr approximations for quasinormal modes in a modified Kerr spacetime by combining 2+1D numerical time evolution of a scalar field with Prony-based QNM extraction. It systematically quantifies modeling uncertainties across multipoles, spins, and near-horizon deformations, and couples these with simple statistical SNR-based errors to define a bias ratio that delineates the domain of validity for approximate methods. Key findings show that eikonal methods capture the prograde/dependent QNM shifts well in many regimes, whereas the first-order post-Kerr expansion often fails at high spin or large deformations unless higher orders or Padé resummation are used. The results have practical implications for high-precision black-hole spectroscopy and motivate extensions to gravitational perturbations and more general beyond-GR spacetimes.

Abstract

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.

Figures (10)

• Figure 1: The solid lines correspond to the prograde ($a>0$) and retrograde ($a<0$) equatorial photon orbits of the Kerr BH ($\epsilon=0$) and the modified Kerr BH ($\epsilon=0.4,1$). The dashed lines denote the position of the outer event horizons \ref{['horizons']} for the same values of $\epsilon$.
• Figure 2: Waveforms obtained setting $\ell=m=2$ and $a=0.3$ for Kerr ($\epsilon=0$) and modified Kerr BHs ($\epsilon=1$). The vertical blue dashed lines identify the starting $t_i=50\,M$ and ending $t_f=180\,M$ times. The shaded region denotes the range of starting times used in the Prony method.
• Figure 3: First row: estimates of the real and imaginary parts of prograde and retrograde modes as a function of the BH spin for $\ell=2$ and $\epsilon=0.4$ from the eikonal method, Prony extraction and different orders in the post-Kerr and Padé approximation. The Kerr values ($\epsilon=0$) are reported as a reference of the unperturbed case. Second row: relative error in $\log_{10}$ scale of post-Kerr and Padé approximations with respect to the eikonal predictions. Third row: relative error in $\log_2$ scale of the eikonal (black dots) and post-Kerr order 1 (blue triangles), post-Kerr order 2 (orange triangles), post-Kerr order 3 (green triangles) estimates with respect to the Prony ones. The Padé estimates are not reported as they overlap with the eikonal ones.
• Figure 4: Fundamental QNMs for different deviation parameters $\epsilon$ for selected values of $\ell$ with $m=\ell$ computed with eikonal (black dots), post-Kerr (blue squares), post-Kerr order 2 (orange triangles), post-Kerr order 3 (green triangles), Padé order $(1,2)$ (light blue) and Prony method (red dots) for spin $a=0.7$ for real (top panels) and imaginary (bottom panels) parts. The Kerr values are shown for comparison (black +). The estimation for Prony method errors is indicated by error bars (very small). The Padè estimates at order $(2,1)$ are not reported as they overlap with order $(1,2)$.
• Figure 5: Plots of $\delta^{\textrm{E/pK}}_R$ and $\delta^{\textrm{E/pK}}_I$ for a given $\epsilon$ as a function of $\ell$ for spins $a=0.3$ (top panels) and $a=0.7$ (bottom panels).