Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation

TL;DR

This work extends the high-order WKB method to compute quasinormal modes of rotating black holes in and beyond general relativity. It establishes a theory-agnostic, beyond-Teukolsky framework and a linearized WKB treatment for small deviations, applying them to Kerr and to higher-derivative gravity (HDG). The Kerr results show good agreement with Leaver data, especially for high angular indices and low overtones, while beyond-GR analyses reveal that WKB can capture linearized QNM shifts and that HDG deviations can dominate over WKB errors for realistic couplings. The findings suggest WKB-based black hole spectroscopy may provide accurate, fast probes of GR with current gravitational-wave observations and guide future tests with next-generation detectors.

Abstract

Exploring gravitational theories beyond general relativity (GR) with black hole (BH) spectroscopy requires accurate and flexible methods for computing their quasinormal mode (QNM) spectrum. A popular method of choice is the higher-order Wentzel-Kramers-Brillouin (WKB) approximation, mostly applied to nonrotating BHs. While previous studies demonstrated that the higher-order WKB method can also be used for Kerr BHs in GR, there has been little work on rotating BHs in modified theories of gravity. In this work, we revive the idea by extending WKB calculations of the Kerr QNM spectrum to higher order and assessing its accuracy against continued-fraction tabulated data. We then apply the WKB approximation beyond GR, comparing it against both linearized and continued fraction calculations in the parametrized beyond-Teukolsky formalism and in higher-derivative gravity (HDG) theories. We find that the frequencies computed by the WKB method in theories beyond GR have better accuracy than the measurement errors for GW250114, the event with the highest ringdown signal-to-noise ratio observed to date.

Figures (8)

• Figure 1: Comparison between WKB approximations (1st to 4th order) and Leaver’s spectral method values for a Kerr BH when changing $(\ell,m,n)$. The vertical axis shows the relative error $\delta \omega _{\ell m n}$ as defined in Eq. \ref{['rel_err_kerr']}, while the horizontal axis spans the spin parameter a from 0 to 0.9. Color indicates WKB orders: first order (dark blue: WKB1), second order (light blue: WKB2), third order (grey: WKB3), fourth order (light red: WKB4), and Kokkotas' WKB at third order Kokkotas:1991vz (green).
• Figure 2: Comparison of the linear coefficients $d^N_\omega$ for $(2,m,0)$ modes computed with $N$-th order WKB and Leaver's method at different spins, $a$. We plot the relative errors $\delta d^N_\omega$ defined in Eq. \ref{['rel_err_d']}. Different columns correspond to different values of $m$, and different rows correspond to different values of $k$. Colors indicates WKB orders, with the same conventions used in Fig. \ref{['fig:kerrgrwkb']}.
• Figure 3: Comparison of QNM relative differences $\delta \omega^\pm_\text{\scshape\tiny{e/o}}$ in a HDG theory with coupling $\lambda_\text{\scshape\tiny{eff}}=0.1$, as defined in Eq. \ref{['rel_err_omega_hdg']}. The left (right) panels refer to the real (imaginary) part of the $(2,2,0)$ mode. Each row selects a combination of polarization ($\pm$) and parity of the theory (even/odd). Orange solid lines show the relative difference between modes in GR and HDG.
• Figure 4: Percentage relative error on the $\ell=m=2$ fundamental QNM frequencies computed with continued fractions and with the $N$-th order WKB approximation, for different values of $\lambda$ and $a$. Dotted lines correspond to $N=1$, dashed lines to $N=2$, solid lines to $N=3$, and dot-dashed lines to $N=4$. Each curve spans values of $\lambda$ for which the relative difference between the linearized QNM calculation and the continued fraction calculation is less than $5\%$.
• Figure 5: Same as Fig. \ref{['fig:kerrgrwkb']}, but for $\ell = 3$.