The parameterized quasinormal modes for modified Teukolsky equations

TL;DR

This work addresses testing general relativity in the strong-field ringdown by framing quasinormal modes (QNMs) of Kerr black holes within a theory-agnostic parameterized Teukolsky framework. It introduces two independent deviation sets, $\{\eta_1^{(k)}\}$ for the radial sector and $\{\eta_2^{(k)}\}$ for the angular sector, and derives linear-in-deviation corrections to the QNM frequency $\omega$ and separation constant $A$ using a continued fraction approach, with spectra determined by $\mathcal{L}_r(\omega,A,\eta_1^{(k)})=0$ and $\mathcal{L}_\theta(\omega,A,\eta_2^{(k)})=0$. The framework is cross-validated against a two-dimensional pseudospectral method in hyperboloidal coordinates, confirming consistency at first order in the deviations and providing quantitative bounds on the deviation parameters. The resulting formalism offers a practical, theory-agnostic tool for analyzing gravitational-wave ringdown data and mapping observational constraints to concrete beyond-GR scenarios in the strong-field regime.

Abstract

We introduce the modified Teukolsky equation within a parameterized framework, analogous to the case of small deviations of potential in spherical symmetry. Both the radial and angular equations acquire modifications described by two independent sets of parameters. We derive the parameterized framework of the quasinormal mode spectra using the continued fraction method. The results are cross-validated with the two-dimensional pseudo-spectral method, demonstrating excellent agreement and ensuring self-consistency. This work establishes a robust foundation for a theory-agnostic interpretation of gravitational-wave ringdown signals, providing a practical tool for probing potential deviations from General Relativity in the strong-field regime.

Figures (3)

• Figure 1: The figure represents the $d$-functions for tensor modes as functions of the parameter $a$ in the range $[0,0.45]$, presented for $l=2$. Different colors represent different values of $m$. Note that $d^{(0)}_{1\omega}$ and $d^{(0)}_{2\omega}$ yield identical results due to the derivative structure in their defining equations when $k=0$.
• Figure 2: The figure displays the migrations of QNM spectra as functions of the deviation parameters for the fundamental modes with $l=2$,$m=2$ and angular momentum parameter $a=0.2$. The first row illustrates the dependence on $\eta_{1}^{(k)}$, while the second row shows the corresponding variations with $\eta_{2}^{(k)}$. In all panels, the blue curve segment corresponds to the trajectory for $\eta_{i}^{(k)}>0$, and the red curve segment to $\eta_{i}^{(k)}<0$. The migration trajectory for $\omega(\eta^{(1)}_1)$ coincides with that for $\omega(\eta_2^{(1)})$.
• Figure 3: Constraints on the deviation parameters obtained by requiring agreement between the continued fraction and pseudo-spectral methods within $1\%$, shown as functions of the black hole spin parameter $a\in[0,0.45]$, where blue lines refer to $\eta_i^{(0)}$, red lines refer to $\eta_i^{(1)}$, yellow lines refer to $\eta_i^{(2)}$, orange lines refer to $\eta_i^{(3)}$, and black lines refer to $\eta_i^{(4)}$.