Computing spectral shifts for Johannsen-Psaltis Black Holes

TL;DR

This work develops a perturbative framework based on a modified Teukolsky equation and eigenvalue perturbation theory to compute beyond-GR QNM spectral shifts for slowly rotating Johannsen–Psaltis black holes. It demonstrates that JP perturbations break isospectrality between even- and odd-parity modes while preserving definite parity, with mode shifts that are approximately linear in the azimuthal number $m$ at slow spin. The authors validate large-ℓ behavior against a scalar WKB approximation and provide detailed contour-integral techniques to extract the shifts, revealing small but potentially detectable deviations in ringdown spectra in upcoming observations. The results lay groundwork for incorporating JP-like deviations into GR tests with gravitational waves and guide future work toward generic spin, full field-equation effects, and data-analysis strategies for strong-field tests of gravity.

Abstract

The growing number of gravitational wave (GW) detections and the increasing sensitivity of GW detectors have enabled precision tests of General Relativity (GR) in the strong-field regime. The recent observation of multiple quasinormal modes (QNMs) in GW250114 marks a major advance for observational black hole spectroscopy. This clear signal, together with the growing number of GW detections, highlights the need for accurate predictions of QNM spectra in beyond-GR theories in order to carry out precision searches for new physics. In this work, we continue to lay the foundation for such predictions using a modified Teukolsky formalism in conjunction with the eigenvalue perturbation method. We compute the spectral shifts of slowly rotating Johannsen-Psaltis black holes for $2 \leq \ell \leq 10$, all $m$, and overtones $n = 0, 1, 2$, and confirm the large-$\ell$ behavior of the modes by comparing with the WKB approximation. We find that these black holes admit definite-parity modes but break the isospectrality between even- and odd-parity QNMs at all spins, and that the shifts depend linearly on $m$ for slow spins. We further derive a general parity condition that any beyond-GR modification to the metric must satisfy to support definite-parity modes, providing new insights into isospectrality breaking and parity structure in gravitational perturbations.

Figures (4)

• Figure 1: The radial integration contours in the complex plane. The radial portions of the spin-weighted scalars have a branch cut at the outer horizon $r_+$, which the contour wraps around. $\mathscr{C}_+$ wraps around the upwards-pointing branch cut, and $\mathscr{C}_-$ wraps around the downwards-pointing branch cut.
• Figure 2: The real (left column) and imaginary (right column) QNM shifts of $(\delta\omega_0)_{\ell n}$ (top row) and $(\delta\omega_1)_{\ell n}$ (bottom row) for modes $3\leq\ell \leq 10$ and $n=0,1,2$. For each value of $\ell$ and $n$, both even and odd parity shifts are shown. All shifts are normalized by $\omega_0$, the corresponding Schwarzschild QNM frequency for each mode. The scalar WKB approximation for each overtone normalized by $\omega_0$ is given by the line of the corresponding color. Note that although the real part of the WKB approximation has no $n$ dependence, the normalization by $\omega_0$ causes a splitting in between the overtones. A small horizontal jitter has also been added to improve the visibility of overlapping points.
• Figure 3: The real (left column) and imaginary (right column) QNM shifts of $(\delta\omega_{0})_{\ell n}$ (top row) and $(\delta\omega_{1})_{\ell n}$ (bottom row) for modes $2\leq\ell \leq 4$ and $n=0,1,2$. For each value of $\ell$ and $n$, both even and odd parity shifts are shown. The scalar WKB approximation for each overtone normalized by $\omega_0$ is given by the line of the corresponding color. Note that although the real part of the WKB approximation has no $n$ dependence, the normalization by $\omega_0$ causes a splitting in between the overtones. All shifts are normalized by $\omega_0$, the corresponding Schwarzschild QNM frequency for each mode. A small horizontal jitter has also been added to improve the visibility of overlapping points.
• Figure 4: Normalized differences between the real (left) and imaginary (right) parts of gravitational modes from the EVP method and the scalar modes from the first-order WKB method for $2\leq \ell\leq 10$, $m=1$, $n=0,1,2$, and $\chi=0.1$. The differences are normalized by the respective Schwarzschild gravitational frequencies, $\omega_0$. The normalized differences are further scaled by $L^2$.