Black hole spectroscopy: from theory to experiment

TL;DR

The paper surveys how black-hole ringdown spectroscopy encodes the strong-field dynamics of general relativity and potential new physics, through the spectrum and amplitudes of quasinormal modes. It combines analytical perturbation theory (Teukolsky formalism), numerical relativity, and data-analysis frameworks to model, extract, and interpret QNMs in GR and beyond, including Kerr-Newman spacetimes, environmental effects, and horizon-scale physics. It introduces and contrasts theory-specific and theory-agnostic approaches for deviations, along with advanced methods (hyperboloidal, pseudospectra, and QQNM calculations) to capture nonlinear and horizon-related phenomena. The review also links QNM physics to practical GW data analysis, detailing how to measure amplitudes, distinguish overtones, and test Kerr with current LVK data while outlining prospects for next-generation detectors and space-based observatories. Overall, it provides a comprehensive toolkit for BH spectroscopy, from foundational perturbation theory to state-of-the-art data-analysis strategies and future horizons (echoes, horizon-scale physics, and beyond-GR tests).

Abstract

The "ringdown" radiation emitted by oscillating black holes has great scientific potential. By carefully predicting the frequencies and amplitudes of black hole quasinormal modes and comparing them with gravitational-wave data from compact binary mergers we can advance our understanding of the two-body problem in general relativity, verify the predictions of the theory in the regime of strong and dynamical gravitational fields, and search for physics beyond the Standard Model or new gravitational degrees of freedom. We summarize the state of the art in our understanding of black hole quasinormal modes in general relativity and modified gravity, their excitation, and the modeling of ringdown waveforms. We also review the status of LIGO-Virgo-KAGRA ringdown observations, data analysis techniques, and the bright prospects of the field in the era of LISA and next-generation ground-based gravitational-wave detectors.

Figures (80)

• Figure 2.1: Examples of the complex mode frequency $M\omega$ for the gravitational ($s=-2$), quadrupolar ($\ell=2$) QNMs. Left panel: sequences for all 5 azimuthal modes ($-2\le{m}\le2$) and for the fundamental mode ($n=0$). The Schwarzschild limit ($a=0$) of each sequence is at $M\omega\approx0.37-0.089i$. Each sequence is parameterized by $a$ over the range $0\le{a}<M$, and markers along each sequence are at intervals of $\Delta{a}=0.05M$. Right panel: sequences for the first 12 overtones (the bottom line is $n=0$) of the $m=2$ axial mode. Note that $0\le{n}\le10$, but that there are two $n=8$ overtones, labeled as $n=8_0$ and $n=8_1$. This overtone multiplet notation is used to make the overtone notation for the sequences consistent with the labeling of the Schwarzschild modes.
• Figure 2.2: Examples of the complex mode frequency $M\omega$ for the gravitational ($s=-2$) QNMs with multipolar indices $\ell=3$ and $\ell=4$. Only the $m=2$ azimuthal modes are plotted. See Fig. \ref{['fig:QNMl2behavior']} for additional details.
• Figure 2.3: Right-pointing triangles are prograde Kerr QNM frequencies computed for $a/M=0.7$, while left-pointing triangles are retrograde modes. Note that prograde and retrograde modes are present both in the right half-plane (ordinary modes) and in the left half-plane (so-called "mirror" modes). Some general trends are visible. As we increase either $\ell$ or $m$, the complex QNM frequencies have larger values of $|\mathop{\mathrm{Re}}\nolimits[M\omega]|$; as we increase $n$, the QNM frequencies move to larger values of $-\mathop{\mathrm{Im}}\nolimits[M\omega]$. Figure adapted from MaganaZertuche:2021syq.
• Figure 2.4: Examples of the exceptional behavior of the complex mode frequency $M\omega$ for the gravitational ($s=-2$), quadrupolar ($\ell=2$) QNM overtones $n=8$ and $9$. For $n=8$, the $m=1$ and $2$ sequences occur as overtone multiplets, and none of these $4$ sequences terminate at the Schwarzschild limit of $M\omega=-2i$. There are also two $m=0$ sequences. The $\omega^+_{208_0}$ sequence does terminate at the Schwarzschild limit, but the $\omega^+_{208_1}$ sequence begins on the NIA with $a\approx0.32M$ as a polynomial mode which is simultaneously a QNM and a TTM${}_L$. For the $n=9$ sequences, all of the $m\ne0$ sequences are nonexceptional. However the $m=0$ sequence is noncontinuous and exceptional. The initial $\omega^+_{209_0}$ portion of the sequence terminates on the NIA as a polynomial QNM at $a\approx0.31M$, then $\omega^+_{209_1}$ re-emerges from the NIA at $a\approx0.40M$ as a polynomial mode which is simultaneously a QNM and a TTM${}_L$. The $\omega^+_{209_1}$ sequence soon begins to undertake many loops, the first $7$ of which become tangent to the NIA. Each of these points of tangency does not represent either a QNM or a TTM.
• Figure 2.5: Examples of the complex mode frequency $M\omega$ for the gravitational ($s=-2$) TTM${}_L$s for $\ell=2$. The left plot with $n=0$ illustrates the family of TTMs which connect to the Schwarzschild TTM frequencies $\Omega_\ell$. The next two plots with $n=1$ and $n=2$ illustrate the 2 families of TTMs whose Schwarzschild limit frequencies are at complex infinity. The insets in each of these plots are log-log plots showing the behavior of the sequences for large $M\omega$. Note that for the first 2 families, only the $m\ge0$ sequences are plotted, because the two mirror-mode families are degenerate.