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Excitation factors and exceptional points of Kerr-de Sitter quasinormal modes

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Dario Rossi, Naritaka Oshita, Leonardo Gualtieri, Emanuele Berti

General Relativity and Quantum Cosmology · Submitted 2 Sep 2026 · 90 references

Pith's verdict: Careful two-method computation of Kerr-de Sitter excitation factors; the (2,2) exceptional point is solid, but the new (3,1) EP is single-method evidence in a numerically hostile corner and needs an error estimate or direct cross-check.

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arXiv ID2609.03009 (version 1)
Pith ID3R4Z4K25, Pith's permanent record of this paper and its review
CitationDario Rossi, Emanuele Berti, Leonardo Gualtieri, Naritaka Oshita. "Excitation factors and exceptional points of Kerr-de Sitter quasinormal modes." arXiv:2609.03009 (2026). https://pith.science/paper/3R4Z4K25
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The paper

The paper claims

Exceptional points—parameter values where two ringdown tones share one complex frequency—are a generic feature of the Kerr-de Sitter spectrum, and the paper locates two of them precisely.

The reading

This paper argues that exceptional points—parameter combinations where two different quasinormal-mode tones of a Kerr-de Sitter black hole ring at exactly the same complex frequency—are a generic feature of the spectrum, not isolated accidents. It locates two of them: the $n=5$ and $n=6$ overtones of the $(\ell,m)=(2,2)$ mode meet at spin $a/M\simeq 0.896$ with cosmological constant $M^2\Lambda\simeq 0.0085$, and the same overtones of the $(\ell,m)=(3,1)$ mode meet at $a/M\simeq 0.9725$ with $M^2\Lambda\simeq 0.000725$. Near these points the two tones exchange identities as one moves around them in parameter space (hysteresis), and the excitation factors—the amplitudes with which each tone rings when the black hole is perturbed—grow sharply while the two tones acquire nearly opposite phases, so the combined signal is not amplified. The authors compute the excitation factors with two independent numerical methods and show that both reduce to the known Kerr results as the cosmological constant vanishes. If the paper is right, gravitational-wave spectroscopists now know where the convenient picture of independent, isolated ringdown tones breaks down.

What carries the argument

Two independent numerical methods carry the calculation, and their agreement is the evidence for the claims. The first writes the solution of the separated perturbation equations as a Heun function—the built-in HeunG of Mathematica used in earlier work—and extracts the excitation factor $E_{\ell mn} = A^{(\rm ref)}_{\ell m}/(2\omega^3_{\ell mn}\, dA^{(\rm inc)}_{\ell m}/d\omega)$ evaluated at the quasinormal frequency, using Wronskians of the ingoing and outgoing solutions at the two horizons. The second, derived in this paper, generalizes the STU/MST technique to expand the solution in a series of hypergeometric functions whose coefficients satisfy a three-term continued-fraction recurrence, fixing a renormalized angular momentum; unlike the original method it needs no Coulomb functions because the integration terminates at the cosmological horizon. The hysteresis argument treats the $(a/M,\,M^2\Lambda)$ plane as a complex plane on which the frequency is multi-valued: a branch cut emerges from each exceptional point, and crossing it swaps the overtone numbers, verified by taking distinct paths between the same two parameter points and finding the two frequencies exchanged. Near the exceptional points the hypergeometric series' convergence radius shrinks, so the paper pushes the series to hundreds of terms and relies on the Heun method where the series becomes noisy.

Extended reading notes

Core claim

The central claim is that exceptional points are a generic property of the Kerr-de Sitter quasinormal-mode spectrum: wherever a one-parameter family of modes shows an avoided crossing, tuning a second parameter (here the cosmological constant alongside the spin) can make the two complex frequencies exactly coincide. The paper confirms the previously identified crossing between the $n=5$ and $n=6$ overtones of the $(\ell,m)=(2,2)$ mode and adds a new one between the same overtones of the $(\ell,m)=(3,1)$ mode. Near each exceptional point the frequency behaves as a multi-valued function on the $(a/M,\,M^2\Lambda)$ plane: a closed loop around the point exchanges the two tones across a branch cut, so the overtone label becomes path-dependent rather than intrinsic. The excitation factors of the individual modes are strongly enhanced near the points, but the two modes acquire nearly opposite phases, so $|E_{225}+E_{226}|$ is far smaller than either $|E_{225}|$ or $|E_{226}|$ (and likewise for the $(3,1)$ modes): the amplification is destructive and no large boost in ringdown amplitude follows. Extrapolating to vanishing $M^2\Lambda$ recovers the Kerr excitation factors to about 1% for spins up to $a/M\simeq 0.99$, with the $(2,2,5)$ mode the known exception in the extremal regime.

Load-bearing premise

The calculation stands on the two numerical methods agreeing inside the very region where their series expansions barely converge: very small cosmological constant and spin above $a/M\simeq 0.97$, where the paper itself sees numerical noise in one of the methods even with hundreds of series terms.

Editorial extensions

If this is right

  • At least two exceptional points exist in the Kerr-de Sitter spectrum, and the authors judge their presence generic, so further ones are likely at other $(\ell,m)$ and higher overtones.
  • Near an exceptional point the overtone number is path-dependent: any numerical code that tracks modes across the $(a/M,\,M^2\Lambda)$ plane must specify its path or it will silently exchange $n=5$ and $n=6$.
  • The enhancement of the individual excitation factors near the exceptional points is destructive—the two tones ring with nearly opposite phases—so no large gravitational-wave amplitude boost is expected from these parameter values.
  • In the $M^2\Lambda\to 0$ limit the Kerr-de Sitter excitation factors match Kerr-spacetime values within about 1% for $a/M\lesssim 0.99$, cross-validating the two computations.
  • The excitation factors computed here supply the mode amplitudes needed to build convergent mode-sum Green's functions for rotating de Sitter black holes, where late-time quasinormal modes and early-time Matsubara contributions must be matched.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • By the same two-parameter logic, exceptional points should appear wherever a black-hole spectrum has both spin and a second continuous parameter—field mass, electric charge, or a cosmological constant—so the avoided crossings already mapped for Kerr-Newman and massive-scalar Kerr are natural hunting grounds for further degeneracies.
  • The path-dependence of overtone labels implies that published frequency tables are only meaningful with the tracking path specified; a convenient convention would be to label modes by their Kerr-limit parent, as this paper does.
  • A concrete testable extension would be a time-domain ringdown simulation of an equatorial plunge into a Kerr-de Sitter black hole near the $(3,1)$ exceptional point: the two tones should appear with nearly equal amplitudes and opposite phases, a signature that would confirm the destructive-interference picture directly in the waveform.
Original abstract

We compute the excitation factors of Kerr-de Sitter black hole quasinormal modes using two different techniques: a Heun function representation and an expansion of the solution in terms of hypergeometric functions. Exceptional points at which the complex frequencies of different overtones coincide are a generic feature of the Kerr-de Sitter quasinormal mode spectrum. Near exceptional points, the modes exhibit a "hysteresis phenomenon" that was previously found for Kerr black holes perturbed by massive fields and the absolute value of the excitation factors of the individual modes is significantly enhanced, while they acquire nearly opposite phases. We extrapolate the excitation factors to small values of the cosmological constant and find good agreement with previous calculations in the Kerr limit.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,710 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We compute the excitation factors of Kerr-de Sitter black hole quasinormal modes using two different techniques: a Heun function representation and an expansion of the solution in terms of hypergeometric functions. Exceptional points at which the complex frequencies of different overtones coincide are a generic feature of the Kerr-de Sitter quasinormal mode spectrum. Near exceptional points, the modes exhibit a "hysteresis phenomenon" that was previously found for Kerr black holes perturbed by massive fields and the absolute value of the excitation factors of the individual modes is significantly enhanced, while they acquire nearly opposite phases. We extrapolate the excitation factors to small values of the cosmological constant and find good agreement with previous calculations in the Kerr limit.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We compute the excitation factors of Kerr-de Sitter black hole quasinormal modes using two different techniques: a Heun function representation and an expansion of the solution in terms of hypergeometric functions. method 0.65 0.96 0.08 asserted 1.00
s1 Exceptional points at which the complex frequencies of different overtones coincide are a generic feature of the Kerr-de Sitter quasinormal mode spectrum. result 0.90 0.45 0.33 asserted 1.00
s2 Near exceptional points, the modes exhibit a "hysteresis phenomenon" that was previously found for Kerr black holes perturbed by massive fields and the absolute value of the excitation factors of the individual modes is significantly enhanced, while they acquire nearly opposite phases. result 0.99 0.59 0.36 asserted 1.00
s3 We extrapolate the excitation factors to small values of the cosmological constant and find good agreement with previous calculations in the Kerr limit. result 0.94 0.94 0.12 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.64 · evidence from simulation or computation 0.93.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. Fundamental KdS QNM frequencies with (
Figure 2. FIG. 2. Same as Fig
Figure 3. FIG. 3. Relative difference in the real (left) and imaginary (right) QNM frequencies of KdS and Kerr spacetimes for the
Figure 4. FIG. 4. Representation of the branch cut (solid line) con
Figure 5. FIG. 5. Real (left) and imaginary (right) parts of the (
Figure 6. FIG. 6. Real (left) and imaginary (right) parts of the (
Figure 7. FIG. 7. Same as Fig
Figure 8. FIG. 8. Same as Fig
Figure 9. FIG. 9. Same as Fig
Figure 10. FIG. 10. Relative difference between the absolute value of the EFs computed with the two different approaches (Heun functions
Figure 11. FIG. 11. Relative difference between the absolute value of the EFs computed with the two different approaches (Heun functions
Figure 12. FIG. 12. Comparison between the absolute value of the
Figure 13. FIG. 13. Absolute value of the EFs for fixed overtone numbers
Figure 14. FIG. 14. Absolute value of the KdS EFs

Figures reproduced from arXiv:2609.03009 by the authors.

References

Discussion

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