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arxiv: 2511.08692 · v2 · pith:OXUBNE55new · submitted 2025-11-11 · 🌀 gr-qc · astro-ph.HE· hep-ph

Excitation factors for horizonless compact objects: long-lived modes, echoes, and greybody factors

Romeo Felice Rosato , Shauvik Biswas , Sumanta Chakraborty , Paolo Pani This is my paper

Pith reviewed 2026-05-17 23:17 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-ph
keywords quasinormal modesexcitation factorsechoesultracompact objectshorizonless compact objectsgreybody factorsgravitational wavesringdown
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The pith

Long-lived modes in horizonless ultracompact objects have excitation factors that scale with their tiny imaginary frequency, suppressing early contributions and confining them to very late echoes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines quasinormal modes of ultracompact horizonless objects, which feature an effective cavity that traps waves and generates long-lived modes with extremely small imaginary frequencies. It demonstrates that the excitation factors for these modes are strongly suppressed in proportion to that imaginary part, so the modes only become relevant at very late times in the gravitational waveform. The resulting hierarchy structures the signal such that the prompt ringdown comes from ordinary light-ring modes, early echoes from moderately damped cavity modes, and only the latest echoes from the longest-lived modes. The authors build a practical waveform model by superposing black-hole-like modes with cavity modes and show that the combination of small excitation and weak damping makes long-lived modes robust to perturbations, while greybody factors stay stable under small potential changes.

Core claim

The excitation factors of long-lived quasinormal modes supported by the cavity between the photon sphere and the interior of ultracompact horizonless objects scale directly with the imaginary part of their complex frequency, which strongly suppresses their amplitude in the waveform until very late times and thereby accounts for the observed structure of echo signals as a sequence of prompt ringdown, early echoes, and late echoes.

What carries the argument

The effective cavity between the photon sphere and the object's interior that traps waves and produces long-lived modes whose excitation factors are computed analytically and numerically.

If this is right

  • The prompt ringdown remains dominated by standard light-ring modes.
  • Early echoes arise from moderately damped cavity modes.
  • Only the latest echoes are governed by the long-lived modes.
  • A practical ringdown waveform can be constructed as a superposition of ordinary black-hole quasinormal modes and cavity modes.
  • Long-lived modes gain robustness against localized perturbations because of their small excitation factors combined with weak damping.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Searches for echoes in gravitational-wave data from ultracompact objects should prioritize very late times rather than the prompt ringdown.
  • Greybody factors may serve as more stable observables than quasinormal frequencies for distinguishing horizonless objects from black holes.
  • The same suppression mechanism could apply to other trapped-mode systems such as those in modified gravity or with different interior structures.

Load-bearing premise

The effective cavity between the photon sphere and the object's interior produces long-lived modes whose excitation factors can be computed analytically and numerically and scale directly with the imaginary frequency part.

What would settle it

A direct numerical computation of the late-time waveform amplitude for a specific ultracompact object model showing whether the coefficient of each long-lived mode is proportional to its imaginary frequency.

Figures

Figures reproduced from arXiv: 2511.08692 by Paolo Pani, Romeo Felice Rosato, Shauvik Biswas, Sumanta Chakraborty.

Figure 1
Figure 1. Figure 1: QNMs and excitation factors for axial gravitational perturbations with [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: All panels refer to the l = 2 mode of a gravitational perturbation for different ECO scenarios. Top panel: imag￾inary part of the fundamental long-lived quasinormal mode for gravitational perturbations as a function of the object’s radius r0, for three different ECO reflectivities: constant re￾flectivity with RECO = 0.9, RECO = 0.5, and a Boltzmann reflectivity profile. Middle panel: corresponding behavior… view at source ↗
Figure 3
Figure 3. Figure 3: Fourier transform of the reflectivity for a wormhole [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mode content and echo reconstruction for a wormhole with throat location at rthroat = (2+10−6 )M Left: representative portion of the cavity-mode spectrum. The longest-lived modes do not dominate the echo onset because their excitation factors are highly suppressed. Right: comparison between the numerical echo signal and QNM-based reconstructions with/without high-frequency modes (defined as those with ω R … view at source ↗
Figure 5
Figure 5. Figure 5: Reconstruction of a wormhole echo with throat [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Response of three QNMs to the localized perturbation in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative variation Glm of the greybody factors under a localized perturbation of the effective potential, as introduced in Eq. (39), shown as a function of its location c/M and defined in Eq. (40). Left panel: wormhole case for different values of the perturbation amplitude ϵ. The variation remains consistently within the same order of magnitude as the perturbation. Right panel: results for three different… view at source ↗
read the original abstract

We present an analytical and numerical investigation of the quasinormal excitation factors of ultracompact horizonless objects. These systems possess long lived quasinormal modes with extremely small imaginary parts, originating from the effective cavity between the photon sphere and the object's interior. We show that the excitation of such modes is strongly suppressed, scaling with the imaginary part of their frequency, and therefore they contribute to the waveform only at very late times. This hierarchy naturally explains the structure of echo signals: the prompt ringdown is dominated by standard light ring modes, the early echoes arise from moderately damped cavity modes, and only the latest echoes are governed by long lived modes. Based on this, we propose a practical ringdown waveform model based on a superposition of ordinary black hole quasinormal modes and cavity modes, which captures the complexity of the ringdown of horizonless ultracompact objects. We further demonstrate that the combination of small excitation factors and weak damping enhances the robustness of long lived modes against localized perturbations, in contrast to the spectral instabilities affecting standard black hole quasinormal modes. Finally, we extend the analysis of greybody factors to exotic compact objects and wormholes, showing that they remain stable under small deformations of the effective potential and thus represent robust observables. Our results provide a unified framework for understanding excitation, stability, and echoes in ultracompact horizonless objects, with direct implications for their spectral properties and gravitational wave signatures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents an analytical and numerical investigation of quasinormal excitation factors for ultracompact horizonless objects (ECOs and wormholes). It claims that long-lived cavity modes, arising from the effective cavity between the photon sphere and the object's interior, have excitation factors that scale directly with the imaginary part of their frequency. This suppression implies that such modes contribute to the waveform only at very late times, naturally explaining the echo hierarchy: prompt ringdown dominated by light-ring modes, early echoes from moderately damped cavity modes, and latest echoes from long-lived modes. The authors propose a practical ringdown model as a superposition of ordinary black-hole QNMs and cavity modes, demonstrate enhanced robustness of long-lived modes against localized perturbations, and extend the analysis to show that greybody factors remain stable under small deformations of the effective potential.

Significance. If the reported scaling of excitation factors with Im(ω) holds, the work provides a unified framework for understanding the structure of echo signals, the relative stability of long-lived modes, and robust observables such as greybody factors in horizonless objects. The use of residue computations via the Wronskian construction together with numerical checks across multiple models is a strength, as is the proposal of a concrete waveform model with direct implications for gravitational-wave data analysis. These results address timely questions in the ringdown and echo phenomenology of exotic compact objects.

major comments (2)
  1. [§3.2] §3.2: The residue computation at the QNM poles using the Wronskian for the modified Regge-Wheeler potential is central to the scaling claim. The manuscript should explicitly verify that the contour-integral representation yields the proportionality to Im(ω) in the small-damping limit without additional assumptions on branch cuts or the analytic continuation of the potential.
  2. [§4] §4: Numerical confirmation of the excitation-factor scaling and echo hierarchy is reported for several ECO models, but the absence of tabulated values or direct plots of excitation factor versus Im(ω) makes it difficult to assess the precision of the claimed linear relation and its robustness across damping rates.
minor comments (3)
  1. The abstract states that the excitation factors 'scale with the imaginary part,' but the introduction could more clearly distinguish the exact analytic result from the small-damping approximation used in the derivation.
  2. Figure captions and axis labels should explicitly indicate which curves correspond to the black-hole limit versus the reflective-boundary or wormhole-throat cases to improve readability of the potential and waveform comparisons.
  3. A brief discussion of the range of validity of the proposed superposition waveform model (e.g., for which echo orders it remains accurate) would help readers apply the results to data analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our work and the recommendation for minor revision. We address each major comment below and have made revisions to enhance the clarity and rigor of the presentation.

read point-by-point responses

  1. Referee: [§3.2] The residue computation at the QNM poles using the Wronskian for the modified Regge-Wheeler potential is central to the scaling claim. The manuscript should explicitly verify that the contour-integral representation yields the proportionality to Im(ω) in the small-damping limit without additional assumptions on branch cuts or the analytic continuation of the potential.

    Authors: We appreciate the referee drawing attention to this aspect of the derivation. The scaling follows from the residue theorem applied to the Green's function constructed via the Wronskian for the modified potential. In the revised manuscript we have added an explicit verification subsection in §3.2. We consider a small semicircular contour around each pole in the lower half-plane and take the limit Im(ω) → 0 while keeping Re(ω) fixed. Under the standard assumption that the effective potential is analytic in a neighborhood of the real axis (consistent with the ECO and wormhole models considered), the integral reduces to 2πi times the residue, which is proportional to Im(ω) with no additional branch-cut contributions crossed by the chosen contour. This confirms the claimed proportionality without further assumptions beyond those already stated for the quasinormal-mode problem. revision: yes

  2. Referee: [§4] Numerical confirmation of the excitation-factor scaling and echo hierarchy is reported for several ECO models, but the absence of tabulated values or direct plots of excitation factor versus Im(ω) makes it difficult to assess the precision of the claimed linear relation and its robustness across damping rates.

    Authors: We agree that direct visual and tabular evidence would strengthen the numerical section. In the revised manuscript we have added a new figure in §4 that plots the modulus of the excitation factors against Im(ω) for the ECO, wormhole, and other models examined, explicitly demonstrating the linear scaling over the range of damping rates considered. We have also included a supplementary table listing representative numerical values of the excitation factors together with the corresponding Im(ω), allowing quantitative assessment of the fit precision and robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central result that excitation factors for long-lived cavity modes scale with Im(ω) is obtained from the residue at the QNM poles using the Wronskian construction on the modified Regge-Wheeler potential (detailed in §3.2) and the standard contour-integral representation in the small-damping limit. This is an independent analytic derivation, not a redefinition or fit of the target quantity itself. Numerical verification across ECO models in §4 supplies external checks. No load-bearing step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled from prior work by the same authors. The derivation remains self-contained against the stated assumptions and does not invoke uniqueness theorems or renamings that collapse the claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated or derivable from the provided text.

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Forward citations

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Reference graph

Works this paper leans on

126 extracted references · 126 canonical work pages · cited by 4 Pith papers · 47 internal anchors

  1. [1]

    nmaxX n=1 ωBH n BBH n e−iωBH n t # +θ(t−2L) 4Re

    In practice, however, this would require a very large number of modes (typically tens to hundreds) [122] ren- dering such an approach phenomenologically impractical. Therefore, along the lines of [115], here we show that it is possible to proceed by modeling the early ringdown phase using standard BH QNMs, and the echo signal using only the oscillation fr...

    work page

  2. [2]

    Regge and J

    T. Regge and J. A. Wheeler, Phys. Rev.108, 1063 (1957)

    work page 1957

  3. [3]

    Chandrasekhar,The mathematical theory of black holes(1985)

    S. Chandrasekhar,The mathematical theory of black holes(1985)

    work page 1985

  4. [4]

    Black hole spectroscopy: from theory to experiment

    E. Bertiet al., (2025), arXiv:2505.23895 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  5. [5]

    Carullo, Gen

    G. Carullo, Gen. Rel. Grav.57, 76 (2025)

    work page 2025

  6. [6]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    Abbottet al., PRL125, 101102 (2020), arXiv:2009.01075 [gr-qc]

    R. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.125, 101102 (2020), arXiv:2009.01075 [gr-qc]

    work page arXiv 2020

  8. [8]

    A. G. Abacet al.(LIGO Scientific, KAGRA, Virgo), Phys. Rev. Lett.135, 111403 (2025), arXiv:2509.08054 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  9. [9]

    (2025), arXiv:2509.08099 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  10. [10]

    Black Hole Spectroscopy: Testing General Relativity through Gravitational Wave Observations

    O. Dreyer, B. J. Kelly, B. Krishnan, L. S. Finn, D. Gar- rison, and R. Lopez-Aleman, Class. Quant. Grav.21, 787 (2004), arXiv:gr-qc/0309007

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  11. [11]

    S. L. Detweiler, Astrophys. J.239, 292 (1980)

    work page 1980

  12. [12]

    On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

    E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73, 064030 (2006), arXiv:gr-qc/0512160

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  13. [13]

    Bayesian model selection for testing the no-hair theorem with black hole ringdowns

    S. Gossan, J. Veitch, and B. S. Sathyaprakash, Phys. Rev. D85, 124056 (2012), arXiv:1111.5819 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  14. [14]

    Tests of General Relativity with GWTC-3

    R. Abbottet al.(LIGO Scientific, VIRGO, KAGRA), (2021), arXiv:2112.06861 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  15. [15]

    Testing General Relativity with Present and Future Astrophysical Observations

    E. Bertiet al., Class. Quant. Grav.32, 243001 (2015), arXiv:1501.07274 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown

    E. Berti, K. Yagi, H. Yang, and N. Yunes, Gen. Rel. Grav.50, 49 (2018), arXiv:1801.03587 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    Testing the nature of dark compact objects: a status report

    V. Cardoso and P. Pani, Living Rev. Rel.22, 4 (2019), arXiv:1904.05363 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  18. [18]

    LISA Definition Study Report

    M. Colpiet al.(LISA), (2024), arXiv:2402.07571 [astro- ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  19. [19]

    Science Case for the Einstein Telescope

    M. Maggioreet al.(ET), JCAP03, 050 (2020), arXiv:1912.02622 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  20. [20]

    Kalogera et al., (2021), arXiv:2111.06990 [gr-qc]

    V. Kalogeraet al., (2021), arXiv:2111.06990 [gr-qc]

    work page arXiv 2021

  21. [21]

    Sensitivity Studies for Third-Generation Gravitational Wave Observatories

    S. Hildet al., Class. Quant. Grav.28, 094013 (2011), arXiv:1012.0908 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  22. [22]

    Science with the Einstein Telescope: a comparison of different designs

    M. Branchesiet al., JCAP07, 068 (2023), arXiv:2303.15923 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  23. [23]

    The Science of the Einstein Telescope

    A. Abacet al., (2025), arXiv:2503.12263 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  24. [24]

    B. P. Abbottet al.(LIGO Scientific), Class. Quant. Grav.34, 044001 (2017), arXiv:1607.08697 [astro- ph.IM]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [25]

    Frequency-Dependent Responses in 3rd Generation Gravitational-Wave Detectors

    R. Essick, S. Vitale, and M. Evans, Phys. Rev. D96, 084004 (2017), arXiv:1708.06843 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Evans et al., (2023), arXiv:2306.13745 [astro-ph.IM]

    M. Evanset al., (2023), arXiv:2306.13745 [astro-ph.IM]

    work page arXiv 2023

  27. [27]

    Spectroscopy of Kerr black holes with Earth- and space-based interferometers

    E. Berti, A. Sesana, E. Barausse, V. Cardoso, and K. Belczynski, Phys. Rev. Lett.117, 101102 (2016), arXiv:1605.09286 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  28. [28]

    Bhagwat, C

    S. Bhagwat, C. Pacilio, E. Barausse, and P. Pani, Phys. Rev. D105, 124063 (2022), arXiv:2201.00023 [gr-qc]

    work page arXiv 2022

  29. [29]

    Bhagwat, C

    S. Bhagwat, C. Pacilio, P. Pani, and M. Mapelli, Phys. Rev. D108, 043019 (2023), arXiv:2304.02283 [gr-qc]

    work page arXiv 2023

  30. [30]

    C. V. Vishveshwara, Nature227, 936 (1970)

    work page 1970

  31. [31]

    K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel.2, 2 (1999), arXiv:gr-qc/9909058

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  32. [32]

    Quasinormal modes of black holes and black branes

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav.26, 163001 (2009), arXiv:0905.2975 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  33. [33]

    R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys.83, 793 (2011), arXiv:1102.4014 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  34. [34]

    M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. Lett.123, 111102 (2019), arXiv:1905.00869 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  35. [35]

    Franchini and S

    N. Franchini and S. H. V¨ olkel, (2023), arXiv:2305.01696 [gr-qc]

    work page arXiv 2023

  36. [36]

    Maggio, L

    E. Maggio, L. Buoninfante, A. Mazumdar, and P. Pani, Phys. Rev. D102, 064053 (2020), arXiv:2006.14628 [gr- qc]

    work page arXiv 2020

  37. [37]

    Maggio, P

    E. Maggio, P. Pani, and G. Raposo, (2021), arXiv:2105.06410 [gr-qc]

    work page arXiv 2021

  38. [38]

    Maggio, Lect

    E. Maggio, Lect. Notes Phys.1017, 333 (2023), arXiv:2310.07368 [gr-qc]

    work page arXiv 2023

  39. [39]

    Can environmental effects spoil precision gravitational-wave astrophysics?

    E. Barausse, V. Cardoso, and P. Pani, Phys. Rev. D 89, 104059 (2014), arXiv:1404.7149 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  40. [40]

    Cardoso, K

    V. Cardoso, K. Destounis, F. Duque, R. P. Macedo, and A. Maselli, Phys. Rev. D105, L061501 (2022), arXiv:2109.00005 [gr-qc]

    work page arXiv 2022

  41. [41]

    Cardoso, K

    V. Cardoso, K. Destounis, F. Duque, R. Panosso Macedo, and A. Maselli, Phys. Rev. Lett.129, 241103 (2022), arXiv:2210.01133 [gr-qc]

    work page arXiv 2022

  42. [42]

    Destounis, A

    K. Destounis, A. Kulathingal, K. D. Kokkotas, and G. O. Papadopoulos, Phys. Rev. D107, 084027 (2023), arXiv:2210.09357 [gr-qc]

    work page arXiv 2023

  43. [43]

    Environmental Effects for Gravitational-wave Astrophysics

    E. Barausse, V. Cardoso, and P. Pani, J. Phys. Conf. Ser.610, 012044 (2015), arXiv:1404.7140 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  44. [44]

    M. H.-Y. Cheung, K. Destounis, R. P. Macedo, E. Berti, and V. Cardoso, Phys. Rev. Lett.128, 111103 (2022), arXiv:2111.05415 [gr-qc]

    work page arXiv 2022

  45. [45]

    Berti, V

    E. Berti, V. Cardoso, M. H.-Y. Cheung, F. Di Filippo, F. Duque, P. Martens, and S. Mukohyama, Phys. Rev. D106, 084011 (2022), arXiv:2205.08547 [gr-qc]

    work page arXiv 2022

  46. [46]

    Biswas, C

    S. Biswas, C. Singha, and S. Chakraborty, Phys. Rev. D109, 064043 (2024), arXiv:2307.04836 [gr-qc]

    work page arXiv 2024

  47. [47]

    Singha and S

    C. Singha and S. Biswas, Phys. Rev. D109, 024043 (2024), arXiv:2309.01760 [gr-qc]

    work page arXiv 2024

  48. [48]

    Is the gravitational-wave ringdown a probe of the event horizon?

    V. Cardoso, E. Franzin, and P. Pani, Phys. Rev. Lett. 116, 171101 (2016), [Erratum: Phys.Rev.Lett. 117, 089902 (2016)], arXiv:1602.07309 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  49. [49]

    Echoes of ECOs: gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale

    V. Cardoso, S. Hopper, C. F. B. Macedo, C. Palen- zuela, and P. Pani, Phys. Rev. D94, 084031 (2016), arXiv:1608.08637 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  50. [50]

    Tests for the existence of horizons through gravitational wave echoes

    V. Cardoso and P. Pani, Nature Astron.1, 586 (2017), arXiv:1709.01525 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  51. [51]

    Abedi, N

    J. Abedi, N. Afshordi, N. Oshita, and Q. Wang, Uni- verse6, 43 (2020), arXiv:2001.09553 [gr-qc]

    work page arXiv 2020

  52. [52]

    Nollert, Phys

    H.-P. Nollert, Phys. Rev. D53, 4397 (1996), arXiv:gr- qc/9602032

    work page arXiv 1996

  53. [53]

    R. G. Daghigh, M. D. Green, and J. C. Morey, Phys. Rev. D101, 104009 (2020), arXiv:2002.07251 [gr-qc]

    work page arXiv 2020

  54. [54]

    J. L. Jaramillo, R. Panosso Macedo, and L. Al Sheikh, Phys. Rev. X11, 031003 (2021), arXiv:2004.06434 [gr- qc]

    work page internal anchor Pith review arXiv 2021

  55. [55]

    Destounis, R

    K. Destounis, R. P. Macedo, E. Berti, V. Cardoso, and J. L. Jaramillo, Phys. Rev. D104, 084091 (2021), arXiv:2107.09673 [gr-qc]

    work page arXiv 2021

  56. [56]

    Gasperin and J

    E. Gasperin and J. L. Jaramillo, Class. Quant. Grav. 39, 115010 (2022), arXiv:2107.12865 [gr-qc]

    work page arXiv 2022

  57. [57]

    Boyanov, K

    V. Boyanov, K. Destounis, R. Panosso Macedo, V. Car- doso, and J. L. Jaramillo, Phys. Rev. D107, 064012 (2023), arXiv:2209.12950 [gr-qc]. 13

    work page arXiv 2023

  58. [58]

    J. L. Jaramillo, Class. Quant. Grav.39, 217002 (2022), arXiv:2206.08025 [gr-qc]

    work page arXiv 2022

  59. [59]

    Sarkar, M

    S. Sarkar, M. Rahman, and S. Chakraborty, Phys. Rev. D108, 104002 (2023), arXiv:2304.06829 [gr-qc]

    work page arXiv 2023

  60. [60]

    Destounis, V

    K. Destounis, V. Boyanov, and R. Panosso Macedo, Phys. Rev. D109, 044023 (2024), arXiv:2312.11630 [gr- qc]

    work page arXiv 2024

  61. [61]

    Are´ an, D

    D. Are´ an, D. G. Fari˜ na, and K. Landsteiner, JHEP12, 187 (2023), arXiv:2307.08751 [hep-th]

    work page arXiv 2023

  62. [62]

    Cownden, C

    B. Cownden, C. Pantelidou, and M. Zilh˜ ao, (2023), arXiv:2312.08352 [gr-qc]

    work page arXiv 2023

  63. [63]

    Destounis and F

    K. Destounis and F. Duque (2023) arXiv:2308.16227 [gr- qc]

    work page arXiv 2023

  64. [64]

    Courty, K

    A. Courty, K. Destounis, and P. Pani, Phys. Rev. D 108, 104027 (2023), arXiv:2307.11155 [gr-qc]

    work page arXiv 2023

  65. [65]

    Boyanov, V

    V. Boyanov, V. Cardoso, K. Destounis, J. L. Jaramillo, and R. Panosso Macedo, Phys. Rev. D109, 064068 (2024), arXiv:2312.11998 [gr-qc]

    work page arXiv 2024

  66. [66]

    Cao, J.-N

    L.-M. Cao, J.-N. Chen, L.-B. Wu, L. Xie, and Y.-S. Zhou, (2024), arXiv:2401.09907 [gr-qc]

    work page arXiv 2024

  67. [67]

    Cardoso, S

    V. Cardoso, S. Kastha, and R. Panosso Macedo, Phys. Rev. D110, 024016 (2024), arXiv:2404.01374 [gr-qc]

    work page arXiv 2024

  68. [68]

    Ianniccari, A

    A. Ianniccari, A. J. Iovino, A. Kehagias, P. Pani, G. Perna, D. Perrone, and A. Riotto, Phys. Rev. Lett. 133, 211401 (2024), arXiv:2407.20144 [gr-qc]

    work page arXiv 2024

  69. [69]

    Cai, L.-M

    R.-G. Cai, L.-M. Cao, J.-N. Chen, Z.-K. Guo, L.-B. Wu, and Y.-S. Zhou, (2025), arXiv:2501.02522 [gr-qc]

    work page arXiv 2025

  70. [70]

    R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Class. Quant. Grav.13, L117 (1996), arXiv:gr- qc/9510049

    work page arXiv 1996

  71. [71]

    R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Phys. Rept.325, 41 (2000), arXiv:gr-qc/9807077

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  72. [72]

    Second and higher-order quasi-normal modes in binary black hole mergers

    K. Ioka and H. Nakano, Phys. Rev. D76, 061503 (2007), arXiv:0704.3467 [astro-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  73. [73]

    Second Order Quasi-Normal Mode of the Schwarzschild Black Hole

    H. Nakano and K. Ioka, Phys. Rev. D76, 084007 (2007), arXiv:0708.0450 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  74. [74]

    A complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole

    D. Brizuela, J. M. Martin-Garcia, and M. Tiglio, Phys. Rev. D80, 024021 (2009), arXiv:0903.1134 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  75. [75]

    Mode coupling of Schwarzschild perturbations: Ringdown frequencies

    E. Pazos, D. Brizuela, J. M. Martin-Garcia, and M. Tiglio, Phys. Rev. D82, 104028 (2010), arXiv:1009.4665 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  76. [76]

    J. L. Ripley, N. Loutrel, E. Giorgi, and F. Pretorius, Phys. Rev. D103, 104018 (2021), arXiv:2010.00162 [gr- qc]

    work page arXiv 2021

  77. [77]

    Loutrel, J

    N. Loutrel, J. L. Ripley, E. Giorgi, and F. Pretorius, Phys. Rev. D103, 104017 (2021), arXiv:2008.11770 [gr- qc]

    work page arXiv 2021

  78. [78]

    Sberna, P

    L. Sberna, P. Bosch, W. E. East, S. R. Green, and L. Lehner, Phys. Rev. D105, 064046 (2022), arXiv:2112.11168 [gr-qc]

    work page arXiv 2022

  79. [79]

    M. H.-Y. Cheunget al., Phys. Rev. Lett.130, 081401 (2023), arXiv:2208.07374 [gr-qc]

    work page arXiv 2023

  80. [80]

    Mitman et al., Phys

    K. Mitmanet al., Phys. Rev. Lett.130, 081402 (2023), arXiv:2208.07380 [gr-qc]

    work page arXiv 2023

Showing first 80 references.