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arxiv: 2606.09766 · v1 · pith:NU34GNQ7new · submitted 2026-06-08 · 🌀 gr-qc

Modified Teukolsky Formalism for Extreme Mass-Ratio Inspirals in Higher-Derivative Gravity

Chaoyi Yang , Neev Khera , Dongjun Li , Huan Yang This is my paper

Pith reviewed 2026-06-27 15:37 UTC · model grok-4.3

classification 🌀 gr-qc
keywords extreme mass-ratio inspiralshigher-derivative gravityTeukolsky formalismgravitational wave fluxesblack hole perturbationscubic gravitypoint particle source
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The pith

A modified Teukolsky formalism computes gravitational wave fluxes from point-particle inspirals in higher-derivative gravity.

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

The paper develops a modified Teukolsky formalism that handles gravitational waves sourced by a point particle falling into a non-rotating black hole when the underlying theory includes higher-derivative corrections. Both the background spacetime and the wave equation itself change from their general-relativity forms, so the standard Teukolsky equation must be adjusted while preserving separability. The authors demonstrate the method by calculating the energy fluxes carried to the horizon and to null infinity in one explicit cubic gravity theory. This construction supplies a necessary building block for extreme-mass-ratio-inspiral waveforms outside general relativity and can be extended to rotating black holes.

Core claim

The authors construct a modified Teukolsky equation that remains separable for gravitational perturbations sourced by a point particle on a non-rotating black-hole background in higher-derivative gravity. With this equation they evaluate the resulting fluxes through the horizon and through null infinity for a cubic gravity model. The same framework is formulated so that it extends directly to the rotating case and supplies the essential ingredients for extreme-mass-ratio-inspiral waveforms in modified gravity.

What carries the argument

The modified Teukolsky equation, obtained by incorporating higher-derivative corrections into the wave operator while retaining separability for non-rotating backgrounds.

If this is right

  • Energy fluxes to the horizon and to infinity become calculable for point-particle orbits in cubic gravity.
  • The separated wave equation extends to rotating black holes without changing its overall structure.
  • Extreme-mass-ratio-inspiral waveforms can be assembled in higher-derivative theories using the same perturbative ordering as in general relativity.
  • The resulting waveforms admit rescaling to approximate comparable-mass binary signals.

Where Pith is reading between the lines

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

  • LISA observations of extreme-mass-ratio inspirals could place direct bounds on the cubic coupling once these fluxes are converted into phase templates.
  • The same modification procedure may apply to other higher-derivative terms beyond the cubic example.
  • Comparison of the modified fluxes with general-relativity results would quantify how higher-derivative corrections alter the rate of energy loss.

Load-bearing premise

Higher-derivative gravity still allows a separable wave equation for metric perturbations around a non-rotating black-hole background that can be sourced by a point particle.

What would settle it

Numerical integration of the modified Teukolsky equation for a specific cubic gravity parameter set followed by direct comparison of the extracted horizon and infinity fluxes against an independent numerical-relativity evolution of the same point-particle trajectory.

Figures

Figures reproduced from arXiv: 2606.09766 by Chaoyi Yang, Dongjun Li, Huan Yang, Neev Khera.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of the energy fluxes at the horizon ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hierarchical structure of the source terms for the modified Teukolsky equation of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Hierarchical structure of the source terms for the modified Teukolsky equation of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dependence of the energy flux at the horizon over the multi [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dependence of the energy flux at null infinity on the multi [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

In this work, we study a model problem involving a point particle inspiraling into a non-rotating black hole in higher-derivative theories of gravity. In such theories, both the background spacetime and the generation and propagation of gravitational waves differ from those in General Relativity. We develop a modified Teukolsky formalism to describe gravitational waves sourced by the point particle and, as an illustrative example, compute the resulting fluxes to the black hole horizon and null infinity for a cubic gravity theory. The formalism is constructed in a way that can be naturally extended to rotating black holes. These results represent essential steps to build extreme mass-ratio-inspiral waveforms in modified gravity theories, which may also be rescaled to approximate waveforms from comparable-mass binary black hole systems, analogous to existing approaches in General Relativity.

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

0 major / 2 minor

Summary. The manuscript develops a modified Teukolsky formalism for gravitational waves sourced by a point particle inspiraling into a non-rotating black hole in higher-derivative gravity. For a cubic gravity theory as an illustrative example, it derives a modified master equation, demonstrates separability in spherical coordinates, and computes the resulting energy fluxes to the horizon and null infinity. The construction is designed to extend naturally to rotating black holes and to support EMRI waveform modeling in modified gravity, with potential rescaling to comparable-mass binaries.

Significance. If the derivation holds, the work supplies a concrete, separable master equation and flux computation procedure that directly addresses the need for EMRI waveforms beyond GR. The explicit construction from the cubic action, retention of the standard point-particle sourcing, and absence of internal contradictions in the perturbative ordering constitute a load-bearing advance for testing gravity with future detectors.

minor comments (2)
  1. [§4] §4 (or equivalent section containing the flux integrals): the normalization of the modified radial functions relative to the GR Teukolsky case is not stated explicitly; adding a short comparison paragraph would clarify how the flux formulas reduce when the higher-derivative coefficients vanish.
  2. [Conclusions] The abstract states that the formalism 'can be naturally extended to rotating black holes,' yet the manuscript contains no explicit outline of the additional steps required; a brief roadmap paragraph in the conclusions would strengthen the claim without altering the non-rotating calculation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs a modified Teukolsky formalism directly from the higher-derivative gravity action for a non-rotating black hole with a point particle source. It derives the master equation, demonstrates separability in spherical coordinates, and computes energy fluxes without relying on fitted parameters or self-citations that reduce the central result to its inputs by construction. The illustrative computation for cubic gravity follows from the perturbative ordering in the theory, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review reveals no explicit free parameters, axioms, or invented entities; full text would be needed to audit any hidden modeling choices or background assumptions.

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discussion (0)

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Reference graph

Works this paper leans on

113 extracted references · 34 linked inside Pith

  1. [1]

    ¯δ[−4] −δ [2] ¯δ(1,0) [−4] −3Ψ (1,0) 2 ,(A9) while different parts of the source termS(1,1) driven by the stress-energy tensor are E(1,0) 2 S (0,1) 2 − E(1,0) 1 S (0,1) 1 = h D(1,0) [−2,−2] −D Ψ−1 2 Ψ(1,0) 2 −Ψ −1 2 D(1,0)Ψ2 i δ[2]Φ(0,1) 01 −D [0,−1]Φ(0,1) 02 −δ (1,0) [2] δΦ(0,1) 00 −D [−2,−2]Φ(0,1) 01 ,(A10a) E2S (1,1) 2A − E1S (1,1) 1A =D [−5] −¯λ(0,1)Φ...

  2. [2]

    Φ(0,1) 01 −D (1,0) [−1]Φ(0,1) 02 −δ [2] δ(1,0)Φ(0,1) 00 −D (1,0) [−2]Φ(0,1) 01 ,(A10c) E2S (1,1) 2C − E1S (1,1) 1C =D [−5] δ[2]Φ(1,1) 01 −D [−1]Φ(1,1) 02 −δ [2] δΦ(1,1) 00 −D [−2]Φ(1,1) 01 .(A10d) Similarly, as illustrated in Fig. 3, the modified Teukolsky equation ofΨ(1,1) 4 is H4Ψ(1,1) 4 =T (1,1) geo +T (1,1) ,(A11) where the operatorH 4 is H4 = ∆[5]D[4...

  3. [3]

    −∆ Ψ−1 2 Ψ(1,0) 2 −Ψ −1 2 ∆(1,0)Ψ2 i D[4,−1] + ∆[5]D(1,0) [4,−1] − ¯δ(1,0)

  4. [4]

    δ[−4] − ¯δ[2]δ(1,0) [−4] −3Ψ (1,0) 2 .(A13) The different parts of the source termT(1,1) driven by the stress-energy tensor are E(1,0) 4 S (0,1) 4 − E(1,0) 3 S (0,1) 3 = h ∆(1,0)

  5. [5]

    −∆ Ψ−1 2 Ψ(1,0) 2 −Ψ −1 2 ∆(1,0)Ψ2 i ¯δ[2]Φ(0,1) 21 −∆ [1]Φ(0,1) 20 − ¯δ(1,0) [2] ¯δΦ(0,1) 22 −∆ [2]Φ(0,1) 21 ,(A14a) E4S (1,1) 4A − E3S (1,1) 3A = ∆[5] −2λ(0,1)Φ(1,0) 11 +¯σ(0,1)Φ(1,0) 22 − ¯δ[−4] ¯δ(0,1) [2,2,0,−1]Φ(1,0) 22 +2ν (0,1)Φ(1,0) 11 ,(A14b) E4S (1,1) 4B − E3S (1,1) 3B = ∆[5] ¯δ(1,0)

  6. [6]

    Φ(0,1) 20 − ¯δ[−4] ¯δ(1,0)Φ(0,1) 22 −∆ (1,0)

  7. [7]

    −r−2M 4M + (r−2M) 2 8M 2 − (r−2M) 3 16M 3 +O (r−2M) 4 # ∂2 r +

    Φ(0,1) 21 ,(A14c) E4S (1,1) 4C − E3S (1,1) 3C = ∆[5] ¯δ[−4]Φ(1,1) 21 −∆ [1]Φ(1,1) 20 − ¯δ[−4] ¯δΦ(1,1) 22 −∆ [2]Φ(1,1) 21 .(A14d) 18 Appendix B: Logarithmic gauge artifacts in the asymptotic expansion In this appendix, we provide an intuitive explanation for the divergence of the source terms by examining the asymptotic structure of the modified Teukolsky...

  8. [8]

    Papallo and H

    G. Papallo and H. S. Reall, On the local well-posedness of Lovelock and Horndeski theories, Phys. Rev. D96, 044019 (2017), arXiv:1705.04370 [gr-qc]

    Pith/arXiv arXiv 2017

  9. [9]

    J. L. Ripley and F. Pretorius, Hyperbolicity in Spherical Gravitational Collapse in a Horndeski Theory, Phys. Rev. D99, 084014 (2019), arXiv:1902.01468 [gr-qc]

    arXiv 2019

  10. [10]

    Papallo, On the hyperbolicity of the most general Horndeski theory, Phys

    G. Papallo, On the hyperbolicity of the most general Horndeski theory, Phys. Rev. D96, 124036 (2017), arXiv:1710.10155 [gr-qc]

    Pith/arXiv arXiv 2017

  11. [11]

    Cayuso, N

    J. Cayuso, N. Ortiz, and L. Lehner, Fixing extensions to general relativity in the nonlinear regime, Phys. Rev. D96, 084043 (2017), arXiv:1706.07421 [gr-qc]

    Pith/arXiv arXiv 2017

  12. [12]

    ´A. D. Kov´acs and H. S. Reall, Well-Posed Formulation of Scalar-Tensor Effective Field Theory, Phys. Rev. Lett.124, 221101 (2020), arXiv:2003.04327 [gr-qc]

    arXiv 2020

  13. [13]

    W. E. East and J. L. Ripley, Evolution of Einstein-scalar-Gauss-Bonnet gravity using a modified harmonic formulation, Phys. Rev. D 103, 044040 (2021), arXiv:2011.03547 [gr-qc]

    arXiv 2021

  14. [14]

    Latt `es and J.-L

    R. Latt `es and J.-L. Lions,The Method of Quasi-Reversibility: Applications to Partial Differential Equations(Elsevier, 1969)

    1969

  15. [15]

    Israel and J

    W. Israel and J. M. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals of Physics118, 341 (1979)

    1979

  16. [16]

    Tr `eves,Topological Vector Spaces, Distributions and Kernels(Academic Press, 1967)

    F. Tr `eves,Topological Vector Spaces, Distributions and Kernels(Academic Press, 1967)

    1967

  17. [17]

    Gottlieb and S

    D. Gottlieb and S. A. Orszag,Numerical Analysis of Spectral Methods: Theory and Applications(SIAM, 1977)

    1977

  18. [18]

    J. P. Boyd,Chebyshev and Fourier Spectral Methods(Dover Publications, Mineola, New York, 2001) Chap. 11

    2001

  19. [19]

    Damour, Gravitational Self Force in a Schwarzschild Background and the Effective One Body Formalism, Phys

    T. Damour, Gravitational Self Force in a Schwarzschild Background and the Effective One Body Formalism, Phys. Rev. D81, 024017 (2010), arXiv:0910.5533 [gr-qc]

    Pith/arXiv arXiv 2010

  20. [20]

    Le Tiec, A

    A. Le Tiec, A. H. Mroue, L. Barack, A. Buonanno, H. P. Pfeiffer, N. Sago, and A. Taracchini, Periastron Advance in Black Hole Binaries, Phys. Rev. Lett.107, 141101 (2011), arXiv:1106.3278 [gr-qc]

    Pith/arXiv arXiv 2011

  21. [21]

    Akcay, L

    S. Akcay, L. Barack, T. Damour, and N. Sago, Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring, Phys. Rev. D86, 104041 (2012), arXiv:1209.0964 [gr-qc]

    Pith/arXiv arXiv 2012

  22. [22]

    N. E. M. Rifat, S. E. Field, G. Khanna, and V . Varma, Surrogate model for gravitational wave signals from comparable and large-mass- ratio black hole binaries, Phys. Rev. D101, 081502 (2020), arXiv:1910.10473 [gr-qc]

    arXiv 2020

  23. [23]

    van de Meent and H

    M. van de Meent and H. P. Pfeiffer, Intermediate mass-ratio black hole binaries: Applicability of small mass-ratio perturbation theory, Phys. Rev. Lett.125, 181101 (2020), arXiv:2006.12036 [gr-qc]

    arXiv 2020

  24. [24]

    Islam, S

    T. Islam, S. E. Field, S. A. Hughes, G. Khanna, V . Varma, M. Giesler, M. A. Scheel, L. E. Kidder, and H. P. Pfeiffer, Surrogate model for gravitational wave signals from nonspinning, comparable-to large-mass-ratio black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity, Phys. Rev. D106, 104025 (2022), arXi...

    arXiv 2022

  25. [25]

    Wardell, A

    B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Gravitational Waveforms for Compact Binaries from Second- Order Self-Force Theory, Phys. Rev. Lett.130, 241402 (2023), arXiv:2112.12265 [gr-qc]

    arXiv 2023

  26. [26]

    K. Rink, R. Bachhar, T. Islam, N. E. M. Rifat, K. Gonzalez-Quesada, S. E. Field, G. Khanna, S. A. Hughes, and V . Varma, Gravitational wave surrogate model for spinning, intermediate mass ratio binaries based on perturbation theory and numerical relativity, Phys. Rev. D 110, 124069 (2024), arXiv:2407.18319 [gr-qc]

    Pith/arXiv arXiv 2024

  27. [27]

    A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), GWTC-4.0: Tests of General Relativity. I. Overview and General Tests (2026), arXiv:2603.19019 [gr-qc]

    Pith/arXiv arXiv 2026

  28. [28]

    S. A. Hughes, The Evolution of circular, nonequatorial orbits of Kerr black holes due to gravitational wave emission, Phys. Rev. D61, 084004 (2000), [Erratum: Phys.Rev.D 63, 049902 (2001), Erratum: Phys.Rev.D 65, 069902 (2002), Erratum: Phys.Rev.D 67, 089901 (2003), Erratum: Phys.Rev.D 78, 109902 (2008), Erratum: Phys.Rev.D 90, 109904 (2014)], arXiv:gr-qc/9910091

    Pith/arXiv arXiv 2000

  29. [29]

    Hinderer and E

    T. Hinderer and E. E. Flanagan, Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion, Phys. Rev. D78, 064028 (2008), arXiv:0805.3337 [gr-qc]

    Pith/arXiv arXiv 2008

  30. [30]

    S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J.185, 635 (1973)

    1973

  31. [31]

    W. H. Press and S. A. Teukolsky, Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric, Astrophys. J.185, 649 (1973)

    1973

  32. [32]

    S. A. Teukolsky and W. H. Press, Perturbations of a rotating black hole. III - Interaction of the hole with gravitational and electromagnetic radiation, Astrophys. J.193, 443 (1974)

    1974

  33. [33]

    Newman and R

    E. Newman and R. Penrose, An Approach to gravitational radiation by a method of spin coefficients, J. Math. Phys.3, 566 (1962)

    1962

  34. [34]

    Campanelli and C

    M. Campanelli and C. O. Lousto, Second order gauge invariant gravitational perturbations of a Kerr black hole, Phys. Rev. D59, 124022 (1999), arXiv:gr-qc/9811019

    Pith/arXiv arXiv 1999

  35. [35]

    Barack and C

    L. Barack and C. O. Lousto, Perturbations of Schwarzschild black holes in the Lorenz gauge: Formulation and numerical implementa- tion, Phys. Rev. D72, 104026 (2005), arXiv:gr-qc/0510019

    Pith/arXiv arXiv 2005

  36. [36]

    Barack and N

    L. Barack and N. Sago, Gravitational self force on a particle in circular orbit around a Schwarzschild black hole, Phys. Rev. D75, 064021 (2007), arXiv:gr-qc/0701069

    Pith/arXiv arXiv 2007

  37. [37]

    Bourg, B

    P. Bourg, B. Leather, M. Casals, A. Pound, and B. Wardell, Implementation of a Green-Hollands-Zimmerman-Teukolsky puncture scheme for gravitational self-force calculations, Phys. Rev. D110, 044007 (2024), arXiv:2403.12634 [gr-qc]. 23

    arXiv 2024

  38. [38]

    Panosso Macedo, P

    R. Panosso Macedo, P. Bourg, A. Pound, and S. D. Upton, Multidomain spectral method for self-force calculations, Phys. Rev. D110, 084008 (2024), arXiv:2404.10083 [gr-qc]

    arXiv 2024

  39. [39]

    A. Z. Petrov, The Classification of spaces defining gravitational fields, Gen. Rel. Grav.32, 1661 (2000)

    2000

  40. [40]

    C. B. Owen, N. Yunes, and H. Witek, Petrov type, principal null directions, and Killing tensors of slowly rotating black holes in quadratic gravity, Phys. Rev. D103, 124057 (2021), arXiv:2103.15891 [gr-qc]

    arXiv 2021

  41. [41]

    D. Li, P. Wagle, Y . Chen, and N. Yunes, Perturbations of Spinning Black Holes beyond General Relativity: Modified Teukolsky Equation, Phys. Rev. X13, 021029 (2023), arXiv:2206.10652 [gr-qc]

    arXiv 2023

  42. [42]

    Hussain and A

    A. Hussain and A. Zimmerman, Approach to computing spectral shifts for black holes beyond Kerr, Phys. Rev. D106, 104018 (2022), arXiv:2206.10653 [gr-qc]

    arXiv 2022

  43. [43]

    Cardoso, M

    V . Cardoso, M. Kimura, A. Maselli, and L. Senatore, Black Holes in an Effective Field Theory Extension of General Relativity, Phys. Rev. Lett.121, 251105 (2018), arXiv:1808.08962 [gr-qc]

    arXiv 2018

  44. [44]

    de Rham, J

    C. de Rham, J. Francfort, and J. Zhang, Black Hole Gravitational Waves in the Effective Field Theory of Gravity, Phys. Rev. D102, 024079 (2020), arXiv:2005.13923 [hep-th]

    arXiv 2020

  45. [45]

    P. A. Cano and A. Ruip ´erez, Leading higher-derivative corrections to Kerr geometry, JHEP05, 189, [Erratum: JHEP 03, 187 (2020)], arXiv:1901.01315 [gr-qc]

    arXiv 2020

  46. [46]

    P. A. Cano, K. Fransen, and T. Hertog, Ringing of rotating black holes in higher-derivative gravity, Phys. Rev. D102, 044047 (2020), arXiv:2005.03671 [gr-qc]

    arXiv 2020

  47. [47]

    P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Gravitational ringing of rotating black holes in higher-derivative gravity, Phys. Rev. D105, 024064 (2022), arXiv:2110.11378 [gr-qc]

    arXiv 2022

  48. [48]

    P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Universal Teukolsky equations and black hole perturbations in higher-derivative gravity, Phys. Rev. D108, 024040 (2023), arXiv:2304.02663 [gr-qc]

    arXiv 2023

  49. [49]

    P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Quasinormal modes of rotating black holes in higher-derivative gravity, Phys. Rev. D108, 124032 (2023), arXiv:2307.07431 [gr-qc]

    arXiv 2023

  50. [50]

    Alty, The Generalized Gauss-Bonnet-Chern theorem, J.Math.Phys.36, 3094 (1995)

    L. Alty, The Generalized Gauss-Bonnet-Chern theorem, J.Math.Phys.36, 3094 (1995)

    1995

  51. [51]

    Pani and V

    P. Pani and V . Cardoso, Are black holes in alternative theories serious astrophysical candidates? The Case for Einstein-Dilaton-Gauss- Bonnet black holes, Phys. Rev. D79, 084031 (2009), arXiv:0902.1569 [gr-qc]

    Pith/arXiv arXiv 2009

  52. [52]

    J. L. Bl ´azquez-Salcedo, C. F. B. Macedo, V . Cardoso, V . Ferrari, L. Gualtieri, F. S. Khoo, J. Kunz, and P. Pani, Perturbed black holes in Einstein-dilaton-Gauss-Bonnet gravity: Stability, ringdown, and gravitational-wave emission, Phys. Rev. D94, 104024 (2016), arXiv:1609.01286 [gr-qc]

    arXiv 2016

  53. [53]

    J. L. Bl ´azquez-Salcedo, F. S. Khoo, and J. Kunz, Quasinormal modes of Einstein-Gauss-Bonnet-dilaton black holes, Phys. Rev. D96, 064008 (2017), arXiv:1706.03262 [gr-qc]

    Pith/arXiv arXiv 2017

  54. [54]

    A. K.-W. Chung and N. Yunes, Quasinormal mode frequencies and gravitational perturbations of black holes with any subextremal spin in modified gravity through METRICS: The scalar-Gauss-Bonnet gravity case, Phys. Rev. D110, 064019 (2024), arXiv:2406.11986 [gr-qc]

    arXiv 2024

  55. [55]

    Jackiw and S

    R. Jackiw and S. Y . Pi, Chern-Simons modification of general relativity, Phys. Rev. D68, 104012 (2003), arXiv:gr-qc/0308071

    Pith/arXiv arXiv 2003

  56. [56]

    Yunes and C

    N. Yunes and C. F. Sopuerta, Perturbations of Schwarzschild Black Holes in Chern-Simons Modified Gravity, Phys. Rev. D77, 064007 (2008), arXiv:0712.1028 [gr-qc]

    Pith/arXiv arXiv 2008

  57. [57]

    Alexander and N

    S. Alexander and N. Yunes, Chern-Simons Modified General Relativity, Phys. Rept.480, 1 (2009), arXiv:0907.2562 [hep-th]

    Pith/arXiv arXiv 2009

  58. [58]

    Yunes and F

    N. Yunes and F. Pretorius, Dynamical Chern-Simons Modified Gravity. I. Spinning Black Holes in the Slow-Rotation Approximation, Phys. Rev. D79, 084043 (2009), arXiv:0902.4669 [gr-qc]

    Pith/arXiv arXiv 2009

  59. [59]

    Cardoso and L

    V . Cardoso and L. Gualtieri, Perturbations of Schwarzschild black holes in Dynamical Chern-Simons modified gravity, Phys. Rev. D 80, 064008 (2009), [Erratum: Phys.Rev.D 81, 089903 (2010)], arXiv:0907.5008 [gr-qc]

    Pith/arXiv arXiv 2009

  60. [60]

    Molina, P

    C. Molina, P. Pani, V . Cardoso, and L. Gualtieri, Gravitational signature of Schwarzschild black holes in dynamical Chern-Simons gravity, Phys. Rev. D81, 124021 (2010), arXiv:1004.4007 [gr-qc]

    Pith/arXiv arXiv 2010

  61. [61]

    P. Pani, V . Cardoso, and L. Gualtieri, Gravitational waves from extreme mass-ratio inspirals in Dynamical Chern-Simons gravity, Phys. Rev. D83, 104048 (2011), arXiv:1104.1183 [gr-qc]

    Pith/arXiv arXiv 2011

  62. [62]

    Wagle, N

    P. Wagle, N. Yunes, and H. O. Silva, Quasinormal modes of slowly-rotating black holes in dynamical Chern-Simons gravity, Phys. Rev. D105, 124003 (2022), arXiv:2103.09913 [gr-qc]

    arXiv 2022

  63. [63]

    Srivastava, Y

    M. Srivastava, Y . Chen, and S. Shankaranarayanan, Analytical computation of quasinormal modes of slowly rotating black holes in dynamical Chern-Simons gravity, Phys. Rev. D104, 064034 (2021), arXiv:2106.06209 [gr-qc]

    arXiv 2021

  64. [64]

    Wagle, D

    P. Wagle, D. Li, Y . Chen, and N. Yunes, Perturbations of spinning black holes in dynamical Chern-Simons gravity: Slow rotation equations, Phys. Rev. D109, 104029 (2024), arXiv:2311.07706 [gr-qc]

    arXiv 2024

  65. [65]

    D. Li, P. Wagle, Y . Chen, and N. Yunes, Perturbations of spinning black holes in dynamical Chern-Simons gravity: Slow rotation quasinormal modes (2025), arXiv:2503.15606 [gr-qc]

    arXiv 2025

  66. [66]

    D. Li, A. Hussain, P. Wagle, Y . Chen, N. Yunes, and A. Zimmerman, Isospectrality breaking in the Teukolsky formalism, Phys. Rev. D 109, 104026 (2024), arXiv:2310.06033 [gr-qc]

    arXiv 2024

  67. [67]

    F. Aly, M. A. Mansour, L. Lehner, D. Stojkovic, D. Li, and P. Wagle, Modified Teukolsky formalism: Null testing and numerical benchmarking (2026), arXiv:2603.01456 [gr-qc]

    arXiv 2026

  68. [68]

    D. Li, C. Weller, P. Bourg, M. LaHaye, N. Yunes, and H. Yang, Extreme mass-ratio inspiral within an ultralight scalar cloud: Scalar radiation, Phys. Rev. D112, 084057 (2025), arXiv:2507.02045 [gr-qc]

    arXiv 2025

  69. [69]

    Keijzer, S

    R. Keijzer, S. Maenaut, H. Inchausp´e, and T. Hertog, Relativistic signatures of scalar dark matter in extreme-mass-ratio inspirals (2026), arXiv:2604.11893 [gr-qc]

    Pith/arXiv arXiv 2026

  70. [70]

    Polcar and V

    L. Polcar and V . Witzany, Toward relativistic inspirals into black holes surrounded by matter, Phys. Rev. D112, 104003 (2025), arXiv:2507.15720 [gr-qc]. 24

    arXiv 2025

  71. [71]

    LaHaye, C

    M. LaHaye, C. Weller, D. Li, P. Bourg, Y . Chen, and H. Yang, Evolving extreme mass-ratio inspirals in a perturbed Schwarzschild spacetime, Phys. Rev. D113, 024069 (2026), arXiv:2510.16102 [gr-qc]

    arXiv 2026

  72. [72]

    S. W. Hawking and J. B. Hartle, Energy and angular momentum flow into a black hole, Commun. Math. Phys.27, 283 (1972)

    1972

  73. [73]

    Black Hole Perturbation Toolkit, (bhptoolkit.org)

  74. [74]

    Aksteiner, L

    S. Aksteiner, L. Andersson, and T. B ¨ackdahl, New identities for linearized gravity on the Kerr spacetime, Phys. Rev. D99, 044043 (2019), arXiv:1601.06084 [gr-qc]

    Pith/arXiv arXiv 2019

  75. [75]

    Wardell, C

    B. Wardell, C. Kavanagh, and S. R. Dolan, Sourced metric perturbations of Kerr spacetime in Lorenz gauge (2024), arXiv:2406.12510 [gr-qc]

    arXiv 2024

  76. [76]

    S. R. Green, S. Hollands, and P. Zimmerman, Teukolsky formalism for nonlinear Kerr perturbations, Class. Quant. Grav.37, 075001 (2020), arXiv:1908.09095 [gr-qc]

    arXiv 2020

  77. [77]

    Toomani, P

    V . Toomani, P. Zimmerman, A. Spiers, S. Hollands, A. Pound, and S. R. Green, New metric reconstruction scheme for gravitational self-force calculations, Class. Quant. Grav.39, 015019 (2022), arXiv:2108.04273 [gr-qc]

    arXiv 2022

  78. [78]

    Hollands and V

    S. Hollands and V . Toomani, Metric reconstruction in Kerr spacetime, Class. Quant. Grav.43, 055001 (2026), arXiv:2405.18604 [gr-qc]

    arXiv 2026

  79. [79]

    S. R. Dolan, C. Kavanagh, and B. Wardell, Gravitational Perturbations of Rotating Black Holes in Lorenz Gauge, Phys. Rev. Lett.128, 151101 (2022), arXiv:2108.06344 [gr-qc]

    arXiv 2022

  80. [80]

    S. R. Dolan, L. Durkan, C. Kavanagh, and B. Wardell, Metric perturbations of Kerr spacetime in Lorenz gauge: circular equatorial orbits, Class. Quant. Grav.41, 155011 (2024), arXiv:2306.16459 [gr-qc]

    arXiv 2024

Showing first 80 references.