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The Nonlinear Tails in Black Hole Ringdown: the Scattering Perspective

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that the nonlinear late-time tail of a Schwarzschild ringdown mode with multipole ℓ decays as $t^{-(2\ell+1)}$ in TT gauge, because quasinormal modes scatter off the black hole's Newtonian potential.

desk verdict A mostly sound in-in scattering re-derivation of the known t^{-(2ℓ+1)} nonlinear tail law, with the decisive selection-rule step borrowed from a same-group preprint and an all-orders claim that outruns the proof. read the letter →

arxiv 2507.17732 v1 pith:QQQ7SGZF submitted 2025-07-23 gr-qc hep-th

A. Ianniccari , L. Lo Bianco , A. Riotto This is my paper
classification gr-qchep-th
keywords blackholeringdownnonlineartailsquasinormalmodesin-informalismgravitationalwavescatteringSchwarzschildPricelawlate-time
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the power-law tails seen at the end of a black hole ringdown are not a separate phenomenon from quasinormal-mode emission: they are what you get when those modes scatter off the Newtonian part of the black hole's gravitational field. Working in the in-in formalism, the authors rederive Price's linear tail $t^{-(2\ell+3)}$ and then show that the dominant nonlinear tail of a mode with multipole $\ell$ decays as $t^{-(2\ell+1)}$ in transverse-traceless gauge, at any order in perturbation theory. The claim matters because it turns a numerical observation—a second-order $\ell=4$ mode decaying roughly as $t^{-9}$—into a simple scattering rule, and it predicts that nonlinear tails decay more slowly than linear ones, so they can dominate the late ringdown. If right, the result gives a direct, parameter-free relation between the multipole of a gravitational wave and its late-time falloff.

What carries the argument

The load-bearing object is the retarded (Price) Green's function in the static limit, $G(r_3,t_3,y)\sim P_{\ell_3}(\chi)/t_3$ with $\chi=[-(t-t_3)^2+(r_3-y)^2]/(2r_3 y)$, imported from Ref. [74]; combined with the cubic graviton vertex (whose tensor contraction $L_{\rm tens}$ reduces to $3\omega_1^2/4$ when the final frequency goes to zero) and the Legendre-polynomial orthogonality rule that makes the radial integral vanish unless the final multipole satisfies $\ell_3\le \ell_1+\ell_2$, it converts a chain of nested commutators in the in-in formalism into the power law $\int d\tau\,dr\,(GM/r)^n P_{\ell_3}(\chi)\sim 1/t^{2\ell_3+1}$.

What would settle it

Run a high-accuracy numerical evolution of a perturbed Schwarzschild black hole and extract the TT-gauge strain of the nonlinear $\ell=4$ mode at a fixed radius: the paper predicts late-time decay $t^{-9}$, while the linear Price law would give $t^{-11}$; the measured exponent settles the claim. A second check is to compute the exact retarded Green's function and test whether its static-limit form really is $P_{\ell}(\chi)/t$.

Watch

Extended reading notes

Core claim

In the paper's own terms, the discovery is that the late-time nonlinear ringdown is controlled by a single scattering process: two quasinormal modes of multipoles $\ell_1=\ell_2$ combine at a cubic graviton vertex into a mode of multipole $\ell_3$, and that mode rescatters off the static Newtonian potential $2M/r$ produced by the black hole. Evaluating the in-in expectation value order by order, the authors find $\langle h\rangle \sim (GM)^{\ell_3}/t^{2\ell_3+1}$ in TT gauge, so an $\ell=4$ mode sourced by two $\ell=2$ modes decays as $t^{-9}$, and in general the exponent is fixed by the propagating mode's multipole, not by the perturbative order. The same machinery reproduces Price's linear tail $t^{-(2\ell+3)}$ as the single-scattering case. The authors argue that the power $t^{-(2\ell+1)}$ holds at every order because every vertex carries two derivatives, so the frequency power of the observed final mode never changes.

Load-bearing premise

The whole argument depends on an imported asymptotic formula for the wave's late-time response that already decays like $1/t$; the target decay law is essentially read out of that formula, so if the approximation fails, the derived exponents collapse.

Editorial extensions

If this is right

  • A nonlinear tail of multipole $\ell$ decays as $t^{-(2\ell+1)}$ in TT gauge regardless of perturbative order; higher-order diagrams change only the amplitude, not the exponent.
  • Nonlinear tails decay more slowly than Price's linear tail $t^{-(2\ell+3)}$, so at sufficiently late times the quadratic effect can dominate the ringdown signal.
  • Gauge matters: the same physical tail appears one power steeper in the Regge-Wheeler gauge ($t^{-10}$ for $\ell=4$) than in TT gauge ($t^{-9}$), so predictions and measurements must be compared in a fixed gauge.
  • The linear and nonlinear tails share one mechanism—backscattering off the Newtonian potential—so the in-in scattering calculation reproduces both Price's law and the newer $t^{-(2\ell+1)}$ result.

Reading between the lines

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

  • A self-contained derivation of the static-limit Green's function asymptotic $G\sim P_{\ell}(\chi)/t$, rather than importing it from earlier work, would be the cleanest check of the chain, since the target exponent is essentially read out of that formula.
  • The same scattering picture may extend to Kerr black holes and to scalar or electromagnetic perturbations; if it does, the multipole would still fix the exponent, with spin entering only through amplitudes and mode frequencies.
  • Because every vertex carries two derivatives, the argument hints that interactions with the external field could be resummed into an effective potential; a resummed calculation would predict not just the exponent but the amplitude of the nonlinear tail, which is what a ringdown measurement would actually constrain.
  • Keeping the three vertices at distinct radii instead of taking the far-away coincidence limit would test whether the simplified radial integral changes only the coefficient or also the exponent.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper uses the in-in (closed-time-path) formalism to give a scattering interpretation of the late-time power-law tails of gravitational waves during black hole ringdown. After rederiving the linear Price tail as scattering off the Newtonian potential, the authors compute the (GM)^2 and (GM)^4 diagrams for two and four external sources, obtaining t^{-5} and t^{-9} tails, and then argue that at any perturbative order the nonlinear tail decays as t^{-(2ℓ+1)} in TT gauge, where ℓ is the multipole of the final propagating mode.

Significance. If the all-orders claim is established, the paper offers a simple and physically transparent scattering picture for nonlinear ringdown tails and a compact derivation of the t^{-(2ℓ+1)} law that has been observed numerically. The explicit two- and four-source calculations reproduce the known t^{-5} and t^{-9} tails, which is a useful consistency check. The in-in formalism is appropriate for the problem, and the paper's diagrammatic organization is clear. However, the nonperturbative generalization rests on a key asymptotic imported from a same-group reference and on several heuristic steps, so the central claim is not yet fully supported.

major comments (4)
  1. [Sec. 4.2, Eq. (4.24)] The static-limit Green's function asymptotic G(r3,t3,y) ~ P_ℓ(χ)/t3 is imported from Ref. [74] without derivation. This asymptotic is load-bearing: together with the Legendre orthogonality selection rule (stated as 'easy to show' after Eq. (4.28)) it is what converts the integrals (4.26)-(4.28) into the claim that the first non-zero contribution is ℓ3 ≤ 2 and hence produces the t^{-(2ℓ+1)} law. Because Ref. [74] is by the same group and already derives the t^{-(2ℓ+1)} tail, the explicit two- and four-source calculations reduce, at this decisive point, to a prior same-group result rather than an independent derivation. Please either prove (4.24) from (4.22) in an appendix or clearly mark it as an input and provide an independent check (e.g., a numerical evaluation of the integrals in a simplified model).
  2. [Sec. 4.2, Eq. (4.27)] The replacement of the three radial integrals by δ(r1 - r2)δ(r2 - r3), described as 'far away from the BH we can confuse the positions of the three vertices', is not a controlled approximation. The vertices r1, r2, r3 are integrated over all radii; the spherical Bessel functions j_{ℓ1}(ω1 r_i) have different arguments, and the Legendre orthogonality is applied only after this reduction. The paper does not show that the delta-function limit gives the leading contribution at late times, nor that the angular selection rule survives the full radial integrals. This step is essential for the selection rule ℓ3 ≤ 2 (Eq. (4.28)) and the analogous rule for ℓ3 ≤ 4 (Eq. (4.41)). Please justify the limit, for instance by performing the r integrals explicitly for small ℓ or by expanding in a small parameter and showing that the delta-function term dominates.
  3. [Sec. 4.4, Eqs. (4.43)-(4.46)] The all-orders extension is heuristic rather than a proof. The statement that 'the only component that provides information on the nonlinear tail' is the inner commutator is asserted without examining whether higher-order time integrals (t4, t5) can introduce additional powers of t or logarithms, or whether re-scatterings can change the observed multipole ℓ3. The argument based on 'the power of the frequency of the final state will not change' because each vertex has two derivatives is not sufficient: the frequency dependence can also enter through the vertex tensors (e.g., L_tens in Eq. (4.6) depends on ω1 and ω3) and through the nested integrals. Since the Abstract claims the nonperturbative law 'at any order in perturbation theory', Sec. 4.4 needs a more rigorous treatment, or the claim should be softened to a conjecture.
  4. [Sec. 4.2, Eqs. (4.21)-(4.23); Sec. 4.3, Eq. (4.39)] The evaluations leading to t^{-5} and t^{-9} are not actually shown. The text says 'Inserting this result ... one gets' and 'We can easily see that the time dependence is (GM)^4/t^9' without specifying the late-time asymptotic of the Green's function (4.22) used, the treatment of the ω1 and radial integrals, or the conditions under which the τ3 integrals converge. These low-order results are the concrete evidence for the general law, so the derivations need to be written out in enough detail for the reader to verify the stated powers.
minor comments (5)
  1. [Sec. 4.2, Eqs. (4.24)-(4.25)] The definition of χ in Eq. (4.25) involves t, but the Green's function in Eq. (4.24) is written as G(r3,t3,y) and earlier as G(r3,τ3,y) with τ3 = t3 - t; please clarify the relation between these variables and the argument of the Legendre polynomial.
  2. [Sec. 4.1, Eqs. (4.10)-(4.12)] The tadpole contribution is said to be 'renormalized to zero by an appropriate counterterm', but the counterterm is never written down. Please specify it or explain why the tadpole expectation value vanishes without an explicit counterterm.
  3. [Sec. 3, Eqs. (3.19)-(3.22)] The contour rotation to the imaginary axis is used to evaluate the frequency integral, but the analyticity domain of the integrand and the convergence of the subsequent r integrals are not discussed. A brief justification would help.
  4. [Sec. 4.2, text after Eq. (4.28)] The phrase 'following the Ref. [74] it is easy to show' delegates a key, non-trivial step. Please write out the orthogonality argument explicitly, or at least give the precise identity used.
  5. [General] There are several typos and informal expressions: 'it to generalize' in Sec. 2, 'the only component that provide information' in Sec. 4.4, and 'confusing the positions of the three vertices' in Sec. 4.2. Also, Eq. (4.7) uses C_{ℓ1ℓ2ℓ3,m1m2m3} without defining the normalization of the Clebsch-Gordan coefficients.

Circularity Check

1 steps flagged · score 4.0 of 10

The generic nonlinear-tail law rests on a same-group citation: the static-limit Green's function Pℓ(χ)/t3 and Legendre selection rule are imported from Ref. [74], which already contains the t^{-(2ℓ+1)} result; low-order explicit examples remain non-circular.

  1. self citation load bearing [Sec. 4.2, Eqs. (4.24)-(4.28) and Sec. 4.3, Eqs. (4.40)-(4.42)]
    "In the static limit t3 ≫ r3 [74] G(r3, t3, y) ∼ Pℓ3(χ)/t3, where we have defined AdS 2 invariant distance χ = −(t − t3)2 + (r3 − y)2/(2r3y), so that we can simplify the integrals ... Now, following the Ref. [74] it is easy to show that from the orthogonality's properties of the Legendre polynomial, the first non-zero contribution will be given by ℓ3 ≤ 2."

    This is the decisive step at which the target t^{-(2ℓ+1)} behavior enters the derivation: the replacement G ∼ Pℓ3(χ)/t3 and the Legendre-orthogonality selection rule convert the in-in radial integrals into the claimed multipole law. The paper does not derive this asymptotic or the selection rule; it cites Ref. [74], by A. Kehagias and A. Riotto, where Riotto is a coauthor of the present paper. The Introduction already credits [72–74] with the law '∼ t−2ℓ−1', so the all-orders claim is imported from a same-group paper rather than obtained from the scattering calculation.

full rationale

The linear-tail section and the explicit two- and four-source in-in computations are not circular: they use the standard Price Green function [76] and produce (GM)^2/t^5 and (GM)^4/t^9, consistent with the known nonlinear tails. The circularity appears in the generalization. In Sec. 4.2 the paper replaces the Price Green function by the static-limit asymptotic G ∼ Pℓ3(χ)/t3, citing Ref. [74], and then 'following the Ref. [74]' invokes Legendre orthogonality to obtain the selection rule ℓ3 ≤ 2. Ref. [74] is by Kehagias and Riotto, a same-group paper, and the Introduction itself states that the decay pattern '∼ t−2ℓ−1' is 'attributable' to [72–74]. Thus the generic and all-orders version of the central claim reduces, at its decisive step, to a self-citation that already contains the result. The delta-function replacement of the three radial integrals in Eq. (4.27) and the heuristic all-orders argument in Sec. 4.4 raise rigor concerns but are not themselves circularity. Because the low-order explicit results are independently computed from an external Green function, the circularity is partial rather than total.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted numerical parameters are introduced. The central claim rests on a chain of domain assumptions: the Newtonian-potential background, TT gauge, the soft outgoing mode limit, the far-field coincidence of vertices, and, crucially, the static-limit Green's function and selection rule imported from the same group's Ref. [74]. No new entities are postulated.

assumptions (7)
  • domain assumption The Schwarzschild background is replaced by its first-order Newtonian potential h_cl ~ diag(2M/r, 2M/r, 0, 0) in Newtonian gauge (Eq. 2.3); all higher-order terms in M/r are dropped.
    Used throughout Secs. 3-4 to define the external-field vertices; valid only far from the BH (r >> M), but the tail calculation is claimed to be dominated by large r.
  • domain assumption The graviton is quantized in TT gauge and the metric expansion g = η + √G h with the cubic action (4.3) in that gauge; the tail power law is gauge-dependent (footnote 1 notes one power difference in RW gauge).
    The central result t^{-(2ℓ+1)} is asserted specifically in TT gauge; the observable implications depend on this choice.
  • domain assumption The initial QNM state is treated as a collection of single-graviton modes with commutation (3.5), and expectation values like ⟨a a†⟩ are taken to be order one; no explicit initial state is defined.
    Used in eqs. (4.10)-(4.12) for the tadpole and the nonlinear expectation values; the QNM is a classical coherent wave, not a single-particle state, so this modeling is not justified in the paper.
  • ad hoc to paper The static-limit Green's function G(r3, t3, y) ~ P_ℓ(χ)/t3 (Eq. 4.24) and the Legendre-polynomial orthogonality selection rule are imported from Ref. [74].
    This is the step that converts the integrals (4.26)-(4.28) into the t^{-(2ℓ+1)} law; it is taken from a same-group paper that already contains the target result.
  • domain assumption The outgoing mode is taken in the soft limit ω3 → 0 after contraction with the external field (Appendix A, around Eq. A.11).
    The tail is dominated by low-frequency backscattered modes; the paper assumes the soft limit captures the leading late-time power without proving it.
  • domain assumption In the far field, the three interaction vertices are taken at the same position r1 ≈ r2 ≈ r3, implemented with δ-functions (Eq. 4.27).
    Used to reduce the radial integrals; the validity of collapsing distinct vertices to a single radius is asserted, not derived.
  • ad hoc to paper At every order in perturbation theory, the final tail's time power is set only by the frequency of the observed mode because each vertex has two derivatives (Sec. 4.4).
    This is the basis of the all-orders claim (4.46); it is a scaling assertion that does not account for the internal frequency integrals or potential cancellations.

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Pith. "Pith review of The Nonlinear Tails in Black Hole Ringdown: the Scattering Perspective." pith.science (2026). https://pith.science/paper/QQQ7SGZF

@misc{pith2026250717732,
  author       = {Pith},
  title        = {Pith review of: The Nonlinear Tails in Black Hole Ringdown: the Scattering Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQQ7SGZF}},
  note         = {Machine review of arXiv:2507.17732}
}
abstract

Black holes regain their static configuration by emitting ringdown gravitational waves, whose amplitude decays in time following a power law at fixed spatial positions. We show that the nonlinear decay power law may be obtained by simple scattering calculations using the in-in formalism and argue that the nonperturbative law should be $t^{-2\ell-1}$, where $\ell$ is the multipole of the propagating spherical gravitational wave.

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Forward citations

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  2. Nonlinear tails of massive scalar fields around a black hole

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Reference graph

Works this paper leans on

76 extracted references · 10 canonical work pages · cited by 2 Pith papers

  1. [74]

    Kehagias and A

    A. Kehagias and A. Riotto, The AdS Perspective on the Nonlinear Tails in Black Hole Ringdown, 2506.14475

    arXiv

  2. [1]

    Berti et al., Testing General Relativity with Present and Future Astrophysical Observations , Class

    E. Berti et al., Testing General Relativity with Present and Future Astrophysical Observations , Class. Quant. Grav. 32 (2015) 243001 [ 1501.07274]

    arXiv 2015

  3. [2]

    Berti, K

    E. Berti, K. Yagi, H. Yang and N. Yunes, Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown , Gen. Rel. Grav. 50 (2018) 49 [ 1801.03587]. – 17 –

    arXiv 2018

  4. [3]

    Franciolini, L

    G. Franciolini, L. Hui, R. Penco, L. Santoni and E. Trincherini, Effective Field Theory of Black Hole Quasinormal Modes in Scalar-Tensor Theories , JHEP 02 (2019) 127 [ 1810.07706]

    arXiv 2019

  5. [4]

    LIGO Scientific, Virgocollaboration, Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog , Phys. Rev. D 103 (2021) 122002 [2010.14529]

    arXiv 2021

  6. [5]

    Berti, V

    E. Berti, V. Cardoso and C.M. Will, On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA , Phys. Rev. D 73 (2006) 064030 [ gr-qc/0512160]

    arXiv 2006

  7. [6]

    KAGRA, LIGO Scientific, Virgocollaboration, Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA , Living Rev. Rel. 19 (2016) 1 [ 1304.0670]

    arXiv 2016

  8. [7]

    LIGO Scientific, Virgocollaboration, Observation of Gravitational Waves from a Binary Black Hole Merger , Phys. Rev. Lett. 116 (2016) 061102 [ 1602.03837]

    arXiv 2016

Show all 76 references
  1. [8]

    KAGRA, VIRGO, LIGO Scientificcollaboration, GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run , Phys. Rev. X 13 (2023) 041039 [ 2111.03606]

    2023 arXiv

  2. [9]

    LIGO Scientific, KAGRA, VIRGOcollaboration, Search for Eccentric Black Hole Coalescences during the Third Observing Run of LIGO and Virgo , Astrophys. J. 973 (2024) 132 [2308.03822]

    2024 arXiv

  3. [10]

    Berti et al., Black hole spectroscopy: from theory to experiment , 2505.23895

    E. Berti et al., Black hole spectroscopy: from theory to experiment , 2505.23895

    arXiv

  4. [11]

    Price, Nonspherical perturbations of relativistic gravitational collapse

    R.H. Price, Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations, Phys. Rev. D 5 (1972) 2419

    1972

  5. [12]

    Price, Nonspherical Perturbations of Relativistic Gravitational Collapse

    R.H. Price, Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields, Phys. Rev. D 5 (1972) 2439

    1972

  6. [13]

    Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys

    E.W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys. Rev. D 34 (1986) 384

    1986

  7. [14]

    Gundlach, R.H

    C. Gundlach, R.H. Price and J. Pullin, Late time behavior of stellar collapse and explosions: 1. Linearized perturbations, Phys. Rev. D 49 (1994) 883 [ gr-qc/9307009]

    1994 arXiv

  8. [15]

    Gundlach, R.H

    C. Gundlach, R.H. Price and J. Pullin, Late time behavior of stellar collapse and explosions: 2. Nonlinear evolution, Phys. Rev. D 49 (1994) 890 [ gr-qc/9307010]

    1994 arXiv

  9. [16]

    Ching, P.T

    E.S.C. Ching, P.T. Leung, W.M. Suen and K. Young, Wave propagation in gravitational systems: Late time behavior , Phys. Rev. D 52 (1995) 2118 [ gr-qc/9507035]

    1995 arXiv

  10. [17]

    Chandrasekhar and S.L

    S. Chandrasekhar and S.L. Detweiler, The quasi-normal modes of the Schwarzschild black hole , Proc. Roy. Soc. Lond. A 344 (1975) 441

    1975

  11. [18]

    Martel and E

    K. Martel and E. Poisson, Gravitational perturbations of the Schwarzschild spacetime: A Practical covariant and gauge-invariant formalism , Phys. Rev. D 71 (2005) 104003 [gr-qc/0502028]

    2005 arXiv

  12. [19]

    Barack, Late time dynamics of scalar perturbations outside black holes

    L. Barack, Late time dynamics of scalar perturbations outside black holes. 2. Schwarzschild geometry, Phys. Rev. D 59 (1999) 044017 [ gr-qc/9811028]

    1999 arXiv

  13. [20]

    Berti, V

    E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes , Class. Quant. Grav. 26 (2009) 163001 [ 0905.2975]

    2009 arXiv

  14. [21]

    Ching, P.T

    E.S.C. Ching, P.T. Leung, W.M. Suen and K. Young, Late time tail of wave propagation on curved space-time, Phys. Rev. Lett. 74 (1995) 2414 [ gr-qc/9410044]

    1995 arXiv

  15. [22]

    Krivan, P

    W. Krivan, P. Laguna and P. Papadopoulos, Dynamics of scalar fields in the background of rotating black holes , Phys. Rev. D 54 (1996) 4728 [ gr-qc/9606003]. – 18 –

    1996 arXiv

  16. [23]

    Krivan, P

    W. Krivan, P. Laguna, P. Papadopoulos and N. Andersson, Dynamics of perturbations of rotating black holes , Phys. Rev. D 56 (1997) 3395 [ gr-qc/9702048]

    1997 arXiv

  17. [24]

    Krivan, Late time dynamics of scalar fields on rotating black hole backgrounds , Phys

    W. Krivan, Late time dynamics of scalar fields on rotating black hole backgrounds , Phys. Rev. D 60 (1999) 101501 [ gr-qc/9907038]

    1999 arXiv

  18. [25]

    Burko and G

    L.M. Burko and G. Khanna, Radiative falloff in the background of rotating black hole , Phys. Rev. D 67 (2003) 081502 [ gr-qc/0209107]

    2003 arXiv

  19. [26]

    Burko and G

    L.M. Burko and G. Khanna, Late-time Kerr tails revisited , Class. Quant. Grav. 26 (2009) 015014 [0711.0960]

    2009 arXiv

  20. [27]

    Hod, How pure is the tail of gravitational collapse? , Class

    S. Hod, How pure is the tail of gravitational collapse? , Class. Quant. Grav. 26 (2009) 028001 [0902.0237]

    2009 arXiv

  21. [28]

    Burko and G

    L.M. Burko and G. Khanna, Late-time Kerr tails: Generic and non-generic initial data sets, ’up’ modes, and superposition , Class. Quant. Grav. 28 (2011) 025012 [ 1001.0541]

    2011 arXiv

  22. [29]

    Racz and G.Z

    I. Racz and G.Z. Toth, Numerical investigation of the late-time Kerr tails , Class. Quant. Grav. 28 (2011) 195003 [ 1104.4199]

    2011 arXiv

  23. [30]

    Zengino˘ glu, G

    A. Zengino˘ glu, G. Khanna and L.M. Burko, Intermediate behavior of Kerr tails , Gen. Rel. Grav. 46 (2014) 1672 [ 1208.5839]

    2014 arXiv

  24. [31]

    Burko and G

    L.M. Burko and G. Khanna, Mode coupling mechanism for late-time Kerr tails , Phys. Rev. D 89 (2014) 044037 [ 1312.5247]

    2014 arXiv

  25. [32]

    Baibhav, M.H.-Y

    V. Baibhav, M.H.-Y. Cheung, E. Berti, V. Cardoso, G. Carullo, R. Cotesta et al., Agnostic black hole spectroscopy: quasinormal mode content of numerical relativity waveforms and limits of validity of linear perturbation theory , 2302.03050

    arXiv

  26. [33]

    Rosato and P

    R.F. Rosato and P. Pani, On the universality of late-time ringdown tail , 2505.08877

    arXiv

  27. [34]

    Gleiser, C.O

    R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin, Second order perturbations of a Schwarzschild black hole , Class. Quant. Grav. 13 (1996) L117 [ gr-qc/9510049]

    1996 arXiv

  28. [35]

    Gleiser, C.O

    R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin, Gravitational radiation from Schwarzschild black holes: The Second order perturbation formalism , Phys. Rept. 325 (2000) 41 [gr-qc/9807077]

    2000 arXiv

  29. [36]

    Campanelli and C.O

    M. Campanelli and C.O. Lousto, Second order gauge invariant gravitational perturbations of a Kerr black hole , Phys. Rev. D 59 (1999) 124022 [ gr-qc/9811019]

    1999 arXiv

  30. [37]

    Garat and R.H

    A. Garat and R.H. Price, Gauge invariant formalism for second order perturbations of Schwarzschild space-times, Phys. Rev. D 61 (2000) 044006 [ gr-qc/9909005]

    2000 arXiv

  31. [38]

    Zlochower, R

    Y. Zlochower, R. Gomez, S. Husa, L. Lehner and J. Winicour, Mode coupling in the nonlinear response of black holes , Phys. Rev. D 68 (2003) 084014 [ gr-qc/0306098]

    2003 arXiv

  32. [39]

    Brizuela, J.M

    D. Brizuela, J.M. Martin-Garcia and G.A. Mena Marugan, Second and higher-order perturbations of a spherical spacetime , Phys. Rev. D 74 (2006) 044039 [ gr-qc/0607025]

    2006 arXiv

  33. [40]

    Brizuela, J.M

    D. Brizuela, J.M. Martin-Garcia and G.A.M. Marugan, High-order gauge-invariant perturbations of a spherical spacetime , Phys. Rev. D 76 (2007) 024004 [ gr-qc/0703069]

    2007 arXiv

  34. [41]

    Nakano and K

    H. Nakano and K. Ioka, Second Order Quasi-Normal Mode of the Schwarzschild Black Hole , Phys. Rev. D 76 (2007) 084007 [ 0708.0450]

    2007 arXiv

  35. [42]

    Brizuela, J.M

    D. Brizuela, J.M. Martin-Garcia and M. Tiglio, A Complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole , Phys. Rev. D 80 (2009) 024021 [0903.1134]

    2009 arXiv

  36. [43]

    Ripley, N

    J.L. Ripley, N. Loutrel, E. Giorgi and F. Pretorius, Numerical computation of second order vacuum perturbations of Kerr black holes , Phys. Rev. D 103 (2021) 104018 [ 2010.00162]. – 19 –

    2021 arXiv

  37. [44]

    Loutrel, J.L

    N. Loutrel, J.L. Ripley, E. Giorgi and F. Pretorius, Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric , Phys. Rev. D 103 (2021) 104017 [ 2008.11770]

    2021 arXiv

  38. [45]

    Pazos, D

    E. Pazos, D. Brizuela, J.M. Martin-Garcia and M. Tiglio, Mode coupling of Schwarzschild perturbations: Ringdown frequencies, Phys. Rev. D 82 (2010) 104028 [ 1009.4665]

    2010 arXiv

  39. [46]

    Sberna, P

    L. Sberna, P. Bosch, W.E. East, S.R. Green and L. Lehner, Nonlinear effects in the black hole ringdown: Absorption-induced mode excitation , Phys. Rev. D 105 (2022) 064046 [ 2112.11168]

    2022 arXiv

  40. [47]

    Redondo-Yuste, G

    J. Redondo-Yuste, G. Carullo, J.L. Ripley, E. Berti and V. Cardoso, Spin dependence of black hole ringdown nonlinearities , Phys. Rev. D 109 (2024) L101503 [ 2308.14796]

    2024 arXiv

  41. [48]

    Mitman et al., Nonlinearities in Black Hole Ringdowns , Phys

    K. Mitman et al., Nonlinearities in Black Hole Ringdowns , Phys. Rev. Lett. 130 (2023) 081402 [2208.07380]

    2023 arXiv

  42. [49]

    Cheung et al., Nonlinear Effects in Black Hole Ringdown , Phys

    M.H.-Y. Cheung et al., Nonlinear Effects in Black Hole Ringdown , Phys. Rev. Lett. 130 (2023) 081401 [2208.07374]

    2023 arXiv

  43. [50]

    Ioka and H

    K. Ioka and H. Nakano, Second and higher-order quasi-normal modes in binary black hole mergers, Phys. Rev. D 76 (2007) 061503 [ 0704.3467]

    2007 arXiv

  44. [51]

    Kehagias, D

    A. Kehagias, D. Perrone, A. Riotto and F. Riva, Explaining nonlinearities in black hole ringdowns from symmetries , Phys. Rev. D 108 (2023) L021501 [ 2301.09345]

    2023 arXiv

  45. [52]

    Khera, A

    N. Khera, A. Ribes Metidieri, B. Bonga, X. Jim´ enez Forteza, B. Krishnan, E. Poisson et al., Nonlinear Ringdown at the Black Hole Horizon , Phys. Rev. Lett. 131 (2023) 231401 [2306.11142]

    2023 arXiv

  46. [53]

    Bucciotti, A

    B. Bucciotti, A. Kuntz, F. Serra and E. Trincherini, Nonlinear quasi-normal modes: uniform approximation, JHEP 12 (2023) 048 [ 2309.08501]

    2023 arXiv

  47. [54]

    Spiers, A

    A. Spiers, A. Pound and J. Moxon, Second-order Teukolsky formalism in Kerr spacetime: Formulation and nonlinear source, Phys. Rev. D 108 (2023) 064002 [ 2305.19332]

    2023 arXiv

  48. [55]

    Ma and H

    S. Ma and H. Yang, Excitation of quadratic quasinormal modes for Kerr black holes , Phys. Rev. D 109 (2024) 104070 [ 2401.15516]

    2024 arXiv

  49. [56]

    Zhu et al., Nonlinear effects in black hole ringdown from scattering experiments: Spin and initial data dependence of quadratic mode coupling , Phys

    H. Zhu et al., Nonlinear effects in black hole ringdown from scattering experiments: Spin and initial data dependence of quadratic mode coupling , Phys. Rev. D 109 (2024) 104050 [2401.00805]

    2024 arXiv

  50. [57]

    Redondo-Yuste, D

    J. Redondo-Yuste, D. Pere˜ niguez and V. Cardoso,Ringdown of a dynamical spacetime , Phys. Rev. D 109 (2024) 044048 [ 2312.04633]

    2024 arXiv

  51. [58]

    Bourg, R

    P. Bourg, R. Panosso Macedo, A. Spiers, B. Leather, B. Bonga and A. Pound, Quadratic quasi-normal mode dependence on linear mode parity , 2405.10270

    arXiv

  52. [59]

    Lagos, T

    M. Lagos, T. Andrade, J. Rafecas-Ventosa and L. Hui, Black hole spectroscopy with nonlinear quasinormal modes, Phys. Rev. D 111 (2025) 024018 [ 2411.02264]

    2025 arXiv

  53. [60]

    Perrone, T

    D. Perrone, T. Barreira, A. Kehagias and A. Riotto, Non-linear black hole ringdowns: An analytical approach, Nucl. Phys. B 999 (2024) 116432 [ 2308.15886]

    2024 arXiv

  54. [61]

    Kehagias and A

    A. Kehagias and A. Riotto, Nonlinear effects in black hole ringdown made simple: Quasinormal modes as adiabatic modes , Phys. Rev. D 111 (2025) L041506 [ 2411.07980]

    2025 arXiv

  55. [62]

    Bucciotti, V

    B. Bucciotti, V. Cardoso, A. Kuntz, D. Pere˜ niguez and J. Redondo-Yuste, Ringdown nonlinearities in the eikonal regime , 2501.17950

    arXiv

  56. [63]

    Bourg, R.P

    P. Bourg, R.P. Macedo, A. Spiers, B. Leather, B. B´ eatrice and A. Pound, Quadratic quasinormal modes at null infinity on a schwarzschild spacetime , 2025

    2025

  57. [64]

    Ben Achour and H

    J. Ben Achour and H. Roussille, Quadratic perturbations of the Schwarzschild black hole: the algebraically special sector, JCAP 07 (2024) 085 [ 2406.08159]. – 20 –

    2024 arXiv

  58. [65]

    Kehagias, D

    A. Kehagias, D. Perrone and A. Riotto, Non-linear Quasi-Normal Modes of the Schwarzschild Black Hole from the Penrose Limit , 2503.09350

    arXiv

  59. [66]

    Perrone, A

    D. Perrone, A. Kehagias and A. Riotto, Nonlinearities in Kerr Black Hole Ringdown from the Penrose Limit, 2507.01919

    arXiv

  60. [67]

    Okuzumi, K

    S. Okuzumi, K. Ioka and M.-a. Sakagami, Possible Discovery of Nonlinear Tail and Quasinormal Modes in Black Hole Ringdown , Phys. Rev. D 77 (2008) 124018 [ 0803.0501]

    2008 arXiv

  61. [68]

    Lagos and L

    M. Lagos and L. Hui, Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole , Phys. Rev. D 107 (2023) 044040 [ 2208.07379]

    2023 arXiv

  62. [69]

    Cardoso, G

    V. Cardoso, G. Carullo, M. De Amicis, F. Duque, T. Katagiri, D. Pereniguez et al., Hushing black holes: Tails in dynamical spacetimes , Phys. Rev. D 109 (2024) L121502 [ 2405.12290]

    2024 arXiv

  63. [70]

    De Amicis et al., Late-time tails in nonlinear evolutions of merging black holes , 2412.06887

    M. De Amicis et al., Late-time tails in nonlinear evolutions of merging black holes , 2412.06887

  64. [71]

    S. Ma, M.A. Scheel, J. Moxon, K.C. Nelli, N. Deppe, L.E. Kidder et al., Merging black holes with Cauchy-characteristic matching: Computation of late-time tails , 2412.06906

    arXiv

  65. [72]

    Kehagias and A

    A. Kehagias and A. Riotto, Nonlinear Tails of Gravitational Waves in Schwarzschild Black Hole Ringdown, 2504.06224

    arXiv

  66. [73]

    S. Ling, S. Shah and S.S.C. Wong, Dynamical nonlinear tails in Schwarzschild black hole ringdown, 2503.19967

    arXiv

  67. [75]

    Chou, Z.-b

    K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu, Equilibrium and Nonequilibrium Formalisms Made Unified, Phys. Rept. 118 (1985) 1

    1985

  68. [76]

    Andersson, Evolving test fields in a black hole geometry , Phys

    N. Andersson, Evolving test fields in a black hole geometry , Phys. Rev. D 55 (1997) 468 [gr-qc/9607064]. – 21 –

    1997 arXiv

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