Perturbations of black holes with primary hair: time evolutions, quasinormal modes and greybody factors

Georgios Antoniou

REVIEW

Perturbations of black holes with primary hair: time evolutions, quasinormal modes and greybody factors

Georgios Antoniou

General Relativity and Quantum Cosmology · Submitted 27 Aug 2026 · 47 references

Pith's verdict: Solid numerical study of axial gravitational perturbations for a specific hairy black hole, with strong internal cross-checks, but the equivalence claim between b(r)=0 and Φ(r)=0 in §3.1 is not accurate as stated and should be corrected.

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CitationGeorgios Antoniou. "Perturbations of black holes with primary hair: time evolutions, quasinormal modes and greybody factors." arXiv:2608.26984 (2026). https://pith.science/paper/XJSENAEW

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Editor's note

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This paper does what the abstract promises: it computes axial QNM frequencies, time-domain waveforms, greybody factors, and absorption cross sections for the p=2 primary-hair DHOST black hole, using direct integration of the full boundary-value problem in both the physical and effective metric formulations. The main new contributions are the direct-integration QNM spectrum in the physical metric (confirming and extending earlier WKB estimates), the first gravitational-sector greybody factors for these solutions, and a clean comparison between time-domain evolutions in the two formulations. The internal cross-checks are genuinely convincing: physical vs effective metric results agree, DI vs time-domain fits agree, and WKB agrees within 0.5% at l=2. The paper is clearly written and the numerical methodology appears sound.

That said, there is a concrete weak point in §3.1 that needs attention. The author states that b(r)=0 is equivalent to Φ(r)=0, but from the explicit coefficient α5 in (A.13) one sees that b(r) also vanishes at the physical horizon A(r)=0 (because α5 ∝ A(r)^2). So the condition b(r)=0 has two distinct branches: the physical horizon r_h and the axial horizon r_g defined by Φ=0. The principal-symbol argument alone does not single out r_g—that choice is motivated by the effective-metric condition (2.26). The numerical agreement with the effective metric suggests the Φ=0 branch is dynamically correct, but the equivalence claim as written is false. This should be corrected to something like 'for r>r_g, b(r)=0 occurs only at r_g, which coincides with Φ=0' or an explicit statement that the equivalence is meant on the exterior domain. It's a minor error in presentation but could mislead a careful reader.

The other soft spots are mostly documentation gaps: the scalar-decoupling assumption is inherited rather than re-derived (low risk, as the stress-test note also notes), error bars and convergence tolerances are missing, and no code or tabulated data are provided. The near-horizon GR-limit nonuniformity is flagged by the author themselves, which is honest but could use more discussion.

Overall the paper is a solid piece of work that adds useful numerical results to the literature. I would recommend sending it to peer review with a request to fix the §3.1 claim and to add quantitative error estimates (or at least a statement of convergence criteria) and ideally release the numerical data.

For a reading group focused on black-hole perturbation theory or modified gravity, this is a worthwhile paper: it cleanly demonstrates how two formulations of the same dynamics agree numerically, and it provides benchmark QNM frequencies and greybody factors for a well-motivated beyond-GR solution.

Referee report

Major objections (1)

[§3.1, Eq. (A.13)] The paper claims 'the condition b(r)=0 is equivalent to Φ(r)=0'. However, from the explicit expression of α5(r) in Eq. (A.13), one sees that α5(r_h)=0 at the physical horizon A(r_h)=0 (provided D(r_h)≠0), so b(r)=0 also at r_h for all ξ. The equivalence is therefore not one-to-one: b(r)=0 has two branches—the physical horizon and the axial horizon defined by Φ=0. The principal-symbol argument does not uniquely single out r_g; the selection of r_g as the relevant inner boundary relies on the effective-metric condition (2.26). This should be corrected or clarified in the text. The numerical results themselves (which place the inner boundary at r_g) are not invalidated, but the conceptual framing in §3.1 needs revision.

Minor objections (5)

[Fig. 2 caption] The caption states 'We also display the fitted real and imaginary parts of the dominant mode in each case', but the values shown appear to be the fit parameters already printed in the figure panels; this could be clarified.
[§3.2] The constraint (2.22) is monitored and said to remain small, but no quantitative bound is given (e.g., RMS or L∞ relative to waveform amplitude). Adding a brief quantitative statement would strengthen the reliability of the time-domain extraction.
[Eq. (2.23)] The notation F(r) denotes both the coupling function F evaluated on the background and the function defined in (2.25); this is clear in context but might benefit from a brief remark to avoid confusion with a new metric coefficient.
[Acknowledgements] Typo: 'sk lodowska' should be 'Skłodowska'.
[§4, page 11] The sentence 'the GR limit ξ→0 cannot be approached smoothly in the physical metric formulation' is discussed; it would be helpful to note explicitly that this is because the expansion point r_g does not approach r_h in a uniform way as ξ→0, which is why the indicial structure must be taken before the limit.

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The paper

The paper claims

For primary-hair black holes, axial gravitational perturbations in physical and effective metric formulations yield consistent QNM spectra, and greybody factors prove more stable under local potential deformations.

The reading

This paper studies axial gravitational perturbations of spherically symmetric black holes with primary scalar hair in shift-symmetric beyond-Horndeski (DHOST) gravity. It demonstrates that the perturbation equations are hyperbolic for hair parameter ξ ≤ 2.5 and that the physical-metric and effective-metric descriptions give the same axial dynamics. By directly integrating the full boundary-value problem in the frequency domain, it obtains quasinormal mode frequencies that agree with sixth-order WKB approximations to within about 0.5% for ℓ=2 and better for higher ℓ. It also computes greybody factors and absorption cross sections, and shows that integrated greybody factors are far less sensitive to localized potential deformations than QNM frequencies, suggesting they may serve as more robust scattering observables.

What carries the argument

The central objects are (i) the two-variable axial perturbation system (2.20)–(2.21) in the physical metric, whose principal symbol determines characteristic speeds and identifies the axial horizon via b(r)=0; (ii) the effective-metric master equation (2.28) with potential V̅_ℓ, which reduces the axial problem to a single Schrödinger-like equation; and (iii) direct integration (DI) of the frequency-domain boundary-value problem, which solves the full radial ODE system without relying on barrier approximations. The greybody factor Γ_ℓ(Ω) = |A_ℓ(Ω)|² is obtained from the same master equation but with scattering boundary conditions.

Extended reading notes

Core claim

The paper finds that the principal symbol of the axial perturbation system yields a characteristic speed condition b(r)=0 which is exactly equivalent to Φ(r)=0, the axial gravitational horizon defined in the effective metric. Time-domain evolutions of the two-variable physical system and of the single master equation in the effective metric produce ringdown waveforms with consistent frequencies. In the frequency domain, direct integration of the physical-metric system (using Frobenius expansions at the horizon and asymptotic series at infinity) and of the effective-metric Schrödinger-like equation gives quasinormal mode frequencies that agree to better than 0.1% for all considered ℓ and ξ. The greybody factors shift to higher frequencies as ξ increases, and their half-maximum frequency tracks the real part of the fundamental QNM. Under a localized Pöschl–Teller deformation of the effective potential, the relative change in the integrated greybody factor remains of order ε, whereas the QNM frequency can change by orders of magnitude, demonstrating that greybody factors are significantly more stable.

Load-bearing premise

The axial perturbations decouple from the scalar field perturbations, meaning the scalar does not affect the gravitational axial dynamics at linear order; this is stated without re-derivation and relies on references [16, 22–25].

Editorial extensions

If this is right

For primary-hair DHOST black holes, axial QNM frequencies can be computed accurately by direct integration, providing templates for ringdown tests with future gravitational-wave detectors such as LISA and the Einstein Telescope.
Sixth-order WKB approximations are reliable for ℓ ≥ 3 but can deviate by up to 0.5% for ℓ = 2; direct integration is recommended for the lowest multipole.
Greybody factors offer a more stable observable than QNM frequencies under small deformations of the effective potential, motivating their use in tests of general relativity.
The physical-metric and effective-metric descriptions are fully equivalent for axial perturbations, so numerical calculations can be performed in either framework without loss of accuracy.

Reading between the lines

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

The restriction ξ ≤ 2.5 for hyperbolicity suggests that stronger primary hair may lead to a loss of hyperbolicity or causality violations in the axial sector, which could bound observationally allowed hair strengths.
The stability of greybody factors under local deformations, previously demonstrated for GR black holes, is here confirmed for a modified-gravity scenario; this may be a generic feature of black hole scattering in metric theories.
The paper does not treat polar perturbations; if polar and axial sectors couple (e.g. through scalar radiation), the ringdown signal could be richer and might require a full coupled analysis.
The agreement between physical and effective metric descriptions implies that the effective metric formalism can be used confidently for gravitational wave data analysis without re-deriving the full perturbation equations.

Original abstract

In this work we study gravitational perturbations of black holes carrying primary scalar hair in shift-symmetric beyond-Horndeski gravity. By focusing on the axial sector, we study the geometric structure of the perturbation equations and perform time-domain evolutions both in the physical and in the effective metric descriptions, in order to quantify differences in the response characteristics of solutions with different hair. In the frequency domain, we then explore the QNM spectrum in the physical-metric formulation, thus verifying and extending existing approximate calculations derived in the effective-metric approach. We also compute the corresponding greybody factors and absorption cross-sections. Finally, we use localized deformations of the effective potential as a diagnostic of the relative stability of quasinormal frequencies and greybody-factor observables.

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

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,302 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

In this work we study gravitational perturbations of black holes carrying primary scalar hair in shift-symmetric beyond-Horndeski gravity. By focusing on the axial sector, we study the geometric structure of the perturbation equations and perform time-domain evolutions both in the physical and in the effective metric descriptions, in order to quantify differences in the response characteristics of solutions with different hair. In the frequency domain, we then explore the QNM spectrum in the physical-metric formulation, thus verifying and extending existing approximate calculations derived in the effective-metric approach. We also compute the corresponding greybody factors and absorption cross-sections. Finally, we use localized deformations of the effective potential as a diagnostic of the relative stability of quasinormal frequencies and greybody-factor observables.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	In this work we study gravitational perturbations of black holes carrying primary scalar hair in shift-symmetric beyond-Horndeski gravity.	contribution 0.95	0.85	0.05	asserted 0.86
s1	By focusing on the axial sector, we study the geometric structure of the perturbation equations and perform time-domain evolutions both in the physical and in the effective metric descriptions, in order to quantify differences in the response characteristics of solutions with different hair.	method 0.80	0.96	0.06	asserted 0.97
s2	In the frequency domain, we then explore the QNM spectrum in the physical-metric formulation, thus verifying and extending existing approximate calculations derived in the effective-metric approach.	contribution 0.64	0.92	0.12	asserted 1.00
s3	We also compute the corresponding greybody factors and absorption cross-sections.	contribution 0.73	0.96	0.05	asserted 1.00
s4	Finally, we use localized deformations of the effective potential as a diagnostic of the relative stability of quasinormal frequencies and greybody-factor observables.	method 0.76	0.95	0.14	asserted 0.99

What kind of paper the classifier reports this is: simulation study 0.82 · evidence from simulation or computation 1.00.

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P abstract

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

References

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Nonreciprocal black holes with primary hair in beyond Horndeski gravity: Exact solutions and stability
gr-qc 2026-10 accept novelty 7.0 of 10

A novel method for constructing non-reciprocal black holes in beyond Horndeski gravity yields exact solutions with primary scalar hair, including regular black holes and solitons.
Axial black hole perturbations in purely-metric Horndeski theories
gr-qc 2026-11 conditional novelty 6.0 of 10

Axial perturbations of curvaturised black holes in purely-metric Horndeski gravity are linearly stable, propagate on a distinct but causally similar effective metric, and their ell=2 ringdown frequencies deviate from ...

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 46 sentences without a confident label

Uses the method or tool of 3 papers

Following [ref] , we compare the sensitivity of the QNM spectrum and the GBF by defining the following quantities

Ringdown stability: greybody factors as stable gravitational-wave observables · Localized deformations and greybody-factor stability · 0.83
Specifically, for the homogeneous $p=2$ branch considered here, and using the notation of Ref. [ref] , the effective line element can be written as

Axial perturbations of black holes with primary scalar hair · Axial perturbations · 0.83
To find the axial absorption cross section we have to sum with respect to the multipole [ref] , i.e.

Scattering and absorption of gravitational plane waves by rotating black holes · Greybody factors and absorption cross section · 0.83

Compares with 1 paper

This comparison shows explicitly how WKB estimates derived in previous works [ref] compare against the direct boundary-value problem.

Axial perturbations of black holes with primary scalar hair · Conclusions · 0.97

Cites as support 5 papers

Finally, GBFs have recently attracted renewed interest for ringdown physics because they define observables which have been shown to be more stable to small deviations in comparison with QNMs [ref] .

Ringdown stability: greybody factors as stable gravitational-wave observables · Introduction · 0.99
Finally, GBFs have recently attracted renewed interest for ringdown physics because they define observables which have been shown to be more stable to small deviations in comparison with QNMs [ref] .

Modeling the frequency-domain ringdown amplitude of comparable-mass mergers with greybody factors · Introduction · 0.99
Finally, GBFs have recently attracted renewed interest for ringdown physics because they define observables which have been shown to be more stable to small deviations in comparison with QNMs [ref] .

Novel Ringdown Tests of General Relativity with Black Hole Graybody Factors · Introduction · 0.99

Cites as background 9 papers

In this subsection we will revisit the background solutions characterized by primary scalar hair which were first discussed in [ref] .

Black holes with primary scalar hair · Background black holes with primary scalar hair · 0.99
Horndeski gravity and its extensions to beyond-Horndeski and degenerate higher-order scalar-tensor (DHOST) theories constitute such frameworks, involving additional scalars while avoiding Ostrogradsky instabilities [ref] .

Exploring gravitational theories beyond Horndeski · Introduction · 0.92
Horndeski gravity and its extensions to beyond-Horndeski and degenerate higher-order scalar-tensor (DHOST) theories constitute such frameworks, involving additional scalars while avoiding Ostrogradsky instabilities [ref] .

Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability · Introduction · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

K.D. Kokkotas and B.G. Schmidt,Quasi-normal modes of stars and black holes,Living Rev. Rel.2(1999) 2 [gr-qc/9909058]

View on Pith Pith review arXiv 1999

[2] [2]

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

View on Pith Pith review arXiv 2009

[3] [3]

E. Berti, V. Cardoso, G. Carullo et al.,Black hole spectroscopy: from theory to experiment, 2505.23895

View on Pith Pith review arXiv

[4] [4]

N. Franchini and S.H. V¨ olkel,Testing General Relativity with Black Hole Quasi-Normal Modes, 2305.01696. – 19 –

View on Pith Pith review arXiv

[5] [5]

E. Barausse, E. Berti, T. Hertog, S.A. Hughes, P. Jetzer, P. Pani et al.,Prospects for Fundamental Physics with LISA,Gen. Rel. Grav.52(2020) 81 [2001.09793]

View on Pith Pith review arXiv 2020

[6] [6]

A. Abac et al.,The Science of the Einstein Telescope,JCAP03(2026) 081 [2503.12263]

View on Pith Pith review arXiv 2026

[7] [7]

G.W. Horndeski,Second-order scalar-tensor field equations in a four-dimensional space,Int. J. Theor. Phys.10(1974) 363

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