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Gravitational odd-parity perturbation of a Horndeski hairy black hole: quasinormal mode and parameter constraint
General Relativity and Quantum Cosmology · Submitted 14 Aug 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is found that this proposal not only cures the superluminal propagation addressed in the previous literature, but also hold the original QNM spectrum of the odd perturbation.
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The abstract, sentence by sentence
During the binary black hole coalescence, gravitational waves emitted at the ringdown stage can be well described by black hole perturbation theory, where the quasinormal modes (QNMs) become the important ingredient in modeling the pattern waveform. In general relativity (GR), the QNMs can be obtained from solving the Regge-Wheeler (RW) equation of a non-rotating black hole. While in Horndeski gravity, the isospectrality between the odd and even parity perturbations is broken due to the scalar field, but the odd perturbation equation can be simplified into a modified RW equation from the perturbed action. In this paper, we propose a new auxiliary field and tortoise coordinate to refine the modified RW equation in Horndeski gravity, and calculate the QNMs frequencies of the odd perturbation of a specific hairy black hole. It is found that this proposal not only cures the superluminal propagation addressed in the previous literature, but also hold the original QNM spectrum of the odd perturbation. Moreover, our results indicate that such a Horndeski hairy black hole is stable under the odd perturbation, which is also verified by the time evolution of the perturbation. In particular, in contrary to GR, the modes with $\ell=2$ can decay faster than modes with $\ell>2$ for a certain range of the Horndeski hair, and the link between the null geodesics and QNM for the odd perturbation in the current theory is violated. Then, we use the ringdown QNMs to preliminarily investigate the signal-to-noise-ratio (SNR) rescaled measurement error of the Horndeski hair. We obtain significant effects of the angular momentum and overtone on the error bound of the hair parameter. We hope that our findings could inspire more theoretical and phenomenological work on the test of the no-hair theorem of black hole using gravitational wave physics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | During the binary black hole coalescence, gravitational waves emitted at the ringdown stage can be well described by black hole perturbation theory, where the quasinormal modes (QNMs) become the important ingredient in modeling the pattern waveform. | background 1.00 | 0.05 | 0.02 | asserted 0.94 |
| s1 | In general relativity (GR), the QNMs can be obtained from solving the Regge-Wheeler (RW) equation of a non-rotating black hole. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s2 | While in Horndeski gravity, the isospectrality between the odd and even parity perturbations is broken due to the scalar field, but the odd perturbation equation can be simplified into a modified RW equation from the perturbed action. | background 1.00 | 0.10 | 0.04 | asserted 1.00 |
| s3 | In this paper, we propose a new auxiliary field and tortoise coordinate to refine the modified RW equation in Horndeski gravity, and calculate the QNMs frequencies of the odd perturbation of a specific hairy black hole. | contribution 0.90 | 0.96 | 0.10 | asserted 1.00 |
| s4 | It is found that this proposal not only cures the superluminal propagation addressed in the previous literature, but also hold the original QNM spectrum of the odd perturbation. | result 0.97 | 0.94 | 0.32 | asserted 1.00 |
| s5 | Moreover, our results indicate that such a Horndeski hairy black hole is stable under the odd perturbation, which is also verified by the time evolution of the perturbation. | result 1.00 | 0.94 | 0.24 | asserted 0.81 |
| s6 | In particular, in contrary to GR, the modes with $\ell=2$ can decay faster than modes with $\ell>2$ for a certain range of the Horndeski hair, and the link between the null geodesics and QNM for the odd perturbation in the current theory is violated. | result 0.97 | 0.87 | 0.17 | asserted 0.97 |
| s7 | Then, we use the ringdown QNMs to preliminarily investigate the signal-to-noise-ratio (SNR) rescaled measurement error of the Horndeski hair. | contribution 0.67 | 0.93 | 0.06 | asserted 0.55 |
| s8 | We obtain significant effects of the angular momentum and overtone on the error bound of the hair parameter. | result 1.00 | 0.96 | 0.09 | asserted 1.00 |
| s9 | We hope that our findings could inspire more theoretical and phenomenological work on the test of the no-hair theorem of black hole using gravitational wave physics. | future work 1.00 | 0.11 | 0.06 | hedged possibility 0.93 |
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Original abstract
During the binary black hole coalescence, gravitational waves emitted at the ringdown stage can be well described by black hole perturbation theory, where the quasinormal modes (QNMs) become the important ingredient in modeling the pattern waveform. In general relativity (GR), the QNMs can be obtained from solving the Regge-Wheeler (RW) equation of a non-rotating black hole. While in Horndeski gravity, the isospectrality between the odd and even parity perturbations is broken due to the scalar field, but the odd perturbation equation can be simplified into a modified RW equation from the perturbed action. In this paper, we propose a new auxiliary field and tortoise coordinate to refine the modified RW equation in Horndeski gravity, and calculate the QNMs frequencies of the odd perturbation of a specific hairy black hole. It is found that this proposal not only cures the superluminal propagation addressed in the previous literature, but also hold the original QNM spectrum of the odd perturbation. Moreover, our results indicate that such a Horndeski hairy black hole is stable under the odd perturbation, which is also verified by the time evolution of the perturbation. In particular, in contrary to GR, the modes with $\ell=2$ can decay faster than modes with $\ell>2$ for a certain range of the Horndeski hair, and the link between the null geodesics and QNM for the odd perturbation in the current theory is violated. Then, we use the ringdown QNMs to preliminarily investigate the signal-to-noise-ratio (SNR) rescaled measurement error of the Horndeski hair. We obtain significant effects of the angular momentum and overtone on the error bound of the hair parameter. We hope that our findings could inspire more theoretical and phenomenological work on the test of the no-hair theorem of black hole using gravitational wave physics.
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References
How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it
Cites it as background 2 papers
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Recent developments have further emphasized the interplay between QNMs, shadows, greybody factors, instability criteria, odd-parity perturbations of hairy geometries, discontinuous effective potentials, and the classification of competing mode families, both in concrete phenomenological models and in more general analyses [ref] .
Echoes of a hairy black hole from gravitational decoupling · Introduction · 0.89
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On the odd-parity side, Horndeski ringdown has recently been revisited with time-domain validation and phenomenological error estimates [ref] , while coupled odd-parity two-field systems have been analyzed in vector-tensor effective field theory [ref] .
Odd-parity perturbations of trace-quadratic $f(R,T)$ black holes with anisotropic matter: admissible branches, axial ringdown, and a coupled-PINN benchmark · Introduction · 0.96
What it cites · 80 sentences without a confident label
Uses the method or tool of 5 papers
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It is noted that due to the $log$ term in the metric function, a redefinition of the radial wave function is crucial when using both methods, as we have addressed in appendix B of our previous work [ref] .
Perturbations of massless external fields in Horndeski hairy black hole · Preparation · 0.81
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* In the time domain, we shall work in tortoise coordinate grids and apply the finite difference method (Readers may refer to [ref] for more details) to discretize the radial coordinate $r=r(r̅_*)=r(jΔr̅_*)=r_j$, thus the perturbation and the potential are discretized as $Ψ(t,r)=Ψ(iΔt,jΔr̅_*)=ψ_i,j$ and $V(r)=V(j Δr̅_*)=V_j$.
Stability of Reissner-Nordstr\"om black hole in de Sitter background under charged scalar perturbation · Preparation · 0.93
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In this section, we will determine the measurement errors of the hair parameter $Q̃$ by the GW detectors, which can be estimated by computing the Fisher information matrix (FIM) [ref] .
On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA · Hair parameter estimation by detection of ringdown wave · 0.99
Compares with 3 papers
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It is noted that comparing against the formulas utilized in [ref] .
Black hole stability under odd-parity perturbations in Horndeski gravity · Odd-parity perturbation and modified Regge-Wheeler equation · 0.98
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It is noted that comparing against the formulas utilized in [ref] .
Stability of Schwarzschild-like solutions in f(R,G) gravity models · Odd-parity perturbation and modified Regge-Wheeler equation · 0.98
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Note that a similar violation was also found in Einstein-Lovelock theory, in which the effective potential of the gravitational perturbation in the eikonal limit was different from radial potential of the null particle [ref] .
Are eikonal quasinormal modes linked to the unstable circular null geodesics? · Numerical results · 0.91
Cites as background 27 papers
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In this subsection, we shall focus on a special subclass of Horndeski theory and review the static hairy black hole constructed in [ref] .
Hairy Black Holes from Horndeski Theory · Review on the action and Horndeski hairy black hole · 0.81
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An interesting new development in the examination of no-hair theorem in Horndeski theories is that the shift-symmetric Horndeski theory and beyond with the action [ref] [equation] allow static and asymptotically flat black holes with a nontrivial static scalar field.
Asymptotically flat black holes in Horndeski theory and beyond · Introduction · 0.81
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Moreover, it was demonstrated in [ref] , that the no-hair theorem cannot hold when a Galileon field is coupled to gravity, but the static spherical solution only admits a trivial Galileon profile.
A no-hair theorem for the galileon · Introduction · 0.86
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