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arxiv: 2501.12213 · v2 · submitted 2025-01-21 · 🌀 gr-qc

Black holes surrounded by dark matter spike: Spacetime metrics and gravitational wave ringdown waveforms

Dong Liu , Yi Yang , Zheng-Wen Long This is my paper

Pith reviewed 2026-05-23 04:48 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black holesdark matter spikesquasinormal modesgravitational wavesringdown waveformsM87Tolman-Oppenheimer-Volkoff
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The pith

Dark matter spikes around black holes shift their ringdown frequencies by up to order 10^{-4}.

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

The paper constructs analytic black hole metrics inside dark matter spikes by solving the Tolman-Oppenheimer-Volkoff equations with the M87 mass model and matching to an exterior vacuum solution. It computes the quasinormal frequencies for axial gravitational perturbations using both time-domain integration and the continued fraction method, then compares the results to the Schwarzschild case while varying the dark matter parameters. The central result is that these parameters alter the frequencies at a level reaching 10^{-4}, with an assessment of whether space-based detectors could register the difference. A sympathetic reader would care because the calculation supplies a concrete, parameter-dependent signature that gravitational wave observations might use to test dark matter distributions near galactic centers.

Core claim

Combining the mass model of M87 with the Tolman-Oppenheimer-Volkoff equations yields analytic interior solutions that match to an exterior vacuum metric. For axial perturbations the resulting quasinormal frequencies differ from the Schwarzschild values by amounts that reach order 10^{-4} and depend on the dark matter parameters; the work evaluates the prospects for detecting these shifts with space-based instruments.

What carries the argument

Analytic black hole metrics obtained from the Tolman-Oppenheimer-Volkoff equations with the M87 dark matter model, together with the quasinormal mode spectra extracted by time-domain integration and continued-fraction methods.

If this is right

  • The dark matter spike produces frequency shifts large enough to be considered for detection by space-based gravitational wave instruments.
  • Different choices of dark matter parameters generate distinct changes in the quasinormal frequencies.
  • The computed shifts supply a concrete signature for exploring dark matter effects on black hole ringdown signals.
  • The analytic construction and perturbation analysis apply directly to the M87 mass model and the axial sector.

Where Pith is reading between the lines

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

  • If the predicted shifts are observed they could be used to place bounds on dark matter density profiles around other supermassive black holes with known mass models.
  • The same matching procedure might be applied to alternative dark matter density profiles or to scalar-field perturbations beyond the axial case.
  • The size of the effect suggests that ringdown data from future detectors could serve as an independent probe complementary to stellar dynamics or weak lensing.

Load-bearing premise

The dark matter spike is accurately described by the chosen M87 mass model together with the Tolman-Oppenheimer-Volkoff equations, allowing an analytic interior solution that matches to an exterior vacuum metric.

What would settle it

A high-precision ringdown measurement from a supermassive black hole merger that shows either no frequency shift at the 10^{-4} level or a shift whose magnitude and parameter dependence disagree with the calculated curves would falsify the central claim.

Figures

Figures reproduced from arXiv: 2501.12213 by Dong Liu, Yi Yang, Zheng-Wen Long.

Figure 1
Figure 1. Figure 1: FIG. 1. The black hole metrics as a function of the radial [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The black hole metrics for the different power-law [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The time evolutions of quasinormal modes in the axial gravitational perturbation with the different angular quantum [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

The supermassive black holes at the centers of galaxies may be surrounded by dark matter spike, which could leave detectable imprints on the gravitational wave signals they emit. In this work, combining the mass model of M87, we present black hole analytical solutions in dark matter spike from the Tolman-Oppenheimer-Volkof equations. Meanwhile, the gravitational waves emitted at the ringdown phase of the black holes under the axial gravitational perturbation is investigated, and compare them with the Schwarzschild black hole. The main research methods are time domain integration method and continued fraction method. Besides, the impact of the different dark matter parameters on the quasinormal frequencies is investigated and its detectability on these impacts based on the space-based detectors. Our results indicate that the impacts of dark matter spike on the quasinormal frequencies of black holes can reach up to the order of $10^{-4}$. These results may provide some help in further exploring the impact of dark matter spike on the black holes and their related gravitational wave phenomena.

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

1 major / 2 minor

Summary. The paper constructs analytic black hole metrics surrounded by dark matter spikes by combining the M87 mass model with interior solutions of the Tolman-Oppenheimer-Volkoff equations, matches them to an exterior Schwarzschild geometry, and computes axial quasinormal modes via time-domain integration and the continued fraction method. It reports frequency shifts relative to Schwarzschild of up to order 10^{-4} and assesses detectability with space-based detectors.

Significance. If the fluid modeling of the spike is appropriate, the work supplies a controlled analytic background on which to quantify dark-matter-induced corrections to ringdown signals at a level that future detectors might probe, though the small size of the reported effect limits immediate observational impact. The analytic metric construction and dual numerical methods for the perturbations are positive features.

major comments (1)
  1. [Metric construction] Metric construction section (abstract and implied methods): the central claim of QNM shifts reaching 10^{-4} rests on modeling the dark matter spike via the TOV hydrostatic equilibrium equations applied to the M87 mass model. This assumes an isotropic fluid with pressure support, which differs from standard collisionless spike profiles (power-law densities with γ≈7/3 obtained from adiabatic growth). No comparison to Gondolo-Silk or equivalent profiles is mentioned; because the background metric and all subsequent perturbation results inherit this choice, the magnitude of the shift may be an artifact of the fluid assumption rather than a generic prediction for realistic spikes.
minor comments (2)
  1. [Abstract] Abstract: 'Tolman-Oppenheimer-Volkof' is a misspelling of 'Volkoff'.
  2. [Abstract] Abstract: the phrase 'impacts ... can reach up to the order of 10^{-4}' would be clearer if it specified the dark-matter parameters or range of models that produce the maximum shift.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comments. Below we respond point-by-point to the major comment on metric construction.

read point-by-point responses

  1. Referee: [Metric construction] Metric construction section (abstract and implied methods): the central claim of QNM shifts reaching 10^{-4} rests on modeling the dark matter spike via the TOV hydrostatic equilibrium equations applied to the M87 mass model. This assumes an isotropic fluid with pressure support, which differs from standard collisionless spike profiles (power-law densities with γ≈7/3 obtained from adiabatic growth). No comparison to Gondolo-Silk or equivalent profiles is mentioned; because the background metric and all subsequent perturbation results inherit this choice, the magnitude of the shift may be an artifact of the fluid assumption rather than a generic prediction for realistic spikes.

    Authors: We chose the TOV equations deliberately because they yield exact analytic interior solutions for the given M87 mass model that can be matched to an exterior Schwarzschild geometry, which is the central technical contribution of the paper. Collisionless adiabatic-growth profiles (Gondolo-Silk or similar) are the standard in many contexts but generally produce non-analytic metrics that must be handled numerically; our fluid model with isotropic pressure is an alternative approximation that permits closed-form expressions while still incorporating a realistic galactic mass distribution. The reported QNM shifts are therefore specific to this solvable setup rather than claimed as universal. We agree that a direct comparison would strengthen the manuscript and will add a short discussion section in the revision that contrasts our density profile with the Gondolo-Silk power-law form, states the modeling assumptions explicitly, and notes the limitations of the fluid approximation. We do not view the 10^{-4} scale as an artifact within the model we solve, but the added context will clarify its scope. revision: partial

Circularity Check

0 steps flagged

No circularity; QNM shift is computed output from external inputs

full rationale

The derivation constructs an interior metric by feeding the external M87 mass model into the TOV equations, matches to exterior Schwarzschild, then solves the axial perturbation equations with time-domain integration and continued-fraction methods. The reported 10^{-4} frequency shift is therefore a numerical output of that pipeline rather than a redefinition, fit, or self-citation reduction of the inputs. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem from the same authors appears in the supplied text. The calculation is self-contained against the stated external benchmarks (TOV hydrostatics, standard perturbation theory).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the TOV equations to a dark matter spike and on the validity of the M87 mass model as an input; no free parameters are explicitly fitted in the abstract, but the dark matter parameters are external inputs whose uncertainty is not quantified here.

axioms (2)
  • domain assumption The dark matter spike can be treated as a static, spherically symmetric fluid obeying the Tolman-Oppenheimer-Volkoff equation of hydrostatic equilibrium.
    Invoked when the authors state they solve the TOV equations with the M87 mass model to obtain analytic black hole solutions.
  • domain assumption Axial gravitational perturbations are sufficient to capture the leading ringdown signal and its sensitivity to the dark matter environment.
    Stated by the choice to restrict the analysis to axial perturbations and compare quasinormal frequencies.

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