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Quasinormal Modes of Schwarzschild Black Holes in the Dehnen-(1, 4, 5/2) Type Dark Matter Halos

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Dark matter halos systematically lower the quasinormal-mode frequencies of Schwarzschild black holes and lengthen their damping time, while preserving stability.

desk verdict The M87* parameter scan is fine but the effective potentials in Eqs. (12) and (23) don't follow from the metric, so the reported QNM frequencies are not those of the SBHD spacetime. read the letter →

arxiv 2505.15540 v1 pith:7KLKDGL2 submitted 2025-05-21 gr-qc

Qi-Qi Liang , Dong Liu , Zheng-Wen Long This is my paper · ORCID
classification gr-qc MSC 83C57 PACS 04.70.-s
keywords quasinormalmodesSchwarzschildblackholeDehnendarkmatterhaloeffectivepotentialWKBapproximationtime-domainmethodM87*shadowradiusstability
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 tries to establish that a Schwarzschild black hole embedded in a Dehnen-(1,4,5/2) dark matter halo—a spherically symmetric density profile used for dwarf galaxies—rings with characteristic quasinormal frequencies that differ systematically from an isolated Schwarzschild black hole. Using the M87* shadow-radius measurement to fix the halo parameters $\rho_s$ and $r_s$, the author derives effective potentials for scalar, electromagnetic, and axial gravitational perturbations and computes the quasinormal-mode frequencies by two independent methods. The central result is that larger halo density or scale radius lowers the effective-potential peak, decreases the real oscillation frequency, and reduces the magnitude of the negative imaginary frequency, so the perturbation rings slower and lasts longer while remaining stable. If correct, the computed frequencies give a concrete template for what ringdown signals from supermassive black holes in such dark matter halos should look like.

What carries the argument

The load-bearing objects are the effective potentials $V(r)$ in the tortoise-coordinate wave equation $d^2\psi/dr_*^2 + (\omega^2 - V)\psi = 0$, where $dr_*/dr = 1/f(r)$. For the SBHD metric, the paper writes Eq. (12) for scalar perturbations, Eq. (13) for electromagnetic perturbations, and Eq. (23) for axial gravitational perturbations; each is built from the metric function $f(r) = 1 - 2M/r - 32\pi\rho_s r_s^3 \sqrt{(r+r_s)/(r_s^2 r)}/r$ multiplied by a field-dependent bracket. These potentials feed the sixth-order WKB formula (25) and the light-cone finite-difference scheme (28), with complex frequencies extracted from the time-domain signal by the Prony method. The trend they encode—peak height decreasing as $\rho_s$ or $r_s$ grows—is what produces the paper's physical conclusions.

What would settle it

Take the metric (2), substitute the scalar and electromagnetic ansätze into Eqs. (8) and (9), and carry out the axial Regge-Wheeler reduction by hand or symbolically; if the resulting potentials differ from Eqs. (12), (13), and (23) in any dark-matter term, recompute the quasinormal-mode frequencies from the corrected potentials and compare with the paper's Table II. A second independent check is a direct full numerical evolution of the perturbation equations with no WKB approximation, to see whether the ringing frequency matches the quoted real part.

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Extended reading notes

Core claim

The paper's central claim is that adding a Dehnen-(1,4,5/2) dark matter halo to a Schwarzschild black hole changes the quasinormal-mode spectrum monotonically: for fixed angular number $l=2$, increasing either the halo central density $\rho_s$ or the scale radius $r_s$ lowers the maximum of the effective potential $V(r)$ for scalar, electromagnetic, and axial gravitational perturbations. Consequently the real part of the quasinormal-mode frequency decreases, the wave oscillation slows down, and the imaginary part stays negative but moves closer to zero, meaning the perturbation damps more slowly; the black hole is stable. The paper also finds that both parameter sets consistent with the 3$\sigma$ M87* shadow-radius window enlarge the event-horizon and photon-sphere radii compared with the Schwarzschild case. The WKB and time-domain/Prony methods agree, which the paper presents as evidence that the quoted complex frequencies are consistent across approximation schemes.

Load-bearing premise

The calculations assume that the effective potentials in Eqs. (12) and (23) are the correct reductions of the Klein-Gordon, Maxwell, and Regge-Wheeler equations for the SBHD metric in Eq. (2); if those reductions are wrong, the computed frequencies describe a different spacetime or different perturbation channel.

Editorial extensions

If this is right

  • Ringdown templates for supermassive black holes in Dehnen-type halos must use lower real frequencies and longer damping times than vacuum Schwarzschild templates.
  • The allowed $(\rho_s, r_s)$ region from the M87* shadow radius maps to a predicted band of quasinormal-mode frequencies, so a future ringdown measurement could test the halo parameters independently.
  • The negative imaginary part in all three perturbation channels means the Dehnen halo does not introduce an instability in scalar, electromagnetic, or axial gravitational perturbations for the studied parameters.
  • The agreement between the WKB and time-domain/Prony methods indicates that the reported frequencies are not an artifact of one numerical scheme.

Reading between the lines

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

  • An implication the paper leaves implicit is that the same pipeline can map other Dehnen indices and higher overtones, producing a catalog that would let observers distinguish halo profiles by their ringdown alone.
  • The negative correlation between $\rho_s$ and $r_s$ under the shadow constraint suggests that ringdown data, which the paper shows respond to both parameters in the same direction, may not cleanly break the degeneracy by itself; combining ringdown with lensing or photon-sphere measurements could.
  • A natural extension is to compute polar gravitational perturbations, which the paper does not treat, and check whether the effective-potential lowering and stability results persist in that channel.
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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

3 major / 5 minor

Summary. The manuscript studies quasinormal modes (QNMs) of a Schwarzschild-like black hole in a Dehnen-(1,4,5/2) dark matter halo. Using the M87* shadow-radius constraint, two sets of halo parameters (rho_s, r_s) are chosen. The authors derive effective potentials for scalar, electromagnetic, and axial gravitational perturbations, compute QNM frequencies with a sixth-order WKB method and a time-domain Prony method, and report that increasing rho_s or r_s lowers the peak of the effective potential and the real part of the QNM frequency, while the imaginary part decreases in magnitude, implying stability. Table II lists QNM frequencies for several multipole numbers.

Significance. If valid, the paper would provide a systematic catalogue of QNM frequencies for this dark-matter model and a possible connection to EHT shadow observations. The use of two independent numerical methods for cross-checking and the explicit parameter constraints from shadow data are strengths. However, the central results depend on effective potentials that are incorrectly derived: the scalar potential in Eq. (12) differs from the exact Klein-Gordon potential for metric (2) by a factor of order 30 at the adopted parameters, and the axial potential raises similar concerns. The quantitative conclusions are therefore not established by the calculations presented.

major comments (3)
  1. [II.B, Eq. (12)] The scalar effective potential is not the Klein-Gordon potential for metric (2). For a massless scalar in a static spherically symmetric metric, the potential is V(r) = f(r)[l(l+1)/r^2 + f'(r)/r]. Inserting f(r) from Eq. (2) and simplifying the dark-matter term H(r) = 32*pi*rho_s*r_s^3*r^(-1)*sqrt((r+r_s)/(r_s^2*r)) gives a correction f'/r = 16*pi*rho_s*r_s^2*(2r+3r_s)/(r^(7/2)*(r+r_s)^(1/2)). Equation (12) instead contains 16*pi*rho_s*r_s^3/(r^(7/2)*(r+r_s)^(1/2)). The ratio of the correct to printed correction at the paper's own parameter values (r ~ 3, r_s = 0.2) is (2r+3r_s)/r_s ~ 33, so this is not a typesetting artifact. Since Eq. (12) is the direct input to the WKB and Prony calculations, the frequencies reported in Figs. 5-7 and Table II are not quasinormal frequencies of the SBHD spacetime described by Eq. (2).
  2. [II.C, Eqs. (21)-(23)] The axial gravitational derivation is internally inconsistent. Equation (21), with the tortoise coordinate defined by dr_* = dr/f(r), does not reduce to a one-dimensional wave equation: after substituting psi = f(r) h1/r and Q = r*psi, the term (f/r) d/dr[f d(r*psi)/dr] produces a second derivative with respect to r_* whose coefficient is f^4/r (or f^5/r for Q), rather than unity, so Eq. (21) is not the standard Regge-Wheeler master equation for this metric. Consequently Eq. (23) does not follow from Eq. (21). The dark-matter term in Eq. (23), after simplification, is -48*pi*rho_s*r_s^3/(r^(7/2)*(r+r_s)^(1/2)), which is not the correct f'- or f''-dependent correction to the Regge-Wheeler potential for a general static spherical metric such as (2). The axial gravitational QNM results in Figs. 5-7 and Table II are therefore unsupported.
  3. [III.B, Figs. 5-7 and Table II] Because the effective potentials in Eqs. (12) and (23) are not derived correctly from metric (2), the central trend claim — that larger rho_s or r_s lowers the QNM frequency and increases damping time — is not established by these computations. The geodesic effective potential in Fig. 2 does decrease with rho_s, but the QNM potential involves f'(r) and f''(r), and with the correct f'(r) from Eq. (2) the correction term has a different magnitude and r-dependence. No calculation is given to show that the claimed monotonic decrease of the real frequency survives with the correct potentials; the conclusion in Section IV therefore rests on the erroneous potentials.
minor comments (5)
  1. [Eqs. (2), (4), (7), (12), (23)] The notation r2s, r3s, and expressions such as r2sr are hard to read; please use proper superscripts and parentheses, e.g., r_s^2 r, r_s^3, and clarify every occurrence.
  2. [Fig. 2] The vertical axis in Fig. 2 is labeled V(r) although the text refers to the effective potential U(r); the labeling should be consistent throughout the figure and caption.
  3. [Table I] The table lists b_ph values but does not define b_ph in the caption; state explicitly that b_ph is the shadow radius in units of M, consistent with the text in Section II.A.
  4. [Section III.B] The phrase 'the shapes overlap very well' should be 'the symbols overlap very well'; also, the legend in Fig. 7 should be clarified so that the reader can identify which color corresponds to scalar, electromagnetic, and axial gravitational fields.
  5. [Section IV] The conclusion that the black hole 'remains stable under perturbations' is inferred only from the fundamental mode for a few multipoles; this does not constitute a proof of stability and should be phrased with appropriate caution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the QNM frequencies are computed from independently cited metric and perturbative potentials, with halo parameters fitted to shadow data rather than to the target QNM trend.

full rationale

The derivation chain is self-contained in the relevant sense. The paper imports the SBHD metric from the external reference [33], fixes halo parameters using M87* shadow-radius data via the geodesic equation, derives effective potentials from the Klein-Gordon, Maxwell, and Regge-Wheeler equations, and then computes quasinormal-mode frequencies with sixth-order WKB and time-domain/Prony methods. Nothing in the QNM computation is fitted to the claimed trend that larger rho_s or r_s lowers frequencies and prolongs damping; that trend emerges only after solving the wave equations. The halo parameter fit to shadow data is an input to the QNM calculation, not a reuse of its output. The authors' own prior works [24, 25] are cited only as general context and do not carry the derivation. The possible algebraic mismatch between the metric (2) and the dark-matter terms in Eqs. (12) and (23), raised as a correctness concern, would mean the computed frequencies do not describe the SBHD spacetime, but that is a derivation error or correctness issue, not an instance of a prediction reducing to its inputs by construction. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the Dehnen halo metric from prior papers, the choice of halo parameter values within the M87* shadow bound, and standard black hole perturbation theory. No new physical entities are introduced, and no constants are fitted to the quasinormal mode results themselves, so the free-parameter ledger is limited to the two halo parameters.

free parameters (2)
  • Dark matter central density ρ_s = 0.01, 0.03, 0.05 (with r_s = 0.2); 0.01 (with r_s = 0.15, 0.3, 0.45)
    Chosen within the range allowed by the M87* shadow radius constraint. The quasinormal mode results and the claimed trends depend directly on these values.
  • Dark matter core radius r_s = 0.15, 0.3, 0.45 (with ρ_s = 0.01); 0.2 (with ρ_s = 0.01, 0.03, 0.05)
    Chosen similarly from the M87* shadow constraint. The paper's central claim about the effect of wider halos on quasinormal mode frequencies is a function of this parameter.
assumptions (4)
  • domain assumption The metric in Eq. (2) is a valid solution for a Schwarzschild black hole embedded in a Dehnen-(1,4,5/2) dark matter halo.
    This metric is taken from Refs. [33,34] without re-derivation. If the metric is not a correct representation of the halo, all subsequent perturbation analysis is moot. The paper does not justify the metric form beyond citing prior work.
  • domain assumption The 3σ shadow radius interval for M87*, 2.546M to 7.846M, from Ref. [37], is a valid constraint on the halo parameters ρ_s and r_s.
    The paper uses this broad interval to select parameter values. The interval is extremely wide and does not uniquely determine the halo parameters; the choices are illustrative rather than constrained in a meaningful way.
  • standard math Standard perturbation equations for massless scalar, electromagnetic, and axial gravitational fields in a static spherical background, including the Regge-Wheeler gauge, apply to this metric.
    The paper invokes the Klein-Gordon equation, Maxwell equations, and Regge-Wheeler gauge as background. These are standard, but the derivation from Eq. (21) to Eq. (23) is not shown, and the resulting potentials appear inconsistent with the metric.
  • domain assumption The sixth-order WKB approximation is sufficiently accurate for the modes considered, including l=0 and l=1.
    The paper uses sixth-order WKB without evaluating its error. For l=0, the WKB result differs from the Prony extraction by about 4 percent in the real part, indicating that the accuracy assumption is questionable for the lowest multipoles.

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Cite this review

Pith. "Pith review of Quasinormal Modes of Schwarzschild Black Holes in the Dehnen-(1, 4, 5/2) Type Dark Matter Halos." pith.science (2026). https://pith.science/paper/7KLKDGL2

@misc{pith2026250515540,
  author       = {Pith},
  title        = {Pith review of: Quasinormal Modes of Schwarzschild Black Holes in the Dehnen-(1, 4, 5/2) Type Dark Matter Halos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KLKDGL2}},
  note         = {Machine review of arXiv:2505.15540}
}
abstract

The Dehnen - type dark matter density distribution model is mainly used for dwarf galaxies. In recent years, researchers have speculated that black holes may exist in this dark matter model and have given the black hole metric solutions. On this basis, this paper conducts a systematic study on the quasinormal modes of a Schwarzschild black hole in a Dehnen - (1,4, 5/2) dark matter halo, revealing the influences of dark matter distribution and perturbation field types on the black hole's quasinormal modes.The research uses the shadow radius data of the M87$^{\ast}$ black hole. Through the geodesic equation, two sets of dark matter halo parameter values of $\rho_{\rm s}$ and $r_{\rm s}$ are determined, and the specific numerical values of the black hole's event horizon radius, photon sphere radius, and shadow radius under the corresponding conditions are obtained. The wave equations and effective potentials of the black hole under the perturbations of the scalar field, electromagnetic field, and axial gravitational were analyzed. It was found that the larger the values of $\rho_{\rm s}$ or $r_{\rm s}$, the smaller the peak value of the effective potential, and the wave function oscillation slows down with a lower frequency. The black hole remains stable under perturbations. These studies provide relevant data for the quasinormal modes of the Schwarzschild black hole in the Dehnen-(1,4, 5/2) type dark matter halo. They also offer crucial evidence for understanding the interaction mechanism between the black hole and the dark matter halo.

Figures

Figures reproduced from arXiv: 2505.15540 by the authors.

Figure 1
Figure 1. FIG. 1. The actual observational data of the 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The curves of the effective formula U(r) and r of black holes under different parameter values, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The variations of the effective potential under different perturbations are presented, with the left-to-right sequence [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The variations of the effective potential under different perturbations are presented, with the left-to-right sequence [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The time-domain profiles from left to right correspond to the scalar field, electromagnetic field, and axial gravitational [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The time-domain profiles from left to right correspond to the scalar field, electromagnetic field, and axial gravitational [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The variations of the real and imaginary parts of the quasinormal mode frequencies under the perturbations of different [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.