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Quasinormal modes and shadow of Schwarzschild black holes embedded in a Dehnen type dark matter halo exhibiting string cloud

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper argues that a Schwarzschild black hole embedded in a Dehnen-(1,4,0) dark-matter halo with a string cloud casts a larger shadow and rings at frequencies that shift with the halo density and string parameter.

desk verdict The shadow analysis is fine but the quasinormal-mode frequencies come from wrong effective potentials and the text contradicts its own tables. read the letter →

arxiv 2412.20037 v1 pith:EQF37L2V submitted 2024-12-28 gr-qc

Ahmad Al-Badawi , Sanjar Shaymatov This is my paper · ORCID
classification gr-qc
keywords blackholeshadowquasinormalmodesDehnendarkmatterhalostringcloudSchwarzschildWKBapproximationEventHorizonTelescopephotonsphere
topics Dark Matter
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

The paper tries to establish that a Schwarzschild black hole embedded in a Dehnen-(1,4,0) dark-matter halo and surrounded by a cloud of strings produces observable imprints: the photon sphere and the shadow radius both grow as the halo central density $\rho_s$ or the string-cloud parameter $a$ increases. It reports sixth-order WKB quasinormal frequencies for scalar, electromagnetic, and gravitational perturbations, tabulated over grids of $\rho_s$ and $a$, and uses Event Horizon Telescope shadow diameters for M87* and Sgr A* to set upper limits on these parameters. If correct, the work turns black-hole shadow imaging into a probe of compact dark-matter halos and string clouds around supermassive black holes, with bounds that are tighter for Sgr A* than for M87*.

What carries the argument

The load-bearing objects are the Dehnen-(1,4,0) halo profile $\rho_D=\rho_s/(1+r/r_s)^4$, the string-cloud metric function $f(r)=1-a-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3(2r+r_s)}{3(r+r_s)^2}$, and the photon-sphere condition $r f'(r)-2f(r)=0$ with shadow radius $R_s=r_{ps}/\sqrt{f(r_{ps})}$. For quasinormal modes, the machinery is the sixth-order WKB formula applied to the effective potentials (13)-(15), which combine the angular factor $(1/2+l)$ with derivative terms $f'$ constructed from the same metric. These potentials determine the peak of the barrier, and the WKB expansion turns the barrier shape into complex frequencies whose real part is the oscillation rate and whose imaginary part is the decay rate.

What would settle it

Recompute the $l=2$ fundamental quasinormal frequencies for the same metric using the standard $l(l+1)$ angular factor in the scalar, electromagnetic, and axial-gravitational potentials; if the values differ materially from Tables I and II, the reported dependence of the QNMs on $\rho_s$ and $a$ is an artifact of the prefactor. A shadow measurement with independently known mass and distance that disagrees with $R_s$ from Eq. (22) would test the shadow part.

Watch

Extended reading notes

Core claim

The central claim is that the metric $f(r)=1-a-\frac{2M}{r}-\frac{4\pi\rho_s r_s^3(2r+r_s)}{3(r+r_s)^2}$ describes a Schwarzschild black hole dressed by a Dehnen-(1,4,0) dark-matter halo and a string cloud, and that the two environment parameters control both the shadow and the ringdown. Solving $r_{ps} f'(r_{ps})-2f(r_{ps})=0$ and $R_s=r_{ps}/\sqrt{f(r_{ps})}$, the authors find in Table III that both $r_{ps}$ and $R_s$ increase monotonically with $\rho_s$ and $a$. For perturbations, they use the effective potentials (13)-(15) and the sixth-order WKB method to produce the fundamental ($l=2$, $n=0$) quasinormal frequencies in Tables I and II; in those tables, increasing $\rho_s$ or $a$ lowers both the real oscillation frequency and the magnitude of the damping rate. From EHT shadow-diameter estimates, the paper derives the parameter bounds $\rho_s<0.5$, $a<0.12$ for M87* and $\rho_s<0.12$, $a<0.032$ for Sgr A*.

Load-bearing premise

Every quasinormal frequency in Tables I and II assumes that the effective potentials (13)-(15), with the angular factor $(1/2+l)$, are the correct perturbation barriers for the scalar, electromagnetic, and gravitational fields; if that factor is wrong, all the reported frequencies shift.

Editorial extensions

If this is right

  • EHT shadow-diameter measurements become direct upper bounds on the environment parameters: $\rho_s<0.5$, $a<0.12$ for M87* and $\rho_s<0.12$, $a<0.032$ for Sgr A*.
  • A detected shadow larger than the Schwarzschild prediction of $3\sqrt{3}M$ could signal either a Dehnen-type dark-matter halo or a string cloud, since both enlarge the shadow in the same direction.
  • The tabulated quasinormal frequencies provide predicted ringdown signatures whose real frequency and damping shift with $\rho_s$ and $a$, so future ringdown measurements could constrain the same halo and string parameters.
  • The metric reduces to Schwarzschild when $\rho_s=0$ and $a=0$, so all predictions connect continuously to standard general relativity and can be tested as smooth deformations of it.

Reading between the lines

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

  • One can read the shadow growth as an effective-mass and deficit-angle effect: at large radius the Dehnen term behaves like an added mass $4\pi\rho_s r_s^3/3$, while $a$ lowers $f(r)$ everywhere; both effects push the photon sphere outward.
  • The monotone decrease of both the real frequency and the damping magnitude with $\rho_s$ and $a$ in the tables matches the behavior expected from a wider, shallower effective-potential barrier; a direct cross-check would be to compare the WKB values against the eikonal estimate $\omega \simeq \ell/R_s$.
  • Because Sgr A* and M87* are rotating, the same constraints should be re-derived with a Kerr-deformed version of the Dehnen/string-cloud metric to test whether spin moves the allowed region.
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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

2 major / 5 minor

Summary. The paper builds a static spherically symmetric black-hole solution by combining a Schwarzschild term with a Dehnen (1,4,0) dark-matter halo profile and a string-cloud parameter a, giving the metric function f(r) = 1 - a - 2M/r - 4πρs rs^3(2r+rs)/(3(r+rs)^2). The authors then write effective potentials for scalar, electromagnetic, and gravitational perturbations and compute l=2 fundamental quasinormal frequencies with the sixth-order WKB approximation (Tables I and II, Figs. 4-7). In the second half, they derive the photon-sphere and shadow radius, compare the shadow radius with EHT data for M87* and Sgr A*, and derive constraints ρs < 0.5, a < 0.12 for M87* and ρs < 0.12, a < 0.032 for Sgr A*. The shadow part is standard and internally consistent; the QNM part is not.

Significance. The shadow constraints are a concrete, falsifiable application of a dark-matter-halo black-hole model, and the paper provides useful tabulated data for the parameter dependence of the shadow. If the QNM results were valid, they would also be a useful complement to the shadow analysis. However, the QNM calculation is built on incorrect effective potentials, so the central advertised quantitative QNM results (Tables I and II and the associated figures) are not reliable and cannot support the abstract or conclusions. The EHT constraint section does not depend on the flawed QNM computation and appears salvageable.

major comments (2)
  1. [Sec. III, Eqs. (13)-(15)] The effective potentials use the angular factor (1/2+l) instead of l(l+1). For the l=2 modes used in all tables, this is 2.5 rather than 6. For a static spherical metric with lapse f, the scalar potential should be V_S = f[l(l+1)/r^2 + f'/r], the electromagnetic potential V_E = f l(l+1)/r^2, and the axial gravitational potential reduces in the Schwarzschild limit to V_G = f[l(l+1)/r^2 - 6M/r^3]. In the Schwarzschild limit the expressions (13)-(15) give (1-2M/r)[2.5/r^2 + 2M/r^3], (1-2M/r)2.5/r^2, and (1-2M/r)[2.5/r^2 - 6M/r^3], respectively, so the centrifugal barrier is wrong in all three cases. Every quasinormal frequency in Tables I and II and every curve in Figs. 4-7 is therefore computed for a different physical system; the heights and locations of the potential barriers are not those of the stated perturbations. This invalidates the QNM claim in the abstract and conclusion.
  2. [Sec. V vs. Tables I-II] The conclusion states that "QNM amplitude and damping increase with ρs and decrease with a" and that Tables I and II support this pattern. The tables show the opposite: as ρs increases from 0 to 1, both Re(ω) and |Im(ω)| decrease for all three fields, and as a increases from 0 to 0.8, both Re(ω) and |Im(ω)| also decrease. Figures 4 and 5 display the same decreasing trend. The verbal summary is inconsistent with the numerical results, so even if the potentials were corrected, the physical interpretation would need to be rewritten.
minor comments (5)
  1. [Figs. 6-7] The captions refer to "Dirac perturbations", but the paper does not analyze Dirac fields; the text and Tables I-II treat scalar, electromagnetic, and gravitational perturbations only.
  2. [Fig. 5] The caption says "Variation of amplitude and damping of QNMs with respect to the central density of the DM halo parameter", but the horizontal axis is the string-cloud parameter a; the caption should be corrected.
  3. [Eq. (19)] The effective potential for null geodesics is written as V_eff = f/r^2 (L^2/E^2 - 1); this form is not used to derive Eq. (20) and appears dimensionally inconsistent. It should be rewritten, for example as the standard radial potential from f L^2/r^2 after fixing E.
  4. [Fig. 7] The caption uses "ρ = 0.5" where the text elsewhere uses ρs = 0.5; the notation should be consistent.
  5. [General] There are numerous typographical errors (e.g., "made op" in the Introduction, "exams" for "examines", "theeffective", "minimun") that should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QNM and shadow results are computed from the stated metric and compared with external EHT data, with no fitted quantity renamed as a prediction.

full rationale

The paper's central derivation chain is self-contained: the metric function f(r) in Eq. (9) is constructed from the Dehnen density profile (Eqs. (2)-(7)) and the string-cloud term via the explicitly cited Xu et al. formalism [44]; the QNM frequencies in Tables I and II are obtained by applying the 6th-order WKB method to the effective potentials (13)-(15), which depend only on f(r) and l, not on any data or on the paper's conclusions; and the shadow radii in Table III and Eqs. (21)-(22) are computed from the same f(r) via the null-geodesic conditions. The EHT comparison (Sec. IV) uses external measured shadow diameters d_sh to constrain the model parameters, which is a genuine test rather than a fits-then-predicts loop. The self-citations that appear (e.g., refs. [29], [30], [37], [38]) are contextual literature citations and do not carry the load of any derivation. No uniqueness theorem or ansatz is imported from the authors' prior work to force a result, and no quantity defined in terms of another predicted quantity is present. I therefore do not find a circular step; separate physical or consistency issues in the effective potentials or the verbal trend statements are correctness concerns, not circularity under the stated criteria.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the Dehnen halo parameters rho_s and r_s, the string-cloud parameter a, the WKB approximation, and an asymptotic-flatness assumption that is actually violated by the string cloud. No new particles or fields are introduced.

free parameters (3)
  • rho_s (Dehnen halo central density) = bounded: rho_s < 0.5 (M87*), rho_s < 0.12 (Sgr A*); scanned up to 6 in Fig. 1
    Varies the dark matter density parameter in metric (9). The paper does not fit it; it scans it and then bounds it against EHT shadow size.
  • a (string cloud parameter) = bounded: a < 0.12 (M87*), a < 0.032 (Sgr A*); scanned up to 0.8
    Varies the string cloud parameter in metric (9). It is a free parameter of the Letelier string cloud model, constrained from shadow diameter.
  • r_s (halo core radius) = set to 0.5 in most tables; constrained to 0.5 <= r_s <= 1 in Fig. 12
    Core radius of the Dehnen profile. It is fixed by hand in the QNM and shadow tables and then varied in the EHT constraint plots.
assumptions (4)
  • domain assumption Dehnen (1,4,0) profile with mass distribution M_D = 4 pi rho_s r_s^3 r^3 / (3 (r + r_s)^3) and tangential velocity v_D^2 = M_D / r
    Assumes the dark matter halo can be described by this double-power-law profile and that the black hole metric can be embedded via the Xu et al. formalism used in Refs. [17,44].
  • standard math 6th-order WKB approximation gives accurate quasinormal frequencies for the stated potentials
    WKB is a standard approximation, but the paper provides no error estimate and no validation against exact Schwarzschild QNM frequencies.
  • domain assumption The spacetime is asymptotically flat so the boundary conditions (16) apply
    In fact f(r) -> 1 - a at infinity when a is nonzero, so the spacetime has a conical deficit and is not asymptotically flat in the usual sense. The paper does not flag this.
  • ad hoc to paper First-order expansion F(r) approximately 1 - 4 pi rho_s r_s^3 (2r + r_s) / (3 (r + r_s)^2) is valid for the parameter range used
    Eq. (7) drops higher-order terms from the exponential. For large rho_s, such as rho_s = 6 in Fig. 1, the exponential is not close to 1, so the truncation is uncontrolled.

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Cite this review

Pith. "Pith review of Quasinormal modes and shadow of Schwarzschild black holes embedded in a Dehnen type dark matter halo exhibiting string cloud." pith.science (2026). https://pith.science/paper/EQF37L2V

@misc{pith2026241220037,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes and shadow of Schwarzschild black holes embedded in a Dehnen type dark matter halo exhibiting string cloud},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQF37L2V}},
  note         = {Machine review of arXiv:2412.20037}
}
abstract

In this paper, we consider a static spherically symmetric black hole (BH) embedded in a Dehnen-(1,4,0) type dark matter (DM) halo in the presence of a cloud string. We examine and present data on how the core density of the DM halo parameter and the cloud string parameter affect BH attributes such as quasinormal modes (QNMs) and shadow cast. To do this, we first look into the effective potential of perturbation equations for three types of perturbation fields with different spins: massless scalar field, electromagnetic field, and gravitational field. Then, using the 6th order WKB approximation, we examine quasinormal modes of the BH disturbed by the three fields and derive quasinormal frequencies. The changes of QNM versus the core density parameter and the cloud string parameter for three disturbances are explored. We also investigate how the core density and the cloud string parameters affect the photon sphere and shadow radius. Interestingly, the study shows that the influence of Dehnen type DM and cloud string increases both photon spheres and shadow radius. Finally, we employ observational data from Sgr $A{^\star}$ and $M87{^\star}$ to set limitations on the BH parameters.

Figures

Figures reproduced from arXiv: 2412.20037 by the authors.

Figure 1
Figure 1. FIG. 1: The function [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Behaviours of BH potential with respect to radial distance [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Behaviours of BH potential with respect to radial distance [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of amplitude and damping of QNMs with respect to the central density [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of amplitude and damping of QNMs with respect to the central density [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Complex frequency plane for the Scalar, EM and Dirac perturbations showing the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Complex frequency plane for the Scalar, EM and Dirac perturbations showing the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Dependence of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Profile of shadows by the Schwarzschild BH embedded in a Dehnen type DM halo [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Constraint values of the DM density [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The density plot of the DM parameter [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Constraint values of the DM density [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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Forward citations

Cited by 1 Pith paper

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

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

    gr-qc 2025-05 reject novelty 4.0 of 10

    Quasinormal mode frequencies of a Schwarzschild black hole in a Dehnen-(1,4,5/2) dark matter halo are computed and found to decrease as halo density or core radius increases, with stability preserved.

Reference graph

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