Accretion Disk Luminosity and Topological Characteristics for a Schwarzschild Black Hole Surrounded by a Hernquist Dark Matter Halo

Farokhnaz Hosseinifar; Hassan Hassanabadi; Kuantay Boshkayev; Luis M. Nieto; Soroush Zare

REVIEW 1 major objections 6 minor 1 cited by

Accretion Disk Luminosity and Topological Characteristics for a Schwarzschild Black Hole Surrounded by a Hernquist Dark Matter Halo

T0 review · 1 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A Schwarzschild black hole immersed in a Hernquist dark matter halo has a larger innermost stable orbit, a reshaped accretion-disk spectrum, and a radiative efficiency above the usual 5.72 percent.

desk verdict A routine but competent application of standard black hole tools to a metric that turns out not to be sourced by the Hernquist profile, so the disk and QNM signatures should not be attributed to Hernquist dark matter without an explicit caveat or proof. read the letter →

arxiv 2507.14305 v1 pith:UNY7ITDO submitted 2025-07-18 gr-qc astro-ph.HE

Luis M. Nieto , Farokhnaz Hosseinifar , Kuantay Boshkayev , Soroush Zare , Hassan Hassanabadi This is my paper · ORCID

classification gr-qcastro-ph.HE

keywords SchwarzschildblackholeHernquistdarkmatterhaloaccretiondiskluminosityinnermoststablecircularorbitradiativeefficiencyquasi-normalmodesshadowtopologicalcharge

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 dark matter is not just a background for a Schwarzschild black hole: a Hernquist-style halo measurably rearranges the spacetime around the hole. Working with the metric $f(r)=1-2M/r-4\pi\rho_s r_s^3/(r+r_s)$, the authors compute how the halo parameters---scale radius $r_s$ and central density $\rho_s$---change the innermost stable circular orbit, the flux, temperature, and luminosity of a thin accretion disk, and the radiative efficiency. They find the ISCO grows beyond $6M$, the peak of the disk emission drops and shifts outward while outer-disk emission strengthens, and the efficiency of converting accreted matter into radiation rises above the Schwarzschild value of about 5.72%. They also link the halo to a larger shadow, lower real quasi-normal frequencies, and a specific topological classification. A sympathetic reader cares because these are concrete, in-principle observable signatures connecting dark matter halo structure to black hole images and spectra.

What carries the argument

The load-bearing object is the static, spherically symmetric metric $ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$ with $f(r)=1-2M/r-4\pi\rho_s r_s^3/(r+r_s)$; this is a superposition of the Schwarzschild potential and the Newtonian Hernquist potential, imported from reference [103] rather than derived in the paper. All later results flow from this lapse function through the effective potential $V_{\rm eff}(r)=f(r)(1+L^2/r^2)/2$, whose extrema define circular orbits and whose second derivative fixes the ISCO, and through the Novikov-Thorne radiative machinery: with $E$, $L$, and $\Omega$ from the geodesic equations, the flux integral, Stefan-Boltzmann temperature, differential luminosity, and spectral luminosity convert the geometry into disk observables. In the eikonal section the same $f$ appears in the Regge-Wheeler potential and in the shadow relation $r_{\rm sh}=r_c/\sqrt{f(r_c)}$, while the thermodynamic and topological analyses use $f$ to define horizon radius, Hawking temperature, and the three potentials whose winding numbers classify the solution.

What would settle it

Compute the Einstein tensor of the metric (4) and compare the density profile it implies with the Hernquist profile (2); if they disagree, the metric is not a faithful general-relativistic representation of the claimed halo, and the same disk calculation repeated with a genuine solution would give different ISCO and efficiency values. A direct check of Eq. (4) against the Einstein equations with a pressure-supported Hernquist fluid would settle whether the predicted efficiency boost is physical.

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Extended reading notes

Core claim

The central discovery is that the halo parameters $\rho_s$ and $r_s$ are not spectators: they change the geodesic structure of the spacetime. The angular velocity $\Omega(r)$, angular momentum $L(r)$, and energy $E(r)$ of circular orbits all shift relative to vacuum Schwarzschild, and the ISCO, obtained from the condition $\partial_r^2 V_{\rm eff}=0$, grows monotonically with both $r_s/M$ and $\rho_s M^2$. Using the Novikov-Thorne model, the paper then shows that the disk's radiative flux $F(r)$, temperature $T(r)$, differential luminosity $dL_\infty/d\ln r$, and spectral luminosity $\nu L_{\nu,\infty}$ all respond to the halo: the maximum flux decreases and shifts to larger radius, low-radius flux is suppressed while outer-disk flux is enhanced, and the radiative efficiency $\eta\simeq 1-E(r_{\rm ISCO})$ exceeds the Schwarzschild limit of about 5.72% as $r_s/M$ increases. In the eikonal limit the same geometry gives a larger shadow radius, a smaller real part of the quasi-normal modes, and a larger greybody factor. Finally, the paper assigns topological charges to the photon sphere, to the Hawking-temperature phase transition, and to the generalized free energy, placing the black hole in the same topological class as Reissner-Nordström.

Load-bearing premise

The entire calculation depends on the imported metric $f(r)=1-2M/r-4\pi\rho_s r_s^3/(r+r_s)$ being a valid spacetime for a Schwarzschild black hole in a Hernquist halo; the paper does not derive it, and if that metric is not an actual solution of the field equations for the Hernquist density, the ISCO, luminosity, QNM, and topological numbers lose their quantitative meaning.

Editorial extensions

If this is right

If the halo is dense enough, the accretion disk starts farther out than $6M$, so measuring the inner edge of a black hole disk becomes a probe of surrounding dark matter.
The radiative efficiency exceeding 5.72% means the same mass accretion rate can produce more luminosity from a black hole embedded in a Hernquist halo than from an isolated Schwarzschild black hole.
The disk spectrum is reshaped: at fixed halo density, larger $r_s$ lowers and outward-shifts the flux peak while raising the outer-disk flux, so broadband spectral fits carry information about the halo.
The larger shadow and lower real quasi-normal frequency predicted in the eikonal limit mean that ringdown and shadow observations would see the halo as a systematic shift rather than as a change in black hole mass alone.
The topological classification (photon-sphere charge $-1$, temperature critical-point charge $-1$, free-energy charges $+1$ and $-1$ summing to zero) places this black hole in the Reissner-Nordström class, indicating that its phase structure is stable under small deformations.

Reading between the lines

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

As an editorial extension: if the imported metric (4) turns out not to satisfy the Einstein equations with the Hernquist stress-energy, the quantitative predictions (efficiency, ISCO shift) should be re-derived with a genuine solution; the qualitative direction—a cuspy halo pulling the ISCO outward—is likely robust because any additional central mass deepens the potential well.
As an editorial extension: the paper's static, spherically symmetric setting leaves open how the same halo affects a rotating black hole; applying the same effective-potential and Novikov-Thorne pipeline to a rotating generalization is a direct next step that would let the predictions be tested against M87* and Sgr A* shadow and continuum data.
As an editorial extension: because the efficiency boost depends on $r_s/M$ and $\rho_s M^2$, a spectral fit to an observed black hole disk could in principle constrain the local dark matter density; the paper computes the curves but does not perform such an observational fit.
As an editorial extension: the eikonal shadow-QNM relation implies a degeneracy between halo parameters and black hole mass, and a joint measurement of shadow size and ringdown frequency could break that degeneracy; the paper does not address this trade-off.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a competent, routine application of the Novikov-Thorne accretion disk, eikonal QNM, greybody, and topological thermodynamics frameworks to the metric f(r)=1-2M/r-4πρs rs^3/(r+rs) imported from Ref. [103]. What is new is the set of numerical curves for ISCO, flux, luminosity, efficiency, shadow, and QNM frequencies for this specific metric, plus the topological classification. The authors correctly check that their expressions reduce to Schwarzschild in the limits rs→0 and ρs→0. The thermodynamic remnant analysis, while simple, is internally consistent.

The soft spot is at the foundation, and it is not minor. The stress-test note is right: for this static, spherically symmetric metric, the effective mass is m(r)=M+2πρs rs^3 r/(r+rs), so the Einstein tensor sources an energy density ρ_eff(r)=ρs rs^4/[2r^2(r+rs)^2], not the Hernquist density ρ_H(r)=ρs rs^4/[r(r+rs)^3]. The two densities differ by a factor (r+rs)/(2r), equal only at r=rs. So the metric does not describe a Schwarzschild black hole immersed in a Hernquist halo; it is a heuristic superposition built from the Newtonian circular-velocity condition. Every quantitative signature in the paper is computed from this f(r), so the paper's central attribution of those signatures to Hernquist dark matter is not established. This should be fixable: either prove the metric is an exact solution, or, more honestly, present it as an effective/phenomenological model and state that the halo density interpretation is only approximate in the weak-field, large-radius regime.

Additional technical issues are secondary but worth fixing. Eq. (18) is typeset in a way that makes the ISCO condition ambiguous. In Eq. (21), MT is introduced as the total mass but then appears inside the spectral luminosity integral; the authors should state explicitly how it is computed and why it is constant. Section V has contradictory descriptions of winding-number signs: the text says clockwise corresponds to −1, while the Fig. 17 caption says clockwise means ω1=1, and the same confusion appears around Fig. 18.

Who this is for: readers who want benchmark calculations for this particular effective metric, or who study how generic dark-matter-inspired terms shift ISCOs and shadows. They will find the derivations straightforward and mostly reproducible after the fixes. The physical interpretation, as it stands, should not be taken at face value.

Recommendation: send to peer review, but expect a major revision. The referee should demand the metric caveat or proof, require cleanup of Eq. (18) and Eq. (21), and ask for the topological signs to be made consistent.

Referee Report

1 major / 6 minor

Summary. The paper studies a Schwarzschild black hole surrounded by a Hernquist dark matter halo using the metric f(r)=1-2M/r-4πρs rs^3/(r+rs) imported from Ref. [103]. It computes thermodynamic quantities (horizon radius, Hawking temperature, remnant radius and mass), geodesic properties and the ISCO, the Novikov-Thorne accretion disk flux, temperature, differential and spectral luminosity, the radiative efficiency, eikonal quasi-normal modes and shadow, greybody factors and absorption cross sections, and finally three topological charges (photon sphere, Hawking temperature, and generalized free energy). All results are stated to reduce to the Schwarzschild limit when rs→0 or ρs→0, and the figures show the claimed dependences on the halo parameters rs and ρs. The topological charges are computed as consistency checks on the same metric.

Significance. If the metric (4) is accepted as a physical spacetime for a black hole embedded in a Hernquist dark matter halo, the paper provides a systematic set of quantitative predictions: ISCO enlargement with rs/M and ρsM^2, shifts in the disk flux and temperature peaks, changes in the spectral luminosity, a radiative efficiency that rises above the Schwarzschild 5.72%, and shifts in the shadow radius and eikonal QNM frequencies. The derivations are mostly standard and transparent, and several formulas were checked analytically in the Schwarzschild limit. However, the paper's central physical interpretation depends entirely on Eq. (4) being a valid spacetime sourced by the Hernquist profile, and that point is not established in the manuscript. The topological charges and QNM relations are internal consistency checks rather than independent validations, so the significance hinges on the status of the imported metric.

major comments (1)

[Section VI, Summary and Conclusions, paragraph 3] The summary states: 'In our study of the differential luminosity, we observed a significant increase with increasing rs, and the maximum now occurs at a larger radius.' This contradicts the body of the paper: Section III and Figure 9 show that as rs/M increases, the maximum of dL∞/dln r decreases and occurs at a smaller r/M. The conclusions should be checked against the numerical results and corrected.

minor comments (6)

[Section II, Eq. (5)] The expression for the horizon radius contains a square root whose argument in the text reads '8Mrs + (rs − 2M − 4πρsrs^3)^2'; please verify the placement of parentheses, as the printed form is ambiguous.
[Section IV, Figure 12 caption] The caption says 'Left panel: shadow radius ... Left panel: the real part of the QNMs'; the second occurrence should be 'Right panel'.
[Section II, paragraph after Eq. (4)] The sentence 'The spacetime metric of pure dark matter can be determined by considering the relationship between the tangential velocity ...' is vague; the derivation is not shown in this paper, and the reader must consult Ref. [103]. A brief summary of that derivation or a clear statement that the metric is assumed would improve the manuscript.
[Figures 1–4 and Section III] The figures use astrophysical units (e.g., ρs in GeV/cm^3 and rs in kpc) while the equations use geometric units with G=ℏ=c=1. The conversion between the two is never stated; please clarify, for instance, how ρs M^2 = 1 is related to the values in Figure 1.
[Section III, after Eq. (19)] The text takes m_dot = 1 for the normalized flux, but Eq. (21) introduces the total mass MT and the normalization T* without explaining how the accretion rate is scaled in the spectral luminosity; a brief clarification would help the reader reproduce the integrals.
[Section V, Figs. 17, 18, 20] The topological charge calculations are presented with specific contour sizes (a=b=0.3) but no error estimate or dependence on the contour size is discussed; the authors should state that the winding numbers are invariant under continuous deformations not crossing zero points, or cite the standard argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all disk, QNM, and topological outputs follow from the imported SBH-HDM metric through standard formulas; the metric is an input, not a prediction.

full rationale

The paper's claimed results—ISCO radius, flux, temperature, differential and spectral luminosity, radiative efficiency, QNM frequencies, shadow, graybody factors, and topological charges—are all obtained by substituting the assumed metric f(r)=1-2M/r-4πρs rs^3/(r+rs) (Eq. 4) into standard geodesic, Novikov-Thorne, Regge-Wheeler, and winding-number formulas. No parameter is fitted to the quantities being predicted, and no output is fed back to define Eq. (4). The central metric is imported from Ref. [103] rather than derived in the paper, which is a substantive correctness concern: the Einstein tensor associated with this metric may not reproduce the Hernquist density profile (Eq. 2), so the attribution of the computed signatures to a Hernquist halo is not automatically justified. However, that is a physical-validity issue, not circularity, because the metric is an input and the disk/topological quantities are genuinely derived consequences of it. Several references are to prior work by the authors, but they are used to cite standard formulas (e.g., Novikov-Thorne flux, luminosity, Hawking temperature) rather than as the sole support for a claim that is defined into existence. The topological charges are consistency checks on the same spacetime model, not independent validations, but this is not equivalent to circular reasoning. No prediction reduces to a fitted parameter, to a self-citation chain, or to a quantity defined in terms of itself.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central results depend on the assumed Hernquist halo profile, the imported effective metric of Ref [103], and the standard Novikov-Thorne, eikonal QNM and topological formalisms. No new entities are introduced. The only hands-tuned quantities are the model parameters used to generate the figures.

free parameters (4)

ρs (Hernquist characteristic density) = ρs M^2 = 0.5, 1.0, 1.5 in figures
Chosen by hand for the plots; controls the halo density and drives the reported trends in ISCO, flux and efficiency.
rs (Hernquist scale radius) = rs/M = 0.0, 0.4, 0.5, and other values
Varied by hand to scan halo size; central to the claimed shifts in accretion disk and QNM observables.
M (black hole mass) = M = 1 in numerical work
Scale parameter; all numerical results are expressed in units of M.
m_dot (mass accretion rate) = m_dot = 1 (normalized)
Set to unity; flux and luminosity are computed per unit accretion rate.

assumptions (6)

standard math Einstein gravity with G = ℏ = c = 1 and the metric ansatz g(r) = f(r), h(r) = r^2
Foundational to all derived quantities; stated at the end of Section I.
domain assumption Hernquist density profile (Eq. 2) provides the dark matter halo
Model assumption for halo structure, taken from Ref [104].
domain assumption The metric f(r)=1-2M/r-4πρs rs^3/(r+rs) (Eq. 4) is a valid spacetime for a BH in a Hernquist halo
Taken from Ref [103]; the paper does not prove it solves the Einstein equations with the Hernquist density, and the Einstein tensor of this metric does not equal that density.
domain assumption Novikov-Thorne thin disk assumptions I-VII (Section III)
Standard accretion disk model; assumes equatorial thin disk, Keplerian orbits, blackbody emission, and constant accretion rate.
standard math Eikonal correspondence between null geodesics and QNMs (Cardoso et al. 2009)
Used in Section IV to infer QNM frequencies and Lyapunov exponent from the photon sphere.
standard math Topological winding number classification (Wei-Liu-Mann formalism)
Used in Section V to assign charges to the photon sphere, Hawking temperature and generalized free energy.

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Cite this review

Pith. "Pith review of Accretion Disk Luminosity and Topological Characteristics for a Schwarzschild Black Hole Surrounded by a Hernquist Dark Matter Halo." pith.science (2026). https://pith.science/paper/UNY7ITDO

@misc{pith2026250714305,
  author       = {Pith},
  title        = {Pith review of: Accretion Disk Luminosity and Topological Characteristics for a Schwarzschild Black Hole Surrounded by a Hernquist Dark Matter Halo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNY7ITDO}},
  note         = {Machine review of arXiv:2507.14305}
}

read the original abstract

In this work, we study some characteristics and gravitational signatures of the Schwarzschild black hole immersed in a Hernquist dark matter halo (SBH-HDM). We determine the black hole's remnant radius and mass, which provide useful residual information at the end of its evaporation. We then explore the luminosity of the accretion disk from the SBH-HDM model. In this way, we determine the key orbital parameters of the test particles within the accretion disk, such as angular velocity, angular momentum, energy, and the radius of the innermost stable circular orbit, based on the dark matter model parameters. We also numerically estimate the accretion disk's efficiency in converting matter into radiation. We also demonstrate that dark matter, which significantly alters the geometry surrounding a Schwarzschild black hole, influences the accretion disk's radiative flux, temperature, differential luminosity, and spectral luminosity. The stability of a black hole spacetime is determined in the eikonal regime. The Lyapunov exponent is also analyzed to quantify the stability of the particle regime and to demonstrate the infall into or escape from the black hole to infinity, as well as the quasi-normal modes. Finally, some properties of black holes are studied from a topological perspective.

Figures

Figures reproduced from arXiv: 2507.14305 by the authors.

**Figure 2.** FIG. 2. Lapse function for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Hawking temperature curves for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (17 more)

**Figure 4.** Figure 4: FIG. 4. Remnant radius of the black hole versus [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: illustrates the variation of the ISCO radius as a function of rs/M, together with the angular velocity, angular momentum, and energy profiles for ρsM2 = 1 and selected values of rs/M, plotted as functions of the radial coordinate r/M. As expected, in the limit of rs → …

**Figure 6.** Figure 6: shows the energy flux and radiation temperature curves considering ρsM2 = 1 for three different values of rs/M. It is observed that as r/M increases, the energy flux reaches a maximum value rs /M=0.0 rs /M=0.4 rs /M=0.5 10 20 30 40 r/M 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ℱ(×10…

**Figure 7.** Figure 7: FIG. 7. Radiation temperature for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. The behavior of the radiation temperature when varying [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: represents the variation of the differential luminosity at infinity and the spectral luminosity for ρsM2 = 1 and three values of rs/M. It is observed that the differential luminosity at infinity exhibits rs /M=0.0 rs /M=0.4 rs /M=0.5 10 20 30 40 50 60 r/M 0.5 1.0 1.5 2…

**Figure 10.** Figure 10: FIG. 10. Radiative efficiency in terms of [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: FIG. 11. Effective potential curves considering [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: FIG. 12. Left panel: shadow radius in terms of [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

**Figure 13.** Figure 13: FIG. 13. Graybody bounds considering [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

**Figure 14.** Figure 14: FIG. 14. Emitted power for the cases [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

**Figure 15.** Figure 15: FIG. 15. Absorption cross section with parameters [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

**Figure 16.** Figure 16: FIG. 16. The potential [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]

**Figure 17.** Figure 17: FIG. 17. Left panel: Vector space of the potential [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

**Figure 18.** Figure 18: FIG. 18. Left panel: Vector space of the potential [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]

**Figure 19.** Figure 19: shows the rh − τ curve obtained from (49) for the values rs = 0.5 and ρs = 1. The sign of the 20 30 40 50 60 70 0 τ 1 2 3 4 5 rh FIG. 19. rh − τ curve (49) for the values rs = 0.5 and ρs = 1. At τ = 18.75922 the sign of the slope of the curve changes. slope of this cu…

**Figure 20.** Figure 20: FIG. 20. Left panel: Vector space of the potential [PITH_FULL_IMAGE:figures/full_fig_p022_20.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

Schwarzschild-like Black Holes Submerged in an Exponential Density Dark Matter Profile
gr-qc 2026-07 conditional novelty 4.0 of 10

An analytic Schwarzschild-like metric with an exponential dark matter halo is constructed and its shadows, quasi-normal modes, and greybody bounds are computed, though several derived expressions have sign errors.

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