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Black hole in the Dekel-Zhao dark matter profile

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A black hole embedded in a Dekel-Zhao dark matter halo acquires a modified metric whose shadow grows with the hole's mass but shrinks as the halo's central density rises.

desk verdict The new metric is a routine Einstein-cluster application, but the shadow analysis is built on a photon-sphere radius that doesn't follow from the paper's own mass profile. read the letter →

arxiv 2501.12559 v2 pith:RD52A7NY submitted 2025-01-22 gr-qc

Ali Övgün , Reggie C. Pantig This is my paper · ORCID
classification gr-qc PACS 95.30.Sf04.70.-s97.60.Lf04.50.+h
keywords generalrelativitydarkmatterblackholeshadowEinsteinclusterDekel-Zhaoprofilegravitationallensingphotonsphere
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 claims that a Schwarzschild black hole surrounded by a Dekel-Zhao dark matter halo can be described by an exact metric whose lapse function gains exponential corrections from the halo, deviating from pure Schwarzschild near the dense core and returning to it at large distances. Using the Einstein-cluster method, the authors derive the mass profile and the two-regime lapse function, then compute the photon sphere, shadow radius, and weak-field deflection angle. The paper's central physical claim is that the shadow radius grows with black hole mass and shrinks as the halo's central density increases, while the deflection angle is sensitive to the halo's characteristic radius. If correct, these relations give a concrete way to infer dark-matter halo parameters from black hole shadow and lensing observations.

What carries the argument

The central machinery is the Einstein-cluster construction, which represents the dark halo as a stationary, collisionless collection of particles on circular geodesics with zero radial pressure. This yields the coupled equations $m'(r)=4\pi r^2 \rho(r)$, $f'(r)/f(r) = 2m(r)/[r(r-2m(r))]$, and $P(r)=m(r)\rho(r)/[2(r-2m(r))]$, so that a chosen density profile fixes the mass profile and then the lapse function by integration. The Dekel-Zhao profile $\rho(r) = \rho_{\rm ch} x^{-a} (1+x^{1/2})^{-2(3.5-a)}$ with $x=r/r_c$ supplies the specific halo. The photon-sphere analysis is carried by the effective potential $V_{\rm eff}=f(r)/r^2$ and the condition $r_{\rm ph} f'(r_{\rm ph}) - 2 f(r_{\rm ph}) = 0$, together with the shadow formula $R_{\rm sh} = r_{\rm ph}/\sqrt{f(r_{\rm ph})}$.

What would settle it

Compute the exact mass profile by numerically integrating $m'(r)=4\pi r^2 \rho(r)$ with the full Dekel-Zhao density (without the $r\ll r_c$ cut), solve for the photon sphere using the exact lapse function, and compare the resulting shadow radius with Eq. (28) at the Figure 3 parameter values ($a=0.1$, $r_c=0.1$, $\rho_{\rm ch}=1$, $M_{\rm BH}=0.15$). If the exact shadow differs substantially, the small-radius approximation is not valid at the photon sphere and the paper's shadow predictions would need revision.

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Extended reading notes

Core claim

The central discovery is the lapse function for a Schwarzschild black hole embedded in a Dekel-Zhao dark matter profile. In the inner regime $r \ll r_c$, the metric is $f(r) = \left(1 - \frac{2M_{\rm BH}}{r}\right) \exp\left[\frac{2M_{\rm BH}}{r} + \frac{8\pi \rho_{\rm ch} r^{2-a}}{(3-a)(2-a) r_c^{a-3}}\right]$, while in the outer regime $r \gg r_c$ the exponential correction decays and the solution reduces to Schwarzschild. The photon sphere lies at $r_{\rm ph} = \left(\frac{3-a}{12\pi \rho_{\rm ch}}\right)^{1/(5-2a)}$, and the shadow radius is $R_{\rm sh} = r_{\rm ph}/\sqrt{f(r_{\rm ph})}$. The paper shows numerically that $R_{\rm sh}$ increases with $M_{\rm BH}$ and decreases with $\rho_{\rm ch}$, and that the deflection angle increases as the characteristic radius $r_c$ shrinks. The outer-regime approximation yields a negative mass profile, so no light ring exists there under that approximation.

Load-bearing premise

The shadow calculation assumes the short-distance approximation of the metric is valid at the photon sphere, but for the parameter values shown in the paper's Figure 3 the photon-sphere radius lies beyond the halo's characteristic radius, so the approximation may not hold where it is used.

Editorial extensions

If this is right

  • For fixed halo parameters, the shadow radius increases monotonically with black hole mass, so heavier black holes present larger apparent shadows to a distant observer.
  • Higher central dark-matter density $\rho_{\rm ch}$ produces a smaller shadow radius, meaning dense halos compress the photon sphere and shrink the observed shadow.
  • The deflection angle of light is larger for smaller characteristic radius $r_c$, so compact halos bend light more strongly.
  • In the large-distance regime the metric reduces to the Schwarzschild metric, so the halo's influence is confined to the inner region and vacuum general relativity is recovered.
  • The outer-regime mass profile becomes negative under the approximation, which implies no photon ring exists far from the hole.

Reading between the lines

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

  • The paper never checks whether its inner-region approximation $r\ll r_c$ actually holds at the photon sphere for the parameters plotted in Figure 3; a direct numerical integration of the exact field equations with the full Dekel-Zhao profile would show whether the predicted shadow radius survives without the approximation.
  • The same Einstein-cluster machinery could be applied to other double power-law halos to see whether the qualitative shadow behavior — larger $M_{\rm BH}$ enlarging the shadow, higher $\rho_{\rm ch}$ shrinking it — is universal across halo shapes.
  • Because the shadow depends on the halo's central density and characteristic radius, precision shadow measurements of supermassive black holes in dense galactic centers could in principle constrain the Dekel-Zhao profile parameters, turning the shadow into a dark-matter probe.
  • The monotonic decrease of the shadow with $\rho_{\rm ch}$ offers a sharp, testable signature that distinguishes this halo model from a vacuum Schwarzschild black hole at the same mass.
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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

4 major / 6 minor

Summary. The manuscript uses the Einstein-cluster method of Cardoso et al. to construct a spherically symmetric black-hole metric whose surrounding matter follows the Dekel-Zhao density profile. It presents separate small-r and large-r approximations for the mass and lapse functions, derives the photon sphere and shadow radius in the small-r regime, and computes a weak-field deflection angle. The paper claims that the shadow radius increases with the black-hole mass M_BH, decreases with the central density rho_ch, and that the deflection angle increases as the characteristic radius r_c decreases.

Significance. The motivation is timely and the Einstein-cluster construction is an appropriate way to model a dark-matter environment around a black hole. The paper gives explicit analytic expressions, discusses asymptotic Schwarzschild recovery, and connects the observables to EHT-type tests, which would be a useful contribution if the derivation were internally consistent. The construction is not circular: the metric is derived from a specified density profile and the observables are consequences. However, the central shadow and lensing results currently rest on a mass-profile formula that does not follow from the density profile, so the claimed predictions are not yet supported.

major comments (4)
  1. [II, Eq. (10)] For r<<r_c, Eq. (3) reduces to rho(r)=rho_ch (r/r_c)^{-a}; integrating Eq. (7) gives m(r)=4*pi*rho_ch*r_c^a*r^{3-a}/(3-a). Equation (10) instead has m(r)=4*pi*rho_ch*r^{3-a}/[(3-a)*r_c^{a-3}] = 4*pi*rho_ch*r_c^{3-a}*r^{3-a}/(3-a), which is dimensionally inconsistent in geometrized units (it has length dimension L^{4-2a} rather than L) and does not follow from the stated density profile. Because this mass profile enters Eq. (8), the error propagates into Y(r) in Eq. (12), the lapse functions in Eqs. (14)-(15), and all subsequent observables.
  2. [III, Eq. (27)] The photon-sphere formula does not follow from either the correct small-r mass or the paper's Eq. (10). Using the correct mass m(r)=4*pi*rho_ch*r_c^a*r^{3-a}/(3-a), the condition r=3m(r) gives r_ph=[(3-a)/(12*pi*rho_ch*r_c^a)]^{1/(2-a)}, which depends on r_c and has exponent 1/(2-a); using Eq. (10) verbatim gives r_ph=[(3-a)*r_c^{a-3}/(12*pi*rho_ch)]^{1/(2-a)}. Neither expression equals Eq. (27), which has no r_c dependence and an exponent of 1/(5-2a). Consequently Eq. (28), Fig. 3, and the claimed monotonic trends of R_sh with M_BH and rho_ch are not consequences of the derived metric.
  3. [II, Eq. (11); III, no-light-ring claim] For r>>r_c, the Dekel-Zhao density behaves as rho(r) about rho_ch*r_c^{3.5}*r^{-3.5}; integrating Eq. (7) gives m(r)=C-8*pi*rho_ch*r_c^{3.5}*r^{-0.5}. The integration constant C is fixed by the central mass and by matching to the inner solution, not by the indefinite integral. Setting C=0, as Eq. (11) implicitly does, produces a negative enclosed mass, so the subsequent statement in Section III that r=3m(r) has no real root in the outer regime is an artifact of that choice and cannot be used to argue that no light ring exists.
  4. [III, Fig. 3] Even accepting Eq. (27), the parameter values used in Fig. 3 (a=0.1, r_c=0.1, rho_ch=1) give r_ph about 0.59, which is larger than r_c=0.1; the small-r metric, Eq. (14), is therefore not valid at the photon sphere. The paper never states or checks the condition r_ph<<r_c, so the plotted shadow radii are outside the regime in which the lapse function was derived.
minor comments (6)
  1. [Fig. 2 caption] The caption appears to swap the two regimes: the in-figure labels say 'Case 1: r<<r_c' for the top panel and 'Case 2: r>>r_c' for the bottom panel, while the caption describes the top as r>>r_c and the bottom as r<<r_c; the caption and figure should be reconciled.
  2. [II, text after Eq. (22)] The statement that the tangential pressure equals rho/2 'for any mass function' is inconsistent with Eq. (9), which gives P_t=m*rho/[2(r-2m)]; the equality holds only at r=3m, not generally.
  3. [II, derivation paragraph] The sentence 'we use this approach approach to derive new black hole solutions' contains a duplicated word and should be edited.
  4. [III, shadow derivation] The phrase 'The radius of shadow to be equal to,' is grammatically incomplete; use 'The shadow radius is'.
  5. [V, Acknowledgments] The phrase 'multi-messenger appronkh' appears to be a typo and should be corrected.
  6. [III, Fig. 4] The deflection-angle section should state which metric regime, Eq. (14) or Eq. (15), is used in the numerical integration, and should confirm that the plotted impact parameters lie in that regime for each r_c value shown.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the metric, shadow, and deflection angle follow from an explicit density-profile ansatz, and the self-citations are non-load-bearing.

full rationale

The paper's derivation chain is not circular. The metric is obtained by integrating the Einstein field equations for a specified Dekel-Zhao density profile, giving the mass profile m(r), the lapse function f(r), and the tangential pressure. The shadow and deflection angle are then computed from the resulting f(r) using standard photon-sphere formulas. No observable is used to set the model parameters, and no fitted quantity is renamed as a prediction. The self-citations to the authors' earlier work (refs. [30]-[32]) appear only in a general list of prior studies on black-hole-shadow and dark-matter research; they are not used to fix a constant, to justify a uniqueness claim, or to supply the central derivation. The method is attributed to Cardoso et al., not to the present authors. A possible concern is that the photon-sphere formula in Eq. (27) appears inconsistent with the mass profile in Eq. (10), but that is a correctness or internal-consistency issue, not circularity: Eq. (27) is not constructed from the shadow output and then used to predict that same output. Hence no circular step can be exhibited, and the appropriate finding is no significant circularity.

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

The central claim rests on the Einstein cluster idealization and on the Dekel-Zhao profile as an assumed density input. Four profile and mass parameters (a, rc, ρ_ch, M_BH) are used as inputs rather than derived. No new entities are introduced. The main unsupported step is the patching of two asymptotic metric solutions and the use of the small-radius branch at radii where it may not be valid.

free parameters (4)
  • a (Dekel-Zhao inner slope) = 0.1 (used in figures)
    Profile exponent controlling the inner density slope; free parameter of the model, not fitted to observable data.
  • rc (characteristic radius) = 0.1, 5, 10, 15, 20 in figures
    Free scale radius of the density profile; controls where the exponential correction dominates.
  • ρ_ch (characteristic density) = 1 in figures
    Free amplitude of the dark matter density; controls the size of the metric correction.
  • M_BH (black hole mass) = 0.15, 2 in figures
    Central black hole mass parameter; an input of the model.
assumptions (4)
  • domain assumption Dark matter is modeled as an Einstein cluster: collisionless particles on circular geodesics, stress-energy diag(-ρ,0,Pt,Pt), Pr=0.
    Section II, Eq (6). This is the key physical idealization that permits an exact solution; real halos have velocity dispersion and radial support.
  • standard math The metric is static, spherically symmetric, with ds² = -f(r)dt² + dr²/(1-2m(r)/r) + r²dΩ².
    Section II, Eq (4). Standard ansatz for spherically symmetric spacetimes in GR.
  • domain assumption The Dekel-Zhao profile with b=2 and g=3.5 (Eq (3)) describes the dark matter halo.
    Section II, Eq (3). The functional form of the halo is assumed, not derived from a more fundamental theory.
  • ad hoc to paper The mass and lapse function can be obtained by patching asymptotic solutions for r << rc and r >> rc.
    Section II, Eqs (10)-(15). The paper switches between two approximations and does not construct a globally valid interpolating metric.

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Cite this review

Pith. "Pith review of Black hole in the Dekel-Zhao dark matter profile." pith.science (2026). https://pith.science/paper/RD52A7NY

@misc{pith2026250112559,
  author       = {Pith},
  title        = {Pith review of: Black hole in the Dekel-Zhao dark matter profile},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RD52A7NY}},
  note         = {Machine review of arXiv:2501.12559}
}
abstract

Motivated by the work of Cardoso et al. [Phys. Rev. D 105 (2022) 6, L061501, https://doi.org/10.1103/PhysRevD.105.L061501] on black holes in galaxies, we derive a new black hole solution surrounded by a Dekel-Zhao (DZ) dark matter profile. The derived metric, influenced by DZ profile parameters, exhibits two distinct regimes: for $r \ll r_{\rm c}$ (\textcolor{black}{$r_{\rm c}$ is a characteristic radius}), exponential corrections dominate, producing significant deviations from the Schwarzschild solution near dense cores, while for $r \gg r_{\rm c}$, these corrections vanish, restoring the Schwarzschild metric at large distances. These findings ensure consistency with general relativity in vacuum. \textcolor{black}{The black hole shadow and deflection angle are analyzed, demonstrating that the shadow radius increases with black hole mass ($M_{\rm BH}$), while higher central densities ($\rho_{\rm ch}$) result in smaller shadows, reflecting the environmental impact of dense dark matter halos.} Photon dynamics reveal how DZ profiles modify critical impact parameters and effective potentials, with gravitational lensing effects highly sensitive to the characteristic radius ($r_{\rm c}$). Smaller $r_{\rm c}$ values lead to larger deflection angles due to stronger gravitational effects near compact cores. This work highlights the significance dark matter profiles in shaping black hole observables, providing a theoretical foundation for future observational studies and advancing the understanding of dark matter-black hole interactions in astrophysical and cosmological contexts.

Figures

Figures reproduced from arXiv: 2501.12559 by the authors.

Figure 1
Figure 1. FIG. 1. The figure shows the visualization of black hole, along [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The above plot above shows the shadow radius [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

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Reviewed August 10, 2026 · model on record in the stance chip above.