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arxiv: 2509.11245 · v4 · pith:V7M7HWANnew · submitted 2025-09-14 · 🌌 astro-ph.HE · astro-ph.GA· gr-qc

Probing Hernquist dark matter through the optical appearance of black holes: A comprehensive study of various accretions

Yuxuan Shi , Hongbo Cheng This is my paper

Pith reviewed 2026-05-21 22:32 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.GAgr-qc
keywords Hernquist dark matter haloblack hole imagesphoton spherecritical curveaccretion modelsbrightness depressionray tracing
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The pith

Hernquist dark matter enlarges the photon sphere around black holes, producing a critical curve 2 to 30 percent larger than in empty space.

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

This paper investigates the visual signatures of a Schwarzschild black hole embedded in a Hernquist dark matter halo under three accretion scenarios. The dark matter halo expands the photon sphere, which in turn enlarges the observable critical curve and alters the central brightness depression in the resulting images. Direct emission dominates for a thin disk, while Doppler de-boosting suppresses brightness in infalling spherical flows. A reader would care because these size and brightness differences supply a concrete way to test dark matter distributions right at galactic centers using future high-resolution observations. The comparisons to the vacuum case isolate the dark matter contribution.

Core claim

The Hernquist dark matter halo significantly enlarges the photon sphere of a Schwarzschild black hole. This enlargement leads to an observable critical curve radius approximately 2% to 30% larger than in the vacuum case. The intensity profiles exhibit general brightness suppression, especially pronounced under Doppler de-boosting in the infalling spherical flow. The size of the central brightness depression and the brightness profile thereby furnish a theoretical framework for constraining dark matter distributions in galactic centers.

What carries the argument

The spacetime metric modified by the Hernquist dark matter density profile, together with ray-tracing calculations performed for three accretion models: geometrically thin disk, static spherical flow, and infalling spherical flow.

If this is right

  • The critical curve radius and photon ring size become direct observables sensitive to the dark matter halo parameters.
  • Brightness profiles show model-dependent suppression, strongest in the infalling spherical case due to Doppler effects.
  • The central brightness depression size varies systematically with the dark matter distribution and can be compared to vacuum predictions.
  • These image features supply a framework for placing constraints on dark matter near supermassive black holes using existing and future telescope data.

Where Pith is reading between the lines

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

  • If the predicted enlargement is detected, the same ray-tracing approach could be applied to other density profiles to differentiate them observationally.
  • Linking to stellar dynamics or weak lensing around the same galaxies would allow cross-checks on the assumed dark matter distribution at small radii.
  • Numerical images generated for specific galaxies such as M87 or the Milky Way center could yield quantitative predictions testable against current Event Horizon Telescope results.

Load-bearing premise

The Hernquist density profile accurately describes the dark matter very close to the black hole without significant back-reaction on the metric or accretion flow, and the three chosen accretion models capture the dominant radiative effects.

What would settle it

A high-precision measurement of the critical curve radius or shadow size for a galactic-center black hole that matches the pure vacuum Schwarzschild prediction within 1% would directly contradict the predicted 2-30% enlargement.

Figures

Figures reproduced from arXiv: 2509.11245 by Hongbo Cheng, Yuxuan Shi.

Figure 1
Figure 1. Figure 1: The effective potential Vph and the impact parameter bp of the Schwarzschild-Hernquist black holes. impossible to ignore the substantial impact that the center density ρc and the core radius rs of the Hernquist DM halo have on the photon paths. III. OBSERVATIONAL FEATURES OF THIN DISK EMISSION In astrophysical settings, black holes are typically surrounded by large amounts of accret￾ing matter. This makes … view at source ↗
Figure 2
Figure 2. Figure 2: The behaviour of light trajectories under various [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The number of orbits n vs the impact parameter b is shown for various Hernquist DM parameters ρc and rs. angle φ, for a photon traveling from the source to the observer. The number of orbits is then given by [40, 75] n(b) = φ 2π , (19) Eq.(19) depends on the impact parameter b. The number of orbits n(b) is determined by how close the impact parameter b is to the critical value bp. It also depends on the me… view at source ↗
Figure 4
Figure 4. Figure 4: Photon behaviour as a function of impact parameter [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The first three transfer functions for the Schwarzschild-Hernquist black hole. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Observational view of the thin disk surrounding the Schwarzschild-Hernquist [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Observational view of the thin disk surrounding the Schwarzschild-Hernquist [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The specific intensity Iobs of a static spherical accretion as observed by a distant observer. Combining Eqs.(27)-(30), the observed intensity for a static observer at infinity can be expressed as Iobs = ˆ γ f(r) 3/2 r 2 s 1 f(r) + r 2  dφ dr 2 dr. (31) The integrand in Eq.(31) represents the intensity contribution as a function of impact pa￾rameters b, which is plotted in Fig.8. Fig.9 displays the shado… view at source ↗
Figure 9
Figure 9. Figure 9: The images of Schwarzschild-Hernquist black holes’ shaodow with [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The specific intensity Iobs of a infalling spherical accretion as observed by a distant observer. [45, 71, 76, 84] R = κµu µ obs κνu ν emi , (32) The four-velocities for a distant static observer and for the infalling accreting matter are uobs = (1, 0, 0, 0) (33) and uemi =  1 f(r) , − p 1 − f(r), 0, 0  , (34) respectively. In Eq.(32), the four-velocities of photons κ released from the accretion disk is… view at source ↗
Figure 11
Figure 11. Figure 11: The images of Schwarzschild-Hernquist black holes’ shaodow with [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
read the original abstract

The observational appearance of a black hole is critically dependent on the surrounding accreting matter, in particular on the central brightness depression and photon ring structure. We perform a systematic comparative analysis of the observational signatures of a Schwarzschild black hole embedded in a Hernquist dark matter (DM) halo under three distinct accretion scenarios: a geometrically thin disk, a static spherical flow, and an infalling spherical flow. For the thin disk model, we find that direct emission dominates the total observed intensity, while the size and brightness of the lensing and photon rings serve as sensitive probes of the Hernquist DM parameters. From a geometric perspective, the Hernquist DM halo significantly enlarges the photon sphere, resulting in an observable critical curve radius approximately $2\%$ to $30\%$ larger than in the vacuum case. Regarding the radiative signatures, the measured intensity profiles, which rely on the particular accretion models, show a general brightness suppression, which is especially affected by the Doppler de-boosting in the infalling scenario. Our results suggest that the size of the central brightness depression and the brightness profile of the black hole image provide a valuable theoretical framework for constraining the distribution of dark matter in galactic centers.

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

2 major / 2 minor

Summary. The manuscript performs a comparative study of the observational appearance of a Schwarzschild black hole embedded in a Hernquist dark matter halo under three accretion scenarios (geometrically thin disk, static spherical flow, and infalling spherical flow). It claims that the Hernquist halo enlarges the photon sphere, producing an observable critical curve radius 2%–30% larger than the vacuum case, and that the resulting intensity profiles exhibit general brightness suppression (most pronounced via Doppler de-boosting in the infalling model), thereby providing a theoretical framework for constraining dark matter distributions near galactic centers.

Significance. If the numerical results hold, the work supplies a concrete theoretical framework linking black-hole image observables (critical curve size and central brightness depression) to Hernquist halo parameters, which could be relevant for interpreting EHT-style observations of galactic-center black holes. The geometric enlargement of the photon sphere follows directly from inserting the enclosed-mass function into the metric and solving the null circular orbit condition; this part is internally consistent under the stated assumptions. The radiative signatures are explicitly conditioned on the three chosen accretion flows, so model dependence is flagged rather than concealed.

major comments (2)
  1. [Methods (ray-tracing and accretion models)] Methods section on ray-tracing and intensity calculation: the emissivity profiles, specific intensity integration along geodesics, and numerical implementation for the three accretion models are only summarized. Full specification (including any analytic expressions for the emissivity and convergence/error estimates) is required to substantiate the reported brightness suppression and the statement that direct emission dominates in the thin-disk case.
  2. [Results (critical curve / photon sphere)] Results on critical curve enlargement: the quoted 2%–30% range for the critical impact parameter is presented without an accompanying table or plot that maps the interval to specific values of the Hernquist scale radius a and central density; because the enlargement is a direct function of these parameters, the range should be derived explicitly from the modified metric function f(r) = 1 – 2M(r)/r.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'the size of the central brightness depression and brightness profile provide a valuable theoretical framework' could be sharpened by identifying which single observable (e.g., depression radius or ring brightness contrast) is most sensitive to the Hernquist parameters.
  2. [Throughout] Notation: ensure that the Hernquist parameters (scale radius and central density) are denoted consistently between the metric definition, the enclosed-mass function, and all figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment point by point below and will revise the manuscript accordingly to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: Methods section on ray-tracing and intensity calculation: the emissivity profiles, specific intensity integration along geodesics, and numerical implementation for the three accretion models are only summarized. Full specification (including any analytic expressions for the emissivity and convergence/error estimates) is required to substantiate the reported brightness suppression and the statement that direct emission dominates in the thin-disk case.

    Authors: We agree that the current Methods section summarizes rather than fully specifies the numerical procedures. In the revised manuscript we will expand this section to provide the explicit analytic forms of the emissivity profiles for the thin-disk, static spherical, and infalling spherical models, the precise line-integral expression for specific intensity along geodesics, and a description of the ray-tracing code including step-size control, convergence tests, and estimated numerical errors. These additions will directly support the reported brightness suppression and the dominance of direct emission in the thin-disk case. revision: yes

  2. Referee: Results on critical curve enlargement: the quoted 2%–30% range for the critical impact parameter is presented without an accompanying table or plot that maps the interval to specific values of the Hernquist scale radius a and central density; because the enlargement is a direct function of these parameters, the range should be derived explicitly from the modified metric function f(r) = 1 – 2M(r)/r.

    Authors: The referee correctly notes that the 2%–30% range is stated without an explicit parameter mapping. This range is obtained by inserting the Hernquist enclosed-mass function into the metric coefficient f(r) = 1 – 2M(r)/r and solving the null circular-orbit condition for representative galactic-center values of the scale radius a and central density. In the revision we will add both a table listing selected (a, density) pairs with the corresponding critical impact parameters and a figure showing the continuous dependence on these parameters, thereby making the derivation fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is direct metric calculation

full rationale

The paper's central geometric result follows from substituting the Hernquist enclosed-mass function directly into the Schwarzschild-like metric function and solving the standard null circular orbit condition; the quoted 2-30% enlargement of the critical curve is a parameter-dependent output of that substitution with M_DM and a as external inputs. Radiative signatures are explicitly conditioned on the three chosen accretion models rather than derived as predictions from fitted quantities. No self-citations, ansatzes, or uniqueness theorems are invoked as load-bearing steps in the provided derivation chain, and the analysis remains self-contained against the stated assumptions without reducing any claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the Hernquist density profile as the description of dark matter, the Schwarzschild metric modified by the halo mass distribution, and standard general-relativistic ray tracing for null geodesics. No new particles or forces are introduced.

free parameters (1)
  • Hernquist scale radius and central density
    These parameters control the DM halo mass distribution and are chosen or varied to produce the reported 2-30% range in critical curve size.
axioms (2)
  • domain assumption The spacetime is described by a Schwarzschild metric with an added Hernquist dark matter halo contribution to the gravitational potential.
    Invoked when setting up the metric for photon ray tracing in all three accretion scenarios.
  • domain assumption Accretion flows can be modeled as geometrically thin disk, static spherical, or infalling spherical without magnetic fields or outflows.
    Used to compute the observed intensity profiles and Doppler effects.

pith-pipeline@v0.9.0 · 5753 in / 1651 out tokens · 32737 ms · 2026-05-21T22:32:49.644447+00:00 · methodology

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