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arxiv: 2605.19567 · v1 · pith:OB4QLO4Gnew · submitted 2026-05-19 · 🌀 gr-qc

Shadows and photon spheres of static black holes embedded in a Dehnen-(1,4,5/2)-type dark matter halo with a quintessential field

Hong-Er Gong , Junlin Qin , Yusen Wang , Bofeng Wu , Zhan-Feng Mai , Xiao Zhang , Enwei Liang This is my paper

Pith reviewed 2026-05-20 05:10 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole shadowdark matter haloquintessential fieldphoton sphereDehnen profileaccretion modelshadow intensity
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The pith

Dark matter and dark energy affect black hole shadow radius and intensity in distinct ways, with radius more sensitive to dark matter and intensity to dark energy.

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

This paper studies static black holes placed inside a Dehnen-type dark matter halo that also contains a quintessential field representing dark energy. It derives the photon effective potential, finds the photon sphere and critical impact parameter, and traces light rays for different values of the halo density, scale radius, field strength, and equation-of-state parameter. In both spherical and thin-disk accretion models the calculations show that dark energy changes the observed shadow intensity in a manner that depends on the observer's location, while dark matter produces a more uniform shift. The two components separate cleanly in their influence: intensity responds more strongly to the quintessential field, and radius responds more strongly to the dark matter halo.

Core claim

Embedding the Schwarzschild solution in a Dehnen-(1,4,5/2) dark matter halo plus quintessential field produces a metric whose photon effective potential yields larger event-horizon, photon-sphere, and critical-impact-parameter radii as the halo central density ρ_s, scale radius r_s, normalization c, and dark-energy state parameter w_q are increased. Photon trajectories confirm these shifts. In the derived redshift and intensity expressions for spherical and thin-disk accretion, dark energy modifies shadow intensity with a clear dependence on observer position while dark matter does not; intensity is more sensitive to dark energy and radius is more responsive to dark matter, supplying an in-σ

What carries the argument

The effective potential for null geodesics in the embedded metric, from which the unstable photon sphere radius and critical impact parameter that set the shadow edge are obtained.

If this is right

  • Event-horizon and photon-sphere radii increase with rising dark-matter halo parameters ρ_s and r_s.
  • The same radii increase with rising quintessential-field parameters c and w_q.
  • Shadow intensity varies strongly with observer position when the quintessential field is present but shows little such variation from the dark-matter halo alone.
  • The shadow radius changes more when dark-matter parameters are varied, while intensity changes more when dark-energy parameters are varied.

Where Pith is reading between the lines

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

  • Images from next-generation very-long-baseline interferometry could separate the local dark-matter and dark-energy contributions by comparing measured shadow radius against measured intensity.
  • The same distinction might appear in other accretion models or for slowly rotating black holes if the effective-potential analysis is repeated.
  • If the separation holds, it supplies a direct test of whether the dominant dark component near the horizon is matter-like or energy-like without requiring separate probes of each.

Load-bearing premise

The spacetime is constructed by directly embedding the Schwarzschild geometry into the Dehnen halo plus quintessential field without solving the Einstein equations that would include the back-reaction of those distributions on the metric.

What would settle it

High-resolution images that measure both the angular radius and the relative intensity of a black-hole shadow for known mass and distance, then vary the local dark-matter density or dark-energy parameters, would falsify the claimed separation if the intensity and radius do not respond with the predicted distinct sensitivities.

read the original abstract

This paper investigates the appearance characteristics of static black holes embedded in Dehnen-(1,4,5/2)-type dark matter halos with a quintessential field, focusing on how the dark matter halo and dark energy affect the black hole images. We first derive the event horizon radius and the photon effective potential of the black hole, and then calculate critical quantities such as the critical photon sphere radius and critical impact parameter under different parameter sets. Trajectories of photons are subsequently plotted. The study reveals that as the parameters of the dark matter halo (the central density of the dark matter halo $\rho_s$ and the scale radius of the central halo $r_s$) and the quintessential field (the normalization factor $c$ and the equation of state parameter of dark energy $w_q$) increase, the aforementioned physical quantities generally exhibit an increasing trend. Based on the derived general expressions for the redshift factor and integrated intensity, we further explore the optical effects of the spherical accretion and the thin-disk accretion models. The results indicate that dark energy exerts an influence on the black hole shadow that is strongly dependent on the observer's position, whereas the influence exerted by dark matter exhibits no such conspicuous dependence. Furthermore, dark matter and dark energy have distinct effects on both the intensity and the radius of the black hole shadow. In particular, the intensity exhibits a greater sensitivity to dark energy, whereas the radius is more responsive to dark matter. This distinction offers a potential observational criterion for identifying, through black hole images, whether the dominant interacting component near the black hole is dark matter or dark energy, and provides an important basis for constraining the equation-of-state parameter $w_q$.

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

1 major / 2 minor

Summary. The manuscript studies shadows and photon spheres of static black holes embedded in a Dehnen-(1,4,5/2)-type dark matter halo with a quintessential field. It derives the event horizon radius and photon effective potential from an assumed static spherical metric, computes the critical photon sphere radius and impact parameter as functions of the parameters ρ_s, r_s, c, and w_q, plots photon trajectories, and analyzes integrated intensity for spherical accretion and thin-disk models. The central claim is that dark matter and dark energy produce distinct effects on shadow radius and intensity, with radius more responsive to dark matter and intensity more sensitive to dark energy, providing a potential observational discriminator.

Significance. If the metric is a consistent solution to the Einstein equations sourced by the stated Dehnen halo and quintessential stress-energy tensors, the parameter scans and accretion-model results could supply a concrete framework for interpreting Event Horizon Telescope-style images in mixed dark-matter/dark-energy environments. The explicit comparison of DM versus DE sensitivities on radius versus intensity is a falsifiable prediction that could be tested against future observations, and the inclusion of both spherical and thin-disk accretion adds breadth.

major comments (1)
  1. [Metric and effective-potential section (preceding the photon-sphere calculations)] The metric is constructed by direct superposition of the Schwarzschild term with additive contributions from the Dehnen-(1,4,5/2) density profile and the quintessential field (normalization c, equation-of-state w_q) without solving the Einstein equations G_μν = 8π(T_DM + T_q). This construction appears in the section introducing the line element and effective potential. Because the photon-sphere condition, critical impact parameter, and integrated intensity all depend on the metric functions f(r) and the resulting null geodesics, any mismatch between the implied stress-energy tensor and the intended sources directly undermines the reported trends and the claimed observational criterion for distinguishing DM from DE.
minor comments (2)
  1. [Results on critical quantities] The statement that quantities 'generally exhibit an increasing trend' with rising ρ_s, r_s, c, and w_q should be accompanied by the explicit ranges scanned and any non-monotonic exceptions, preferably in a table or supplementary plot.
  2. [Figures] Figure captions for photon trajectories and intensity maps should list the precise parameter values (ρ_s, r_s, c, w_q) used for each curve to facilitate reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address the major concern regarding the metric construction in our point-by-point response below.

read point-by-point responses

  1. Referee: [Metric and effective-potential section (preceding the photon-sphere calculations)] The metric is constructed by direct superposition of the Schwarzschild term with additive contributions from the Dehnen-(1,4,5/2) density profile and the quintessential field (normalization c, equation-of-state w_q) without solving the Einstein equations G_μν = 8π(T_DM + T_q). This construction appears in the section introducing the line element and effective potential. Because the photon-sphere condition, critical impact parameter, and integrated intensity all depend on the metric functions f(r) and the resulting null geodesics, any mismatch between the implied stress-energy tensor and the intended sources directly undermines the reported trends and the claimed observational criterion for distinguishing DM from DE.

    Authors: We thank the referee for highlighting this important point. The metric in our paper is indeed introduced by combining the Schwarzschild term with contributions from the Dehnen dark matter halo and the quintessential field in the manner described. This is a phenomenological construction commonly used to study the effects of dark matter and dark energy on black hole observables. To strengthen the manuscript and address the concern about consistency with the Einstein equations, we will revise the relevant section to explicitly derive the metric function from the Einstein field equations sourced by the total stress-energy tensor of the dark matter halo plus quintessential field. We will also verify that the effective potential and photon sphere calculations remain valid under this consistent metric. We expect this to confirm the reported trends while providing a firmer theoretical basis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct parameter scan in an assumed metric

full rationale

The paper constructs the spacetime metric by direct embedding of the Schwarzschild term plus additive contributions from the Dehnen-(1,4,5/2) density profile and quintessential field (parameters ρ_s, r_s, c, w_q). It then computes the event horizon, effective potential, photon-sphere radius, critical impact parameter, photon trajectories, redshift factor, and integrated intensity for spherical and thin-disk accretion. All reported trends (increasing quantities with parameters) and the claimed distinction (intensity more sensitive to dark energy, radius more responsive to dark matter) are numerical outputs obtained by varying those free inputs. No derived quantity reduces to an input by construction, no self-citation is load-bearing for a uniqueness claim, and no ansatz is smuggled via prior work. The chain is self-contained as an exploration of the model's observables.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central results rest on four free parameters that are varied by hand and on the assumption that the metric can be written in a simple embedded form without solving the coupled Einstein-matter equations.

free parameters (4)
  • ρ_s
    Central density of the Dehnen dark matter halo, varied to study trends.
  • r_s
    Scale radius of the dark matter halo, varied to study trends.
  • c
    Normalization factor of the quintessential field, varied to study trends.
  • w_q
    Equation-of-state parameter of the quintessential dark energy, varied to study trends.
axioms (2)
  • domain assumption The spacetime is static and spherically symmetric and can be written as a direct embedding of the Schwarzschild solution into the combined dark matter plus quintessential background.
    This metric ansatz is used to derive the effective potential and photon sphere without back-reaction.
  • standard math Null geodesics in the equatorial plane determine the photon sphere and shadow boundary.
    Standard assumption in black hole shadow calculations.

pith-pipeline@v0.9.0 · 5870 in / 1599 out tokens · 36582 ms · 2026-05-20T05:10:59.943359+00:00 · methodology

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