Black holes immersed in modified Chaplygin-like dark fluid and cloud of strings: geodesics, shadows, and images

Bum-Hoon Lee; Hao-Peng Yan; Xiang-Qian Li; Xiao-Jun Yue; Yoonbai Kim

REVIEW 3 major objections 6 minor 129 references

Black holes immersed in modified Chaplygin-like dark fluid and cloud of strings: geodesics, shadows, and images

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A static black hole surrounded by a cloud of strings and a modified Chaplygin-like dark fluid reproduces the observed shadow radii of Sgr A* and M87* while predicting that the photon ring in thin-disk images is negligible.

desk verdict Solid phenomenology, but the EHT constraints rest on a unit conversion error — Tables II and III need to be redone or dropped. read the letter →

arxiv 2505.12077 v1 pith:GRIR52JY submitted 2025-05-17 gr-qc

Xiang-Qian Li , Yoonbai Kim , Bum-Hoon Lee , Hao-Peng Yan , Xiao-Jun Yue This is my paper · ORCID

classification gr-qc

keywords blackholeshadowmodifiedChaplygingascloudofstringsgeodesicstructureaccretiondiskimagephotonringISCO/OSCOcosmologicalconstantconstraint

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 builds a static, spherically symmetric black hole whose spacetime is sourced by a cloud of strings and a modified Chaplygin-like dark fluid with equation of state $p=A\rho-B/\rho^{\beta}$. It argues that this one metric controls the full observational chain: the radii of stable circular orbits, the photon sphere, the shadow seen by a distant observer, and the image of a thin accretion disk. The central quantitative claim is that identifying the fluid's asymptotic contribution with the cosmological constant makes the predicted shadow radius compatible with the measured shadows of Sgr A* and M87*, thereby placing bounds on the fluid and string parameters. A further claim is that in the computed images the direct disk emission dominates, the lensing ring is weak, and the photon ring is effectively invisible. A sympathetic reader would take the paper as showing that a single Chaplygin-like dark-fluid environment can connect black hole observables to the cosmic expansion.

What carries the argument

The load-bearing object is the combined lapse function of Eq. (9), a linear superposition of the string-cloud term $1-a-2M/r$ and the modified Chaplygin-like dark fluid term with Gauss hypergeometric function $F(r)={}_2F_1(\ldots)$. The two derived tools that carry the argument are the effective potential $V_{\rm eff}(r)=(\delta+L^2/r^2)f(r)$, whose extrema locate the photon sphere, ISCO, and OSCO, and the transfer functions $r_m(b)$ from the ray classification, whose slope $dr_m/db$ is the demagnification factor that suppresses the lensing and photon rings. The shadow formula $r_{\rm sh}=r_{\rm ph}\sqrt{f(r_O)/f(r_{\rm ph})}$ is the link between the metric and the observed shadow radii.

What would settle it

A next-generation horizon-scale image that resolves a photon ring as bright as the lensing ring would falsify the claim that the photon ring is negligible; alternatively, an independent cosmological measurement showing that $(B/(1+A))^{1/(1+\beta)}$ differs from the measured $\Lambda$ would invalidate the parameter constraints derived from Eq. (39).

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Extended reading notes

Core claim

The central claim is that the metric function $f(r)=1-a-2M/r-(r^2/3)(B/(1+A))^{1/(1+\beta)}F(r)$ with $F(r)$ a Gauss hypergeometric function describes a two-horizon, asymptotically de Sitter black hole. In this spacetime the timelike geodesics can have a band of stable circular orbits bounded by an innermost and an outermost stable circular orbit; the size of this band is controlled by the dark-fluid parameters $B$, $\beta$, $a$ and, non-monotonically, by $A$. For light, the photon sphere is unstable and the shadow radius for a static observer is $r_{\rm sh}=r_{\rm ph}\sqrt{f(r_O)/f(r_{\rm ph})}$. By equating the asymptotic fluid density with the cosmological constant, the predicted shadow radii fall inside the measured $1\sigma$ and $2\sigma$ bounds for Sgr A* and M87* for the parameter ranges given in Tables II and III. In thin-disk images the direct emission dominates the brightness, the lensing ring contributes little, and the photon ring is extremely demagnified; when stable circular orbits exist, their outer boundary can appear as an outer edge in the direct and lensing-ring images.

Load-bearing premise

The parameter constraints rest on equating $(B/(1+A))^{1/(1+\beta)}$ with the observed cosmological constant $\Lambda$, so the dark fluid around the black hole is assumed to be the same medium that drives cosmic acceleration and to be static with no expansion effect on light; if that identification fails, the bounds in Tables II and III do not follow, though the geodesic and image calculations would survive.

Editorial extensions

If this is right

The shadow data for Sgr A* and M87* narrow the allowed values of the dark-fluid and string-cloud parameters, with the Kottler black hole recovered as the $a=0$ limiting case.
When the parameters allow stable circular orbits, the images of a thin disk acquire outer edges set by the OSCO, so the observed brightness profile can carry a direct geometric signature of the dark fluid.
The photon ring should be effectively invisible in these images, so a clean, bright ring around the shadow is not expected for this accretion geometry.
Larger string-cloud parameter $a$ shrinks the outer communication region, moves all image rings outward, and lowers their peak brightness.
The MCDF intensity parameter $Q$ barely changes the shadow radius across the studied range, so shadow observations alone constrain $Q$ only weakly.

Reading between the lines

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

If the same fluid parameters were tested against independent cosmological probes such as supernova distances or cosmic microwave background angular scales, a mismatch with the value of $\Lambda$ assumed in Eq. (39) would sever the link between these local shadow bounds and cosmic acceleration while leaving the geodesic and image computations intact.
The static-observer approximation is the paper's own stated limitation; a fully time-dependent cosmological background would likely introduce redshift-dependent corrections to the shadow radius and ring brightness that could shift or widen the allowed parameter bands.
Applying multi-peaked or turbulent disk emission profiles instead of the single-peaked profiles used here would test whether the direct-emission dominance and photon-ring suppression persist for more realistic accretion flows.
A rotating generalization of this spacetime would produce an asymmetric shadow and a brighter photon ring on one side; horizon-scale movies of M87* could then distinguish the rotating case from the static model presented here.

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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

Two things to know about arXiv:2505.12077. First, it is a competent, thorough application of the standard geodesic/shadow/image pipeline to a static spherically symmetric black hole built from a modified Chaplygin fluid plus Letelier's cloud of strings. The geodesic analysis (ISCO/OSCO structure, epicyclic frequencies, photon sphere) and the thin-disk images are internally consistent and clearly presented. Second, the paper's central observational claim — that EHT shadow radii constrain the model parameters — does not survive inspection, because Eq. (39) is implemented with a unit mismatch.

The problem is concrete. The authors set M = 1 in their numerical code (as the figure captions and Table I imply), so all lengths are in units of the gravitational radius. In those code units, the physical cosmological constant Λ = 1.47×10^-52 m^-2 becomes Λ M² ≈ 5.5×10^-33 for Sgr A*. Eq. (39) therefore requires B_code = (1+A)(Λ M²)^{1+β} ≈ 10^-49 for β=0.5, not Log10B ≈ -77.65 as plotted. Alternatively, reading the plotted B as a code value gives an effective asymptotic MCDF term ~10^-71 m^-2 in physical units, about 19 orders below the observed Λ. Either way, Tables II and III do not follow from the EHT data. The same issue affects the M87* analysis. This is not a minor detail; it is the load-bearing part of the paper's claim to have constrained anything observationally.

What survives is the phenomenology. The effective-potential and circular-orbit analysis is careful, and the finding that OSCOs can imprint outer edges on direct and lensed images is a genuine observation, even if the features are similar to those in the authors' earlier CDF work [101]. The image classification by impact parameter follows Gralla–Lupsasca–Wald correctly, and the figures are informative.

Weaker points beyond the unit error: the spacetime metric is imported from prior papers without re-derivation; the constraint analysis fixes Q = 1; and no code or data is released. The authors' own Section V properly flags the deeper question of whether a static spherical MCDF can be identified with the cosmic dark energy, so the on-the-books math is not being hidden. The unit error is the one thing that turns a weak-but-legitimate constraint into an invalid one.

Who gets value: someone working on black hole shadow phenomenology in modified gravity or dark fluid backgrounds will find the geodesic and imaging sections useful as a template, but should not cite Tables II and III. I would not cite this paper in the next year.

Recommendation: send it to a referee. The calculations deserve an airing, but the referee must require a corrected Eq. (39) with explicit unit bookkeeping. If the constraints cannot be redone consistently, the authors should remove them and present the paper as pure phenomenology.

Referee Report

3 major / 6 minor

Summary. The manuscript studies a static, spherically symmetric black hole in Einstein gravity surrounded by a cloud of strings and a modified Chaplygin-like dark fluid (MCDF) with equation of state p = Aρ − B/ρ^β. Using the metric imported from earlier work (Eqs. (6)–(9)), it computes timelike and null geodesics, identifies ISCO/OSCO regions and the photon sphere, derives the shadow radius via Eq. (38), and compares it with EHT shadow-radius bounds for Sgr A* and M87* after identifying the asymptotic MCDF term with the cosmological constant through Eq. (39). It then computes thin-disk images with three GLM emission profiles and classifies the contributions of direct emission, the lensing ring, and the photon ring. The central claims are that the MCDF and string-cloud parameters control stable circular orbit structure and that EHT observations constrain those parameters, with direct emission dominating the image.

Significance. If the results are correct, the paper offers a useful phenomenological extension of black-hole shadow and thin-disk image studies to a Chaplygin-like dark fluid with a string cloud. The geodesic analysis, the ISCO/OSCO classification, the parameter-space maps (Figs. 2–7), and the transfer-function framework (Figs. 12–17) are carefully carried out and are of interest to the shadow-imaging community. The paper does not provide code or machine-checked derivations, but the numerical tables are extensive and the forward ray-tracing calculation is standard. The EHT-based parameter constraints are, however, conditional on the model assumption embodied in Eq. (39), and the unit-handling issue discussed below currently undermines the published bounds.

major comments (3)

[III.C, Eq. (39), Figs. 10–11, Tables II–III] The numerical implementation of Eq. (39) is dimensionally inconsistent as reported. In Eq. (39), Λ = 1.47×10^-52 m^-2 is an SI quantity, and [B] = L^{-2(1+β)}. For β = 0.5 and A ≈ 0.245, Eq. (39) gives B ≈ 2.2×10^-78 m^-3, i.e., log10(B/m^-3) ≈ -77.65, which is the value quoted in Table II. But in the M = 1 code, the radial coordinate is dimensionless, so the metric (9) requires the dimensionless code value B_code = B_phys M^{2(1+β)}. For Sgr A* (M ≈ 6.1×10^9 m) this gives B_code ≈ 5×10^-49, not 10^-77. If the code instead uses B = 10^-77 directly as a dimensionless parameter, then the asymptotic coefficient in Eq. (12) is (10^-77/(1+A))^{2/3} M^{-2} ≈ 10^-71 m^-2, about 19 orders of magnitude below the observed Λ. In that case Eq. (39) is not satisfied in physical units and the EHT bounds in Tables II and III do not follow. The manuscript does not document which convention is used. Please state explicitly how B and rO are rescaled to M = 1 units, and rerun the constraints with the correct conversion.
[III.C and Section V] The claim that EHT shadow radii constrain the MCDF parameters depends on two unquantified modeling assumptions: the identification in Eq. (39) of the static black-hole fluid with the cosmic dark fluid, and the static-observer formula (38). Section V correctly acknowledges that the relation between the FLRW dark-energy equation of state and its black-hole analog is unclear, yet the abstract and conclusions present the bounds as constraints on MCDF parameters without this caveat. Please move this limitation into the abstract/conclusions and quantify the static-observer error for the stated observer distances rO = 2.55×10^20 m and 5.06×10^23 m, or justify that it is negligible for the reported bounds.
[II, Eq. (9)] The metric is imported from Refs. [97]–[99] without a derivation. Since the paper's shadow and image calculations are forward computations from this metric, the central results would be more self-contained if the field equations and the assumptions behind the linear superposition of the cloud-of-strings and MCDF energy-momentum tensors were stated explicitly. At minimum, the authors should verify that the summed energy-momentum tensor satisfies the Einstein equations for the summed lapse function (9).

minor comments (6)

[III.C] The phrase 'electric charge Q' on page 11 is inaccurate: Q in Eq. (6) is the MCDF normalization/intensity parameter, not an electric charge.
[IV.C and Eq. (46)] The GLM emission profile in Eq. (46) uses β as a width parameter, which conflicts with the MCDF equation-of-state exponent β used throughout the paper. This creates ambiguity in Cases I–III; please rename one of them.
[V] In the conclusions, 'extreme demagnetization effects' should read 'extreme demagnification effects'.
[Table I and Section IV.A] The quantities b^±_1, b^±_2, b^±_3 in Table I are used before they are defined in Section IV.A. Please define them in the table caption or move the definition earlier.
[Figs. 10–11 and Eq. (38)] The figure captions give rO in light-years or Mpc, while Eq. (38) requires rO and rph in the same length units. Please state the conversion used for the M = 1 computations; this is related to Major Comment 1.
[References] Reference [70] contains a typographical double comma, and Refs. [104]–[106] are arXiv preprints without publication status; please update or mark them accordingly.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the shadow and image results are forward computations from the assumed metric Eq. (9), and the EHT constraints are external data; the self-citations are not load-bearing.

full rationale

The derivation chain starts from the assumed MCDF equation of state p = A rho - B / rho^beta and the string-cloud EMT (Eq. (2)), solves Einstein's equations to obtain Eq. (9), and then forward-computes geodesics, photon-sphere radii, shadow radii via Eq. (38), and disk images via Eq. (45). The EHT shadow-radius bounds in Eqs. (40)-(41) are external observational inputs used to constrain parameters, not quantities that are re-injected as predictions. Eq. (39) is an explicit modeling identification, not a hidden fit: the paper states it as a prerequisite, and A is then derived from B and beta through that relation; the resulting bounds in Tables II-III are consistency statements obtained by comparing the computed rsh with the external bounds. The cited prior work [99] for the MCDF energy-momentum tensor and solution is by an overlapping author, but the metric is restated in Eqs. (6)-(9) and the geodesic/shadow/image analysis is carried out in this paper, so the self-citation is not load-bearing. Section V explicitly flags the static-spacetime approximation and the unclear FLRW-to-black-hole equation-of-state relation as limitations, which is consistent with a non-circular, assumption-based analysis. No step reduces to its inputs by construction; any unit-handling concern would be a correctness issue rather than circularity.

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

The central claims rest on the imported metric Eq. (9), the linear superposition assumption, the cosmological-constant identification Eq. (39), and the static observer approximation. These are stated in the paper, but only Eq. (39) and the static approximation are acknowledged as limitations. The free parameters (A, B, beta, Q, a, and the GLM profile parameters) are either from the dark fluid model or chosen by hand; none are fit in a least-squares sense, but the shadow constraints select allowed ranges.

free parameters (6)

B = Log10 B in [-77.663, -77.601] for Sgr A* with a=0.08 at 2 sigma; lower bounds near -77.654 for a=0
Positive EoS parameter; constrained via shadow radius under Eq. (39).
A = A in [0.218, 0.406] for Sgr A* with a=0.08 at 2 sigma; lower bound near 0.245 for a=0
Positive EoS parameter; in the constraint section derived from B and beta via Eq. (39), not fitted independently.
beta = beta in [0.4872, 0.4884] for Sgr A* with a=0.08 at 2 sigma
Free exponent in the EoS, constrained by shadow data.
Q = Q = 1.0 (chosen by hand)
Fixed to 1.0 in the constraints; the paper notes Q has negligible influence on shadow radius (Figs. 10 and 11 right panels), so it is effectively unconstrained.
a = Values 0 to 0.24 explored; allowed by shadow fit
Dimensionless string cloud parameter; constrained by shadow radius bounds for fixed Q and beta.
GLM disk profile parameters (gamma, alpha, beta_GLM) = Case I: gamma=-2, alpha=rISCO, beta_GLM=M/4; Case II: gamma=-2, alpha=rph, beta_GLM=M/8; Case III: gamma=-3, alpha=rh…
Chosen by hand to model three accretion scenarios; results for images depend on these arbitrary profile shapes.

assumptions (6)

domain assumption The spacetime is static, spherically symmetric, with metric ansatz Eq. (1).
The paper assumes no rotation, no time dependence, and a 4D Einstein gravity background.
domain assumption The MCDF energy-momentum tensor from Ref. [99] (Eqs. 4-5) is a valid source in Einstein gravity.
The metric Eq. (7) is imported from the authors' prior Lovelock-gravity solution; this paper does not re-derive it from the action.
domain assumption The cloud of strings and the MCDF do not interact, so their energy-momentum tensors superpose linearly.
Stated before Eq. (9); if there are interaction terms, the combined metric Eq. (9) is not a solution.
ad hoc to paper The MCDF that surrounds the black hole has the same parameters as the cosmic dark fluid, and its asymptotic density equals the cosmological constant, Eq. (39).
The identification (B/(1+A))^{1/(1+beta)} = Lambda, with Lambda = 1.47e-52 m^-2, is the basis for the parameter constraints in Section III.C. It connects a local black hole environment to cosmology without a derived mechanism.
domain assumption A static observer at the Earth's position measures the shadow radius via Eq. (38), neglecting cosmic expansion effects.
The authors use observer distances 26996 ly (Sgr A*) and 16.4 Mpc (M87*) and acknowledge in the conclusions that the dynamical expansion is not modeled.
domain assumption The GLM emission profile Eq. (46) with the three parameter cases represents plausible accretion disk emission.
The images in Section IV are computed for these assumed profiles; the observed intensity maps depend on these choices.

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Cite this review

Pith. "Pith review of Black holes immersed in modified Chaplygin-like dark fluid and cloud of strings: geodesics, shadows, and images." pith.science (2026). https://pith.science/paper/GRIR52JY

@misc{pith2026250512077,
  author       = {Pith},
  title        = {Pith review of: Black holes immersed in modified Chaplygin-like dark fluid and cloud of strings: geodesics, shadows, and images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRIR52JY}},
  note         = {Machine review of arXiv:2505.12077}
}

abstract

This study investigates a black hole surrounded by a cloud of strings and a cosmological dark fluid characterized by a modified Chaplygin-like equation of state (MCDF), $p=A\rho-B/\rho^{\beta}$. We analyze its geodesic structure, shadow, and optical appearance. Analysis of the effective potential and epicyclic frequencies reveals that the existence of innermost/outermost stable circular orbits (ISCOs/OSCOs) for timelike particles is controlled by the parameters of the MCDF and the cloud of strings. The behavior of orbital conserved quantities and the Keplerian frequency are also examined. By equating the influence of the MCDF on the spacetime metric at spatial infinity with that of a cosmological constant, we constrain the MCDF parameters using the observed shadow radii of Sgr A* and M87*. We investigate the effects of the cloud of strings and MCDF on the black hole's shadows and optical images, assuming various thin disk accretion profiles. Using the method developed by Wald and collaborators, light trajectories are classified by their impact parameters into direct emission, the lensing ring, and the photon ring. The presence of OSCOs can lead to the existence of outer edges in the direct emission and lensing ring images. Observed brightness primarily originates from direct emission, with a minor contribution from the lensing ring, while the photon ring's contribution is negligible due to extreme demagnification. The influence of the cloud of strings and MCDF parameters on all results is analyzed throughout the study.

Figures

Figures reproduced from arXiv: 2505.12077 by the authors.

**Figure 2.** FIG. 2. The profile of the effective potential curves of timelike particles for black holes without (left) and with (right) SCOs. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Parameter space of black holes without (red region) and with (blue region) SCOs. The gray region represents parameter [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: FIG. 4. The radial profiles of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. The radial profiles of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Radial dependence of Keplerian frequencies of test particles around a black hole for different values of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Radial (left) and vertical (right) epicyclic frequencies for black holes with ( [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. The profile of the effective potential (blue lines) for black holes without (left) and with (right) SCOs. The dashed lines [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Trajectories of light rays around the black hole for [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10. Shadow radius in [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11. Shadow radius in [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

**Figure 12.** Figure 12: FIG. 12. Behavior of photons as a function of impact parameter [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 13.** Figure 13: FIG. 13. Observational appearances of a geometrically and optically thin disk with different emissivity profiles near black hole [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

**Figure 14.** Figure 14: FIG. 14. Observational appearances of a geometrically and optically thin disk with different emissivity profiles near black hole [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

**Figure 15.** Figure 15: FIG. 15. Observational appearances of a geometrically and optically thin disk with different emissivity profiles near black hole [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

**Figure 16.** Figure 16: FIG. 16. Observational appearances of a geometrically and optically thin disk with different emissivity profiles near black hole [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]

**Figure 17.** Figure 17: FIG. 17. Observational appearances of a geometrically and optically thin disk with different emissivity profiles near black hole [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

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When 0 < L2 < L2 IS, the potential exhibits only one maximum, corresponding to a single unstable circular orbit (e.g., point I)
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When L2 = L2 IS, the particle can occupy both an unstable circular orbit (e.g., point J) and the ISCO (e.g., point K)
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When 0 < L2 < L2 IS, the potential exhibits only one maximum, corresponding to a single unstable circular orbit (e.g., point I)

[5] [2]

When L2 = L2 IS, the particle can occupy both an unstable circular orbit (e.g., point J) and the ISCO (e.g., point K)

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When L2 IS < L2 < L2 OS, the potential shows two maxima and one minimum, indicating one stable circular orbit (e.g., point M) and two unstable cir- cular orbits (e.g., points N and L)

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When L2 =L2 OS, the particle possesses an unstable circular orbit (e.g., point O) and the OSCO (e.g., point P)

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When L2 >L 2 OS, the potential once again features only one maximum, indicating a single unstable or- bit (e.g., point Q). For black holes concerned in this work, the existence of SCOs is primarily influenced by the parametersB,β, and a, whereas Q andA are more crucial for determining the presence of the event horizon and thus the existence of the black h...