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This paper shows that the shadow sizes of M87* and Sgr A* constrain the dark-matter parameter of a Bardeen black hole to a narrow range, predicting ~0.27–2.67 g/cm³ just outside Sgr A*'s shadow.

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2026-08-03 19:21 UTC pith:LD3MBULU

load-bearing objection Standard shadow+disk computation for the PFDM-Bardeen metric, but the abstract contradicts the body on blueshift and the promised DM-profile comparison is absent; the density claim also misses an 8π normalization. the 5 major comments →

arxiv 2512.00824 v2 pith:LD3MBULU submitted 2025-11-30 astro-ph.HE gr-qc

Images of shadow and thin accretion disk around Bardeen black hole surrounded by perfect fluid dark matter

Haiyuan Feng , Ziqiang Cai , Hao-Peng Yan , Rong-Jia Yang , Jinjun Zhang This is my paper
classification astro-ph.HE gr-qc
keywords black hole shadowBardeen black holeperfect fluid dark matterEvent Horizon TelescopeM87*Sgr A*accretion disk imagingdark matter density
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to show that the shadow diameters of M87* and Sgr A*, as measured by the Event Horizon Telescope, can be used to fix the strength of perfect fluid dark matter around a Bardeen black hole. Matching the photon-sphere impact parameter to the observed shadow sizes forces the dark-matter parameter b/M into a narrow window, about 10^-1–10^-2 for M87* and 10^-2–10^-3 for Sgr A*, and implies a local dark-matter density near Sgr A*'s shadow of roughly 0.27–2.67 g/cm³, falling to 10^-24–10^-25 g/cm³ at 100 pc. The paper further argues that dark matter, rather than the magnetic charge responsible for the Bardeen regularity, dominates the geometry and the disk image: increasing b enlarges the shadow and suppresses the disk brightness, whereas g produces negligible changes. Comparing four dark-matter halo profiles (PFDM, NFW, Dehnen-type, Moore), it finds distinct densities at the shadow scale and at 100 pc, and reports redshift patterns that could discriminate the models. If the argument holds, black-hole shadow observations become a direct probe of dark-matter density at horizon scales.

Core claim

On the paper's own terms, the central discovery is a constraint chain: for a static, spherically symmetric Bardeen black hole immersed in perfect fluid dark matter, the shadow radius equals the critical impact parameter b̄_c = r_ph/√f(r_ph) determined by the photon-sphere radius r_ph, with f(r) = 1 − 2Mr²/(r²+g²)^{3/2} − (b/r) ln(r/|b|). Comparing this to the EHT-inferred shadow diameters d_sh = (11 ± 1.5)M for M87* and (9.77 ± 0.67)M for Sgr A* confines b/M to O(10^-1–10^-2) for M87* and O(10^-2–10^-3) for Sgr A*. From ρ = b/r³, this implies a PFDM density of 0.27–2.67 g/cm³ at the Sgr A* shadow scale (R_sh ~ 5M), dropping to 10^-24–10^-25 g/cm³ at 100 pc. The same analysis shows that incre

What carries the argument

The load-bearing object is the critical impact parameter b̄_c = r_ph/√f(r_ph), which maps the photon-sphere radius to the apparent shadow diameter and is matched against EHT measurements. It is evaluated in the PFDM-Bardeen metric f(r) = 1 − 2Mr²/(r²+g²)^{3/2} − (b/r) ln(r/|b|), whose dark-matter content is encoded in the perfect-fluid stress-energy tensor with density profile ρ = −p_r = b/r³ and tangential pressures p_θ = p_φ = b/(2r³). The imaging side uses transfer functions that sum the observed intensity from successive disk crossings, together with the Novikov–Thorne flux formula and the redshift factor 1+z = (1+Ω b̄ sinθ cosα)/√(−g_tt − Ω²g_φφ), to produce primary and secondary images

Load-bearing premise

The whole argument depends on the assumption that the spacetime around M87* and Sgr A* is described by the Bardeen metric with the perfect-fluid dark-matter profile ρ = b/r³; if that profile is wrong, the shadow-derived bounds on b and the density prediction do not follow.

What would settle it

Take a sub-percent measurement of Sgr A*'s shadow diameter (for example, with space-based very-long-baseline interferometry): the PFDM-Bardeen model with b/M in the derived 10^-2–10^-3 range predicts a specific diameter interval, so a value outside it would refute the metric. Alternatively, a precision pulsar-orbit measurement of the local density near Sgr A* would independently test the 0.27–2.67 g/cm³ prediction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the metric is right, dark matter at the Sgr A* shadow scale has density ~0.27–2.67 g/cm³, orders of magnitude above typical galactic-halo values, so horizon-scale dark matter is dynamically significant.
  • Shadow diameter and disk brightness together pin b/M; sharper future shadow measurements will shrink the allowed interval and sharpen the density prediction.
  • Because the magnetic charge g is masked by PFDM, current shadow and disk images cannot distinguish the Bardeen regular black hole from a PFDM-Schwarzschild black hole of the same b.
  • Different DM halo models (PFDM, NFW, Dehnen-type, Moore) predict distinct densities at the shadow radius and at 100 pc, making two-scale density measurements a potential model discriminator.
  • The redshift analysis gives a qualitative test: the models predict a specific redshift/blueshift pattern with inclination, so observing significant blueshifted emission at low inclination would challenge all four DM models.

Where Pith is reading between the lines

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

  • The density prediction is independently testable: a pulsar in a tight orbit around Sgr A* whose periastron precession fixes the local density would either confirm or exclude the 0.27–2.67 g/cm³ value, independent of shadow fitting.
  • The analysis assumes a non-rotating metric and a distant observer; a rotating PFDM-Bardeen spacetime would shift the shadow diameter by a few percent, comparable to the 1σ bands, so the quoted b/M bounds may need revision once spin is included.
  • The same shadow-matching pipeline could be applied to the central black holes in other galaxies with future space-VLBI measurements, turning any single shadow diameter into a local dark-matter density estimate.
  • A cleaner falsifier than the shadow diameter itself is the predicted relation between b and the 100-pc density: measuring the large-scale density independently and checking consistency with the shadow-derived b would test the assumed b/r³ profile.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper studies the shadow and thin-disk images of a static, spherically symmetric Bardeen black hole immersed in perfect fluid dark matter (PFDM), with metric given by Eq. (6). The authors derive null geodesics, classify photon trajectories into direct, lensing, and photon-ring families, compute transfer functions and images for three emission models, and use EHT shadow-diameter measurements of M87* and Sgr A* to constrain the PFDM parameter b. The abstract further claims a PFDM density near the shadow of about 0.27–2.67 g/cm^3 for Sgr A* and a comparison with NFW, Dehnen-type, and Moore DM profiles. The numerical machinery is standard, but the manuscript contains several internal inconsistencies and missing derivations that affect the stated claims.

Significance. If fully supported, the paper would provide a useful EHT-based bound on a phenomenological PFDM parameter and demonstrate that the magnetic charge of the Bardeen model is subdominant for shadow and disk observables. The use of transfer functions, photon-ring classification, and redshift maps follows established methods. However, the headline density prediction is not derived in the body, the normalization of the PFDM stress-energy tensor is inconsistent with the metric, the promised comparison with NFW/Dehnen/Moore profiles is absent, and the blueshift statements in the different abstracts and in the body contradict each other. These issues must be resolved before the scientific claims can be assessed.

major comments (5)
  1. [Sec. II, Eqs. (4) and (6)] The PFDM stress-energy tensor in Eq. (4) is not consistent with the metric in Eq. (6) under standard G=c=1 units. For the metric f=1-2Mr^2/(r^2+g^2)^{3/2}-(b/r)ln(r/|b|), the Einstein equation G^t_t=8πT^t_t with T^t_t=-ρ gives a PFDM contribution to G^t_t of -b/r^3, so ρ should be b/(8π r^3), not b/r^3. Equivalently, the paper must explicitly state that it uses 8π=1 throughout. This is not a bookkeeping detail: the abstract's density range 0.27–2.67 g/cm^3 is computed from ρ=b/r^3 and is too large by a factor 8π. The shadow fit itself is unaffected because b/M is dimensionless, but the density claim is wrong as written. Additionally, the numerical density values are never derived in the body of the paper.
  2. [Abstract vs. Sec. V and Figs. 14–15] The abstract supplied with the paper states that 'blueshift appears in the primary image as inclination increases,' while the full-text abstract and Sec. V state that 'No blueshifted regions appear in any configuration' and 'all images show exclusively redshifted emission.' Figures 14 and 15 show colorbars labeled z with values starting at 0.75; if z denotes the redshift, values below 1 indicate blueshift. The paper must clarify whether the colorbar is z or 1+z and must reconcile the contradictory statements in the abstract and the body.
  3. [Abstract and Secs. II–VI] The abstract promises a comparison with NFW, Dehnen-type, and Moore dark-matter profiles and claims distinct densities at the shadow radius and at 100 pc. It also advertises a concrete PFDM density prediction near the shadow scale. None of these calculations appears in Sections II–VI. The body only analyzes Bardeen, PFDM-Schwarzschild, and PFDM-Bardeen metrics. Missing promised content is not a minor presentation issue when the abstract's central 'prediction' and 'distinguishing signature' rely on it.
  4. [Sec. III.B and Eq. (4)] The 'prediction' of the PFDM density is circular in an important sense: Eq. (4) defines ρ=b/r^3, and b is fitted from the shadow diameter. Substituting the fitted b into the assumed density profile gives a rearrangement of the fit, not an independent prediction. The statement that the four DM models exhibit distinct densities at the shadow radius is a comparison of model assumptions, not a falsifiable test, unless the PFDM density profile itself is constrained by independent data. The paper should be reframed accordingly.
  5. [Sec. III.B and Table I] The constraint on Sgr A* is essentially one-sided. For b≥0, the PFDM term in Eq. (6) makes the shadow larger than the Schwarzschild value of about 10.39M, whereas the central EHT value for Sgr A* is 9.77M. Table I confirms that only upper bounds are given for Sgr A* at 1σ; the allowed interval includes b=0. The abstract's wording that b is restricted to a 'narrow range' O(10^-2–10^-3) is therefore an overstatement. The paper should state explicitly that the Sgr A* data provide an upper limit rather than a two-sided constraint.
minor comments (5)
  1. [Notation] The symbol b is used both for the PFDM parameter and for the impact parameter in equations such as (31). This creates confusion, especially in Section V where the redshift factor is introduced.
  2. [Refs.] Reference [52] is incomplete ('6 2025'), and several references have formatting inconsistencies. Please check the bibliography.
  3. [Figs. 4, 12, 13] Several figures lack axis labels or have labels such as 'Y’ X’' that are not defined. The contour plots in Fig. 4 would benefit from explicit axis captions.
  4. [Eq. (17)] The definition n(b̄)=(2m-1)/4 for m∈Z^+ is not directly connected to the intervals 1/4<n<3/4, 3/4<n<5/4, etc. Please clarify the indexing or use a more transparent parametrization.
  5. [Sec. IV] The text says the observer is placed along the polar axis (face-on view), but later figures and the redshift analysis consider inclinations such as 40° and 80°. The setup should be stated more carefully.

Circularity Check

1 steps flagged

EHT shadow fit is a legitimate external constraint, but the headline PFDM-density 'prediction' is the fitted b converted via the assumed profile ρ=b/r^3, so the prediction reduces by construction.

specific steps
  1. fitted input called prediction [Abstract (density claim); Sec. II Eq. (4); Sec. III Eqs. (12)-(14)]
    "Abstract: 'From these constraints we derive a rough prediction for the PFDM density near the shadow scale (R_sh∼5M): ρ_PFDM ∼ 0.27-2.67 g/cm^3 for Sgr A*.' Sec. II Eq. (4): 'ρ = −p_r = b/r^3, pθ = pφ = b/2r^3.' Sec. III: '¯bc = rph/√f(rph)' (Eq. 12), fitted to 'dM87∗sh = (11 ± 1.5)M, dSgr.A∗sh = (9.77 ± 0.67)M' (Eq. 14)."

    The EHT shadow diameter fixes the single free parameter b through b̄c = r_ph/√f(r_ph). Eq. (4) then defines the PFDM density as ρ=b/r^3, so the quoted density range at R_sh≈5M is exactly the fitted b rewritten in density units with the assumed r^-3 profile. No independent observable is introduced after the fit; the 'prediction' is statistically forced by the shadow constraint and would change only if the assumed profile (not fit to any density data) were changed. Thus the headline claim reduces to a refit of the same parameter.

full rationale

The core shadow computation is not circular: Eq. (12) is a genuine geometric relation, and the comparison with EHT diameters in Eq. (14) uses external observational data to constrain b. The image/disk parts (Secs. IV-V) are forward-model consequences of the assumed metric and standard ray-tracing, not circular. Circularity is localized to the abstract's density 'prediction': since the model postulates ρ=b/r^3 (Eq. 4), the fitted b determines ρ completely, so the density is the fit relabeled. The metric and PFDM source are adopted from Refs. [103,104] rather than derived; that is a model-dependence/correctness issue, not circularity. Self-citations (e.g., Refs. [18,19,52,119,120]) are used only for textbook Novikov-Thorne formulas and are not load-bearing. As a separate correctness caveat, Eqs. (4) and (6) are mutually consistent only if one uses the unstated 8π=1 normalization; in standard G=c=1 units the density would be b/(8πr^3), rescaling the quoted g/cm^3 range. That caveat affects the numerical density claim but is not itself a circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim rests on the assumed PFDM-Bardeen metric and stress-energy tensor, which are imported from prior work. The only fitted parameter is b; the density prediction is a direct transformation of this fit. No new entities are introduced.

free parameters (2)
  • PFDM parameter b = b/M ~ 0.12-0.33 (M87*, 1-2σ), ~0.0015-0.092 (Sgr A*)
    Appears in metric (Eq. 6) and stress-energy (Eq. 4); constrained by matching shadow diameter to EHT data in Section III.B and Table I.
  • Magnetic charge g
    Model parameter; the paper scans g/M = 0, 0.1, 0.15, 0.3, 0.6, 0.7, 0.8 to study its effect, but it is not fitted; the derived b range depends on the assumed g.
axioms (5)
  • domain assumption The PFDM-Bardeen metric (Eq. 6) and the PFDM stress-energy tensor (Eqs. 3-4) correctly describe a Bardeen BH embedded in PFDM.
    Taken from refs [103,104]; the paper does not derive this from an action or independent microphysics. All subsequent shadow and disk calculations rest on this metric.
  • domain assumption Weak energy condition b ≥ 0
    Stated after Eq. (4) to justify the sign of b.
  • domain assumption EHT shadow diameter corresponds to the photon-sphere impact parameter of the static, spherically symmetric metric
    Section III.B uses d_theo_sh = 2b̄_c and matches to observed diameters; relies on the cited GRMHD studies [108-110] that spin effects are negligible.
  • domain assumption Novikov-Thorne thin disk assumptions: geometrically thin, optically thick, Keplerian circular orbits, local blackbody radiation
    Section V.C lists these assumptions before Eq. (28).
  • standard math Liouville's theorem and the transfer-function decomposition (direct/lensing/photon rings) from Gralla et al. [114]
    Used in Section IV to compute observed intensities (Eqs. 18-20).

pith-pipeline@v1.3.0-alltime-deepseek · 25887 in / 21570 out tokens · 192376 ms · 2026-08-03T19:21:03.702199+00:00 · methodology

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read the original abstract

We investigate the shadow and optical appearance of Bardeen black hole (BH) immersed in perfect fluid dark matter (PFDM). Using EHT observations of M87* and Sgr A*, we constrain the DM parameter to a narrow range $b/M \sim \mathcal{O}(10^{-1}-10^{-2})$ for M87* and to $\mathcal{O}(10^{-2}-10^{-3})$ for Sgr A*. From these constraints we derive a rough prediction for the PFDM density near the shadow scale ($R_{\mathrm{sh}}\sim5M$): $\rho_{\mathrm{PFDM}} \sim 0.27$-$2.67\,\mathrm{g/cm^3}$ for Sgr A*, dropping to $\sim10^{-24}$-$10^{-25}\,\mathrm{g/cm^3}$ at 100 pc. Moreover, increasing $b$ substantially enlarges the photon sphere, impact parameter, shadow radius, and suppresses the observed disk brightness, while the magnetic charge $g$ produces only negligible corrections completely masked by PFDM on macroscopic scales. Subsequently, we investigate the primary/secondary images, flux, and redshift profiles for the PFDM-Bardeen BH using the Novikov-Thorne disk model, and compare these quantities with those of NFW, Dehnen-type and Moore DM BHs. The four BH types exhibit distinct densities at the shadow radius and at 100 pc, offering a potential distinguishing signature. Furthermore, for all DM BH models, blueshift appears in the primary image as inclination increases, while the secondary image remains redshift dominated even at high inclinations. Hence, if significant blueshifted emission were detected at low inclination, the predictions of these four DM models would be seriously challenged.

Figures

Figures reproduced from arXiv: 2512.00824 by Haiyuan Feng, Hao-Peng Yan, Jinjun Zhang, Rong-Jia Yang, Ziqiang Cai.

Figure 1
Figure 1. Figure 1: FIG. 1: On the horizontal axis, we have dimensionless DM parameter [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: depicts the dependence of G(u) on u for magnetic charge g/M = 0.5 and DM parameter b/M = 1. The plot clearly reveals the existence of a critical impact parameter b¯ c that distinguishes different photon trajectories. For b¯ > b¯ c , photons arriving from spatial infinity are gravitationally deflected, reaching a minimum radial distance rmin before escaping back to infinity (where rmin denotes the closest r… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The two pictures illustrate the dependence of photon sphere radius on the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The figure illustrates the contour plots of the shadow diameter within the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The photon dynamics for the Bardeen, PFDM-Sch, and PFDM-Bardeen BHs are [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The first three transfer functions for BHs with different parameter choices are [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The face-on optical appearance of the thin disk is displayed for three distinct [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: In each row, the black, green, and red panels correspond to configurations with a [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The deflection [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The top two rows display the primary and secondary images for fixed [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: This figure presents the distribution of [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The figure shows the cases with fixed DM parameter [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: This figure illustrates the distribution of the redshift [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Three representative viewing angles are fixed together with [PITH_FULL_IMAGE:figures/full_fig_p035_15.png] view at source ↗

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Reference graph

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