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EHT shadows of Sgr A* and M87* bound the central density of an Einasto dark-matter halo to ρ₀ ≲ 10⁻¹¹ M⊙/pc³ for Sgr A*.

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2026-07-10 19:30 UTC pith:ATJG7BEZ

load-bearing objection New usable 1σ density bound for Einasto around Sgr A* from EHT d_sh plus stellar mass prior, but every number rests on an unvalidated Padé mass approximation. the 4 major comments →

arxiv 2607.07752 v1 pith:ATJG7BEZ submitted 2026-07-08 gr-qc

Observational Limits on Einasto Dark Matter Parameters from Event Horizon Telescope Images of Sgr A^(*) and M87^(*)

A. Errehymy , S. Hansraj , C. Hansraj This is my paper
classification gr-qc
keywords black-hole shadowEinasto profiledark matter haloEvent Horizon TelescopeSgr A*M87*photon spherefuzzy dark matter
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.

The paper asks how much dark matter can sit near the photon sphere of a supermassive black hole before the black-hole shadow measured by the Event Horizon Telescope would look wrong. It embeds a static black hole in an Einasto halo, builds an approximate metric that joins the horizon to the outer halo, and computes the resulting shadow diameter. Matching that diameter to the published EHT values for Sgr A* and M87*, while using independent stellar-orbit mass priors, yields an upper limit on the central density: for Sgr A* one obtains ρ₀ ≲ 10⁻¹¹ solar masses per cubic parsec at 1σ. The Einasto index itself is only weakly constrained, showing that present shadow precision mainly senses the mass enclosed near the photon sphere rather than the detailed slope of the density profile. The result supplies a horizon-scale density bound that is complementary to galactic-scale dynamical limits and can be mapped onto fuzzy-dark-matter particle-mass windows.

Core claim

When a static spherically symmetric black hole is surrounded by an Einasto dark-matter halo whose metric is constructed from a Padé-type mass interpolation, Bayesian comparison of the predicted dimensionless shadow diameter with EHT data, conditioned on stellar-dynamical mass priors, restricts the Einasto central density to ρ₀ ≲ 10⁻¹¹ M⊙/pc³ at 1σ for Sgr A*; the same data leave the Einasto index only weakly constrained.

What carries the argument

The interpolating metric function f(r)=1−2M/r+2M∞ g̃(r) obtained from the approximate enclosed-mass formula M(r)≈M∞ r³/(r³+α̃³). This single function carries the entire photon-orbit and shadow calculation that is later compared with EHT diameters.

Load-bearing premise

The exact incomplete-gamma mass integral of the Einasto profile is replaced by a simple Padé interpolation claimed to be accurate to a few percent near the photon sphere; every subsequent orbit and shadow result rests on that approximation.

What would settle it

Recompute the photon-sphere radius and shadow diameter with the exact incomplete-gamma mass function (or a high-resolution numerical integration of the true Einasto density) and check whether the resulting 1σ upper bound on ρ₀ for Sgr A* still lies near 10⁻¹¹ M⊙/pc³.

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

If this is right

  • Horizon-scale imaging can be combined with stellar-orbit mass priors to place quantitative density caps on any extended mass component near the photon sphere.
  • The Einasto index remains essentially free at current EHT precision, so future tighter diameter measurements or higher-order photon-ring detections will be needed to constrain profile slope.
  • Under the solitonic-core scaling of fuzzy dark matter the Sgr A* density bound maps to a lower limit mψ ≳ 10⁻²⁰ eV for kiloparsec-scale cores.
  • M87* yields weaker density limits mainly because of larger distance uncertainty; improved distance measurements will tighten its Einasto constraints.

Where Pith is reading between the lines

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

  • Because the bound is driven by enclosed mass near the photon sphere, any other cored density profile with the same enclosed mass at a few gravitational radii should produce a comparable numerical limit.
  • Spin and axisymmetry, deliberately omitted here, will re-open a degeneracy between black-hole spin and halo density once Kerr+halo metrics are analysed.
  • Next-generation EHT arrays that resolve the n=1 photon ring could convert the present density upper limit into a genuine measurement of the inner-halo contribution.

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

4 major / 6 minor

Summary. The manuscript constructs a static, spherically symmetric black-hole metric embedded in an Einasto dark-matter halo by replacing the exact incomplete-gamma enclosed mass with a Padé interpolant M(r)≈M_∞ r³/(r³+α̃³), derives the associated photon-sphere condition and shadow radius, and performs a Bayesian comparison to the EHT dimensionless shadow diameters of Sgr A* and M87* (with stellar-dynamical mass priors). The headline result is an upper bound ρ₀ ≲ 10^{-11} M_⊙/pc³ (1σ) on the Einasto central density for Sgr A*, with the index ν̃ only weakly constrained; weaker bounds are reported for M87*. The analysis is restricted to the non-spinning limit and includes theoretical side calculations of the energy-emission rate and null-geodesic stability.

Significance. Horizon-scale imaging as a probe of the innermost dark-matter distribution is a timely and legitimate complement to stellar-dynamical and Lyman-α constraints. The Bayesian formulation that combines d_sh with an external mass prior, and the explicit acknowledgment of the M–ρ₀ degeneracy, are methodological strengths. If the metric construction and unit conversion are placed on a firm footing, the pipeline could be reused for other halo profiles and for next-generation EHT data. At present the quantitative bound and the FDM mapping rest on an unvalidated approximation and on a scale identification that is not internally consistent, so the claimed significance is not yet secured.

major comments (4)
  1. Sect. 3, Eqs. (13)–(14): The exact Einasto mass (incomplete gamma) is replaced by the Padé form M(r)≈M_∞ r³/(r³+α̃³) with the assertion that the error is “a few percent” near the photon sphere. No residual plot, relative-error table, or comparison of r_ph or R_sh between the two mass functions is supplied. All subsequent geodesic, shadow, and Bayesian results inherit this approximation; an uncontrolled bias of even a few percent in M(r) near r∼3M propagates directly into the quoted ρ₀ limit. A quantitative validation (or replacement by the exact mass) is required before the bound can be trusted.
  2. Sects. 3–7 and Figs. 1–7 versus abstract/Sect. 8.3: Throughout the calculations α̃ is O(1)–O(few) in units of M, so the dark-matter distribution is horizon-scale. The astrophysical interpretation as a galactic Einasto halo or an FDM soliton with r_c∼O(kpc) is then inconsistent: for true galactic α̃≪r_ph the enclosed DM mass is negligible for any reasonable ρ₀, while the plotted geometric densities convert to enormous physical densities. The paper must state clearly whether it constrains ultra-compact horizon-scale DM or galactic halos, and the analysis and FDM mapping must be redone consistently with that choice.
  3. Abstract, Sect. 8, Fig. 7 caption: The headline bound ρ₀ ≲ 10^{-11} M_⊙/pc³ is said to follow by converting the dimensionless posterior values shown in Fig. 7. Those values are ρ₀∼10^{-3} (geometric); a standard conversion for Sgr A* yields ∼10^{23} M_⊙/pc³, not 10^{-11}. Either a different region of parameter space is being converted, or the unit conversion is erroneous. A transparent, equation-level conversion (geometric ρ₀, α̃ → physical M_⊙/pc³ and pc) must be provided and must match the quoted number.
  4. Sect. 3, Eqs. (18)–(23): The indefinite integral that defines g̃(r) approaches a non-zero constant as r→∞, so exp(2M_∞ g̃) does not tend to 1 and the linearization 1+2M_∞ g̃ does not recover the claimed Schwarzschild asymptotics. The integral should be taken from infinity (or the asymptotic constant subtracted) so that g(∞)=1 exactly; the present form leaves the asymptotic mass and the normalization of f(r) ambiguous.
minor comments (6)
  1. Title page: “Event Horizon T elescope” contains a spurious space; the draft date “July 10, 2026” is in the future relative to the arXiv stamp.
  2. Keywords list “Dark energy (351)” although the paper concerns dark matter; this should be corrected.
  3. Fig. 1–4 captions quote widely different fiducial (ρ₀, α̃, ν̃) sets with no single reference table; a compact parameter table would aid reproducibility.
  4. Sect. 4 correctly labels Hawking emission as unobservable for SMBHs, yet the section and Fig. 2 occupy substantial space; consider moving them to an appendix to keep the observational narrative focused.
  5. Eq. (9) and the surrounding reparameterization of (ρ_s, r_s, d_ν̃) into (ρ₀, α̃, ν̃) are dense; a one-line dictionary of all Einasto conventions used would reduce reader friction.
  6. Sect. 8.1: the prior ranges ρ₀∈[0,10^{-8}], α̃∈[0.1,5], ν̃∈[0.5,10] are stated without units; specify geometric versus physical units explicitly.

Circularity Check

0 steps flagged

No significant circularity; the Einasto-parameter bounds are obtained by feeding an approximate but explicitly constructed metric into standard null-geodesic equations and then confronting the resulting shadow diameter with external EHT data plus independent stellar-dynamical mass priors.

full rationale

The derivation chain begins with the Einasto density (Eq. 10), replaces the exact incomplete-gamma enclosed mass (Eq. 13) by a Padé interpolant (Eq. 14) that is introduced as an explicit modeling choice following the external reference Xu et al. (2018), builds the metric function f(r) (Eq. 23), solves the standard photon-sphere condition (Eq. 55/58) for the critical impact parameter, and converts it into the dimensionless shadow diameter d_sh. This theoretical d_sh is then compared, via a Bayesian likelihood (Eq. 65), to the published EHT values d_sh^{M87*}=11.0±1.5 and d_sh^{SgrA*}=9.5±1.4 together with Gaussian mass priors taken from stellar-orbit and VLBI measurements that are independent of the present analysis. Because the mass prior is external, the final upper limit ρ₀ ≲ 10^{-11} M_⊙ pc^{-3} is not forced by construction; it is the ordinary result of marginalizing a degeneracy band against an outside datum. Minor citations to earlier papers co-authored by Hansraj (e.g., Amir et al. 2019) appear only as background remarks on wormhole shadows and do not enter the likelihood or the metric construction. No self-definitional loop, fitted-input-as-prediction, uniqueness theorem imported from the authors, or renaming of a known result is present. The Padé approximation itself is a potential source of systematic error (correctness risk), but it is not circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central observational bound rests on three free parameters of the Einasto profile (fitted/constrained by the shadow data), the Padé mass approximation introduced for analytic tractability, the assumption of a static non-spinning geometry, and the identification of the EHT ring diameter with the critical-curve diameter within the quoted error bars. No new particles or forces are postulated.

free parameters (4)
  • ρ₀ (Einasto central density) = ≲ 10^{-11} M⊙/pc^{3} (1σ, Sgr A*)
    Primary parameter constrained by the shadow data after marginalization over mass; the quoted upper limit is the main result.
  • α̃ (Einasto scale radius)
    Free shape parameter scanned in the Bayesian analysis; only weakly constrained.
  • ν̃ (Einasto index)
    Free curvature index; paper states it is only weakly constrained by current EHT precision.
  • M (black-hole mass) = Sgr A* (4.297±0.013)×10^6 M⊙; M87* (6.5±0.7)×10^9 M⊙
    Treated as a free parameter with a Gaussian prior from stellar dynamics; the prior is essential to break the M–ρ₀ degeneracy.
axioms (4)
  • domain assumption Spacetime is static and spherically symmetric (non-spinning limit).
    Stated in Sect. 1 and used throughout the metric construction and geodesic analysis; both target black holes are known to spin.
  • ad hoc to paper The exact Einasto enclosed-mass integral may be replaced by the Padé approximant M(r)≈M∞ r^{3}/(r^{3}+α̃^{3}) with only a few-percent error near the photon sphere.
    Introduced in Sect. 3 (Eq. 14) following Xu et al. (2018) but never validated against the incomplete-gamma expression inside r~3M.
  • domain assumption The observed EHT ring diameter equals the critical-curve diameter within the published 1σ error bars after accretion-model marginalization.
    Explicitly adopted in Sect. 8; residual 5–10% model dependence is absorbed into σ_d.
  • standard math General relativity holds; the metric is of the form f(r)=1-2M/r+2M∞ g̃(r).
    Standard Einstein equations plus the interpolated halo contribution.

pith-pipeline@v1.1.0-grok45 · 29803 in / 3289 out tokens · 39507 ms · 2026-07-10T19:30:14.100629+00:00 · methodology

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read the original abstract

The Event Horizon Telescope (EHT) has provided images of the supermassive black-holes Sgr A$^{*}$ and M87$^{*}$, enabling direct tests of gravity. Any extended mass-distribution, such as a dark-matter halo, perturbs null-geodesics in the photon-ring regime, making shadow-measurements a probe of inner-halo structure. In this work, we investigate static, spherically-symmetric black-holes surrounded by Einasto-type dark-matter halos and derive constraints from EHT shadow-data. Starting from the Einasto density-profile with parameters ${\varrho_0, \tilde{\alpha}, \tilde{\nu}}$, we construct a metric-function $f(r)=1-2M/r+2M_\infty \tilde{g}(r)$ that interpolates between the black-hole horizon and the asymptotic halo, following the approach of Xu et al. (2018) but adapted specifically to the Einasto scenario. We analyze the photon-potential, null-geodesics, and shadow-radius as functions of black-hole mass $M$, in the non-spinning limit. Using the dimensionless shadow-diameter $d_{\text{sh}}\equiv D\theta/M$ measured by the EHT -- $d_{\text{sh}}^{M87*}=11.0\pm1.5$ and $d_{\text{sh}}^{SgrA*}=9.5\pm1.4$ -- we perform Bayesian parameter-estimation to identify allowed regions in the Einasto parameter-space. Combined with the independently-measured black-hole masses from stellar-dynamics, our results place constraints on the inner dark-matter distribution: for Sgr A$^{*}$, adopting the stellar-orbit mass-prior, we find $\varrho_0 \lesssim 10^{-11},M_\odot/\text{pc}^3$ at $1\sigma$ confidence, while for M87$^{*}$ the bounds are weaker due to distance-uncertainties. The Einasto-index $\tilde{\nu}$ is weakly-constrained, indicating that EHT precision primarily limits the mass enclosed near the photon-sphere rather than the profile-slope. Future EHT observations will refine these constraints and distinguish between competing dark-matter descriptions.

Figures

Figures reproduced from arXiv: 2607.07752 by A. Errehymy, C. Hansraj, S. Hansraj.

Figure 1
Figure 1. Figure 1: The lapse function f(r) is analyzed in the presence of an Einasto dark matter distribution, with its behavior governed by the parameters ˜ν, ˜α, ϱ0, and the black-hole mass M. The leftmost panel illustrates the effect of varying ˜ν = 6.28, 6.31, 6.34, 6.37, and 6.40, while keeping ˜α = 0.95, ϱ0 = 10−16.75, and M = 1.0 fixed. The second panel from the left shows how changes in ˜α = 0.5, 0.8, 1.2, 1.6, and 2… view at source ↗
Figure 2
Figure 2. Figure 2: The energy emission rate d 2E dω dt is examined in the framework of an Einasto dark matter distribution, where its behavior is regulated by the parameters ˜ν, ˜α, ϱ0, and the black-hole mass M. The leftmost panel displays the response of the emission rate to variations in ˜ν = 0.1, 1.3, 1.5, 1.65, and 1.75, while ˜α = 0.5, ϱ0 = 10−3 , and M = 1.0 are kept fixed. The second panel from the left illustrates t… view at source ↗
Figure 3
Figure 3. Figure 3: The photon effective potential is examined in the presence of an Einasto dark matter distribution, with its profile determined by the parameters ˜ν, ˜α, ϱ0, and the black-hole mass M. The leftmost panel shows the response of the effective potential to variations in ν˜ = 0.6 and 1.2, while ˜α = 0.6, ϱ0 = 10−16.75 , M = 1.0, and the angular momentum L = 5.0 are kept fixed. The second panel from the left illu… view at source ↗
Figure 4
Figure 4. Figure 4: The black-hole shadow is investigated in the presence of an Einasto dark matter distribution, with its characteristics determined by the parameters ˜ν, ˜α, ϱ0, and the black-hole mass M. The leftmost panel illustrates the effect of varying ˜ν = 6.28, 6.31, 6.34, 6.37, and 6.40, while keeping ˜α = 0.95, ϱ0 = 10−16.75, and M = 1.0 fixed. The second panel from the left shows how modifications in ˜α = 0.5, 0.8… view at source ↗
Figure 5
Figure 5. Figure 5: The left panel shows the phase portrait of null geodesics around a black-hole with mass M = 1.357, metric parameters α˜ = 0.9, ˜ν = 1.0, background density ϱ0 = 0.001, and angular momentum L = 5.0. The vertical green dashed line marks the horizon at rhorizon ≈ 2.02, while magenta points indicate the saddle points of the effective potential Veff(r) = L2f(r)/r2 , where dVeff/dr = 0. Small disks and labels sh… view at source ↗
Figure 6
Figure 6. Figure 6: The black-hole shadow radius (solid black curve) is shown in the presence of an Einasto dark matter distribution, with its dependence governed by the parameters ˜ν, ˜α, ϱ0, and the black-hole mass M. The leftmost panel illustrates how the shadow radius responds to variations in ˜ν, while ˜α = 0.5, ϱ0 = 10−3 , and M = 1.0 are kept fixed. The second panel from the left demonstrates the effect of changing ˜α,… view at source ↗
Figure 7
Figure 7. Figure 7: Shaded regions in the three panels illustrate the ranges of key parameters and black-hole mass M that are consistent with the observed shadow sizes of Sgr A∗ and M87∗. The left panel shows (˜α, M), the middle panel (˜ν, M), and the right panel (ϱ0, M). In all panels, dark red and blue regions correspond to the 1σ confidence intervals for Sgr A∗ and M87∗, respectively, while lighter shades indicate the 2σ i… view at source ↗

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