Optical Signatures of Sgr A* and M87* with Dark Matter Halos

Mubasher Jamil; Muhammad Ali Paracha; Noraiz Tahir

REVIEW 4 major objections 6 minor 82 references

Optical Signatures of Sgr A* and M87* with Dark Matter Halos

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that Sgr A* and M87* shadows cannot yet distinguish a vacuum black hole from dark-matter-halo spacetimes, but that the two systems should differ in their lensing caustic topology.

desk verdict The caustic extension to SFDM halos is genuinely new and worth a referee, but the EHT comparison rests on an unexamined ring-vs-shadow identification, and the M87* SFDM parameters are missing. read the letter →

arxiv 2608.06924 v1 pith:VFJ6OVXU submitted 2026-08-07 astro-ph.CO gr-qc

Noraiz Tahir , Muhammad Ali Paracha , Mubasher Jamil This is my paper · ORCID

classification astro-ph.COgr-qc

keywords blackholeshadowSgrA*M87*darkmatterhaloscalarfieldNFWprofilegravitationallensingcaustics

topics Dark Matter

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 asks whether the dark matter halos presumably surrounding Sgr A* and M87* leave measurable imprints in the images already taken. It computes shadows, lensing observables, and caustic structures for two contrasting halo models, a cuspy cold-dark-matter halo and a cored scalar-field halo, alongside the bare vacuum black hole. All three spacetime models predict shadow diameters within $1.2\sigma$ of the published measurements, with $\Delta\chi^2 \lesssim 1.2$, so the current data cannot yet tell the models apart. The paper's sharper claim is topological: Sgr A* should retain both tangential and radial critical curves, while the more massive M87* may show only tangential critical curves, giving future observations a discrete signature to look for.

What carries the argument

The load-bearing objects are the two halo-modified metric functions $f_{\rm SFDM}(r)$ and $f_{\rm CDM}(r)$ (eqs. 2.4 and 2.7), which embed, respectively, a solitonic scalar-field core and a cuspy cold-dark-matter profile into a Schwarzschild-like line element. The photon sphere condition $r f'(r)-2f(r)=0$ and the critical impact parameter $b_c=r_{\rm ph}/\sqrt{f(r_{\rm ph})}$ convert each metric into a shadow diameter through $d_{\rm sh}=2b_c/D_l$. The caustic analysis uses a dimensionless lens equation for a point mass of strength $\kappa_P$ embedded in a halo with convergence $\kappa_c$; the disappearance of the radial critical curve above a critical $\kappa_P$ is what carries the topological prediction.

What would settle it

Measure the shadow diameters at about 10 µas resolution: for Sgr A*, a value below 50 µas favors CDM, near 52.2 µas favors the vacuum metric, and above 53 µas favors SFDM; for M87*, the corresponding values are 38.6, 39.7, and 40.7 µas. Separately, resolve the critical curves around M87*: the prediction is that only the tangential critical curve exists, so detecting a radial critical curve there would falsify the central topological claim.

Watch

Extended reading notes

Core claim

Working with spherically symmetric metrics that solve the Einstein equations for an SFDM soliton core and an NFW cusp, the paper finds that the halo pushes the photon sphere in opposite directions: SFDM moves it outward ($r_{\rm ph}/M = 3.0485$ for Sgr A*, $3.0613$ for M87*) while CDM moves it inward ($2.9397$ and $2.9637$), compared with $3$ for the vacuum case. The resulting shadow diameters are $49.81$-$53.16\,\mu$as for Sgr A* and $38.63$-$40.74\,\mu$as for M87*, all within $1.2\sigma$ of the measured values ($48.7\pm7.0$ and $42.0\pm3.0\,\mu$as). The best fits are CDM for Sgr A* ($+0.16\sigma$) and SFDM for M87* ($-0.42\sigma$), but the inter-model differences are only a few microarcseconds. In the caustic analysis, the dimensionless point-mass strength $\kappa_P$ for M87* ($2.0\times10^{-4}$ to $3.0\times10^{-4}$) sits near or above the critical threshold $\kappa_{\rm crit}$ ($2.71\times10^{-4}$ for CDM, $2.58\times10^{-4}$ for SFDM), while Sgr A* ($1.0\times10^{-4}$ to $1.5\times10^{-4}$) stays below it; hence the paper predicts tangential-only critical curves for M87* and both critical curve types for Sgr A*.

Load-bearing premise

The conclusion that all models fit within $1.2\sigma$ rests on identifying the measured bright emission ring with the theoretical shadow diameter $d_{\rm sh}=2\theta_\infty$; if the known offset between the emission ring and the true shadow boundary, or calibration systematics, is significant, the all-models-fit ranking changes.

Editorial extensions

If this is right

Current horizon-scale shadow measurements cannot statistically distinguish vacuum, cuspy-halo, or cored-halo spacetimes for either black hole, since all models are within $1.2\sigma$ and $\Delta\chi^2\lesssim1.2$.
A shadow measurably larger than the vacuum prediction would favor a cored scalar-field halo, while a smaller shadow would favor the cuspy cold-dark-matter halo, because the two halos shift the photon sphere in opposite directions.
Sgr A* should show both tangential and radial critical curves, whereas M87* should show only tangential critical curves, a discrete and falsifiable difference for future horizon-scale observations.
The predicted inter-model shadow differences ($\sim3.3\,\mu$as for Sgr A*, $\sim2.1\,\mu$as for M87*) are near the resolution next-generation very-long-baseline arrays aim for, so the degeneracy may be broken soon.
Relativistic-image separations of order $10^{-5}\,\mu$as are unobservable, but the predicted time delays between the first two images ($11.0$-$11.7$ minutes for Sgr A*, about $1.7\times10^4$ minutes for M87*) offer an independent timing channel.

Reading between the lines

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

Beyond the paper, the two halos bracket what other cored profiles should do: any profile shallower than a cusp should land between the CDM and SFDM shadow sizes, so a future measurement cannot uniquely identify a particle model from shadow size alone.
Because $\kappa_P$ grows with black hole mass for fixed halo parameters, the paper's topology rule extrapolates: heavier supermassive black holes should show tangential-only critical curves, and lighter ones should show both; this hierarchy is testable without resolving the few-microarcsecond ring offsets.
The paper leaves out adiabatic compression of the halo by the growing black hole; including it would steepen the central density, which should shrink the CDM shadow further and sharpen the Sgr A* preference for CDM.

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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

Short version: this is a careful, mostly derivative application of known DM-halo metrics to Sgr A* and M87*, and its one genuinely new piece—the caustic structure for SFDM halos—is worth a referee's time. The headline 'all models within 1.2σ' is real but shakier than it looks, for two fixable reasons.

The novel part is Section 4. Extending Ref. [77]'s NFW caustic analysis to solitonic SFDM halos is a legitimate extension, and the paper reproduces the CDM critical point-mass κ_P^crit = 2.714×10^-4 from [77], which is a good check. The prediction that M87* sits near or above the tangential/radial threshold while Sgr A* sits below is interesting and testable with ngEHT-scale VLBI. The tables are thorough and the weak/strong lensing numbers look internally consistent.

Now the soft spots. First, the EHT comparison in Sec 3.2.4 equates d_sh = 2θ_∞ with the measured shadow diameter. That ignores the known offset between the bright emission ring and the shadow boundary, and ignores EHT calibration systematics. The paper quotes EHT diameters of 48.7±7 and 42±3 μas as if they were direct geometric shadow measurements. They are not; they come from model-dependent reconstructions. The 7 μas (Sgr A*) and 3 μas (M87*) uncertainties are of the same order as the model spread, so the Δχ² ≲ 1.2 ordering could easily change if the offset is ~a few μas. The qualitative conclusion—current EHT can't distinguish these halo models—probably survives, but the quantitative best-fit claims (CDM preferred for Sgr A*, SFDM for M87*) are not supported by the data as presented.

Second, the M87* SFDM halo parameters are missing. Sec 3.1 gives ρ_c,S and r_c,S for Sgr A* and then says 'and M87* respectively' without listing a second pair. Without those numbers, the M87* SFDM row in Table 4 cannot be reproduced. This is a fixable typo, but it must be fixed before publication.

Third, the caustic κ_P ranges in Table 5 are quoted but not derived from the halo parameters and black hole mass. The text says κ_P ∝ M, but the explicit conversion would let a reader verify why M87* is in the 2.0–3.0×10^-4 range. As written, it's an assumption, not a computed result.

Overall: the paper is a workmanlike extension of an established program, and the caustic section is a genuine contribution. The central EHT comparison is unfortunately the weakest part of the paper. It deserves peer review, and the referee should ask for the ring-vs-shadow discussion, the missing M87* SFDM parameters, and an explicit derivation of κ_P.

Referee Report

4 major / 6 minor

Summary. The paper studies the gravitational lensing signatures of Schwarzschild black holes surrounded by two dark-matter halo models (cored scalar-field DM, SFDM, and cuspy cold DM, CDM) for Sgr A* and M87*, using the static spherically symmetric metrics of Ref. [27]. It computes photon-sphere radii, shadow diameters, weak- and strong-lensing observables, Einstein-ring constraints, and caustic critical curves. The central comparison claims that all three spacetime models predict shadow diameters within 1.2σ of the EHT measurements (Δχ²≲1.2, Table 4), and that the caustic analysis predicts a topological distinction: Sgr A* should retain both tangential and radial critical curves, while M87*, having a larger mass and hence larger κ_P, may show only tangential critical curves (Section 4, Table 5).

Significance. If the comparison is valid, the paper would provide a useful, physically motivated benchmark for DM effects on strong-field observables, with the merit of not fitting the shadow predictions to the EHT data (the halo parameters come from Ref. [27], and the EHT comparison is external). The analytic expansions for the DM corrections to the critical impact parameter (Eqs. 3.10-3.15) and the reproduction of the known CDM caustic threshold of Ref. [77] (κ_crit^P = 2.714×10^-4) are concrete technical contributions. The predicted caustic-topology difference between Sgr A* and M87* is falsifiable by future VLBI observations. However, the statistical claim and the caustic prediction currently rest on an unjustified identification of the EHT bright-ring diameter with the theoretical shadow diameter, on missing M87* SFDM parameters, and on an un-derived κ_P mapping; these need to be repaired before the results can be taken as established.

major comments (4)

The central statistical claim equates the predicted shadow diameter d_sh = 2θ∞, with θ∞ = b_c/D_l, directly to the EHT-measured angular diameter (48.7±7.0 μas for Sgr A*, 42.0±3.0 μas for M87*). The EHT values are the diameters of the bright emission ring, which is known to be offset from the shadow/photon-ring boundary in a way that depends on the astrophysical emission model, and the EHT collaboration's shadow-size estimates carry additional calibration and model systematics. The paper applies no ring-to-shadow offset and adds no systematic error term. Since the model predictions differ by only ~3.3 μas (Sgr A*) and ~2.1 μas (M87*), an unmodeled offset at the level of the quoted statistical uncertainties can reorder the model preferences and invalidate the Δχ²≲1.2 conclusion. The authors should either use the EHT collaboration's published shadow-size estimates (including their systematics) or introduce a ring-to-shadow offset with a prior informed by the EHT imaging papers, and show how the χ² values in Table 4 change under this treatment.
The text states 'Also ρ_c,S = 3.43×10^7 M⊙/kpc^3, r_c,S = 15.7 kpc for Sgr A*, and M87* respectively', but only one set of SFDM parameters is given. No values for the M87* SFDM halo are listed. Consequently the M87* SFDM row in Table 4 (d_sh = 40.74 μas) and the corresponding photon-sphere and strong-lensing entries in Table 3 cannot be reproduced or checked. The authors must supply the M87* SFDM (ρ_c,S, r_c,S) values, or explicitly state that they are taken to be identical to the Sgr A* values. Without this fix, a key part of the central comparison is unverifiable.
The caustic prediction depends on the quoted κ_P ranges: Sgr A* (0.8–1.5)×10^-4 and M87* (2.0–3.0)×10^-4, with M87* placed near or above the critical threshold κ_crit^P. However, the paper never defines the relation between κ_P and the halo parameters and black hole mass, nor does it show how the ranges are computed from the models of Sec. 2. The assertion 'κ_P ∝ M for fixed halo parameters' is insufficient, and the ranges appear to be imposed rather than derived. Because the claimed topological difference between Sgr A* and M87* rests entirely on these κ_P values, the authors should provide the explicit formula for κ_P, evaluate it using the Sec. 3.1 halo parameters and the adopted masses, and recompute the ranges; if the ranges change, the caustic conclusion must be revised accordingly.
The Einstein-ring analysis in Sec. 3.2.2 is presented as providing constraints on DM parameters, but the resulting best-fit parameters (Table 2) are not propagated into the shadow predictions used in Table 4, which instead adopt the Ref. [27] values. The text's later claim that these Einstein-ring constraints 'provide independent validation of our DM halo models' is therefore misleading: the comparison is not a joint fit and the validation is only qualitative. The authors should either clarify that the Einstein-ring fits are not used in the shadow comparison or explicitly quantify the consistency between the Table 2 bounds and the adopted halo parameters.

minor comments (6)

The text reads 'his discrepancy persists even after accounting for baryonic feedback'; 'his' should be 'This'.
The text refers to 'Table 3 (rows 4-8)' for weak-lensing observables, but the table is labeled 'Table 1'. Please correct the cross-reference.
The caption and text state that Table 2 includes 'the corresponding χ² values for each model', but the printed table has no χ² column; either add it or revise the caption and sentences describing it.
The caption says 'Solid curves correspond to the Schwarzschild case (dashed black), SFDM halo (orange), and CDM halo (blue)'; the parenthetical contradicts the line-style description. Please specify one line style per model.
The expressions for β_0^CDM and β_0^SFDM contain unmatched parentheses and unclear denominator structures. Please re-typeset them with unambiguous bracketing.
The conversion to physical units in Eq. (3.45) is unclear: the factor '2π b_c^i / 60' with dimensions of minutes/60 is not obviously equivalent to Eq. (3.44). Please spell out the units of b_c and the conversion.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the shadow predictions are computed from externally adopted halo metrics and benchmarked against EHT, not fitted to it.

full rationale

The central shadow predictions in Table 4 are not fitted to the EHT shadow diameters. The halo metrics (2.4) and (2.7) and the halo parameters are taken from Ref. [27], and the shadow boundary is computed from the photon-sphere condition (3.5) and the critical impact parameter (3.6). The EHT comparison in Sec. 3.2.4 uses the external measured diameters 48.7±7.0 μas (Sgr A*) and 42.0±3.0 μas (M87*) only in the χ² statistic (3.52), so the agreement is an external benchmark rather than a reconstructed input. The Einstein-ring fit in Sec. 3.2.2 is a consistency check and is not used to set the shadow predictions; the paper states that its best-fit parameters are 'consistent with the values used in our analysis.' The caustic analysis follows the external formalism of Ref. [77] and reports recovering its CDM critical value κ_P^crit = 2.714×10^-4 as numerical validation. The only author-overlapping reference that is cited, Ref. [34], is background context and is not load-bearing for any derivation. The ring-vs-shadow identification d_sh = 2θ∞ and the underived κ_P ranges in Table 5 are correctness or reproducibility concerns, not circularity, because neither the halo parameters nor the caustic thresholds are defined in terms of the EHT shadow diameters or the paper's conclusions.

Assumptions & free parameters 10 free parameters · 5 assumptions · 0 invented entities

The shadow and lensing results depend on halo normalization parameters adopted from Ref [27], not fitted in this paper. Two of the M87* SFDM parameters are missing entirely. The caustic conclusion adds estimated κ_P ranges. No new particles or forces are introduced.

free parameters (10)

ρ_{c,N} (Sgr A*) = 1.936×10^7 M⊙/kpc^3
NFW characteristic density adopted from Ref [27]; affects all Sgr A* CDM results.
r_{c,N} (Sgr A*) = 17.46 kpc
NFW scale radius adopted from Ref [27].
ρ_{c,N} (M87*) = 0.008×10^{7.5} M⊙/kpc^3 ≈ 2.5×10^5 M⊙/kpc^3
NFW characteristic density for M87* adopted from Ref [27]; notation in Sec 3.1 is unusual.
r_{c,N} (M87*) = 130 kpc
NFW scale radius for M87* adopted from Ref [27].
ρ_{c,S} (Sgr A*) = 3.43×10^7 M⊙/kpc^3
SFDM core density adopted from Ref [27].
r_{c,S} (Sgr A*) = 15.7 kpc
SFDM core radius adopted from Ref [27].
ρ_{c,S} (M87*)
Not specified in Sec 3.1; the sentence gives Sgr A* values only and says 'and M87* respectively'.
r_{c,S} (M87*)
Not specified; M87* SFDM results in Tables 3 to 5 cannot be reproduced.
κ_P range (Sgr A*) = (0.8 to 1.5)×10^-4
Dimensionless point-mass strength ranges used in the caustic topology argument (Table 5); not derived from the halo parameters used elsewhere.
κ_P range (M87*) = (2.0 to 3.0)×10^-4
Places M87* near or above the critical threshold; the caustic conclusion depends on these estimated ranges.

assumptions (5)

domain assumption Static, spherically symmetric spacetime with f(r)=g(r) for both halo models (Eq 2.1).
Excludes black hole spin, halo triaxiality, accretion and backreaction; acknowledged in Sec 5 as a limitation.
domain assumption The metric functions for SFDM (Eq 2.4) and CDM (Eq 2.7) are taken from Ref [27].
The paper does not re-derive these metrics; all subsequent observables inherit this input.
domain assumption The EHT-measured ring diameter is identified with the theoretical shadow diameter d_sh=2θ∞ (Sec 3.2.4).
No correction for the ring-to-shadow offset; this is the load-bearing mapping for the χ² comparison.
standard math Bozza strong lensing formalism applies to these spacetimes.
Used in Sec 3.2.3; the paper notes it may need extension for spinning black holes.
ad hoc to paper The κ_P ranges in Table 5 place Sgr A* below and M87* near/above the critical threshold.
The caustic topology claim depends on these estimated ranges, which are not derived from the halo parameters.

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Cite this review

Pith. "Pith review of Optical Signatures of Sgr A* and M87* with Dark Matter Halos." pith.science (2026). https://pith.science/paper/VFJ6OVXU

@misc{pith2026260806924,
  author       = {Pith},
  title        = {Pith review of: Optical Signatures of Sgr A* and M87* with Dark Matter Halos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFJ6OVXU}},
  note         = {Machine review of arXiv:2608.06924}
}

abstract

The event horizon telescope (EHT) has opened a new window onto the strong-field regime by imaging the shadows of the supermassive black holes (SMBHs) Sgr A* and M87*. These observations provide a unique laboratory for probing the dark matter (DM) distribution around black holes. In this work we systematically investigate the imprints of two distinct DM halo models, the cold DM (CDM), and the cored scalar field DM (SFDM) profiles on the gravitational lensing signatures of Sgr A* and M87*. We compute the photon spheres, shadows, weak and strong lensing observables, and caustic structures for both models. We then compared the obtained values of the shadow diameters with the EHT data, using $\chi^2$ statistics. We found that all the models are within $1.2\sigma$ of the measured shadow diameters with the small $\chi^2$ differences, $\Delta\chi^2 \lesssim 1.2$. The caustic analysis reveals distinct topological regimes, Sgr A* retains both tangential and radial critical curves, while M87* may show only tangential critical curves due to its larger mass. This topological difference provides a clear observational signature for future observations.

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