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REVIEW 4 major objections 6 minor 60 references

Gravitational Lensing by a Dark Compact Object in Modified Gravity and Observational Constraints from Einstein Rings

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

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Pith's one-line read The paper derives explicit α-dependent corrections to light bending in modified gravity (MOG) and shows that Einstein-ring, shadow, and time-delay observations bound the MOG parameter α near zero.

desk verdict Solid deflection-angle derivations for MOG, but the galaxy-scale Einstein-ring constraints are circular and the M87* constraint is not independent. read the letter →

arxiv 2502.06313 v1 pith:UNELCV7K submitted 2025-02-10 gr-qc astro-ph.GA

Nagina Rehman , Zoe C. S. Chan , Mubasher Jamil , Mustapha Azreg-Aïnou This is my paper · ORCID
classification gr-qcastro-ph.GA
keywords modifiedgravityMOGgravitationallensingweakdeflectionanglestronglimitEinsteinringblackholeshadowM87*
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

This paper works out how gravitational lensing changes if gravity is governed by the scalar-tensor-vector modified gravity (MOG) rather than general relativity. It derives the weak-field deflection angle to second order in the small ratio $M/b$, obtaining $\hat\alpha = 4M/b + 15\pi M^2/(4b^2) + \alpha(4M/b + 27\pi M^2/(4b^2)) + 3\pi M^2\alpha^2/b^2$ plus higher-order terms, and the strong-field logarithmic deflection with $\alpha$-dependent coefficients; setting $\alpha=0$ returns the Schwarzschild results. Using these formulas, the paper computes Einstein-ring radii, magnifications, image distortions, and time delays for the supermassive black holes M87* and Sgr A*, and compares them with observed galaxy-scale Einstein rings. The resulting constraints place $\alpha$ in narrow windows around zero, for example $-0.142 \lesssim \alpha \lesssim 0.338$ for M87*, and the paper argues that horizonless MOG compact objects would produce secondary images of opposite parity, giving an observational route to distinguish them from black holes.

What carries the argument

The central object is the MOG metric function $f(r)=1-2(1+\alpha)M r^2(r^2+\alpha(1+\alpha)M^2)^{-3/2}+\alpha(1+\alpha)M^2 r^2(r^2+\alpha(1+\alpha)M^2)^{-2}$, whose large-$r$ expansion $f(r)=1-2(1+\alpha)M/r+\alpha(1+\alpha)M^2/r^2+O(M^3,\alpha^3)$ feeds every lensing calculation. This single function determines the optical metric for the weak-field geometric deflection angle and determines the photon-sphere radius, impact parameter, and critical impact parameter for the strong-deflection logarithm. The dimensionless MOG parameter $\alpha$—which rescales the effective gravitational constant, $G=G_N(1+\alpha)$—is the knob that shifts every lensing observable, with $\alpha=0$ recovering Schwarzschild.

What would settle it

Measure the lens galaxy's dynamical mass independently, for example from stellar velocity dispersion, and compare it with $\theta_E^2 D/4$ for the Clone Einstein ring (the system with the tightest reported mass error). If the dynamical mass matches the general-relativistic value within the reported uncertainty, then $|\alpha|\lesssim 0.03$ and the paper's wide allowed intervals are excluded; if the dynamical mass is significantly larger, the MOG correction is ruled out.

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Extended reading notes

Core claim

For a static, spherically symmetric MOG spacetime with metric function $f(r)=1-2(1+\alpha)M/r+\alpha(1+\alpha)M^2/r^2+O(M^3,\alpha^3)$, the photon deflection angle in the weak field is $\hat\alpha = 4M/b + 15\pi M^2/(4b^2) + \alpha(4M/b + 27\pi M^2/(4b^2)) + 3\pi M^2\alpha^2/b^2$ to second order in $M/b$. In the strong deflection limit the same metric yields the coefficients $a = 1 + \alpha/9 - 7\alpha^2/162 + O(M^3,\alpha^3)$ and a constant term $b = -\pi + \log 6 + 2\log\bigl(6(2-\sqrt{3})\bigr) + O(\alpha)$, which enter the logarithmic deflection $\hat\alpha = a\log(b/b_c - 1) + b$ near the photon sphere. These coefficients drive the observables: the outermost relativistic image $\theta_\infty = b_c/D_{OL}$, the separation $s=\theta_1-\theta_\infty$, and the time delay $\Delta T = 2\pi b_c$. The paper's central claim is that, for fixed mass and distance, a positive $\alpha$ enlarges the Einstein ring, increases the image separation, and lengthens the time delay, while a negative $\alpha$ does the opposite; comparing these predictions against the M87* and Sgr A* shadows and four galaxy-scale Einstein rings yields the quoted constraint intervals.

Load-bearing premise

For the galaxy-scale Einstein ring constraints, the paper assumes the lens can be modeled as a point mass and that the reported 'mass enclosed' was measured independently of the ring; if that mass was instead derived from the same Einstein radius under general relativity, the comparison is circular and forces $\alpha = 0$.

Editorial extensions

If this is right

  • At a fixed impact parameter, a positive $\alpha$ increases the weak deflection angle by the term $\alpha(4M/b + 27\pi M^2/(4b^2)) + 3\pi M^2\alpha^2/b^2$, so precision astrometry of light bending can bound $\alpha$ without invoking strong-field observables.
  • The Einstein ring angular radius scales as $\theta_E = \sqrt{4DM(1+\alpha)/D_{OL}}$, so for a fixed lens mass and distances, a positive $\alpha$ enlarges the ring and a negative $\alpha$ shrinks it; measured ring radii therefore translate directly into allowed $\alpha$ intervals.
  • For the supermassive lenses M87* and Sgr A*, the outermost relativistic image $\theta_\infty$ and its separation $s$ from the other images vary monotonically with $\alpha$, giving specific microarcsecond targets that very-long-baseline observations can test.
  • The time delay between the first two relativistic images changes by roughly $\pm 1$ minute for Sgr A* when $\alpha$ goes from $-0.1$ to $0.1$, so timing measurements provide a sign-sensitive probe of $\alpha$.

Reading between the lines

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

  • The galaxy-scale constraints in Table 3 are likely degenerate: the reported 'mass enclosed' is normally derived from the same Einstein radius under general relativity, so the comparison $M=\theta_E^2 D/(4(1+\alpha))$ against $M_{\rm obs}=\theta_E^2 D/4$ tends to force $\alpha\simeq 0$; independent dynamical masses are needed to make these rings informative.
  • The same $\alpha$-dependent deflection formula implies that microlensing light curves, especially caustic-crossing events, should show a characteristic shift in the Einstein-radius crossing time; existing microlensing surveys could be reanalyzed to place independent bounds on $\alpha$ at the $\sim 0.1$ level.
  • Because the strong-field coefficient $a$ has two roots in $\alpha$, the image separation $s$ is multi-valued in $\alpha$; a future measurement of $s$ would have to identify which branch of the $\alpha$ relation is being probed, as the paper's graphs show three separate branches.
  • For $\alpha > \alpha_{\rm crit}$, the MOG object is horizonless, and the equations predict secondary images with opposite parity; deep imaging that looks for such inverted images could separate a horizonless MOG object from a Schwarzschild black hole without relying on shadow size alone.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies gravitational lensing by the static, spherically symmetric MOG/STVG spacetime of Moffat. It derives the weak-field deflection angle using the Gauss-Bonnet method, computes weak-field magnifications and distortion parameters, then derives strong-deflection-limit coefficients and uses them to model Einstein rings for M87* and Sgr A*. It also attempts to constrain the MOG parameter α using four galaxy-scale Einstein rings and computes time delays between relativistic images for a set of nearby galaxies. The central analytic formulas (weak deflection angle, strong-field coefficients a and b, critical impact parameter) reduce to the Schwarzschild results at α=0 and appear internally consistent within the stated α-expansions.

Significance. If the analytic lensing formulas are correct, the paper provides a useful reference calculation for testing MOG with lensing and shadow observables. The derivation of the weak and strong deflection angles is standard and self-contained, and the α→0 limits recover known Schwarzschild lensing results. However, the headline observational constraints are the paper's main weakness. The galaxy-scale Einstein-ring constraints in Table 3 are essentially tautological because the comparison masses are the GR point-mass Einstein masses derived from the same ring radii. The M87* constraint is also largely a re-parametrization of the EHT mass, which itself is inferred from the shadow under GR. The point-mass model for galaxy-scale lenses and the inversion of small-α expansions to large α further undermine the quoted constraints. The analytic part of the paper can stand, but the observational-constraints claims need substantial revision or removal before publication.

major comments (4)
  1. [Section 6, Eq. (70), Table 3] The galaxy-scale constraints are tautological. The column M in Table 3 is the GR point-mass Einstein mass M_obs = θ_E²D/4; for instance, the first row with θ_E=3.32 arcsec and D=993.67 h⁻¹ Mpc gives 1.35×10¹² h⁻¹ M☉, exactly the quoted value. Equation (70) is M₂ = θ_E²D/[4(1+α)]. Equating M₂ to M_obs forces 1+α=1, i.e. α≈0 up to the reported mass errors. The intervals in Table 3 therefore merely re-express the measurement uncertainty of the same Einstein radius and contain no independent information about MOG. The paper should either use genuinely independent mass estimates or present Eq. (70) only as a consistency relation, not as a constraint.
  2. [Section 6, Eqs. (67)-(69)] The M87* constraint is likewise not an independent test. The mass 6.5×10⁹ M☉ from reference [51] is inferred from the EHT shadow angular diameter assuming GR; inserting that same angular size into Eq. (67) and solving for α is an algebraic re-parametrization of the measurement, not a new constraint on MOG. For Sgr A* the situation is partially better because reference [52] incorporates stellar-orbit mass, but the quoted combined mass is not shadow-independent. As written, the abstract's and conclusion's claim that Einstein-ring observations constrain MOG is not established; the Sgr A* orbital-mass constraint should be isolated and analyzed separately.
  3. [Section 6 and Section 8] The point-mass model used for Table 3 and Eq. (70) is not adequate for galaxy-scale lenses. Galaxy-scale Einstein rings are produced by the total projected mass distribution of stars and the dark-matter halo, not by a point mass; the Einstein radius depends on the density profile (SIS, NFW, etc.). The acknowledgment in Section 8 that a simple circularly symmetric point-mass model is assumed does not rescue the analysis: the derived α ranges are strongly model-dependent and cannot be interpreted as constraints on MOG unless an explicit, observationally motivated mass profile is fitted.
  4. [Section 6, Eqs. (45)-(53), (68), (69)] Inverting the O(α²) expansions to produce branches at |α|≈25 and α<-1 is invalid. The critical impact parameter in Eq. (45) and the strong-field coefficients a and b in Eqs. (49) and (53) are truncated at second order in α; the polynomial equations used to generate branches with α≈25-28 (e.g. Eqs. (68b)-(68d) and (69b)-(69d)) lie far outside the radius of validity of these expansions. Those branches should be removed or rederived from the full α dependence of the metric functions.
minor comments (6)
  1. [Section 3] The symbol α is used both for the MOG parameter and for the deflection angle in Eq. (15), which is confusing; please rename one of them (the deflection angle is later called α̂, so Eq. (15) should be adjusted accordingly).
  2. [Sections 4 and 6] The symbol D is defined as the dimensionless ratio D_LS/D_OS in Section 4, but in Section 6 and Eq. (70) it denotes the effective angular diameter distance D_OL D_OS/D_LS; this inconsistency makes Eq. (70) appear to contradict Eq. (35) unless the reader notices the redefinition. Please use distinct symbols.
  3. [Eq. (14)] The displayed horizon radii r_± = M(1+α ± √(1+α)) do not appear to reproduce the quoted critical value α_crit=0.674; as written the expression has two positive roots for all α>0 and never merges. Please verify this formula against the exact horizon condition of the metric (12).
  4. [Table 4] In the NGC 1374 row, the time delay at α=0 (1709.08 min) is larger than at α=-0.1 (1446.07 min), yet the text states that positive α increases the delay; please check this entry.
  5. [Throughout] There are several typos and formatting issues: 'weak filed' in Section 1, 'diamter' in Section 6, 'Table 6' in the Conclusion should be 'Table 4', and references [35]-[37] and [51] are duplicated.
  6. [Figures 2-4] The sub-panels labeled (a)-(i) are not referenced in the text, and some axis labels use what appears to be σ where D is intended; please clarify the captions and refer to the panels in the body.

Circularity Check

2 steps flagged · score 7.0 of 10

The Einstein-ring constraints on α are largely tautological: for the galaxy-scale lenses and for M87*, the 'estimated mass' is derived from the same observed angular scale under GR, so comparing it with the MOG mass formula forces α≈0 and does not independently test MOG.

  1. fitted input called prediction [Section 6, Eq. (70) and Table 3]
    "We can also constrain the metric parameter α against past observational data to investigate how far the spherically symmetric MOG deviates from Schwarzschild spacetime. We will conduct it mainly using the measured radius of Einstein rings along with the estimated total mass enclosed within the ring. ... The total estimated mass enclosed by a given Einstein ring in MOG can be determined by simply rearranging (35), M2 = θ2 ED / 4(1 + α) (70)"

    The 'estimated total mass enclosed within the ring' for each galaxy-scale Einstein ring is the standard GR point-mass lens mass M_GR = θ_E^2 D_eff / 4, derived from the same observed Einstein radius θ_E. Equation (70) is M2 = θ_E^2 D_eff / [4(1+α)]. Setting M2 equal to the literature mass M_obs = θ_E^2 D_eff / 4 gives 1/(1+α) = 1, hence α = 0 identically. The intervals in Table 3 therefore only propagate the measurement errors in θ_E and M_obs; they do not provide an independent test of MOG. The point-mass assumption adds no separate constraint.

  2. fitted input called prediction [Section 6, Eq. (67) and constraints (68)]
    "For a start, we assume that the observed Einstein radius equals the outermost image value, (59). Substituting (45) into (59) and solving for M gives M1 = θEDOL/(1+en)(3√3 + 5/2 √3 α − 7/24 √3 α^2) + O(M^3, α^3) (67). Taking M87∗ as the lensing body and using observational results of [51], we have the following constraints up to the 1−σ level"

    The M87* mass M = 6.5×10^9 M_sun from EHT [51] is inferred from the measured shadow angular size assuming GR, with the GR relation bc = 3√3 M. Equation (67) inverts the MOG critical impact parameter bc(α) = 3√3 M + (5/2)√3 M α + ... for the same observed angular scale θ∞. Equating the resulting M1 to the GR-inferred mass forces the α-dependent corrections to vanish, so α ≈ 0 up to the EHT uncertainties. The quoted interval −0.142 ≲ α ≲ 0.338 is therefore a re-expression of the GR mass uncertainty rather than an independent MOG constraint. The Sgr A* constraint is less affected because its mass comes from stellar orbits, but the M87* and galaxy-ring constraints are circular.

full rationale

The analytic core of the paper — the weak-field deflection angle (27) and the strong-field coefficients (49), (53), (56) — is a self-contained calculation from the MOG metric (12) using standard Gibbons-Werner and Tsukamoto methods. These formulas are not circular: they are derived from the assumed line element and reduce to Schwarzschild at α=0, and the Sgr A* constraint can in principle use an independent stellar-dynamical mass. The circularity enters only in the observational constraints on α. For the galaxy-scale Einstein rings, Eq. (70) rearranges the lens equation into M2 = θ_E^2 D_eff/[4(1+α)], while the literature masses in Table 3 are themselves Einstein-ring masses M_GR = θ_E^2 D_eff/4 obtained from the same θ_E under GR; equating the two forces α=0 by construction. For M87*, Eq. (67) uses the EHT shadow-derived GR mass as input, so the constraint range (68a) merely maps the GR mass error into α. Thus the claim of new Einstein-ring constraints is substantially weakened, though not the underlying deflection-angle derivation. Score 7 reflects that most of the paper's observational constraints reduce by construction, while the lensing formulas themselves retain independent content.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central lensing derivations introduce no new entities and only one model parameter (α). The main epistemic load is carried by the assumed MOG metric and by the observational assumptions, two of which are ad hoc: the point-mass galaxy model and the reuse of the Einstein radius for both mass and α.

free parameters (1)
  • α (MOG parameter) = constrained ranges, e.g., -0.142 to 0.338 (M87*, 1σ)
    Dimensionless MOG parameter controlling the effective gravitational constant and the metric; the paper derives constraints on it from Einstein ring and shadow data.
assumptions (7)
  • domain assumption The MOG metric (12) with parameters α and mass M is the correct spacetime for a dark compact object in STVG theory
    Taken from Moffat [8,41]; the paper does not derive it. If the metric is wrong, all lensing results change.
  • standard math The Gibbons-Werner Gauss-Bonnet method yields the weak deflection angle for asymptotically flat static spherically symmetric spacetimes
    Used in Section 3 to derive (27).
  • standard math The Tsukamoto strong-field limit expansion applies, with a single photon sphere r_m and critical impact parameter b_c
    Used in Section 5 to derive (45), (49), (53).
  • ad hoc to paper The lens can be treated as a circularly symmetric point mass in flat cosmology
    Assumed in Section 6 for galaxy-scale Einstein rings (Table 3); unrealistic for galaxies and not valid for dark matter halos.
  • ad hoc to paper For the galaxy rings, the 'measured mass' in the cited literature is an independent estimate not already assuming α=0
    Eq. (70) reuses the same θ_E to define M; if M_obs comes from GR lens modeling, the constraint α≈0 is imposed by construction.
  • domain assumption Flat ΛCDM with ΩM=0.3, ΩΛ=0.7 and H0=100h km/s/Mpc
    Used in Eq. (71)-(72) to compute angular diameter distances for Table 3; standard but not exact.
  • domain assumption The EHT shadow angular radius can be identified with the strong-field limit θ∞ = b_c/DOL
    Used in Section 6 to derive constraints (68)-(69). This is standard but approximate; the observed ring is the photon ring, not exactly the shadow boundary.

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Cite this review

Pith. "Pith review of Gravitational Lensing by a Dark Compact Object in Modified Gravity and Observational Constraints from Einstein Rings." pith.science (2026). https://pith.science/paper/UNELCV7K

@misc{pith2026250206313,
  author       = {Pith},
  title        = {Pith review of: Gravitational Lensing by a Dark Compact Object in Modified Gravity and Observational Constraints from Einstein Rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNELCV7K}},
  note         = {Machine review of arXiv:2502.06313}
}
abstract

In this manuscript, we provide a comprehensive study of gravitational lensing by dark compact objects predicted by a Modified Gravity (MOG) based on the Scalar-Vector-Tensor action, and the aim is to analyze new insights into the nature of gravitational interactions. We compute weak and strong deflection angles for the specified static, spherically symmetric MOG spacetime. Additionally, we dedicate a section to explore observational implications in the weak field limit. By employing a supermassive galactic black hole as a gravitational lens, we compare various parameters in MOG with those of the Schwarzschild black hole as lens in strong-field scenarios. Specifically, we model the black holes M87${^*}$ and Sgr A${^*}$ as lenses within the MOG framework, calculating the corresponding lensing coefficients and distortion parameters in the weak field regime.

Figures

Figures reproduced from arXiv: 2502.06313 by the authors.

Figure 1
Figure 1. Graph of weak deflection angle for different values of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The tangential, radial, and total magnifications are plotted against different parameters for primary [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. For the secondary images, the tangential, radial, and total magnifications are plotted as functions [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: The Distortion parameter is plotted for different values of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Strong deflection angle for the different values of dimensionless parameter [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: a and b is plotted against the different values of α. Two α values correspond to a single a or b value except at the turning point of each graphs; α = 9 7 for a and α = −3.04245 for b. Magenta straight lines represent the a and b values in the Schwarzschild case. To sh…
Figure 7
Figure 7. Figure 7: The Einstein rings for different values of [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: The magnification of the Einstein ring is plotted against various values of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: The behavior of the lensing coefficients: [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Constraints of α taking M87 as a lens. In the left figure, the right branches represent positive θ1 values whilst the left branches are the negative ones. It is the reversed way in the right figure. The shaded regions contains the values of the estimated mass up to 1 …

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