Statistical Strong Lensing as a Test of Conformal Gravity

Da-Ming Chen; Li-Xue Yue

REVIEW 4 major objections 5 minor 39 references

Statistical Strong Lensing as a Test of Conformal Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Conformal gravity's key constant fails strong-lensing data

desk verdict The rotation-curve gamma* fails against strong lensing statistics, but the paper's mass-dependent gamma* claim rests on an unquantified point-mass approximation. read the letter →

arxiv 2506.21019 v1 pith:BUSXBELK submitted 2025-06-26 astro-ph.CO

Li-Xue Yue , Da-Ming Chen This is my paper · ORCID

classification astro-ph.CO

keywords conformalgravitystatisticalstronglensingprobabilitylinearpotentialgalaxystellarmassfunctiondarkmatteralternativequasarlenssearch

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

Conformal Gravity replaces dark matter with an extra linear term in the gravitational potential, whose strength is set by a single universal constant calibrated on spiral-galaxy rotation curves. This paper asks whether that same constant can reproduce the observed statistics of strong gravitational lensing by galaxies and clusters. Using a large sample of lens systems and the SDSS Quasar Lens Search, the authors find that the rotation-curve value predicts far too few lenses. Refitting the constant to lensing data yields values roughly seven orders of magnitude larger, and a best-fit constant still underpredicts wide-separation lenses. The paper concludes that Conformal Gravity cannot simultaneously explain rotation curves and strong lensing statistics unless its linear-potential parameter is allowed to depend on mass, which contradicts its foundational premise.

What carries the argument

The engine of the test is the conformal-gravity metric $B(r) = 1 - 2\beta/r + \gamma r$, whose extra linear term $\gamma r$ adds a $\gamma$-dependent piece to the photon deflection angle, $\hat{\alpha} = 4GM/c^2 r_0 + 2GM\gamma/c^2 + \cdots$. The universal parameter $\gamma^*$ enters through $\gamma = (M/M_\odot)\gamma^* + \gamma_0$, with $\gamma_0$ fixed from the cosmological background. The authors combine this deflection angle with a Hernquist luminous-mass profile, a Schechter or double-Schechter galaxy stellar mass function, and the magnification-bias-weighted cross section to predict the probability that a quasar at $z_S = 1.57$ is split into images separated by more than $\Delta\theta$ with flux ratio below $3.16$; matching that prediction to the observed sample yields both the mass-dependent fit and the best-fit constant.

What would settle it

Measure $\gamma^*$ for the same lens systems using an extended-mass lens model that includes the full stellar and gas distribution instead of only the mass inside the Einstein radius; if the apparent decrease of $\gamma^*$ with $M_*$ disappears or inverts, the claimed mass dependence is an artifact of the point-mass shortcut.

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Extended reading notes

Core claim

The central claim is that the universal linear-potential parameter of Conformal Gravity, $\gamma^*$, fails the statistical strong-lensing test. Fitting $\gamma^*$ system-by-system to the Einstein radii of a compiled galaxy and cluster lens sample gives a mass-dependent relation $\gamma^*_t = 4.57 \times 10^{-15} (M_*/M_\odot)^{-1.51}\,\mathrm{m}^{-1}$, decreasing with stellar mass, rather than the constant value required by the theory. When this mass-dependent form is used in the lensing probability calculation, the prediction overshoots at separations below about 3 arcseconds and undershoots badly at larger separations; a best-fit constant $\gamma^*_c = 3.50 \times 10^{-32}\,\mathrm{m}^{-1}$ still falls below the observed lensing probability at large separations. The conclusion is that the current formulation of Conformal Gravity cannot provide a self-consistent account of both galactic rotation curves and strong lensing statistics.

Load-bearing premise

The analysis assumes that each lens can be treated as a point mass whose visible mass inside the Einstein radius is the only source of the linear potential, so that mass outside that radius merely raises the fitted value of $\gamma^*$ uniformly, without changing how it scales with lens mass.

Editorial extensions

If this is right

The rotation-curve value of $\gamma^*$ predicts a strong-lensing probability far below the observed one at all separations, so the same parameter cannot serve both galaxy dynamics and lensing.
The best-fit constant $\gamma^*_c = 3.50 \times 10^{-32}\,\mathrm{m}^{-1}$ improves the small-separation prediction but still leaves a deficit at separations above about 3 arcseconds, indicating the failure is not cured by simple renormalization.
If the mass-dependent $\gamma^*_t$ is used, low-mass galaxies receive a much larger linear potential, boosting the predicted number of small-separation lenses and suppressing wide-separation lenses.
The same SDSS lensed-quasar sample, when modeled with SIS+NFW dark halos in General Relativity, also underpredicts large-separation lenses, so the large-separation deficit is not unique to Conformal Gravity.
The results imply that strong-lensing statistics can discriminate between Conformal Gravity and dark-matter-based models, at least for the current generation of lens surveys.

Reading between the lines

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

A natural next step would be to fit the same lens sample with an extended-mass model that includes stellar mass outside the Einstein radius; if the apparent decrease of $\gamma^*$ with $M_*$ persists, the mass dependence is physical, but if it flattens, it is an artifact of the point-mass shortcut.
The mass dependence $\gamma^*_t \propto M_*^{-1.51}$ has the same functional flavor as a dark-matter halo profile, so the linear-potential term may be mimicking the missing-mass scaling that Conformal Gravity was intended to remove.
Strong-lensing time delays, which depend on the absolute gravitational potential rather than only the deflection angle, could break the degeneracy between $\gamma^*$ and the assumed mass profile and provide a sharper test than image separations 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.

Desk Editor's Note

read the letter

Colleague—read this one if you want a concrete statistical test of conformal gravity on lensing. The headline: the standard rotation-curve gamma* fails against strong lensing statistics by orders of magnitude, and that part is probably solid. But the paper's additional claim that gamma* cannot be constant—that it must be mass-dependent—does not yet hold. The per-system gamma* values are forced from a point-mass lens model, and the exterior-mass contribution is never actually computed.

What's genuinely new: they use the corrected deflection angle that includes the 2Mγ term and re-fit gamma* on the large Oguri sample, not just the few clusters used in earlier work. That gives gamma*_t(M*) ∝ M*^-1.51, and they then compare the predicted lensing probability against the SDSS SQLS sample. The comparison is straightforward, and the conclusion that the rotation-curve gamma* under-predicts the observed lensing frequency is consistent with earlier studies and visible by eye in Figure 3. They are also honest in the conclusions: they call it "difficulties" rather than a refutation.

The soft spots are real. The per-system gamma*_t values in Table 1 are solutions of the lens equation given the Einstein radius, so their scatter is not an independent test. The exterior-mass effect is asserted after Eq. (31) to "not alter the order of magnitude," but no calculation is shown. In conformal gravity the linear-potential parameter is an integral over the full mass distribution, so if exterior mass enters with a different weight per unit mass for low- and high-mass lenses, the -1.51 slope could be an artifact. Second, Eq. (35) is an empirical fit to the same sample used later to predict the lensing probability at small separations, so that part of the curve is inherited, not predicted. Third, there are no error bars on the fitted gamma* values, and the probability fits have no uncertainties, so it is hard to know how much to weight the large-separation shortfall. Finally, the paper notes that GR with SIS+NFW also under-predicts large-separation lenses, which makes one wonder whether the shortfall is a missing-mass issue rather than a specific CG failure.

The citation pattern is fine. The paper builds on Cutajar and Zarb Adami, uses standard lensing samples, and is careful to distinguish their deflection angle from earlier ones.

Who this is for: anyone following modified-gravity tests with lensing statistics. It deserves a serious referee, but a referee should ask for a quantitative treatment of the exterior-mass contribution and error propagation before the mass-dependence claim is accepted. The central rotation-curve tension is probably robust.

Referee Report

4 major / 5 minor

Summary. The paper tests conformal gravity (CG) by confronting its universal linear-potential parameter \gamma^* with statistical strong lensing observations. The authors use the Oguri et al. strong-lensing sample to infer a value of \gamma^* for each system under a point-mass lens model, with the stellar mass taken as the Hernquist-enclosed mass within the Einstein radius, and obtain a mass-dependent empirical relation \gamma^*_t = 4.57\times10^{-15}(M_*^{\rm enc}/M_\odot)^{-1.51} m^{-1} (Eq. 35). They then compute the lensing probability for the SQLS/Inada et al. quasar-lens sample using the galaxy stellar mass function, finding that the rotation-curve value \gamma^*=5.42\times10^{-39} m^{-1} severely underpredicts lensing, that the mass-dependent relation overpredicts at small separations and underpredicts at large separations, and that even a best-fit constant value \gamma^*_c=3.50\times10^{-32} m^{-1} (Eq. 45) fails at large separations. The paper concludes that CG cannot self-consistently explain both galactic rotation curves and the observed statistics of strong lensing.

Significance. If the result is correct, it is a significant multi-scale test of conformal gravity: the same parameter that fits rotation curves would be ruled out by strong lensing statistics, and the apparent mass dependence of \gamma^* would contradict the theory's foundational premise of a universal constant. The paper is valuable in using larger, well-defined samples than earlier single-cluster studies, in including the second-order \sim GM\gamma term in the deflection angle, and in making its fitted formulas explicit and reproducible from the published sample. However, the central falsification claim is currently carried by an unvalidated point-mass/exterior-mass reduction, and the probability calculation contains a mismatch between the enclosed mass used in the fitted relation and the total stellar mass sampled by the mass function. The conclusion may survive those issues, but the manuscript does not yet demonstrate that it does.

major comments (4)

[Section 4, paragraph after Eq. (31)] The manuscript asserts that luminous mass outside the Einstein radius contributes non-locally to the deflection and that fitting with only M_*(\theta_E) therefore overestimates \gamma^*, but that this overestimation 'does not alter the order of magnitude of \gamma^*.' No calculation is provided for this assertion. This is load-bearing: the per-system values in Table 1, the mass-dependent fit Eq. (35), and the constant-\gamma^*_c comparison in Eq. (45) all inherit the point-mass reduction. A concrete estimate should be given, e.g., by recomputing the inferred \gamma^*_t under alternative assumptions such as using the total stellar mass M_* rather than M_*(\theta_E), or by a genuine extended-source calculation in CG. Without this, the paper's strongest claim—that no constant \gamma^* can fit the strong-lensing statistics—is not established.
[Section 6, Eq. (44)] There is a mismatch in the mass variable between the fitted relation and the probability integral. Equation (35) is a function of the enclosed stellar mass M_*(\theta_E), while Eq. (44) integrates over the total stellar mass M drawn from the galaxy stellar mass function. The text never specifies how M_*(\theta_E) is obtained from M and \theta_E in the probability calculation; no effective-radius or size-mass relation appears in Section 6. If the authors set M = M_*(\theta_E), this contradicts Eq. (34) and overestimates the linear-potential contribution for low-mass systems; if they use total M, then Eq. (35) is misapplied. The dash-dotted curve in Figure 3 is therefore not a well-defined prediction of the model, and the large-separation underprediction attributed to \gamma^*_t needs to be recomputed with a consistent mass prescription.
[Section 6 / Figure 3] The claim that the theoretical predictions fall 'significantly below' the observed distribution at large image separations is not quantified. The observed probability is built from only 26 lensed quasars, so the large-separation bins carry substantial Poisson uncertainties, yet Figure 3 shows no error bars, and no goodness-of-fit statistic (e.g., a Kolmogorov-Smirnov test or a likelihood) is reported. In addition, the procedure used to obtain the best-fit constant value \gamma^*_c=3.50\times10^{-32} m^{-1} is not described: there is no statement of the fitted quantity, the binning, the treatment of magnification bias, or the confidence interval. The central falsification claim should be supported by a quantitative statistical comparison.
[Section 4, Eq. (35)] The mass-dependent relation Eq. (35) is a least-squares fit to values of \gamma^*_t that are, by construction, forced to reproduce the observed Einstein radius of each system through the lens equation. The scatter in \gamma^*_t is therefore not an independent prediction of conformal gravity; the apparent anti-correlation with mass may reflect the assumed Hernquist profile and the mass-concentration relation rather than the theory. This should be stated explicitly, and the relation should be presented as an empirical parametrization, not as a 'derived' formula, in the abstract and conclusions.

minor comments (5)

[Section 2, Eq. (14)] The cosmological Hubble parameter in Eq. (14) contains \bar\Omega_{M0}, but the text specifies only \bar\Omega_{K0}=0.67 and \bar\Omega_{\Lambda0}=0.33; the adopted value of \bar\Omega_{M0} should be stated explicitly.
[Figure 1] The axis label 'log M* (log M_\odot)' is ambiguous; it should be written as log10[M_*/M_\odot] for clarity.
[Eq. (35)] The fit in Eq. (35) should be reported in log-space with uncertainties on the slope and normalization, since the displayed form implies a precision in the prefactor that the least-squares fit almost certainly does not warrant.
[Abstract and Conclusions] The abstract states that the authors 'derived a formula for \gamma^*' as a function of stellar mass; because Eq. (35) is an empirical fit to lensing data, the wording should be softened to 'fit' or 'parameterized.'
[Table 1] The subscript in \gamma^*_t is not defined in the table header; the text should specify that 't' denotes the value fitted from strong lensing, as opposed to the rotation-curve value \gamma^*.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mass-dependent gamma*_t is fitted to the Oguri sample and then applied to the separate Inada SQLS sample, while gamma*_c is a best-fit parameter whose poor large-separation agreement is a genuine goodness-of-fit result, not a forced prediction.

full rationale

The paper's two central quantitative claims are (1) that gamma*_t follows a power-law in M*(theta_E) with slope -1.51 (Eq. 35), and (2) that no constant gamma* reproduces the observed SQLS lensing probability. Step (1) is a least-squares fit to the Oguri et al. sample (Table 1), not a prediction: the Inada et al. SQLS sample used for the probability comparison in Section 6 is an independent data set, so inserting Eq. (35) into Eq. (44) is an out-of-sample application, not a recycled input. Step (2) is a parameter estimation: gamma*_c = 3.50e-32 m^-1 (Eq. 45) is obtained by fitting Eq. (44) to the SQLS distribution and then evaluating whether the best-fit model reproduces that distribution. A model that fails in the large-separation bins despite being tuned to the full distribution is a genuine lack of fit, not a circular prediction. The load-bearing point-mass/enclosed-mass reduction in Section 4 (after Eq. 31) and the unsupported assertion that exterior mass only elevates gamma* without changing its order of magnitude are correctness or robustness risks, not circularity: they do not make any equation equal to its own input. The only self-citations (Refs. 35, 36, 38, 39) concern standard lensing-probability formulas and a future-work remark; they are not load-bearing. Under the stated rules, no circular step can be exhibited.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central claim rests on several external inputs: the MK vacuum solution, the adopted deflection angle, the Hernquist point-mass model, the GSMF, and the magnification-bias slope. The main free parameters are the two coefficients of the gamma*_t(M*) fit and the tuned constant gamma*_c. The mass-dependent gamma*_t is best understood as an invented, empirically fitted modification of the theory rather than a derived consequence.

free parameters (4)

Normalization of gamma*_t mass relation = 4.57e-15 m^-1
Least-squares amplitude in Eq. (35) obtained from the strong lensing systems in Table 1. This is an empirical fit, not a derived CG prediction.
Slope of gamma*_t mass relation = -1.51
Power-law index in Eq. (35) fitted to the forced gamma*_t values. It encodes the claimed anti-correlation between gamma* and stellar mass.
Constant gamma*_c = 3.50e-32 m^-1
Best-fit constant linear potential parameter in Eq. (45), tuned to the observed lensing probability from the Inada et al. sample. Fitted, not predicted.
gamma0 = 3.06e-28 m^-1
Universal linear potential term from rotation-curve fits [3], adopted as a fixed input in all gal = N*gamma* + gamma0 calculations. Not fit in this paper, but the central claim depends on it.

assumptions (6)

domain assumption The Mannheim-Kazanas vacuum solution B(r) = 1 - 2*beta/r + gamma*r describes static spherically symmetric lenses, with the linear potential term generated by luminous mass.
The paper restricts to vacuum solutions where W_mu_nu = T_mu_nu = 0 and adopts Eq. (24). The validity of this solution for real galaxies and clusters is assumed.
domain assumption The deflection angle formula Eq. (31), including the 2*GM*gamma/c^2 term, is the correct expression for light bending in conformal gravity.
The paper notes that previous formulas disagree [12-17] and adopts the result of [17]. If this formula is wrong, all fitted gamma* values shift.
ad hoc to paper The lens can be treated as a point mass with a Hernquist-enclosed luminous mass M*(theta_E), and external mass contributes non-locally to the fitted gamma*.
Section 4 uses the point-mass lens equation with M*(theta_E) and asserts that external mass only elevates gamma* without changing its order of magnitude. This assertion is not derived.
domain assumption The galaxy stellar mass function from Davidzon et al. and Li and White describes the comoving number density of lenses.
Used in Eq. (44). Systematic uncertainties in the GSMF are not propagated into the probability comparison.
domain assumption The source luminosity function is a power law with slope 2.1, giving magnification bias B approximately mu^1.1.
Adopted from Rusin and Tegmark [34]. This affects the normalization of the predicted lensing probability.
domain assumption The conformal cosmology background parameters Omega_K0 = 0.67, Omega_Lambda0 = 0.33, and H0 = 69.3 km/s/Mpc from [7] are correct.
These parameters enter the angular diameter distances used in Eqs. (15) and (32). They are taken from the cited literature without independent verification.

invented entities (1)

Mass-dependent gamma*_t(M*) relation
purpose: Introduced to reconcile conformal gravity's linear potential with strong lensing data after the constant gamma* fails; it replaces the universal constant required by CG with an ad hoc mass scaling.
The formula of Eq. (35) is a least-squares fit to lensing data, not a prediction of conformal gravity. No independent observable is predicted outside the fitted sample, and it contradicts the theory's requirement that gamma* be a universal constant.

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Cite this review

Pith. "Pith review of Statistical Strong Lensing as a Test of Conformal Gravity." pith.science (2026). https://pith.science/paper/BUSXBELK

@misc{pith2026250621019,
  author       = {Pith},
  title        = {Pith review of: Statistical Strong Lensing as a Test of Conformal Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BUSXBELK}},
  note         = {Machine review of arXiv:2506.21019}
}

abstract

As an alternative gravitational theory to General Relativity (GR), Conformal Gravity (CG) can be verified through astronomical observations. Currently, Mannheim and Kazanas have provided vacuum solutions for cosmological and local gravitational systems, and these solutions may resolve the dark matter and dark energy issues encountered in GR, making them particularly valuable. For static, spherically symmetric systems, CG predicts an additional linear potential generated by luminous matter in addition to the conventional Newtonian potential. This extra potential is expected to account for the observations of galaxies and galaxy clusters without the need of dark matter. It is characterized by the parameter $\gamma^*$, which corresponds to the linear potential generated by the unit of the solar mass, and it is thus a universal constant. The value of $\gamma^\ast$ was determined by fitting the rotation curve data of spiral galaxies. These predictions of CG should also be verified by the observations of strong gravitational lensing. In this study, building upon the previous research, we tested CG via strong lensing statistics. We used a well-defined sample that consisted of both galaxies and galaxy clusters. This allowed us to test CG through statistical strong lensing in a way similar to the conventional approach in GR. As anticipated, our results were consistent with previous studies, namely that the fitted $\gamma^*$ is much larger than that from rotation curves. Intriguingly, we further discovered that, in order to fit the strong lensing data of another sample, the value of $\gamma^*$ cannot be a constant, as is required in CG. Instead, we derived a formula for $\gamma^*$ as a function of the stellar mass $M_*$ of the galaxies or galaxy clusters. It was found that $\gamma^*$ decreases as $M_*$ increases.

Figures

Figures reproduced from arXiv: 2506.21019 by the authors.

**Figure 1.** A plot showing the relation between the Mannheim–Kazanas linear potential parameter γ ∗ and the stellar mass M∗(θE) within the Einstein radius, which was derived from strong gravitational lensing observations. The data points represent individual lensing systems, and the dotted line denotes the empirical best-fit result. In the following section, we will perform a statistical analysis of the lensing probability dist… view at source ↗

**Figure 2.** The lensing Equation (32) for a point luminous mass of M = 1011M⊙. The parameters were set as γ ∗ = 1.5 × 10−31 m−1 , zS = 1.57, and zL = 0.1. The horizontal dashed line represents the allowed value of βqr = 0.82, which is constrained by the largest flux density ratio qr = 3.16 of the sample that was used in ref. [30]. In this study, we utilized the strong lensing sample presented in reference Inada et al. [30]. In … view at source ↗

**Figure 3.** Lensing probability with image separations > ∆θ and flux density ratios < qr = 3.16. The thick histogram shows the observed distribution from the Inada et al. [30] sample. The theoretical predictions are shown for the following: a standard rotation-curve-derived γ ∗ = 5.42 × 10−39 m−1 (thin solid line; [3]); obtained from strong gravitational lensing fits γ ∗ t = 4.57 × 10−15 M∗(θE) M⊙ −1.51 m−1 (dash-dotted line)… view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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