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REVIEW 4 major objections 4 minor 35 references

Test of conformal gravity as an alternative to dark matter from the observations of elliptical galaxies

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Conformal gravity's linear term fails on elliptical galaxies.

desk verdict A genuinely new test of conformal gravity on elliptical scales, but the headline gamma*-M* correlation is a fitting artifact; the core mismatch deserves a serious but demanding referee. read the letter →

arxiv 2506.03955 v1 pith:JBBOPE7N submitted 2025-06-04 astro-ph.GA gr-qc

Li-Xue Yue , Da-Ming Chen This is my paper
classification astro-ph.GAgr-qc
keywords conformalgravitydarkmatteralternativeellipticalgalaxiesdwarfspheroidalJeansequationvelocitydispersionlinearpotentialgamma-staruniversality
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 tests conformal gravity (CG) against the velocity dispersions of elliptical and dwarf spheroidal galaxies, where CG predicts an extra linear potential term beyond the Newtonian one. The authors extend the Jeans equation by substituting the CG potential for the Newtonian potential and fit the theory's parameters to three galaxy samples. They find that the linear-potential strength required by dwarf spheroidals is about four orders of magnitude larger than the value inferred from spiral rotation curves, meaning the CG extra potential is not sufficient to explain ellipticals without some dark matter. They also find that the fitted parameter $\gamma^*$ decreases with increasing stellar mass, a variation that contradicts CG's prediction that $\gamma^*$ should be a universal constant. If correct, this would close off conformal gravity as a dark-matter replacement, at least for pressure-supported galaxies.

What carries the argument

The load-bearing object is the non-relativistic CG potential of a point mass, $V(r) = -\beta^*c^2 N^*/r + \gamma^* c^2 N^* r/2$, augmented by the universal background term $\gamma_0 c^2 r/2$. Substituting this potential into the spherical, isotropic, constant-dispersion Jeans equation yields a direct relation between the observed surface-brightness profile, $\sigma_*$, and the parameters $(\gamma^*, \gamma_0)$; using Plummer profiles for dwarf spheroidals and Sersic profiles for bright ellipticals, the equations are evaluated at $r=R_e$ and least-square fitted to each galaxy. This machinery converts measured velocity dispersions into a per-galaxy $\gamma^*$, allowing the paper to compare ellipticals with spirals and to search for a mass dependence.

What would settle it

Measure the full line-of-sight velocity dispersion profile and velocity anisotropy of a dwarf spheroidal using resolved stellar proper motions, then solve the anisotropic Jeans equation with the CG potential; if $\gamma^*$ then converges to $5.42\times10^{-39}\,{\rm m}^{-1}$ across galaxies, the paper's failure claim would be overturned.

Watch

Extended reading notes

Core claim

The paper's central claim is that the non-relativistic conformal-gravity potential, $V(r)= -N^*\beta^* c^2/r + N^*\gamma^* c^2 r/2 + \gamma_0 c^2 r/2$, fails to account for elliptical galaxy dynamics once tested with velocity dispersion instead of rotation curves. Applying the CG potential in the spherical Jeans equation to 43 dwarf spheroidals and two bright-elliptical samples gives $\gamma^*_{\rm dSph}=1.22\times10^{-35}\,{\rm m}^{-1}$, roughly four orders of magnitude larger than the spiral-galaxy value $\gamma^*=5.42\times10^{-39}\,{\rm m}^{-1}$. Treating $\gamma^*$ as a free parameter with $\gamma_0$ fixed produces a strong $\gamma^*$--$M^*$ anticorrelation in dwarf spheroidals, $\gamma^*_{\rm dSph}=2.75\times10^{-28}(M^*/M_\odot)^{-0.963}\,{\rm m}^{-1}$, which mirrors the dark-matter-to-stellar-mass trend in Newtonian fits. Since CG demands $\gamma^*$ be a universal constant independent of galaxy, the authors conclude that the variation violates the theory's core prediction and that elliptical galaxies are better described by Newtonian gravity with dark matter.

Load-bearing premise

The central claim depends on treating each galaxy as spherical, isotropic, and with constant velocity dispersion, and on evaluating the Jeans equation at the single radius $r=R_e$; if velocity anisotropy or dispersion gradients are significant, the inferred $\gamma^*$ values could change.

Editorial extensions

If this is right

  • If the spiral value $\gamma^*=5.42\times10^{-39}\,{\rm m}^{-1}$ is correct, CG's linear potential supplies only a tiny fraction of the extra force needed in dwarf spheroidals, so dark matter or another mechanism is still required.
  • The strong $\gamma^*$--$M^*$ anticorrelation in dwarf spheroidals contradicts CG's requirement that $\gamma^*$ be a universal constant, removing a key predictive advantage of the theory.
  • The same anticorrelation pattern reappears, more weakly, in bright ellipticals from the SDSS and SLACS samples, indicating that the problem is not limited to dwarf galaxies.
  • In the Newtonian comparison, the dark-matter fraction $M_{\rm DM}/M_*$ within $R_e$ falls as stellar mass rises, closely matching the $\gamma^*$--$M^*$ trend and suggesting the CG parameter is absorbing the role normally played by dark matter.

Reading between the lines

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

  • A natural extension would be to let the velocity-dispersion anisotropy $\beta(r)$ vary freely; if that removes the factor-of-$10^4$ mismatch, the failure would sit in the simplified Jeans modeling rather than in CG itself.
  • If $\gamma^*$ truly must vary as a power of $M_*$, CG loses its status as a parameter-free alternative, because the theory would need a new mass-dependent rule to explain why the coupling changes from dwarfs to spirals.
  • The fitted slope of the $\gamma^*$--$M_*$ relation, roughly $-0.96$ for dwarf spheroidals, could be compared with dark-matter scaling relations from independent mass measurements, such as gravitational lensing, to see which framework more economically explains the trend.
  • A decisive test could use galaxies with resolved stellar proper motions to measure the full velocity-dispersion tensor and check whether the CG linear potential or a dark halo reproduces the enclosed mass profile without fine-tuning.
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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 / 4 minor

Summary. The manuscript extends the Jeans equation to the non-relativistic conformal-gravity potential V = -GM/r + (N*gamma* + gamma0)c^2 r/2 and fits the linear-potential parameters gamma* and gamma0 to the velocity dispersions of 43 dwarf spheroidal galaxies (Plummer profile), 76 SDSS early-type galaxies, and 73 SLACS lenses (Sersic profile). It reports a best-fit gamma*_dSph about four orders of magnitude larger than the spiral value, and a log-log correlation between per-galaxy gamma* and stellar mass with slope -0.96, which it interprets as a violation of conformal gravity's prediction that gamma* is universal. It also compares the inferred dark-matter-to-stellar-mass ratios in Newtonian dynamics.

Significance. If the claims were correct, the paper would provide a sharp observational falsification of conformal gravity as a dark-matter substitute and identify a scaling relation that any modified-gravity theory must reproduce. The paper is also useful for compiling kinematic and structural parameters of dSph and early-type galaxy samples. However, the two central quantitative results are not supported by the analysis as presented: the fit has no error budget, and the gamma*-M* correlation is a mathematical artifact of the boundary choice for gamma0. The negative result for conformal gravity may survive a more careful fit, but it is not established here.

major comments (4)
  1. [Section 3.3, Eq. (51)] The chi-squared in Eq. (51) sets sigma_obs = 1 and evaluates the Jeans equation only at r = Re, so the reported values gamma*_dSph = 1.22 x 10^-35 m^-1 and gamma0 = 5.27 x 10^-28 m^-1 come with no uncertainties and no goodness-of-fit measure. The order-of-magnitude comparison with the spiral value (5.42 x 10^-39 m^-1) therefore has no statistical basis. The fits should be redone with the full observational errors in sigma, Re, and M* propagated, and with a fit over the observed radial profile where available.
  2. [Section 3.3, Eqs. (39), (52), (53)] The correlation in Fig. 1 is generated by the fitting procedure rather than by the data. With gamma0 fixed at the boundary value 3.97 x 10^-29 m^-1, solving Eq. (39) for gamma*_i at r = Re gives an expression whose denominator is proportional to the stellar mass M_i and which contains an explicit -C gamma0/M_i term, so an inverse scaling with M_i is built in. The reported slope -0.963 is therefore not evidence against a universal gamma*. A constant-gamma* model should be fitted to the same data and compared via an information criterion or F-test before making any universality claim.
  3. [Section 3.2, Eqs. (35), (39), (43)] The Jeans model assumes isotropy (beta = 0), a constant velocity dispersion per galaxy, and is evaluated at the single radius r = Re. For dwarf spheroidals, anisotropy and radial dispersion gradients can change the inferred potential by factors of order unity, and the same assumption is applied to the Newtonian comparison. The quantitative mismatch between dSphs and spirals should be accompanied by a systematic-error estimate due to these modeling choices.
  4. [Section 3.3] The choice of gamma0 = 3.97 x 10^-29 m^-1 is made from the same dSph data after noting that the best-fit gamma0 is 5.27 x 10^-28 m^-1. Using a data-selected boundary to 'ensure gamma* cannot be negative' biases the subsequent per-galaxy gamma* values and makes the universality test circular. The test should instead adopt an independent gamma0 (e.g., the spiral value in Eq. (31)) or sample gamma0 from a prior.
minor comments (4)
  1. [Throughout] There are several typographical errors, including 'lang-standing' in the Introduction, 'aplly' and 'patern' in Section 3.3, and 'dSpshs' in Section 3.2; these should be corrected.
  2. [Table 1, Table 3] Table 1 contains a final row labeled 'average' that is not a galaxy and should be moved to the caption or omitted; Table 3 has a column header that reads 'gamma*_SDSS' where the text and figure use 'gamma*_SLACS'.
  3. [Eq. (54)] The aperture correction sigma_e = sigma_SDSS(1.5''/theta_e)^0.05 is stated without specifying the exact definitions of theta_e and the original source for the exponent; please provide the reference and units.
  4. [Figures 5 and 6] The caption of Fig. 5 begins with a lowercase 'the' and the notation for gamma* is inconsistent (SLACS vs SDSS) across captions and tables; please harmonize the notation.

Circularity Check

2 steps flagged · score 8.0 of 10

The reported γ*–M* anticorrelation is forced by the fitting procedure and by the definition γ = N*γ*, so it is not an independent test of CG's universal-γ* prediction.

  1. fitted input called prediction [Section 3.3, paragraph beginning 'To achieve this, we fix γ0...' and Equation (53)]
    "To achieve this, we fix γ0 to be a smaller value of 3 .97 × 10−29 m−1 (as opposed to the optimized value of γ0,dSph = 5.27 × 10−28 m−1), and keep γ∗ as a free parameter to be determined. This fixed value of γ0 is obtained by setting γ∗ = 0 for all galaxies and fitting the value of γ0 according to the Jeans equation (39), then finding the smallest one. ... For our selected sample of 43 dSphs, by applying Equation (39) to Equation (52) we find an empirical formula γ∗ dSph = 2.75 × 10−28(M∗/M⊙)−0.963 m−1."

    The constant γ0 is not fixed by independent spiral-galaxy data or by an external CG calculation; it is chosen from the same dSphs sample as the smallest value that keeps all per-galaxy γ* non-negative. The per-galaxy γ* values are then residuals left after subtracting this boundary value, and the regression of those residuals against M* is presented as a discovered correlation. Because the boundary choice and the per-galaxy γ* values come from the same fitting operation, the resulting γ*–M* relation is a property of the fitting procedure rather than an independent prediction of CG that can be used to test the theory's universality.

  2. self definitional [Section 3.2, Equations (26) and (39); Section 3.3, Equation (53)]
    "If we denote N ∗ = M M⊙ , β = N ∗β∗ and γ = N ∗γ∗, then for any point mass M , the expression for its potential shown in Equation (24) can be rewritten as V (r) = Vβ + Vγ = − N ∗β∗c2 r + N ∗γ∗c2r 2 . ... as is evident from Equation (30), for a given galaxy, the combination of γ0 and N ∗γ∗ must remain a constant. Thus, a decrease in the value of γ0 necessarily implies an increase in the value of γ∗."

    Solving Equation (39) at r = Re for γ* divides the residual left after subtracting the fixed γ0 term by N* = M*/M_sun, because the CG linear potential is written as N*γ* c²r/2. The same M* is then placed on the abscissa of the correlation fitted in Equation (53). Hence the fitted γ* carries an explicit, built-in 1/M* dependence: for an approximately constant residual, the regression of log10 γ* on log10 M* will mechanically produce a slope close to −1. The claimed variation of γ* with M* is therefore imposed by the definitional relation γ = N*γ* and by the chosen boundary value of γ0, not an observed property that independently falsifies CG.

full rationale

The paper's basic negative result—that the CG linear potential parameter calibrated on spiral galaxies (γ* ≈ 5.42×10⁻³⁹ m⁻¹) is too weak to explain dSphs dynamics, which prefer γ* ≈ 1.22×10⁻³⁵ m⁻¹—is an empirical comparison against external rotation-curve values and is not circular. The circularity lies in the separate, headline-grabbing claim that the fitted γ* values correlate strongly with stellar mass and that this correlation violates CG's prediction of a universal γ*. That correlation is not an independent observable: each γ* is obtained by solving the Jeans equation at r = Re with γ0 fixed to the boundary value from the same sample that keeps all γ* non-negative, and the solution contains an explicit factor 1/N* = M_sun/M* from the definition γ = N*γ*. Regressing log10 γ* against log10 M* therefore has a built-in near-slope of −1 from construction, as seen in Equation (53)'s exponent −0.963. The self-citations to Chen (2022) and Chen & Wang (2024) appear only as a speculative alternative at the end and are not load-bearing. Because a central claim reduces by construction to the fitting procedure and definition, the circularity score is 8, although the four-orders-of-magnitude mismatch with the spiral-calibrated γ* retains independent empirical content.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities. The free parameters are the conformal gravity coefficients gamma* and gamma0, which are fitted to the galaxy data, plus the parameters of the empirical power-law correlation. The main load-bearing assumptions are the simplified Jeans model and the form of the conformal gravity potential inherited from Mannheim's framework.

free parameters (7)
  • gamma* (dSph global fit) = 1.22e-35 m^-1
    Fitted to dSph velocity dispersions via Equation (39) at r=Re, with gamma0 free.
  • gamma0 (dSph global fit) = 5.27e-28 m^-1
    Fitted simultaneously with gamma* using the same dSph data.
  • gamma0 (fixed for correlation) = 3.97e-29 m^-1
    Chosen as the smallest gamma0 that keeps all fitted gamma* non-negative after setting gamma*=0; a post hoc selection that shapes the gamma*-M* correlation.
  • per-galaxy gamma* (43 dSphs) = range about 1e-29 to 1e-36 m^-1
    Each galaxy's gamma* computed from Equation (39) at r=Re with the fixed gamma0.
  • a and b in gamma*-M* power law = a = -0.963, b = -27.56
    Least squares fit to log10 gamma* = a log10(M*/Msun) + b for the 43 dSphs, Equation (52)-(53).
  • gamma*_SDSS per galaxy = around 1e-35 m^-1
    Computed from the Sersic Jeans equation (43) for the SDSS DR10 sample.
  • gamma*_SLACS per galaxy = around 1e-36 to 1e-38 m^-1
    Computed from the Sersic Jeans equation (43) for the SLACS sample.
assumptions (6)
  • domain assumption The non-relativistic limit of conformal gravity is a Newtonian potential plus a linear potential V_gamma = (1/2) gamma c^2 r, with gamma = N* gamma* for a source of N* solar masses.
    Invoked in Equations (24)-(26) from Mannheim (2006); this is the theory being tested.
  • standard math The collisionless Boltzmann equation and Jeans equation apply with the modified Hamiltonian H = 1/2 v^2 + V(x).
    Section 3.2, Equations (32)-(34); standard tool in stellar dynamics.
  • domain assumption Systems are spherically symmetric, isotropic (beta=0), and have constant velocity dispersion sigma^2_r = sigma^2_*.
    Stated just before Equation (35); simplifies the Jeans equation. Generally not true for ellipticals and can bias the inferred potential.
  • domain assumption The luminous mass distribution follows the surface brightness profile with a constant mass-to-light ratio, and no dark matter is included in the Newtonian part.
    Used to derive rho(r) from I(R) in Equations (38) and (42); variations in M/L would change V'_beta and thus the fitted gamma*.
  • domain assumption The potential of an extended source is a superposition of point-mass conformal gravity potentials, so V_gamma(r) = (gamma* c^2 / 2) integral dM |r - r'|.
    Equation (45); linearity follows from the fourth-order Poisson equation, but the coefficient gamma* per unit mass is assumed universal.
  • domain assumption A universal background linear potential gamma0 c^2 r / 2 from the rest of the universe applies to every local system.
    Equation (30) and the fitted gamma0; this is part of the conformal gravity model being tested.

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Pith. "Pith review of Test of conformal gravity as an alternative to dark matter from the observations of elliptical galaxies." pith.science (2026). https://pith.science/paper/JBBOPE7N

@misc{pith2026250603955,
  author       = {Pith},
  title        = {Pith review of: Test of conformal gravity as an alternative to dark matter from the observations of elliptical galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBBOPE7N}},
  note         = {Machine review of arXiv:2506.03955}
}
abstract

As an alternative gravitational theory to General Relativity (GR), the Conformal Gravity (CG) has recently been successfully verified by observations of Type Ia supernovae (SN Ia) and the rotation curves of spiral galaxies. The observations of galaxies only pertain to the non-relativistic form of gravity. In this context, within the framework of the Newtonian theory of gravity (the non-relativistic form of GR), dark matter is postulated to account for the observations. On the other hand, the non-relativistic form of CG predicts an additional potential: besides the Newtonian potential, there is a so-called linear potential term, characterized by the parameter $\gamma^*$, as an alternative to dark matter in Newtonian gravity. To test CG in its non-relativistic form, much work has been done by fitting the predictions to the observations of circular velocity (rotation curves) for spiral galaxies. In this paper, we test CG with the observations from elliptical galaxies. Instead of the circular velocities for spiral galaxies, we use the velocity dispersion for elliptical galaxies. By replacing the Newtonian potential with that predicted by non-relativistic form of CG in Hamiltonian, we directly extend the Jeans equation derived in Newtonian theory to that for CG. By comparing the results derived from the ellipticals with that from spirals, we find that the extra potential predicted by CG is not sufficient to account for the observations of ellipticals. Furthermore, we discover a strong correlation between $\gamma^*$ and the stellar mass $M^*$ in dwarf spheroidal galaxies. This finding implies that the variation in $\gamma^*$ violates a fundamental prediction of Conformal Gravity (CG), which posits that $\gamma^*$ should be a universal constant.

Figures

Figures reproduced from arXiv: 2506.03955 by the authors.

Figure 1
Figure 1. The correlation between the stellar mass [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. The correlation between MDM and M∗ for the sample of dSphs. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The correlation between γ ∗ SDSS and M∗ based on the sample of SDSS DR 10 (Saulder, Christoph et al. 2015). The first sample is composed of 76 compact, high velocity-dispersion, early-type galaxies from the Sloan Digital Sky Survey (SDSS) with 0.05 < z < 0.2. We denote this sample as SDSS DR 10. This sample was established in reference (Saulder, Christoph et al. 2015) by employing de Vaucouleurs model (Sersic profil… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The correlation between MDM and M∗ derived from the sample of SDSS DR 10 (Saulder, Christoph et al. 2015). log γ = -1.34 log (M* / M⊙) - 21.90 10.4 10.6 10.8 11.0 11.2 11.4 11.6 11.8 -38.0 -37.5 -37.0 -36.5 -36.0 log M* (log M⊙) log γ* (log m-1 ) [PITH_FULL_IMAGE:figu…
Figure 5
Figure 5. Figure 5: the correlation between γ ∗ SLACS(Re) and M∗(Re) based on sample SLACS. 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 -0.2 0.0 0.2 0.4 0.6 0.8 log M*(Re) (log M) log(MDM/M*)( Re) [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The correlation between MDM and M∗ derived from the sample SLACS. than that for sample dSphs and that for sample SDSS DR 10. The parameters for sample SLACS are also presented in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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