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This paper claims that flat galaxy rotation curves—usually seen as evidence for dark matter—can be reproduced without any dark matter if the covarying coupling constant α is allowed to vary locally inside a galaxy, with the derived baryonic

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2026-08-04 16:41 UTC pith:BYYWAKFM

load-bearing objection A credible but circular attempt to replace dark matter with a locally varying CCC parameter; the galaxy fit is a re-parameterization of the observed curve, not a prediction. the 4 major comments →

arxiv 2509.11896 v1 pith:BYYWAKFM submitted 2025-09-15 astro-ph.CO astro-ph.GA

Testing CCC+TL Cosmology with Galaxy Rotation Curves

Rajendra P. Gupta This is my paper
classification astro-ph.CO astro-ph.GA
keywords galaxy rotation curvesdark matter alternativescovarying coupling constantstired lightalpha-matterSPARCmodified Friedmann equationsbaryonic Tully-Fisher relation
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 attempts to show that the flat rotation curves of galaxies, long cited as evidence for dark matter, can be reproduced without any dark matter. The author works in a 'covarying coupling constants plus tired light' (CCC+TL) cosmology in which fundamental constants evolve with cosmic time. The central move is to let the CCC parameter α—written as X = −α/H_X—vary locally inside a galaxy, so that in dense baryonic regions only baryons contribute to gravity while in the outskirts effective 'α-matter' and 'α-energy' densities take over. By fitting a single free parameter, the turn-off density ρ_t, the derived baryonic rotation curves match the SPARC database curves for seven galaxies. If correct, this would mean no new particle or modified-gravity physics is needed to explain galactic rotation dynamics.

Core claim

The core claim is that dark matter and dark energy in ΛCDM are emergent effects of the covarying coupling constant α appearing in modified Friedmann equations. Expanding the modified Friedmann equation generates two extra density terms—α-matter and α-energy—that act like dark matter and dark energy. While α is constant cosmologically, the paper argues that X(r) = −α(r)/H_X can vary locally between 0 (pure baryons, inside a turn-off density ρ_t) and 1 (no baryons, far outside). Using the observed rotation curve and assuming the density stays at ρ_t beyond the turn-off radius, the paper inverts the spherical mass integral to derive X(R) and from it the baryonic mass profile; the resulting bary

What carries the argument

The central object is the local field X(r) = −α(r)/H_X, which controls the split between baryonic, α-matter, and α-energy densities through the identity 1 = (1−X)² + X² + 2X(1−X). The workhorse equations are the modified Friedmann equation expanded as H² = (8πG/3c²)(ε + ε_αe + ε_αm), and the spherical mass integral M_o(R) = 4π∫ ρ(r)(1−X(r))² r² dr. The parameter X is assumed to be 0 inside the turn-off density ρ_t and to rise toward 1 outside; Equation (17) then yields X(R) from the observed rotation curve under the assumption that ρ stays at ρ_t for R > R_t. This converts a dark-matter fitting problem into a one-parameter extraction of the baryonic mass distribution.

Load-bearing premise

The derivation rests on allowing the cosmological parameter α to become a local function X(r) inside a galaxy, with the homogeneous Friedmann decomposition applied point by point; if α is truly constant or this local decomposition is invalid, the rotation-curve explanation collapses.

What would settle it

A direct falsifier: measure high-resolution rotation curves of disk galaxies at z ≈ 1–2. The model predicts that the α-matter contribution relative to baryons drops with redshift (roughly from 8 at z=0 to 3 at z=2), so outer rotation curves should become progressively more Keplerian/declining. If high-z galaxies show flat or rising outer rotation curves with large dark-matter-like fractions, the prediction fails. Alternatively, an exact numerical reconstruction of X(r) from full SPARC data with uncertainties would settle whether one universal ρ_t exists across galaxies.

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

If this is right

  • If rotation curves are generated by α-matter and α-energy, then no dark matter particles or modified gravity (MOND, f(R), etc.) are required to explain galaxy dynamics.
  • The derived baryonic mass profile falls as R⁻⁴ for flat outer rotation curves, matching Hernquist-profile behavior, while the total density falls as R⁻².
  • The baryonic Tully–Fisher relation M_B ∝ V⁴ follows directly from the model's scaling relations at large radius.
  • At higher redshift the ratio of α-matter to baryonic density decreases, so galaxy rotation curves should tend toward Keplerian (declining) shapes, consistent with some high-z observations.
  • The same α-matter/α-energy framework, applied to a galaxy cluster density profile from weak lensing, yields a baryonic density curve in visual agreement with published data.

Where Pith is reading between the lines

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

  • A sharp test: the local-X mechanism predicts a specific radial acceleration relation whose scatter should be explained by baryonic density alone; because the paper finds turn-off accelerations near MOND's a0, a combined analysis could distinguish CCC+TL from MOND.
  • The one-free-parameter fits are visual; quantifying goodness-of-fit and propagating SPARC uncertainties over the full 175-galaxy sample would reveal whether ρ_t is truly a universal parameter or a per-galaxy fudge.
  • The local variation of α is introduced ad hoc; deriving X(r) from the underlying action rather than from the observed curve would strengthen the model and connect it to galaxy formation simulations.

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 paper proposes that the covarying coupling constants (CCC+TL) model, in which a parameter α (or X = −α/H_X) encodes the variation of physical constants, can explain galaxy rotation curves without dark matter. The author rewrites the Friedmann equation to define effective 'α-matter' and 'α-energy' densities, then extends the cosmological decomposition to galaxies by allowing X to vary locally: X=0 in high-density baryonic regions and X→1 at large radii. Using seven SPARC galaxies, the paper derives a baryonic velocity curve V_bX from the observed rotation curve under an assumed constant turn-off density ρ_t, and compares it visually to SPARC baryonic curves, reporting good agreement. It also applies the same procedure to a galaxy cluster density profile. The central conclusion is that observed rotation curves can be used to derive baryonic matter distributions, removing the need for dark matter or modified gravity.

Significance. If the paper's procedure were a genuine first-principles derivation, it would constitute a significant alternative to dark matter and MOND. The algebraic rewriting in Eq. (9) is internally consistent, and the scaling relations for flat rotation curves (ρ_o ∼ R^{-2}, ρ_bX ∼ R^{-4}) follow logically from the stated assumptions. The use of public SPARC data and the explicit acknowledgment of the 'proof-of-concept' nature are positive aspects. However, the significance is severely limited by two load-bearing gaps: the local variation of X is asserted rather than derived from the CCC action, and the baryonic curve V_bX is obtained by inverting the observed curve through an ad hoc constant-density assumption. As presented, the comparison with SPARC is a re-parameterization of the same data, not a falsifiable prediction. The paper does not provide a quantitative goodness-of-fit, relying instead on visual inspection of a few galaxies with a per-galaxy free parameter. Thus, the central claim that dark matter is unnecessary is not supported by the analysis.

major comments (4)
  1. [Section 3, Eqs. (15)–(17); Discussion after Eq. (14)] The transition from the homogeneous cosmological Friedmann equation to a point-by-point local gravitational law inside galaxies is assumed, not derived. The paper states after Eq. (14) that X 'can change locally due to the baryon accretion' and in Section 4 that 'By allowing the variation of X...' but provides no inhomogeneous field equations from the covarying-coupling action. Since the CCC+TL model is defined by a homogeneous Friedmann equation, the step X → X(r) inside a galaxy is a new physical assumption. If X is a true constant of Nature, the galaxy model collapses. This is load-bearing and requires derivation from the action, not an ad hoc ansatz.
  2. [Section 3, Eqs. (16)–(17) and the inversion procedure] The procedure for R > R_t is circular as a test of the model. Equation (16) determines ρ_o(R) from the observed V_o(R); Equation (17) then assumes ρ(R) = ρ_t (constant) for all R > R_t and solves for X(R) from the same V_o(R). Thus the total (observed) density curve is reproduced exactly by construction. The only nontrivial comparison is V_bX, but V_bX is obtained by replacing (1−X)^2 with (1−X)^4 in Eq. (15), a substitution introduced without justification. Since X(R) is already fixed by V_o, the baryonic curve is a transformation of the observed curve, not an independent prediction. The assumption of a constant baryonic density out to large radii is also unphysical for a real galaxy. These issues undermine Conclusion 2.
  3. [Section 3, Table 1 and Figures 3–11] The fit quality is assessed only visually, with no statistical measure (e.g., chi-square, RMS residual) and no treatment of SPARC uncertainties. The turn-off density ρ_t is a free parameter chosen per galaxy (Table 1), and the remarks ('Good overall fit', 'Not a good fit for R ≤ 15 kpc') are subjective. Because X(R) can accommodate rising, flat, or declining rotation curves by adjusting ρ_t and the local profile, the agreement with SPARC baryonic curves is a consistency check of a one-parameter fit rather than a falsifiable test. The paper's own 'proof-of-concept' caveat (end of Section 3) reinforces this limitation.
  4. [Section 2, Eq. (9) and Eq. (13); Section 4] The α-matter and α-energy terms are defined by an algebraic rearrangement of the Friedmann equation; treating them as independent physical densities with their own local behavior (e.g., 'α-energy' remaining constant while 'α-matter' varies) is not derived from the modified field equations. The paper acknowledges in Section 4 that 'α-matter is not dark matter; it is the effect of covarying coupling constants and nothing physical,' yet it is assigned a local energy density that gravitates. This conceptual tension is acceptable if the effective description is derived, but no such derivation is presented. Consequently, the claim that the model is 'cosmologically and astrophysically consistent' (Conclusion 2) is premature.
minor comments (6)
  1. [Section 2, after Eq. (9)] Typo: 'Eqauation' should be 'Equation'.
  2. [Section 3, text before Eq. (15)] The notation is confusing: the subscript 'o' is used both for 'observed' and for 'current value' (e.g., ρ_o vs ρ_0). Clarify or use distinct symbols.
  3. [Section 3, Eq. (19)] The derivation of X(R) = 1 − R^{-1} assumes V_o is exactly constant and sets (1−X)^2 = 1/R^2 without specifying units for R. State that R is in units of R_t.
  4. [Figure 12 and Section 4] The galaxy cluster application uses the same inversion approach with a visual fit (ρ_t = 2.0×10^{-24} g cm^{-3}) but no quantitative comparison. The claim of a 'good visual fit' should be supported by residuals or a goodness-of-fit statistic.
  5. [Section 3, paragraph on mass-to-light ratios] The adopted disk and bulge mass-to-light ratios (0.5 and 0.7) are taken from a private communication and [66]; this choice should be justified more explicitly, as the resulting V_b comparison depends on it.
  6. [Section 5, Conclusion 1] The statement that dark matter and dark energy 'may be considered emerging from the weakening of the forces of nature' is a reinterpretation, not a derivation. Wording could be softened to match the proof-of-concept nature of the paper.

Circularity Check

2 steps flagged

The galaxy rotation-curve 'prediction' reduces to a fit: X(R) is extracted from the observed V_o and then used to rebuild the baryonic curve, so the SPARC agreement is a consistency check, not an independent test; the α-matter/α-energy identification is also definitional.

specific steps
  1. fitted input called prediction [Section 3, 'Galaxy Rotation Curves' (paragraph beginning 'We wish to emphasize')]
    "Our method uses the observed rotation curves to determine the parameter X(R) and then use this X(R) to fit the baryonic rotation curve with one free parameter and compare it with a composite baryonic matter curve using the SPARC database. We show that the predicted baryonic rotation curves are close to those from the database."

    The parameter X(R) is not derived from an independent model; it is solved from the same V_o that the 'prediction' is meant to explain. Equation (15) defines M_o(R) from V_o, Eq. (17) then inverts this to obtain X(R) while holding ρ_t constant outside R_t, with ρ_t chosen per galaxy by visual fit. The baryonic mass M_bX is then obtained by replacing (1-X)^2 with (1-X)^4. Therefore V_bX is a transform of the input rotation curve, not a prediction of it. The agreement with the SPARC baryonic curve is a consistency check of a one-parameter re-parameterization, and cannot by itself establish that dark matter is unnecessary, since X(r) can be adjusted to accommodate rising, flat, or declining curves.

  2. self definitional [Section 2, after Eq. (9) and Figures 1–2]
    "Since the second and the third terms emerge from 𝛼, we have labeled them accordingly —𝜀𝛼𝑒 as the 𝛼-energy density and 𝜀𝛼𝑚 as the 𝛼-matter energy density. As 𝛼 turns out to be negative, the contribution of 𝛼-matter is positive."

    Equation (9) is obtained solely by algebraically expanding Equation (2); the terms ε_αe and ε_αm are labels for expansion terms, not independent physical components. Their 'evolution' in Figures 1–2 follows from this algebraic definition. Concluding that dark matter and dark energy 'emerge' from α is therefore a definitional renaming of the Friedmann-equation cross-term, not an independent derivation. This does not fully control the galaxy-curve comparison, but it makes the broad 'illusion of dark matter' claim self-definitional.

full rationale

The central galaxy section is partially circular: the baryonic curve V_bX is constructed from the observed rotation curve V_o through Eq. (15)–(17), using one free parameter ρ_t per galaxy, so the agreement with SPARC's photometric baryonic curve is a consistency check rather than a prediction. The paper is transparent about this method, even calling it 'proof-of-concept,' but the phrase 'predicted baryonic rotation curves' overstates the epistemic status. The cosmological identification of α-matter and α-energy is also a definitional rewriting of Eq. (2), not a newly derived component. The self-citations to earlier CCC+TL papers are not the main circular element, since the galaxy fit does not use the fitted cosmological H0 and α; the load-bearing issue is that the new local X(r) variation is not derived from the action and is instead calibrated to the data being 'explained.' Overall the paper has real external content (the SPARC V_b comparison is independent), so it is not a 8 or 10, but the central no-dark-matter claim is substantially weaker than a genuine prediction.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 2 invented entities

The central claim rests on the ad hoc local variation of X(r) inside galaxies and the constant-density outer-baryon assumption, neither of which is derived from the underlying field equations. The cosmological input values come from the author's own prior fits, and the galaxy-model entities are labels for algebraic terms, not physical degrees of freedom with independent evidence.

free parameters (4)
  • Turn-off density rho_t = Varies per galaxy; Table 1, typical 2-8 x 10^-24 g/cm3
    One free parameter per galaxy, chosen by visual fit of V_bX to V_b (Section 3, Table 1).
  • Mass-to-light ratios for disk and bulge = 0.5 for disk, 0.7 for bulge
    Taken from McGaugh (private communication, [66]) to convert SPARC velocities to baryon velocities; used in Section 3.
  • Local profile X(r) = Derived from observed V_o(R) via Equations (15)-(17)
    A free function determined by the data, used to reconstruct the baryonic mass distribution.
  • Cosmological CCC parameter X = 0.8
    Taken from the author's earlier Pantheon+ fit (ref. [58]) and used as an anchor for the local variation in this paper.
axioms (6)
  • domain assumption The modified FLRW metric (Equation 1) and Friedmann equations (Equations 2-4) correctly describe the CCC cosmology.
    The paper builds on the author's prior work (refs. 58-61) and does not re-derive these equations.
  • domain assumption The evolution function f(t) = exp(alpha(t-t0)) with c(t)=c0 f(t), G=G0 f^3(t), etc., holds.
    Stated in Section 2 as the defining assumption of covarying coupling constants.
  • ad hoc to paper The energy-density decomposition of Equation (13) applies locally inside a galaxy with X replaced by X(r).
    Introduced after Equation (14) as the bridge from cosmological densities to galactic dynamics; not derived from field equations.
  • ad hoc to paper For R > R_t, the baryonic density rho(R) remains constant and equal to rho_t.
    Equation (17) assumes rho(R_t) is constant for R > R_t; a modeling simplification with no physical justification.
  • domain assumption Galaxies are spherically symmetric and obey Newtonian (Keplerian) dynamics.
    Section 3, 'Under spherical galaxy approximation...' and Equation (15).
  • domain assumption Numerical differentiation of sparse, noisy SPARC data yields reliable density profiles.
    The paper acknowledges numerical errors (Section 4) but relies on differentiations and integrations of sparse data.
invented entities (2)
  • alpha-matter (effective energy density epsilon_alpha_m) no independent evidence
    purpose: Mimics dark matter in galaxy dynamics; a positive contribution from the cross term -2 alpha (8 pi G0 epsilon / 3 c0^2)^(1/2) in Equation (9).
    The paper labels it as not physical: 'alpha-matter is not something physical.' It is a mathematical artifact of rewriting the Friedmann equation, with no independent observable beyond the rotation-curve behavior it is used to fit.
  • alpha-energy (effective energy density epsilon_alpha_e) no independent evidence
    purpose: Mimics dark energy; equals (3 c0^2 / 8 pi G0) alpha^2, a constant term analogous to a cosmological constant.
    Defined by the rewriting in Equation (9). No independent handle; it is the same parameter alpha that is fitted cosmologically.

pith-pipeline@v1.3.0-alltime-deepseek · 16516 in / 22354 out tokens · 229458 ms · 2026-08-04T16:41:31.599562+00:00 · methodology

0 comments
read the original abstract

This paper aims to explore whether astrophysical observations, primarily galaxy rotation curves, result from covarying coupling constants (CCC) rather than from dark matter. We have shown in earlier papers that cosmological observations, such as supernovae type 1a (Pantheon+), the small size of galaxies at cosmic dawn, baryon acoustic oscillations (BAO), the sound horizon in the cosmic microwave background (CMB), and time dilation effect, can be easily accounted for without requiring dark energy and dark matter when coupling constants are permitted to evolve in an expanding Universe, as predicted by Dirac, and the redshift is considered jointly due to the Universe's expansion and Zwicky's tired light (TL) effect. Here, we show that the CCC parameter {\alpha} is responsible for generating the illusion of dark matter and dark energy, which we call {\alpha}-matter and {\alpha}-energy, and is influenced by the baryonic matter density distribution. While cosmologically {\alpha} is a constant determined for the homogenous and isotropic Universe, e.g., by fitting Pantheon+ data, it can vary locally due to the extreme anisotropy of the matter distribution. Thus, in high baryonic density regions, one expects {\alpha}-matter and {\alpha}-energy densities to be relatively low and vice versa. We present its application to a few galaxy rotation curves from the SPARC database and find the results promising.

Figures

Figures reproduced from arXiv: 2509.11896 by Rajendra P. Gupta.

Figure 4
Figure 4. Figure 4: Density determined from the galaxy rotation curves for NGC3198 (Sc) with turn-off den￾sity 𝜌𝑡 = 2.09 × 10−24 g cm−3 at the turn-off radius 𝑅𝑡 =3.37 kpc. Here 𝜌𝑜 designates the density determined from the observed rotational velocity curve and 𝜌𝑏𝑋 is the baryon density estimated with Equations (15) and (16). The waviness in the curves is an artifact of numerical differentiation (see also Criss and Hofmesite… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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