Testing Covarying Coupling Constants (CCC) against the full SPARC rotation-curve sample: a like-for-like comparison with MOND and NFW

Journal: MNRAS 000, 1–16 (2026), Preprint 13 August 2026, arXiv:2608.11575v1

The Covarying Coupling Constants (CCC) framework, developed to account for high-redshift JWST observations, contains a mechanism – a covarying-constant virtual mass field keyed to local density – that modifies galactic dynamics without particle dark matter. The authors test it against the full Spitzer Photometry and Accurate Rotation Curves (SPARC) sample of 175 disc galaxies, extending an earlier study of a few objects. Working in an inverse formulation, in which each model predicts the baryonic rotation curve from the observed one, they compare CCC against Modified Newtonian Dynamics (MOND) and one- and two-parameter Navarro–Frenk–White (NFW) haloes on identical footing, using the reduced χ2ν. They show that the published sharp density turn-off in the earlier study is unphysical and replace it with a smooth transition – the density-space analogue of the MOND interpolating function, introducing no new parameter. One-parameter smooth-CCC then performs comparably to galaxy-by-galaxy fitted MOND (the lower χ2ν in 56 per cent of galaxies, with near-equal mean χ2ν of 2.58 versus 2.65; the paired difference is not statistically significant), while the two-parameter NFW halo shows a substantially broader fit-quality distribution and a larger tail of poor or boundary-limited fits (mean χ2ν ≈ 7). The CCC turn-off density is not universal (scatter 0.82 dex) and correlates with galaxy size, qualitatively consistent with the expected effect of a spherical reconstruction applied to flattened disc systems. Recast as an acceleration, however, at = V2flat/Rt has scatter 0.33 dex (on the 91-galaxy resolved subset) – matching the MOND scale a0 (0.34 dex) – and comparable magnitude of order 2×10−10 m s−2, with its size correlation removed. CCC, though not designed for galactic dynamics, describes rotation curves as well as galaxy-by-galaxy fitted MOND, while the two-parameter dark-matter halo shows a broader distribution of fit quality and frequent boundary-limited solutions.

The paper begins by reviewing the historical development of galaxy rotation curve studies, from early spectroscopy of M31 (Babcock 1939) through Rubin & Ford (1970) and subsequent work establishing extended, approximately flat rotation curves in spiral galaxies. The modern mass-modelling problem is expressed as:

V2obs(R) = V2gas(R) + Υd V2disc(R) + Υb V2bulge(R) + V2x(R)

where Υd and Υb are stellar mass-to-light ratios, and Vx represents either a dark halo or an effective modification of the gravitational field.

The SPARC database (Lelli et al. 2016a) transformed galaxy mass modelling by combining homogeneous 3.6-µm surface photometry with published H i and Hα rotation curves for 175 nearby disc galaxies. The sample spans morphologies from S0 to irregular galaxies, approximately five decades in luminosity, and four decades in surface brightness. A commonly adopted value is Υ3.6 ≃ 0.5 M⊙/L⊙.

The paper discusses empirical regularities revealed by SPARC, including the baryonic Tully–Fisher relation (BTFR), Mbar ∝ V4f, and the radial acceleration relation (RAR), which correlates gobs(R) = V2obs(R)/R with gbar(R) = V2bar(R)/R. Below a characteristic acceleration of order g† ∼ 10−10 m s−2, the discrepancy increases systematically.

The Covarying Coupling Constants (CCC) plus Tired Light (TL) framework (Gupta 2023) was developed to address tensions in the standard cosmological model, particularly the abundance and maturity of high-redshift galaxies reported by JWST. The galaxy application permits the cosmological parameter α to vary locally with the strongly inhomogeneous baryonic density and defines:

X(R) = −α(R)/H, 0 ≤ X(R) ≤ 1

The adopted phenomenology suppresses the effective component in high-density regions and activates it outside a galaxy-dependent turn-off density ρt.

In the spherical approximation, the total (observed) mass density is:

4πρo(R) = (1/G R2) [V2o/R + 2VoV′o]

The CCC cosmology creates the effective mass field X(R) such that the gravitating baryonic density is ρbX = ρ(1−X)4 while the density inferred dynamically is ρo = ρ(1−X)2. The transition is governed by the local density: below the turn-off the field is inactive (X = 0), so ρo = ρbX = ρ and the dynamics are purely Newtonian; the field activates where the underlying density falls through the turn-off value.

The paper derives a new closed form: ρ2o = ρt ρbX, which gives ρo = √(ρt ρbX) for ρbX < ρt. This is described as the density-space analogue of the deep-MOND relation g = √(gN a0), both being geometric means with a single constant scale.

Inverse formulation: Each model predicts Vb from Vobs, and the prediction is compared with the SPARC Vb. This places the cleanly-measured quantity on the independent axis and the uncertain quantity on the dependent axis.

Objective fitting: The analysis minimises χ2 = Σ [VbX(Ri) − Vb(Ri)]2/σ2i over the single parameter ρt, by a logarithmic grid scan refined with Brent's method. The observed density ρo(R) is reconstructed by differentiating an error-weighted smoothing spline fit to Vobs(R).

Error model: The dominant term is the propagation of eVobs through each model's own inverse chain, evaluated by Monte Carlo; the robust 16–84 percentile half-width of the Monte Carlo ensemble is used, plus a 5 per cent photometric floor on Vb and a 1 km s−1 numerical floor.

Mass-to-light degeneracy: Υ is treated as an explicit nuisance parameter with a 0.1 dex Gaussian prior.

Smooth turn-off (this work): The published sharp transition is replaced with smooth transitions using the two standard MOND interpolation forms transcribed into density space:

Standard (McGaugh et al. 2016): ρo = ρbX / [1 − exp(−√(ρbX/ρt))]

Simple (Famaey & Binney 2005): ρo = ρbX/2 + √(ρ2bX/4 + ρbX ρt)

Both reduce to the sharp published law in the deep limit and to ρo = ρbX in the Newtonian limit, introducing no new parameter.

Comparison models: MOND is represented by the RAR interpolation function applied algebraically and pointwise. NFW haloes have the circular-velocity profile V2NFW(R) = V2200 [µ(cx)/(x µ(c))], with one-parameter (free V200, c fixed by c–M200 relation) and two-parameter (independent V200 and c) versions.

Applying the objective smooth-transition fit to the seven galaxies of Gupta (2025) returns turn-off densities that agree with published values to within a factor of about two. The reconstructed density profiles reproduce the analytically-predicted asymptotic slopes ρo ∝ R−2 and ρbX ∝ R−4.

Applying the pipeline to all 165 galaxies (of 175, retaining those with at least six velocity measurements and a determined flat rotation speed) yields turn-off densities with a median of 2.9 × 10−24 g cm−3 and a scatter of 0.82 dex.

Fitting MOND with a sharp acceleration threshold gt shows the sharp threshold is biased ∼40 per cent high relative to the smooth a0 (gt/a0 = 1.42), and χ2/ν degrades by a factor ∼1.9. The smooth transition returns ρt at 0.61× the sharp value.

Table 3 gives per-galaxy reduced χ2ν statistics:

Model k mean median f10

CCC (smooth, standard ν) 1 2.58 0.89 63% 7%

CCC (smooth, simple ν) 1 2.59 0.90 65% 5%

MOND (smooth) 1 2.65 1.50 63% 7%

CCC (sharp; published) 1 2.92 1.17 61% 5%

NFW (2-parameter) 2 7.24 2.00 50% 24%

NFW (1-parameter) 1 7.81 3.48 32% 24%

Head-to-head, CCC has the lower χ2ν in 56 per cent of galaxies against MOND. This edge is not statistically significant: a sign test gives p = 0.16, a Wilcoxon signed-rank test p = 0.24, and a bootstrap 95 per cent confidence interval on the mean paired difference, [−0.54, +0.41], includes zero. Against the two-parameter NFW halo, CCC has the lower χ2ν in 68 per cent of galaxies. Of the 165 two-parameter NFW fits, 25 (15 per cent) reach the imposed lower concentration boundary c = 1 and 7 (4 per cent) the upper V200 = 600 km s−1 boundary, so 31 solutions (19 per cent) are boundary-limited.

The cumulative distribution of per-galaxy reduced χ2ν shows CCC and MOND are nearly coincident, while both show a substantially tighter distribution than the two NFW halo models. CCC tends to fit high-surface-brightness spirals better and MOND the gas-rich, low-surface-brightness dwarfs, with a Spearman rank correlation between the fit-quality difference and effective surface brightness of ρS ≈ +0.35 (p ∼ 10−5).

CCC and MOND have essentially identical widths (0.088 vs 0.097 dex), with CCC marginally the tighter and the only model with a near-symmetric, mildly positive-skewed distribution. All four distributions are sharply peaked with heavy tails, well fit by a Lorentzian (Cauchy) profile rather than a Gaussian. The Lorentzian half-widths γ are only 0.026–0.046 dex.

Constructing a forward density–density relation yields a scatter of 0.33 dex for CCC against 0.23 dex for MOND. The gap is partly a differentiation artefact; MOND's native relation in acceleration space has a scatter of only 0.18 dex.

ρt correlates significantly and only with size and mass indicators – most strongly with effective radius (ρS = −0.36, surviving false-discovery-rate control) – with a negative sign: larger galaxies have lower ρt. A robust (Theil–Sen) slope gives ρt ∝ R−α with α = 0.6 (Reff) to 0.57 (Rdisk). The correlation survives a partial rank correlation controlling simultaneously for distance and linear resolution (partial ρS = −0.26, p < 10−3).

Defining at ≡ V2flat/Rt converts CCC's turn-off scale into a quantity directly comparable with a0. On the resolved sample of 91 galaxies:

Quantity scatter (dex) robust (dex) median

CCC ρt (density) 0.48 0.49 2.1 × 10−24 g cm−3

CCC at (accel.) 0.33 0.25 2.1 × 10−10 m s−2

MOND a0 (accel.) 0.34 0.27 9.5 × 10−11 m s−2

The transformation from ρt to at also removes the size correlation: while ρt correlates strongly with effective radius (ρS = −0.47, p ∼ 10−6), at does not (ρS = −0.16, p = 0.13, not significant).

Bulge-dominated galaxies have ρt scatter of 0.42 dex against 0.82 dex for bulgeless galaxies – a halving. The same split in a0 is milder (0.29 vs 0.55 dex).

CCC makes an analytic prediction for the BTFR with slope b = 4. All three models satisfy the BTFR with slopes close to the predicted value: CCC yields b = 3.70, MOND gives 3.55, and two-parameter NFW 3.68, against a data value of 3.45.

The paper discusses several important considerations:

Single mass-to-light ratio: The fiducial analysis assumes a universal stellar mass-to-light ratio (Υdisk = 0.5 and Υbul = 0.7 M⊙/L⊙ at [3.6] µm). A profiled-Υ variant with a 0.1 dex prior mildly prefers larger stellar masses (median scaling ∼1.2), but Υ is weakly constrained.

Parameter scatter comparison: MOND's a0 has a smaller raw scatter (0.52 dex) than CCC's ρt (0.82 dex). However, density is a local, differential quantity; acceleration is a cumulative, integrated one. The decisive demonstration is that recast as an acceleration, the CCC turn-off scale at has the same scatter as a0 (0.33 vs 0.34 dex) and loses its size correlation.

Data-quality limitation: CCC's one structural disadvantage relative to MOND is numerical – it requires differentiating and re-integrating a sparsely-sampled, noisy rotation curve. This limitation is concentrated in high-surface-brightness spirals with steeply-declining curves.

Geometry: The spherical approximation is applied within the CCC reconstruction. CCC's effective extra mass, following the baryons through equation (15), is disc-like rather than spherical – a genuine physical distinction from NFW, shared with MOND.

The paper's seven main conclusions are:

(i) Extended from a handful of galaxies to the full SPARC sample with an objective fitting procedure, the CCC galactic mechanism describes disc rotation curves well: with a smooth density turn-off it achieves a mean χ2/ν = 2.58 in the inverse formulation, comparable to MOND (2.65).

(ii) On a strictly like-for-like footing, one-parameter CCC is statistically comparable to galaxy-by-galaxy fitted MOND (the lower χ2ν in 56 per cent of galaxies, mean χ2ν of 2.58 versus 2.65; the paired difference is not statistically significant). The two-parameter NFW halo shows a substantially broader fit-quality distribution (24 per cent of fits with χ2ν > 10, against 7 per cent for CCC and MOND) and frequent boundary-limited solutions (19 per cent), with CCC giving the lower χ2ν in 68 per cent of galaxies.

(iii) The published sharp density turn-off is an unphysical simplification. Replacing it with a smooth transition – the density-space analogue of MOND's interpolating function, introducing no new parameter – is what brings CCC to parity with MOND, and simultaneously resolves a normalisation offset in the fitted ρt.

(iv) The turn-off density is not a universal constant (scatter 0.82 dex), but its scatter is partly geometric: ρt correlates robustly with galaxy size (ρS = −0.36; ρt ∝ R−0·6), qualitatively consistent with the expected effect of applying a spherical reconstruction to flattened disc systems, and its scatter halves in bulge-dominated galaxies where the spherical approximation is most valid.

(v) Expressed in its natural form as an acceleration, at = V2flat/Rt, the CCC turn-off scale is as universal as MOND's a0 (scatter 0.33 versus 0.34 dex on the resolved sample) and of comparable magnitude (median 2.1 × 10−10 m s−2, of order a0). The size correlation present in ρt vanishes in at.

(vi) That a framework constructed to address high-redshift JWST cosmology reproduces galactic rotation curves as well as the phenomenology purpose-built for them is a non-trivial success of the CCC programme. Within the common inverse reconstruction adopted here, CCC performs comparably to galaxy-by-galaxy fitted MOND, while the two-parameter NFW model exhibits a substantially broader distribution of fit quality and a larger population of poor or boundary-limited fits.

Based on this paper, here are specific improvements to AI systems and what the improved systems can do:

Improvement: Implement a standardized like-for-like comparison protocol that evaluates models on identical footing (same data, same error propagation, same fitting procedure, same number of parameters).

Capability: An AI system can now objectively compare competing physical theories (e.g., modified gravity vs. dark matter) without bias from differing methodologies. It can automatically detect when apparent model superiority is an artifact of asymmetric treatment (e.g., one model gets free parameters, another doesn't).

Improvement: Replace sharp threshold/discontinuity functions with smooth interpolating functions that reduce to the sharp limit asymptotically, introducing no new parameters.

Improvement: Train models to predict the well-measured quantity from the uncertain one (inverse formulation) rather than the reverse.

Improvement: Implement a correction module that accounts for systematic biases when a spherical model is applied to flattened/disky systems, including a size-dependent scaling law.

Improvement: Add a transformation layer that converts local, differential quantities (like density) into cumulative, integrated quantities (like acceleration) before assessing universality.

Improvement: Implement automated detection and flagging of fits that hit parameter boundaries, with statistical reporting of what fraction of solutions are boundary-limited.

Improvement: Integrate multiple non-parametric tests (sign test, Wilcoxon signed-rank, bootstrap confidence intervals) for model comparison, not just mean χ2 values.

Improvement: Include morphological classification as a conditioning variable in parameter estimation, with automatic detection of when parameter scatter halves for symmetric systems.

Improvement: Add a module that derives analytic predictions from model equations (like the BTFR slope of 4) and automatically checks them against empirical fits.

Improvement: Implement error propagation that explicitly accounts for numerical differentiation of noisy data, using Monte Carlo ensembles and robust percentile-based error estimates.

The Covarying Coupling Constants (CCC) framework, developed to account for high-redshift JWST observations, contains a mechanism -- a covarying-constant effective mass field keyed to local density -- that modifies galactic dynamics without particle dark matter. We test it against the full Spitzer Photometry and Accurate Rotation Curves (SPARC) sample of 175 disc galaxies, extending an earlier study of a few objects. Working in an inverse formulation, in which each model predicts the baryonic rotation curve from the observed one, we compare CCC against Modified Newtonian Dynamics (MOND) and one- and two-parameter Navarro-Frenk-White (NFW) haloes on identical footing, using the reduced chi 2 nu. We show that the published sharp density turn-off in the earlier study is unphysical and replace it with a smooth transition -- the density-space analogue of the MOND interpolating function, introducing no new parameter. One-parameter smooth-CCC then performs comparably to galaxy-by-galaxy fitted MOND (the lower chi 2 nu in 56 per cent of galaxies, mean chi 2 nu of 2.58 versus 2.65; the paired difference is not significant), while two-parameter NFW shows a substantially broader fit-quality distribution and a larger tail of poor or boundary-limited fits (mean chi 2 nu about 7). The CCC turn-off density is not universal (scatter 0.82 dex) and correlates with galaxy size, qualitatively consistent with a spherical reconstruction applied to flattened disc systems. Recast as an acceleration, however, a t = V flat 2/R t has scatter 0.33 dex (on the 91-galaxy resolved subset) -- matching the MOND scale a 0 (0.34 dex) -- and comparable magnitude of order 2 times 10-10 m/s squared, with its size correlation removed. Though not designed for galactic dynamics, CCC describes rotation curves as well as galaxy-by-galaxy fitted MOND.