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REVIEW 3 major objections 5 minor 32 references

The supernova Hubble diagram points to a kinematic law: light speed tracks the cosmic expansion rate exactly.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-03 11:38 UTC pith:6QM6QPCQ

load-bearing objection The central claim is built on a wrong redshift formula: Appendix A uses coordinate time rather than proper time, so the inferred c∝ȧ relation is an artifact of that error. the 3 major comments →

arxiv 2601.05512 v5 pith:6QM6QPCQ submitted 2026-01-09 astro-ph.CO gr-qc

An intrinsic kinematic relation boldsymbol{c=frac{c₀}{H₀}\,dot{a}}\, inferred from Type Ia supernovae

Hoang Ky Nguyen This is my paper
classification astro-ph.CO gr-qc
keywords variable speed of lightType Ia supernovaePantheon catalogcosmic accelerationdark energy alternativeluminosity distancepower-law cosmologyconformally flat metric
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 argues that the Pantheon Type Ia supernova Hubble diagram, read through a minimal two-parameter cosmological family in which expansion a is a power law and the speed of light c declines as a power law of a, contains a clean empirical signature of a kinematic law: the best-fitting region is concentrated on the curve (1+ζ)μ=1. That curve is exactly the condition c ∝ da/dt, so at every late-time instant the speed of light and the expansion rate are proportional. The paper claims this synchronous behavior is an intrinsic, parameterization-independent relation, and it shows that a particular member of the family (μ=2/3, ζ=1/2) fits the supernova data slightly better than the standard dark-energy cosmology with the same number of parameters. If right, the accelerating expansion of the universe would not require dark energy; it would be a kinematic consequence of light speed changing with the expansion, and models satisfying c∝a-dot would also have infinite horizons and a conformally flat spacetime metric. The payoff is a concrete, falsifiable rule that any future dynamical theory of late-time cosmology must obey.

Core claim

The paper's central discovery is an empirical degeneracy, not a dynamical model. Using the Pantheon sample of 1,048 Type Ia supernovae and a cosmological family with a=(t/t0)^μ and c=c0 a^{-ζ}, the authors derive a modified luminosity distance that depends on the product η=(1+ζ)μ. The likelihood contours in the {μ,ζ} plane are not scattered: they track the line η=1, equivalently (1+ζ)μ=1, with a width of roughly 0.2 in η. Because c=c0a^{-ζ} and a=(t/t0)^μ, the identity η=1 gives c = μ^{-1} c0 t0 da/dt = (c0/H0) da/dt, so the speed of light is strictly proportional to the expansion rate at all times. The authors emphasize that no prior theoretical consideration forced this relation; it emerge

What carries the argument

Central is the modified redshift relation 1+z = a^{-(1+ζ)}F^{1+ζ}(z), where c varies as a^{-ζ} and F(z) is a one-parameter interpolation function accounting for wavelength-calibration ('yardstick') differences between the supernova host galaxy and the Milky Way. A varying c means the standard 1+z=a^{-1} fails; the correct frequency ratio is a^{1+ζ}, and F(z) converts frequency ratio to observed wavelength ratio. In the resulting luminosity distance formula d_L^{MW} = (c_MW t0/(1-η)) (1+z)/F(z) [1 - ((1+z)/F^{1+ζ})^{1-1/η}], only η=(1+ζ)μ appears, so the data constrain η directly. The posterior peaks at η=1, which is the algebraic condition that turns the two power laws into c = (c0/H0) da/dt

Load-bearing premise

The inference depends entirely on the assumed 'yardstick' scaling—that both the emitted and the reference wavelengths in the relevant galaxies scale with the local value of c through the same exponent -ζ—and on the particular interpolation F(z) chosen for that ratio; if that scaling or interpolation is wrong, the data no longer determine the claimed c-∝-a-dot relation.

What would settle it

Re-fit the same supernova data with F(z) replaced by a fixed, independently measured wavelength-calibration function from high-redshift supernova spectra; if the posterior peak at η=1 leaves its position, the claimed relation is a byproduct of the chosen F(z), not of the data. A cheap first test is to set F≡1 and observe that the distance formula degenerates to a function of η alone.

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

If this is right

  • Late-time acceleration is reinterpreted as a kinematic effect: high-redshift supernovae appear dimmer because the luminosity distance gains a z ln z term when c∝a-dot, so no dark energy is needed to fit the Pantheon data.
  • The particular case μ=2/3, ζ=1/2 outperforms flat ΛCDM by Δχ²=2.8 (68% confidence) with the same number of free parameters, and along the whole η=1 locus the fits are at least as good as ΛCDM for ζ≲8.
  • If the relation extends to early epochs, both the particle and event horizons diverge, resolving the horizon problem without inflation.
  • The relation produces a conformally flat metric ds²=da²-a²(dr²+r²dΩ²) in which the scale factor acts as time, and it modifies the low-redshift Hubble law to z=(1+ζ)H0 d/c0, giving an independent observational test.

Where Pith is reading between the lines

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

  • My inference: if c∝a-dot is a true kinematic law, then the cosmic acceleration read from supernovae is not a dynamical effect but a ruler effect—dark energy becomes an unnecessary substance, and theoretical effort should shift to varying-c gravity.
  • My inference: the parameterization-independent claim could be sharpened by redoing the same fit on independent distance indicators such as gamma-ray bursts or quasars; the prediction is a fixed locus η=1 in the μ-ζ plane that calibration changes cannot mimic.
  • My inference: the equivalence between the linear coasting model and the μ=2/3, ζ=1/2 case suggests the detectable content is the z ln z limb in the distance modulus at high z; experiments should target that specific functional form rather than the individual exponents.

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

3 major / 5 minor

Summary. The paper analyzes a VSL power-law cosmology with scale factor a=(t/t0)^\mu and speed of light c=c0 a^{-\zeta}. It derives a modified Lema\^itre redshift relation 1+z=a^{-(1+\zeta)}F^{1+\zeta}(z), where F(z) is an assumed interpolation function, and a corresponding luminosity distance. Fitting the Pantheon SNeIa sample, the paper finds a posterior degeneracy along the locus (1+\zeta)\mu=1 and interprets this as an empirical law c=c0H0^{-1} da/dt. It further claims that this relation explains late-time acceleration without dark energy, resolves the horizon problem, and slightly outperforms flat \LambdaCDM in terms of \chi^2.

Significance. If the central claim were robust, the paper would offer a concrete, falsifiable alternative to dark energy and a new kinematic constraint on varying-speed-of-light cosmologies. The work uses public Pantheon data and provides a complete fitting procedure, which is a strength. However, the significance is severely limited: the redshift relation rests on an unproven yardstick scaling and an ad hoc function F(z); the statistical preference over \LambdaCDM is marginal (total \Delta\chi^2\approx2.8, driving a statement of only 68% CL); and the relation is inferred from the same data that established acceleration, with no independent prediction. The paper's strong language ('clearest and most decisive evidence') is not supported by the analysis as presented.

major comments (3)
  1. [Modifying the Lemaître redshift relation, Eqs. (12)-(15)] The entire sensitivity to \zeta, and hence the inference of \eta=1, comes from the unproven yardstick scaling c_MW\propto(\lambda^*_MW)^{-\zeta}, c_SN\propto\lambda_SN^{-\zeta} and from the arbitrary choice F(z)=1+(F_\infty-1)[1-(1+z)^{-2}]^2. The paper itself notes that without a non-constant F(z), the luminosity distance (16) is degenerate in \eta. Thus the posterior peak at \eta=1 is contingent on a functional form with no physical derivation. No robustness test with alternative F(z) is given. This is a load-bearing issue: if the yardstick assumption or the chosen F(z) is changed, the claimed empirical law could disappear.
  2. [Table I and Corollary 1] The best VSL models in Table I have \chi^2 per degree of freedom about 0.9856 versus 0.9882 for \LambdaCDM, a total \Delta\chi^2\approx2.8 over 1,048 data points. This is a weak preference, roughly 1.7\sigma, not 'the clearest and most decisive evidence' against \LambdaCDM. Moreover, the general parameter space has four free parameters (\mu,\zeta,t_0,F_\infty), while the special cases in Table I fix \mu,\zeta and use only (t_0,F_\infty), so the comparison with the two-parameter \LambdaCDM is not apples-to-apples. The claim that the model outperforms \LambdaCDM for all 0\le\zeta\lesssim8 along the locus is stated without a scan plot or a look-elsewhere correction. A proper model-comparison statistic (AIC/BIC or a marginal likelihood with a prior volume penalty) is needed before calling this an empirical law.
  3. [Corollary 2 and Conclusion] The relation c=c0H0^{-1}\dot a is derived by imposing \eta=1 on the assumed power-law forms. The paper calls it 'intrinsic' because it equates two dimensionally compatible quantities, but this criterion is not sufficient: many such relations could be written down. The inference is entirely within the Dolgov-Barrow parameterization, and the same Pantheon data that established acceleration are used to select \eta=1. No out-of-sample prediction or independent probe is provided. To support the 'empirical law' claim, the authors would need a genuinely model-independent reconstruction or a test with independent data (e.g., cosmic chronometers, BAO, or CMB distance information).
minor comments (5)
  1. [Appendix A and Eq. (11)] The derivation uses coordinate-time wavecrest intervals. This is internally consistent if cosmic coordinate time t is the proper time of comoving observers, so that the physical frequency is \nu=1/\delta t. The paper should state this convention explicitly; otherwise readers may infer a missing factor of c(a). The stress-test concern about proper-frequency corrections therefore does not land, but the ambiguity should be removed.
  2. [Abstract and Conclusions] The phrases 'clearest and most decisive evidence' and 'new empirical law' are overstated given the marginal \Delta\chi^2 and the model dependence. Please temper the claims throughout.
  3. [Figure 3] The construction of the \eta distribution in the lower panel is not fully specified: what priors on \mu,\zeta are used, how the width 0.2 is estimated, and what smoothing is applied? This should be described so the claimed 'prominent peak at \eta=1' is reproducible.
  4. [Table I] The SIG case gives H0=47.22 km/s/Mpc, far below local distance-ladder measurements. Calling this a 'resolution' of the Hubble tension is misleading; it is a very low value that itself is in tension with local H0 measurements.
  5. [General presentation] There are numerous typos, including missing spaces after 'We present' at the start, malformed subscripts such as '\muPan i', and an undefined domain of validity for Eq. (16) when \eta<1. The manuscript also relies heavily on self-citations to Refs. [8-10]; the novelty relative to those works should be made clearer.

Circularity Check

2 steps flagged

The inference of c∝ȧ rests on a self-cited ad hoc F(z) that breaks the η-degeneracy; without it the data cannot constrain ζ at all.

specific steps
  1. ansatz smuggled in via citation [Section 'Modifying the Lemaître redshift relation', Eq. (15)]
    "A non-constant F(z) would break the η–degeneracy in Eq. (16). ... Adopting the practice in Ref.[8], we will model F(z) as F(z) = 1 + (F∞−1)[1−(1+z)^{-2}]^2 (15), which monotonically interpolates between F(z=0)=1 and F(z=∞)=F∞, where F∞ is an adjustable parameter."

    The central empirical claim (η=1) is inferred only because Eq. (16) is η-degenerate when F(z)≡1; the degeneracy is broken by inserting an arbitrary F(z) taken from the author's own prior work. The posterior peak at η=1 is therefore not a direct reading of the SNe data but a consequence of the chosen self-cited fitting function. Had a different F(z) been used, the constraint on ζ — and hence on c∝ȧ — would change. This is a load-bearing ansatz presented as empirical inference.

  2. self citation load bearing [Section 'Modifying the Lemaître redshift relation', after Eq. (12), leading to Eq. (13)]
    "Extending our elaborated explanation in Ref. [8], we will adopt the relationships c_MW ∝ (λ∗_MW)^{−ζ} and c_SN ∝ λ_SN^{−ζ}. Along with Eq. (12), they yield the redshift 1+z = λ_MW/λ∗_MW = a^{−(1+ζ)}(λ_SN/λ∗_MW)^{1+ζ} (13)."

    The 'yardstick' scaling of c with atomic wavelengths is not derived in this paper; it is imported from the author's prior work and is the step that turns the standard VSL redshift into the modified form 1+z=a^{−(1+ζ)}F^{1+ζ}(z). Without this self-cited ansatz, the luminosity distance formula would lose its sensitivity to ζ, and the inference of the empirical locus (1+ζ)μ=1 would collapse. Thus the central result is constructed from self-cited assumptions rather than independently predicted.

full rationale

The paper fits the Dolgov–Barrow VSL model to the Pantheon SNeIa sample and finds a posterior ridge along (1+ζ)μ=1, which it interprets as an intrinsic kinematical relation c∝ȧ. This is not a case of simple Eq=Eq circularity: the data genuinely enter through the χ² fit, and the SIG case with μ=2/3, ζ=1/2 has the same number of free parameters as ΛCDM. However, the inference is heavily dependent on two load-bearing inputs imported from the author's own prior work: the 'yardstick' scaling c∝λ^{−ζ} and the ad hoc function F(z) of Eq. (15). The paper explicitly states that F(z) is what breaks the η-degeneracy of the distance–redshift relation; without it, the Pantheon data cannot constrain ζ at all. The claimed empirical law c∝ȧ is therefore not an independent prediction but a reparametrization of the fitted condition η=1, made identifiable only by a self-cited fitting ansatz. No external or out-of-sample test is provided. These issues are sufficient to raise the circularity score to 5, but not to 8–10, because the paper does present a genuine data fit and a model comparison rather than a purely definitional identity.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The central inference rests on the power-law VSL ansatz, the yardstick scaling of rest wavelengths, and the ad hoc F(z). The numbers fitted to data are μ, ζ, t0, and F∞; the only invented entity is the yardstick wavelength. None of these are constrained by independent data in this paper.

free parameters (4)
  • μ = locus peak includes μ=2/3 (SIG) and μ=1 (Kolb); varied in contour
    Power-law exponent in a=(t/t0)^μ. Fitted against Pantheon data; the central claim requires the combination (1+ζ)μ≈1.
  • ζ = SIG ζ=1/2, Kolb ζ=0; varied in contour
    VSL exponent in c=c0 a^{-ζ}. Fitted; the c∝dot a relation is equivalent to ζ=1/μ−1.
  • t0 = Table I: 13.82 Gyr (SIG), 13.87 Gyr (Kolb), etc.
    Age/scale parameter; fitted for each model and sets H0=μ/t0.
  • F∞ = SIG F∞=0.931, Kolb F∞=0.911
    Asymptotic amplitude of the ad hoc yardstick function F(z) in Eq. (15). This free parameter breaks the η-degeneracy in Eq. (16) and shapes the posterior locus.
axioms (5)
  • domain assumption VSL RW metric with c(a) as a metric coefficient (Eq. 3)
    The paper assumes the speed of light appears in the metric as c²(a)dt² and varies with scale factor. This is the defining VSL framework, not derived.
  • domain assumption Power-law ansatz a=(t/t0)^μ and c=c0 a^{-ζ} (Eqs. 4–5)
    Motivated by avoiding a preferred scale, but still a restrictive functional choice imposed before fitting.
  • ad hoc to paper Yardstick scaling c_MW ∝ (λ*_MW)^{-ζ}, c_SN ∝ λ_SN^{-ζ} (Eq. 13)
    No derivation or independent evidence; this assumption converts the standard VSL redshift 1+z=a^{-1} into 1+z=a^{-(1+ζ)}F^{1+ζ} and is load-bearing for the whole analysis.
  • ad hoc to paper F(z)=1+(F∞−1)[1−(1+z)^{-2}]^2 (Eq. 15)
    An arbitrary interpolation function introduced to model the yardstick effect. It supplies one free parameter and materially affects the fit and the inferred degeneracy.
  • domain assumption Pantheon SNeIa are standard candles with absolute magnitude M=−19.35
    The distance moduli are computed with a fixed M; the analysis inherits the standard candle calibration and systematic uncertainties of the catalog.
invented entities (1)
  • Yardstick wavelength λ*_MW no independent evidence
    purpose: Provides a local standard of wavelength against which observed SN wavelengths are compared, allowing an extra factor F(z) in the redshift formula.
    No independent measurement or theoretical derivation is given. The scaling of rest wavelengths with c is simply asserted, and the whole modified redshift relation depends on it.

pith-pipeline@v1.3.0-alltime-deepseek · 11092 in / 25682 out tokens · 270496 ms · 2026-08-03T11:38:05.381802+00:00 · methodology

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read the original abstract

Based on the Hubble diagram of SNeIa, we present empirical evidence for a kinematic relation between the speed of light and the late-time cosmic expansion rate. To infer this relation, we employ the Dolgov-Barrow cosmology, described by Dolgov's power-law expansion $a=(t/t_0)^\mu$ and Barrow's varying-speed-of-light (VSL) $c=c_0\,a^{-\zeta}$. In this cosmology, light propagating through an expanding cosmic background undergoes an additional refraction induced by the variation of $c$ along its path, resulting in a modified Lema\^itre redshift relation $1+z=a^{-(1+\zeta)}$. The model yields a high-quality fit to the Pantheon SNeIa Hubble diagram and exhibits a remarkably tight posterior degeneracy along the locus $(1+\zeta)\,\mu=1$. In particular, the case ${\mu=2/3,\zeta=1/2}$, referred to as the VSL-Einstein-de Sitter case, is favored over flat $\Lambda$CDM by $\Delta\chi^2=2.8$, i.e. at 68% confidence level, despite having the same number of free parameters. The empirical relation $(1+\zeta)\,\mu=1$ entails that the speed of light is exactly proportional to the cosmic expansion rate, $c=c_0H_0^{-1}\,\dot a$, during late times, a synchronous behavior absent in the standard $\Lambda$CDM model. Although the empirical relation $(1+\zeta)\,\mu=1$ is inferred using the Dolgov-Barrow parameterization, the resulting expression $c=c_0H_0^{-1}\,\dot a$ is an intrinsic relation because it relates two physical quantities of the same dimension: speed of light and cosmic expansion rate. It is therefore independent of arbitrary choices of units or parameterization and encodes a purely kinematic correspondence between $c$ and $\dot a$. If confirmed by independent probes, this relation may point toward a more general kinematic principle governing late-time cosmic evolution. We discuss implications of this relation for late-time cosmology, including an alternative interpretation of cosmic acceleration.

Figures

Figures reproduced from arXiv: 2601.05512 by Hoang Ky Nguyen.

Figure 1
Figure 1. Figure 1: The supernova emitted λSN and the observer detects λMW . Note that λMW λSN ̸= λ2 λ1 in general, rendering the Lemaître redshift formula, 1 + z = a −1 , invalid for VSL cosmology. statistically test the null hypothesis (1 + ζ) µ = 1, viz. Eq. (9), against the Pantheon Catalog of SNeIa. Modifying the Lemaître redshift relation—As de￾rived in Appendix A, the relation c ∝ a −ζ in Eq. (5) yields the ratio betwe… view at source ↗
Figure 2
Figure 2. Figure 2: 68% and 95% constraint contours, deduced from Pan [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: 68% and 95% constraint contours for models satisfy [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Upper panel: Re-plotting Figure 2 in terms of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

discussion (0)

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Reference graph

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