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REVIEW 3 major objections 4 minor 50 references

Comparison of $R_h=ct$ and $\Lambda$CDM using DESI DR1 measurements

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper finds that DESI DR1 baryon acoustic oscillation data decisively disfavour the parameter-free $R_h=ct$ cosmology over flat $\Lambda$CDM, with the verdict carried by the Lyman-$\alpha$ measurement at redshift 2.33.

desk verdict A clean, standard BAO test that disfavors Rh=ct with DESI DR1, but the decisive result rests on a single Ly-alpha point and the paper's 'rules out' wording overreaches. read the letter →

arxiv 2506.18445 v1 pith:7K66QGIL submitted 2025-06-23 astro-ph.CO

Kunj Panchal , Shantanu Desai This is my paper
classification astro-ph.CO PACS 98.80.-k
keywords R_h=ctcosmologyflatLambdaCDMDESIDR1baryonacousticoscillationsBayesianmodelselectionBayesfactorLyman-alphaforestquasarsample
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 the $R_h=ct$ universe, a cosmology with no free parameters, against flat $\Lambda$CDM using the first DESI baryon acoustic oscillation data release. The authors work with the ratio of the transverse comoving distance to the Hubble distance, which cancels the sound-horizon scale and the Hubble constant, and compare the two models with Bayesian model selection. Up to redshift $z=1.3$ the two models are equally favoured, but the Lyman-$\alpha$ quasar measurement at $z=2.33$ is discrepant with $R_h=ct$ by $4.1$-$4.5\sigma$ and produces Bayes factors above 100 for that single point, reaching about 21,000 in the most favourable case. For the combined data the Bayes factor ranges from roughly 800 to 78,000 against $R_h=ct$. The comparison matters because $R_h=ct$ is a genuine, parameter-free alternative to the standard model, and the ratio test used here sidesteps the sound-horizon assumptions that have complicated earlier BAO comparisons.

What carries the argument

The load-bearing object is the dimensionless ratio $D_M(z)/D_H(z)$ of the transverse comoving distance to the Hubble distance, reconstructed from the DESI DR1 measurements of $D_M/r_d$ and $D_H/r_d$ via error propagation. Because only this ratio is used, the sound-horizon scale $r_d$ and the Hubble constant $H_0$ cancel out, removing the model dependence that enters when absolute BAO distances are analysed. For $R_h=ct$, with $H(z)=H_0(1+z)$, there are no free parameters, so the Bayesian evidence is simply the likelihood; for flat $\Lambda$CDM, the evidence integrates the likelihood over $\Omega_m$ under two priors, a uniform prior $U(0,1)$ and a Planck-informed Gaussian $N(0.315,0.007)$. The evidences are computed with nested sampling, and their ratio, the Bayes factor, is read against the Jeffreys' scale.

What would settle it

Recompute the $z\approx2.33$ Lyman-$\alpha$ value of $D_M/D_H$ from an independent sample or pipeline, for instance the DESI DR2 release the paper cites, and re-run the Bayes factor. If the updated ratio moves onto the $R_h=ct$ curve (a downward shift comparable to the current 4.1-4.5$\sigma$ discrepancy) or its uncertainty grows enough to push the Bayes factor below 100, the paper's central claim collapses; the authors themselves flag that such a revision would change their conclusion.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the DESI DR1 ratio measurements of $D_M/D_H$ rule out $R_h=ct$ relative to flat $\Lambda$CDM, decisively on the Jeffreys' scale, but only because of one data point. Every tracer at $z\le1.3$ yields a Bayes factor near unity, meaning the BAO samples below that redshift cannot tell the two models apart; the $z=2.33$ Lyman-$\alpha$ measurement, however, lies on the $\Lambda$CDM curve and 4.1-4.5$\sigma$ away from $R_h=ct$, driving Bayes factors of 760-21,500 by itself and 788-77,677 for the combined sample. This reverses the conclusion of an earlier analysis that applied the same ratio test to eBOSS Lyman-$\alpha$ data and found $R_h=ct$ favoured. The authors state plainly that the verdict hinges on the single high-redshift point: if that measurement were revised, or its systematics doubted, the conclusion would change.

Load-bearing premise

The entire rejection of $R_h=ct$ rests on the accuracy of one DESI Lyman-$\alpha$ measurement at $z=2.33$; the authors state in their conclusion that if this point were revised or its systematics questioned, their conclusion would change.

Editorial extensions

If this is right

  • BAO tracers at $z\le1.3$ cannot distinguish the two models: every galaxy and quasar sample below that redshift gives a Bayes factor of order 1.
  • The single Lyman-$\alpha$ point at $z=2.33$ carries the entire discrimination, disagreeing with $R_h=ct$ by 4.1-4.5$\sigma$ while agreeing with flat $\Lambda$CDM.
  • The combined DESI DR1 analysis gives Bayes factors from about $8\times10^2$ to $7.8\times10^4$ in favour of $\Lambda$CDM, which the Jeffreys' scale labels decisive; the ranking is stable under both priors on $\Omega_m$ and with or without the measurement correlations.
  • The result reverses the earlier eBOSS-based ratio test, which had favoured $R_h=ct$ over Planck $\Lambda$CDM with the same $D_M/D_H$ methodology.
  • Flat $\Lambda$CDM does not fit the ratio data cleanly ($\chi^2/{\rm dof}\approx8$-$10$, $p\approx0.05$-$0.09$), so the authors note that extended models such as $w$CDM or time-varying dark energy are plausible targets for the same test.

Reading between the lines

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

  • If the $z=2.33$ point survives scrutiny, the favourable earlier verdict for $R_h=ct$ from eBOSS data most likely stemmed from that dataset's systematics rather than from the shared $D_M/D_H$ method, and the two BAO datasets are now in direct tension.
  • The same ratio test can be run immediately on the DESI DR2 release that this paper cites; a DR2 Lyman-$\alpha$ value that replicates DR1 would harden the rejection, while a shift toward the $R_h=ct$ curve would overturn it.
  • Because the discrimination is concentrated in a single high-redshift bin, more and better Lyman-$\alpha$-forest data at $z\gtrsim2$, not additional galaxy BAO samples, are the practical route to settling this model.
  • The marginal $p$-value of the $\Lambda$CDM fit hints that the ratio data may prefer something beyond flat $\Lambda$CDM; extending the same Bayesian comparison to $w$CDM would show whether a varying dark-energy equation of state beats the standard model by an even larger margin.
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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

3 major / 4 minor

Summary. The paper uses DESI DR1 BAO measurements of the ratio D_M/D_H (normalized by the sound horizon, which cancels in the ratio) to compare flat ΛCDM against the R_h=ct model. R_h=ct has no free parameters, while flat ΛCDM has one free parameter, Ω_m, which is marginalized over either a uniform prior or a Planck-informed Gaussian prior. The authors compute Bayes factors with Dynesty for each redshift bin and for the combined sample, with and without accounting for the reported correlation between D_M and D_H. They find that most low-redshift bins give Bayes factors they describe as close to 1, while the Lyman-α QSO bin at z=2.33 gives Bayes factors between roughly 760 and 21,500 favoring ΛCDM, and the combined sample gives values up to about 77,700. The abstract concludes that DESI DR1 measurements rule out R_h=ct, albeit driven by the z=2.33 point, and Section V explicitly notes that the conclusion would change if this measurement were revised or its systematics questioned.

Significance. If the central claim were robust, this would be a valuable addition to the model-comparison literature for R_h=ct, since it uses a geometric quantity independent of r_d and H_0 and applies a standard Bayesian evidence calculation with two priors. The paper's method is transparent and the use of both correlated and uncorrelated cases is a useful cross-check. However, the decisive result rests on a single DESI Lyman-α measurement, and the authors themselves concede this fragility in Section V. The absence of any systematic-sensitivity analysis, together with an internal inconsistency between the text and Table I regarding the low-redshift Bayes factors, means the paper in its current form overstates the strength of the evidence.

major comments (3)
  1. [Sec. IV, Table I] The statement that for most observables up to z=1.3 the Bayes factors are close to 1 is not supported by the table: LRG2 has B=5.6 (Normal/correlation) and 8.3 (Normal/no correlation), LRG3+ELG1 has 2.9–3.6, while ELG2 has 0.1–0.3 under the uniform prior. These are not 'close to 1' in the Jeffreys' scale sense; 5.6 indicates moderate evidence for ΛCDM and 0.3 actually indicates moderate evidence for R_h=ct. Please revise the claim and discuss the per-redshift pattern, since the current wording is contradicted by the paper's own table.
  2. [Sec. V] The sentence 'In case this measurement gets revised or if systematics of this data point questioned, it would change our conclusion' concedes that the central result is entirely driven by the Lyα point at z=2.33. Since the abstract states that DESI DR1 measurements 'rule out' R_h=ct, the analysis needs a quantitative robustness test. For example, inflate the Lyα measurement error by plausible systematic contributions (continuum fitting, SiIII absorption, broadband distortions) or shift the central value and recompute the Bayes factors. Without such a test, 'decisively favored' and 'rule out' overstate the evidence.
  3. [Sec. IV, combined row of Table I] The combined Bayes factor is computed by multiplying likelihoods across the six redshift bins, which treats the DESI DR1 measurements as statistically independent. DESI's BAO measurements from overlapping tracers (e.g., LRG3 and ELG1 are combined into one point, and the Lyα and QSO samples may share large-scale structure and systematic effects) can have non-negligible cross-correlations. Please justify the independence assumption or include available covariance between redshift bins and show how the combined Bayes factor changes.
minor comments (4)
  1. [Abstract and Secs. IV–V] Several typos should be corrected: 'tranvserse' in the abstract, 'rules our' in Sec. IV, and 'one conclusions' and 'prveious' in Sec. V.
  2. [Sec. IV] The error-propagation formula used to obtain the uncertainty on D_M/D_H from the DESI-reported D_M/r_d and D_H/r_d values is not shown. Please include the formula and state explicitly how the correlation coefficient r enters the propagated error.
  3. [Fig. 1] The figure would be much more informative if the data points had error bars and the theoretical curves were labeled with the model names and prior choices. As it stands, the reader cannot visually verify the quoted 4.5σ discrepancy.
  4. [Sec. IV, Table I] The text says the results are 'agnostic to the choice of priors', but several low-redshift rows differ by factors of 2–3 between the uniform and Normal priors (e.g., LRG2 from 2.2 to 5.6 with correlation). This statement should be softened or qualified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: DESI DR1 external ratios are compared to parameter-free R_h=ct and to ΛCDM via Bayesian evidence; no fitted quantity is renamed as a prediction.

full rationale

The derivation chain is self-contained against external data and contains no step in which a claimed prediction is equivalent to its input by construction. The observables are the DESI DR1 ratios D_M/D_H (formed from public D_M/r_d and D_H/r_d values), and the model predictions follow from Eqs. (2)–(5): H(z)=H0(1+z) for R_h=ct and H(z)=H0[Omega_m(1+z)^3+1-Omega_m]^{1/2} for flat LambdaCDM. No parameter is fitted to the target claim and then renamed a prediction; R_h=ct has no free parameters, and the LambdaCDM parameter Omega_m is marginalized over stated priors (uniform and Planck-informed). The same qualitative conclusion holds under both priors, so the Planck prior is not load-bearing for the central result. The citations to the authors' own prior work (e.g., [38], [48]) are general references and are not used to force the model choice or the interpretation. The caveat in Section V that the conclusion would change if the z=2.33 Lyman-alpha measurement is revised is a statement of data sensitivity, not a circular reduction: the decisive Bayes factor is computed from an external DESI measurement, not from the model being tested. No step satisfies any of the enumerated circularity patterns.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

All ingredients are standard cosmological models and published DESI DR1 data. No new entities are introduced. The analysis depends on the model-independence of the distance ratio, on the accuracy of the DESI measurements, and critically on the Lyman-alpha point at z=2.33.

free parameters (1)
  • Omega_m (flat LambdaCDM) = 0.32 (with correlation), 0.31 (without), using Normal prior; marginalised over uniform prior in other case
    The only free parameter in flat LambdaCDM; its posterior and evidence are computed via nested sampling. The best-fit values are reported in Fig. 1.
assumptions (3)
  • domain assumption The ratio DM/DH derived from DESI BAO measurements is independent of the fiducial cosmology and of rd and H0.
    Invoked in Section IV via reference [49]; the paper does not re-derive this.
  • domain assumption The DESI DR1 BAO measurements are unbiased and their reported errors and correlation coefficients are accurate.
    The likelihood is constructed from these published values with no additional systematic budget.
  • domain assumption The Lyman-alpha BAO measurement at z=2.33 is reliable and free of unmodeled systematics.
    The authors explicitly state that if this measurement is revised, the conclusion changes (Section V).

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Cite this review

Pith. "Pith review of Comparison of $R_h=ct$ and $\Lambda$CDM using DESI DR1 measurements." pith.science (2026). https://pith.science/paper/7K66QGIL

@misc{pith2026250618445,
  author       = {Pith},
  title        = {Pith review of: Comparison of $R_h=ct$ and $\Lambda$CDM using DESI DR1 measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7K66QGIL}},
  note         = {Machine review of arXiv:2506.18445}
}
abstract

We use DESI DR1 BAO measurements of the ratio of tranvserse comoving distance to Hubble distance in order to test the compatibility of $R_h=ct$ over flat $\Lambda$CDM. For this purpose, we used Bayesian model selection to evaluate the efficacy of these models given the observed data. When we consider the BAO measurements up to redshift of 1.3, both models are equally favored. However, when we consider the Lyman-$\alpha$ QSO measurement at redshift of 2.33, we find Bayes factors of greater than 100 for flat $\Lambda$CDM over $R_h=ct$ using two different priors for $\Omega_m$, indicating that $\Lambda$CDM is decisively favored over $R_h=ct$. The same is the case when we combine all the measurements. Therefore, the DESI DRI measurements rule out $R_h=ct$ cosmology, albeit this is driven by the Lyman-$\alpha$ QSO measurement at $z=2.33$.

Figures

Figures reproduced from arXiv: 2506.18445 by the authors.

Figure 1
Figure 1. FIG. 1: The best-fit plots for the ratio [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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