Pith. sign in

REVIEW 3 major objections 4 minor 26 references

A New Consistency Test for the $\Lambda$CDM Model using Radial and Transverse BAO Measurements

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Flat ΛCDM survives a calibration-free consistency test built from ratios of BAO distances measured by DESI.

desk verdict The radial–radial inversion is misprinted (Eq. 6 missing the reciprocal, Eq. 7 gives negative ΩM for all their pairs), so the tables aren't reproducible from the text; the mixed ratios and covariance treatment are still worth a referee. read the letter →

arxiv 2601.07075 v3 pith:6KYLCQLM submitted 2026-01-11 astro-ph.CO

Xing-Han A. Zhao , Zheng Cai This is my paper · ORCID
classification astro-ph.CO
keywords baryonacousticoscillationsflatLambdaCDMconsistencytestOmdiagnosticsoundhorizondarkenergyDESIdistanceratios
topics Dark Energy
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 tries to establish a simple, calibration-free internal consistency test for the standard flat ΛCDM cosmology. The test forms ratios of BAO distances—radial, transverse, and volume-averaged—so that the unknown sound-horizon scale cancels, and then converts each observed ratio into the matter density Ω_M that a flat ΛCDM universe would need to reproduce it. Flat ΛCDM predicts that this inferred Ω_M should be the same at every redshift and for every kind of ratio. Applied to DESI Data Release 1 and Data Release 2 BAO measurements with full covariance propagation, the inferred Ω_M values are broadly consistent with a redshift-independent constant within current uncertainties. The result matters because it provides a transparent, low-assumption check that can be sharpened as future BAO data arrive.

What carries the argument

The central objects are three ratio diagnostics—R_HH, R_HM, and R_HV—each defined as a ratio of BAO distances so that the sound horizon scale cancels exactly, with the overall H_0 normalization also dropping out. Each measured ratio is mapped to an effective flat-ΛCDM matter density Ω_M^Λ by equating it to the model prediction and solving for Ω_M, analytically for the radial–radial case and numerically for the mixed ratios. For integrated distances, the integral mean value theorem supplies effective line-of-sight redshifts and conservative redshift intervals, allowing transverse and isotropic BAO information to be placed on the same footing as direct radial measurements.

What would settle it

Recompute the constant-Ω_M^Λ fits using the full covariance matrix of the derived ratios—or of the underlying BAO measurements—instead of symmetrized independent errors; if the apparent redshift constancy disappears or a family disagreement becomes statistically significant, the paper's central claim fails. Alternatively, a future BAO data release with tighter errors that shows Ω_M^Λ trending with redshift, or a clear offset between the R_HH and R_HM families, would falsify flat ΛCDM under this test.

Watch

Extended reading notes

Core claim

The central claim is that flat ΛCDM predicts a single effective matter density parameter, Ω_M^Λ, that is independent of redshift and independent of which BAO distance ratio is used, and that DESI DR1 and DR2 BAO data satisfy this prediction at current precision. Three families of ratios are constructed: a purely radial ratio D_H(z_i)/D_H(z_j), a radial–transverse ratio D_H(z_i)/D_M(z_i), and a radial–isotropic ratio D_H(z_i)/D_V(z_j). For ratios involving the integrated distances D_M and D_V, the paper uses the integral mean value theorem to associate each measurement with a well-defined effective redshift interval. All derived Ω_M^Λ values are obtained by propagating the full published BAO

Load-bearing premise

The conclusion that Ω_M^Λ is redshift-independent rests on constant-Ω_M fits that treat each derived value as an independent measurement with symmetrized 68% errors, even though the ratios are built from heavily overlapping BAO measurements and are therefore strongly correlated.

Editorial extensions

If this is right

  • If the constancy claim holds, BAO data can check flat ΛCDM without any calibration of the sound horizon or H_0, making the test insensitive to a whole class of systematic uncertainties.
  • Radial and transverse BAO information can be combined in a single null test, extending the popular Om diagnostic to integrated distance measurements.
  • A statistically significant redshift dependence of Ω_M^Λ within a single ratio family, or a disagreement between families built from the same dataset, would signal a departure from flat ΛCDM and motivate further investigation.
  • The framework generalizes to extended background models such as wCDM or models with curvature, and can be combined with other late-time probes to diagnose the origin of any detected inconsistency.
  • Future DESI data releases and other wide-area surveys will tighten the test by providing more precise distance measurements over finer redshift bins and more ratio combinations.

Reading between the lines

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

  • The current constancy conclusion is weaker than it appears because the derived ratios heavily reuse the same underlying BAO measurements: the same D_H appears in multiple R_HH and R_HV ratios, and the same D_M anchors both R_HM and R_HV. A full covariance treatment of the derived ratios, rather than treating them as independent, could strengthen or weaken the apparent consistency.
  • The effective-redshift mappings depend weakly on the assumed Ω_M when assigning redshift intervals; as redshift bins become finer, the mixed ratios become more local and the test gains discriminating power.
  • The same ratio construction could be applied to other cosmological distance probes, or to future BAO releases, to produce an evolving 'BAO-Om ladder' that isolates redshift-dependent departures from flat ΛCDM without invoking a calibrated distance scale.
Share X Bluesky LinkedIn Reddit HN

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 proposes a calibration-free null test of spatially flat ΛCDM using ratios of BAO distances: D_H(z_i)/D_H(z_j), D_H(z_i)/D_M(z_i), and D_H(z_i)/D_V(z_j). Each ratio is mapped to an effective matter-density parameter Ω_M^Λ, defined as the value that reproduces the observed ratio in flat ΛCDM. The sound horizon and H_0 cancel in the ratios. The method is applied to DESI DR1 and DR2 BAO data with full covariance propagation via Monte Carlo. The paper concludes that the inferred Ω_M^Λ values are broadly consistent with a redshift-independent constant, providing an internal consistency check of flat ΛCDM.

Significance. If the result holds, this is a useful, model-dependent-but-calibration-free diagnostic that combines radial and transverse BAO information in a single framework. The ratio construction cleanly cancels r_d and H_0, and the use of the integral mean value theorem to assign effective redshifts is a transparent way to handle integrated distances. The Monte Carlo propagation of the full DESI covariance matrices is a strength, as is the public availability of the analysis code. However, the central algebraic derivation contains an error that undermines the reproducibility of the printed results.

major comments (3)
  1. [§2.2.1, Eq. (6)–(7)] Eq. (6) is not the standard flat-ΛCDM Hubble distance. With H(z)/H_0 = sqrt(Ω_M(1+z)^3 + 1−Ω_M), the correct relation is D_H = (c/H_0) [Ω_M(1+z)^3 + 1−Ω_M]^{-1/2}; Eq. (6) omits the reciprocal. Consequently, Eq. (7), derived from Eq. (6), gives the inversion for the inverse ratio R_HH → 1/R_HH, not for R_HH = D_H(z_i)/D_H(z_j) as defined in Eq. (5). For standard flat ΛCDM with z_i < z_j and R_HH > 1, Eq. (7) as printed yields the wrong sign. The R_HH values in Tables 1 and 2 therefore cannot be reproduced from the text as written, and the same reciprocal error propagates into the numerical inversions for R_HM and R_HV that use Eq. (6) (§2.2.2, §2.2.3). The authors should correct Eqs. (6)–(7), re-run the analysis with the correct formula, and verify that the reported tables and conclusions remain valid. This is load-bearing because the R_HH results feed into the constant-Ω_M fits and the
  2. [§3.5] The constant-Ω_M fits treat the derived Ω_M^Λ values as independent measurements with symmetrized 68% uncertainties. This is acknowledged in the text, but the quoted χ²/dof values (e.g., 1.14 for DR1 DH/DH, 1.45 for DR1 DH/DM) are not valid significance statistics because the ratios heavily reuse the same D_H and D_M data across redshift pairs, inducing strong correlations. The abstract's statement that the values are 'broadly consistent with a redshift-independent constant' rests partly on these fits. A full covariance propagation into the derived Ω_M^Λ vector is needed to claim quantitative consistency; as it stands, the qualitative conclusion is driven by the very large 68% intervals, not by the fit statistics.
  3. [§3.5/§4] The within-family best-fit constants differ substantially between diagnostics: for DR1, Ω_M^Λ ≈ 0.146 for DH/DH versus ≈ 0.293 for DH/DM, with a similar spread in DR2. The paper notes 'differences between diagnostic families motivate the discussion' but does not quantify whether these differences are statistically significant. Given that the test is explicitly designed to check consistency across ratio families, the authors should either assess this cross-family tension with the full covariance or soften the claim of overall consistency in the abstract.
minor comments (4)
  1. [References] The reference 'P. Collaboration et al. 2020' should be formatted as 'Planck Collaboration et al. 2020' for consistency with standard usage.
  2. [Table 2] For the R_HV row with z_i = 0.510 and z_j = 0.295, z_j < z_i; the text does not specify whether the ratio convention requires z_i > z_j. Please clarify the ordering convention for all ratios.
  3. [General] The notation 'Λ−→' is unconventional and visually confusing; standard arrows would improve readability.
  4. [§1] The phrase 'relative distances' and 'Om statistic' would benefit from consistent capitalization and a brief definition of the Om variable at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ΩM^Λ values are one-parameter inversions of observed ratios, and the test's content is the cross-ratio/redshift consistency check, which is non-circular.

full rationale

The derivation chain is self-contained and non-circular. The paper defines ΩM^Λ as the flat-ΛCDM parameter that maps each observed BAO distance ratio (Eqs. 5–10) to an inferred matter density. This is a one-parameter inversion, not a prediction of new data: the test's content lies in checking whether the same ΩM^Λ reproduces all ratios across redshift and across ratio families. That is the standard logic of the Om diagnostic, and it is a consistency test rather than a circular derivation. No load-bearing argument depends on the authors' own prior work or on an imported uniqueness theorem. The phrase 'Flat ΛCDM predicts that ΩM^Λ should be independent of redshift' is a genuine model consequence: under flat ΛCDM with matter density ΩM, each ratio equals the corresponding ΛCDM prediction at ΩM, so inversion returns ΩM for every ratio. The paper explicitly labels the Sec. 3.5 constant fits as descriptive ('this is intended as a compact summary rather than a full likelihood analysis (which would require the complete covariance of the derived ratios)') and therefore does not disguise a fit as an independent prediction. Statistical concerns about correlated ratios and the algebraic sign/reciprocal issue in Eqs. (6)–(7) are correctness/reproducibility limitations, not circularity. Accordingly no circular steps are identified and the score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced; ΩM^Λ and effective redshifts are mathematical re-parameterizations of existing DESI BAO data. The central claim rests on the listed standard assumptions and on the accuracy of the published covariance; the per-ratio ΩM^Λ inversions are fitted outputs, not independent inputs.

free parameters (3)
  • Per-ratio effective matter density ΩM^Λ from RHH = DR1: 0.069, 0.228, 0.598, 0.286; DR2: 0.298, 0.153, 0.296, 0.900, 0.277
    Inverted from each observed DH(zi)/DH(zj) using Eq. 7; these are fitted outputs whose constancy is being tested, not external calibrations.
  • Per-ratio effective matter density ΩM^Λ from RHM = DR1: 0.648, 0.209, 0.268, 0.330, 0.361; DR2: 0.465, 0.353, 0.271, 0.273, 0.339, 0.304
    Obtained by numerically solving R^Λ_HM(z;ΩM)=observed DH/DM (Sec 2.2.2).
  • Per-ratio effective matter density ΩM^Λ from RHV = DR1: 0.454, 0.382; DR2: 0.381
    Obtained by numerically solving R^Λ_HV(zi,zj;ΩM)=observed DH/DV (Sec 2.2.3).
assumptions (6)
  • domain assumption Flat ΛCDM Hubble distance: D_H^Λ(z)=c/H0 [ΩM(1+z)^3+(1−ΩM)]^{-1/2} (Eq. 6)
    Used to map every observed ratio to ΩM^Λ; any model violation would appear as scatter in the derived ΩM^Λ.
  • standard math Standard BAO distance definitions DM(z)=∫_0^z DH dz' and DV=(z DM^2 DH)^{1/3} (Eqs. 3–4)
    Adopted from BAO literature; needed to form the ratios in Eqs. 5, 8, 10.
  • domain assumption Sound horizon rd cancels in all ratios and H0 drops out
    Requires DESI reported distances to share a common sound-horizon calibration; residual systematics in different BAO statistics would not cancel.
  • standard math Integral mean value theorem guarantees existence of effective redshifts z_c, z_d (Sec 2.3, Appendix A)
    Requires DH continuous and positive over the integration range; used for quoting redshift intervals, not for the ΩM^Λ inversion itself.
  • domain assumption Published DESI DR1/DR2 BAO measurements and covariance matrices are accurate and approximately Gaussian (Sec 2.6)
    MC draws from a multivariate Gaussian with the published covariance; non-Gaussian tails or underestimated covariance would bias the quoted intervals.
  • ad hoc to paper Constant-ΩM^Λ fits treat entries as independent with symmetrized errors (Sec 3.5)
    Used to produce descriptive χ2/dof values; invalid as significance statistics because ratios reuse the same BAO measurements. The authors acknowledge this limitation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A New Consistency Test for the $\Lambda$CDM Model using Radial and Transverse BAO Measurements." pith.science (2026). https://pith.science/paper/6KYLCQLM

@misc{pith2026260107075,
  author       = {Pith},
  title        = {Pith review of: A New Consistency Test for the $\Lambda$CDM Model using Radial and Transverse BAO Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KYLCQLM}},
  note         = {Machine review of arXiv:2601.07075}
}
abstract

We present a calibration-free consistency test of spatially flat $\Lambda$CDM based on baryon acoustic oscillation (BAO) distance measurements. The method forms ratios of BAO distances including the Hubble distance, the comoving angular diameter distance, and the volume-averaged distance, so that the sound horizon scale cancels, and then maps each observed ratio to an effective flat-$\Lambda$CDM matter density parameter, ${\Omega}_{\rm M}^{\Lambda}$, defined as the value of $\Omega_{\rm M}$ that reproduces the measured ratio within $\Lambda$CDM. Flat $\Lambda$CDM predicts that ${\Omega}_{\rm M}^{\Lambda}$ should be independent of redshift and of the particular ratio used. For ratios involving the integrated distances, we associate them with well-defined effective line-of-sight redshift intervals based on the integral mean value theorem. We apply the test to BAO measurements from the Dark Energy Spectroscopic Instrument (DESI) Data Release~1 and Data Release~2, propagating the full published BAO covariance matrices into all derived ratios and ${\Omega}_{\rm M}^{\Lambda}$ constraints. Within current uncertainties, the inferred ${\Omega}_{\rm M}^{\Lambda}$ values are broadly consistent with a redshift-independent constant, providing an internal consistency check of flat $\Lambda$CDM that can be strengthened straightforwardly as BAO measurements improve.

Figures

Figures reproduced from arXiv: 2601.07075 by the authors.

Figure 1
Figure 1. Summary of BAO distances and ratio-based [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Effective-redshift mapping implied by DM(z) = z DH(zc) (left) and DV(z) = z DH(zd) (right) in flat ΛCDM for representative values of ΩM. For each z, the mapped redshifts satisfy 0 < zc < zd < z. Increasing ΩM shifts both zc and zd to lower values, so ΩM = 1 provides a conservative lower bound for the effective redshift associated with a given DM(z) or DV(z). Alam, S., Aubert, M., Avila, S., et al. 2021, Physical Rev… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 1 canonical work pages

  1. [1]

    2025, Phys

    Abdul Karim, M., et al. 2025, Phys. Rev. D, 112, 083515, doi: 10.1103/tr6y-kpc6

    doi:10.1103/tr6y-kpc6 2025

  2. [2]

    G., et al

    Adame, A. G., et al. 2025a, JCAP, 04, 012, doi: 10.1088/1475-7516/2025/04/012

    doi:10.1088/1475-7516/2025/04/012 2025

  3. [3]

    G., et al

    Adame, A. G., et al. 2025b, JCAP, 01, 124, doi: 10.1088/1475-7516/2025/01/124

    doi:10.1088/1475-7516/2025/01/124 2025

  4. [4]

    Akeson, R., et al. 2019, arXiv e-prints, doi: 10.48550/arXiv.1902.05569 9 0.0 0.5 1.0 1.5 2.0 2.5 zDM 0.0 0.5 1.0 1.5 2.0 2.5 3.0 zDH(Ω M) Ω M = 0.0 Ω M = 0.3153 Ω M = 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 zDV Figure 2.Effective-redshift mapping implied byD M(z) =z D H(zc) (left) andD V(z) =z D H(zd) (right) in flat ΛCDM for representative values of Ω M. For ea...

    arXiv doi:10.48550/arxiv.1902.05569 2019

  5. [5]

    2021, Physical Review D, 103, 083533, doi: 10.1103/PhysRevD.103.083533

    Alam, S., Aubert, M., Avila, S., et al. 2021, Physical Review D, 103, 083533, doi: 10.1103/PhysRevD.103.083533

    doi:10.1103/physrevd.103.083533 2021

  6. [6]

    L., Banday, A

    Bennett, C. L., Banday, A. J., G´ orski, K. M., et al. 1996, Astrophysical Journal Letters, 464, L1, doi: 10.1086/310075

    doi:10.1086/310075 1996

  7. [7]

    R., Dave, R., & Steinhardt, P

    Caldwell, R. R., Dave, R., & Steinhardt, P. J. 1998, Physical Review Letters, 80, 1582, doi: 10.1103/PhysRevLett.80.1582

    doi:10.1103/physrevlett.80.1582 1998

  8. [8]

    Carroll, S. M. 2001, Living Reviews in Relativity, 4, 1, doi: 10.12942/lrr-2001-1

    doi:10.12942/lrr-2001-1 2001

Show all 26 references
  1. [9]

    J., Peacock, J

    Cole, S., Percival, W. J., Peacock, J. A., et al. 2005, Monthly Notices of the Royal Astronomical Society, 362, 505, doi: 10.1111/j.1365-2966.2005.09318.x

    2005

  2. [10]

    2020, Astronomy & Astrophysics, 641, A6, doi: 10.1051/0004-6361/201833910 de Bernardis, P., Ade, P., Bock, J., et al

    Collaboration, P., Aghanim, N., Akrami, Y., Ashdown, M., et al. 2020, Astronomy & Astrophysics, 641, A6, doi: 10.1051/0004-6361/201833910 de Bernardis, P., Ade, P., Bock, J., et al. 2000, Nature, 404, 955, doi: 10.1038/35010035

    2020 doi

  3. [11]

    J., Zehavi, I., Hogg, D

    Eisenstein, D. J., Zehavi, I., Hogg, D. W., et al. 2005, The Astrophysical Journal, 633, 560, doi: 10.1086/466512

    2005 doi

  4. [12]

    W., Leitch, E

    Halverson, N. W., Leitch, E. M., Pryke, C., et al. 2002, Astrophysical Journal, 568, 38, doi: 10.1086/338879

    2002 doi

  5. [13]

    2011, arXiv e-prints, doi: 10.48550/arXiv.1110.3193

    Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, doi: 10.48550/arXiv.1110.3193

    2011 doi arXiv

  6. [14]

    Peebles, P. J. E. 1980, The Large-Scale Structure of the Universe (Princeton, NJ: Princeton University Press)

    1980

  7. [15]

    1999, The Astrophysical Journal, 517, 565, doi: 10.1086/307221

    Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, The Astrophysical Journal, 517, 565, doi: 10.1086/307221

    1999 doi

  8. [16]

    Ratra, B., & Peebles, P. J. E. 1988, Physical Review D, 37, 3406, doi: 10.1103/PhysRevD.37.3406

    1988 doi

  9. [17]

    G., Filippenko, A

    Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, The Astronomical Journal, 116, 1009, doi: 10.1086/300499

    1998 doi

  10. [18]

    1976, Principles of Mathematical Analysis, 3rd edn

    Rudin, W. 1976, Principles of Mathematical Analysis, 3rd edn. (New York: McGraw-Hill)

    1976

  11. [19]

    Sahni, V., Shafieloo, A., & Starobinsky, A. A. 2008, Physical Review D, 78, 103502, doi: 10.1103/PhysRevD.78.103502

    2008 doi

  12. [20]

    Sahni, V., Shafieloo, A., & Starobinsky, A. A. 2014, The Astrophysical Journal, 793, L40, doi: 10.1088/2041-8205/793/2/L40

    2014 doi

  13. [21]

    Shafieloo, A., Sahni, V., & Starobinsky, A. A. 2012, Physical Review D, 86, 103527, doi: 10.1103/PhysRevD.86.103527

    2012 doi

  14. [22]

    F., Bennett, C

    Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., et al. 1992, Astrophysical Journal Letters, 396, L1, doi: 10.1086/186504

    1992 doi

  15. [23]

    2015, arXiv e-prints, doi: 10.48550/arXiv.1503.03757

    Spergel, D., Gehrels, N., Baltay, C., et al. 2015, arXiv e-prints, doi: 10.48550/arXiv.1503.03757

    2015 doi arXiv

  16. [24]

    2014, Publications of the Astronomical Society of Japan, 66, R1, doi: 10.1093/pasj/pst019

    Takada, M., et al. 2014, Publications of the Astronomical Society of Japan, 66, R1, doi: 10.1093/pasj/pst019

    2014 doi

  17. [25]

    1989, Reviews of Modern Physics, 61, 1, doi: 10.1103/RevModPhys.61.1

    Weinberg, S. 1989, Reviews of Modern Physics, 61, 1, doi: 10.1103/RevModPhys.61.1

    1989 doi

  18. [26]

    2008, Physical Review Letters, 101, 181301, doi: 10.1103/PhysRevLett.101.181301

    Zunckel, C., & Clarkson, C. 2008, Physical Review Letters, 101, 181301, doi: 10.1103/PhysRevLett.101.181301

    2008 doi

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.