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New tests of cosmic distance duality relation with DESI 2024 BAO observations

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Using ratio tests that cancel the supernova absolute magnitude and the BAO sound horizon, this paper finds that the cosmic distance duality relation holds at low redshift but deviates from unity by more than 3σ at z=2.33 in the DESI 2024…

desk verdict A clean low-redshift null test of the CDDR wrapped around an overinterpreted high-redshift hint that sits on Lyα BAO points with known systematics. read the letter →

arxiv 2506.12759 v1 pith:7DY6NKG6 submitted 2025-06-15 astro-ph.CO

Qiumin Wang , Shuo Cao , Jianyong Jiang , Kaituo Zhang , Xinyue Jiang , Tonghua Liu , Chengsheng Mu , Dadian Cheng This is my paper
classification astro-ph.CO
keywords cosmicdistancedualityrelationEtheringtonreciprocitytheorembaryonacousticoscillationsDESItypeIasupernovaeartificialneuralnetworkreconstructionsoundhorizontwo-pointdiagnostic
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 cosmic distance duality relation, the theorem that luminosity distance equals angular diameter distance times $(1+z)^2$, by combining the latest baryon acoustic oscillation measurements (including DESI Year 1 and Year 2) with type Ia supernova samples. The analysis uses ratios of the relation at two redshifts, a construction in which the sound horizon from BAO and the supernova absolute magnitude cancel exactly, so no cosmological model or calibration is needed. At low redshifts the ratios are consistent with unity, but at the two highest measured redshifts, $z=2.33$ and $z=2.334$, the paper reports deviations from unity at more than $3\sigma$ and $2\sigma$ confidence, respectively, and calls this positive evidence that the relation may break down there. If the result is right, either photons behave unexpectedly at high redshift or the distance measurements carry a redshift-dependent systematic that mimics a violation.

What carries the argument

The machinery is the two-point diagnostic ratio $$\eta_{ij}=\frac{\eta(z_i)}{\eta(z_j)}=\frac{\$\theta$(z_i)}{\$\theta$(z_j)}\frac{1+z_j}{1+z_i}\,10^{\$\Delta$ m/5},$$ where $\theta(z)$ is the observed BAO angular scale and $\Delta m$ is the difference in corrected supernova apparent magnitudes between the two redshifts. Because $r_d$ and $M_B$ appear only as multiplicative factors that cancel, the ratio is free of both nuisance parameters. At each BAO redshift the analysis anchors $N-1$ such ratios and combines them with the mean, the weighted mean, and the median absolute deviation, while an artificial neural network reconstruction of the Pantheon sample supplies supernova magnitudes at the exact BAO redshifts, including the high-redshift Lyman-$\alpha$ points.

What would settle it

Recompute the $\eta_{ij}$ statistics with the Lyman-$\alpha$ BAO points removed: if no deviation above $2\sigma$ remains, the claimed CDDR failure is driven entirely by those two measurements. Alternatively, fit the same data allowing a separate sound-horizon scale for the Lyman-$\alpha$ tracer; if the $z\approx2.33$ deviation vanishes when that scale is free, the CDDR violation is not robust.

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Extended reading notes

Core claim

The paper's central claim is that an unbiased, model-independent test of the Etherington relation, $\eta(z)=D_L(z)/[D_A(z)(1+z)^2]$, can be built from the ratio $\eta_{ij}=\eta(z_i)/\eta(z_j)$, and that this ratio is exactly insensitive to the sound horizon $r_d$ and the absolute magnitude $M_B$. Applying this to 2D and 3D BAO data from BOSS, eBOSS, DES and DESI together with DESY5 and ANN-reconstructed Pantheon supernovae, the paper finds that nearly all $\eta_{ij}$ agree with unity within $2\sigma$, but the two Lyman-$\alpha$ BAO points at $z=2.33$ and $z=2.334$ produce $\eta_{ij}\neq 1$ at $>3\sigma$ and $>2\sigma$ under all three statistical treatments (mean, weighted mean, and median absolute deviation). The paper therefore concludes there is positive evidence of deviation from the cosmic distance duality relation at those high redshifts, while emphasizing that the deviation is not yet definitive and could reflect systematics.

Load-bearing premise

The test assumes all BAO measurements share one sound horizon that cancels exactly in the ratio, so any systematic offset in the angular scale of the high-redshift Lyman-$\alpha$ measurements — from continuum fitting, tracer bias, or redshift-space distortions — would be misread as a violation of the cosmic distance duality relation.

Editorial extensions

If this is right

  • If the high-redshift deviations are real, the cosmic distance duality relation fails at $z\gtrsim 2.3$, which would mean the simple assumptions behind distance measurements — photon number conservation, unique null geodesics, metric gravity — are incomplete at those redshifts.
  • The nuisance-parameter-free ratio method gives a direct, model-independent way to screen future BAO and supernova datasets for violations, so the same test can be rerun on every new data release without re-calibrating $r_d$ or $M_B$.
  • The implied sound horizon $r_d=122.5\pm14.7$ Mpc and $r_d=127.4\pm15.6$ Mpc at the two deviant redshifts sit about $1.3$–$1.7\sigma$ away from the cosmic-microwave-background calibration, so the anomaly is not explained by the standard $r_d$ tension alone.
  • If instead the deviation comes from systematics in the Lyman-$\alpha$ BAO measurements, the result would indicate that high-redshift BAO angular scales are currently not reliable enough for precision cosmology, which matters for dark-energy and curvature constraints.

Reading between the lines

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

  • Because the two deviant points are both Lyman-$\alpha$ BAO measurements, a direct test that allows a separate sound-horizon scale for the Lyman-$\alpha$ tracer — rather than one shared $r_d$ for all BAO — would separate a genuine CDDR violation from a tracer-specific systematic; the paper does not perform this test.
  • The paper acknowledges but does not apply multiple-comparison corrections across the $N$ anchored statistical evaluations; applying a formal correction would likely lower the nominal significance of the $z\approx2.33$ deviations, so the true detection strength is probably weaker than the headline $>3\sigma$.
  • The same $\eta_{ij}$ construction could be applied to other distance indicators, such as strong lensing time delays or gravitational-wave standard sirens at high redshift; if the apparent violation appears only for BAO-based angular distances, that would point toward BAO systematics rather than new physics affecting photon propagation.
  • A future DESI data release or an independent Lyman-$\alpha$ measurement near $z=2.33$ with a different continuum-fitting pipeline would provide a sharp test: if the deviation tracks the effective redshift of the Ly$\alpha$ correlation rather than a physical distance, it is a systematic.
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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 / 5 minor

Summary. The paper tests the cosmic distance duality relation (CDDR) using the two-point diagnostic ratio eta_ij = eta(z_i)/eta(z_j), constructed from BAO angular scale measurements and SN Ia apparent magnitudes. The method cancels the SN absolute magnitude M_B and the BAO sound horizon r_d analytically in Eq. (6). The authors use DESY5 and Pantheon SN samples (the latter reconstructed with an ANN to match BAO redshifts) and BAO data from SDSS/BOSS/eBOSS and DESI DR1/DR2. They report consistency with the CDDR at low redshifts across all statistical treatments, but find deviations exceeding 2 sigma at z=2.33 and z=2.334, which they interpret as positive evidence for a possible CDDR violation. They also acknowledge the multiple comparison problem and known tensions between high-redshift Ly-alpha BAO and Planck-LambdaCDM.

Significance. If the high-redshift deviations are real, this would be an important result: the CDDR is a fundamental consequence of the Etherington reciprocity theorem, and a robust violation would indicate new physics or unrecognized systematics. The paper's main methodological strength is the clean cancellation of M_B and r_d in Eq. (6), which removes the dominant nuisance parameters. The low-redshift analysis is careful and demonstrates consistency across DESY5, Pantheon, and multiple BAO releases, which is a useful and credible result. However, the central high-redshift claim is not yet established: it rests entirely on the two Ly-alpha BAO points, which are known to be in tension with Planck-LambdaCDM, and on an ANN reconstruction that is extrapolated beyond the Pantheon redshift range. The authors are transparent about these limitations, but the paper does not currently contain the robustness tests needed to support the claimed >2 sigma evidence.

major comments (3)
  1. [Section 3, Figs. 5 and 6] The claimed >2 sigma deviations at z=2.33 and z=2.334 rest entirely on the two Ly-alpha BAO measurements. The cancellation of a common sound horizon in Eq. (6) is valid only if all BAO tracers measure the same acoustic scale with no differential offset. The authors themselves note that the z=2.33 Ly-alpha point shows a 2.3 sigma tension with Planck-LambdaCDM, reduced to 1.5 sigma after combining BOSS and eBOSS. Any systematic offset in the Ly-alpha angular scale, for example from continuum fitting or UV background modeling, would propagate linearly into eta_ij through theta(z_i)/theta(z_j). The paper does not provide a test that distinguishes an apparent CDDR violation from a Ly-alpha BAO scale systematic. A robustness check that removes these two points or models a redshift-dependent scale offset is essential before the high-redshift claim can be accepted.
  2. [Section 2.2, Fig. 4] The ANN reconstruction of m_B is trained on the Pantheon sample, which extends only to z ~ 2.3, while the reconstructed values at z=2.33 and z=2.334 are extrapolations beyond the training range. The uncertainty from this extrapolation is not propagated into the quoted error bars on eta_ij. The authors should either quantify the reconstruction uncertainty at these redshifts (for example by cross-validation or by comparing alternative interpolators) or restrict the central claim to redshifts that lie within the SN Ia sample coverage. Without this, the reported significance of the high-redshift deviations is not fully characterized.
  3. [Section 3, paragraph beginning 'We emphasize that'] The authors correctly acknowledge the multiple comparison problem, noting that the analysis involves N distinct statistical evaluations and that formal multiplicity corrections are not applied. Since the high-redshift deviations are selected from 26-27 eta_ij pairs, the nominal >2 sigma and >3 sigma levels do not represent experiment-wide significance. The manuscript should apply a multiple-comparison correction (for example Benjamini-Hochberg) and report adjusted significance levels. Without this, the statement that the results provide 'positive evidence of deviation from the CDDR' at these redshifts is overstated.
minor comments (5)
  1. [Equation (6) and surrounding text] The sentence 'where i and j denote the indices of BAO and SN Ia, respectively' is ambiguous because theta and m_B are both evaluated at z_i and z_j. The indices appear to label redshift values rather than datasets; please clarify the notation.
  2. [Fig. 4 caption] The caption states that the ANN reconstruction provides a continuous representation of m_B up to z ~ 2.4, but the Pantheon data end near z ~ 2.3 and the reconstruction at z=2.334 is an extrapolation. Please state explicitly which part of the curve is extrapolated.
  3. [Section 2.2, ANN description] The ANN architecture, activation function, learning rate, number of epochs, and training-validation split are not reported. Since the method is central to the high-redshift result, these details are needed for reproducibility.
  4. [Throughout] There are several typographical errors, including 'with with' in Section 2.1 and 'the the presence' in Section 3. In Section 3 the text also uses 'DDR' instead of 'CDDR' in one instance.
  5. [Section 3, sensitivity test] The sensitivity test that fixes M_B=-19.25 or r_d=147.09 Mpc to derive the other parameter is not a decisive diagnostic of a CDDR violation, because it assumes one of the two nuisance parameters a priori. Reporting these tensions as 1.7 sigma and 1.6 sigma is useful but should not be presented as evidence for or against the CDDR; please frame it as an illustrative consistency check.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the eta_ij ratio cancels M_B and r_d by construction and the high-redshift deviation is a direct data comparison, not a fitted or self-cited result.

full rationale

The paper's derivation chain is self-contained. Equation (1) defines D_A from the BAO angular scale and a common sound horizon; Eq. (2) defines D_L from the SN apparent magnitude and a common absolute magnitude; Eq. (4) defines eta; Eq. (5) is direct substitution; Eq. (6) algebraically cancels M_B and r_d, leaving a ratio of measured angles and measured (or ANN-reconstructed) magnitudes. No free parameter is fitted to the target, so the central claim is a direct data comparison rather than a fitted input relabeled as a prediction. The only self-citation is to Liu et al. (2023) as provenance for the two-point ratio; the same formula is re-derived in the present paper and is simple algebra, so the citation is not load-bearing and does not import an unverified uniqueness theorem. The ANN reconstruction is trained on the Pantheon magnitudes themselves, but it serves as an interpolator returning m_B at BAO redshifts; it does not encode a CDDR value, and the final eta_ij test remains a direct comparison. The paper explicitly flags the multiple-comparison issue and the previously known 2.3-sigma tension of the Lyman-alpha BAO point with Planck, which are honesty statements about systematic and statistical caveats rather than circular steps. I therefore find no equivalence between inputs and outputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central test is parameter-free after cancellation of r_d and M_B, but it rests on the assumption that all BAO measurements share one sound horizon and that the SN absolute magnitude does not evolve with redshift. The ANN reconstruction at the high-redshift edge adds an assumption tied to this paper's specific implementation.

free parameters (1)
  • ANN training configuration (architecture, learning rate, epochs)
    Chosen by hand and not specified in the paper. The reconstructed m_B at z=2.33 depends on these settings. The ANN weights are fitted to the Pantheon data, though they are not physical parameters.
assumptions (4)
  • domain assumption All BAO measurements share a single sound horizon r_d that cancels in the ratio
    Eq. (6) cancels r_d only if every BAO angular scale is generated by the same physical r_d. Any redshift-dependent systematic in BAO calibration would masquerade as a CDDR violation.
  • domain assumption SN Ia absolute magnitude M_B is redshift-independent after standardization
    The ratio test removes a constant M_B, but any evolution of M_B with redshift would be absorbed into eta_ij and appear as a CDDR violation. This degeneracy is not broken by the data.
  • ad hoc to paper The ANN reconstruction accurately represents the underlying m_B(z) relation, including at z=2.33 near the edge of the Pantheon sample
    The high-redshift BAO points lie at or above the maximum Pantheon redshift (z=2.3), so the reconstructed magnitudes are near-extrapolations whose bias is not quantified.
  • domain assumption Individual BAO measurements in the combined sample are independent
    The paper calls the method 'uncorrelated', but multiple measurements from the same survey (e.g., DESI DR2) may share systematics. Ignoring correlations can bias the reported significance.

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Cite this review

Pith. "Pith review of New tests of cosmic distance duality relation with DESI 2024 BAO observations." pith.science (2026). https://pith.science/paper/7DY6NKG6

@misc{pith2026250612759,
  author       = {Pith},
  title        = {Pith review of: New tests of cosmic distance duality relation with DESI 2024 BAO observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DY6NKG6}},
  note         = {Machine review of arXiv:2506.12759}
}
abstract

In this paper, we test the cosmic distance duality relation (CDDR), as required by the Etherington reciprocity theorem, which connects the angular diameter distance and the luminosity distance via the relation \( D_{\rm L}(z) = D_{\rm A}(z)(1+z)^2 \). Our analysis is based on the latest baryon acoustic oscillation (BAO) measurements provided by the Dark Energy Survey (DES), the Baryon Oscillation Spectroscopic Survey (BOSS)/Extended BOSS (eBOSS), and the Dark Energy Spectroscopic Instrument (DESI) surveys. Specifically, an unbiased test of the CDDR is performed through a novel, model-independent method inspired by the two-point diagnostic approach, with DES-SN5YR and Pantheon type Ia supernova (SN Ia) sample reconstructed using the Artificial Neural Network (ANN) technique. This methodology effectively eliminates all nuisance parameters, including the sound horizon scale \( r_{\rm d} \) from BAO and the absolute magnitude \( M_{\rm B} \) from SN Ia. A set of \( N-1 \) independent CDDR ratios \( \eta_{ij} \) are constructed for statistical analysis. At the current observational level, no significant deviation from the CDDR is observed at low redshifts, whereas we find positive evidence ($>2\sigma$ C.L.) of deviation from the CDDR at two high redshifts ($z=2.33$ and $z=2.334$). Therefore, our results confirm that the BAO measurement provides a powerful tool to test such fundamental relation in modern cosmology.

Figures

Figures reproduced from arXiv: 2506.12759 by the authors.

Figure 1
Figure 1. The complete BAO sample used in this analysis, following the relation of θ(z) [rad] = rd/ [DA(z)(1 + z)]. The blue and orange points represent the 2D BAO and 3D BOSS measurements, the solid red points correspond to 3D DESI DR1 data, and the hollow light-red points denote the 3D DESI DR2 data. The inner panel illustrates the data points and associated error bars within the redshift range of 0.4 < z < 1. with 3D DR1, … view at source ↗
Figure 2
Figure 2. A 2D bubble plot of the ηij two-point diagnostics derived from 21 data pairs between DESY5 and the combined BAO datasets (3D BAO and 3D DR1). The color of each bubble indicates the value of ηij , while the bubble size is proportional to the corresponding error bar (left panel). The results obtained from three different statistical methods (anchored at each of the 21 zi), are also presented (right panel). 0.2 0.4 0.6… view at source ↗
Figure 3
Figure 3. The same as [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The Pantheon dataset and the reconstructed ap￾parent magnitude (mB) using ANN. The blue shaded area represents the 1σ confidence region of the ANN reconstruc￾tion, while the black points with error bars show the Pan￾theon data points. The ANN reconstruction closely fol…
Figure 5
Figure 5. Figure 5: A 2D bubble plot of the ηij two-point diagnostics derived from 26 data pairs between ANN-reconstructed Pantheon sample and the combined BAO datasets (3D BAO and 3D DR1). The color of each bubble indicates the value of ηij , while the bubble size is proportional to the …
Figure 6
Figure 6. Figure 6: The same as [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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