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This paper combines gamma-ray bursts with five other cosmic datasets to test whether the cosmic distance duality relation holds, and finds no violation with a Hubble constant consistent with the early-universe Planck value.

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2026-08-04 19:26 UTC pith:PWJAYM57

load-bearing objection The no-violation result is probably a byproduct of multiplying the BAO distances by η(z); removing that factor is the key test. the 4 major comments →

arxiv 2509.09247 v1 pith:PWJAYM57 submitted 2025-09-11 astro-ph.CO gr-qc

Investigating the cosmic distance duality relation with gamma-ray bursts

Anna Chiara Alfano , Carlo Cafaro , Salvatore Capozziello , Orlando Luongo , Marco Muccino This is my paper
classification astro-ph.CO gr-qc PACS 98.80.Es98.70.Rz
keywords cosmic distance duality relationEtherington relationgamma-ray burstsBézier interpolationHubble constant tensionbaryon acoustic oscillationscosmic chronometersmodel-independent cosmology
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 asks whether the cosmic distance duality relation—the prediction that luminosity distances and angular diameter distances are related by (1+z)²—holds up to very high redshifts. It answers yes: across three different parametrizations of possible violations and two different gamma-ray burst correlations, the deviation parameter η₀ is always consistent with zero at 1 sigma. The same analysis returns a Hubble constant h₀ around 0.68, in line with the Planck early-universe measurement and not with the higher local supernova value. If true, this means gamma-ray bursts can extend the test of cosmic transparency and distance duality to redshift ~9, and the Hubble tension is not explained by a distance-duality violation.

Core claim

The central claim is that, using a model-independent Bézier reconstruction of the expansion history, the ratio D_L/[D_A(1+z)²] shows no statistically significant departure from unity. For all three adopted parametrizations (power-law, logarithmic, and Padé of order (1,2)), and for both the Amati and Combo gamma-ray burst correlations, the fitted η₀ is compatible with zero at the 1-sigma level. In the same global fit, the dimensionless Hubble constant h₀ converges to about 0.68, preferring the Planck value over the SH0ES local measurement. The authors interpret this as evidence that no CDD violation is needed and that gamma-ray bursts can serve as high-redshift probes without invoking new phy

What carries the argument

The analysis leans on Etherington's reciprocity law, which states D_L = (1+z)² D_A for any metric theory where photons travel on null geodesics and are conserved. Violations are parametrized by η(z) in three forms: (1+z)^η₀, exp[η₀ ln(1+z)], and a Padé rational function P(1,2)(z). To stay model-independent, the Hubble rate H(z) is reconstructed with a second-order Bézier polynomial from cosmic-chronometer data, and this reconstruction is propagated into distances for galaxy clusters, supernovae, BAO, and GRB correlations. The gamma-ray burst correlations (E_iso–E_p and L0–E_p–T) are used at high redshift; the Bézier reconstruction breaks their circularity problem, allowing them to constrain

Load-bearing premise

The conclusion rests on treating BAO distances as if they carry the same CDD violation factor η(z); if BAO is a purely geometric probe that should not be rescaled, the joint fit would be biased toward η₀ = 0.

What would settle it

Re-fit the same data with the BAO likelihood altered so that the measured D_M/r_d and D_V/r_d are not multiplied by η(z); if η₀ then shifts away from zero by more than 1 sigma, the reported no-violation conclusion is an artifact of that modeling choice. Alternatively, find direct luminosity and angular diameter distances to the same sources at z > 1 and compute the ratio D_L/[D_A(1+z)²] without assuming any parametrization.

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

If this is right

  • The cosmic distance duality relation holds within 1 sigma to redshifts of at least 2.3 (BAO) and potentially ~9 (GRBs), so no correction to photon number conservation or cosmic transparency is required by these data.
  • The combined low- and high-redshift datasets favor h₀ near 0.68, matching the Planck early-universe value and disagreeing with the SH0ES local value, so the Hubble tension persists under this model-independent treatment.
  • Gamma-ray bursts calibrated through the Bézier reconstruction can be used as standardizable distance indicators at z > 1, extending the redshift range of CDD tests beyond what SNe Ia and BAO alone provide.
  • The result is insensitive to which gamma-ray burst correlation is used, since both the Amati and Combo analyses give η₀ consistent with zero.

Where Pith is reading between the lines

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

  • If BAO distances are truly geometric and independent of the CDD factor, then multiplying them by η(z) in Eq. (16) may artificially pull η₀ toward zero; a re-analysis omitting this factor would test whether the no-violation conclusion is an artifact of that modeling choice.
  • The same pipeline could be applied to standard sirens from gravitational-wave events, which have an independent absolute distance calibration; a larger high-redshift sample would tighten η₀ and could distinguish a small violation from zero.
  • Because the three parametrizations agree, any real CDD violation must be smaller than current sensitivity; future GRB samples at z > 2 could push constraints on η₀ below 0.01 and settle the question more decisively.

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

4 major / 4 minor

Summary. The paper tests the cosmic distance duality relation by fitting three parametrizations of the violation parameter η(z) (power-law, logarithmic, and Padé) in a supposedly model-independent framework. The Hubble expansion is reconstructed with a second-order Bézier polynomial, and the resulting distances are used to fit a combination of OHD, SZ galaxy clusters, Pantheon SNe Ia, DESI DR2 BAO, and two GRB correlations (Amati and Combo). Two MCMC analyses are run (A and C). The authors report η0 consistent with zero at 1σ for all parametrizations and both GRB samples, and h0 consistent with the Planck value and inconsistent with the SH0ES value, concluding that there is no evidence for CDD violation and a preference for Planck's H0.

Significance. If the analysis were correct, the paper would provide a useful multi-probe, model-independent test of the CDD relation at higher redshifts than most previous work, and it makes an explicit connection to the H0 tension. The use of a Bézier reconstruction to calibrate GRB correlations is a practical way to address the circularity problem, and the paper is transparent about the likelihoods and data sets. However, the central conclusion is compromised by a load-bearing modeling error in the BAO likelihood: the geometric BAO distances are multiplied by η(z), even though BAO measures angular-diameter/transverse distances directly and should not depend on luminosity-distance modifications such as cosmic opacity. This artificially pulls η0 toward zero. A subsidiary circularity in fixing r_d to the Planck value weakens the H0 claim. The paper's useful ingredients do not rescue the central claim as written; a corrected analysis would be needed before the conclusions can be accepted.

major comments (4)
  1. [Sec. 3, Eq. (16a) and (16c)] The BAO likelihood is misspecified. In Eqs. (16a) and (16c), the transverse comoving distance and the volume-averaged distance are written as D_M/r_d = η(z)D_2(z)(1+z)/r_d and D_V/r_d ∝ [η(z)D_2(z)(1+z)]^{2/3}. But D_2(z) in Eq. (10) is already the angular diameter distance D_A(z), and BAO measures the angular scale of the sound horizon, θ_s = r_d/D_M, where D_M = (1+z)D_A. This is a purely geometric quantity: it does not depend on the luminosity distance or on cosmic transparency, and it should not carry the factor η(z) defined in Eq. (2). By inserting η(z) into D_M and D_V, the DESI BAO data—which are precise at the percent level—strongly force η(z)≈1, i.e. η0≈0, and thereby manufacture the paper's central 'no violation' conclusion. The factor η^{2/3} in D_V is likewise unmotivated. The analysis must be rerun with D_M/r_d = (1+z)D_2(z)/r_d and the corresponding correction to D_V, and t
  2. [Sec. 3, 'DR2 DESI-BAO'] The paper fixes the sound horizon to r_d = (147.09±0.26) Mpc from Planck before the fit, in order to break the r_d–h0 degeneracy. Because the BAO distances are expressed in units of r_d, this prior effectively converts the BAO data into distances that are consistent with Planck's early-universe calibration. The later statement that h0 agrees with Planck at 1σ is therefore partly circular and should not be presented as an independent confirmation. The authors should either marginalize over r_d with a broad prior, or show results with BAO removed, or otherwise demonstrate that the h0 preference does not come from the imposed Planck calibration.
  3. [Sec. 3, OHD likelihood, Eq. (8)] The OHD analysis adds statistical and systematic errors in quadrature rather than using the full covariance matrix. The text notes that the published covariance matrix covers only the original 15 points, but those points are part of the present 34-point sample. Ignoring the off-diagonal systematic terms (stellar population synthesis, stellar libraries, etc.) can bias the reconstructed Bézier coefficients and their error bars. Given that h0 and the distance scale D_2 are derived from this reconstruction, the impact on the central claims should be assessed, for example by using the 15-point covariance for that subsample and checking whether the h0 and η0 posteriors shift.
  4. [Sec. 5, Tables 4 and 5] The reported 'no violation' conclusion is based only on the fact that η0 is within 1σ of zero for each fitted parametrization. Since η0 is a free parameter fit simultaneously with all others, this is not a model-independent statement: the same data set is used to constrain the null hypothesis and to claim consistency with it. The authors should also report a model-comparison statistic (e.g., Δχ^2 or an information criterion) relative to η0=0, and should state, for each parametrization, what the upper limit on η0 actually is. This would make the strength of the conclusion quantitative rather than qualitative.
minor comments (4)
  1. [Abstract] 'well-estabilished' should be 'well-established'. Also, 'Our results seem to point towards ...' is too vague for a quantitative claim; the abstract should state the measured η0 and h0 values with uncertainties.
  2. [Sec. 2, Eq. (4)] The LOG parametrization is simply the first-order Taylor expansion of the power-law form. This is fine, but the text should state explicitly that the LOG model is a small-η0 approximation and therefore is not independent of the power-law model; it will necessarily give very similar constraints.
  3. [Sec. 3, Eq. (12)] Eq. (12) uses σ_DAi only, ignoring any covariance between cluster angular-diameter-distance measurements. If such covariance is negligible this is fine, but it should be stated; otherwise the GC likelihood is incomplete.
  4. [Sec. 5, Figures 1 and 2] The 1D posteriors for h0 and η0 are helpful, but the figures would be more informative if they also showed the 2D contours and the full parameter correlations, especially given the degeneracies among α0, α1, α2 and η0.

Circularity Check

2 steps flagged

BAO likelihood inserts η into geometric distances, forcing η0≈0; h0 preference is partly calibrated by fixing r_d to Planck.

specific steps
  1. self definitional [Section 3, DR2 DESI-BAO, Eqs. (16a)-(16c)]
    "Considering the violation of the CDD relation the three distances are written as DM(z)/rd = η(z)D2(z)(1+z)/rd, (16a), DH(z)/rd = c/rdH2(z), (16b), DV(z)/rd = 1/rd[zc/H2(z)]^{1/3}[η(z)D2(z)(1+z)]^{2/3}. (16c)"

    The BAO transverse and volume-averaged distances are geometric: they measure the angular scale of the sound horizon, D_M = (1+z)D_A. In this model D2(z) is already the angular diameter distance, while η(z) parameterizes the ratio D_L/[D_A(1+z)^2] for luminosity-based probes. By writing D_M/r_d and D_V/r_d with an explicit factor η(z), the likelihood effectively defines the BAO observable to contain the very parameter being tested. Since BAO data are precise, the fit is pulled toward η≈1 (η0≈0) by construction. The paper's concluding 'no violation' is therefore not an independent inference from BAO geometry; it is partly built into the definition of the fitted distance.

  2. fitted input called prediction [Section 3, DR2 DESI-BAO; Section 6, Final outlooks]
    "To break the r_d−h_0 degeneracy (DESI Collaboration, 2024) and constrain h_0, we fix the sound horizon at the baryon drag epoch to r_d =(147.09±0.26) Mpc (Planck Collaboration, 2020)."

    The BAO distances are dimensionless ratios D_M/r_d, D_H/r_d, D_V/r_d. Fixing r_d to the Planck-calibrated value transfers the Planck sound-horizon scale into the fit. The fitted h_0 then naturally comes out near the Planck value, because the BAO part of the likelihood is anchored to Planck through r_d. The abstract's 'preference towards Planck's value of H_0' is thus not a blind prediction: it is partially inherited from the Planck input used to break the r_d–h_0 degeneracy. Some independent H_0 information remains in the OHD and cluster distances, so this is partly circular rather than fully tautological.

full rationale

The paper's central result, no CDD violation, rests on fitting a free parameter η0 and finding it consistent with zero. That alone is not circular—fitting a parameter is a legitimate empirical exercise. The circularity enters through the BAO likelihood: Eqs. (16a,c) insert η(z) into the transverse and volume-averaged BAO distances, even though BAO is a geometric standard ruler that should be independent of the luminosity-distance duality violation parameter. By defining the BAO observable to contain η, the precise DESI DR2 data are made to enforce η0≈0, so the 'no violation' claim is at least partly an artifact of the model definition. The h0 preference is also partly inherited from the Planck value of r_d that is fixed before the fit. The paper does not rely on any load-bearing self-citation chain, and the Bézier interpolation is a standard cosmographic tool; the main problems are the BAO model specification and the Planck calibration of r_d.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The central CDD test relies on one fitted deviation parameter eta0 plus the Bezier coefficients that set the distance scale, and on several domain assumptions. The most consequential is the ad hoc inclusion of eta in the geometric BAO distances, which can predetermine the no-violation result.

free parameters (7)
  • eta0 = 0.003 +/- 0.016 (PL, Analysis A); similar for LOG and Pade
    CDD deviation parameter; fitted to the combined data. The central result is that it is consistent with zero.
  • alpha0 (h0) = 0.683 +/- 0.010 (Analysis A, PL)
    Bezier coefficient at z=0, interpreted as the reduced Hubble constant.
  • alpha1 = 1.054 +/- 0.027 (Analysis A, PL)
    Bezier coefficient controlling the linear term of the H(z) reconstruction.
  • alpha2 = 2.002 +/- 0.024 (Analysis A, PL)
    Bezier coefficient controlling the quadratic term of the H(z) reconstruction.
  • a (GRB slope) = 0.854 (Analysis A, Amati), 0.821 (Analysis C, Combo)
    Slope of the GRB correlation, fitted jointly with the cosmological parameters.
  • b (GRB intercept) = 1.618 (Analysis A), 49.702 (Analysis C)
    Intercept of the GRB correlation, fitted jointly with the cosmological parameters.
  • sigma (GRB intrinsic scatter) = 0.327 (Analysis A), 0.368 (Analysis C)
    Intrinsic dispersion of the GRB correlation, included as a free parameter.
axioms (5)
  • domain assumption Flat universe (Omega_k = 0)
    The analysis sets Omega_k = 0 in Eq. (10) so the distance formula reduces to the flat case; a nonzero curvature could change the distance scale and the eta0 constraints.
  • domain assumption Second-order Bezier polynomial adequately reconstructs H(z) from OHD
    The paper uses Eq. (6) as a model-independent H(z) and propagates it to all probes. A 3-parameter quadratic may not capture the true expansion history over the full redshift range.
  • domain assumption Validity of GRB correlations as distance indicators
    E_iso and L_0 are assumed to obey the Amati and Combo correlations with a single slope and intercept, with an intrinsic scatter; if the correlations evolve with redshift, the test is biased.
  • ad hoc to paper BAO distances should be multiplied by eta(z)
    Eqs. (16a,c) insert eta into the geometric BAO distances without physical justification; this can force eta0 toward zero.
  • domain assumption OHD covariance approximated as diagonal
    The OHD systematic terms are added in quadrature, neglecting correlations between points.

pith-pipeline@v1.3.0-alltime-deepseek · 22376 in / 21217 out tokens · 238871 ms · 2026-08-04T19:26:49.827906+00:00 · methodology

0 comments
read the original abstract

Deviations from the so-called {\it cosmic distance duality relation} may result from systematic errors in distance measurements or, more interestingly, hint at new physics. Further, it can also be related to the Hubble constant tension between early and local measurements of $H_0$. Based on this, we test validity of this relation through a model-independent parameterization of the Hubble rate via the well-estabilished B\'ezier polynomials approach. We seek for possible departures from the relation considering three parametrizations, i) a power-law correction, ii) a logarithmic correction and iii) a Pad\'e series $P_{n,m}(z)$ of order (1;2) with $n=1$ being the order of the numerator while $m=2$ is the order of the denominator. Then, assuming a flat scenario, we test them through Monte Carlo -- Markov chain analyses that combine low- and intermediate/high-$z$ data sets, such as observational Hubble data, the Pantheon catalog of type Ia supernovae, galaxy clusters, the second data release from the DESI Collaboration and gamma-ray bursts. In particular, we distinguish between \emph{Analysis A} and \emph{Analysis C}, depending whether the prompt emission $E_{iso}-E_p$ or the prompt-afterglow $L_0-E_p-T$ gamma-ray burst correlations, respectively, is fit together with the other probes previously described. Our results seem to point towards a \emph{no violation} of the cosmic distance duality relation and a preference towards Planck's value of $H_0$.

Figures

Figures reproduced from arXiv: 2509.09247 by Anna Chiara Alfano, Carlo Cafaro, Marco Muccino, Orlando Luongo, Salvatore Capozziello.

Figure 1
Figure 1. Figure 1: 1D posterior distributions for the parameters [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Same 1D posteriors and color code of Fig. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Forward citations

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