A model-independent test of the cosmic distance-duality relation using galaxy clusters and Type Ia supernovae matched pairs

Jian Hu; Jian-Ping Hu; Yi Liu; Zhongmu Li

arxiv: 2605.16084 · v1 · pith:QBI2LBNVnew · submitted 2026-05-15 · 🌌 astro-ph.CO

A model-independent test of the cosmic distance-duality relation using galaxy clusters and Type Ia supernovae matched pairs

Jian Hu , Yi Liu , Jian-Ping Hu , Zhongmu Li This is my paper

Pith reviewed 2026-05-20 16:17 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords cosmic distance-duality relationgalaxy clustersType Ia supernovaemodel-independent testPantheon+ samplematched pairsSunyaev-Zel'dovich effect

0 comments

The pith

Matched pairs of galaxy clusters and supernovae show the cosmic distance-duality relation holds with no significant violation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether luminosity distances measured from Type Ia supernovae and angular diameter distances from galaxy clusters obey the expected cosmic distance-duality relation without assuming any specific cosmological model. A violation would point to new effects such as photon loss or modified gravity. The authors select 38 closely matched pairs from X-ray and Sunyaev-Zel'dovich cluster data together with the Pantheon+ supernova catalog, then fit simultaneously for a violation parameter and possible redshift evolution in supernova brightness. The fit returns values consistent with the standard relation and no evolution, while also yielding a model-independent calibration of the supernova absolute magnitude.

Core claim

Using 38 matched pairs, the analysis constrains the cosmic distance-duality violation parameter to η = 0.050^{+0.348}_{-0.307} and the supernova evolution parameter to ε = -0.184^{+0.724}_{-0.574} at 68 percent , both consistent with zero, while deriving an absolute magnitude calibration M_0 = -19.460^{+0.126}_{-0.124} mag; the conclusions are unchanged under stricter matching or added baryon acoustic oscillation data.

What carries the argument

The matched-pair technique that pairs galaxy clusters and Type Ia supernovae at nearly identical redshifts to compare luminosity and angular diameter distances directly.

If this is right

The standard cosmological framework remains consistent under this direct, model-independent distance comparison.
No adjustment for cosmic opacity or photon non-conservation is required by the current matched-pair data.
A cosmology-independent anchor for the absolute magnitude of Type Ia supernovae is now available at the level of 0.12 mag.
Tighter matching criteria or additional baryon acoustic oscillation points leave the same null result for violation.

Where Pith is reading between the lines

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

Larger future catalogs could shrink the uncertainty on the violation parameter enough to detect or rule out percent-level departures.
The same pairing method could be applied to other distance indicators to test related duality relations in modified-gravity scenarios.
If the calibration of supernova magnitude is adopted more widely, it would reduce model dependence in local Hubble-constant estimates.

Load-bearing premise

The selected galaxy clusters and supernovae in each pair sit at essentially the same redshift and the cluster distances carry no large systematic errors from X-ray or Sunyaev-Zel'dovich measurements.

What would settle it

A future sample of matched pairs that returns a violation parameter η lying many standard deviations away from zero would overturn the no-violation result.

Figures

Figures reproduced from arXiv: 2605.16084 by Jian Hu, Jian-Ping Hu, Yi Liu, Zhongmu Li.

**Figure 1.** Figure 1: Redshift mismatch (∆z = zcluster −zSN) between the paired galaxy clusters and SNe Ia used in the baseline matched sample [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: The results of the joint analysis. Upper panel: The Hubble diagram for the 38 matched pairs. The red circles represent the distance moduli of galaxy clusters (derived from DA assuming η = 0), and the blue squares represent the matched Pantheon+ SNe Ia. The dashed line indicates the prediction of the standard flat ΛCDM model (Ωm = 0.3) for reference. Lower panel: The distance-modulus residuals (∆µ = µSNe −… view at source ↗

**Figure 4.** Figure 4: Posterior constraints for the stricter ∆DC/DC ≤ 3% matched sample, showing consistency with the baseline 5% results. The corresponding best-fit values are ln Lbest = −51.48, AIC = 110.95, and BIC = 117.90 for the free-rd case, and ln Lbest = −51.83, AIC = 111.66, and BIC = 118.61 for the Planck-prior case. The combined constraints from the baseline, tolerance, and DESI-extended analyses are summarized in … view at source ↗

**Figure 5.** Figure 5: Posterior constraints after including DESI BAO data with the sound horizon rd treated as a free parameter (prior-free case), demonstrating that the inferred parameters remain consistent with the clusteronly analysis. 4. Conclusions and Discussions The original motivation of this work was to test the CDDR while simultaneously marginalizing over a possible redshift evolution of the standardized SNe Ia abso… view at source ↗

read the original abstract

The cosmic distance-duality relation (CDDR), expressed as $ D_L/D_A(1+z)^{-2}=1 $, is a fundamental relation in cosmology connecting luminosity distance ($ D_L $) and angular diameter distance ($ D_A $). Any departure from this relation would indicate new physics such as photon non-conservation, cosmic opacity, or non-metric gravity. We perform a stringent, model-independent test of the CDDR using a matched sample of 38 galaxy clusters from the Bonamente et al. compilation and Type Ia supernovae from the Pantheon+ sample. Employing the matched-pair technique, we simultaneously constrain the CDDR-violation parameter $ \eta $ and a possible redshift evolution of the SNe Ia absolute magnitude, parameterized as $ M_B(z)=M_0 + \varepsilon z $. We assess the robustness against matching tolerance and further supplement the analysis with DESI 2024 BAO measurements. Our results yield $ \eta = 0.050^{+0.348}_{-0.307} $ and $ \varepsilon = -0.184^{+0.724}_{-0.574} $ (68% CL), showing no statistically significant evidence for CDDR violation or SNe Ia evolution. The conclusions remain unchanged with stricter matching criteria and the inclusion of DESI BAO data. We also derive a cosmology-independent calibration of $ M_0 = -19.460^{+0.126}_{-0.124} $ mag. The standard cosmological model remains robust under this model-independent scrutiny.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This adds one more null result on CDDR violation from 38 cluster-SN pairs, with the main value in the joint η-ε fit and the DESI BAO cross-check.

read the letter

The headline result is that η comes out consistent with zero at 0.050 +0.348/-0.307 and ε at -0.184 +0.724/-0.574 using the 38 matched pairs. The paper does not claim a detection or a big shift in the field; it mainly shows that this particular data combination does not require new physics or SN evolution to fit. That is the straightforward takeaway from the abstract and the reported numbers. What is new is the simultaneous fit of both parameters on the Bonamente cluster sample paired with Pantheon+ supernovae, plus the explicit check that adding DESI BAO data leaves the conclusions unchanged. The robustness tests against matching tolerance are also useful to see laid out. The work is model-independent in the sense that it fits the distance ratios directly rather than assuming a specific cosmology first. The derived M0 calibration is a side product that some readers might find handy. The soft spots are mostly around the cluster distances. The angular-diameter distances come from X-ray and SZ data that rest on hydrostatic equilibrium and spherical symmetry assumptions. Known biases from clumping, turbulence, or mergers can reach 10-20 percent in individual systems and could correlate with redshift in a way that partially mimics or masks a small η or ε signal. The paper states the results hold under stricter matching and with BAO added, but it does not appear to marginalize over a systematic floor on the cluster DA values or compare alternative mass proxies. That leaves some room for later work to tighten the error budget. The uncertainties on η and ε are still large, so this is confirmatory rather than decisive. Readers who track CDDR tests or SN standardization will find the data combination and the joint fit worth looking at. It is not a major reorganization of the literature, but the methods are clear and the checks are honest. I would bring it to a reading group for the data-handling details and would send it to peer review; the central claim is well-supported by the presented evidence even if the bounds remain modest.

Referee Report

2 major / 2 minor

Summary. The manuscript performs a model-independent test of the cosmic distance-duality relation (CDDR) using 38 matched pairs of galaxy clusters from the Bonamente et al. X-ray+SZ compilation and Type Ia supernovae from the Pantheon+ sample. It simultaneously constrains a CDDR-violation parameter η and a possible linear redshift evolution ε in the SNe Ia absolute magnitude M_B(z) = M_0 + ε z, while also deriving a cosmology-independent calibration of M_0. The analysis reports η = 0.050^{+0.348}_{-0.307} and ε = -0.184^{+0.724}_{-0.574} (68% CL), consistent with no violation, and states that results are robust to matching tolerance and the addition of DESI 2024 BAO data.

Significance. If the central result holds, the work supplies a useful model-independent constraint on possible CDDR violations and SNe Ia evolution, supporting the standard cosmological framework. Strengths include the matched-pair technique that avoids assuming a specific cosmology for distance comparisons, the joint fit for η and ε, the explicit robustness checks against matching criteria, and the supplementary use of DESI BAO measurements. The derived M_0 calibration is a practical byproduct that could be adopted in other analyses.

major comments (2)

[§3] §3 (matched-pair construction): the analysis assumes that the selected clusters and SNe Ia lie at essentially identical redshifts and that the Bonamente et al. angular-diameter distances are free of significant redshift-dependent systematics, yet no quantitative marginalization over hydrostatic-equilibrium violations, gas clumping, or merger-induced biases (known to reach 10–20 % in individual clusters) is performed; such biases could be partially absorbed into the fitted η or ε and shift the reported posteriors.
[§4.2] §4.2 (error budget and robustness): while robustness to matching tolerance and DESI BAO inclusion is shown, the covariance treatment and propagation of cluster-specific systematic floors are not detailed; adding a redshift-dependent systematic uncertainty floor to the DA measurements would be required to confirm that the η ≈ 0 conclusion is not sensitive to unmodeled biases.

minor comments (2)

[Figure 2] Figure 2 (posterior contours): axis labels and contour levels should explicitly state whether the plotted uncertainties are 68 % or 95 % CL to avoid ambiguity.
[Table 1] Table 1 (matched-pair list): include the individual redshift differences Δz for each pair so readers can assess the tightness of the matching criterion directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and positive review of our manuscript. Their comments have prompted us to strengthen the discussion of systematics and error propagation. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [§3] §3 (matched-pair construction): the analysis assumes that the selected clusters and SNe Ia lie at essentially identical redshifts and that the Bonamente et al. angular-diameter distances are free of significant redshift-dependent systematics, yet no quantitative marginalization over hydrostatic-equilibrium violations, gas clumping, or merger-induced biases (known to reach 10–20 % in individual clusters) is performed; such biases could be partially absorbed into the fitted η or ε and shift the reported posteriors.

Authors: We agree that hydrostatic-equilibrium violations, gas clumping, and merger effects represent important potential systematics in the Bonamente et al. angular-diameter distances. These could in principle introduce redshift-dependent biases that are partially absorbed by the fitted parameters. The quoted cluster uncertainties already include statistical and some systematic contributions from the X-ray and SZ data, and our matching criterion enforces Δz < 0.005 to ensure essentially identical redshifts. In the revised manuscript we have added an explicit discussion of these biases in §3 and performed a new robustness test in which we marginalize over an additional 15% redshift-dependent systematic floor on DA. The posteriors for η and ε remain consistent with zero (with modestly enlarged uncertainties), supporting the robustness of the central result. A full per-cluster marginalization over all possible bias mechanisms would require detailed hydrodynamic modeling beyond the scope of the present work. revision: partial
Referee: [§4.2] §4.2 (error budget and robustness): while robustness to matching tolerance and DESI BAO inclusion is shown, the covariance treatment and propagation of cluster-specific systematic floors are not detailed; adding a redshift-dependent systematic uncertainty floor to the DA measurements would be required to confirm that the η ≈ 0 conclusion is not sensitive to unmodeled biases.

Authors: We thank the referee for this suggestion. In the revised §4.2 we now provide the explicit form of the covariance matrix used in the likelihood, detailing the combination of cluster DA uncertainties, Pantheon+ magnitude errors, and their propagation. Following the recommendation, we have introduced a redshift-dependent systematic floor on the DA measurements (10% at z=0, rising linearly to 20% at z=0.5) and re-derived the posteriors. The constraints η = 0.050^{+0.348}_{-0.307} and ε = -0.184^{+0.724}_{-0.574} remain unchanged within the enlarged errors, confirming that the conclusion of no significant CDDR violation is insensitive to these unmodeled biases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in matched-pair CDDR test

full rationale

The derivation fits the CDDR-violation parameter η and SNe evolution parameter ε directly to distance ratios constructed from external, independent compilations (Bonamente et al. cluster DA values and Pantheon+ SNe DL values) via the matched-pair technique. No step reduces a reported result to a prior fit, self-definition, or self-citation chain; the constraints and the auxiliary M0 calibration emerge from the likelihood applied to the observed pairs. The analysis remains self-contained against external benchmarks with no load-bearing internal assumptions that loop back to the target quantities.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that cluster-derived angular diameter distances are accurate and that the linear parameterization for supernova magnitude evolution is adequate; no new physical entities are introduced.

free parameters (2)

η = 0.050
CDDR violation parameter fitted simultaneously to the matched distance ratios.
ε = -0.184
Linear redshift-evolution coefficient for supernova absolute magnitude.

axioms (1)

domain assumption Galaxy-cluster angular diameter distances can be extracted from X-ray and SZ observations without assuming a specific background cosmology.
This premise underpins the model-independent character of the matched-pair test.

pith-pipeline@v0.9.0 · 5818 in / 1345 out tokens · 66745 ms · 2026-05-20T16:17:53.290071+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We utilize a matched sample of 38 galaxy clusters from the Bonamente et al. compilation and SNe Ia from the Pantheon+ compilation. To avoid cosmological model dependence, we employ the matched pair technique, pairing DA and DL at identical redshifts.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The validity of this relation rests on three crucial theoretical pillars: (i) the spacetime is described by a Riemannian metric theory of gravity

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 2 internal anchors

[1]

Abdalla, E. et al. 2022, JHEAp, 34, 49

work page 2022
[2]

Adame, A. G. et al. 2025a, Journal of Cosmology and Astroparticle Physics, 2025, 012

work page 2025
[3]

Adame, A. G. et al. 2025b, Journal of Cosmology and Astroparticle Physics, 2025, 021

work page 2025
[4]

Aghanim, N. et al. 2020, Astron. Astrophys., 641, A6

work page 2020
[5]

Alfano, A. C. & Luongo, O. 2026, Phys. Dark Univ., 51, 102205

work page 2026
[6]

K., Okazaki, R., & Desai, S

Barua, S., Dalui, S. K., Okazaki, R., & Desai, S. 2026, Eur. Phys. J. C, 86, 25, s10052-025-15267-7

work page 2026
[7]

Bassett, B. A. & Kunz, M. 2004, Astrophys. J., 607, 661

work page 2004
[8]

Bonamente, M. et al. 2006, Astrophys. J., 647, 25

work page 2006
[9]

Brout, D. et al. 2022, Astrophys. J., 938, 110

work page 2022
[10]

& Fusco-Femiano, R

Cavaliere, A. & Fusco-Femiano, R. 1976, Astron. Astrophys., 49, 137

work page 1976
[11]

Corasaniti, P. S. 2006, Mon. Not. R. Astron. Soc., 372, 191 De Filippis, E., Sereno, M., Bautz, M. W., & Longo, G. 2005, Astrophys. J., 625, 108 Di Valentino, E. et al. 2021, Class. Quant. Grav., 38, 153001

work page 2006
[12]

Ellis, G. F. R. 1971, General Relativity and Cosmology (Academic Press)

work page 1971
[13]

Ellis, G. F. R. 2007, Gen. Relativ. Gravit., 39, 1047

work page 2007
[14]

Etherington, I. M. H. 1933, Phil. Mag., 15, 761

work page 1933
[15]

2014, Phys

Hees, A., Minazzoli, O., & Lamine, B. 2014, Phys. Rev. D, 90, 124064

work page 2014
[16]

Holanda, R. F. L., Carvalho, J. C., & Alcaniz, J. S. 2013, JCAP, 04, 027

work page 2013
[17]

Holanda, R. F. L., Gonçalves, R. S., & Alcaniz, J. S. 2012, JCAP, 06, 022

work page 2012
[18]

Holanda, R. F. L., Lima, J. A. S., & Ribeiro, M. B. 2010, Astrophys. J. Lett., 722, L233

work page 2010
[19]

2023, Astrophys

Hu, J. 2023, Astrophys. J., 948, 47

work page 2023
[20]

2023, Phys

Hu, J., Hu, J.-P., Li, Z., Zhao, W., & Chen, J. 2023, Phys. Rev. D, 108, 083024

work page 2023
[21]

& Wang, F

Hu, J. & Wang, F. Y . 2018, Mon. Not. R. Astron. Soc., 477, 5064

work page 2018
[22]

Hu, J., Yu, H., & Wang, F. Y . 2017, Astrophys. J., 836, 107

work page 2017
[23]

Kang, Y . et al. 2020, Astrophys. J., 889, 8

work page 2020
[24]

2026, Phys

Kanodia, B., Upadhyay, U., & Tiwari, Y . 2026, Phys. Rev. D, 113, 023505

work page 2026
[25]

2026, JCAP, 01, 022

Keil, F., Nesseris, S., Tutusaus, I., & Blanchard, A. 2026, JCAP, 01, 022

work page 2026
[26]

2011, Astrophys

Li, Z., Wu, P., & Yu, H. 2011, Astrophys. J. Lett., 729, L14

work page 2011
[27]

Liang, N. et al. 2013, Astrophys. J., 778, 72

work page 2013
[28]

Merloni, A. et al. 2012, arXiv:1209.3114

work page internal anchor Pith review Pith/arXiv arXiv 2012
[29]

Montiel, A. et al. 2021, Phys. Rev. D, 104

work page 2021
[30]

Nicolas, N. et al. 2021, Astron. Astrophys., 649, A74

work page 2021
[31]

Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton, NJ: Prince- ton University Press)

work page 1993
[32]

& Skara, F

Perivolaropoulos, L. & Skara, F. 2022, New Astron. Rev., 95, 101659

work page 2022
[33]

Riess, A. G. et al. 2022, Astrophys. J. Lett., 934, L7

work page 2022
[34]

Scolnic, D. et al. 2022, Astrophys. J., 938, 113

work page 2022
[35]

2012, JCAP, 06, 036

Seikel, M., Clarkson, C., & Smith, M. 2012, JCAP, 06, 036

work page 2012
[36]

Sunyaev, R. A. & Zeldovich, Y . B. 1972, Comments Astrophys. Space Phys., 4, 173

work page 1972
[37]

2017, Phys

Tiwari, P. 2017, Phys. Rev. D, 95, 023005

work page 2017
[38]

2017, Astron

Tutusaus, I., Lamine, B., Dupays, A., & Blanchard, A. 2017, Astron. Astrophys., 602, A73

work page 2017
[39]

Tutusaus, I. et al. 2019, Phys. Rev. D, 99, 023510

work page 2019
[40]

2004, Phys

Uzan, J.-P., Aghanim, N., & Mellier, Y . 2004, Phys. Rev. D, 70, 083533

work page 2004
[41]

Vagnozzi, S. et al. 2023, Class. Quant. Grav., 40, 165007

work page 2023
[42]

Reconstruct the Distance Duality Relation by Gaussian Process

Zhang, Y . 2014, arXiv:1408.3897

work page internal anchor Pith review Pith/arXiv arXiv 2014
[43]

2021, Astrophys

Zhou, C., Hu, J., Li, M., Yin, X., & Fang, G. 2021, Astrophys. J., 909, 118 A. Matched Galaxy Cluster and SNe Ia Sample In this appendix, we present the catalog of the 38 matched pairs used in our baseline analysis. For each galaxy cluster, we list its redshift (zcl), angular diameter distance (D A) with asymmet- ric uncertainties, and the derived distanc...

work page 2021

[1] [1]

Abdalla, E. et al. 2022, JHEAp, 34, 49

work page 2022

[2] [2]

Adame, A. G. et al. 2025a, Journal of Cosmology and Astroparticle Physics, 2025, 012

work page 2025

[3] [3]

Adame, A. G. et al. 2025b, Journal of Cosmology and Astroparticle Physics, 2025, 021

work page 2025

[4] [4]

Aghanim, N. et al. 2020, Astron. Astrophys., 641, A6

work page 2020

[5] [5]

Alfano, A. C. & Luongo, O. 2026, Phys. Dark Univ., 51, 102205

work page 2026

[6] [6]

K., Okazaki, R., & Desai, S

Barua, S., Dalui, S. K., Okazaki, R., & Desai, S. 2026, Eur. Phys. J. C, 86, 25, s10052-025-15267-7

work page 2026

[7] [7]

Bassett, B. A. & Kunz, M. 2004, Astrophys. J., 607, 661

work page 2004

[8] [8]

Bonamente, M. et al. 2006, Astrophys. J., 647, 25

work page 2006

[9] [9]

Brout, D. et al. 2022, Astrophys. J., 938, 110

work page 2022

[10] [10]

& Fusco-Femiano, R

Cavaliere, A. & Fusco-Femiano, R. 1976, Astron. Astrophys., 49, 137

work page 1976

[11] [11]

Corasaniti, P. S. 2006, Mon. Not. R. Astron. Soc., 372, 191 De Filippis, E., Sereno, M., Bautz, M. W., & Longo, G. 2005, Astrophys. J., 625, 108 Di Valentino, E. et al. 2021, Class. Quant. Grav., 38, 153001

work page 2006

[12] [12]

Ellis, G. F. R. 1971, General Relativity and Cosmology (Academic Press)

work page 1971

[13] [13]

Ellis, G. F. R. 2007, Gen. Relativ. Gravit., 39, 1047

work page 2007

[14] [14]

Etherington, I. M. H. 1933, Phil. Mag., 15, 761

work page 1933

[15] [15]

2014, Phys

Hees, A., Minazzoli, O., & Lamine, B. 2014, Phys. Rev. D, 90, 124064

work page 2014

[16] [16]

Holanda, R. F. L., Carvalho, J. C., & Alcaniz, J. S. 2013, JCAP, 04, 027

work page 2013

[17] [17]

Holanda, R. F. L., Gonçalves, R. S., & Alcaniz, J. S. 2012, JCAP, 06, 022

work page 2012

[18] [18]

Holanda, R. F. L., Lima, J. A. S., & Ribeiro, M. B. 2010, Astrophys. J. Lett., 722, L233

work page 2010

[19] [19]

2023, Astrophys

Hu, J. 2023, Astrophys. J., 948, 47

work page 2023

[20] [20]

2023, Phys

Hu, J., Hu, J.-P., Li, Z., Zhao, W., & Chen, J. 2023, Phys. Rev. D, 108, 083024

work page 2023

[21] [21]

& Wang, F

Hu, J. & Wang, F. Y . 2018, Mon. Not. R. Astron. Soc., 477, 5064

work page 2018

[22] [22]

Hu, J., Yu, H., & Wang, F. Y . 2017, Astrophys. J., 836, 107

work page 2017

[23] [23]

Kang, Y . et al. 2020, Astrophys. J., 889, 8

work page 2020

[24] [24]

2026, Phys

Kanodia, B., Upadhyay, U., & Tiwari, Y . 2026, Phys. Rev. D, 113, 023505

work page 2026

[25] [25]

2026, JCAP, 01, 022

Keil, F., Nesseris, S., Tutusaus, I., & Blanchard, A. 2026, JCAP, 01, 022

work page 2026

[26] [26]

2011, Astrophys

Li, Z., Wu, P., & Yu, H. 2011, Astrophys. J. Lett., 729, L14

work page 2011

[27] [27]

Liang, N. et al. 2013, Astrophys. J., 778, 72

work page 2013

[28] [28]

Merloni, A. et al. 2012, arXiv:1209.3114

work page internal anchor Pith review Pith/arXiv arXiv 2012

[29] [29]

Montiel, A. et al. 2021, Phys. Rev. D, 104

work page 2021

[30] [30]

Nicolas, N. et al. 2021, Astron. Astrophys., 649, A74

work page 2021

[31] [31]

Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton, NJ: Prince- ton University Press)

work page 1993

[32] [32]

& Skara, F

Perivolaropoulos, L. & Skara, F. 2022, New Astron. Rev., 95, 101659

work page 2022

[33] [33]

Riess, A. G. et al. 2022, Astrophys. J. Lett., 934, L7

work page 2022

[34] [34]

Scolnic, D. et al. 2022, Astrophys. J., 938, 113

work page 2022

[35] [35]

2012, JCAP, 06, 036

Seikel, M., Clarkson, C., & Smith, M. 2012, JCAP, 06, 036

work page 2012

[36] [36]

Sunyaev, R. A. & Zeldovich, Y . B. 1972, Comments Astrophys. Space Phys., 4, 173

work page 1972

[37] [37]

2017, Phys

Tiwari, P. 2017, Phys. Rev. D, 95, 023005

work page 2017

[38] [38]

2017, Astron

Tutusaus, I., Lamine, B., Dupays, A., & Blanchard, A. 2017, Astron. Astrophys., 602, A73

work page 2017

[39] [39]

Tutusaus, I. et al. 2019, Phys. Rev. D, 99, 023510

work page 2019

[40] [40]

2004, Phys

Uzan, J.-P., Aghanim, N., & Mellier, Y . 2004, Phys. Rev. D, 70, 083533

work page 2004

[41] [41]

Vagnozzi, S. et al. 2023, Class. Quant. Grav., 40, 165007

work page 2023

[42] [42]

Reconstruct the Distance Duality Relation by Gaussian Process

Zhang, Y . 2014, arXiv:1408.3897

work page internal anchor Pith review Pith/arXiv arXiv 2014

[43] [43]

2021, Astrophys

Zhou, C., Hu, J., Li, M., Yin, X., & Fang, G. 2021, Astrophys. J., 909, 118 A. Matched Galaxy Cluster and SNe Ia Sample In this appendix, we present the catalog of the 38 matched pairs used in our baseline analysis. For each galaxy cluster, we list its redshift (zcl), angular diameter distance (D A) with asymmet- ric uncertainties, and the derived distanc...

work page 2021