Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Model-Independent Probe of Cosmic Distance Duality Relation

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read This paper tests the cosmic distance duality relation without fixing the expansion history, and finds its violation parameter consistent with zero out to redshift about 2.3.

desk verdict A competent CDDR test with Pantheon+ and SGL, but the 'model-independent' framing is weakened by the joint fit; the null result is believable but not as precise as claimed. read the letter →

arxiv 2501.15086 v1 pith:USAUYZCN submitted 2025-01-25 gr-qc astro-ph.CO

Savita Gahlaut This is my paper · ORCID
classification gr-qcastro-ph.CO
keywords cosmicdistancedualityrelationEtheringtonreciprocitystronggravitationallensingPantheon+supernovaedimensionlesscomovingsumruleFLRWmetricmodel-independentcosmology
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 (CDDR), the theorem $\eta(z)=D_L(z)/[D_A(z)(1+z)^2]=1$ that any Riemannian, photon-number-conserving spacetime must satisfy, using 102 strong gravitational lens systems and the Pantheon+ Type Ia supernova sample. It does this without fixing a background cosmological model: the FLRW metric is assumed, and a dimensionless comoving distance function $d(z)$ is reconstructed from lensing via the distance sum rule, with SNe supplying luminosity distances at the same redshifts. The central result is that the violation parameter is consistent with zero: $\eta_0=-0.0051^{+0.0677}_{-0.0621}$ in flat space and $\eta_0=-0.0330^{+0.0749}_{-0.0702}$ in non-flat space for the linear parametrization, with the nonlinear parametrization also showing no significant deviation. The paper concludes that CDDR is valid out to $z\approx2.3$, a regime where many cosmological analyses silently assume it. This matters because an established model-independent confirmation protects distance measurements from hidden new-physics or systematic-error contamination.

What carries the argument

The carrying identity is the distance sum rule for FLRW null geodesics, $d_{ls}=d_s\sqrt{1+\Omega_K d_l^2}-d_l\sqrt{1+\Omega_K d_s^2}$, which converts the observed strong-lensing ratio $D(z_l,z_s)=d_{ls}/d_s$ into constraints on a continuous dimensionless comoving distance $d(z)$. The paper parametrizes $d(z)=z+a_1z^2+a_2z^3$ (the cubic polynomial ansatz), fits its coefficients together with $\eta_0$, $\Omega_K$, the SIS velocity-dispersion factor $f_E$, and the supernova absolute-magnitude nuisance parameter $M$, and then reads off whether $\eta(z)$ deviates from 1. The distance sum rule is what allows the two distance measures to be compared at the same redshift without ever choosing $H(z)$.

What would settle it

Add a quartic term $a_3 z^4$ to the distance function in equation (10), rerun the same MCMC on the same 102 lens systems and Pantheon+ sample, and compare the recovered $\eta_0$; if the best fit shifts by more than its 68 percent uncertainty, the cubic ansatz, not the data, is carrying the CDDR result.

Watch

Extended reading notes

Core claim

The paper's central claim is that the CDDR violation function $\eta(z)$ stays consistent with 1 when the dimensionless comoving distance $d(z)$ is built from strong-lensing data rather than from any cosmological model. With $d(z)=z+a_1z^2+a_2z^3$ fitted to the SGL distance ratios through the distance sum rule, and with the Pantheon+ covariance matrix used for SNe, the joint MCMC gives $\eta_0=-0.0051^{+0.0677}_{-0.0621}$ (flat, linear), $\eta_0=-0.0330^{+0.0749}_{-0.0702}$ (non-flat, linear), and correspondingly non-significant deviations for $\eta(z)=1+\eta_0 z/(1+z)$. The paper argues this establishes CDDR at high confidence in flat space and within $1\sigma$ in curved space, with $\eta_0$ only mildly dependent on $\Omega_K$, and extends the validity of the duality relation out to $z\approx2.3$.

Load-bearing premise

The load-bearing assumption is that the true dimensionless comoving distance out to $z\approx2.3$ is exactly the cubic formula $d(z)=z+a_1z^2+a_2z^3$ with only two free coefficients; if the real distance-redshift curve bends differently, the fitted $\eta_0$ and curvature parameter will absorb that shape error and the zero result would not be a genuine test of the duality relation.

Editorial extensions

If this is right

  • The flat-space bound on $\eta_0$ is tighter than earlier SGL plus SNe combinations, so future distance measurements can safely adopt CDDR as a prior out to $z\approx2.3$.
  • The non-flat fit returns a curvature parameter consistent with zero as well, showing the CDDR result is not an artifact of assuming flatness.
  • The agreement persists for both linear and nonlinear $\eta(z)$ parametrizations, so the conclusion is not tied to one functional form for the violation.
  • A weak residual correlation between $\eta_0$ and $\Omega_K$ remains, so future CDDR tests should continue to fit curvature and duality violation jointly.

Reading between the lines

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

  • An external reader would predict that replacing the cubic ansatz with a non-parametric $d(z)$ reconstruction, using the same distance sum rule and the same two catalogs, would be the decisive robustness test; a stable $\eta_0\approx0$ there would make the zero result geometric rather than functional.
  • The distance-sum-rule pipeline could be inverted to forecast how many additional SGL systems with $z_s\gtrsim1.5$ would be needed to detect a violation at the $|\eta_0|\approx0.02$ level, a sensitivity useful for photon-nonconservation and modified-propagation scenarios.
  • High-redshift quasars and gravitational-wave standard sirens are future probes the paper itself names; applying the same model-independent estimator to those data would extend the CDDR test beyond $z\approx2.3$ toward the last scattering surface.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 tests the cosmic distance duality relation (CDDR) by combining 102 strong gravitational lensing (SGL) systems from Chen et al. (2019) with the Pantheon+ type Ia supernova sample. The lensing data are used, through the distance sum rule in FLRW geometry, to build a dimensionless comoving distance function d(z) modeled as a cubic polynomial d(z)=z+a1 z^2+a2 z^3, and the supernova data are used to constrain the CDDR violation parameter η(z) for two parametrizations, in both flat and non-flat universes. The main results are η0=−0.0051+0.0677/−0.0621 (flat, linear), η0=−0.0330+0.0749/−0.0702 (non-flat, linear, as listed in Table 1), and consistent-with-zero values for the non-linear parametrization. The author concludes that CDDR holds to very high confidence out to z∼2.3.

Significance. If the claimed model-independence were fully established, this would be a valuable addition to the CDDR literature: it uses the largest current SGL and SNe samples, avoids redshift-coincidence binning by constructing a continuous d(z), and extends the test to non-flat geometries with the distance sum rule. The central null result is plausible because the posterior intervals on η0 are broad. However, the 'model-independent' and 'very high level of confidence' claims are not supported by the analysis as presented, because the polynomial coefficients are fitted jointly with η0 to the same SNe data and no truncation-error or SGL-only analysis is provided. The paper openly mentions the lens-mass-model limitation but does not address the more serious parametric degeneracy, which is the main barrier to the advertised conclusions.

major comments (3)
  1. [Section 3, Eqs. (15), (17), (18)] The SNe likelihood (Eq. 17) constrains the distance modulus through the product (1+η0 z)d(z) (Eq. 12). With d(z)=z+a1 z^2+a2 z^3, a nonzero η0 is partially degenerate with the polynomial coefficients: at leading order the SNe data constrain combinations such as a1+η0 and a2+a1 η0. Because a1 and a2 are free parameters in the same joint likelihood (Eq. 18), the separation of η0 from d(z) relies entirely on the SGL data via Eq. (15). The paper does not report SGL-only constraints on a1 and a2, nor any demonstration that 102 lensing systems alone pin these coefficients tightly enough to prevent a genuine CDDR violation from being absorbed into d(z). This is a load-bearing gap for the claim that the test is model-independent and that the result is at 'very high level of confidence'.
  2. [Section 2.2, Eq. (10)] The cubic ansatz d(z)=z+a1 z^2+a2 z^3 is an ad hoc functional form, and the paper provides no estimate of its truncation error nor any comparison with non-parametric reconstructions (e.g., Gaussian processes) or higher-order polynomials. If the true d(z) bends differently in the redshift range z∼0.1–2.3, the inferred η0 and ΩK could be biased. The authors should either justify the cubic form quantitatively or show that the conclusions are robust to the choice of parametrization; without this, the 'model-independent' claim is not fully supported.
  3. [Section 4, Table 1] There is an inconsistency between the text and Table 1 for the non-flat linear case: Section 4 states η0=0.033+0.0749−0.0702, while Table 1 lists η0=−0.0330+0.0749−0.0702 for the same row. The authors should correct the sign typo and ensure that all quoted values match the table entries.
minor comments (4)
  1. [Section 3, Eq. (15)] The theoretical distance ratio Dth in Eq. (15) depends on ΩK through the distance sum rule (Eq. 9), but the parameter list in Eq. (15) is written as (a1,a2) only; ΩK should be included explicitly in that expression for the non-flat fits.
  2. [Section 3, Sec. 3] The MCMC analysis is described only as using emcee; the priors, number of walkers, chain length, and convergence checks are not reported. These details are needed for reproducibility and to judge the robustness of the marginalized uncertainties.
  3. [Abstract and Section 2] The abstract says 'the only a priori assumption is that the Universe is described by the FLRW metric,' but the cubic polynomial form for d(z) is also an a priori assumption. The wording should distinguish between a dynamical background model (which is indeed avoided) and the kinematic parametrization of d(z).
  4. [Throughout] There are several typos and formatting issues, e.g., 'Friedmann-Lemaˆitre-Robertson-Walker' missing a closing parenthesis in the abstract and 'catlog' instead of 'catalog' in Section 3. These should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CDDR test is an explicit joint fit with independent SGL leverage, not a derivation that reduces to its own inputs.

full rationale

The paper does not claim to derive CDDR from first principles; it fits the violation parameter η0 together with the distance-function coefficients, the curvature parameter, the lens-velocity parameter fE, and the magnitude offset M. The combined likelihood (Eqs. 14-18) includes the SGL chi-square, which depends on a1 and a2 through the distance ratios, so the SNe data do not alone determine d(z) before η0 is extracted. The cubic ansatz d(z)=z+a1z^2+a2z^3 (Eq. 10) is an explicit modeling choice, and the joint re-fitting of a1,a2 is a model-dependence/correctness concern rather than circularity: a true η0≠0 is not forced to zero by construction. The only self-citation, Gahlaut (2024), concerns the standard practice of treating fE as a free parameter and is not load-bearing. The results are also compared with independent external estimates in Table 2. No equation reduces to an input by definition, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The analysis rests on FLRW geometry, the distance sum rule, the SIS lens model, a truncated polynomial for d(z), and several nuisance calibrations. The central target eta0 is fitted, not derived; the main unquantified burden is the polynomial ansatz and the SGL selection cuts.

free parameters (6)
  • a1 = -0.2942^{+0.0641}_{-0.0664} (flat, linear)
    Shape parameter of d(z) fitted from the combined SGL+SNe likelihood.
  • a2 = 0.0471^{+0.0176}_{-0.0154} (flat, linear)
    Second shape parameter of d(z), sourced from the polynomial ansatz.
  • eta0 = -0.0051^{+0.0677}_{-0.0621} (flat, linear)
    CDDR deviation parameter; the central fitted quantity of the paper.
  • f_E = 1.0033 +/- 0.0109 (flat, linear)
    Correction between SIS velocity dispersion and observed sigma0; fitted within limits from prior work.
  • M = 23.8226^{+0.0063}_{-0.0066} (flat, linear)
    Absolute magnitude nuisance, degenerate with H0.
  • OmegaK = 0.0567^{+0.0313}_{-0.0374} (non-flat, linear)
    Spatial curvature density, fitted only in the non-flat case.
assumptions (5)
  • domain assumption The Universe is described by the FLRW metric (eq. 2).
    This is the only stated a priori assumption, but it excludes non-FLRW and inhomogeneous geometries from the start.
  • standard math The distance sum rule along null geodesics (eq. 8) holds, relating d_ls to d_l and d_s.
    Derived from FLRW geometry; used to build d(z) from lensing distance ratios.
  • ad hoc to paper d(z)=z+a1 z^2+a2 z^3 is an adequate parametrization of the comoving distance out to z~2.3.
    Chosen for simplicity; no truncation error or comparison with non-parametric reconstructions is given.
  • domain assumption Strong lens systems follow the Singular Isothermal Sphere model with sigma_SIS = f_E sigma_0.
    Standard but a simplification; f_E is fitted within limits but the model cannot capture all mass profile complexity.
  • domain assumption Lens galaxies are spherically symmetric with no significant substructure or companions.
    Used to justify SIS; sample selection enforces this but residual asymmetries remain.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Model-Independent Probe of Cosmic Distance Duality Relation." pith.science (2026). https://pith.science/paper/USAUYZCN

@misc{pith2026250115086,
  author       = {Pith},
  title        = {Pith review of: Model-Independent Probe of Cosmic Distance Duality Relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/USAUYZCN}},
  note         = {Machine review of arXiv:2501.15086}
}
abstract

In this paper, cosmic distance duality relation is probed without considering any background cosmological model. The only \textit{a priori} assumption is that the Universe is described by the Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) metric The strong gravitational lensing (SGL) data is used to construct the dimensionless co-moving distance function $d(z)$ and latest type Ia supernovae (SNe Ia) Pantheon+ data is used to estimate luminosity distances at the corresponding redshifts $z$. Using the distance sum rule along null geodesics of the FLRW metric, the CDDR violation is probed in both flat and non-flat space-time by considering two parametrizations for $\eta(z)$, the function generally used to probe the possible deviations from CDDR. The results show that, CDDR is compatible with the observations at a very high level of confidence for linear parametrization in flat Universe. In non-flat Universe too, CDDR is valid within $1\sigma$ confidence interval with a mild dependence of $\eta$ on the curvature density parameter $\Omega_{K}$ . The results for non-linear parametrization also show no significant deviation from CDDR.

Figures

Figures reproduced from arXiv: 2501.15086 by the authors.

Figure 1
Figure 1. From the catalog of 161 SGL systems, only those systems which show best consistency with the SIS lens model (for which 200kms−1 ≤ σap ≤ 300kms−1 ) are chosen. Further, the lens systems with the observed distance ratio Dob > 1 (being unphysical) are excluded [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. 1-D posterior probability distributions and 2- D confidence regions of the parameters in flat spacetime with η(z) = 1 + η0z [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. 1-D posterior probability distributions and 2- D confidence regions of the parameters in flat spacetime with η(z) = 1 + η0(z/1 + z) [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: 1-D posterior probability distributions and 2- D confidence regions of the parameters in non- flat spacetime with η(z) = 1 + η0z [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: 1-D posterior probability distributions and 2- D confidence regions of the parameters in non-flat spacetime with η(z) = 1 + η0(z/1 + z) [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Growth, geometry, and early-universe split of the matter density parameter $\Omega_{\rm m}$

    astro-ph.CO 2026-07 conditional novelty 5.5 of 10

    Splitting Ω_m into geometry, growth, and early-universe regimes yields mutually compatible values, yet ΔΩ_m^{geo,early} is 2σ from zero under combined DES, Planck (scale-cut), DESI, Pantheon+, and RSD data.

  2. Probing the Cosmic Distance Duality Relation via Non-Parametric Reconstruction for High Redshifts

    astro-ph.CO 2025-09 conditional novelty 4.0 of 10

    A Gaussian process reconstruction of the cosmic distance duality parameter eta(z) from BAO, galaxy clusters, supernovae, and quasars finds consistency with eta=1 at the 2-sigma level out to z about 2.33.

Reference graph

Works this paper leans on

34 extracted references · 32 canonical work pages · cited by 2 Pith papers

  1. [1]

    P., et al., 2016, Phys

    Abbot B. P., et al., 2016, Phys. Rev. Lett., 116, 061102. 7 Abbot B. P., et al., 2016, Phys. Rev. Lett., 116, 241103. 7 Abbot B. P., et al., 2017, Phys. Rev. Lett., 118, 221101. 7 Arjona R., et al., 2021, Phys. Rev. D 103, 103513. 8 Bassett B. A. & Kunz M., 2004, Phys. Rev. D 69, 101305. 1 Brout D., et al., 2022, ApJ, 938,110. 5 Cai R. G. & Yang T., 2017,...

    work page 2016

  2. [2]

    & Bellido J

    3 Nesseris S. & Bellido J. G., 2012, JCAP 11,

    work page 2012

  3. [6]

    A., 2013, JCAO, 04 ,

    2, 5 Shafieloo A., Majumdar S., Sahni V .& Starobinsky A. A., 2013, JCAO, 04 ,

    work page 2013

  4. [10]

    2 R¨as¨anen S., Bolejko K.& Finoguenov A., 2015, Phys. Rev. Lett., 115,101301. 4 Ratnatunga K. U., Griffiths R. E. & Ostrander E. J., 1999, AJ, 117,

    work page 2015

  5. [16]

    3 Cao S., Biesiada M., Yao M., & Zhu Z.-H., 2016, MNRAS, 461,

    5 Cao S., et al., 2015, ApJ, 806,185. 3 Cao S., Biesiada M., Yao M., & Zhu Z.-H., 2016, MNRAS, 461,

    work page 2015

  6. [22]

    5 Holanda R. F. L. & Busti V . C., 2014, Phys. Rev. D 89, 103517. 2 Holanda R. F. L., Lima J. A. S. & Ribeiro M. B., 2011, Astron. & Astrophys., 528, L14. 1 Holanda R. F. L., Lima J. A. S. & Ribeiro M. B. 2012, Astron. Astrophys. 538, A131. 1 Holanda R. F. L., Carvalho J. C. & Alcaniz J. S., 2013, JCAP, 04,

    work page 2014

  7. [26]

    2 Rana A. et. al., 2017, JCAP,

    work page 2017

  8. [27]

    1 Jorgensen I., Franx M., & Kjaergaard P., 1995, MNRAS, 273,

    work page 1995

Show all 34 references
  1. [31]

    3 Schutz B

    2, 6, 7 Schneider P., Kochanek C.,& Wambsganss J., 2006, :”Gravitational Lensing: Strong, Weak and Micro”, Springer-Verlag Berlin Heidelberg, eBook-ISBN 978-3-540-30310-7. 3 Schutz B. F., 1986, Nature, 323,

    2006

  2. [32]

    2 Melia F., Wei J.-J.& Wu X.-F., 2015, AJ, 149,

    2, 6, 7 Melia F., 2018, arXive:1804.09906. 2 Melia F., Wei J.-J.& Wu X.-F., 2015, AJ, 149,

    2018 arXiv

  3. [33]

    O., Rix H

    2 Ofek E. O., Rix H. W. & Maoz D., 2003, MNRAS 343,

    2003

  4. [34]

    7 Zheng X., et al., 2020 ApJ 892,

    2020

  5. [42]

    2 Treu T., et al.,2006, ApJ, 650,

    2006

  6. [56]

    S., 1992, ApJ 397,381

    5 Kochanek C. S., 1992, ApJ 397,381. 5 Kochanek C. S., et al., 2000, ApJ, 543,

    1992

  7. [74]

    Z., Li Z

    2, 6, 7 Lyu M. Z., Li Z. X.& Xia J.-Q., 2020, ApJ, 888,

    2020

  8. [115]

    6, 7 Yang J., et al., 2023, ApJS, 269,

    2023

  9. [127]

    4 Xu B., Wang Z., Zhang K., et al., 2022, ApJ, 939,

    2022

  10. [131]

    3 Model-Independent Probe of CDDR 9 Koopmans L. V . E., et al., 2006, ApJ, 649,

    2006

  11. [191]

    1 Ellis G. F. R., et al., 2013, Phys. Rev. D 87, 103530. 1, 2 Etherington I., 1933, The London Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 15,

    2013

  12. [281]

    1 Cappellari M., Bacon R., Bureau M., et. al. , 2006, MNRAS, 366,

    2006

  13. [306]

    Gahlaut S., 2024, Pramana - J

    6 S. Gahlaut S., 2024, Pramana - J. Phys., 98,

    2024

  14. [310]

    8 Scolnic D., et al., 2022, ApJ, 938, 113

    2022

  15. [599]

    5 Koopmans L. V . E., et al., 2006, ApJ, 703, L51. 5 Lewis A., 2019, ”GetDist: a Python package for analysing Monte Carlo samples”, arXive e-prints [1910.13970]. 6 Li Z., Wu P.& Yu H., 2011, ApJL 729, L14. 2 Li Z., et al., 2013, Phys. Rev. D 87, 103013. 2 Liao K., et al., 2016...

    2006 arXiv

  16. [639]

    E., 1993, ”Principals of Physical Cosmology”, Princeton University Press

    5 Peebles P, J. E., 1993, ”Principals of Physical Cosmology”, Princeton University Press. 4 Qi J.- Z., et al., 2021, MNRAS, 503,

    1993

  17. [702]

    & Melia F., 2020, ApJ 897,

    6, 7 Wei J.- J. & Melia F., 2020, ApJ 897,

    2020

  18. [761]

    W., Lang D

    1 Foreman-Mackey D., Hogg D. W., Lang D. , Goodman J., 2013, PASP, 125,

    2013

  19. [1097]

    4, 5 Jorgensen I., Franx M., & Kjaergaard P., 1995 MNRAS, 276,

    1995

  20. [1126]

    4 Chen Y ., Li R.& Shu Y ., 2019, MNRAS, 488,

    2019

  21. [1219]

    5 Wang M., Fu X., Xu B., et al., 2024, Eur. Phys. J. C 84,

    2024

  22. [1341]

    & Scolnic D., 2017, ApJ, 836,

    4, 5 Kessler R. & Scolnic D., 2017, ApJ, 836,

    2017

  23. [2010]

    G., et al., 2022, ApJ Lett., 934, L7

    3 Riess A. G., et al., 2022, ApJ Lett., 934, L7. 5 Ruan C.-Z., Melia F. & Zhang T.-J., 2018, ApJ, 866,

    2022

  24. [2179]

    4 Rana A., et al., 2016, JCAP ,07 ,

    2016

  25. [2192]

    5 Cao S. & Zhu Z. H., 2011, Sci. China Phys. Mech. Astron. 54, 2260–2264. 1 Cao S., Biesiada M., Zheng X., & Zhu Z.-H., 2016, MNRAS, 457,

    2011

  26. [3745]

    2, 4 Collet T., Montanari F.& R¨as¨anen S., 2019, Phys. Rev. Lett., 123, 231101. 4 Corasaniti P. S., 2006, MNRAS, 372,

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.