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

Armando Bernui; Camila Franco; Felipe Avila; Fernanda Oliveira; Maria Lopes; Rafael C. Nunes; Rodrigo Holanda

REVIEW 3 major objections 4 minor 5 cited by

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

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The cosmic distance duality relation remains consistent with η = 1 within 2σ across 0.023 ≤ z ≤ 2.33 when tested with Gaussian Process reconstruction.

desk verdict The high-redshift CDDR claim is weakened by a LambdaCDM-based quasar cut, but the paper is a competent GP consistency check worth engaging with. read the letter →

arxiv 2509.07848 v1 pith:3DAGV4PO submitted 2025-09-09 astro-ph.CO

Felipe Avila , Fernanda Oliveira , Camila Franco , Maria Lopes , Rodrigo Holanda , Rafael C. Nunes , Armando Bernui This is my paper · ORCID

classification astro-ph.CO

keywords cosmicdistancedualityrelationGaussianprocessregressionnon-parametricreconstructionangulardiameterluminosityquasarstypeIasupernovaebaryonacousticoscillations

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 whether the cosmic distance duality relation—the identity $D_L(z) = (1+z)^2 D_A(z)$ linking luminosity distance to angular diameter distance—continues to hold at redshifts beyond the supernova range. It combines angular diameter distances from transverse baryon acoustic oscillations and galaxy clusters with luminosity distances from Type Ia supernovae and quasars, then reconstructs the ratio $\eta(z) = D_L/[(1+z)^2 D_A]$ with Gaussian Process regression. The central result is that $\eta(z)$ stays within $2\sigma$ of unity across $0.023 \le z \le 2.33$ in every dataset pairing tested. If correct, this supports the three assumptions behind the relation—metric gravity, null geodesics, and photon-number conservation—at high redshifts where quasars become the only standard candles.

What carries the argument

The central object is the duality parameter $\eta(z) = D_L(z)/[(1+z)^2 D_A(z)]$, with $\eta = 1$ marking exact CDDR validity. The argument is carried by a Gaussian Process regression with a squared-exponential kernel, which reconstructs $D_A(z)$ and $D_L(z)$ from sparse, unevenly sampled measurements and propagates $1\sigma$ and $2\sigma$ bands into $\eta(z)$. For galaxy clusters, the paper uses the relation $D_A^{\rm cluster}(z) = \eta^2(z) D_A(z)$ to rewrite the test as $\eta(z) = D_A^{\rm cluster}(z)(1+z)^2 / D_L(z)$, avoiding circular use of the relation itself. Quasar luminosity distances are first filtered against the $1\sigma$ dispersion of supernova residuals around flat $\Lambda$CDM and then binned into 25 points ($\Delta z = 0.1$) to suppress their large intrinsic scatter.

What would settle it

Re-run the $\eta(z)$ reconstruction without the $\Lambda$CDM-based quasar filter, or with a model-independent robust cut, and compare the binned quasar-only $\eta$ values at $z > 1$ with the paper's filtered result; a mean deviation above $2\sigma$ in the unfiltered sample would contradict the claim that the CDDR holds at high redshifts.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that no statistically significant deviation from the cosmic distance duality relation appears when non-parametric Gaussian Process reconstruction is applied to combined BAO/galaxy-cluster angular diameter distances and supernova/quasar luminosity distances. Three reconstructions are performed: a low-redshift pairing of galaxy clusters with supernovae plus quasars, a high-redshift pairing of BAO with supernovae plus quasars, and a pure angular-distance cross-check of clusters against BAO. In all three, $\eta(z) = D_L(z)/[(1+z)^2 D_A(z)]$ is consistent with 1 at the $2\sigma$ level over the full interval $0.023 \le z \le 2.33$. The paper therefore concludes that the CDDR remains valid under current observations even at high redshifts, while stressing that the quasar selection step explicitly assumes flat $\Lambda$CDM.

Load-bearing premise

The analysis assumes flat $\Lambda$CDM is correct when it rejects quasars whose distance-modulus residuals fall outside the $1\sigma$ supernova band; if the CDDR is actually violated at high redshift, that filter removes the very objects carrying the signal and biases $\eta(z)$ toward 1.

Editorial extensions

If this is right

Exotic photon-number-violating physics, such as axion-like photon mixing, is not needed to explain the distance measurements out to $z \approx 2.33$.
Filtered and binned quasar distances can serve as high-redshift standard candles, extending cosmography beyond the Type Ia supernova horizon.
BAO-derived angular diameter distances provide the tightest high-redshift constraint, while galaxy clusters act as a low-redshift consistency check.
Improved quasar luminosity-distance precision would sharpen the same test from a $2\sigma$ consistency into a few-percent constraint on $\eta(z)$.

Reading between the lines

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

The $\Lambda$CDM-based quasar filter may bias the test toward $\eta = 1$: if the CDDR is violated at high redshift, the excluded quasars would preferentially carry the deviation. Repeating the analysis with a model-independent outlier cut would show whether the consistency survives.
Gaussian Process smoothing has a characteristic correlation length; a narrow or oscillatory deviation in $\eta(z)$ shorter than that length would be smeared into the $2\sigma$ band. A redshift-bin-by-bin $\chi^2$ test against unity would probe that regime.
Treating the quasar filter threshold as a free parameter and marginalizing over it would convert the present conditional test into a genuinely model-independent statement.
If future high-redshift samples reduce quasar distance errors below the current intrinsic scatter, the same pipeline could detect a redshift-dependent $\eta(z)$ at the few-percent level.

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.

Desk Editor's Note

read the letter

You should know upfront: the abstract overstates the high-redshift result. The quasar sample is filtered by requiring each quasar's distance modulus to lie within the 1-sigma scatter of SNe around flat LambdaCDM. Since flat LambdaCDM satisfies the CDDR by construction, the high-z eta(z) reconstruction is largely built from data that already agrees with the relation being tested. That is a real circularity, and the paper admits it in the conclusions, though the abstract does not. The angular-only eta_d combination (GC vs BAO) avoids that problem, but it does not test the distance duality in any meaningful sense; it is just a cross-check of two angular-diameter distance catalogs.

What is actually new: combining Pantheon+ SNe, Lusso et al. quasars, DESI DR2 transverse BAO, and De Filippis clusters into one GP reconstruction, with a binning pipeline, is a new data combination. The GP machinery is standard and competently executed. The low-redshift eta_l (clusters vs SNe+QSO over 0.023

The soft spots are proportionate to the claim. The central high-z null result is not an independent validation of the CDDR; removing the filter would be the obvious robustness check, and varying the 1-sigma threshold would quantify how much the conclusion is driven by selection. The Lusso QSO distance moduli are also not stated to be model-independent; the usual Risaliti-Lusso calibration fits the X-ray/UV relation against an assumed cosmology, so the D_L values likely inherit that dependence. A short paragraph addressing this would be enough. Finally, the data availability line says "on request," which is weak for a paper whose pipeline is otherwise reproducible in principle.

Bottom line: this is a solid consistency check, not a decisive high-z test. A serious referee should see it; with revisions, including an unfiltered quasar analysis, a threshold variation, and a clearer statement of what is actually being tested, it could be a useful paper. I would not cite it as evidence for the CDDR at high z in its current form, but I would bring it to reading group and I would referee it myself.

Referee Report

3 major / 4 minor

Summary. The paper tests the cosmic distance duality relation (CDDR), η(z) = D_L(z)/[(1+z)^2 D_A(z)], by combining angular diameter distances from transverse BAO and galaxy clusters with luminosity distances from Pantheon+ SNe Ia and a quasar catalog. Quasar distances are filtered by retaining only objects whose residuals relative to flat ΛCDM lie within the 1σ dispersion of the SNIa residuals, then binned; Gaussian Process regression is used to reconstruct η(z). Three combinations are considered: GC+SNIa+QSO at low z, BAO+SNIa+QSO at high z, and an angular-only GC+BAO combination. All reconstructions are reported to be consistent with η=1 at 2σ, and the authors conclude that the CDDR is supported up to z≈2.33.

Significance. If the high-redshift test were independent, extending CDDR verification to z≈2.3 with quasars would be a valuable complement to SNIa-only analyses. The paper uses standard Gaussian Process machinery, provides a clear data summary in Table I, and explicitly acknowledges in the Conclusions that the quasar filter assumes ΛCDM. These are strengths. However, the central high-redshift claim is weakened by the model-dependent pre-selection of the quasar sample and by the unstated cosmology dependence of the quasar distance moduli. The paper is therefore more a consistency check within ΛCDM than an independent test of the CDDR, and the abstract overstates the evidential weight of the result.

major comments (3)

[III A and Fig. 2] The quasar filtering procedure selects only quasars whose distance-modulus residuals lie within the 1σ dispersion of SNIa residuals around flat ΛCDM. Because flat ΛCDM satisfies the CDDR by construction, this filter preferentially removes quasars that would carry a CDDR-violation signal, and the surviving high-z D_L sample is preselected to agree with η=1. Consequently, the high-redshift η_h reconstruction in Fig. 4 cannot be presented as an independent confirmation of the CDDR; it is partly a consequence of the selection. The Conclusions admit the model dependence, but the Abstract's claim that the results 'support its validity even at high redshifts' is not justified by the analysis as presented. A concrete remedy would be to repeat the analysis without the ΛCDM-based filter (e.g., using robust binning or a heavier-tailed likelihood) and to show the redshift-dependent retention fraction, or to inject a simulated η≠1 and demonstrate that the pipeline can recover it.
[II D] The manuscript does not state whether the quasar distance moduli taken from the Lusso et al. catalog are cosmology-independent. The standard Risaliti-Lusso method calibrates the L_X–L_UV relation by fitting the Hubble diagram under an assumed cosmological model, so the quoted D_L values generally inherit that model dependence. If the catalog distances are already ΛCDM-calibrated, the subsequent 1σ ΛCDM filter compounds the circularity. The authors should either demonstrate that the adopted D_L values are insensitive to the calibration cosmology or explicitly restrict their conclusions to 'consistency with ΛCDM-based quasar distances.'
[IV, Fig. 4 (η_d)] The angular-only combination η_d, constructed from GC and BAO D_A measurements alone, does not involve D_L and therefore cannot test the CDDR as defined in Eq. (15). It tests only the mutual consistency of two angular-diameter-distance datasets. The text and figure caption should state this limitation explicitly; otherwise the reader may infer that η_d provides evidence about the duality relation, which it does not.

minor comments (4)

[III A, Eq. (11)] The binning uncertainty in Eq. (11) is the standard deviation of the binned values, not the standard error of the mean, and individual measurement errors are not propagated into the binned points. The authors should justify this choice or use inverse-variance weighting, since it directly affects the width of the GP confidence intervals.
[Fig. 2 caption] The caption says the red dashed lines represent the '1σ dispersion of the SNIa distribution,' but the plotted lines appear to be constant offsets in μ. Please clarify whether the dispersion is computed globally or as a function of redshift, and state the numerical value used.
[II D and Table I] Table I lists 2195 quasars, while the filtered sample contains 1160 quasars. Adding the filtered count and the retention fraction to the table or the text would improve transparency.
[Throughout] There are minor typographical and grammatical issues, including 'FlAT-ΛCDM' and 'e Ωk' in Section IV, and the phrase 'Enabling access to the robustness of our analyses regarding dependence on priors' in the Conclusions is unclear and should be rewritten.

Circularity Check

1 steps flagged · score 6.0 of 10

Quasar selection is based on LambdaCDM residuals, so the high-redshift eta(z)~1 result is partly built into the input sample; the paper itself concedes the filter assumes LambdaCDM.

self definitional [Section III A / Fig. 2 upper-panel caption; Section V Conclusions]
"Upper panel: Residuals of the luminosity distance measurements relative to the LambdaCDM predictions for quasars (gray squares) and SNIa (blue dots). The red dashed lines represent the 1-sigma dispersion of the SNIa distribution. Only quasars falling within this 1-sigma region were retained for further analysis."

The high-redshift D_L sample is defined by requiring quasars to lie within the 1-sigma SNIa dispersion around flat LambdaCDM, a model that satisfies the CDDR with eta=1 by construction. Any quasar whose D_L would generate a genuine high-z CDDR violation is preferentially removed before the GP reconstruction of eta(z). The abstract's statement that the results support the CDDR 'even at high redshifts' therefore recovers, on the luminosity-distance side, the model used to select the sample. The paper itself acknowledges this in Sec.

full rationale

The main circular step is the quasar filtering. The paper keeps only QSOs whose distance-modulus residuals fall within the 1-sigma SNIa dispersion around flat LambdaCDM, and the same filtered sample supplies the high-redshift D_L used to reconstruct eta(z). Since flat LambdaCDM obeys the CDDR, this filter preferentially removes the high-z objects that would carry a CDDR-violation signal on the luminosity-distance side, so the reconstructed eta close to 1 at z>~1 is not an independent prediction. The Conclusions explicitly state: 'We emphasize that the quasar filtering process explicitly assumes the LambdaCDM model.' The BAO angular-diameter-distance side and the low-z SNIa/GC combination are independent of this filter, so the circularity is partial rather than total; a true deviation in D_A from BAO would still appear in eta(z). Nevertheless, the abstract's claim that the results 'support' the CDDR 'even at high redshifts' rests on the filtered D_L sample and overstates the evidential weight of the high-z QSO data. I find no load-bearing self-citation or imported-uniqueness issue: the cited prior work is used for standard GP methodology and robustness checks, not to forbid alternative reconstructions. The BAO scale r_d=147.05 Mpc does come from Planck-LambdaCDM, but the paper argues the dependence is weak and this is a model-dependence caveat rather than a circular step. The score reflects one central, explicitly acknowledged partial circularity, not a fully forced derivation.

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

The analysis rests on several externally supplied inputs: LambdaCDM parameters for the quasar filter, the fiducial sound horizon for BAO, and the SZE/X-ray cluster modeling assumption. The quasar filter is the most consequential because it can impose the null result at high redshift. No new physical entities are introduced.

free parameters (4)

GP kernel hyperparameters (sigma_f, l) = optimized during reconstruction
The squared exponential kernel hyperparameters are optimized by the GaPP code and control the smoothness of the reconstructed distance functions and hence of eta(z).
QSO retention threshold = 1 sigma dispersion of SNIa residuals
Quasars lying outside the 1 sigma SNIa dispersion band around LambdaCDM are discarded. This threshold directly determines the high-redshift D_L sample and the resulting eta(z).
Binning width Delta z = 0.1
The bin width was chosen after successive tests; the paper states it cannot be too large or too small, and the choice affects the binned D_L inputs to the Gaussian process.
Fiducial sound horizon r_d = 147.05 Mpc
BAO measurements of D_A/r_d are converted to D_A in Mpc using this Planck-based value. The paper argues that a 5 percent variation has no significant impact, so the parameter enters with modest weight.

assumptions (4)

ad hoc to paper The flat LambdaCDM model with standard parameters is the correct fiducial for computing SNIa and QSO residuals used in the quasar filter.
Section III A and Figure 2: quasars are retained only if they lie within 1 sigma of the SNIa residuals around LambdaCDM predictions. This assumes the model under test is correct before the test is performed.
domain assumption The SZE plus X-ray method yields D_A^cluster = eta^2 D_A, so Equation (7) directly gives eta from cluster and luminosity distances.
Section II B, Equations (5) and (6). This depends on isothermal elliptical beta-model assumptions for the intracluster gas; systematic errors are added in quadrature.
domain assumption Transverse BAO measurements provide D_A/r_d with weak cosmological dependence.
Section II A, Equation (1). The conversion to physical D_A uses the fiducial sound horizon r_d.
domain assumption The squared exponential kernel is flexible enough for the D_A and D_L GP reconstructions, and other kernels give compatible results.
Section III B. The reconstruction depends on the chosen kernel; the paper argues kernel variation is equivalent for this data set.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Probing the Cosmic Distance Duality Relation via Non-Parametric Reconstruction for High Redshifts." pith.science (2026). https://pith.science/paper/3DAGV4PO

@misc{pith2026250907848,
  author       = {Pith},
  title        = {Pith review of: Probing the Cosmic Distance Duality Relation via Non-Parametric Reconstruction for High Redshifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DAGV4PO}},
  note         = {Machine review of arXiv:2509.07848}
}

abstract

We test the validity of the cosmic distance duality relation (CDDR) by combining angular diameter distance and luminosity distance measurements from recent cosmological observations. For the angular diameter distance, we use data from transverse baryon acoustic oscillations and galaxy clusters. On the other hand, the luminosity distance is obtained from Type Ia supernovae in the Pantheon+ sample and from quasar catalogs. To reduce the large dispersion in quasar luminosity distances, we apply a selection criterion based on their deviation from the $\Lambda$CDM model and implement a binning procedure to suppress statistical noise. We reconstruct the CDDR using Gaussian Processes, a non-parametric supervised machine learning method. Our results show no significant deviation from the CDDR within the $2\sigma$ confidence level across the redshift range explored, supporting its validity even at high redshifts.

Figures

Figures reproduced from arXiv: 2509.07848 by the authors.

**Figure 2.** FIG. 2. Upper panel: Residuals of the luminosity distance [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Reconstruction of the luminosity distance, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. These plots show the statistical GP reconstruction of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

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

A First Observational Assessment of Cosmic Backreaction Over an Extended Redshift Range
astro-ph.CO 2026-04 unverdicted novelty 7.0 of 10

First model-light constraints on Ω_R + 3Ω_Q from Pantheon+ and BAO reconstructions are consistent with flat FLRW yet too broad to exclude percent-level backreaction.
Hubble tension: the shape wall
astro-ph.CO 2026-07 accept novelty 6.0 of 10

Late-time modifications to the expansion history can raise H0 by at most about 2% (conservative) to 3.7% (permissive) if the CMB acoustic scale is fixed.
Neutrino mass constraints in the Schwarzschild-de Sitter black-hole dark energy model with ACT DR6 and DESI DR2 data
astro-ph.CO 2026-07 conditional novelty 4.5 of 10

SdSDE dark energy yields a ~0.16–0.21 eV positive neutrino-mass preference with DESI+ACT data, but χ² strongly favors extended ΛCDM and the shift is likely compensation.
Alleviating the $H_0$ tension through the interacting dark energy model from quantum gravitational field theory in light of DESI DR2
astro-ph.CO 2025-10 conditional novelty 4.0 of 10

With DESI DR2 BAO plus CMB and a SH0ES prior, the two-parameter eeΛCDM model gives δΛ=-0.41±0.14 and H0=71.9±1.0, easing the Hubble tension to 0.8σ, but SN datasets erase the signal.
Crosschecking Cosmic Distances from DESI BAO and DES SNe
astro-ph.CO 2025-10 conditional novelty 4.0 of 10

For DESI DR2 BAO versus DES SNe, the comoving-distance ratio is flat but the inverse-Hubble-distance ratio falls with redshift at 2.3–2.5σ, which the authors attribute to systematics.

Reference graph

Works this paper leans on

118 extracted references · 13 canonical work pages · cited by 5 Pith papers

[1]

Di Valentinoet al., (2025), arXiv:2504.01669 [astro- ph.CO]

E. Di Valentinoet al., (2025), arXiv:2504.01669 [astro- ph.CO]

arXiv 2025
[2]

R. R. Caldwell and M. Kamionkowski, Annual Re- view of Nuclear and Particle Science59, 397 (2009), 9 arXiv:0903.0866 [astro-ph.CO]

arXiv 2009
[3]

D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess, and E. Rozo, Phys. Rep.530, 87 (2013), arXiv:1201.2434 [astro-ph.CO]

arXiv 2013
[4]

Perivolaropoulos and F

L. Perivolaropoulos and F. Skara, nar95, 101659 (2022), arXiv:2105.05208 [astro-ph.CO]

arXiv 2022
[5]

Vagnozzi, Universe9, 393 (2023), arXiv:2308.16628 [astro-ph.CO]

S. Vagnozzi, Universe9, 393 (2023), arXiv:2308.16628 [astro-ph.CO]

arXiv 2023
[6]

I. M. H. Etherington, Philosophical Magazine15, 761 (1933)

1933
[7]

G. F. R. Ellis, General Relativity and Gravitation39, 1047 (2007)

2007
[8]

J. Qin, F. Melia, and T.-J. Zhang, MNRAS502, 3500 (2021), arXiv:2101.05574 [astro-ph.CO]

arXiv 2021

Show all 118 references

[9]

B. A. Bassett and M. Kunz, PRD69, 101305 (2004), arXiv:astro-ph/0312443 [astro-ph]

2004 arXiv
[10]

A. Hees, O. Minazzoli, and J. Larena, PRD90, 124064 (2014), arXiv:1406.6187 [astro-ph.CO]

2014 arXiv
[11]

R. P. L. Azevedo and P. P. Avelino, PRD104, 084079 (2021), arXiv:2104.01209 [gr-qc]

2021 arXiv
[12]

Levi Said, J

J. Levi Said, J. Mifsud, D. Parkinson, E. N. Saridakis, J. Sultana, and K. Zarb Adami, JCAP2020, 047 (2020), arXiv:2005.05368 [astro-ph.CO]

2020 arXiv
[13]

R. F. L. Holanda and K. N. N. O. Barros, PRD94, 023524 (2016), arXiv:1606.07923 [astro-ph.CO]

2016 arXiv
[14]

R. F. L. Holanda, S. H. Pereira, V. C. Busti, and C. H. G. Bessa, Classical and Quantum Gravity34, 195003 (2017), arXiv:1705.05439 [gr-qc]

2017 arXiv
[15]

Kumar, A

D. Kumar, A. Rana, D. Jain, S. Mahajan, A. Mukher- jee, and R. F. L. Holanda, JCAP2022, 053 (2022), arXiv:2107.04784 [astro-ph.CO]

2022 arXiv
[16]

R. F. L. Holanda, S. H. Pereira, and D. Jain, PRD97, 023538 (2018), arXiv:1801.04344 [astro-ph.CO]

2018 arXiv
[17]

Combes, nar48, 583 (2004), arXiv:astro-ph/0308144 [astro-ph]

F. Combes, nar48, 583 (2004), arXiv:astro-ph/0308144 [astro-ph]

2004 arXiv
[18]

Shafieloo, S

A. Shafieloo, S. Majumdar, V. Sahni, and A. A. Starobinsky, JCAP2013, 042 (2013), arXiv:1212.1277 [astro-ph.CO]

2013 arXiv
[19]

J. F. Jesus, R. F. L. Holanda, and M. A. Dantas, General Relativity and Gravitation49, 150 (2017), arXiv:1605.01342 [astro-ph.CO]

2017 arXiv
[20]

Z. Li, P. Wu, H. Yu, and Z.-H. Zhu, PRD87, 103013 (2013), arXiv:1304.7317 [astro-ph.CO]

2013 arXiv
[21]

K. Liao, Z. Li, J. Ming, and Z.-H. Zhu, Physics Letters B718, 1166 (2013), arXiv:1212.6612 [astro-ph.CO]

2013 arXiv
[22]

Rasanen, J

S. Rasanen, J. Valiviita, and V. Kosonen, JCAP04, 050 (2016), arXiv:1512.05346 [astro-ph.CO]

2016 arXiv
[23]

Renzi, N

F. Renzi, N. B. Hogg, and W. Giar` e, Mon. Not. Roy. Astron. Soc.513, 4004 (2022), arXiv:2112.05701 [astro- ph.CO]

2022 arXiv
[24]

Qi, Y.-F

J.-Z. Qi, Y.-F. Jiang, W.-T. Hou, and X. Zhang, Astro- phys. J.979, 2 (2025), arXiv:2407.07336 [astro-ph.CO]

2025 arXiv
[25]

F. Keil, S. Nesseris, I. Tutusaus, and A. Blanchard, (2025), arXiv:2504.01750 [astro-ph.CO]

2025
[26]

A. Rana, D. Jain, S. Mahajan, and A. Mukherjee, JCAP07, 026 (2016), arXiv:1511.09223 [astro-ph.CO]

2016 arXiv
[27]

Gahlaut, Res

S. Gahlaut, Res. Astron. Astrophys.25, 025019 (2025), arXiv:2501.15086 [gr-qc]

2025 arXiv
[28]

Mukherjee and A

P. Mukherjee and A. Mukherjee, Mon. Not. Roy. Astron. Soc.504, 3938 (2021), arXiv:2104.06066 [astro-ph.CO]

2021
[29]

S. More, H. Niikura, J. Schneider, F. P. Schuller, and M. C. Werner, (2016), arXiv:1612.08784 [astro-ph.CO]

2016 arXiv
[30]

R. F. L. Holanda, J. A. S. Lima, and M. B. Ribeiro, ApJL722, L233 (2010), arXiv:1005.4458 [astro-ph.CO]

2010 arXiv
[31]

Avgoustidis, C

A. Avgoustidis, C. Burrage, J. Redondo, L. Verde, and R. Jimenez, JCAP2010, 024 (2010), arXiv:1004.2053 [astro-ph.CO]

2010 arXiv
[32]

R. F. L. Holanda, V. C. Busti, and J. S. Alcaniz, JCAP 2016, 054 (2016), arXiv:1512.02486 [astro-ph.CO]

2016 arXiv
[33]

R. F. L. Holanda, J. A. S. Lima, and M. B. Ribeiro, aap528, L14 (2011), arXiv:1003.5906 [astro-ph.CO]

2011 arXiv
[34]

R. F. L. Holanda, Astroparticle Physics84, 78 (2016), arXiv:1605.09415 [astro-ph.CO]

2016 arXiv
[35]

Z. Li, P. Wu, and H. Yu, ApJL729, L14 (2011), arXiv:1101.5255 [astro-ph.CO]

2011 arXiv
[36]

R. F. L. Holanda, R. S. Gon¸ calves, and J. S. Al- caniz, JCAP2012, 022 (2012), arXiv:1201.2378 [astro- ph.CO]

2012 arXiv
[37]

G. F. R. Ellis, R. Poltis, J.-P. Uzan, and A. Weltman, PRD87, 103530 (2013), arXiv:1301.1312 [astro-ph.CO]

2013 arXiv
[38]

P. Wu, Z. Li, X. Liu, and H. Yu, PRD92, 023520 (2015)

2015
[39]

T. Yang, R. F. L. Holanda, and B. Hu, Astroparticle Physics108, 57 (2019)

2019
[40]

B. Xu, K. Zhang, and Q. Huang, European Physical Journal C80, 838 (2020)

2020
[41]

W. J. C. da Silva, R. F. L. Holanda, and R. Silva, PRD 102, 063513 (2020), arXiv:2005.04131 [astro-ph.CO]

2020 arXiv
[42]

Santos-da-Costa, V

S. Santos-da-Costa, V. C. Busti, and R. F. L. Holanda, JCAP2015, 061 (2015), arXiv:1506.00145 [astro-ph.CO]

2015 arXiv
[43]

T. Liu, S. Cao, S. Zhang, X. Gong, W. Guo, and C. Zheng, European Physical Journal C81, 903 (2021), arXiv:2110.00927 [astro-ph.CO]

2021 arXiv
[44]

E. M. Teixeira, W. Giar` e, N. B. Hogg, T. Montandon, A. Poudou, and V. Poulin, (2025), arXiv:2504.10464 [astro-ph.CO]

2025 arXiv
[45]

R. F. L. Holanda, International Journal of Mod- ern Physics D21, 1250008-1-1250008-10 (2012), arXiv:1202.2309 [astro-ph.CO]

2012 arXiv
[46]

Afroz and S

S. Afroz and S. Mukherjee, arXiv e-prints , arXiv:2504.16868 (2025), arXiv:2504.16868 [astro- ph.CO]

2025
[47]

de Carvalho, A

E. de Carvalho, A. Bernui, G. C. Carvalho, C. P. Novaes, and H. S. Xavier, JCAP4, 064 (2018), arXiv:1709.00113 [astro-ph.CO]

2018 arXiv
[48]

DESI Collaboration, arXiv e-prints , arXiv:2503.14738 (2025), arXiv:2503.14738 [astro-ph.CO]

2025 arXiv
[49]

S´ anchez, A

E. S´ anchez, A. Carnero, J. Garc´ ıa-Bellido, E. Gazta˜ naga, F. de Simoni, M. Crocce, A. Cabr´ e, P. Fosalba, and D. Alonso, MNRAS411, 277 (2011), arXiv:1006.3226 [astro-ph.CO]

2011 arXiv
[50]

Menote and V

R. Menote and V. Marra, MNRAS513, 1600 (2022), arXiv:2112.10000 [astro-ph.CO]

2022 arXiv
[51]

de Carvalho, A

E. de Carvalho, A. Bernui, F. Avila, C. P. Novaes, and J. P. Nogueira-Cavalcante, aap649, A20 (2021), arXiv:2103.14121 [astro-ph.CO]

2021 arXiv
[52]

Planck Collaboration, aap641, A6 (2020), arXiv:1807.06209 [astro-ph.CO]

2020 arXiv
[53]

Bonilla, S

A. Bonilla, S. Kumar, and R. C. Nunes, Eur. Phys. J. C81, 127 (2021), arXiv:2011.07140 [astro-ph.CO]

2021 arXiv
[54]

R. A. Sunyaev and Y. B. Zeldovich, Comments on As- trophysics and Space Physics4, 173 (1972)

1972
[55]

Birkinshaw, Phys

M. Birkinshaw, Phys. Rep.310, 97 (1999), arXiv:astro- ph/9808050 [astro-ph]

1999
[56]

C. L. Sarazin, Reviews of Modern Physics58, 1 (1986)

1986
[57]

E. D. Reese, J. E. Carlstrom, M. Joy, J. J. Mohr, L. Grego, and W. L. Holzapfel, apj581, 53 (2002), 10 arXiv:astro-ph/0205350 [astro-ph]

2002 arXiv
[58]

J. E. Carlstrom, G. P. Holder, and E. D. Reese, araa 40, 643 (2002), arXiv:astro-ph/0208192 [astro-ph]

2002 arXiv
[59]

J.-P. Uzan, N. Aghanim, and Y. Mellier, PRD70, 083533 (2004), arXiv:astro-ph/0405620 [astro-ph]

2004 arXiv
[60]

R. F. L. Holanda, J. S. Alcaniz, and J. C. Car- valho, JCAP2013, 033 (2013), arXiv:1303.3307 [astro- ph.CO]

2013 arXiv
[61]

R. F. L. Holanda, J. V. Cunha, L. Marassi, and J. A. S. Lima, JCAP2012, 035 (2012), arXiv:1006.4200 [astro- ph.CO]

2012 arXiv
[62]

L. R. Cola¸ co, M. S. Ferreira, R. F. L. Holanda, J. E. Gonzalez, and R. C. Nunes, arXiv e-prints , arXiv:2310.18711 (2023), arXiv:2310.18711 [astro- ph.CO]

2023 arXiv
[63]

De Filippis, M

E. De Filippis, M. Sereno, M. W. Bautz, and G. Longo, apj625, 108 (2005), arXiv:astro-ph/0502153 [astro-ph]

2005 arXiv
[64]

Bonamente, M

M. Bonamente, M. K. Joy, S. J. LaRoque, J. E. Carl- strom, E. D. Reese, and K. S. Dawson, apj647, 25 (2006), arXiv:astro-ph/0512349 [astro-ph]

2006 arXiv
[65]

R. F. L. Holanda, J. A. S. Lima, and M. B. Ribeiro, aap538, A131 (2012), arXiv:1104.3753 [astro-ph.CO]

2012 arXiv
[66]

Broutet al., Astrophys

D. Broutet al., Astrophys. J.938, 110 (2022), arXiv:2202.04077 [astro-ph.CO]

2022 arXiv
[67]

Risaliti and E

G. Risaliti and E. Lusso, apj815, 33 (2015), arXiv:1505.07118 [astro-ph.CO]

2015 arXiv
[68]

Risaliti and E

G. Risaliti and E. Lusso, Nature Astronomy3, 272 (2019), arXiv:1811.02590 [astro-ph.CO]

2019 arXiv
[69]

Lusso, G

E. Lusso, G. Risaliti, E. Nardini, G. Bargiacchi, M. Benetti, S. Bisogni, S. Capozziello, F. Civano, L. Eggleston, M. Elvis, G. Fabbiano, R. Gilli, A. Mar- coni, M. Paolillo, E. Piedipalumbo, F. Salvestrini, M. Signorini, and C. Vignali, aap642, A150 (2020), arXiv:2008.08586 [...

2020 arXiv
[70]

Raffai, A

P. Raffai, A. Pataki, R. L. B¨ ottger, A. Karsai, and G. D´ alya, apj979, 51 (2025), arXiv:2412.15717 [astro- ph.CO]

2025 arXiv
[71]

X. Li, R. E. Keeley, and A. Shafieloo, apj983, 141 (2025), arXiv:2408.15547 [astro-ph.CO]

2025 arXiv
[72]

Seikel, C

M. Seikel, C. Clarkson, and M. Smith, JCAP2012, 036 (2012), arXiv:1204.2832 [astro-ph.CO]

2012 arXiv
[73]

Seikel and C

M. Seikel and C. Clarkson, arXiv e-prints , arXiv:1311.6678 (2013), arXiv:1311.6678 [astro-ph.CO]

2013 arXiv
[74]

Yang, Z.-K

T. Yang, Z.-K. Guo, and R.-G. Cai, PRD91, 123533 (2015), arXiv:1505.04443 [astro-ph.CO]

2015 arXiv
[75]

G´ omez-Valent and L

A. G´ omez-Valent and L. Amendola, JCAP2018, 051 (2018), arXiv:1802.01505 [astro-ph.CO]

2018 arXiv
[76]

Zhang and H

M.-J. Zhang and H. Li, European Physical Journal C 78, 460 (2018), arXiv:1806.02981 [astro-ph.CO]

2018 arXiv
[77]

Perenon, M

L. Perenon, M. Martinelli, R. Maartens, S. Camera, and C. Clarkson, Phys. Dark Univ.37, 101119 (2022), arXiv:2206.12375 [astro-ph.CO]

2022 arXiv
[78]

Bonilla, S

A. Bonilla, S. Kumar, R. C. Nunes, and S. Pan, Mon. Not. Roy. Astron. Soc.512, 4231 (2022), arXiv:2102.06149 [astro-ph.CO]

2022 arXiv
[79]

Abedin, G.-J

M. Abedin, G.-J. Wang, Y.-Z. Ma, and S. Pan, (2025), arXiv:2505.04336 [astro-ph.CO]

2025 arXiv
[80]

J. F. Jesus, R. Valentim, A. A. Escobal, and S. H. Pereira, JCAP2020, 053 (2020), arXiv:1909.00090 [astro-ph.CO]

2020 arXiv
[81]

Mukherjee and N

P. Mukherjee and N. Banerjee, arXiv e-prints , arXiv:2007.15941 (2020), arXiv:2007.15941 [astro- ph.CO]

2020 arXiv
[82]

Perenon, M

L. Perenon, M. Martinelli, S. Ili´ c, R. Maartens, M. Lochner, and C. Clarkson, Phys. Dark Univ.34, 100898 (2021), arXiv:2105.01613 [astro-ph.CO]

2021 arXiv
[83]

Avila, A

F. Avila, A. Bernui, A. Bonilla, and R. C. Nunes, European Physical Journal C82, 594 (2022), arXiv:2201.07829 [astro-ph.CO]

2022 arXiv
[84]

Calder´ on, B

R. Calder´ on, B. L’Huillier, D. Polarski, A. Shafieloo, and A. A. Starobinsky, Phys. Rev. D108, 023504 (2023), arXiv:2301.00640 [astro-ph.CO]

2023 arXiv
[85]

L’Huillier, A

B. L’Huillier, A. Shafieloo, D. Polarski, and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc.494, 819 (2020), arXiv:1906.05991 [astro-ph.CO]

2020 arXiv
[86]

Oliveira, F

F. Oliveira, F. Avila, C. Franco, and A. Bernui, Physics of the Dark Universe49, 101996 (2025), arXiv:2507.00779 [astro-ph.CO]

2025 arXiv
[87]

Avila, A

F. Avila, A. Bernui, R. C. Nunes, E. de Car- valho, and C. P. Novaes, MNRAS509, 2994 (2022), arXiv:2111.08541 [astro-ph.CO]

2022 arXiv
[88]

Yin and H

Z.-Y. Yin and H. Wei, Science China Physics, Mechanics, and Astronomy62, 999811 (2019), arXiv:1808.00377 [astro-ph.CO]

2019 arXiv
[89]

Mu, E.-K

Y. Mu, E.-K. Li, and L. Xu, Classical and Quantum Gravity40, 225003 (2023)

2023
[90]

Oliveira, F

F. Oliveira, F. Avila, A. Bernui, A. Bonilla, and R. C. Nunes, European Physical Journal C84, 636 (2024), arXiv:2311.14216 [astro-ph.CO]

2024 arXiv
[91]

L. A. Escamilla, O. Akarsu, E. Di Valentino, E. ¨Oz¨ ulker, and J. A. Vazquez, (2025), arXiv:2503.12945 [astro- ph.CO]

2025 arXiv
[92]

M. A. Sabogal, O. Akarsu, A. Bonilla, E. Di Valentino, and R. C. Nunes, Eur. Phys. J. C84, 703 (2024), arXiv:2407.04223 [astro-ph.CO]

2024 arXiv
[93]

S. Gao, Q. Gao, Y. Gong, and X. Lu, (2025), arXiv:2503.15943 [astro-ph.CO]

2025 arXiv
[94]

Jiang, D

J.-Q. Jiang, D. Pedrotti, S. S. da Costa, and S. Vagnozzi, Phys. Rev. D110, 123519 (2024), arXiv:2408.02365 [astro-ph.CO]

2024 arXiv
[95]

B. R. Dinda and R. Maartens, JCAP01, 120 (2025), arXiv:2407.17252 [astro-ph.CO]

2025 arXiv
[96]

B. R. Dinda, R. Maartens, S. Saito, and C. Clarkson, (2025), arXiv:2504.09681 [astro-ph.CO]

2025 arXiv
[97]

Y. Yang, Q. Wang, C. Li, P. Yuan, X. Ren, E. N. Sari- dakis, and Y.-F. Cai, (2025), arXiv:2501.18336 [astro- ph.CO]

2025
[98]

B. R. Dinda and N. Banerjee, PRD107, 063513 (2023), arXiv:2208.14740 [astro-ph.CO]

2023 arXiv
[99]

Ruiz-Zapatero, C

J. Ruiz-Zapatero, C. Garc´ ıa-Garc´ ıa, D. Alonso, P. G. Ferreira, and R. D. P. Grumitt, MNRAS512, 1967 (2022), arXiv:2201.07025 [astro-ph.CO]

2022 arXiv
[100]

L. A. Escamilla, ¨O. Akarsu, E. Di Valentino, and J. A. Vazquez, JCAP2023, 051 (2023), arXiv:2305.16290 [astro-ph.CO]

2023 arXiv
[101]

W. Sun, K. Jiao, and T.-J. Zhang, apj915, 123 (2021), arXiv:2105.12618 [astro-ph.CO]

2021 arXiv
[102]

´O Colg´ ain and M

E. ´O Colg´ ain and M. M. Sheikh-Jabbari, Eur. Phys. J. C81, 892 (2021), arXiv:2101.08565 [astro-ph.CO]

2021 arXiv
[103]

Ahlstr¨ om Kjerrgren and E

A. Ahlstr¨ om Kjerrgren and E. M¨ ortsell, MNRAS518, 585 (2023), arXiv:2106.11317 [astro-ph.CO]

2023 arXiv
[104]

Renzi and A

F. Renzi and A. Silvestri, arXiv e-prints , arXiv:2011.10559 (2020), arXiv:2011.10559 [astro- ph.CO]

2020 arXiv
[105]

Calder´ on, B

R. Calder´ on, B. L’Huillier, D. Polarski, A. Shafieloo, and A. A. Starobinsky, Phys. Rev. D106, 083513 11 (2022), arXiv:2206.13820 [astro-ph.CO]

2022 arXiv
[106]

B. R. Dinda, European Physical Journal C84, 402 (2024), arXiv:2311.13498 [astro-ph.CO]

2024 arXiv
[107]

L. A. Escamilla, E. ¨Oz¨ ulker, O. Akarsu, E. Di Valentino, and J. A. V´ azquez, (2024), arXiv:2408.12516 [astro- ph.CO]

2024
[108]

J. d. J. Vel´ azquez, L. A. Escamilla, P. Mukher- jee, and J. A. V´ azquez, Universe10, 464 (2024), arXiv:2410.02061 [astro-ph.CO]

2024 arXiv
[109]

G´ omez-Vargas, R

I. G´ omez-Vargas, R. M. Esquivel, R. Garc´ ıa-Salcedo, and J. A. V´ azquez, Eur. Phys. J. C83, 304 (2023), arXiv:2104.00595 [astro-ph.CO]

2023 arXiv
[110]

R. E. Keeley, A. Shafieloo, G.-B. Zhao, J. A. Vazquez, and H. Koo, Astron. J.161, 151 (2021), arXiv:2010.03234 [astro-ph.CO]

2021 arXiv
[111]

Zheng, D.-C

J. Zheng, D.-C. Qiang, Z.-Q. You, and D. Kumar, (2025), arXiv:2507.17113 [astro-ph.CO]

2025
[112]

Li, G.-H

T.-N. Li, G.-H. Du, P.-J. Wu, J.-Z. Qi, J.-F. Zhang, and X. Zhang, (2025), arXiv:2507.13811 [astro-ph.CO]

2025
[113]

F. Yang, X. Fu, B. Xu, K. Zhang, Y. Huang, and Y. Yang, Eur. Phys. J. C85, 186 (2025), arXiv:2502.05417 [astro-ph.CO]

2025 arXiv
[114]

Cosmai, G

L. Cosmai, G. Fanizza, M. Gasperini, and L. Tedesco, Class. Quant. Grav.30, 095011 (2013), arXiv:1303.5484 [gr-qc]

2013 arXiv
[115]

M. Wang, X. Fu, B. Xu, Y. Huang, Y. Yang, and Z. Lu, Eur. Phys. J. C84, 702 (2024), arXiv:2407.12250 [astro- ph.CO]

2024 arXiv
[116]

Hwang, B

S.-g. Hwang, B. L’Huillier, R. E. Keeley, M. J. Jee, and A. Shafieloo, JCAP2023, 014 (2023), arXiv:2206.15081 [astro-ph.CO]

2023 arXiv
[117]

Zhang, Y.-C

H. Zhang, Y.-C. Wang, T.-J. Zhang, and T. Zhang, apjs266, 27 (2023), arXiv:2304.03911 [astro-ph.CO]

2023 arXiv
[118]

C. E. Rasmussen and C. K. I. Williams,Gaussian pro- cesses for machine learning., Adaptive computation and machine learning (MIT Press, 2006) pp. I–XVIII, 1–248

2006

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Di Valentinoet al., (2025), arXiv:2504.01669 [astro- ph.CO]

E. Di Valentinoet al., (2025), arXiv:2504.01669 [astro- ph.CO]

arXiv 2025

[2] [2]

R. R. Caldwell and M. Kamionkowski, Annual Re- view of Nuclear and Particle Science59, 397 (2009), 9 arXiv:0903.0866 [astro-ph.CO]

arXiv 2009

[3] [3]

D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess, and E. Rozo, Phys. Rep.530, 87 (2013), arXiv:1201.2434 [astro-ph.CO]

arXiv 2013

[4] [4]

Perivolaropoulos and F

L. Perivolaropoulos and F. Skara, nar95, 101659 (2022), arXiv:2105.05208 [astro-ph.CO]

arXiv 2022

[5] [5]

Vagnozzi, Universe9, 393 (2023), arXiv:2308.16628 [astro-ph.CO]

S. Vagnozzi, Universe9, 393 (2023), arXiv:2308.16628 [astro-ph.CO]

arXiv 2023

[6] [6]

I. M. H. Etherington, Philosophical Magazine15, 761 (1933)

1933

[7] [7]

G. F. R. Ellis, General Relativity and Gravitation39, 1047 (2007)

2007

[8] [8]

J. Qin, F. Melia, and T.-J. Zhang, MNRAS502, 3500 (2021), arXiv:2101.05574 [astro-ph.CO]

arXiv 2021