Model independent test of the FLRW metric and the curvature in light of DESI DR2

Benjamin L'Huillier; Cl\'ea Millard; Marian Douspis

REVIEW 4 major objections 6 minor 1 cited by

Model independent test of the FLRW metric and the curvature in light of DESI DR2

T0 review · 4 major / 6 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 Using model-independent supernova distance reconstructions and DESI DR2 BAO data, this paper tests whether the FLRW metric and spatial flatness hold without assuming any dark energy model.

desk verdict A careful null test of FLRW with DESI DR2, but the headline curvature medians depend on an unvalidated p-value filter and shift with smoothing scale; the broad flatness-consistency conclusion is likely solid. read the letter →

arxiv 2601.20293 v2 pith:YPRCDORZ submitted 2026-01-28 astro-ph.CO

Cléa Millard , Benjamin L'Huillier , Marian Douspis This is my paper · ORCID

classification astro-ph.CO

keywords FLRWmetricspatialcurvatureiterativesmoothingDESIDR2BAOPantheon+DESY5Okdiagnosticmodel-independentcosmology

topics Dark Energy

open problems Dark Energy

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

Flatness of the universe and the Friedmann-Lemaitre-Robertson-Walker metric are foundational assumptions of standard cosmology. This paper asks whether current data, taken on their own, can rule them out. The authors reconstruct supernova distances and their derivatives with an iterative smoothing algorithm, then combine these with DESI DR2 baryon acoustic oscillation measurements to build the $\mathcal{O}_k$ diagnostic, a quantity that must be constant and equal to $\Omega_{k,0}$ in an FLRW universe. They find that after filtering out reconstructions whose $\mathcal{O}_k$ is not consistent with a constant, the median curvature is close to zero for all three data combinations: $0.035^{+0.046}_{-0.079}\pm0.037$ for Pantheon+ & DESI DR2, $0.092^{+0.055}_{-0.132}\pm0.064$ for the same supernovae cut at $z=1.13$, and $-0.119^{+0.113}_{-0.047}\pm0.043$ for DES Y5 & DESI DR2. So the FLRW metric is not ruled out and most reconstructions are consistent with flatness within about $3\sigma$.

What carries the argument

The central object is the $\mathcal{O}_k$ diagnostic, defined as $\mathcal{O}_k(z) = (\Theta^2(z)-1)/D^2(z)$ with $\Theta(z)=h(z)D'(z)$, where $D(z)$ is the dimensionless comoving distance, $D'(z)$ its derivative, and $h(z)$ the dimensionless Hubble parameter; in an FLRW universe this quantity is identically $\Omega_{k,0}$. The reconstruction side uses iterative smoothing of the supernova distance modulus to obtain $D$ and $D'$ at the BAO redshifts, while the BAO side supplies transverse and radial mode ratios $d_M/r_d$ and $d_H/r_d$. The key trick is writing $\Theta(z)$ as the ratio of these two BAO mode ratios times $D'/D$, which cancels the unknown $H_0$ and sound-horizon scale, allowing a genuinely model-independent reconstruction. A p-value filter based on the chi-squared of fitting a constant to the $\mathcal{O}_k$ points decides which reconstructions are treated as FLRW-consistent and therefore interpretable as measurements of $\Omega_{k,0}$.

What would settle it

Run the full pipeline on mock data generated from a flat FLRW model with the same redshift distribution, masks, and covariance matrices: if the p-value filter rejects more than 5 percent of reconstructions, or if the accepted median $\Omega_{k,0}$ deviates from zero by more than the reported uncertainty, the filter is miscalibrated. Equivalently, compute the eigenvalue spectrum of the correlation matrix of the $\mathcal{O}_k$ points and count the modes above noise to measure the effective number of independent degrees of freedom and compare it to $N_{\rm BAO}-1$.

Watch

Extended reading notes

Core claim

The paper claims that a data-driven reconstruction of the expansion history, independent of any dark energy model, produces an $\mathcal{O}_k$ diagnostic that is consistent with the FLRW prediction of a constant value for most reconstructions. In an FLRW universe, $\mathcal{O}_k(z)\equiv\Omega_{k,0}$, and the authors treat deviations from a constant as evidence against the metric. After applying a 95\% p-value filter to select reconstructions that are FLRW-consistent, and keeping only those that improve the fit relative to the best flat $\Lambda$CDM model, the median curvature values are the ones quoted above: a slight positive preference from Pantheon+ data and a slight negative preference from DES Y5 data, with uncertainties comfortably overlapping flatness. The high-redshift Pantheon+ data, where the sample is sparse, are identified as the source of apparent FLRW inconsistency, and the authors find that truncating the supernovae at $z=1.13$ restores full consistency.

Load-bearing premise

The p-value filter assumes that the chi-squared of fitting a constant to the $\mathcal{O}_k$ points follows a chi-squared distribution with $N_{\rm BAO}-1$ degrees of freedom, even though the $\mathcal{O}_k$ points share correlated errors from the same smoothing reconstruction and the same supernova covariance matrix, so the effective number of independent degrees of freedom is likely smaller.

Editorial extensions

If this is right

The FLRW metric, a core assumption of the concordance model, is not rejected by this combined SNIa+BAO dataset, so dark energy models built on that metric remain viable.
The results place model-independent constraints on spatial curvature that are independent of the Hubble constant and the sound-horizon scale, complementing parametric fits.
The apparent high-redshift deviation from FLRW in Pantheon+ is driven by sparse and less reliable supernovae at $z>1$, rather than by a genuine breakdown of the metric.
The sign of the preferred curvature depends on the supernova sample, with Pantheon+ leaning positive and DES Y5 leaning negative, yet both are statistically consistent with flatness for most reconstructions.
The recovery of near-zero curvature under the flatness-assuming selection criterion serves as an internal consistency check of the method.

Reading between the lines

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

The p-value filter assumes that the chi-squared of the constant fit has $N_{\rm BAO}-1$ independent degrees of freedom, but the $\mathcal{O}_k$ points share correlated errors from the same smoothing reconstruction and the same supernova covariance matrix; if the effective degrees of freedom are smaller, the 95\% filter would be miscalibrated, and this specific test is not performed here.
A direct way to test that calibration would be to run the full pipeline on mock realizations of a flat FLRW universe with the same redshift masks and covariances, checking whether exactly about 5\% of reconstructions are rejected and whether the accepted median $\Omega_{k,0}$ remains unbiased; this would also reveal whether the reported spread is inflated or deflated.
The dataset-dependent sign of the curvature preference suggests that residual systematics in the two supernova compilations, especially high-redshift selection effects, could easily shift the median by amounts comparable to the quoted uncertainties, meaning the sign is not cosmologically informative until those systematics are better understood.

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

Two things you should know. The paper extends the Ok null test to DESI DR2 BAO combined with Pantheon+ and DES Y5 supernovae, and its broad conclusion—FLRW and flatness are not ruled out—is probably right. But the specific curvature medians in the abstract and Table 1 are conditional on a p-value filter whose calibration isn't demonstrated, and the values shift noticeably with reasonable analysis choices.

What's genuinely new is modest but real: same iterative-smoothing plus Ok machinery from L'Huillier and Shafieloo and later work, applied to the newest BAO data, with a p-value gate intended to select reconstructions whose Ok is constant, plus a post-hoc low-redshift cut for Pantheon+. The paper is careful about its own limitations: it admits the DESI BAO pipeline assumes flatness, treats the Delta chi^2_tot selection as a consistency check rather than a genuine test, and includes a smoothing-scale appendix.

The soft spot is the Section 3.2 p-value filter. The paper assumes chi^2_Ok has a chi-squared distribution with N_BAO - 1 degrees of freedom. But the Ok points at different redshifts come from the same smoothing reconstruction and share SNIa covariance, so they're correlated; the effective degrees of freedom are smaller. No Monte Carlo or analytic calibration is given. That matters because under the Delta chi^2_SNIa selection, only 1.12% of Pantheon+ & DESI DR2 reconstructions pass, so the median is defined by a tiny, possibly arbitrary subset. Under Delta chi^2_SNIa+dM the pass rate is 82%, but the filter still affects the result. The appendix reinforces the fragility: for Pantheon+ low-z, the median Omega_k jumps from 0.172 at smoothing scale 0.2 to 0.011 at 0.4—a range not reflected in the quoted error bars.

The low-z cut is post hoc, but the paper is transparent about it. The Ok diagnostic itself is measured, not assumed, so there's no circularity in the main numbers. The citation pattern looks fine; they reference the relevant DESI DR2 BAO papers and earlier null-test work.

If you work on curvature tests or DESI, this is a useful consistency check, and the broad flatness result is probably robust. I wouldn't quote the specific medians. I'd send it to a serious referee, with the request that the p-value filter be either calibrated or dropped, and that the dependence on the smoothing scale be presented as part of the headline result.

Referee Report

4 major / 6 minor

Summary. The paper performs a model-independent test of the FLRW metric and spatial curvature by reconstructing the dimensionless comoving distance D(z) and its derivative from Type Ia supernovae (Pantheon+ or DES Y5) with an iterative smoothing algorithm, and combining these reconstructions with DESI DR2 BAO transverse and radial mode measurements. Using the O_k diagnostic, which is constant and equal to Omega_k0 in an FLRW universe, the authors fit a constant to the reconstructed O_k(z) at BAO redshifts, apply a p-value filter to select reconstructions consistent with FLRW, and report the median fitted constant as an estimate of Omega_k0. The headline results are Omega_k0 med = 0.035 (+0.046, -0.079) +/- 0.037 for Pantheon+ & DESI DR2, 0.092 (+0.055, -0.132) +/- 0.064 with the Pantheon+ data cut at z=1.13, and -0.119 (+0.113, -0.047) +/- 0.043 for DES Y5 & DESI DR2. The paper concludes that the O_k diagnostic does not rule out the FLRW metric and that most reconstructions are consistent with flatness within 3 sigma, with a slight preference for positive curvature for Pantheon+ and negative for DES Y5.

Significance. If the methodology were fully validated, this would be a valuable addition to the literature on null tests of the FLRW metric, providing a data-driven, Dark-Energy-model-independent measurement of spatial curvature using current DESI DR2 BAO and SNIa data. The paper is transparent about many limitations: it acknowledges the flatness assumption in the DESI BAO pipeline (Section 2.2), warns that the Delta chi2_tot selection is not a genuine litmus test (Section 2.4), and provides an appendix documenting sensitivity to the smoothing scale. The use of two SNIa compilations and multiple Delta chi2 selection criteria is a strength. However, the central claim rests on a p-value filter whose statistical null distribution is assumed rather than calibrated, and the headline numbers in the abstract are not reproduced in Table 1. These issues must be addressed before the results can be taken at face value.

major comments (4)

[Section 3.2] The p-value filter assumes that the statistic chi^2_Ok for a constant fit to the reconstructed O_k(z) follows a chi^2 distribution with nu = N_BAO - 1. This assumption is not justified. The O_k values at different BAO redshifts are constructed from the same iterative-smoothing reconstruction of D(z) and D'(z), which inherits the full SNIa covariance matrix, and the transverse and radial BAO modes are correlated within each bin. The effective number of independent degrees of freedom is therefore likely smaller than N_BAO - 1, yet no analytic or Monte Carlo calibration of the null distribution is provided. The filter is load-bearing: for Pantheon+ & DESI DR2 with Delta chi^2_SNIa < 0, only 1.12% of reconstructions survive; for Delta chi^2_SNIa+dM < 0, 82.34% survive, and the reported medians in Table 1 and the abstract are conditional on this selection. Without a calibrated null distribution, the 95% consistency claim is unsubstantiated. I recommend adding a simulation-based calibration of the p-values (e.g., mock SNIa+BAO realisations from a known FLRW model through the full pipeline) and, if necessary, using the full covariance of the O_k estimates rather than treating the points as independent.
[Abstract vs Table 1] The abstract quotes Omega_k0 med = 0.035 (+0.046, -0.079) +/- 0.037 for Pantheon+ & DESI DR2 and 0.092 (+0.055, -0.132) +/- 0.064 for the low-z cut, but Table 1 contains no such values. The closest entries are 0.033 (+0.047, -0.072) +/- 0.037 and 0.091 (+0.055, -0.131) +/- 0.063 in the Delta chi^2_SNIa+dM < 0 rows, and 0.058 (+0.043, -0.107) +/- 0.038 and 0.098 (+0.048, -0.134) +/- 0.064 in the Delta chi^2_SNIa < 0 rows. The central values and asymmetric errors differ beyond rounding. Since the abstract is the primary statement of the result, this inconsistency must be resolved before publication; the authors should either update the abstract to quote the Table 1 values or explain the exact selection criterion and reconstruction set used for the abstract numbers.
[Appendix A and Section 4] The results depend strongly on the smoothing scale Delta. For example, for Pantheon+ & DESI DR2 (low-z) with Delta chi^2_SNIa+dM < 0, Omega_med varies from 0.172 at Delta = 0.2 to 0.011 at Delta = 0.4, compared with 0.091 at Delta = 0.3; for DES Y5 & DESI DR2, it varies from -0.176 to -0.066. These variations are as large as or larger than the reported 'spread' uncertainties and the median 1-sigma errors. The paper nevertheless describes the results as 'robust' (Section 4). The authors should either include the smoothing-scale variation as a systematic uncertainty in the headline results, or soften the robustness claim to reflect the demonstrated sensitivity.
[Section 3.2 and Discussion] The cut at z < 1.13 is introduced after observing that the high-redshift Pantheon+ points drive the O_k reconstructions away from FLRW, and the abstract presents the resulting values (e.g., 0.092) as one of the main results. Because the cut is motivated by the outcome it removes, the resulting Omega_k0 medians should be framed as an exploratory consistency test rather than a primary measurement. The authors should prespecify the redshift range or treat the cut as a robustness check with an explicit discussion of the selection effect.

minor comments (6)

[Section 3.2] The sentence 'Only 1.12% the Pantheon+ & DESI DR2 reconstructions pass the p-value test. The remaining reconstructions are consistent with the FLRW metric' is logically contradictory; it should read 'these reconstructions' instead of 'the remaining reconstructions'.
[Section 3.2] The statistic chi^2_Ok is not explicitly defined. Please provide the formula and specify whether the full covariance matrix of the O_k estimates is used or only the diagonal errors.
[Table 1 and text] The first uncertainty on Omega_k0 med is described as the spread (max - min) of the central values over reconstructions, not a standard 1-sigma confidence interval. Please clarify the notation (e.g., use brackets for the range) and state explicitly that the asymmetric errors are not 1-sigma errors.
[Section 4 vs Section 3.2] The percentage of reconstructions passing the p-value test is given as 1.12% in Section 3.2 and 1.11% in Section 4; please make these consistent.
[Section 1 and 2.2] The statement that the litmus test results 'do not depend on the values of H0 and rd' should be qualified by the fact that the BAO pipeline assumes a flat fiducial cosmology, as acknowledged in Section 2.2; the H0/rd independence is exact only at the level of the O_k construction.
[References] Reference [16] appears incomplete; please provide the full bibliographic details for the DES Y5 cosmology paper.

Circularity Check

1 steps flagged · score 2.0 of 10

Headline Omega_k medians are not circular; the only by-construction loop is the explicitly labeled Delta-chi^2_tot flatness-consistency check, plus an uncalibrated p-value null that is a statistical risk rather than a circular step.

self definitional [Section 2.4 (eqs. 2.2c, 2.10a, 2.10b) and Section 4; Table 1]
"It is important to note that this equation assumes flatness. ... Using the selection criteria Δχ²_tot<0 filters out reconstructions inconsistent with flatness. Such a selection can be used as a self consistency check of the Ok diagnostic, but the results should not be used as a genuine litmus test of the FLRW metric and flatness. ... we recover an Ok diagnostic highly consistent with Ωk,0=0 (within 1.5σ) ... after biasing the collection of selected reconstructions toward flatness."

The dH/rd normalization used in χ²_tot is computed with eq. (2.2c), which the paper states assumes flatness. The Δχ²_tot<0 cut therefore selects reconstructions that agree with a flatness-assuming radial BAO conversion; reporting Ωk≈0 for that subsample is a consequence of the selection, not a test of curvature. The paper explicitly labels it a self-consistency check, so it is disclosed rather than hidden, and the abstract's headline numbers come from Δχ²_SNIa/Δχ²_SNIa+dM selections, which do not use eq. (2.2c) in the same way.

full rationale

The main Ωk medians are produced by the Ok diagnostic: D and D' are reconstructed from SNIa by iterative smoothing, combined with DESI DR2 BAO ratios, and a constant fit to Ok is interpreted as Ωk only after a p-value check of constancy. This is the intended logic of the Clarkson-Bassett-Lu test and is not circular: the quoted constants are fitted quantities, not inputs. The Δχ²_tot selection is a genuine by-construction loop (flatness-assuming eq. 2.2c defines the radial BAO prediction used in the selection, and the selected sample is then found to be flat), but the paper states plainly that those results should not be used as a litmus test, so the circularity is limited and disclosed. The p-value filter assumes χ²_Ok ~ χ²_{NBAO-1}; since the Ok points share one smoothing realization and the SNIa covariance, the effective number of degrees of freedom may be smaller, making the 95% FLRW-consistency gate potentially miscalibrated. That is a statistical/correctness risk, not an equivalence-by-construction, so it is not counted as a circular step. The self-citations to [9-11] supply the reconstruction and error-propagation method rather than a uniqueness theorem, and the appendix shows smoothing-scale sensitivity, which the authors acknowledge. Overall, the central claim has independent content and the derivation is largely self-contained.

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

The central measurement rests on the Ok identity from FLRW, on the DESI BAO measurements (which assume a flat fiducial model), and on the iterative smoothing reconstructions. The smoothing scale, the flat ΛCDM baseline, the low-z cut and the p-value threshold are chosen by hand or post hoc and affect the results. No new particles, mediators, or forces are introduced.

free parameters (5)

smoothing scale Δ = 0.3 (0.2 and 0.4 in Appendix A)
Controls the Gaussian kernel width in iterative smoothing; results shift with Δ (Table A.1), so the central Ωk estimates depend on this hand-chosen value.
flat ΛCDM baseline parameters Ωm,0 and MB = Not reported (fit to each dataset)
Used to define Δχ² and select reconstructions; H0 fixed to Planck 2018. The selection of which reconstructions are kept depends on this fitted baseline.
low-z redshift cut z<1.13 (Pantheon+ only) = 1.13
Applied post hoc after observing high-redshift tension; changes the Pantheon+ median Ωk substantially (e.g. 0.058 to 0.098 for Δχ²_SNIa selection).
p-value threshold α = 0.05
Defines which reconstructions are FLRW-consistent and therefore included in the Ωk medians; a different α would change the sample.
initial guess flat ΛCDM grid = Various (not enumerated)
The collection of reconstructions and hence the spread in Ωk depends on the set of initial cosmologies used; the paper does not specify the grid.

assumptions (5)

standard math Ok identity Θ²=1+Ωk D² holds for FLRW metrics
Eq. (2.1) is derived from the FLRW metric; used to interpret a constant Ok as Ωk.
domain assumption DESI DR2 BAO distance measurements are valid despite the pipeline assuming a flat FLRW universe
Section 2.2 states 'a flat FLRW universe is assumed in the whole BAO pipeline'; if this biases dM/rd or dH/rd, the Ok test is contaminated.
ad hoc to paper χ²_Ok follows a χ² distribution with ν=NBAO-1
Section 3.2 introduces this assumption for the p-value filter without correcting for correlations among smoothed reconstructions.
domain assumption Iterative smoothing reconstructions converge to a representative set of plausible expansion histories
Section 2.3 treats the ensemble of reconstructions as representative, but no convergence or coverage proof is given.
domain assumption Etherington distance-duality relation dL=(1+z)dM holds
Used implicitly in eq. (1.3) to convert SNIa luminosity distances to comoving distances; violations would mimic curvature.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Model independent test of the FLRW metric and the curvature in light of DESI DR2." pith.science (2026). https://pith.science/paper/YPRCDORZ

@misc{pith2026260120293,
  author       = {Pith},
  title        = {Pith review of: Model independent test of the FLRW metric and the curvature in light of DESI DR2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPRCDORZ}},
  note         = {Machine review of arXiv:2601.20293}
}

abstract

We perform a data-driven test of the FLRW metric and the flatness of the Universe, independently of any Dark Energy model, and in light of the latest DESI DR2 results. We use Pantheon+ and DESY5 SNIa data to reconstruct the distance modulus, dimensionless comoving distance and Hubble parameter, using an iterative smoothing algorithm. Then, combining the various reconstructions with the recent BAO measurements from DESI DR2, we perform the $\mathcal{O}_k$ diagnostic, a litmus test of the FLRW metric and the flatness of the Universe. We obtain robust results that do not depend on Dark Energy models and test some of the underlying hypotheses of the concordance model. We find that when the reconstructed $\mathcal{O}_k$ diagnostic is consistent with the FLRW metric, then the median value of $\Omega_{k,0}$ over all reconstructions that provide an improved fit relative to the flat $\Lambda$CDM model are: ${\Omega}_{k,0}^\text{med} = 0.035 ^{+0.046}_{-0.079}\pm 0.037$ for the Pantheon+ \& DESI DR2 data combination, ${\Omega}_{k,0}^\text{med} = 0.092 ^{+0.055}_{-0.132} \pm 0.064$ for the same data but with the Pantheon+ SNIa cut at redshift $z=1.13$, which is the maximum redshift of the DES~Y5 data, and ${\Omega}_{k,0}^\text{med} = -0.119^{+0.113}_{-0.047}\pm 0.043$ for DES~Y5 \& DESI DR2. The first uncertainties correspond to the spread in $\Omega_{k,0}$ over all reconstructions, followed by the median 1$\sigma$ error.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Intertwined Constraints in Extended Cosmologies: Dark Energy, Curvature, Neutrinos, and Inflation
astro-ph.CO 2026-07 unverdicted novelty 4.0 of 10

Dynamical dark energy remains preferred across extended models while curvature, neutrino mass and inflation parameters show strong model dependence, with no resolution of the H0 tension.

Reference graph

Works this paper leans on

29 extracted references · 9 canonical work pages · cited by 1 Pith paper

[1]

Riess, A.V

A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant,The Astronomical Journal116(1998) 1009–1038

work page 1998
[2]

Perlmutter, G

S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro et al., Measurements ofωandλfrom 42 high-redshift supernovae,The Astrophysical Journal517 (1999) 565–586

work page 1999
[3]

Aghanim, Y

Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., Planck 2018 results. VI. Cosmological parameters, A&A641(2020) A6 [1807.06209]

arXiv 2020
[4]

Abdul-Karim, J

DESI Collaboration, M. Abdul-Karim, J. Aguilar, S. Ahlen, S. Alam, L. Allen et al.,DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints, arXiv e-prints(2025) arXiv:2503.14738 [2503.14738]

arXiv 2025
[5]

Di Valentino, J.L

E. Di Valentino, J.L. Said, A. Riess, A. Pollo, V. Poulin, A. Gómez-Valent et al.,The cosmoverse white paper: Addressing observational tensions in cosmology with systematics and fundamental physics,Physics of the Dark Universe49(2025) 101965

work page 2025
[6]

Di Valentino, O

E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al.,In the realm of the Hubble tension-a review of solutions,Classical and Quantum Gravity38(2021) 153001 [2103.01183]

arXiv 2021
[7]

Clarkson, B

C. Clarkson, B. Bassett and T.H.-C. Lu,A General Test of the Copernican Principle, Phys. Rev. Lett.101(2008) 011301 [0712.3457]

arXiv 2008
[8]

Shafieloo and C

A. Shafieloo and C. Clarkson,Model independent tests of the standard cosmological model, Phys. Rev. D81(2010) 083537 [0911.4858]

arXiv 2010

Show all 29 references

[9]

L’Huillier and A

B. L’Huillier and A. Shafieloo,Model-independent test of the FLRW metric, the flatness of the – 17 – Universe, and non-local estimation of H0 rd, J. Cosmology Astropart. Phys.2017(2017) 015 [1606.06832]

2017 arXiv
[10]

Shafieloo, B

A. Shafieloo, B. L’Huillier and A.A. Starobinsky,FalsifyingΛCDM : Model-independent tests of the concordance model with eBOSS DR14Q and Pantheon, Phys. Rev. D98(2018) 083526 [1804.04320]

2018 arXiv
[11]

L’Huillier, A

B. L’Huillier, A. Mitra, A. Shafieloo, R.E. Keeley and H. Koo,Litmus tests of the flatΛCDM model and model-independent measurement of H0 r d with LSST and DESI, J. Cosmology Astropart. Phys.2025(2025) 030 [2407.07847]

2025 arXiv
[12]

Montanari and S

F. Montanari and S. Räsänen,Backreaction and frw consistency conditions,Journal of Cosmology and Astroparticle Physics2017(2017) 032–032

2017
[13]

Dinda, R

B.R. Dinda, R. Maartens, S. Saito and C. Clarkson,Improved null tests ofλcdm and flrw in light of desi dr2,Journal of Cosmology and Astroparticle Physics2025(2025) 018

2025
[14]

Collett, F

T. Collett, F. Montanari and S. Räsänen,Model-Independent Determination of H0 andΩ K0 from Strong Lensing and Type Ia Supernovae, Phys. Rev. Lett.123(2019) 231101 [1905.09781]

2019 arXiv
[15]

Scolnic, D

D. Scolnic, D. Brout, A. Carr, A.G. Riess, T.M. Davis, A. Dwomoh et al.,The Pantheon+ Analysis: The Full Data Set and Light-curve Release, ApJ938(2022) 113 [2112.03863]. [16]DEScollaboration,The dark energy survey: Cosmology results with 1500 new high-redshift type ia supernov...

2022 arXiv
[17]

Shafieloo, U

A. Shafieloo, U. Alam, V. Sahni and A.A. Starobinsky,Smoothing supernova data to reconstruct the expansion history of the Universe and its age, MNRAS366(2006) 1081 [astro-ph/0505329]

2006 arXiv
[18]

Shafieloo,Model-independent reconstruction of the expansion history of the Universe and the properties of dark energy, MNRAS380(2007) 1573 [astro-ph/0703034]

A. Shafieloo,Model-independent reconstruction of the expansion history of the Universe and the properties of dark energy, MNRAS380(2007) 1573 [astro-ph/0703034]

2007 arXiv
[19]

L’Huillier, A

B. L’Huillier, A. Shafieloo and H. Kim,Model-independent cosmological constraints from growth and expansion, MNRAS476(2018) 3263 [1712.04865]

2018 arXiv
[20]

H. Koo, A. Shafieloo, R.E. Keeley and B. L’Huillier,Model-independent Constraints on Type Ia Supernova Light-curve Hyperparameters and Reconstructions of the Expansion History of the Universe, ApJ899(2020) 9 [2001.10887]

2020 arXiv
[21]

H. Koo, A. Shafieloo, R.E. Keeley and B. L’Huillier,Model selection and parameter estimation using the iterative smoothing method, J. Cosmology Astropart. Phys.2021(2021) 034 [2009.12045]

2021 arXiv
[22]

Koo, R.E

H. Koo, R.E. Keeley, A. Shafieloo and B. L’Huillier,Bayesian vs frequentist: comparing Bayesian model selection with a frequentist approach using the iterative smoothing method, J. Cosmology Astropart. Phys.2022(2022) 047 [2110.10977]

2022 arXiv
[23]

L’Huillier, A

B. L’Huillier, A. Shafieloo, E.V. Linder and A.G. Kim,Model independent expansion history from supernovae: Cosmology versus systematics, MNRAS485(2019) 2783 [1812.03623]

2019 arXiv
[24]

Abdul Karim, J

M. Abdul Karim, J. Aguilar, S. Ahlen, S. Alam, L. Allen, C.A. Prieto et al.,Desi dr2 results. ii. measurements of baryon acoustic oscillations and cosmological constraints,Physical Review D112(2025)

2025
[25]

Y. Liu, S. Cao, T. Liu, X. Li, S. Geng, Y. Lian et al.,Model-independent constraints on cosmic curvature: Implication from updated hubble diagram of high-redshift standard candles,The Astrophysical Journal901(2020) 129

2020
[26]

Favale, A

A. Favale, A. Gómez-Valent and M. Migliaccio,Revisiting model-independent constraints on spatial curvature and cosmic ladders calibration: updated and forecast analyses,arXiv e-prints (2025) arXiv:2511.19332 [2511.19332]. – 18 –

2025
[27]

Fortunato, W.S

J.A.S. Fortunato, W.S. Hipólito-Ricaldi and G.E. Romero,Probing Cosmic Curvature with Fast Radio Bursts and DESI DR2,arXiv e-prints(2025) arXiv:2506.01504 [2506.01504]

2025
[28]

S. Cao, J. Qi, Z. Cao, M. Biesiada, J. Li, Y. Pan et al.,Direct test of the FLRW metric from strongly lensed gravitational wave observations,Scientific Reports9(2019) 11608 [1910.10365]

2019 arXiv
[29]

Medeiros dos Santos, J

F.B. Medeiros dos Santos, J. Morais, S. Pan, W. Yang and E. Di Valentino,A New Window on Dynamical Dark Energy: Combining DESI-DR2 BAO with future Gravitational Wave Observations,arXiv e-prints(2025) arXiv:2504.04646 [2504.04646]

2025 arXiv
[30]

Liao,Constraints on cosmic curvature with lensing time delays and gravitational waves, Phys

K. Liao,Constraints on cosmic curvature with lensing time delays and gravitational waves, Phys. Rev. D99(2019) 083514 [1904.01744]. – 19 –

2019 arXiv

Pith tools

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

[1] [1]

Riess, A.V

A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant,The Astronomical Journal116(1998) 1009–1038

work page 1998

[2] [2]

Perlmutter, G

S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro et al., Measurements ofωandλfrom 42 high-redshift supernovae,The Astrophysical Journal517 (1999) 565–586

work page 1999

[3] [3]

Aghanim, Y

Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., Planck 2018 results. VI. Cosmological parameters, A&A641(2020) A6 [1807.06209]

arXiv 2020

[4] [4]

Abdul-Karim, J

DESI Collaboration, M. Abdul-Karim, J. Aguilar, S. Ahlen, S. Alam, L. Allen et al.,DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints, arXiv e-prints(2025) arXiv:2503.14738 [2503.14738]

arXiv 2025

[5] [5]

Di Valentino, J.L

E. Di Valentino, J.L. Said, A. Riess, A. Pollo, V. Poulin, A. Gómez-Valent et al.,The cosmoverse white paper: Addressing observational tensions in cosmology with systematics and fundamental physics,Physics of the Dark Universe49(2025) 101965

work page 2025

[6] [6]

Di Valentino, O

E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al.,In the realm of the Hubble tension-a review of solutions,Classical and Quantum Gravity38(2021) 153001 [2103.01183]

arXiv 2021

[7] [7]

Clarkson, B

C. Clarkson, B. Bassett and T.H.-C. Lu,A General Test of the Copernican Principle, Phys. Rev. Lett.101(2008) 011301 [0712.3457]

arXiv 2008

[8] [8]

Shafieloo and C

A. Shafieloo and C. Clarkson,Model independent tests of the standard cosmological model, Phys. Rev. D81(2010) 083537 [0911.4858]

arXiv 2010