Is cosmological data suggesting a nonminimal coupling between matter and gravity?

Miguel Barroso Varela; Orfeu Bertolami

REVIEW 4 major objections 4 minor 3 cited by

Is cosmological data suggesting a nonminimal coupling between matter and gravity?

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that current cosmological data favor a nonminimal coupling between matter and curvature over flat $\Lambda$CDM.

desk verdict New constraints from DESY5 and DESI/eBOSS are useful and the analysis is clean, but the abstract's claim of 'moderate to strong evidence ... for all dataset combinations' is contradicted by the paper's own Table 2. read the letter →

arxiv 2412.09348 v2 pith:OWGLRYJA submitted 2024-12-12 astro-ph.CO gr-qc

Miguel Barroso Varela , Orfeu Bertolami This is my paper · ORCID

classification astro-ph.COgr-qc

keywords ModifiedgravityNonminimallycoupledCosmologicaldataHubbletensiondarkenergybaryonacousticoscillationstypeIasupernovaemodelselection

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 asks whether current cosmological data point to a modified theory of gravity in which matter is coupled directly to spacetime curvature, rather than coupled minimally as in general relativity. It compares a one-parameter nonminimal coupling model, with no cosmological constant, against flat $\Lambda$CDM using type Ia supernova distances, Cepheid-calibrated distance anchors, and baryon acoustic oscillation measurements from several surveys. The authors report moderate to strong statistical evidence, by AIC and BIC criteria, that at least one version of the nonminimally coupled theory is preferred over $\Lambda$CDM for every dataset combination they consider. If the result holds, a single extra interaction term in the gravitational action could account for the late-time acceleration attributed to dark energy and ease the Hubble tension, although a mismatch with BAO data remains.

What carries the argument

The central object is the action $S = \int d^4x \, \sqrt{-g} \left[(1/2\kappa^2)R + (1+f_2(R))\mathcal{L}_m\right]$, with $\mathcal{L}_m=-\rho$ and $f_2(R)=(R_n/R)^n$. The inverse power law makes the coupling negligible at high curvature and increasingly significant at low curvature, so CMB-era $\Lambda$CDM values can be used as fixed initial conditions and the theory has a single free parameter, $R_n$. The modified Friedmann and Raychaudhuri equations are integrated numerically using $F_2 \tilde{\rho}$ as the dynamical variable, and the resulting expansion history $H(z)$ is scored against the data with $\chi^2$, AIC, and BIC statistics.

What would settle it

Measure the BAO sound horizon from early-universe physics without assuming any late-time cosmology and check whether it equals $r_d = 147.46 \pm 0.28$ Mpc; a significant deviation would remove the anchor on which the NMC model's BAO predictions rest.

Watch

Extended reading notes

Core claim

The central claim is that current cosmological data suggest the presence of a nonminimal coupling of matter and curvature over the minimally coupled standard theory. In the paper's own terms, the model with $f_2(R)=(R_n/R)^n$ and $n=4,6,10$ fits Cepheid-calibrated supernova distances as well as or better than $\Lambda$CDM while returning a Hubble constant consistent with both CMB-era initial conditions and local distance-ladder measurements, and when BAO data are added the $n=6$ version is preferred over $\Lambda$CDM with moderate to strong evidence. The $n=10$ version is strongly disfavoured by both information criteria, and the $n=4$ version fits supernovae alone well but degrades when BAO data are included. The authors conclude that the data as a whole favour the nonminimally coupled theory, while explicitly acknowledging an unresolved BAO tension.

Load-bearing premise

The comparison assumes the nonminimal coupling fully switches off at high redshift, so early-universe $\Lambda$CDM values for the matter density, expansion rate, and BAO sound horizon can be used as fixed anchors; if the coupling still matters at the CMB epoch, the model is fitted with biased inputs.

Editorial extensions

If this is right

If the preference holds, late-time cosmic acceleration can be reproduced without a cosmological constant, because the inverse-power coupling mimics dark energy at low curvature.
The model bridges part of the Hubble tension: with CMB-anchored early conditions and a late-time deviation, its fitted $H_0$ stays compatible with Cepheid-calibrated supernova distances.
A BAO tension persists: adding baryon acoustic oscillation data pulls $H_0$ down toward $69$ km/s/Mpc, several sigma away from the supernova-only value, so the theory is not yet coherent across all probes.
The $n=10$ version is strongly ruled out by both AIC and BIC, while the $n=6$ version is the most consistently preferred variant when BAO data are included.
Ongoing and future BAO and supernova surveys can discriminate the model from $\Lambda$CDM because its distance-redshift relation is fixed by a single parameter.

Reading between the lines

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

A natural extension the paper does not carry out is treating $n$ as a free parameter: the results suggest $n=6$ is the compromise between the smooth $n=4$ and sharp $n=10$ behaviours, so allowing $n$ to vary could sharpen the comparison at the cost of a second parameter.
Because the NMC model is anchored to $\Lambda$CDM before recombination, a direct search for modified-gravity effects in CMB temperature and polarization spectra would test the foundation of the comparison; none is performed here.
The remaining BAO tension might be relieved by adding positive powers of $R$ to $f_2$, which act at early times; the paper calls this remote given how well CMB data match $\Lambda$CDM, but it is a concrete modification to test.
Independent late-time probes of the expansion rate, such as gravitational-wave standard sirens, could confirm or challenge the predicted $H_0$ without relying on the Cepheid or BAO assumptions used here.

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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

Here's my read of the Varela–Bertolami NMC paper.

The genuinely new content: they run their established nonminimally coupled curvature-matter model against DES-SN5YR, DESI BAO and eBOSS, and they add the n=6 case. None of that appeared in their earlier work, and the resulting parameter constraints for Rn and derived H0, Omega_m are useful. The analysis is well done: cobaya MCMC, proper covariance matrices, and a sensible calibration check (their LCDM fit to Pantheon+ matches the original). They are also upfront that the model does not resolve the BAO tension, which is honest.

The soft spot is the headline. The abstract and conclusions claim 'moderate to strong evidence for a preference of the nonminimally coupled theory ... for all dataset combinations.' That is not what Table 2 shows. n=10 is strongly disfavored by both AIC and BIC in every combination. n=4 is disfavored by BIC for PS+DESI and PS+eBOSS. Only n=6 is preferred, and its ΔBIC values are between -1.25 and -8.84, i.e. weak to moderate, with mixed signs in ΔAIC. So the evidence is neither strong nor universal. It rests on choosing n=6 after comparing three exponents, without a multiple-model penalty, and on treating Planck-derived anchors (Omega_m*, H0*, rd) as exact numbers. If the uncertainties on rd and the Planck parameters were propagated, the effective parameter count of the NMC model would rise and the BIC advantage would shrink. The authors should either soften the abstract or do the extra work of marginalizing over the anchors.

None of this is fatal. The model is what it is, the data handling looks clean, and the constraints are a legitimate contribution. What needs fixing is the inference from Table 2 to the conclusion. I'd send it to peer review—the referee should push for a revised abstract and a discussion of the post-hoc n=6 selection and the fixed-anchor assumption, but the paper deserves engagement rather than desk rejection.

For a reading group, it's a decent example of how model-selection claims can outrun the tables. I'd bring it if we want to talk about BIC vs AIC and fixed external parameters.

Referee Report

4 major / 4 minor

Summary. The paper compares a nonminimally coupled curvature-matter gravity model, with f2(R)=(Rn/R)^n and f1=R, to flat ΛCDM using Pantheon+SH0ES supernovae (PS), DES-SN5YR supernovae, and DESI and eBOSS BAO data. The NMC model is initialized with Planck ΛCDM parameters at high redshift, leaving Rn (and, for PS, MB) as the fitted parameter; n is fixed to 4, 6, or 10. The authors compute χ2, AIC, and BIC model-comparison statistics and report best-fit H0 and Ωm values. They conclude that current cosmological data suggest a preference for the NMC theory, with 'moderate to strong evidence' for all dataset combinations, while acknowledging an unresolved BAO-related tension.

Significance. If the statistical claim were supported, the paper would be significant: a one-parameter extension of GR would simultaneously remove the cosmological constant, reproduce SNIa distances, and relax the Hubble tension. The work is also commendably transparent: it uses recent public datasets, calibrates the ΛCDM baseline against published Pantheon+ results, quotes convergence criteria, and provides parameter tables. However, the headline conclusion is not supported by the paper's own Table 2, and the evidence for the NMC model is at most weak-to-moderate and only for the post-hoc selected n=6 variant. The core value lies in the parameter constraints and in demonstrating that n=10 is strongly ruled out, which gives the framework falsifiable content.

major comments (4)

[Section 4.2 and Section 5, Table 2] The abstract and conclusions state that there is 'moderate to strong evidence' for a preference of the nonminimally coupled theory 'for all dataset combinations.' Table 2 contradicts this. For n=4, the NMC model is disfavoured by both AIC and BIC for PS+DESI (ΔAIC=12.56, ΔBIC=7.12) and PS+eBOSS (ΔAIC=16.67, ΔBIC=11.23); for n=10, it is strongly disfavoured in every combination (ΔBIC > 9). Only the n=6 variant is preferred, with ΔBIC values between -1.25 and -8.84 and mixed ΔAIC signs (+4.69, -3.40, -2.39, +0.13, -0.18). The data therefore support, at most, a weak-to-moderate preference for the n=6 variant, not for the nonminimally coupled theory as a whole, and not with consistent evidence from both information criteria.
[Table 2 versus Table 1] Table 2 lists for the PS+eBOSS rows the same χ2 values as the PS+DESI rows for n=4 (1575.92) and n=10 (1581.60), but Table 1 gives χ2=1574.26 and χ2=1570.45 for those models, respectively. The reported ΔAIC and ΔBIC values for these two rows are therefore internally inconsistent with Table 1. For example, using Table 1, the PS+eBOSS n=10 χ2 difference from ΛCDM is 13.20, which would give ΔBIC ≈ 5.76 rather than the reported 16.91. Since these rows are used to argue that n=4 is disfavoured when BAO data are added, the corrected values must be recomputed and may materially weaken that specific conclusion.
[Section 4.2, choice of n=6] The authors compare n=4, 6, and 10 and then single out n=6 as the preferred representative of the NMC model, but n is not included in the model-comparison statistics. No trials factor, prior over integer n, or marginalization over n is applied. This makes the reported evidence for the NMC framework a post-hoc selection from three exponents. A proper treatment of this model-selection step is required before the 'preference for the NMC theory' conclusion can be drawn; the raw ΔBIC values for n=6 overstate the evidence for the framework as a whole.
[Section 2.3, Section 3.3, Section 4.2] The NMC model is initialized with Planck ΛCDM values H*0=67.4 km/s/Mpc and Ω*m=0.315, and the Planck-calibrated sound horizon rd=147.46 Mpc is used as an exact input. The quoted uncertainties on these anchors (δH0=0.5 km/s/Mpc, δΩm=0.007, δrd=0.28 Mpc) are never propagated into the analysis. This is load-bearing for the BIC comparison because the NMC model is assigned k=1 while the anchors effectively carry information that should either be marginalized over or included as informative priors, both of which would reduce the parsimony advantage that drives much of the reported BIC preference. The paper should also demonstrate quantitatively that f2(R) is negligible at recombination for the fitted Rn values, for example by evaluating (Rn/R)^n at z≈1100, since the claimed consistency with Planck rests on this decoupling assumption.

minor comments (4)

[Throughout] There are several typographical issues, including 'E ffectively' at the start of Section 4 and 'overperforming' in Section 4.2; these should be corrected.
[Section 4.2] The sentence 'this model’s fit to the PS sample is weakly disfavoured ... when considering their respective AIC values' understates the sign inconsistency in Table 2: for PS, n=6 has ΔAIC=+4.69, which is a weak-to-moderate disfavour by the cited criteria, while the BIC gives no clear preference; the text should be more precise about this mixed evidence.
[Section 3] The paper does not state the number of data points N used in each BIC calculation. Since the BIC penalty scales with ln N and the PS and DESYR5 samples have different sizes, listing N for each dataset combination would improve reproducibility.
[Section 4.1] The comparison of the NMC model with the DESI BAO constraints in Ref. [31] could be made more explicit, since that reference also studies a nonminimally coupled gravity model against DESI data and would help contextualize the present results.

Circularity Check

2 steps flagged · score 4.0 of 10

CMB match and H0 prediction reduce to inputs, but the central Lambda-CDM vs NMC fit comparison is independent.

self definitional [Section 2.3, Eq. (9); abstract and Section 5 conclusions]
"As the theory decouples matter and curvature at high redshifts, we can assume that the Universe is governed by GR in the distant past, namely around the CMB epoch. This means that the theory behind early-time measurements, such as those carried out by the Planck experiment, is precisely the same as predicted by the standard ΛCDM model. ... We then write the decoupled initial conditions of the cosmological evolution as [30] H2(zi) = H∗2 0 Ω∗ m(1 + zi)3, R = ˜ρ(zi) = 3H∗2 0 Ω∗ m(1 + zi)3, (9)"

The abstract credits the model with 'matching early-time observations from the cosmic microwave background,' but Eq. (9) sets H2(zi) and R(zi) exactly equal to Planck-ΛCDM extrapolated values, and Section 2.3 states that this is because the NMC theory 'decouples' at high redshift. CMB agreement is therefore an input assumption, not an output of the fit: any model evolved from these anchors inherits Planck-ΛCDM early-time behaviour by construction. The later claim of 'fixing the tension between conclusions drawn from supernovae and CMB data' leans on this imposed match rather than on an independent CMB prediction.
fitted input called prediction [Section 3.1, Eq. (17); Section 5 conclusions]
"This allows us to constrain both MB and Rn (which directly determines H0 given the assumptions discussed in Section 2) for each NMC model ... We found that two of these models fit the Cepheid-calibrated Pantheon+ SNIa sample at a level superior to ΛCDM, predicting H0 values within error of the standard model and the model-independent cosmographic approach from the SH0ES collaboration [15], thus effectively fixing the tension between conclusions drawn from supernovae and CMB data."

Rn is the one free parameter of the NMC model, and Eq. (10) makes H0 = H(z=0) a deterministic function of Rn. The Pantheon+ likelihood (Eq. 17) includes SH0ES Cepheid host distances that calibrate the SNIa absolute magnitude and thus the Hubble scale; fitting Rn to that likelihood is statistically equivalent to fitting H0. The conclusions then describe the resulting H0 as a 'prediction' that effectively fixes the Hubble tension, but this value is forced by the Cepheid-anchored data and the one-to-one Rn-to-H0 mapping, not independently derived. The ΔAIC/ΔBIC comparison with ΛCDM remains a genuine fit comparison and is not circular.

full rationale

The central model-selection claim does not reduce to its inputs: both ΛCDM and the NMC model are fitted to the same SNIa and BAO likelihoods, and the ΔAIC/ΔBIC values in Table 2 are genuine comparisons of independent fits. No load-bearing self-citation chain or imported uniqueness theorem is present; Ref. [30] supplies the numerical method and the decoupling assumption, but the latter is stated explicitly in Section 2.3 and is physically motivated by the inverse power-law form of f2(R). Two supporting claims are circular as detailed above: the CMB 'match' is imposed via Eq. (9), and the H0 'prediction' is a re-description of the Rn fit to Cepheid-calibrated data. The advertised 'moderate to strong evidence ... for all dataset combinations' is not supported by Table 2, where only n=6 is preferred and with mixed AIC signs, while n=4 and n=10 are often disfavoured; however, that is an evidence-versus-conclusion mismatch rather than a circularity. Score 4 reflects partial circularity in supporting claims while the central comparison retains independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on four main assumptions: the flat FLRW background, the choice Lm = -rho (which is essential for the late-time effect), the inverse power-law form of f2(R), and the decoupling of the NMC at high redshift so that Planck-Lambda-CDM inputs can be used. The last two are model-specific and carry the most risk. The only fitted parameters are Rn and, for the Pantheon+ fits, MB; n is a hand-chosen discrete index.

free parameters (3)

Rn (scale in f2(R)) = 3.11 to 4.54 x 10^-7 Mpc^-2 depending on n and dataset
The single dynamical parameter of the NMC model, fitted to supernova and BAO data in each run. It directly controls the strength of the nonminimal coupling at late times.
n (exponent in f2(R)) = Fixed at 4, 6, or 10
Chosen by hand among integer values; n=6 was selected as a 'middle ground' after a qualitative analysis of intermediate values. This discrete choice is not penalized as a fitted parameter in the model comparison.
MB (SNIa absolute magnitude) = -19.241 to -19.309 (PS fits only)
Fitted for the Pantheon+ SH0ES analysis; degenerate with H0 but anchored by Cepheid host distances. Not fitted for DES data, where it is marginalized over.

assumptions (5)

domain assumption Flat FLRW metric (Eq. 4) is the correct background for late-time cosmology.
Used throughout the derivation of the modified Friedmann and Raychaudhuri equations. Standard in cosmological model comparison.
ad hoc to paper The perfect fluid Lagrangian density is Lm = -rho, not Lm = p.
The choice is load-bearing because with Lm = p, the NMC modifications vanish for dust (p=0), removing the late-time effect entirely. The paper cites prior literature for this choice, but it remains an assumption that is not independently tested here.
ad hoc to paper The coupling function f2(R) = (Rn/R)^n with inverse power law is the correct functional form.
Chosen so that the coupling is negligible at early times (high R) and grows at late times (low R). This form is not derived from a deeper principle, and the paper explicitly frames it as one possible term in a series.
domain assumption Nonminimal coupling fully decouples at the CMB epoch, so Planck-Lambda-CDM initial conditions (Eq. 9) and rd from Ref. [52] are valid inputs.
This assumption lets the paper anchor the NMC model to Planck parameters and use the Lambda-CDM sound horizon for BAO predictions. If residual coupling at high z exists, all subsequent comparisons are biased.
standard math Radiation density is negligible for the redshift range of the numerical integration.
The initial redshift is chosen below matter-radiation equality, so neglecting radiation is a standard approximation. The paper states this explicitly.

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Cite this review

Pith. "Pith review of Is cosmological data suggesting a nonminimal coupling between matter and gravity?." pith.science (2026). https://pith.science/paper/OWGLRYJA

@misc{pith2026241209348,
  author       = {Pith},
  title        = {Pith review of: Is cosmological data suggesting a nonminimal coupling between matter and gravity?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OWGLRYJA}},
  note         = {Machine review of arXiv:2412.09348}
}

abstract

Theoretical predictions from a modified theory of gravity with a nonminimal coupling between matter and curvature are compared to data from recent cosmological surveys. We use type Ia supernovae data from the Pantheon+ sample and the recent 5-year Dark Energy Survey (DES) data release along with baryon acoustic oscillation measurements from the Dark Energy Spectroscopic Instrument (DESI) and extended Baryon Oscillation Spectroscopic Survey (eBOSS) to constrain the modified model's parameters and to compare its fit quality to the Flat-$\Lambda$CDM model. We find moderate to strong evidence for a preference of the nonminimally coupled theory over the current standard model for all dataset combinations. Although the modified model is shown to be capable of matching early-time observations from the cosmic microwave background and late-time supernovae data, we find that there is still some incoherence with respect to the conclusions drawn from baryon acoustic oscillation observations.

Figures

Figures reproduced from arXiv: 2412.09348 by the authors.

**Figure 1.** Posterior distributions for the n = 4 NMC model fit to the Pantheon+SH0ES Cepheid-calibrated distance moduli. The Rn axis is in units of 10−7 Mpc−2 for simplicity. We show the 1σ and 2σ regions in the 2D posterior and the 1σ region in the marginalised 1D posteriors. 2.4. Numerical Method The methodology of the numerical evolution of the cosmological parameters is described in detail in Ref. [30]. Never3 [PITH_FU… view at source ↗

**Figure 2.** Posterior distributions of H0 for the n = 4 NMC model fit to different dataset combinations. 4.2. Fit quality For each model and dataset, we present the calculated values for χ 2 and two criteria were selected to compare the quality of fit of each model. The first of these is the Akaike Information Criterion (AIC), defined by AIC = 2k − 2 ln L max , (25) where k is the number of fitted parameters in the model and Lm… view at source ↗

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Forward citations

Cited by 3 Pith papers

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