Bayesian and Machine-Learning Analyses of Nonminimal $f(Q)$ Gravity and $H_0$ Tension

Mridul Patel; Simran Arora

REVIEW 5 major objections 4 minor 80 references

Bayesian and Machine-Learning Analyses of Nonminimal $f(Q)$ Gravity and $H_0$ Tension

T0 review · 5 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A nonminimally coupled f(Q) gravity model, fitted to late-time data, returns H0 ≈ 68 km/s/Mpc — between Planck and SH0ES — and the authors argue this partially alleviates the Hubble tension.

desk verdict A competent but overclaimed constraint paper: the f(Q) model fits late-time data about as well as ΛCDM, the BIC disfavors it, and the abstract promises CMB constraints the analysis never uses. read the letter →

arxiv 2511.06332 v2 pith:SXIZU3EK submitted 2025-11-09 gr-qc

Simran Arora , Mridul Patel This is my paper · ORCID

classification gr-qc MSC 83D0583F05 PACS 04.50.Kd98.80.Es95.36.+x

keywords f(Q)gravitynonminimalmattercouplingHubbletensionH0DESIBAODR2TypeIasupernovaeMCMCmachinelearning

topics The Hubble Tension

open problems The Hubble Tension

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

The paper attempts to show that a nonminimal coupling between matter and the nonmetricity scalar Q in symmetric teleparallel gravity can shift the late-time Hubble constant to an intermediate value near 68 km/s/Mpc, reducing the gap between early- and late-universe measurements without degrading the fit to cosmic chronometers, DESI BAO DR2, and Type Ia supernovae. A specific model is constructed with f1(Q) = -Q + αQ² and f2(Q) = 1 + βQ, its background equations are derived, and the parameter space is constrained with MCMC alongside machine-learning reconstructions of H(z). The authors read the result as a partial alleviation of the H0 tension, while cautioning that the model is only fitted to late-time probes, so the comparison with Planck rests on external input rather than a joint early-plus-late fit.

What carries the argument

The central object is the nonminimal coupling f2(Q)L_m in the action S = ∫ d⁴x √−g [½ f1(Q) + f2(Q)L_m], which directly couples the matter Lagrangian to the nonmetricity scalar Q (Q = 6H² in a flat FLRW background). The authors choose f1(Q) = -Q + αQ² and f2(Q) = 1 + βQ, leading to a modified Friedmann equation and an energy density ρ = (3αQ² - Q)/(2(βQ - 1)); the resulting nonlinear Hubble equation is solved numerically and sampled with MCMC against CC + DESI BAO DR2 + SNe combinations. The machine-learning section (SVR with RBF kernel, random forest, linear regression) reconstructs H(z) from the best-fit curves and is used to corroborate the model's predictive performance.

What would settle it

Run the same MCMC with Planck CMB likelihoods (e.g., Planck 2018 TT,TE,EE+lowE) jointly with the late-time data; if the posterior for H0 shifts to about 67–68 km/s/Mpc or if the model's minimum χ² worsens significantly relative to ΛCDM, the claimed partial alleviation is not robust.

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Extended reading notes

Core claim

On the paper's own terms: in the symmetric teleparallel framework, a nonminimal matter–geometry coupling of the form f2(Q)L_m produces modified Friedmann equations; for the power-law choice f1(Q)=-Q+αQ² and f2(Q)=1+βQ, the model fits all late-time probes with reduced chi-squared near unity and yields H0 in the range 67.7–69.0 km/s/Mpc across four data combinations. The authors interpret this as a partial alleviation of the Hubble tension, with inferred H0 sitting between the Planck value and the SH0ES distance-ladder value, while the deceleration parameter, effective equation of state, and Om(z) diagnostic remain close to ΛCDM.

Load-bearing premise

The claim of partial alleviation rests on comparing a late-time-only fit's H0 to the Planck value from outside the fit; if early-universe data were included as an actual constraint, the same parameters would have to satisfy both, and the intermediate H0 could disappear.

Editorial extensions

If this is right

If the claim holds, late-time cosmic acceleration can be accommodated without a cosmological constant, with the nonminimal coupling supplying the extra degrees of freedom.
The model predicts H0 ≈ 68 km/s/Mpc, so it points to a mild resolution of the Hubble tension rather than a full one; future late-time datasets should keep H0 in this range if the model is correct.
The parameters α and β are constrained to small values, so deviations from general relativity are tiny at early times; the model effectively reduces to ΛCDM at high redshift, which is why BAO and CC fits remain good.
The stability of rd ≈ 147 Mpc across all dataset combinations suggests the model does not disturb the sound-horizon scale, keeping consistency with CMB-based determinations of the baryon drag scale.

Reading between the lines

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

Because the fit deliberately excludes early-universe likelihoods (despite the abstract mentioning CMB), the 'partial alleviation' is a comparison, not a joint constraint; a full CMB + late-time fit could shift α and β and pull H0 back to the Planck value, potentially erasing the claimed alleviation.
The machine-learning analysis reconstructs H(z) from already-fitted theoretical curves, so its high R² scores largely reflect interpolation of model output rather than independent evidence about f(Q) gravity; a stronger test would train on raw data and predict out-of-sample redshifts.
A natural next test is to compute the growth rate fσ8 or the ISW effect for this model, since nonminimal matter coupling generically modifies the continuity equation; the authors work in a gauge where the standard conservation law is recovered, so perturbation-level consistency deserves scrutiny.

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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

Honestly, this is a workmanlike data-analysis paper overclaiming its result. The model is not new — f1 = -Q + αQ², f2 = 1+βQ is already in the authors' earlier work — but the fit to DESI DR2 plus Pantheon+, DESY5, and Union3 is a legitimate exercise. The background equations check out, the MCMC setup is standard, and the parameter constraints are plausible on their own terms.

The core claim doesn't hold up. The f(Q) H0 values in Table I (67.7–69.0) are essentially the same as the ΛCDM fits to the same data (68.6–69.9). So the 'partial alleviation' is just the usual late-time H0, not a distinctive prediction. The ΔBIC values are all positive (+7 to +12), which is strong evidence against the extra parameters; calling the model 'promising' contradicts its own statistics. And the abstract says CMB is used, but Section IV explicitly restricts the analysis to late-time probes and Eq. (35) contains no CMB term. Planck enters only as an external comparison point.

The ML section is circular. Table II is titled 'Comparison of Machine Learning Models on Theoretical H(z)' — the algorithms are trained on the paper's own best-fit H(z), then presented as validating the model. That's not a check on the gravity theory.

There is a salvageable paper here: the fits are honest as fits, and the DESI DR2 constraints on this model are a potentially useful data point for the f(Q) community. But the H0-tension claim is not established, and the abstract must be corrected. I'd send it to peer review rather than desk reject — referee time could force the authors to fix the framing — but in its current form it should not be accepted.

Referee Report

5 major / 4 minor

Summary. The paper constructs a nonminimally coupled f(Q) gravity model with f1(Q) = -Q + αQ² and f2(Q) = 1 + βQ, derives the background Friedmann-like equations, and fits the model to cosmic chronometers, DESI BAO DR2, and three Type Ia supernova samples (Pantheon+, DESY5, Union3) using MCMC. It reports H0 ≈ 67.7–69.0 km s⁻¹ Mpc⁻¹ and interprets this as a partial alleviation of the H0 tension. The paper also applies linear regression, support vector regression, and random forest to 'theoretical H(z)' data. The central claims are that f(Q) gravity is promising for late-time cosmology and that it partially alleviates the H0 tension.

Significance. If the claimed partial alleviation of the H0 tension were robust, this would be a useful contribution to the modified-gravity literature. However, as presented the analysis is restricted to late-time probes (explicitly stated in Section IV), the derived H0 values are statistically indistinguishable from ΛCDM fits to the same data, and the reported ΔBIC values strongly disfavor the f(Q) model relative to ΛCDM. The machine-learning section is circular, training on the model's own predictions. The theoretical derivation also contains an apparent algebraic error in Eq. (17). The paper does provide a transparent MCMC setup and clear tables/figures, but the main interpretive claims are not supported by the evidence in the manuscript.

major comments (5)

[Section IV, Eq. (35)] The abstract lists CMB among the data probes, but Section IV states 'we restrict our analysis to late-time probes: CC, DESI BAO DR2, and Type Ia supernovae' and the total likelihood in Eq. (35) contains no CMB term. Planck H0 enters only through the post-hoc comparison in Fig. 6. Consequently α and β are not constrained by early-universe physics; a joint CMB fit could shift H0 and erase the claimed alleviation. The central H0-tension claim is therefore not supported by the analysis actually performed.
[Table I] The f(Q) H0 values (67.7–69.0 km s⁻¹ Mpc⁻¹) are statistically indistinguishable from the ΛCDM fits to the same data (68.6–69.9 km s⁻¹ Mpc⁻¹). The heat map (Fig. 6) shows 2.5–3.9σ tension with Planck. The purported 'partial alleviation' is thus the usual late-time H0 value, not a distinctive prediction of nonminimal f(Q). The abstract's claim of alleviation is not established relative to a ΛCDM baseline.
[Table I, Section V] The reported ΔBIC values (+6.9 to +12.2) are strong evidence against the f(Q) model relative to ΛCDM on the Kass–Raftery scale, and ΔAIC is positive for three of four data combinations. The text describes the model as providing 'a fit of comparable statistical quality to ΛCDM' and 'promising', which misrepresents the paper's own model-selection statistics. This is a load-bearing interpretational error.
[Section VI, Table II] Table II is explicitly titled 'Comparison of Machine Learning Models on Theoretical H(z)'. The ML models are trained on the best-fit f(Q) model's H(z) predictions, not on independent observational data. The near-perfect R²≈0.9998 for SVR (RBF) is therefore expected and provides no evidence for the model's predictive power. The ML section does not validate the gravity model and is largely circular.
[Eq. (17)] In the GR limit (f1=-Q, f2=1, F=-1), Eq. (17) reduces to \dot H = 6H² - p, whereas Eq. (23) gives \dot H = -(ρ+p)/2 (and GR requires \dot H = -1.5H² for pressureless matter). This indicates a sign/algebraic error in the printed field equation. The numerical analysis appears to use Eq. (23), so the results may be unaffected, but the theoretical derivation needs correction.

minor comments (4)

[Table I] The f(Q) H0 error for CC+DESI is listed as 69.0±0.027, which is likely a typo for 69.0±1.6. The table column headers are also misaligned.
[Section VI] Section heading reads 'MACHINE LEANING TECHNIQUES' instead of 'MACHINE LEARNING'.
[Section IV] The 'PP Data' bullet appears to be a formatting artifact; the text is missing a bullet point and runs into the following line.
[Abstract] The abstract lists CMB as a probe but the conclusion and analysis do not use CMB data; the abstract should be aligned with the late-time-only scope.

Circularity Check

2 steps flagged · score 6.0 of 10

Two self-referential loops: H0 is a fitted free parameter presented as a tension 'alleviation' without any CMB likelihood, and the ML section trains and tests on H(z) generated from the same best-fit model.

fitted input called prediction [Section IV (Eq. 35) and Section V (Table I, text after Table I)]
"In this work, we restrict our analysis to late-time probes: CC, DESI BAO DR2, and Type Ia supernovae (SNe). The total likelihood is constructed as −2 lnL=χ 2 CC +χ 2 BAO +χ 2 SNe,(35), and is used to constrain the model characterized by the free parameters{α, β, γ, Mb, H0, rd}. ... The Hubble constant lies in the rangeH0 ≃67–69 km s−1 Mpc−1, and intermediate between the CMB and local distance-ladder determinations, suggesting a mild amelioration of theH0 tension."

H0 is a free parameter of the late-time likelihood, not a prediction of the theory. The posterior H0 values in Table I are direct fit outputs of CC+BAO+SNe, with no CMB term in Eq. (35), despite the abstract saying CMB is used. The 'partial alleviation of the H0 tension' is therefore only a restatement that the fitted late-time H0 sits between Planck and SH0ES; it is not an independent test. Adding CMB constraints would jointly constrain α, β, H0 and rd and could erase the claimed intermediate value. The paper's central headline claim thus reduces to a redescription of a fitted parameter.
fitted input called prediction [Section VI, Table II and surrounding text; Fig. 7/8 captions]
"The observational datasets employed in this analysis correspond to the cosmological models outlined in Section IV. ... TABLE II: Comparison of Machine Learning Models on TheoreticalH(z). ... This indicates an exceptional capacity to capture the variance in the theoreticalH(z)data (obtained from each combination of data considered)."

The machine-learning models are trained and tested on H(z) values generated from the best-fit f(Q) model itself, as the table title 'On Theoretical H(z)' makes explicit. An 80/20 train/test split of these generated points measures how well a regressor interpolates a smooth theoretical curve, not whether the cosmological model predicts independent observations. The near-perfect R2 of SVR (RBF) is therefore a self-consistency check, not an external validation of f(Q) gravity or of the H0-tension claim. The ML 'predictive performance' reduces by construction to fitting the model's own output.

full rationale

Most of the gravitational derivation is self-contained: the action (5), field equations (12)-(13), and Friedmann equations (21)-(22) follow algebraically from the assumed f1(Q), f2(Q) forms, and the MCMC fit to CC/BAO/SNe is a standard likelihood analysis. The functional choice f1=-Q+αQ², f2=1+βQ is an explicit ansatz, and the citations, including the authors' own Ref. [55], are background rather than load-bearing; no uniqueness theorem is imported to force the result. However, the two headline validations are closed loops. First, H0 is a free parameter of the late-time likelihood in Eq. (35), while Section IV explicitly states 'we restrict our analysis to late-time probes'; the claimed 'partial alleviation of the H0 tension' is therefore a verbal repackaging of the fitted H0, not a theoretical prediction. The abstract says CMB is included, but no CMB term appears in Eq. (35), so the comparison with Planck is post hoc and could be undone by a joint early-universe fit. The positive ΔBIC values in Table I (+6.9 to +12.2) and the heat map's ~3σ residual tension with Planck further weaken the 'promising resolution' language. Second, the ML section trains on theoretical H(z) generated from the same best-fit model (Table II) and reports near-perfect R2 for SVR; this demonstrates interpolation, not independent predictive power. These two loops make the 'partial alleviation' and 'ML robustness' claims partially circular by construction, while the core field-equation derivation remains non-circular.

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

The central claim rests on six fitted parameters (α, β, γ, Mb, H0, rd). The model itself is not derived here but imported from Refs [53,55]; the only new in-theory input is the chosen power-law ansatz and the data. The prior on β fixes the sign of the matter coupling, and the late-time-only dataset selection determines the H0 result.

free parameters (6)

α = ≈ 0.105 ± 0.09 (CC+DESI+DESY)
Coefficient of Q² in f1(Q); prior U[-5,5]; fitted to all datasets; controls deviation from GR at background level.
β = ≈ -0.24 to -0.265 ± ~0.01
Coefficient of Q in f2(Q) (matter coupling); prior U[-1,0]; fitted; the prior fixes the sign of the coupling.
γ = ≈ 0.949–0.987
Barotropic index (p = (γ-1)ρ); prior U[0,2]; fitted; consistent with dust γ = 1.
Mb = -19.98 (PP data set)
Absolute magnitude of SNe Ia; nuisance parameter fitted to calibrate the distance modulus.
H0 = ≈ 67.7–69.0 km/s/Mpc
Present-day Hubble constant; fitted; the central claim about the H0 tension depends directly on this value.
rd = ≈ 147 Mpc
Sound horizon at drag epoch; treated as free parameter in BAO likelihood, helping absorb the H0-rd degeneracy.

assumptions (6)

domain assumption Coincident-gauge symmetric teleparallel geometry with Q = 6H² for flat FLRW.
Section III: the background equations use the coincident-gauge condition and the FLRW line element (15).
domain assumption Matter Lagrangian L_M = -ρ, which makes the matter continuity equation standard.
Section III, Eq. (18)→(19): the nonminimal coupling's effect on matter conservation vanishes under this choice.
ad hoc to paper Power-law ansatz f1(Q) = -Q + αQ², f2(Q) = 1 + βQ.
Eq. (28): chosen for tractability; not derived from a more fundamental principle.
domain assumption Spatially flat, homogeneous, isotropic FLRW background.
Eq. (15): the analysis is restricted to background cosmology; perturbations are not considered.
ad hoc to paper Prior β ∈ U[-1,0] restricts the sign of the nonminimal coupling.
Section IV: no physical justification is given for excluding positive β; this can bias the posterior.
domain assumption The public datasets and their covariance matrices are used as released.
Section IV: the analysis relies on the reliability of CC, DESI DR2 BAO, Pantheon+, DESY5, and Union3 data products.

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Cite this review

Pith. "Pith review of Bayesian and Machine-Learning Analyses of Nonminimal $f(Q)$ Gravity and $H_0$ Tension." pith.science (2026). https://pith.science/paper/SXIZU3EK

@misc{pith2026251106332,
  author       = {Pith},
  title        = {Pith review of: Bayesian and Machine-Learning Analyses of Nonminimal $f(Q)$ Gravity and $H_0$ Tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXIZU3EK}},
  note         = {Machine review of arXiv:2511.06332}
}

abstract

In this study, the cosmological implications of nonminimally coupled $f(Q)$ gravity are examined within the metric-affine formalism, in which the nonmetricity scalar $Q$ couples directly to the matter Lagrangian. Within the symmetric teleparallel framework, a representative $f(Q)$ model is constructed, and the corresponding background cosmological equations are derived. The analysis aims to test whether this geometric formulation yields more consistent realizations of nonminimal matter-geometry couplings. A comprehensive statistical MCMC analysis is performed using cosmic chronometers, DESI BAO DR2, and Type~Ia supernovae from the Pantheon+, DESY5, and Union3 samples and CMB. To complement the statistical study, we employ machine learning methods, such as linear regression, support vector regression (SVR), and random forest algorithms, to evaluate the predictive performance and robustness of the data. The results indicate that a partial alleviation of the $H_0$ tension can be achieved for a broad range of parameter choices. Nonetheless, $f(Q)$ gravity emerges as a promising and flexible framework for late-time cosmology, motivating further exploration of extended models consistent with all observations.

Figures

Figures reproduced from arXiv: 2511.06332 by the authors.

**Figure 3.** FIG. 3: Evolution of the deceleration parameter [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** FIG. 4: The panel displays the cosmological parameters plotted alongside DESI BAO data. The first plot shows the behavior of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5: Evolution of the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: Heat map of the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7: Comparison of theoretical predictions for the Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8: Performance comparison of different machine-learning models trained on the theoretical [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

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