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Alleviating the $H_0$ tension in Rastall gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Rastall gravity with a mildly closed universe lowers the Hubble-constant discrepancy with SH0ES from about 4σ to about 2σ.

desk verdict A strong MCMC effort undermined by a fatal algebraic error: the R-LambdaCDM model actually fitted is not Rastall gravity as defined by Eq. (10). read the letter →

arxiv 2508.20129 v2 pith:WAPGXAG3 submitted 2025-08-26 gr-qc

R. Mohebi , Kh. Saaidi T. Golanbari , K. Karami This is my paper
classification gr-qc MSC 83F0583D05 PACS 98.80.Es04.50.Kd
keywords HubbletensionRastallgravityΛCDMspatialcurvatureH0MCMCparameterestimationfσ8growthdatacosmologicalmodelselection
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 sets out to show that extending ΛCDM to Rastall gravity—a modified gravity in which the energy-momentum tensor is not conserved and a single new parameter ε measures the matter–geometry coupling—can soften the long-standing Hubble-constant tension. Fitting the model to three dataset combinations that mix supernovae, H(z), BAO, BBN, fσ8 growth data, and (in two cases) CMB distance priors, the authors find that the closed-geometry version returns H0 ≈ 68.7–69.7 km s−1 Mpc−1. That places the disagreement with the SH0ES local value at 1.97σ–2.18σ, roughly half the 4.17σ tension of flat ΛCDM, while remaining within about 1.8σ of Planck. The paper also reports that the σ8 clustering tension is not resolved, and model-selection criteria give mixed support: AIC mildly favors the closed Rastall model in some datasets, while BIC still prefers the simpler ΛCDM.

What carries the argument

The load-bearing object is the Rastall-modified Friedmann equation (Eq. 20), built from a field equation in which the covariant divergence of the energy-momentum tensor is proportional to the gradient of the Ricci scalar. Its work is to change the background expansion: each density parameter is rescaled by ε-dependent factors (Eqs. 21–23), and matter density scales as (1+z)^{3/[1−ε(1+6ε)]} rather than (1+z)^3. When combined with curvature Ω_k0 and fit via MCMC to the late-time and CMB-distance-prior datasets, this modified expansion history is what shifts H0 upward relative to the flat ΛCDM value and produces the reported 2σ tension reduction.

What would settle it

Check the algebra: substituting 8πG=1 into Eq. (10) gives 6ε κ² − (4ε+1)κ + 1 = 0, whereas the paper's Eq. (11) is 12ε²κ² + (2ε−1)κ + 1 = 0. Re-running the MCMC with the correct quadratic—or, equivalently, fitting the closed R-ΛCDM with the full Planck CMB likelihood instead of distance priors—would show whether the ~2σ H0 reduction survives.

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Extended reading notes

Core claim

The paper's central claim is that a minimally modified gravity theory can absorb part of the Hubble tension through the combination of a non-conserved energy-momentum tensor and negative spatial curvature. In the Rastall ΛCDM model, the parameter ε rescales the effective densities of matter, radiation, and the cosmological constant and changes how matter density scales with redshift. The MCMC fits yield a small negative ε (≈ −0.008) and a mildly closed geometry (Ω_k0 ≈ −0.024) for the datasets that include CMB priors, with best-fit H0 around 69 km s−1 Mpc−1. The paper argues that this is the most balanced compromise among the geometries tested: it reduces the SH0ES discrepancy to about 2σ wh

Load-bearing premise

The load-bearing premise is that the paper's derivation of the Rastall gravitational coupling is correct; if the algebraic relation between the coupling and the Rastall parameter is wrong, the fitted model is not actually Rastall gravity.

Editorial extensions

If this is right

  • If the closed R-ΛCDM result is correct, a mildly curved universe with a small matter–geometry coupling can reconcile CMB-based and distance-ladder H0 estimates without invoking early dark energy or extra relativistic species.
  • The model places H0 in the narrow band 68.7–69.7 km s−1 Mpc−1 across dataset choices that include or omit CMB distance priors, implying the reduction is not driven only by the late-time dataset.
  • The σ8 tension remains at roughly the same level, so the same mechanism that eases H0 does not solve the structure-growth discrepancy; future growth measurements would need to detect a genuinely lower σ8.
  • Model selection is split: AIC supports the closed R-ΛCDM over closed ΛCDM for datasets with CMB priors, while BIC still penalizes the extra Rastall parameter, so the model is a viable but not decisively favored alternative.

Reading between the lines

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

  • The paper treats ε and Ω_k0 as independent, but its D2 fit (no CMB) drives both ε and Ω_k0 to larger negative values simultaneously; a natural next step is to check whether ε is partly absorbing curvature information, using independent curvature measurements such as BAO angular-diameter distances at z≈1.
  • Because Eq. (11) does not follow algebraically from Eq. (10) when 8πG=1, the numerical constraints are conditional on that relation; re-deriving the modified Friedmann equation from the corrected coupling could change the inferred H0.
  • The closed R-ΛCDM model predicts a definite spatial curvature Ω_k0 ≈ −0.02; a future measurement that bounds curvature closer to zero at high precision would directly test the geometry that carries the tension reduction.
  • If the model is right, it suggests that phenomenological matter–geometry couplings are a viable alternative to changing the early Universe; this could be tested by looking for small deviations from geodesic motion or non-conservation in high-precision solar-system or binary-pulsar experiments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper constrains ΛCDM and Rastall-ΛCDM (R-ΛCDM) in flat, open, and closed spatial geometries using MCMC over combined datasets D1–D4 (BBN, H(z), Pantheon SNe, BAO, Planck CMB distance priors, and fσ8). For flat ΛCDM it reproduces the known Planck–SH0ES tension; for closed R-ΛCDM it reports H0 ≈ 68.7–69.7 km/s/Mpc and a reduced SH0ES tension of about 1.97–2.29σ across datasets, with σ8 tension unresolved. The central theoretical input is a modified Friedmann equation for Rastall gravity derived from a κ(ε) relation, and this derivation is algebraically flawed.

Significance. If valid, a modified-gravity model in which spatial curvature and a small Rastall parameter reduce the H0 tension to ~2σ would be of interest to the cosmology community. The paper's strengths include a broad compilation of cosmological datasets, transparent parameter tables, and explicit AIC/BIC model comparison. However, the central claim is not supported by the derivation as written: Eq. (11) does not follow from Eq. (10), and all R-ΛCDM constraints inherit this error. In addition, the fσ8 analysis uses unmodified ΛCDM perturbation equations, so the growth-rate constraints are not self-consistent. These are load-bearing problems, not presentation issues, and they affect all of the reported R-ΛCDM results.

major comments (3)
  1. [II.A, Eqs. (10)-(11)] Eq. (11) is not the algebraic consequence of Eq. (10). With 8πG=1, cross-multiplying (10) gives κ(1−6εκ)=1−4εκ, i.e. 6εκ²−(1+4ε)κ+1=0, not 12ε²κ²+(2ε−1)κ+1=0. The two disagree already at O(ε²); for ε=0.1, the quadratic from Eq. (10) has discriminant 1−16ε+16ε²<0 while Eq. (11) admits real κ. Therefore the interval (12), the expansion κ≃1+2ϵ, and Eqs. (13)-(15) are not consequences of Eq. (10). Since Eq. (20) and all R-ΛCDM constraints in Tables VIII-XII depend on this κ(ε), the headline H0-tension reduction is unsupported. The model being constrained is not the model defined by Eq. (10), unless Eq. (11) is taken as an independent definition, in which case the connection to Rastall gravity is unsubstantiated.
  2. [III.F and V.B (Tables VIII-XII)] The fσ8 likelihood (52) is evaluated using the coupled perturbation equations (55)-(56), whose coefficients A_m, A_Λ, B_Λ are the standard ΛCDM expressions; no Rastall ϵ corrections appear. Since R-ΛCDM modifies the background expansion, the matter conservation equation (17), and the gravitational coupling, the linear growth equations must be re-derived from the modified field equations (9)/(13). Using unmodified ΛCDM perturbation theory makes the σ8 and fσ8 constraints, and hence the D1 and D2 H0 posteriors that include fσ8, not self-consistent. A derivation or explicit justification that Rastall perturbations reduce to ΛCDM at the relevant order is required.
  3. [III.D and D3 (Eqs. (42)-(44), Table XII)] The CMB distance priors (R, la, Ωb h²) and the covariance (44) are adopted from Planck 2018 as ΛCDM-derived values. In R-ΛCDM the sound horizon, recombination redshift, and expansion history differ, so using these fixed priors in the χ²_CMB term for R-ΛCDM is not justified without recomputation or marginalization over the model's early-universe parameters. The D3 result in Table XII, which contributes to the claimed 2.18σ SH0ES tension, therefore inherits a ΛCDM assumption. This should be discussed and, if possible, the priors recomputed in R-ΛCDM.
minor comments (4)
  1. [Abstract and Section VI] The abstract and conclusions state the closed R-ΛCDM SH0ES tension is 1.97σ–2.18σ, but Section V.B.3 and Table XII report 2.29σ for D1. Please reconcile the headline numbers throughout the paper.
  2. [II.A, Eq. (16)] The step from Eq. (15) to Eq. (16) is not shown; the origin of the 8ϵ²(1−3ω_i) term and the rescaling (19) should be made explicit.
  3. [Section VI] The conclusions mention "CMB data from Planck 2018 and ACT 2025" as used in the analysis, but the dataset definitions in Section IV use only Planck distance priors and the D4 Planck table; ACT data are not part of D1-D3. Please correct this statement.
  4. [Throughout] There are numerous formatting and typographical issues (e.g., 'RΛCDM' vs 'R-ΛCDM', missing parentheses around the exponent in Eq. (20), inconsistent use of H0 symbols) that should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the H0 constraints and SH0ES tension are fitted results against external data, not predictions that reduce to their inputs.

full rationale

I walked the derivation chain from Section II.A through Section V. The paper constructs R-LambdaCDM from Rastall's field equations, then fits the model parameters to external datasets (BBN, H(z), SNe Ia, BAO, CMB distance priors, f sigma8) via MCMC, and finally compares the inferred H0 posterior to the external SH0ES value. No step in this chain equates an input with an output by construction: H0 is a free parameter, not a derived prediction, and the SH0ES value 73.2 ± 1.3 is not included in any of the likelihoods. The CMB distance priors in Eq. (43) are external Planck 2018 compressed values, and the model is fit to them; this makes the quoted agreement with Planck in D3 non-independent, but the headline SH0ES tension, especially for D2 (which has no CMB data), remains a comparison with an external reference. The Rastall field equations are attributed to the literature and are not justified by a uniqueness theorem from the present authors; one shared-author reference in the citation list is not load-bearing. There is therefore no reduction of a claimed result to its own inputs. I note separately, as a correctness matter rather than a circularity, that Eq. (11) does not obviously follow algebraically from Eq. (10): with 8piG = 1, cross-multiplication of Eq. (10) gives 6*epsilon*kappa^2 - (1 + 4*epsilon)*kappa + 1 = 0, not the quadratic displayed in Eq. (11). This is a derivation/consistency concern that affects the model's validity, but it is not a circularity because the paper's results are not built by renaming an input as a prediction.

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

The model depends on the Rastall coupling epsilon and curvature as free parameters, on an algebraically unsupported derivation step, and on transferring LambdaCDM-derived CMB priors and growth equations into the modified theory.

free parameters (4)
  • epsilon (Rastall parameter) = -0.00202 to -0.115 (flat), 0.0038 to -0.125 (open), -0.0076 to -0.102 (closed)
    Controls the matter-geometry coupling in Rastall gravity; fit values depend strongly on dataset and drive the H0 shift.
  • Omega_k0 (curvature density) = open: 0.0048 to 0.066; closed: -0.0018 to -0.083
    Spatial curvature is fit per dataset and geometry; the closed-universe preference is central to the main claim.
  • H0 = about 68.3 to 69.7 km/s/Mpc for R-LambdaCDM depending on dataset
    The Hubble constant is a free parameter in the fit; the paper's central claim is about its value relative to external measurements.
  • sigma8 = about 0.63 to 0.69
    Amplitude of matter fluctuations, fitted but modeled with unmodified LambdaCDM growth equations.
assumptions (4)
  • domain assumption Rastall field equations with non-conserved energy-momentum tensor (Eq. 9)
    The whole model rests on accepting Rastall's modification of GR; this is not derived within the paper.
  • ad hoc to paper Coupling relation Eq. (10) and its solution Eq. (11)
    The paper presents Eq. (11) as following from Eq. (10), but algebraically it does not; the modified Friedmann equation depends on this.
  • domain assumption Standard LambdaCDM perturbation equations (55)-(56) apply to R-LambdaCDM growth
    The f sigma8 predictions for R-LambdaCDM use equations without epsilon, with no justification.
  • domain assumption Planck CMB distance priors (R, l_a, Omega_b h^2) from Eqs. (43)-(44) are valid constraints in R-LambdaCDM
    These priors were calibrated within LambdaCDM; using them to constrain a modified expansion model is an unverified transfer.

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Cite this review

Pith. "Pith review of Alleviating the $H_0$ tension in Rastall gravity." pith.science (2026). https://pith.science/paper/WAPGXAG3

@misc{pith2026250820129,
  author       = {Pith},
  title        = {Pith review of: Alleviating the $H_0$ tension in Rastall gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WAPGXAG3}},
  note         = {Machine review of arXiv:2508.20129}
}
abstract

The persistent discrepancy between local determinations of the Hubble constant $H_0$ and the Planck 2018 value ($67.4 \pm 0.5~{\rm km\,s^{-1}\,Mpc^{-1}}$) within $\Lambda$CDM remains a central challenge in precision cosmology. We investigate the Hubble tension in $\Lambda$CDM and its Rastall extension (R-$\Lambda$CDM) for flat, open, and closed geometries. We analyze three primary dataset combinations: D$_1$ (late-time probes: SN + $H(z)$ + $f\sigma_8$), D$_2$ (late-time probes combined with DESI DR2 BAO and BBN), and D$_3$ (late-time probes combined with BAO and Planck 2018 CMB distance priors). Parameters are constrained via Markov Chain Monte Carlo sampling, and tensions with SH0ES ($73.2 \pm 1.3~{\rm km\,s^{-1}\,Mpc^{-1}}$) and Planck are expressed in units of the combined uncertainty. In addition, we include a Planck-only configuration (D$_4$) as a reference baseline to isolate early-Universe constraints on $H_0$. Within $\Lambda$CDM, D$_1$ and D$_2$ yield $H_0 \simeq 70.75$-$71.43~{\rm km\,s^{-1}\,Mpc^{-1}}$, reducing the SH0ES discrepancy to $1.11\sigma$--$1.63\sigma$ while maintaining a $3.62\sigma$-$4.23\sigma$ tension with Planck. Including CMB distance priors (D$_3$) shifts the result to $H_0 \simeq 67.18$--$67.55~{\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.09\sigma$-$0.38\sigma$ but increasing the SH0ES discrepancy to $4.25\sigma$-$4.51\sigma$.

Figures

Figures reproduced from arXiv: 2508.20129 by the authors.

Figure 1
Figure 1. FIG. 1: One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

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    The constraints on the cosmological parameters of the six-parameter flat ΛCDM model derived from these datasets are summarized in Table IV and illustrated in Figure 1

    Flat ΛCDM model We perform a comprehensive comparison of the datasets D 1, D 2, D 3, and D 4 within the framework of the flat ΛCDM model. The constraints on the cosmological parameters of the six-parameter flat ΛCDM model derived from these datasets are summarized in Table IV and illustrated in Figure 1. This table reports the 68% confidence intervals for...

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    The corresponding results are presented in Tables V and VII, and illustrated in Figure 2

    Open ΛCDM model The open ΛCDM model involves eight free parameters, constrained using different datasets. The corresponding results are presented in Tables V and VII, and illustrated in Figure 2. The results indicate that the Hubble constant H0 in the open ΛCDM model, for datasets D1, D2, and D3, lies within the range of approximately 68 .37 to 68 .7 km s...

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    The Hubble parameter H0 lies in the range of approximately 67 to 69.3 km s −1 Mpc−1

    Closed ΛCDM model Cosmological parameter constraints for the seven parameter of closed ΛCDM model, derived from various datasets, are summarized in Tables VI and VII, and are shown in Figure 3. The Hubble parameter H0 lies in the range of approximately 67 to 69.3 km s −1 Mpc−1. These values exhibit small deviations from the Planck 2018 result (67.4 ± 0.5)...

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    The differences are expressed in terms of the statistical deviation ( σ) between dataset D 1 and the other combinations

    Assessment of geometrical models based on parameter consistency: The ΛCDM model A comparative examination of the cosmological parameter differences between datasets D1, D2, and D3 in Table VII allows us to evaluate the relative internal consistency of the flat, open, and closed ΛCDM scenarios. The differences are expressed in terms of the statistical devi...

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    R- ΛCDM model in a flat universe In the flat R-ΛCDM framework, the deviation parameter ϵ, commonly referred to as the Rastall parameter, in- troduces a modification to the standard ΛCDM dynamics. This modification originates from the non-conservation of the matter energy-momentum tensor due to a non-minimal coupling between matter and geometry (for furthe...

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    The Hubble constant H0 is estimated to be 68 .92 ± 0.38, 68.8+1.2 −0.86, and 68 .3 ± 1.2 km s −1 Mpc−1 for datasets D 1, D2, and D3, respectively

    R- ΛCDM model in an open universe The cosmological parameter estimates for the eight-parameter open R-ΛCDM model, obtained using datasets D 1, D2, and D 3, are summarized in Table X and illustrated in Figure 5. The Hubble constant H0 is estimated to be 68 .92 ± 0.38, 68.8+1.2 −0.86, and 68 .3 ± 1.2 km s −1 Mpc−1 for datasets D 1, D2, and D3, respectively....

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    These results are derived from three independent observational datasets (D1, D2, and D 3) and are also illustrated in Figure 6

    R- ΛCDM model in a closed universe Table XII summarizes the cosmological parameter estimates of the eight parameter R-ΛCDM model assuming a closed universe geometry (negative curvature). These results are derived from three independent observational datasets (D1, D2, and D 3) and are also illustrated in Figure 6. The estimated values of the Hubble constan...

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.