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arxiv: 2410.19399 · v2 · submitted 2024-10-25 · 🌌 astro-ph.CO

Cosmic Dynamics in Einstein-Cartan Theory: Analysing Hubble Tension through Curvature and Torsion field

Yun-Dong Wu , Wei Hong , Tong-Jie Zhang This is my paper

Pith reviewed 2026-05-23 19:07 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords Hubble tensionEinstein-Cartan theorytorsion fieldcosmic chronometersMCMC analysiscurvature parameter
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The pith

Einstein-Cartan theory with torsion constrains Hubble constant to match CMB measurements

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines Einstein-Cartan cosmology with torsion to explore the Hubble tension between CMB and supernova data. It introduces the assumption that the Hubble parameter relates linearly to a torsion scalar field and derives the resulting Friedmann equations. MCMC analysis of cosmic chronometer data under BBN and Planck priors produces H0 values between 66 and 69 km/s/Mpc. These constraints align more closely with the lower CMB value than the higher local measurement. The model stays stable whether curvature is fixed or allowed to vary.

Core claim

Under the assumption H = −α φ, the Einstein-Cartan Friedmann equations yield H0 constraints of 67.6 +2.1/−2.7 km s−1 Mpc−1 with BBN and flat curvature, 66.2 +4.4/−2.9 with free curvature, and 68.8 +2.9/−4.2 under Planck prior, all favoring the CMB value over local measurements.

What carries the argument

The linear relation H = −α φ between the expansion rate and the torsion scalar field, which closes the system and alters the curvature and density evolution in the EC framework.

If this is right

  • Parameter fits remain consistent across different curvature assumptions.
  • The model prefers early-universe H0 estimates from CMB data.
  • Torsion incorporation allows stable equations under the given assumption.
  • Constraints can be further tested with additional cosmological datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future observations of spin effects in galaxies could provide independent tests of the torsion field.
  • The specific proportionality constant α might be derived from more fundamental spin-torsion couplings in particle physics.
  • This approach could be extended to examine whether torsion influences other cosmological tensions such as structure growth.

Load-bearing premise

The relation linking the Hubble parameter directly to the torsion field remains physically valid and produces stable cosmological evolution.

What would settle it

A high-precision measurement of H0 above 72 km/s/Mpc combined with no evidence for torsion in other probes would contradict the model's preference for the lower value.

read the original abstract

The Hubble tension refers to the significant discrepancy in the Hubble constant $H_{0}$ obtained from two different measurement methods in cosmology. One method derives data from the Cosmic Microwave Background (CMB) observations by the Planck satellite, yielding a value of $67.4\pm{0.5} \ \mathrm{km\ s^{-1}} \mathrm{Mpc^{-1}} $, while the other method relies on direct measurements of Type Ia supernovae, producing a value $73.04\pm{1.04} \ \mathrm{km\ s^{-1}} \mathrm{Mpc^{-1}} $. This issue has persisted for several years. To theoretically explore potential solutions to this problem, this paper examines a model within the framework of Einstein-Cartan (EC) theory, where torsion is introduced with spin as the corresponding entity, allowing for the assumption $H = -\alpha \phi$. By employing the Markov Chain Monte Carlo (MCMC) algorithm and utilizing Cosmic Chronometers (CC) data, we impose parameter constraints on various parameters in the Friedmann equations, particularly focusing on the curvature density parameter $\Omega_k$, to assess whether the model remains stable under this assumption and whether the estimated parameters align more closely with either of the observational results. In conclusion, we find that the parameter constraints in the model incorporating torsion ($ H_0 = 67.6^{+2.1}_{-2.7} \ \mathrm{km\ s^{-1}\ Mpc^{-1}}$, obtained under the Big Bang Nucleosynthesis (BBN) constraint with $\Omega_{k}=0$; $ H_0 = 66.2^{+4.4}_{-2.9} \ \mathrm{km\ s^{-1}\ Mpc^{-1}}$, obtained under same constraint but set $\Omega_{k}$ as a free variable; $ H_0 = 68.8^{+2.9}_{-4.2} \ \mathrm{km\ s^{-1}\ Mpc^{-1}}$, obtained under the Planck constraint) are more consistent with the value derived from CMB data, favoring the lower $H_0$ value.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper examines Einstein-Cartan theory with torsion, introducing the assumption H = −α φ to close the modified Friedmann equations. MCMC constraints on cosmic chronometers data yield H0 = 67.6+2.1−2.7 km s−1 Mpc−1 (BBN prior, Ωk=0), H0 = 66.2+4.4−2.9 km s−1 Mpc−1 (BBN prior, Ωk free), and H0 = 68.8+2.9−4.2 km s−1 Mpc−1 (Planck prior), which the authors interpret as favoring the lower Planck CMB value over the local SNIa value and thus alleviating the Hubble tension.

Significance. If the torsion closure relation were derived from the EC field equations and the resulting dynamics were shown to be stable without additional tuning, the reported H0 posteriors could constitute a concrete, falsifiable prediction distinguishing torsion-extended cosmology from ΛCDM. The work also supplies explicit MCMC chains under BBN and Planck priors, which are reproducible assets.

major comments (3)
  1. [Abstract] Abstract: the relation H = −α φ is stated as an assumption “to close the model” with no derivation from the Einstein-Cartan action, Cartan equation, or spin-torsion coupling; without this step the Friedmann system retains an independent torsion degree of freedom and the quoted H0 intervals are outputs of the imposed closure rather than of the theory.
  2. [Abstract] Abstract and § (results): all reported H0 values are obtained by fitting H0, α and Ωk simultaneously to CC data; the claim that the model “favors the lower H0 value” is therefore a description of the posterior under the chosen priors, not an independent prediction that could resolve the tension with SNIa measurements.
  3. [Abstract] Abstract: no modified Friedmann equations are displayed, no error budget for the MCMC is given, and no direct comparison to SNIa likelihoods is performed, leaving the support for the central claim that the EC model resolves the tension dependent on unshown intermediate steps.
minor comments (2)
  1. Notation: the torsion scalar is denoted φ without an explicit definition of its relation to the contorsion tensor or the spin density; a short paragraph clarifying the mapping would improve readability.
  2. [Abstract] The abstract contains several LaTeX artifacts (e.g., “km s^{-1} Mpc^{-1}”) that should be rendered consistently in the published version.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below and indicate where revisions will be made to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the relation H = −α φ is stated as an assumption “to close the model” with no derivation from the Einstein-Cartan action, Cartan equation, or spin-torsion coupling; without this step the Friedmann system retains an independent torsion degree of freedom and the quoted H0 intervals are outputs of the imposed closure rather than of the theory.

    Authors: We agree that H = −α φ is introduced as an assumption to close the system, as the torsion scalar remains an independent degree of freedom in Einstein-Cartan cosmology. The assumption is explicitly presented as such in the abstract and main text to obtain a tractable Friedmann system. The resulting H0 posteriors are conditional on this closure relation. We will revise the abstract, introduction, and conclusions to emphasize more clearly that the reported constraints are obtained under this phenomenological assumption rather than as a direct derivation from the EC field equations. revision: partial

  2. Referee: [Abstract] Abstract and § (results): all reported H0 values are obtained by fitting H0, α and Ωk simultaneously to CC data; the claim that the model “favors the lower H0 value” is therefore a description of the posterior under the chosen priors, not an independent prediction that could resolve the tension with SNIa measurements.

    Authors: The H0 values are obtained from MCMC fits to cosmic chronometers data with H0, α, and Ωk varied simultaneously under the stated priors. The interpretation that the posteriors favor the lower (Planck) value is therefore a statement about the location of the posterior means relative to the two observational anchors. We do not claim this constitutes an independent prediction capable of resolving the tension with SNIa data. We will revise the abstract and discussion sections to remove any implication of direct resolution and to stress that the results demonstrate consistency with the lower H0 under the adopted closure and CC constraints. revision: partial

  3. Referee: [Abstract] Abstract: no modified Friedmann equations are displayed, no error budget for the MCMC is given, and no direct comparison to SNIa likelihoods is performed, leaving the support for the central claim that the EC model resolves the tension dependent on unshown intermediate steps.

    Authors: The modified Friedmann equations with torsion are derived and shown in Section II; we will ensure they are explicitly highlighted and reproduced in the abstract or a dedicated equation block in the revision. We will expand the methods section and add an appendix with MCMC convergence diagnostics, chain lengths, and error-budget details. The analysis was restricted to CC data; we will include a supplementary comparison of the model predictions against SNIa likelihoods as an additional consistency check to strengthen the discussion of the Hubble tension. revision: yes

Circularity Check

1 steps flagged

H0 posteriors and 'favoring CMB value' reduce to MCMC fit under the imposed H = −αφ closure assumption

specific steps
  1. fitted input called prediction [Abstract (conclusion paragraph)]
    "In conclusion, we find that the parameter constraints in the model incorporating torsion ( H_0 = 67.6^{+2.1}_{-2.7} km s^{-1} Mpc^{-1}, obtained under the Big Bang Nucleosynthesis (BBN) constraint with Ω_k=0; H_0 = 66.2^{+4.4}_{-2.9} ... ; H_0 = 68.8^{+2.9}_{-4.2} ... ) are more consistent with the value derived from CMB data, favoring the lower H_0 value."

    All three H0 intervals are the direct output of MCMC parameter fitting (including α) to Cosmic Chronometers data once the model is closed by the assumption H = −αφ. The statement that the model 'favors the lower H0' therefore describes the statistical outcome of that fit rather than a prediction derived from the unmodified EC equations or torsion dynamics.

full rationale

The paper explicitly introduces the relation H = −αφ as an assumption 'to close the model' (abstract) and then runs MCMC on CC data to obtain the quoted H0 intervals. The central claim that these intervals 'are more consistent with... CMB data, favoring the lower H0 value' is therefore a restatement of the fit result rather than an independent output of the EC field equations. This matches the fitted-input-called-prediction pattern at the level of the headline conclusion, producing partial circularity (score 6) while the underlying EC setup and data choice remain independent.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 1 invented entities

The claim rests on the ad-hoc H = -α φ relation, standard EC theory, and multiple parameters fitted directly to data; torsion is treated as an extension whose specific dynamics are postulated for this analysis.

free parameters (3)
  • α
    Proportionality constant in the H = -α φ assumption, fixed by MCMC
  • Ω_k
    Curvature density, either fixed at zero or varied as a free parameter
  • H0
    Hubble constant itself obtained as a fitted output under different priors
axioms (2)
  • domain assumption Einstein-Cartan theory supplies the correct gravitational framework once torsion is included
    Used to write the modified Friedmann equations
  • ad hoc to paper The linear relation H = -α φ holds for the torsion field in the late universe
    Explicitly introduced to close the model
invented entities (1)
  • torsion field φ no independent evidence
    purpose: To modify the cosmic expansion rate via spin-torsion coupling
    Postulated through the H-φ assumption; no independent falsifiable signature is given

pith-pipeline@v0.9.0 · 5934 in / 1576 out tokens · 40878 ms · 2026-05-23T19:07:24.907146+00:00 · methodology

discussion (0)

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Reference graph

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