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REVIEW 3 major objections 4 minor 2 cited by

Teleparallel dark energy in a nonflat universe

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Teleparallel dark energy with a vanishing scalar potential is decisively preferred over the standard cosmological model in a combined fit of supernovae, chronometers, and growth data.

desk verdict A clean, honest extension of teleparallel dark energy to non-flat FLRW, whose headline preference over Lambda CDM rests on an asserted rather than derived curved-space Geff and on a growth sector the authors themselves flag as incomplete. read the letter →

arxiv 2505.21359 v1 pith:RJ4ZGXPD submitted 2025-05-27 gr-qc astro-ph.CO

Rocco D'Agostino , Francesco Bajardi This is my paper
classification gr-qcastro-ph.CO MSC 83D0583F0583B05 PACS 04.50.Kd95.36.+x98.80.-k
keywords teleparallelgravitydarkenergynonminimalscalar-torsioncouplingspatialcurvatureHubbletensionS8MarkovchainMonteCarlodynamicalsystems
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 tries to establish that teleparallel dark energy, a scalar field nonminimally coupled to torsion rather than to curvature, remains viable once spatial curvature is included, and that its simplest version with a zero scalar potential is statistically preferred over the standard $\Lambda$CDM model. Fitting Pantheon+SH0ES supernovae, cosmic chronometers, and growth-rate data jointly, the authors find $\Delta\mathrm{AIC} = -10.4$ and $\Delta\mathrm{DIC} = -14.1$ for the vanishing-potential nonflat scenario, a margin conventionally read as decisive. All three potential choices give $h \approx 0.717$, matching local SH0ES measurements, and $S_8 \approx 0.77\text{--}0.80$, lower than the Planck value and closer to low-redshift structure probes. The posterior distributions mildly favor an open geometry, with positive $\Omega_{k,0}$, while remaining consistent with a flat universe at the $1\sigma$ level.

What carries the argument

The load-bearing object is the nonminimal scalar-torsion coupling $\xi T\phi^2$ inside the action. For the diagonal tetrad on FLRW, the torsion scalar is $T = -6\left(H^2 - k/a^2\right)$, so the same coupling enters the background and the perturbation equations. The scalar field acts as an effective dark-energy fluid with density and pressure given by Eqs. (20)–(21), and its equation of state $w_{\mathrm{DE}}$ can cross the phantom divide. The growth sector is governed by Eq. (43) with the effective gravitational coupling $$\frac{G_{\mathrm{eff}}}{G} = \frac{1 - $3x_1^{2}$ + \xi $x_4^{2}$}{(1 + \xi $x_4^{2}$)^2},$$ where $x_1$ and $x_4$ are dynamical-system variables related to the scalar kinetic term and the field value. Because $\xi$ is constrained to negative values around $-0.34$, this $G_{\mathrm{eff}}$ differs from Newton's constant and suppresses the growth of matter fluctuations, which is what makes $S_8$ come out low. Curvature enters through the dynamical variable $x_5 = \sqrt{1 + k/(a^2H^2)}$, and the autonomous system of dimensionless variables converts the field equations into a form suitable for Markov chain Monte Carlo sampling.

What would settle it

Deriving the full linear scalar perturbations in covariant teleparallel gravity on a nonflat FLRW background would settle the matter: if the Poisson equation acquires a correction of order $k/(a^2H^2)$ relative to Eq. (26), the reported $S_8$ and $\Delta\mathrm{AIC}$ values would shift; alternatively, re-running the MCMC without the growth-rate data tests whether the decisive $\Delta\mathrm{AIC}=-10.4$ preference survives without the assumed $G_{\mathrm{eff}}$.

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Extended reading notes

Core claim

The central claim is that in a Friedmann-Lemaître-Robertson-Walker universe with arbitrary spatial curvature $k$, the teleparallel dark energy action $$S = \int $d^{4}$x\, e\left[\frac{T}{2\$kappa^{2}$} + \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi + \xi T\$phi^{2}$ - V(\phi) + \mathcal{L}_m\right]$$ produces an effective dark-energy fluid that fits the combined Pantheon+SH0ES+CC+GRF data better than $\Lambda$CDM, with the vanishing potential $V(\phi)=0$ as the standout case: $\Delta\mathrm{AIC}=-10.4$ and $\Delta\mathrm{DIC}=-14.1$ relative to $\Lambda$CDM, while exponential and power-law potentials remain statistically comparable. The fit for $V=0$ yields $h=0.717^{+0.009}_{-0.009}$, $\Omega_{m,0}=0.333$, $\Omega_{k,0}=0.003$, $\xi=-0.342$, $\sigma_8=0.760$, and $S_8=0.801\pm0.150$; the other models give similar $h$ and $S_8\approx0.785$ and $0.773$. The authors read these results as showing that the negative torsion-scalar coupling, rather than curvature alone, drives the improvement, and that nonflat teleparallel dark energy can ease both the Hubble and the $S_8$ tensions relative to Planck-$\Lambda$CDM.

Load-bearing premise

The growth-rate and $S_8$ results rest on assuming the flat-space quasi-static formula $G_{\mathrm{eff}}$ remains valid in a spatially curved FLRW background whenever the curvature scale is much larger than the perturbation scale; the paper asserts rather than derives this, and its own final remark concedes that strong-coupling issues may make linear perturbation analyses around symmetric teleparallel backgrounds incomplete.

Editorial extensions

If this is right

  • If the vanishing-potential result holds, $\Lambda$CDM would be statistically outperformed by a torsion-based scalar field with nonzero spatial curvature, by a margin conventionally read as decisive.
  • The Hubble constant in all three teleparallel dark energy scenarios sits near $h \approx 0.717$, in $1\sigma$ agreement with SH0ES and more than $4\sigma$ away from the Planck value, so the Hubble tension would be substantially eased without invoking early-universe physics.
  • The inferred $S_8$ values fall below Planck's $0.832$ and overlap better with low-redshift large-scale-structure measurements, easing the growth-tension side as well.
  • The mild preference for $\Omega_{k,0}>0$ means curvature is not required by the data; the negative torsion-scalar coupling $\xi$ is the parameter doing the main work.
  • The exponential- and power-law-potential variants remain statistically comparable to $\Lambda$CDM, so the strong preference is specific to the zero-potential scenario rather than generic to the framework.

Reading between the lines

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

  • The decisive $\Delta\mathrm{AIC}$ for the vanishing-potential scenario should be tested against an extended dataset that includes CMB distance priors or BAO, since those probes constrain curvature and expansion at redshifts where the torsion-scalar coupling may behave differently.
  • If the full Hamiltonian analysis called for in the paper's final remark reveals additional strong-coupled degrees of freedom on FLRW backgrounds, those modes could back-react on the background and alter the posteriors, so the observational success is not guaranteed to survive the more complete theory.
  • A direct extension would replace the flat-space quasi-static $G_{\mathrm{eff}}$ with a self-consistent curved-background derivation and run N-body or higher-order perturbation simulations; the predicted $f\sigma_8(z)$ at $z\lesssim1$ could then be compared directly with upcoming redshift-space distortion surveys.
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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 extends teleparallel dark energy (TDE) with a nonminimally coupled scalar field to a spatially curved FLRW background. The authors build a dynamical-system formulation for three potentials (zero, exponential, power-law), then run an MCMC analysis on the joint Pantheon+SH0ES+CC+GRF likelihood. They report that the zero-potential nonflat TDE model is very strongly preferred over ΛCDM (ΔAIC=-10.4, ΔDIC=-14.1), that all TDE models yield h≈0.717 and S8≈0.77-0.80, and that these values are closer to local H0 and low-redshift growth measurements than Planck-ΛCDM predictions.

Significance. If the central claim holds, the paper demonstrates that a minimal torsion-scalar action with spatial curvature can simultaneously move H0 toward the local distance-ladder value, lower S8 relative to Planck, and statistically outperform ΛCDM on the chosen data combination. The dynamical-system construction is transparent, the MCMC pipeline is internally consistent, and the paper provides clear tables of best-fit values and information criteria. The significance is nonetheless conditional: the growth-rate sector relies on a flat-space quasi-static effective gravitational coupling that is asserted rather than derived for curved FLRW, and the reported H0 compatibility is partly circular because the SH0ES calibration is inside the fitted likelihood.

major comments (3)
  1. [Sec. II.B, Eq. (26); Sec. IV, Eq. (43)] The effective gravitational coupling Geff in Eq. (26) is imported from flat-space quasi-static linear perturbation theory and used in the nonflat growth equation (43) without a derivation. The manuscript asserts that the expression holds when the curvature scale is much larger than the perturbation scale, but it provides no quantitative criterion or check against the posterior values of Ωk,0. Because the GRF data enter the joint likelihood and thus the reported ΔAIC and ΔDIC in Table II, the headline preference for the V=0 model is conditional on this unverified extension. The final remark in Sec. V concedes that linear perturbation analyses around FLRW may be incomplete in teleparallel gravity, which further underscores the need to either derive Geff for nonflat backgrounds or demonstrate explicitly that curvature corrections are negligible at the scales probed by the growth data.
  2. [Sec. IV.A, Table I and prior on ξ] The prior on ξ is restricted to the interval [-1,0], so the posterior means ξ≈-0.34 lie in the middle of the allowed range and the statement that the coupling is "robustly constrained to negative values" is a restatement of the prior support. Because ξ=0 is excluded a priori, the data cannot discriminate between a minimal coupling and the negative values reported. To support the claim of a significant nonminimal coupling, the analysis should be repeated with a prior that includes ξ=0 (e.g., [-1,1]) or with a model-comparison statistic that properly accounts for the prior volume.
  3. [Sec. IV.B and Sec. V (H0 compatibility)] The agreement between the fitted h≈0.717 and the SH0ES local value is not an independent confirmation of the model: h is a free parameter and the SH0ES Cepheid host distances are part of the Pantheon+SH0ES likelihood used in Eq. (45). The reported value is therefore a fit to, not a prediction of, the local distance ladder. The interpretation that the model alleviates the H0 tension should be reframed as consistency with the SH0ES calibration included in the data, rather than as a resolution of the tension.
minor comments (4)
  1. [Abstract] The abstract refers to "Bayesian information criteria," but the paper computes the Akaike information criterion (AIC) and the deviance information criterion (DIC); the Bayesian information criterion (BIC) is not used. The wording should be changed to "information criteria" or the specific criteria should be named.
  2. [Sec. V] There is a typo "FRLW" in the concluding section; it should be "FLRW."
  3. [Sec. IV.A] The phrase "the the null potential" contains a duplicated article and should be corrected.
  4. [Table I and Sec. IV.B] For the V=0 model, σ8 has a 1σ uncertainty of ±0.118, yielding S8=0.801±0.150, which is consistent with the Planck value 0.832±0.013 at about 1σ. The statement that the model is in better agreement with low-redshift growth data should be tempered by this large uncertainty, since the difference is not statistically significant.

Circularity Check

1 steps flagged · score 3.0 of 10

Minor fitted-input overstatement: H0 and S8 compatibility claims partly restate the data used to fit h and σ8, while the central AIC/DIC model comparison remains an independent fit comparison.

  1. fitted input called prediction [Sec. IV (data description), Sec. IV B, and Sec. V (H0 and S8 compatibility claims)]
    "we probe the background evolution through the most recent sample of type Ia supernova (SN) data, the Pantheon+ set [78, 79] ... in combination with the SH0ES Cepheid host distances [80] // Vice versa, our results are in agreement at the 1σ level with the model-independent estimate based on local Cepheids calibrations by the SH0ES team [80], h = 0.73 ± 0.01."

    The Hubble constant h is a free parameter of the MCMC, and the SH0ES Cepheid host distances are part of the Pantheon+SH0ES likelihood used to constrain it. The reported posterior h ≈ 0.717 is therefore anchored to the SH0ES calibration, so the 1σ agreement with the SH0ES estimate is a restatement of an input rather than an out-of-sample prediction. Similarly, σ8 is a fitted parameter in the same likelihood that includes GRF data, so the 'better agreement with low-redshift structure formation data' is partly forced by the fit, although the growth-shape comparison retains non-trivial model dependence. This does not invalidate the central AIC/DIC model comparison, which is a separate statistical comparison of the same datasets under different models.

full rationale

The core derivations are self-contained: the action, background Friedmann equations, autonomous dynamical system, MCMC likelihood, and AIC/DIC comparison form a coherent chain in which the V = 0 preference is determined by the joint fit, not by a parameter defined to equal the target statistic. The effective gravitational coupling Geff in Eq. (26) is taken from prior flat-space perturbation theory [50,55]; it is a parameter-free derived quantity with stated approximations, so citing it is not circular. Its extension to nonflat FLRW is asserted in Sec. II B rather than derived, and the final remark concedes possible strong-coupling incompleteness of linear perturbation theory; these are physical-assumption risks, not circular reductions. The self-citations to [56] and [50] provide a numerical strategy and a derived formula, respectively, and are not used as uniqueness arguments or as substitutes for evidence. The only circularity-adjacent element is the presentation of H0 and, to a lesser extent, S8 compatibility as a success: because h and σ8 are fitted to data that include SH0ES and GRF, those agreements are statistically forced rather than independent confirmations. This is a partial, secondary circularity in the tension-relief narrative, but the central model-selection result does not reduce to its inputs, so the overall score is modest.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

Six numbers are fitted to the data (h, Omega_m0, Omega_k0, xi, sigma_8, plus n in the power-law case), so the reported profile is a constrained fit, not a prediction. The theoretical input includes standard cosmological assumptions plus three fragile extras: tetrad and covariance equivalence for curved space, the transfer of the flat-space quasi-static Geff to nonflat space, and hand-picked initial conditions plus a negative-only xi prior. No new particles, forces, or dimensions are introduced.

free parameters (6)
  • h = 0.717 (68% CI 0.708-0.726) for V=0
    Fitted to the joint likelihood; the SH0ES agreement is partly built in because SH0ES Cepheid distances are included in Pantheon+SH0ES.
  • Omega_m,0 = 0.333 (68% CI 0.306-0.360) for V=0
    Fitted matter density parameter; consistent with Planck within the stated uncertainties.
  • Omega_k,0 = 0.003 (68% CI -0.026 to 0.033) for V=0
    Fitted curvature parameter; small positive mean gives the mild open preference, but flat is allowed at 1 sigma.
  • xi = -0.342 (68% CI -0.348 to -0.336) for V=0
    Fitted nonminimal coupling; constrained negative, with the prior restricted to [-1,0] and no stated positive-side motivation.
  • sigma_8 = 0.760 (68% CI 0.648-0.878) for V=0
    Fitted present-day matter fluctuation amplitude; drives the S8 comparison with Planck.
  • n = 0.95 (68% CI 0.84-1.09) for V=V0(kappa phi)^n
    Fitted power-law index only in the power-law potential model; consistent with n=1 at 1 sigma.
assumptions (5)
  • domain assumption The diagonal tetrad in Weitzenbock gauge for the nonflat FLRW metric yields the same field equations as the covariant tetrad-spin-connection formulation.
    Invoked after Eq. (14) to justify using the pure tetrad formalism for k not equal to 0; relies on [49] and is not re-derived here.
  • domain assumption The flat-space quasi-static subhorizon formula for Geff (Eq. 26) remains valid for nonflat FLRW backgrounds.
    Stated in Sec. II B with the condition that the curvature scale is much larger than the perturbation scale; no quantitative check is provided, and the final remark concedes a possible strong-coupling problem in linear perturbation theory.
  • ad hoc to paper The evolution starts from hand-chosen initial conditions at a_in=10^-2 with Omega_m,in=0.999, Omega_k,in=10^-6, x2,in=10^-6, x4,in=10^-6.
    Sec. IV numerical strategy; these conditions are not derived from the model and could influence the late-time solution and parameter constraints.
  • ad hoc to paper The coupling xi is restricted to negative values by the flat prior xi in [-1,0].
    Sec. IV priors; no theoretical or observational justification is given for excluding positive couplings, which can bias the posterior and the model comparison.
  • domain assumption The growth-rate data can be corrected for the fiducial Lambda CDM cosmology following [84].
    Sec. IV before Eq. (45); the correction procedure is standard but model-dependent and not independently validated in this paper.

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Cite this review

Pith. "Pith review of Teleparallel dark energy in a nonflat universe." pith.science (2026). https://pith.science/paper/RJ4ZGXPD

@misc{pith2026250521359,
  author       = {Pith},
  title        = {Pith review of: Teleparallel dark energy in a nonflat universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJ4ZGXPD}},
  note         = {Machine review of arXiv:2505.21359}
}
abstract

In this paper, we investigate the cosmological dynamics of teleparallel dark energy in the presence of nonzero spatial geometry. Extending previous analyses of nonminimal scalar-tensor theories in the torsion-based framework, we consider different scalar field potentials and examine the resulting background evolution and linear perturbations. Adopting a dynamical systems approach, we reformulate the field equations and constrain the model parameters via a Markov chain Monte Carlo analysis combining updated datasets from Pantheon+SH0ES supernovae, cosmic chronometers, and growth rate measurements. Our results suggest a mild preference for an open geometry, although all models remain consistent with a flat universe at the $1\sigma$ level. Notably, Bayesian information criteria indicate that the nonflat teleparallel scenario with a vanishing potential is strongly favored over the standard $\Lambda$CDM model. Furthermore, all teleparallel scenarios are compatible with local determinations of the Hubble constant and exhibit better agreement with low-redshift structure formation data compared to $\Lambda$CDM. These findings highlight the potential of nonflat teleparallel gravity to address current observational tensions and motivate its further investigation as a viable alternative to standard cosmology.

Figures

Figures reproduced from arXiv: 2505.21359 by the authors.

Figure 1
Figure 1. FIG. 1. 68% and 95% regions of the nonflat TDE models, along with posterior distributions, obtained from the MCMC analysis [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Growth rate of fluctuations as a result of the mean [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. 68% and 95% contours for the standard FLRW cosmologies, obtained from the MCMC analysis on the Pan [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

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