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REVIEW 4 major objections 4 minor 89 references

Testing the viability of $f(T, \mathcal{T})$ gravity models via effective equation of state constraints

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A linear torsion-matter coupling in $f(T,\mathcal{T})$ gravity fits the joint cosmic-chronometer and Pantheon+ data, with an acceleration transition at $z\approx 0.57$.

desk verdict A clean but largely circular f(T,T) background fit: the assumed effective EoS is exactly the ΛCDM total EoS, so the headline values are derived from the ansatz, not independent tests of modified gravity. read the letter →

arxiv 2501.14908 v1 pith:XYC6GHLB submitted 2025-01-24 astro-ph.CO gr-qc

M. Koussour , O. Donmez , S. Bekov , A. Syzdykova , S. Muminov , A.I. Ashirova This is my paper · ORCID
classification astro-ph.COgr-qc
keywords f(TT)gravitytorsionscalarcosmicaccelerationdarkenergyequationofstateMCMCquintessenceobservationalcosmology
topics Dark Energy
open problems Dark Energy
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 aims to show that the minimal extension of teleparallel gravity with $f(T,\mathcal{T}) = T + \beta\mathcal{T}$ stays viable against the main late-universe distance and expansion data. The authors close the field equations with an effective equation of state that depends on redshift, integrate to a closed-form Hubble law, and fit it to 31 cosmic-chronometer measurements plus 1701 Pantheon+ supernovae. Their best-fit parameters are $H_0 = 68.04 \pm 0.64$, $\beta = 0.14 \pm 0.17$, and $\gamma = 0.96^{+0.38}_{-0.69}$, from which they derive $q_0 = -0.51$, $z_t = 0.57$, and $\omega_0 = -0.76$. A sympathetic reader would care because the model reproduces the observed deceleration-to-acceleration transition and a quintessence-like dark-energy equation of state without putting a cosmological constant into the action.

What carries the argument

The machinery is the linear functional form $f(T,\mathcal{T}) = T + \beta\mathcal{T}$, where $T$ is the torsion scalar of teleparallel geometry and $\mathcal{T}$ is the trace of the energy-momentum tensor, together with the redshift-dependent effective equation of state $\omega(z) = -3/(\gamma(z+1)^3+3)$ used to close the generalized Friedmann equations. This ansatz is what converts the field equations into a single first-order equation for $H(z)$, yielding the closed-form solution and the derived expressions for $q(z)$, $\rho(z)$, and $\omega(z)$ that are later compared with data. The same equation of state has the limiting behavior $\omega \to 0$ at high redshift (matter-like) and $\omega \to -1$ as $z \to -1$ (cosmological-constant-like), which anchors the model's claim to describe both the matter era and late-time acceleration.

What would settle it

Fit the same two datasets to the $\beta=0$ limit of the model with the same priors and compare the statistical evidence; if the evidence does not favor $\beta\neq 0$, the viability claim would lack support. Independently, replace the adopted $\omega(z)$ with a model-agnostic two-parameter dark-energy form; if the best-fit $q_0$ and $z_t$ shift significantly from $-0.51$ and $0.57$, the headline numbers are artifacts of the assumed equation of state.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the linear model $f(T,\mathcal{T}) = T + \beta\mathcal{T}$, together with the assumed effective equation of state $\omega(z) = -3/(\gamma(1+z)^3+3)$, produces a Hubble parameter of the closed form $H(z)=H_0\left[\frac{-12\beta+(\beta+2)\gamma(1+z)^3+6}{-12\beta+(\beta+2)\gamma+6}\right]^{\frac{2\beta+3}{3\beta+6}}$, and that this law fits the joint $H(z)$+Pantheon+ data with the quoted parameter values. The same solution gives $q_0=-0.51$, a transition redshift $z_t=0.57$, a current effective equation of state $\omega_0=-0.76$, and a positive energy density at every redshift. The paper presents these as evidence that $f(T,\mathcal{T})$ gravity can mimic late-time cosmic acceleration and act as a geometric alternative to dark energy.

Load-bearing premise

The load-bearing premise is the assumed pressure-to-density ratio of the cosmic fluid, $\omega(z) = -3/(\gamma(1+z)^3+3)$; since this is exactly the pressure-to-density ratio of a $\Lambda$CDM universe, the derived expansion history and headline numbers are pre-shaped to look like $\Lambda$CDM no matter what the $f(T,\mathcal{T})$ extension does.

Editorial extensions

If this is right

  • A nonzero but small $\beta$ is compatible with the joint dataset, so the matter-trace coupling does not spoil the successful late-time expansion history.
  • The derived transition redshift $z_t=0.57$ places the deceleration-to-acceleration crossover close to $\Lambda$CDM expectations, which is what a viable dark-energy alternative must reproduce.
  • With $\omega_0=-0.76$, the model's effective fluid sits in the quintessence regime rather than the phantom regime, so it avoids a $\omega<-1$ singularity at the background level.
  • Because the energy density stays positive across the fitted redshift range, the homogeneous solution is free of the sign-flip pathologies that can appear in modified-gravity models.
  • If the claim is right, $f(T,\mathcal{T})$ deserves perturbative stability and structure-formation tests, as the paper itself notes, since background fits alone do not establish full viability.

Reading between the lines

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

  • A consequence the paper leaves implicit is that, because the assumed $\omega(z)$ is exactly the total equation of state of a $\Lambda$CDM background, the fitted $q_0$, $z_t$, and $\omega_0$ are largely dictated by the ansatz; a truly independent test of $f(T,\mathcal{T})$ gravity would need a reconstructed $\omega(z)$ rather than the $\Lambda$CDM one.
  • The $1\sigma$ interval for $\beta$ includes zero, so the current data do not require the torsion-matter coupling; a formal model comparison against the $\beta=0$ limit would show whether the extra parameter is justified.
  • Re-running the same closed-form pipeline with a different two-parameter dark-energy equation of state would show whether the quoted parameter values are robust or an artifact of the $\Lambda$CDM-shaped $\omega(z)$.
  • The stability result reported is only for the homogeneous background energy density; perturbative stability and growth-rate data remain the open ground on which the model's viability must be decided.
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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

4 major / 4 minor

Summary. The paper studies the cosmological viability of f(T, T) gravity with the linear form f(T, T) = T + βT in a flat FLRW universe. To close the field equations, the authors adopt an effective equation-of-state parameter ω(z) = −3/[γ(1+z)^3+3], derive the Hubble parameter H(z) in Eq. (18), and fit the model to a joint H(z)+Pantheon+ dataset using MCMC. They report best-fit parameters H0 = 68.04 ± 0.64, β = 0.14 ± 0.17, and γ = 0.96^{+0.38}_{−0.69}, and then compute q0 = −0.51, zt = 0.57, ω0 = −0.76, and a positive energy density, concluding that the f(T, T) model is a viable framework for cosmic acceleration.

Significance. If the central claim were supported, the paper would provide useful observational constraints on a specific f(T, T) model. The algebraic derivation of H(z) from the field equations and the adopted ansatz is internally consistent, and the MCMC analysis with a joint H(z)+Pantheon+ dataset is standard and reproducible in structure. These are genuine strengths. However, the significance is severely limited because the assumed effective equation of state is exactly the total equation of state of ΛCDM, so the fit essentially constrains the parameters of a preset ΛCDM-like expansion history rather than testing whether the β coupling is required. The reported values of q0, zt, and ω0 are deterministic functions of the ansatz and the fitted parameters, not independent cosmological findings, and no baseline comparison or perturbation analysis is provided. The paper is better viewed as a reconstruction exercise than as a test of f(T, T) gravity in its current form.

major comments (4)
  1. [Sec. III, Eqs. (15)–(19)] The assumed effective EoS, ω(z) = −3/[γ(1+z)^3+3], is exactly the total equation of state of a ΛCDM universe with Ω_m/Ω_Λ = γ/3. Since the total EoS determines H(z) in any FLRW background, Eq. (18) is a ΛCDM-like template, and for β = 0 Eq. (19) is precisely the ΛCDM Hubble rate. The data therefore constrain the parameters of a presupposed expansion history, not whether f(T, T) = T + βT modifies gravity. A concrete test would be a model-comparison statistic (Δχ², AIC, or BIC) against the β = 0 ΛCDM limit; with β = 0.14 ± 0.17 the data are fully consistent with β = 0, so the current analysis provides no evidence for the β coupling.
  2. [Sec. V, Eq. (21) and Figs. 2–4] The claims q0 = −0.51, zt = 0.57, and ω0 = −0.76 are not independent results: they are deterministic functions of the assumed ω(z) ansatz and the fitted β and γ. Moreover, no uncertainties are propagated to these derived quantities, so there is no statistical error bar on any of them. The authors should either propagate the MCMC posteriors to q0, zt, and ω0 and report their full marginalized distributions, or explicitly state that these are point estimates obtained from the best-fit parameters of an assumed template.
  3. [Sec. V, Fig. 3 and Sec. VI] The statement that a positive cosmic fluid energy density 'reinforces stability' is not a stability analysis. Positivity of ρ at the background level does not address ghost instabilities, gradient instabilities, or the growth of perturbations. The paper itself, in Sec. VI, acknowledges that 'the stability of the f(T, T) theory should be rigorously tested' and that future studies could 'detail stability analyses'; this directly contradicts the earlier stability claim. The stability statement should be removed or replaced with a genuine perturbation-level analysis.
  4. [Sec. IV] No goodness-of-fit or model-selection statistic is reported. The paper states that the best-fit parameters 'align well with current observational data,' but without a reduced χ², AIC, BIC, or a comparison to the ΛCDM limit, this statement is not quantitatively supported. Given that the model reduces to ΛCDM at β = 0, reporting such a comparison is essential for assessing whether the additional parameter β is justified by the data.
minor comments (4)
  1. [Sec. III, notation] The partial derivatives f_T and f_{\mathcal{T}} are both rendered with the same symbol in the text (e.g., 'fT = 1, fT = β'), which is confusing; the two derivatives should be distinguished with clear notation throughout.
  2. [Abstract and Sec. V] The abstract and Sec. V present q0, zt, and ω0 as headline findings without noting that they are fixed by the assumed form of ω(z) and the fitted parameters; a caveat should be added to avoid overinterpretation.
  3. [Fig. 1] The corner plot shows the 1σ and 2σ contours, but the text does not report the correlation coefficients between H0, β, and γ; reporting these would help the reader understand parameter degeneracies.
  4. [Sec. IV] The prior ranges are stated as H0 ∈ [50, 100], β ∈ [−1, 1], and γ ∈ [−1, 1], but the best-fit γ = 0.96 lies very close to the prior edge, and the posterior is asymmetric; the sensitivity of the results to the prior width should be discussed.

Circularity Check

2 steps flagged · score 7.0 of 10

The assumed EoS ansatz ω(z)=−3/[γ(1+z)^3+3] is exactly the flat-ΛCDM total equation of state, so the fitted expansion history and the derived q0, zt, and ω0 reduce by construction to the input ansatz; β is consistent with zero.

  1. renaming known result [Sec. III, before Eq. (17) and Eqs. (18)-(19)]
    "we introduce an effective EoS expressed in terms of redshift as ω(z) = − 3 γ(z+1)3+3, where γ is a constant [59]. ... When we set β = 0, the model simplifies to f (T, T ) = f (T) = T, establishing a direct connection to the ΛCDM model. Consequently, the equation for the Hubble parameter H is reduced to H(z) =H0 [ γ(1 + z)3 + 3 / γ + 3 ]^{1/2}, which closely resembles the ΛCDM model [65]."

    The adopted ω(z) is exactly the total (effective) equation of state of flat ΛCDM: w_ΛCDM(z) = -Ω_Λ/[Ω_m(1+z)^3+Ω_Λ] = -1/[1+(Ω_m/Ω_Λ)(1+z)^3]. Setting γ = 3Ω_m/Ω_Λ makes it identical to the paper's ω(z) = -3/[γ(1+z)^3+3]. Inserting this input into Eq. (15) forces H(z) into the ΛCDM form; Eq. (19) is precisely H² = H0²[Ω_m(1+z)^3+Ω_Λ] with Ω_m = γ/(γ+3). The 'viability' of f(T,T) is therefore not tested against the data: the fitted γ only relabels the ΛCDM matter density, and the best-fit β = 0.14 ± 0.17 is statistically consistent with zero. The known ΛCDM success is re-presented as a feature of the modified-gravity model.

  2. fitted input called prediction [Sec. III Eq. (21); Sec. V (q0, zt, ω0)]
    "At z = 0, the current redshift, the effective EoS takes the form ω(0) = − 3 γ+3. ... In our f (T, T ) gravity model, we obtain q0 = −0.51 ... The model also yields a transition redshift zt = 0.57 ... the current value of the EoS parameter (at z = 0) is ω0 = −0.76 [85]."

    ω0 = -3/(γ+3) is the input ansatz evaluated at z=0; with the fitted γ=0.96 it is -0.76 by the paper's own formula, not a new result. q0 and zt are the value and root of q(z) in Eq. (21), which depend only on the fitted β and γ. These are deterministic re-expressions of the assumed EoS plus best-fit parameters; presenting them as 'findings' that validate f(T,T) gravity is reporting the fit's input as an independent prediction. No likelihood comparison with the β=0/ΛCDM limit is provided, so the derived ω0, q0, and zt add no evidence beyond the ansatz.

full rationale

The paper's algebra from the f(T,T) field equations to Eqs. (13)-(15) is internally consistent, and the MCMC fit to H(z)+Pantheon+ is a legitimate external-data exercise. The circularity lies in the load-bearing ansatz: the effective EoS ω(z) = -3/[γ(1+z)^3+3] chosen in Sec. III is exactly the total EoS of flat ΛCDM, so the resulting H(z) (Eq. 19 at β=0) is the standard ΛCDM Hubble rate with Ω_m = γ/(γ+3). The data therefore constrain the parameters of a presupposed ΛCDM-like expansion history. Moreover, the headline results q0 = -0.51, zt = 0.57, and ω0 = -0.76 are obtained by evaluating the assumed ansatz and its analytic consequences at the best-fit parameters; they carry no independent information about the f(T,T) coupling. The fit itself shows β = 0.14 ± 0.17, consistent with zero, which further illustrates that the ΛCDM-like input, not the β-coupling, drives the reported success. I did not find load-bearing self-citation: the ansatz is attributed to an external reference [59], and no uniqueness theorem from the present authors is invoked. The central derivation is not circular in the sense of being logically invalid, but the paper's presentation converts an assumed ΛCDM-equivalent EoS and fitted parameters into claimed predictions of f(T,T) gravity, warranting a high circularity score.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central setup has three fitted numbers (H0, beta, gamma) and one ad hoc ansatz for the equation of state. The 'predictions' advertised in the abstract are algebraic functions of these fitted inputs, so the ledger contains no independent predictive entity.

free parameters (3)
  • H0 = 68.04 +/- 0.64
    Present-day Hubble parameter, free parameter in H(z) and fit to H(z)+Pantheon+ data.
  • beta = 0.14 +/- 0.17
    Coupling parameter in f(T,T)=T+beta*T; fitted via MCMC.
  • gamma = 0.96 (+0.38/-0.69)
    Parameter in the assumed EoS ansatz omega(z)=-3/(gamma*(z+1)^3+3); fitted.
assumptions (4)
  • ad hoc to paper The effective EoS has the form omega(z) = -3/(gamma*(z+1)^3+3).
    Introduced in Sec. III to close the system; taken from Mukherjee & Banerjee, not derived from f(T,T) gravity.
  • domain assumption The gravitational action is f(T,T)=T+beta*T.
    Assumed in Sec. III; motivated by linearity but not derived.
  • domain assumption Universe is flat FLRW with a perfect fluid.
    Standard cosmological background assumption, Sec. II.
  • standard math Field equations (10)-(11) from Harko et al. are correct.
    Used as the starting point; not re-derived.

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Pith. "Pith review of Testing the viability of $f(T, \mathcal{T})$ gravity models via effective equation of state constraints." pith.science (2026). https://pith.science/paper/XYC6GHLB

@misc{pith2026250114908,
  author       = {Pith},
  title        = {Pith review of: Testing the viability of $f(T, \mathcalT)$ gravity models via effective equation of state constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XYC6GHLB}},
  note         = {Machine review of arXiv:2501.14908}
}
abstract

This paper rigorously examines the potential of the $f(T, \mathcal{T})$ theory as a promising framework for understanding the dark sector of the universe, particularly in relation to cosmic acceleration. The $f(T, \mathcal{T})$ theory extends gravitational dynamics by incorporating both the torsion scalar $T$ and the trace of the energy-momentum tensor $\mathcal{T}$. Further, we explore the functional form $f(T, \mathcal{T}) = T + \beta \mathcal{T}$, where $\beta$ is a free parameter that modulates the matter's influence on spacetime evolution. To evaluate this model, we employ an effective EoS parameter dependent on redshift $z$, to solve the field equations and analyze the evolution of the Hubble parameter $H(z)$. Using a joint dataset ($H(z)+Pantheon^+$) and the Markov Chain Monte Carlo (MCMC) method with Bayesian analysis, we obtain the best-fit parameter values: $H_0 = 68.04 \pm 0.64$, $\beta = 0.14 \pm 0.17$, and $\gamma = 0.96^{+0.38}_{-0.69}$, which align well with current observational data. Our findings indicate a deceleration parameter of $q_0 = -0.51$, supporting a present-day accelerated expansion phase, with a transition redshift $z_t = 0.57$ marking the universe's shift from deceleration to acceleration. Moreover, we confirm a positive cosmic fluid energy density, reinforcing stability, and find an EoS parameter value of $\omega_0 = -0.76$, consistent with quintessence-driven acceleration. These results underscore the viability of $f(T, \mathcal{T})$ as a robust framework for addressing the accelerating universe and dark energy dynamics, paving the way for future investigations into its cosmological implications.

Figures

Figures reproduced from arXiv: 2501.14908 by the authors.

Figure 1
Figure 1. FIG. 1: The plot shows the best-fit values of the model [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The plot shows the variation of the energy [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The plot shows the variation of the deceleration [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The plot shows the variation of the effective EoS [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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