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Cosmological Dynamics of Accelerating Model in $f(T)$ Gravity with Special Forms of Deceleration Parameter

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two deceleration-parameter models in f(T) gravity, fitted to chronometer and supernova data, place the start of cosmic acceleration near z ≈ 0.6 and give a universe age of 13–14 Gyr.

desk verdict The f(T) sector is a costume: with alpha=1, n=1 the torsion term cancels, so every dark-energy result is bookkeeping from the assumed q(z), not a prediction of modified gravity. read the letter →

arxiv 2506.03756 v3 pith:IHAMA2B2 submitted 2025-06-04 gr-qc

Shivank Pal , Romanshu Garg , Gyan Prakash Singh , Gauree Shanker This is my paper
classification gr-qc MSC 83D0583F05 PACS 04.50.Kd95.36.+x98.80.Es
keywords f(T)gravityteleparalleldecelerationparameterparameterizationdarkenergyequationofstatecosmicchronometerdataPantheonsupernovatransitionredshiftuniverseage
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 sets out to show that two simple analytic parameterizations of the deceleration parameter, each fitted to cosmic chronometer and Pantheon supernova data, can be realized inside power-law $f(T)$ teleparallel gravity while remaining consistent with the observed late-time acceleration. With the median fitted parameters, both models return a negative present-day deceleration parameter ($q_0 \approx -0.62$ to $-0.70$), a transition from decelerated to accelerated expansion near redshift $z \approx 0.6$, and a dark-energy equation of state that is quintessence-like today and crosses to phantom in the future. Both fits also give a present universe age of 13.07–13.78 Gyr, close to the standard $\Lambda$CDM age, and $H_0 \approx 69$ km/s/Mpc. The paper's conclusion is that both models demonstrate consistency with observational evidence for the current epoch of accelerated cosmic expansion.

What carries the argument

The argument runs through four linked pieces. First, teleparallel $f(T)$ gravity — a modified gravity whose action is a function of the torsion scalar $T$ of the Weitzenböck connection instead of the Ricci curvature of general relativity — supplies the background: for the flat FLRW metric $T = -6H^2$, and the Lagrangian $T + f(T)$ with $f(T) = \alpha(-T)^n$ produces the modified Friedmann equations (11)–(12). Second, each deceleration-parameter ansatz fixes the expansion history through the identity $\dot{H} = -(1+q)H^2$, yielding closed-form Hubble functions (21) and (23) in terms of $H_0$ and the ansatz parameters. Third, the reconstruction identities (15)–(16) convert the chosen $H(z)$ and $f(T)$ into a dark-energy density $\rho_{\mathrm{DE}}$, pressure $p_{\mathrm{DE}}$, and equation-of-state parameter $\omega_{\mathrm{DE}}$ given by (17), which the paper then diagnoses with energy conditions and the statefinder pair $(r,s)$. Fourth, a Bayesian Markov chain Monte Carlo likelihood fit to 31 cosmic chronometer points and to the joint chronometer-plus-Pantheon sample determines the median values of $H_0$, $q_0$, $q_1$, $q_2$, $N$, and the supernova absolute-magnitude nuisance parameter $M$ that drive every conclusion. The displayed dynamical plots are evaluated at the paper's stated choice $\alpha = 1$ and $n = 1$.

What would settle it

Set $\alpha = 1$ and $n = 1$ as the paper does: the action $T+f(T)$ vanishes, equations (11)–(12) force $\rho = p = 0$, and equation (34) must reduce to the kinematic identity $\omega_{\mathrm{DE}}(z) = (2q(z)-1)/3$ at every redshift. Evaluating that identity at the Model-1 cosmic-chronometer median parameters ($q_0 = -0.648$, $q_1 = -0.557$, $N = 22$) gives $\omega_0 \approx -0.765$, whereas the paper reports $\omega_0 = -0.6857$; recomputing the curves of Figures 6 and 7 from equations (21) and (34) at the tabulated medians would settle whether the discrepancy is an arithmetic or typographical inconsistency or evidence that the plotted dark-energy quantities are not the $f(T)$ reconstruction described.

Watch

Extended reading notes

Core claim

The central claim is that two redshift-dependent parameterizations of the deceleration parameter — $q(z) = q_0 + q_1(\ln(N+z)/(z+1) - \ln N)$ and $q(z) = \frac{1}{2} + (q_1 z + q_2)/(1+z)^2$ — embedded in the teleparallel model $f(T) = \alpha(-T)^n$ and constrained by Bayesian Monte Carlo fits to cosmic chronometer and joint (chronometer + Pantheon) data, reproduce the observed late-time acceleration. For the median parameters, the universe decelerates in the matter-dominated past and switches to acceleration at $z_t \approx 0.6$, the present $q_0$ is negative (about $-0.62$ to $-0.70$), and $\omega_{\mathrm{DE}}$ lies in the quintessence band today (roughly $-0.67$ to $-0.80$) before crossing the cosmological-constant line to phantom values at late times, behavior the authors call quintom dark energy. The reconstructed dark-energy density stays positive while the pressure turns negative, the energy conditions show the pattern expected for acceleration (SEC violated today; NEC, WEC, and DEC satisfied up to the present and violated in the phantom future), and the statefinder trajectories pass through the $\Lambda$CDM fixed point. The integrated age of the universe comes out at 13.07–13.78 Gyr depending on model and dataset, close to the $\Lambda$CDM value, so the authors conclude that both models are consistent with the observational evidence for current accelerated expansion.

Load-bearing premise

Every displayed dark-energy result is computed with the theory's free parameters fixed by hand to $\alpha = 1$ and $n = 1$, the choice that makes the action $T+f(T)$ identically zero, so the reported dark-energy behavior is a repackaging of the assumed expansion history rather than a prediction of $f(T)$ gravity, and fitting $n$ instead could alter the conclusions.

Editorial extensions

If this is right

  • If the paper is right, both $q(z)$ parameterizations are observationally viable descriptions of the expansion history, with a matter-to-acceleration transition near $z \approx 0.6$ and a negative present-day $q_0$, so these kinematic forms alone are sufficient to fit the chronometer and supernova data.
  • The dark-energy equation of state is predicted to lie in the quintessence window today and to cross into the phantom region in the future, implying that the NEC, WEC, and DEC — satisfied up to the present epoch — will eventually be violated while the SEC is already violated, a testable signature for upcoming surveys.
  • The models predict a present universe age of 13.07–13.78 Gyr, consistent with the $\Lambda$CDM age, so they do not suffer from the age problem that plagues some accelerating cosmologies.
  • The fitted $H_0 \approx 69$ km/s/Mpc is stable across both parameterizations and both dataset choices, and the statefinder trajectories pass through the $\Lambda$CDM fixed point $(r,s) = (1,0)$, so the models are kinematically close to $\Lambda$CDM in these diagnostics.

Reading between the lines

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

  • Because the displayed results fix $\alpha = 1$ and $n = 1$, in which case $f(T) = -T$ and the gravitational action $T+f(T)$ vanishes identically, the field equations force $\rho = p = 0$ and the reported $\rho_{\mathrm{DE}}$, $p_{\mathrm{DE}}$, and $\omega_{\mathrm{DE}}$ equal exactly the kinematic identities $3H^2$, $-2\dot{H}-3H^2$, and $(2q-1)/3$ relabeled as dark energy; the quintessence-to-ph
  • The parameter fit never varies $n$ or $\alpha$, so the $f(T)$ sector is not genuinely constrained by the data; marginalizing over $n$, or fitting it together with $q_0$, $q_1$, $q_2$, and $H_0$, would change the reconstructed $\omega_{\mathrm{DE}}$, the energy-condition verdicts, and possibly the transition redshift.
  • A decisive cross-check, not performed in the paper, would rerun the same pipeline with $f(T) = 0$ (pure general-relativistic kinematics with the same two $q(z)$ ansätze) and compare the fitted $H_0$, $q_0$, $z_t$, and age; close agreement would show that the $f(T)$ dressing is observationally inert for these models.
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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 / 5 minor

Summary. The paper studies power-law teleparallel gravity with f(T)=α(-T)^n in a flat FLRW universe, adopting two redshift-dependent deceleration-parameter ansätze q(z). It derives analytic H(z) expressions, fits the parameters to cosmic-chronometer and Pantheon data using MCMC/χ² minimization, and then displays effective dark-energy density, pressure, equation of state, energy conditions, statefinder diagnostics, and the age of the universe. The authors conclude that both models are consistent with the observed late-time accelerated expansion.

Significance. If the central claim were established, the paper would provide observational constraints on two q(z) parameterizations within f(T) gravity, including transition redshifts near z≈0.6, current quintessence-like EoS, and ages near 13–14 Gyr. The MCMC machinery and the analytic H(z) integrals are standard and appear internally consistent. However, the f(T) content is not actually tested: all displayed dynamical results use α=1 and n=1, for which the gravitational action T+f(T) vanishes identically, so the inferred dark-energy quantities are kinematic reconstructions from the assumed q(z) rather than predictions of f(T) gravity. The paper's central conclusion is therefore unsupported as written.

major comments (4)
  1. [Section 5.1, Eqs. (15)–(17)] The stated illustrative choice α=1, n=1 turns f(T)=α(-T)^n into f(T)=-T, making the action in Eq. (7) identically zero. Substituting f=-T, f_T=-1, and f_TT=0 into the field equations (11)–(12) gives ρ=0 and p=0 for any H(z). The effective dark-energy quantities (15)–(16) then reduce to ρ_DE=3H^2 and p_DE=-2H˙-3H^2, which are just the Einstein-tensor terms rearranged as a dark fluid. They carry no information about f(T), so the EoS behavior, phantom crossing, and energy-condition violations in Section 5 are not predictions of f(T) gravity.
  2. [Section 5.1, Tables 1–2] Under the stated n=1 choice, Eq. (34) simplifies to ω_DE=(2q-1)/3. Inserting the fitted values q0=-0.648 (CC) and q0=-0.622 (joint) gives ω0=-0.765 and -0.748, respectively, not the tabulated values -0.6857 and -0.6722 for Model-1. The reported EoS values therefore cannot be reproduced from the equations in the text, indicating either an algebraic error in Eq. (34) or the use of unstated parameter choices.
  3. [Section 4, Eqs. (24) and (29)] The MCMC likelihoods depend on H(z) only through the assumed q(z) parameterizations; the f(T) parameters α and n never enter the fits. Since Section 5 then fixes α and n by hand, the data constrain only the kinematic q-parameters, and the conclusion that these are f(T) gravity models consistent with observations is not established. A genuine dynamical test of f(T) would require constraining α and n through the field equations (11)–(12) or using independent matter-density data.
  4. [Section 5.4] The age calculation t0=∫ dz/[(1+z)H(z)] uses the same assumed H(z) and yields 13–14 Gyr. Because this result is independent of f(T), it cannot be cited as evidence in favor of the f(T) model; it is a kinematic consequence of the q(z) ansatz and the fitted H0.
minor comments (5)
  1. [Eq. (4)] The contorsion tensor expression appears to have an index-placement error; it should be compared with the standard definition, e.g., in reference [62].
  2. [Throughout] The text contains repeated OCR-like artifacts such as 'redshi/f_t', 'ma/t_ter', and 'a/t_tracted'; these should be corrected in the published version.
  3. [Section 5.3] The statefinder parameter s is defined without parentheses; it should read s=(r-1)/[3(q-1/2)].
  4. [Tables 1–2] The supernova absolute magnitude M is listed for the joint analysis but not for the CC-only fit; please clarify whether M was marginalized over in the CC-only analysis or treated as a nuisance parameter only for the Pantheon likelihood.
  5. [Figures 10–11] The statefinder trajectories would be easier to interpret with arrows indicating the direction of increasing time along the curve.

Circularity Check

2 steps flagged · score 6.0 of 10

The dark-energy results of Section 5 reduce by construction: for the plotted choice α=1, n=1 the f(T) action term T+f(T) vanishes, so ρ_DE, p_DE, ω_DE and the energy-condition/statefinder claims are kinematic consequences of the assumed q(z), not predictions of f(T) gravity.

  1. self definitional [Section 5.1, Eqs. (30)-(35); with Section 2 Eqs. (7), (15)-(17)]
    "We take α = 1 and n = 1 for these graphical illustrations."

    For f(T)=α(−T)^n, α=1,n=1 give f(T)=−T, so the action (7) has T+f(T)=0. Then (11)-(12) give ρ=p=0 identically, and (15)-(16) reduce to ρ_DE=3H^2, p_DE=−2H_dot−3H^2. Thus ω_DE=p_DE/ρ_DE depends only on H and H_dot; with H_dot=-(1+q)H^2, ω_DE=(2q−1)/3. Since H(z) is constructed from the assumed q(z) via (19), all Section 5 ρ_DE/p_DE/ω_DE, energy-condition, and statefinder plots are kinematic bookkeeping of the q(z) ansatz. The parameters α,n do not appear in the Section 4 χ^2 and are unconstrained; the f(T) model makes no independent contribution to the claimed dynamics.

  2. fitted input called prediction [Section 3, following Eqs. (20)-(23); Tables 1-2]
    "The negative values indicate that the universe is currently undergoing an accelerated expansion at (z = 0). It has been demonstrated that the deceleration parameter supports the present phase of accelerated expansion."

    q0 is a free parameter of the adopted q(z) ansatz (20) or (22), and the Hubble parameter is defined from that same ansatz by Eq. (19): H(z)=H0 exp[∫(1+q(z)) d ln(1+z)]. The MCMC analysis in Section 4 directly fits q0 (and q1, q2, N) to H(z) data. Reporting q0<0 and a transition redshift z_t is therefore restating the fitted parameters and their analytic consequences, not an independent gravitational prediction. The sentence 'the deceleration parameter supports the present phase of accelerated expansion' is equivalent, by construction, to the input parametrization that was fitted.

full rationale

The statistical fits to CC and Pantheon data are legitimate, and the paper is largely self-contained apart from standard external references for the q(z) parametrizations [71,72] and f(T) formalism [31,32]. However, the f(T) content of the derived dynamics is eviscerated by the hand-set choice α=1, n=1 in Section 5.1. For these values, T+f(T)=0 and the modified field equations (11)-(12) reduce to ρ=p=0; the reconstructed dark-energy density and pressure (15)-(16) are just the Einstein-tensor terms rearranged as a dark fluid. Consequently, the central conclusions—quintessence-like ω_DE now crossing to phantom, energy-condition violations, and accelerating expansion—are consequences of the assumed q(z) and the fitting of its free parameters, not of the power-law f(T) model. This is partial circularity: the model's prediction reduces to its kinematic input, although the data analysis itself is a genuine constraint on H(z). A non-trivial f(T) test would require constraining or fixing α,n to values where T+f(T) does not cancel and where the f(T) terms in (15)-(16) are not identically zero.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The model rests on the assumed q(z) ansatze and on hand-fixed alpha,n. All fitted parameters (q0, q1, N or q1, q2, H0, M) are free. No new entities are introduced. The most consequential choice is n=1, which cancels the torsion term in the action.

free parameters (9)
  • alpha (f(T) coupling) = 1 (hand-fixed)
    Set to 1 in Section 5.1 for all plots; never constrained by MCMC, yet it enters rho_DE and p_DE.
  • n (f(T) power) = 1 (hand-fixed)
    Set to 1 for graphical illustrations; with n=1, f(T)=alpha(-T)^1=-T (for alpha=1), so T+f(T)=0 in the action.
  • q0 (Model-1 present deceleration parameter) = -0.648 (CC), -0.622 (joint)
    Fitted parameter in eq. (20); sets present acceleration.
  • q1 (Model-1) = -0.557 (CC), -0.53 (joint)
    Fitted coefficient in eq. (20).
  • N (Model-1) = 22 (CC), 22 +/- 3 (joint)
    Fitted scale parameter in eq. (20).
  • q1 (Model-2) = -0.12 (CC), -0.22 (joint)
    Fitted coefficient in eq. (22).
  • q2 (Model-2) = -1.20 (CC), -1.17 (joint)
    Fitted coefficient in eq. (22); determines present q0=1/2+q2.
  • H0 = 69.21 (CC), 68.9 (joint) for Model-1; 69.3, 69.1 for Model-2
    Fitted Hubble constant in both models.
  • M (supernova absolute magnitude) = 23.804 (joint Model-1), 23.801 (joint Model-2)
    Nuisance parameter marginalized in Pantheon likelihood.
assumptions (5)
  • standard math The action S = (1/2 kappa^2) integral d^4x e [T+f(T)] + L_m yields field equations (8), (11), (12) for the diagonal FLRW tetrad.
    Standard f(T) teleparallel gravity framework taken from refs [31,32]; the paper does not re-derive it.
  • standard math The torsion scalar for flat FLRW is T=-6H^2.
    Used in eqs. (11)-(12) and (15)-(17).
  • ad hoc to paper The deceleration parameter ansatze (20) and (22) are assumed a priori from refs [71,72] rather than derived from f(T) dynamics.
    This is the entire dynamical input; H(z) follows by integration. This is a phenomenological reconstruction assumption.
  • ad hoc to paper alpha=1 and n=1 are fixed by hand for the derived rho_DE, p_DE, omega_DE and energy conditions.
    Section 5.1 states 'We take alpha = 1 and n = 1 for these graphical illustrations.' With n=1, T+f(T)=0.
  • domain assumption The CC and Pantheon datasets are treated as independent, with the quoted covariance matrix for Pantheon.
    Standard observational datasets; priors and chain details are not fully specified.

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Cite this review

Pith. "Pith review of Cosmological Dynamics of Accelerating Model in $f(T)$ Gravity with Special Forms of Deceleration Parameter." pith.science (2026). https://pith.science/paper/IHAMA2B2

@misc{pith2026250603756,
  author       = {Pith},
  title        = {Pith review of: Cosmological Dynamics of Accelerating Model in $f(T)$ Gravity with Special Forms of Deceleration Parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IHAMA2B2}},
  note         = {Machine review of arXiv:2506.03756}
}
abstract

In this paper, the dynamical behavior of the accelerated expansion of the universe is discussed within the framework of $f(T)$ gravity, considering power law functional form of $ f(T)=\alpha (-T)^{n}$. Two distinct redshift-dependent parameterization of the deceleration parameter such as $q(z)=q_{0}+ q_{1}\left(\frac{ \ln(N+z)}{z+1}-\ln N \right)$ and $ q(z)=\frac{1}{2}+ \frac{q_{1}z+ q_{2}}{(1+z)^2} $ are considered. We have derived the Hubble parameter in terms of redshift and discussed its effect on other cosmological parameters. Using Bayesian statistical analysis and $\chi^2$-minimization, the median values of the model parameters for both the cosmic chronometer(CC) and the joint(CC + Pantheon) dataset have been determined. Further, energy density, pressure, the equation of state for Dark Energy, Energy condition and statefinder diagnostics are analyzed. The current age of the universe is also computed for these models.

Figures

Figures reproduced from arXiv: 2506.03756 by the authors.

Figure 1
Figure 1. For Model-1: q(z) with z [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. For Model-2: q(z) with z ‘e analysis of the deceleration parameter for the CC and the joint data sets is pre￾sented in figures (1) and (2). Figures (1) and (2) demonstrate the transition of the uni￾verse’s evolution from a decelerated to an accelerated expansion phase. For median val￾ues of model parameters, the current values of the deceleration parameter for model-1 are q0 = −0.648, q0 = −0.622 for the CC data and… view at source ↗
Figure 3
Figure 3. ‘e best-fit H(z) curve for the present models is compared with the ΛCDM model. 4.2 ‡e Pantheon Data ‘e Pantheon sample, comprising 1048 Type Ia supernova (SNIa) data points within the red￾shi‰ range 0.01 < z < 2.26[81], is utilized in this study. ‘e SNIa dataset is assembled from multiple surveys, including CfA1–CfA4 [82, 83], the Pan-STARRS1 Medium Deep Survey [81], SDSS [84], SNLS[85] and the Carnegie Supernova Pr… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: For Model-1: Marginalized 1D and 2D posterior contour map with median values of H0, q0, q1 and N using Joint data set [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: For Model-2: Marginalized 1D and 2D posterior contour map with median values of H0, q1 and q2 using Joint data set. 5 Cosmic Dynamics and the Physical Behavior 5.1 ‡e Physical Behavior of Models We examine the physical properties exhibited by fundamental quantities suc…
Figure 6
Figure 6. Figure 6: For Model-1: ωDE with z 00 0 0  0  0     0 0 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: For Model-2: ωDE with z ‘e graphical representation of ωDE for both models’ parameters is shown in figures (6) and (7). For model-1, the value of the EoS parameter is ωDE = −0.6857 and ωDE = −0.6722 for CC data and Joint data, respectively, at z = 0. Similarly, for mod…
Figure 8
Figure 8. Figure 8: For Model-1: Components of energy conditions ‘e SEC consists of the inequality ρef f + 3pef f ≥ 0, which may be connected to the Raychaudhuri equation for determining whether the universe’s expansion is accelerating or decelerating[95]. During a decelerating phase, the…
Figure 9
Figure 9. Figure 9: For Model-2 Components of energy conditions 5.3 Statefinder Diagnostic It is well understood that geometric parameters can provide insights into the cosmological dynamics of a model. To explore alternative dark energy models beyond ΛCDM, it is crucial to examine additi…
Figure 10
Figure 10. Figure 10: For Model-1: s and r plane Using the Hubble parameters H(z) (from equation (21) and (23)), we numerically evaluate this integral at the present epoch (z = 0) to determine the current age of the universe. For the model-1, the present age is t0 (at z = 0) = 13.07 Gyr an…
Figure 11
Figure 11. Figure 11: For Model-2: s and r plane Additionally, we analyzed the behavior of physical parameters. ‘e energy density for DE (ρDE) remains positive throughout cosmic evolution while pressure for DE (pDE) becomes negative in the recent due to dominance of dark energy. ‘e model d…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.