Comparative Study of $f(T)$ Gravity Models with Observational Constraints from \textit{OHD} and \textit{Pantheon+ datasets}

Abstract: The late-time acceleration of the universe remains one of the most significant open problems in modern cosmology. Modified gravity frameworks such as $f(T)$ gravity provide a geometric alternative to dark energy by attributing cosmic acceleration to torsional effects. In this study, we present a comparative analysis of three different forms of $f(T)$ models: (i) a simple power-law form $f(T) = η(-T)<sup>{n}$, (ii) the exponential form $$f(T) = βT_{0}\left(1-e^{-p \sqrt{T/T_{0}}}\right)$$ and (iii) a logarithmic form $f(T) = γT \ln!\left(\frac{T}{T_{0}}\right)$. Using parameterization of the deceleration parameter $q(z)$ and the corresponding $H(z)$ expression, we constrain the model parameters with the recent Hubble parameter and BAO data through a Markov Chain Monte Carlo (MCMC) approach. The physical behavior of the effective energy density, equation of state parameter, squared sound speed, cosmological $Om(z)$ diagnostics, and energy conditions (NEC, DEC, SEC) were investigated for all three models. Our comparative analysis shows that all models asymptotically approach the $Λ$CDM behavior at late times, while they differ in stability properties and energy condition behaviors. In particular, the violation of the strong energy condition (SEC) has emerged as a common feature consistent with current accelerated expansion. This study highlights how different $f(T)$ functional forms can yield distinct cosmological dynamics while maintaining consistency with observational data.

• The paper compares three f(T) gravity forms through an Om(z)-based reconstruction constrained by OHD and Pantheon+, obtaining H₀ = 70.54⁺⁰·⁷⁵₋⁰·⁷⁸ km s⁻¹ Mpc⁻¹ from the combined fit.
• The exponential model provides the closest match to ΛCDM, with wT(0) ≈ −1.02 ± 0.02, smooth evolution, and a stable de Sitter attractor, while power-law and logarithmic models allow richer or milder evolving behavior.
• All models satisfy NEC and DEC, violate SEC as expected for late-time acceleration, and pass basic ghost and gradient-stability tests near their best-fit regions, although the analysis indirectly constrains Lagrangian parameters rather than fitting them directly.

Overview and motivation

$f(T)$ gravity generalizes the teleparallel equivalent of general relativity (TEGR) by replacing the torsion scalar $T$ in the action with an arbitrary function $f(T)$. Because the resulting field equations remain second-order—unlike the fourth-order equations of $f(R)$ gravity—the framework offers a mathematically tractable geometric route to late-time acceleration, with torsional corrections playing the role of an effective dark energy fluid (2605.04113). The paper under discussion addresses a specific gap in this literature: while power-law, exponential, and logarithmic forms of $f(T)$ have each been studied individually, a systematic comparison of all three under identical parameterizations and dataset combinations had not previously been carried out.

The authors analyze three representative models: the power-law form $f(T)=\eta(-T)^n$, the exponential form $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$, and the logarithmic form $f(T)=\gamma T\ln(T/T_0)$. For each, they derive the effective torsional energy density $\rho_T$, pressure $p_T$, and equation-of-state parameter $T$0, then evaluate energy conditions, ghost-free and gradient-stability criteria, and consistency with observational data.

Methodology: kinematic reconstruction rather than direct Lagrangian constraints

A methodological point that bears directly on how the results should be interpreted is that the background expansion history is not obtained by solving the modified Friedmann equations of any particular $T$1 Lagrangian. Instead, the authors adopt a phenomenological reconstruction based on the $T$2 diagnostic,

$T$3

with a logarithmic ansatz $T$4, yielding

$T$5

This choice is motivated by the high nonlinearity of the $T$6 field equations, which precludes closed-form or numerically stable solutions for realistic models. The reconstructed $T$7 determines the torsion scalar via $T$8, which is then substituted into the modified Friedmann equations to evaluate the derived physical quantities for each model. The authors are explicit about the consequence: the observational likelihood constrains only $T$9; the fundamental Lagrangian parameters $f(T)$0 are not statistically fitted. Observational constraints on $f(T)$1 propagate indirectly by restricting the parameter ranges in which the derived quantities remain physically viable. This is an important caveat—the paper provides a viability test of the models against an observationally supported expansion history, not a fully self-consistent Bayesian estimation at the Lagrangian level.

Observational constraints

Two datasets are employed: a compilation of 77 OHD measurements from cosmic chronometers and BAO over $f(T)$2, and the Pantheon+ sample (1701 light curves, 1550 spectroscopically confirmed SNe Ia, $f(T)$3). Parameter estimation uses Gaussian chi-square likelihoods with Monte Carlo sampling (100,000 realizations), with model comparison via AIC and BIC.

Parameter	OHD	OHD + PP
$f(T)$4	$f(T)$5	$f(T)$6
$f(T)$7	$f(T)$8	$f(T)$9

The joint fit gives AIC = 762.61 and BIC = 770.19 for the model parameters. The inferred $f(R)$0 and distance modulus $f(R)$1 tracks are close to flat $f(R)$2CDM with $f(R)$3, $f(R)$4. The $f(R)$5 value from the combined fit ($f(R)$6 km s$f(R)$7 Mpc$f(R)$8) sits between the Planck and local-sh ladder determinations, though the paper does not claim resolution of the Hubble tension.

Cosmological dynamics of each model

Power-law model $f(R)$9: the effective energy density is positive and monotonically decreasing, the pressure negative throughout. The torsional equation-of-state stays close to $f(T)$0 at low redshift with quintessence-like or mildly phantom behavior depending on $f(T)$1; the present value $f(T)$2 is consistent with Planck + BAO + Pantheon bounds. Among the three models it exhibits the strongest deviation from SEC violation patterns and allows the richest phenomenology (both phantom crossings and quintessence-like phases).

Exponential model $f(T)$3: this yields the smoothest evolution of the three. The equation of state remains closest to $f(T)$4 across the full redshift range, with $f(T)$5, comfortably within the combined constraint $f(T)$6. The model asymptotically approaches a de Sitter attractor, and the paper identifies it as providing the best observational match and dynamical stability among the cases examined.

Logarithmic model $f(T)$7: an intermediate case. The equation of state evolves from $f(T)$8 at low redshift toward the cosmological-constant limit at late times, with $f(T)$9. This describes a mild dynamical dark energy whose small deviations from $f(T)=\eta(-T)^n$0CDM could in principle be distinguished by future precision datasets.

Energy conditions and stability

Across all three models the pattern is uniform: NEC ($f(T)=\eta(-T)^n$1) and DEC ($f(T)=\eta(-T)^n$2) are satisfied throughout the observed redshift range, while SEC ($f(T)=\eta(-T)^n$3) is violated at low redshift ($f(T)=\eta(-T)^n$4). The SEC violation is not a defect but the expected signature of accelerated expansion driven by negative torsional pressure; its magnitude differs across models—strongest for the power-law form, mildest and steadiest for the exponential, intermediate for the logarithmic. Satisfaction of NEC and DEC indicates no ghost-like exotic behavior and causal energy transport in the effective torsional fluid.

The stability analysis imposes two additional criteria: non-negativity of the squared sound speed $f(T)=\eta(-T)^n$5 (absence of gradient instabilities) and positivity of the kinetic factor $f(T)=\eta(-T)^n$6 (ghost avoidance). Under the best-fit parameters, both conditions hold and the models admit a stable late-time attractor with $f(T)=\eta(-T)^n$7. However, the paper's own comparative summary table marks several of these criteria as parameter-dependent ("$f(T)=\eta(-T)^n$8" entries) for the power-law and hybrid forms—meaning the stated stability conclusions apply to the best-fit region rather than to the full parameter space, and case-by-case checks are required elsewhere.

Limitations and open questions

Several caveats deserve emphasis. First, as noted above, the analysis constrains the kinematic parameters $f(T)=\eta(-T)^n$9 directly but treats the Lagrangian constants $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$0 only through viability screening; the authors concede that a self-consistent parameter estimation would require solving the nonlinear field equations explicitly. Second, the logarithmic $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$1 ansatz is phenomenological—it is not derived from a fundamental Lagrangian, and its validity is restricted to the redshift range probed by low- and intermediate-redshift observations; extrapolation to very high redshift is unwarranted without further theoretical input. Third, the perturbative sector is treated only at the level of background sound-speed and ghost criteria; a full treatment including matter growth rates and gravitational-wave propagation remains outside the scope of this work. Fourth, the paper notes that the hybrid power-logarithmic variant offers flexibility but carries the greatest risk of degeneracy and unconstrained parameters, and lacks a dedicated Hubble + Pantheon analysis. Finally, the qualitative summary concedes that satisfaction of WEC and DEC for the power-law form is conditional on parameter choices, so the blanket statement of physical viability applies strictly to the constrained best-fit region.

Conclusion

The paper delivers a unified observational and theoretical assessment of three standard $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$2 functional forms using OHD and Pantheon+ data through a kinematic reconstruction of the expansion history. All three models converge toward $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$3CDM behavior at late times, satisfy NEC and DEC, violate SEC consistently with acceleration, and pass basic ghost and gradient-stability criteria near their best-fit regions. The exponential model emerges as the most observationally compatible and dynamically stable case, with $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$4 and a de Sitter attractor; the power-law form permits richer quintessence/phantom phenomenology; the logarithmic form occupies an intermediate position. The principal contribution lies less in new attractor solutions than in the integrated framework combining phenomenological reconstruction, direct data confrontation, and stability diagnostics. The natural next steps identified are perturbative tests against structure formation, extension to $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$5 and $f(T)=\beta T_0(1-e^{-p\sqrt{T/T_0}})$6 classes, and tighter constraints from DESI chronometers, LSST supernovae, and future gravitational-wave standard sirens—which together could discriminate among these torsional scenarios on quantitative grounds.