Probing cosmic dynamics in $f(T)$ teleparallel gravity: Constraints from logarithmic and log-periodic deceleration ansatzes

Abstract: In this study, we probe the cosmological evolution of the universe within the framework of modified teleparallel gravity by considering a power-law form of the function $f(T)=α(-T)<sup>{n}$. To characterize the expansion dynamics, we employ logarithmic and log-periodic parametrizations of the deceleration parameter. These specific parametrizations provide a flexible and well-structured description of the cosmic expansion history across different cosmological epochs. The corresponding Hubble parameter is obtained as a function of redshift, facilitating a systematic investigation of the background dynamics. The model parameters are constrained using cosmic chronometer (CC) and joint (CC+Pantheon) datasets through a Bayesian analysis based on the $χ<sup>{2}$-minimization approach. The evolution of key cosmological quantities, including the deceleration parameter, energy density, pressure, equation of state parameter and energy conditions, is examined in detail. The geometrical diagnostics indicate a clear departure from the standard cosmological constant behavior, pointing toward a dynamically evolving dark energy scenario. Further, the proposed models remain thermodynamically consistent and the estimated age of the universe is found to be compatible with observational constraints, thereby reinforcing the robustness and viability of the framework.

• The paper introduces modified f(T) gravity models incorporating novel logarithmic and log-periodic deceleration ansatzes to capture cosmic expansion features.
• It employs Bayesian MCMC analysis on cosmic chronometer and SNIa data, yielding robust constraints on H0 and deceleration parameters.
• Results indicate viable quintom behavior with oscillatory dynamics, supporting deviations from ΛCDM while remaining thermodynamically consistent.

Probing Cosmic Dynamics in $f(T)$ Teleparallel Gravity: Constraints from Logarithmic and Log-Periodic Deceleration Ansatzes

Overview

This work analyzes cosmological dynamics in the context of modified teleparallel gravity, specifically $f(T)$ gravity with a power-law form $f(T)=\alpha(-T)^n$, employing two nonstandard parametric forms for the deceleration parameter $q(z)$. These are the logarithmic model, $q(z)=q_0+q_1\log(1+z)$, and the log-periodic model, $q(z)=q_0+q_1\sin[\log(1+z)]$. The chosen ansatzes enable the capture of both smooth and oscillatory features during cosmic expansion, thus providing a versatile framework for confronting the late-time acceleration of the universe with observational data.

$f(T)$ Gravity Formalism and Background Evolution

The $f(T)$ framework extends the teleparallel equivalent of GR (TEGR) by generalizing the Lagrangian to an arbitrary function of the torsion scalar $T$. Unlike $f(R)$ gravity, which modifies the Ricci curvature formulation, $f(T)$0 gravity is based on the torsion produced by the Weitzenböck connection.

For spatially flat FLRW spacetime, the torsion scalar simplifies as $f(T)$1. The modified Friedmann equations are recast as effective matter plus DE contributions, with dark energy components arising purely from the nontrivial $f(T)$2 terms:

• $f(T)$3
• $f(T)$4

These lead to an effective dark energy EoS depending on $f(T)$5, its derivatives, and the selected $f(T)$6.

Parametric Deceleration and Hubble Expansion

Motivated by the limitations of common $f(T)$7 parametrizations (restricted ranges, divergence issues), the paper adopts two robust alternative forms:

• Logarithmic: $f(T)$8, yielding smooth, well-behaved tracking of acceleration across all relevant $f(T)$9.
• Log-periodic: $f(T)=\alpha(-T)^n$0, introducing controlled oscillatory behavior, allowing for possible nonmonotonicities such as multiple acceleration-deceleration transitions.

By integrating $f(T)=\alpha(-T)^n$1, analytic expressions for $f(T)=\alpha(-T)^n$2 are derived for each case, yielding closed-form solutions that are directly compared to cosmic chronometer ($f(T)=\alpha(-T)^n$3) data and SNIa luminosity distances.

Bayesian Parameter Constraints

Constraints on $f(T)=\alpha(-T)^n$4, $f(T)=\alpha(-T)^n$5, and $f(T)=\alpha(-T)^n$6 are established using a Bayesian MCMC approach:

• Cosmic chronometers (CC): 31 $f(T)=\alpha(-T)^n$7 measurements in $f(T)=\alpha(-T)^n$8 via the differential age technique.
• Pantheon SNIa: 1048 SNIa data points in $f(T)=\alpha(-T)^n$9.

The likelihood analysis yields posterior distributions for the parameters with robust credible intervals. Both models reproduce $q(z)$0 values consistent with CMB/distance-ladder inferences (e.g., $q(z)$1), transition redshifts $q(z)$2, and $q(z)$3 values matching Planck and independent chronometer calibrations.

Key numerical results:

• Model-1 (logarithmic): $q(z)$4 (CC), $q(z)$5 (CC+Pantheon)
• Model-2 (log-periodic): $q(z)$6 (CC), $q(z)$7 (CC+Pantheon)

Both models' $q(z)$8 closely approach $q(z)$9CDM at $q(z)=q_0+q_1\log(1+z)$0 but can deviate at higher/lower $q(z)=q_0+q_1\log(1+z)$1, especially in the log-periodic case.

Dynamical Features and Cosmological Diagnostics

Energy Density, Pressure, and EoS

The reconstructed DE energy density $q(z)=q_0+q_1\log(1+z)$2 remains positive-definite throughout cosmic history, while $q(z)=q_0+q_1\log(1+z)$3 transitions negative at low $q(z)=q_0+q_1\log(1+z)$4, signifying the onset of acceleration.

• At $q(z)=q_0+q_1\log(1+z)$5, both models yield $q(z)=q_0+q_1\log(1+z)$6 to $q(z)=q_0+q_1\log(1+z)$7, i.e., in the quintessence regime.
• At future $q(z)=q_0+q_1\log(1+z)$8, trajectories cross the $q(z)=q_0+q_1\log(1+z)$9 line, entering the phantom regime for some parameter values, hence supporting quintom phenomenology.

Energy Conditions

Both models robustly satisfy NEC, WEC, and DEC up to the present epoch. SEC violation emerges as necessary for late-time acceleration. NEC and DEC are broken in the future ($q(z)=q_0+q_1\sin[\log(1+z)]$0) when the EoS becomes more negative than $q(z)=q_0+q_1\sin[\log(1+z)]$1, typifying phantom dark energy.

Statefinder and Om Diagnostics

• Statefinder: The evolutionary trajectories in the $q(z)=q_0+q_1\sin[\log(1+z)]$2 plane demonstrate passage from Chaplygin gas-like dynamics at early $q(z)=q_0+q_1\sin[\log(1+z)]$3 through the $q(z)=q_0+q_1\sin[\log(1+z)]$4CDM fixed point at $q(z)=q_0+q_1\sin[\log(1+z)]$5, $q(z)=q_0+q_1\sin[\log(1+z)]$6, then into a unified dark sector and eventually to a late-time regime differing from standard $q(z)=q_0+q_1\sin[\log(1+z)]$7.
• Om($q(z)=q_0+q_1\sin[\log(1+z)]$8) diagnostic: The function’s non-constancy directly indicates deviation from $q(z)=q_0+q_1\sin[\log(1+z)]$9CDM (cosmological constant), with the sign of the slope distinguishing quintessence versus phantom regimes.

Thermodynamic Consistency

The models remain thermodynamically consistent:

• Temperature $f(T)$0 declines monotonically with decreasing $f(T)$1, mirroring standard thermal history, leveling off in the future.
• Entropy density $f(T)$2 is highest in early epochs, decreasing as the universe expands, but the generalized second law holds (total entropy increases).
• The relation $f(T)$3 is recovered, linking the equation of state to entropy scaling.

Age of the Universe

Both models yield present ages $f(T)$4 compatible with globular cluster and CMB constraints ($f(T)$5–$f(T)$6 Gyr, depending on dataset and parametrization), signaling viability relative to standard cosmological models.

Implications and Outlook

The results reinforce the adaptability of $f(T)$7 models in reproducing observational background dynamics while offering systematically controlled departures from $f(T)$8CDM. Both the logarithmic and log-periodic ansatzes provide frameworks for exploring dynamical (rather than strictly constant) dark energy, including oscillatory or future-phantom regimes not accessible in traditional parametrizations. Notably, permitted crossings of $f(T)$9 indicate the possibility of quintom behavior within physically and thermodynamically viable modified gravity settings.

On the practical side, the compatibility with $f(T)$0 and SNIa data, along with proper DE dominance and transition epochs, demonstrates that such $f(T)$1 models are competitive alternatives to standard GR-based DE models at the level of background evolution.

Future theoretical directions include perturbative studies (structure formation, growth index), consistency with CMB/lensing constraints, and the extension of parameter space to richer $f(T)$2 forms. The oscillatory $f(T)$3 forms also motivate further exploration of potential signatures in high-precision time-domain cosmological probes.

Conclusion

This study provides a comprehensive comparative analysis of modified gravity cosmologies with noncanonical deceleration parametrizations. The $f(T)$4 power-law models with logarithmic and log-periodic $f(T)$5 are shown to satisfy all observational and physical criteria for a viable late-time cosmology while enabling systematic examination of physics beyond $f(T)$6CDM. The results validate $f(T)$7 gravity as a robust, flexible geometric framework for dynamic dark energy phenomenology and motivate further detailed investigation of its implications at both background and perturbation levels.