Cosmological Parameters in $f(T)$ Gravity: Theoretical and Observational Analysis

Abstract: The $f(T)$ gravity is one of the extensions of teleparallel equivalent of general relativity, in which more general functions of the torsion scalar $T$ can be described. With the proposed functional form of $f(T) = αT - βu^{-n} + γu<sup>m$, where $u = (-T/6)$, we have analyzed the cosmological parameters using dynamical system analysis and cosmological datasets. The dynamical behavior of this model is analyzed with phase-space analysis by transforming the cosmological equations into an autonomous system. Critical points are identified, and their stability conditions examined, enabling the classifications of the early and late-time evolutionary phases of the Universe. The stability conditions are further demonstrated by phase-portrait diagrams that highlight transitions between radiation, matter, and dark-energy-dominated epochs. Then we used the Markov Chain Monte Carlo statistical technique to constrain the model parameters with the recent observational dataset, such as DESI DR2 BAO, and its combination with the Hubble and Pantheon+SH0ES data. The best-fit values for the model parameters were obtained by data analysis, $m \equiv 0.91<sup>{+0.07}_{-0.09}$ and $n \equiv 0.69<sup>{+0.09}_{-0.08}$, and are well within the stability range obtained ($m&lt;1\land n&gt;-1$) through dynamical system analysis. The combined theoretical and observational analysis shows that the proposed $f(T)$ gravity model successfully reproduces the observed cosmic expansion history of the Universe.

• The paper presents an f(T) gravity model extending TEGR with non-linear torsion terms to reproduce cosmic acceleration.
• It employs phase-space dynamical analysis and MCMC observational fitting to tightly constrain model parameters like m and n.
• The study demonstrates smooth cosmic transitions through radiation, matter, and de Sitter epochs, aligning well with ΛCDM predictions.

Cosmological Parameters in $f(T)$ Gravity: Theoretical and Observational Insights

Introduction and Theoretical Framework

The study systematically investigates a well-motivated class of $f(T)$ gravity models, extending the teleparallel equivalent of General Relativity (TEGR) via the torsion scalar $T$. The functional form $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$ (with $u = -T/6$) encompasses the effects of non-linear torsion contributions, allowing a flexible description of gravitational phenomena. The model is structured to address both theoretical and observational constraints, with $\alpha, \beta, \gamma, n, m$ parameterizing deviations from TEGR and introducing new dynamical features relevant for cosmic acceleration.

The framework takes vierbein fields as primary dynamical variables, constructing modified Friedmann equations within a flat FLRW background. The resultant field equations naturally yield effective dark energy (DE) density and pressure terms, with the equation-of-state (EoS) parameter for DE acquiring non-trivial redshift evolution. As is standard in $f(T)$ cosmology, the violation of local Lorentz invariance is present but does not affect the field equations' general structure in the background cosmology considered.

Dynamical Systems and Phase Space Analysis

The model is recast into an autonomous dynamical system using dimensionless variables $(x, y, r)$, encoding the effective densities corresponding to the $\beta$, $\gamma$ extensions and the radiation sector, respectively. Critical points of the system, corresponding to radiation, matter, and dark energy dominated phases, are analytically determined, and their stability is interrogated via the eigenvalues of the Jacobian matrix.

The analysis identifies three fundamental critical points:

• $f(T)$0: A de Sitter attractor with $f(T)$1, corresponding to a late-time dark energy-dominated Universe; the stability analysis reveals two negative eigenvalues and one zero eigenvalue, confirming a stable, non-isolated attractor.
• $f(T)$2: Radiation-dominated ($f(T)$3), generically unstable or a saddle depending on the $f(T)$4 parameter regime.
• $f(T)$5: Matter-dominated ($f(T)$6), typically a saddle except for a subset of parameter choices.

These results establish that the $f(T)$7 model robustly accommodates the entire sequence of cosmic evolution—radiation, matter, and accelerated expansion—as fixed points with the appropriate (in)stability properties, provided the model parameters satisfy $f(T)$8, $f(T)$9 for dynamical viability.

Figure 1: Phase space diagram of the autonomous system, illustrating the location and stability properties of the critical points $T$0 (de Sitter), $T$1 (radiation), and $T$2 (matter).

The redshift-dependent evolution of $T$3, $T$4, density parameters, and the deceleration parameter further corroborates this structure. Notably, $T$5 crosses the phantom divide ($T$6 at $T$7), matching recent CMB and supernova data.

Figure 2: Evolution of EoS parameters, density components, and the deceleration parameter as functions of redshift, demonstrating transitions through radiation, matter, and dark energy dominated eras and consistent with $T$8CDM at late times.

Observational Constraints and MCMC Data Analysis

To anchor the model in observational reality, the analysis employs Markov Chain Monte Carlo (MCMC) methods on multiple cosmological datasets: $T$9 expansion rate measurements, the Pantheon+SH0ES Type Ia supernovae sample, and DESI DR2 BAO data. The parameter space $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$0 is simultaneously fit to the combined likelihood, with dimensional priors fixed by theoretical considerations.

Contour plots from these analyses reveal tight constraints, particularly from the combined dataset, with the best-fit values:

• $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$1,
• $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$2,
• $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$3,
• $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$4,

where the $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$5 values are consistent with the theoretically required stability regime.

Figure 3: Two-dimensional parameter contours derived from DESI DR2 BAO analysis, showing the constraint region for $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$6.

Figure 4: Combined constraints from $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$7, Pantheon+SH0ES, and DESI DR2 BAO, illustrating the overlap region and the precision reached for the $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$8 model parameters.

A further cross-dataset comparison indicates that the $f(T) = \alpha T - \beta u^{-n} + \gamma u^m$9 model's best-fit parameters are consistently compatible with current expansion history and distance modulus observations. Hubble diagrams and distance modulus residuals confirm close alignment with the standard $u = -T/6$0CDM prediction.

Figure 5: Left: Redshift evolution of the Hubble parameter with error bars. Right: Distance modulus as a function of redshift, both compared to observational data.

Implications and Future Directions

The paper establishes that the specific $u = -T/6$1 model considered efficiently reproduces the observed cosmic acceleration while providing a more general theoretical foundation than the cosmological constant. Its ability to transition through all cosmic epochs, reproduce the observed Hubble rate, and match density parameter values ($u = -T/6$2, $u = -T/6$3) underscores its phenomenological plausibility. The non-trivial evolution of $u = -T/6$4, including phantom crossing, may offer signatures distinguishable from $u = -T/6$5CDM in future, higher-precision surveys.

From a theoretical standpoint, the simultaneous satisfaction of dynamical system stability and data-driven parameter constraints highlights the appropriateness of the chosen $u = -T/6$6 form. The methodology—dynamical analysis followed by high-dimensional MCMC observational fitting—is generalizable to broader $u = -T/6$7 and related torsional gravity models.

Prospective research directions include:

• Examining structure growth and perturbation spectrum consistency,
• Extension to non-flat cosmologies or inclusion of non-minimal couplings,
• Investigation of potential small-scale anomalies or distinguishing signatures relative to $u = -T/6$8 gravity,
• Deeper comparison with upcoming Stage IV survey data.

Conclusion

By integrating thorough dynamical systems analysis with multifaceted observational constraints, the study demonstrates that the considered $u = -T/6$9 gravity model is both theoretically viable and observationally consistent as an alternative to $\alpha, \beta, \gamma, n, m$0CDM. The parameter space supporting cosmological transitions and late-time acceleration is sharply delineated, and empirical datasets validate the model’s efficacy in fitting key cosmological observables. The work therefore reinforces the case for $\alpha, \beta, \gamma, n, m$1 gravity as a credible extension framework, meriting continued investigation with forthcoming high-precision cosmological data.