The dynamical behavior of $f(T)$ theory

Abstract: Recently, a new model obtained from generalizing teleparallel gravity, named $f(T)$ theory, is proposed to explain the present cosmic accelerating expansion with no need of dark energy. In this paper, we analyze the dynamical property of this theory. For a concrete power law model, we obtain that the dynamical system has a stable de Sitter phase along with an unstable radiation dominated phase and an unstable matter dominated one. We show that the Universe can evolve from a radiation dominated era to a matter dominated one, and finally enter an exponential expansion phase.

• The paper demonstrates that f(T) theory offers a viable alternative to dark energy by producing cosmic acceleration via a power-law modification.
• It employs a dynamical analysis with dimensionless variables to identify three critical points corresponding to radiation, matter, and de Sitter phases.
• Numerical results show that subtle parameter shifts yield realistic cosmic evolutions that align with observational constraints.

An Analysis of the Dynamical Behavior of $f(T)$ Theory

The research paper under consideration presents an in-depth analysis of a novel gravitational model known as $f(T)$ theory, developed as a modification of teleparallel gravity. In contrast to traditional dark energy models, $f(T)$ theory posits that the cosmic acceleration can be explained without the introduction of dark energy. The focal point of the paper involves examining the dynamical properties of this theory, particularly through the utilization of a concrete power-law model.

Theoretical Foundations

Within the framework of $f(T)$ theory, the scalar torsion $T$ from teleparallel gravity is replaced with a generalized function, $T + f(T)$, extending the action similar to the $f(R)$ modification of general relativity. This approach transforms the conventional Friedmann equations, introducing modifications characterized through the torsion scalar. The paper provides a detailed derivation of these modified equations, representing the dynamical behavior through a system of dimensionless variables: $x$, $y$, and $z$, which correspond to the energy density parameters and are used to describe the universe's dynamical evolution.

Dynamical Analysis

The paper identifies three critical points in the phase space: Point A corresponds to a radiation-dominated era, Point B signifies a matter-dominated era, and Line C represents a de Sitter (dark energy dominated) phase. Analyzing the eigenvalues of the linearized system at these points, the research highlights:

• Point A is unstable, indicating the universe can exit this phase.
• Point B is similarly unstable, allowing the transition from the matter-dominated era.
• Line C, in contrast, is stable. This suggests the universe can eventually enter a stable de Sitter phase, indicative of exponential expansion.

These results are indicative of a universe evolving consistently from radiation dominance into a matter-dominated phase, culminating ultimately in an accelerated expansion phase without dark energy's intervention.

Implications and Numerical Results

The focus on a power-law model, represented by $f(T) = \alpha (-T)^n$, elucidates the conditions under which typical cosmic behavior is observed. Specifically, for $n \neq 1$, the universe transitions accurately between radiation and matter phases to an accelerating universe. Empirical consistency requires $|n| \ll 1$, aligning with previous observational constraints.

The numerical results presented demonstrate how minute initial parameters for $x$ and $y$ lead to realistic cosmic evolutions, emphasizing early universe behavior's sensitivity to these model parameters. Such findings underscore the capability of $f(T)$ theory to replicate observed cosmic patterns while providing an alternative to dark energy explanations.

Future Directions and Theoretical Implications

The study enhances the understanding of modified gravities like $f(T)$ theory by showcasing its potential to describe cosmic acceleration without exotic dark energies. These implications, both theoretical and practical, suggest pathways for further research. Future investigations could explore parameter space more comprehensively, engage in observational testing, and assess the implications on quantum gravity frameworks.

In conclusion, the dynamical behavior of $f(T)$ theory presents a compelling alternative to dark energy-centric cosmic acceleration models, fostering a deeper understanding of gravitational physics. The study's rigorous analytical and numerical methodologies lay a robust foundation for ongoing and future explorations into modified gravity theories.