Reconstruction of $f(T)$ gravity: Rip cosmology, finite-time future singularities and thermodynamics (1202.4057v3)

Abstract: We demonstrate that there appear finite-time future singularities in $f(T)$ gravity with $T$ being the torsion scalar. We reconstruct a model of $f(T)$ gravity with realizing the finite-time future singularities. In addition, it is explicitly shown that a power-low type correction term $T^\beta$ ($\beta>1$) such as a $T^2$ term can remove the finite-time future singularities in $f(T)$ gravity. Moreover, we study $f(T)$ models with realizing inflation in the early universe, the $\Lambda$CDM model, Little Rip cosmology and Pseudo-Rip cosmology. It is demonstrated that the disintegration of bound structures for Little Rip and Pseudo-Rip cosmologies occurs in the same way as in gravity with corresponding dark energy fluid. We also discuss that the time-dependent matter instability in the star collapse can occur in $f(T)$ gravity. Furthermore, we explore thermodynamics in $f(T)$ gravity and illustrate that the second law of thermodynamics can be satisfied around the finite-time future singularities for the universe with the temperature inside the horizon being the same as that of the apparent horizon.

• The paper investigates the framework of f(T) gravity, exploring the possibility and classification of finite-time future singularities and proposing reconstruction methods.
• Authors show that finite-time singularities can arise in power-law f(T) models under certain conditions, and propose correction terms to remove these singularities.
• The study reconstructs f(T) models for various cosmological scenarios, including inflation, Lambda CDM, and Rip cosmologies, and discusses thermodynamic viability near singularities.

Overview of Reconstruction of f(T) Gravity and Cosmological Implications

The paper under discussion meticulously investigates the framework of modified teleparallel gravity, specifically focusing on $f(T)$ gravity, where $T$ represents the torsion scalar. The authors explore the possibility of finite-time future singularities within this gravitational theory and propose a reconstruction of $f(T)$ gravity that admits such singularities. The paper also explores mechanisms to address these singularities, including the introduction of a correction term to the $f(T)$ models. This investigation is contextualized within various cosmological scenarios, including inflation in the early universe, the $\Lambda$CDM model, Little Rip cosmology, and Pseudo-Rip cosmology.

Finite-Time Future Singularities and Their Classification

The authors classify finite-time future singularities according to the behavior of the Hubble parameter, extending the classification system known from $f(R)$ gravity. They identify conditions under which these singularities may arise in $f(T)$ gravity, focusing on four primary types, ranging from the "Big Rip" singularity to more tempered scenarios such as Type IV singularities, where higher derivatives of the Hubble parameter diverge. The classification is pivotal for understanding the robustness of $f(T)$ gravity against future singularities.

Reconstruction of f(T) Gravity

The authors employ a reconstruction method analogous to those used in $f(R)$ gravity to build $f(T)$ models where finite-time future singularities either occur or are circumvented. A power-law $f(T)$ model is shown to potentially exhibit such singularities, provided certain conditions on the model parameters are met. Additionally, they explore correction terms like $T^\beta$ (with $\beta > 1$) that effectively remove these singularities, thus stabilizing the theory against undesirable future behaviors.

Examination of Various Cosmological Models

In terms of practical and theoretical implications, the paper reconstructs $f(T)$ models for various cosmological scenarios. These include:

• Inflation in the Early Universe: Utilizing a power-law form of $f(T)$ which allows for accelerated expansion, mimicking inflationary models.
• $\Lambda$CDM Model: Demonstrated equivalence to the $\Lambda$CDM model is achieved, indicating the potential of $f(T)$ gravity to reproduce known cosmological results.
• Rip Cosmologies: Both Little Rip and Pseudo-Rip scenarios are explored. The authors show that these scenarios in $f(T)$ gravity can lead to the dissolution of cosmic structures at some future time, akin to phantom energy-dominated models in standard cosmology.

Thermodynamics and Matter Instability

A significant part of the investigation pertains to the thermodynamic viability of $f(T)$ gravity. It is argued that the second law of thermodynamics holds near finite-time future singularities if the temperature conditions across the horizon are met. Furthermore, the paper discusses the potential instability arising in stellar collapse, drawing analogies to findings in $f(R)$ gravity to anticipate the occurrence of singularities within collapsing stars, which question the soundness of the theory regarding matter stability.

Conclusions and Future Directions

The paper concludes by emphasizing the feasibility of constructing $f(T)$ gravity models that either exhibit or evade finite-time singularities, showing this to be a significant test for modified theories of gravity. The results suggest that inclusion of specific functional forms into the teleparallel action—such as power-law or corrective terms—could offer a method to reconcile $f(T)$ gravity with observations and theoretical constraints. The prospect of extending these findings and developing new models are highlighted as promising directions for future research, particularly the consideration of non-minimal couplings and correction terms to better model the cosmic acceleration without singularity issues. The ongoing challenge, however, is to derive an $f(T)$ theory that consistently matches the predictive success of general relativity while offering novel insights into cosmic inflation and late-time acceleration.