Over recent decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f (T) gravity, where f (T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f (T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f (T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, alternative to a cosmological constant, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, for a certain class of f (T) models, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations-or, alternatively, the Big Bang singularity can be avoided at even earlier moments due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f (T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f (T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f (R) gravity, trying to illuminate the subject of which formulation, or combination of formulations, might be more suitable for quantization ventures and cosmological applications.

f(T) teleparallel gravity and cosmology

Over the past decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f(T) gravity, where f(T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f(T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f(T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations, or alternatively, the Big Bang singularity can be avoided due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f(T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f(T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f(R) gravity, trying to enlighten the subject of which formulation might be more suitable for quantization ventures and cosmological applications.

Teleparallel gravity and its popular generalization gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity. Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models. While inertial spin connections are commonplace in special relativity, and not something inherent to teleparallel gravity, the role of the inertial spin connection in removing the spurious inertial effects within a given frame of reference is emphasized here. The careful consideration of the inertial spin connection leads to the construction of a fully invariant theory of teleparallel gravity and its generalizations. Indeed, it is the nature of the spin connection that differentiates the relationship between what have been called good tetrads and bad tetrads and clearly shows that, in principle, any tetrad can be utilized. The field equations for the fully invariant formulation of teleparallel gravity and its generalizations are presented and a number of examples using different assumptions on the frame and spin connection are displayed to illustrate the covariant procedure. Various modified teleparallel gravity models are also briefly reviewed.

https://iopscience.iop.org/article/10.1088/1361-6382/ab2e1f/pdf

Teleparallel theories of gravity: illuminating a fully invariant approach

We investigate the main features of the flat Friedmann-Lema\^{\i}tre-Robertson-Walker cosmological models in the $f(T)$ modified gravity regime. In particular, a general approach to find out exact cosmological solutions in $f(T)$ gravity is discussed. Instead of taking into account phenomenological models, we consider as a selection criterion, the existence of Noether symmetries in the cosmological $f(T)$ pointlike Lagrangian. We find that only the $f(T)={f}_{0}{T}^{n}$ model admits extra Noether symmetries. The existence of extra Noether integrals can be used in order to simplify the system of differential equations (equations of motion) as well as to determine the integrability of the $f(T)={f}_{0}{T}^{n}$ cosmological model. Within this context, we can solve the problem analytically and thus we provide the evolution of the main cosmological functions such as the scale factor of the Universe, the Hubble expansion rate, the deceleration parameter, and the linear matter perturbations. We show that the $f(T)={f}_{0}{T}^{n}$ cosmological model suffers from two basic problems. The first problem is related to the fact that the deceleration parameter is constant which means that it never changes sign, and therefore the Universe always accelerates or always decelerates depending on the value of $n$. Second, we find that the clustering growth rate remains always equal to unity implying that the recent growth data disfavor the $f(T)={f}_{0}{T}^{n}$ gravity. Finally, we prove that the $f(T)={f}_{0}{T}^{n}$ gravity can be cosmologically equivalent with the $f(R)={R}^{n}$ gravity model and the time varying vacuum model $\ensuremath{\Lambda}(H)=3\ensuremath{\gamma}{H}^{2}$ (for ${n}^{\ensuremath{-}1}=1\ensuremath{-}\ensuremath{\gamma}$) because the above cosmological scenarios share exactly the same Hubble expansion, despite the fact that the three models have a different geometrical origin. Finally, some important differences with power-law $f(R)$ gravity are pointed out.

Noether symmetries and analytical solutions in f(T)-cosmology: A complete study

We use the local value of the Hubble constant recently measured with 2.4% precision, as well as the latest compilation of cosmic chronometers data, together with standard probes such as Supernovae Type Ia and Baryon Acoustic Oscillation distance measurements, in order to impose constraints on the viable and most used f(T) gravity models, where T is the torsion scalar in teleparallel gravity. In particular, we consider three f(T) models with two parameters, out of which one is independent, and we quantify their deviation from ΛCDM cosmology through a sole parameter. Our analysis reveals that for one of the models a small but non-zero deviation from ΛCDM cosmology is slightly favored, while for the other models the best fit is very close to ΛCDM scenario. Clearly, f(T) gravity is consistent with observations, and it can serve as a candidate for modified gravity.

https://iopscience.iop.org/article/10.1088/1475-7516/2016/08/011/pdf

New observational constraints on f(T) gravity from cosmic chronometers

Teleparallel gravity is an equivalent formulation of general relativity in which instead of the Ricci scalar R, one uses the torsion scalar T for the Lagrangian density. Recently teleparallel dark energy has been proposed by Geng et al. (in Phys. Lett. B 704, 384, 2011). They have added quintessence scalar field, allowing also a non-minimal coupling with gravity in the Lagrangian of teleparallel gravity and found that such a non-minimally coupled quintessence theory has a richer structure than the same one in the frame work of general relativity. In the present work we are interested in tachyonic teleparallel dark energy in which scalar field is responsible for dark energy in the frame work of torsion gravity. We find that such a non-minimally coupled tachyon gravity can realize the crossing of the phantom divide line for the effective equation of state. Using the numerical calculations we display such a behavior of the model explicitly.

https://www.tpbin.com/Uploads/Subjects/7373e347-f4d6-4176-8d3e-4aa03adabef8.pdf

Tachyonic teleparallel dark energy

Teleparallel gravity is an equivalent formulation of general relativity in which instead of the Ricci scalar $R$, one uses the torsion scalar $T$ for the Lagrangian density. Recently teleparallel dark energy has been proposed by Geng et al. in (Geng et al., 2011). They have added quintessence scalar field, allowing also a non-minimal coupling with gravity in the Lagrangian of teleparallel gravity and found that such a non-minimally coupled quintessence theory has a richer structure than the same one in the frame work of general relativity. In the present work we are interested in tachyonic teleparallel dark energy in which scalar field is responsible for dark energy in the frame work of torsion gravity. We find that such a non-minimally coupled tachyon gravity can realize the crossing of the phantom divide line for the effective equation of state. Using the numerical calculations we display such a behavior of the model explicitly.

B. Fazlpour

Papers

Tachyonic teleparallel dark energy

Tachyonic Teleparallel Dark Energy