Teleparallel gravity: from theory to cosmology

Abstract

Teleparallel gravity has significantly increased in popularity in recent decades, bringing attention to Einstein’s other theory of gravity. In this Review, we give a comprehensive introduction to how teleparallel geometry is developed as a gauge theory of translations together with all the other properties of gauge field theory. We also related this form of geometry to the broader metric-affine approach to forming gravitational theories where we describe a systematic way of constructing consistent teleparallel theories that respect certain physical conditions such as local Lorentz invariance. We first use teleparallel gravity to formulate a teleparallel equivalent of general relativity which is dynamically equivalent to general relativity but which may have different behaviors for other scenarios, such as quantum gravity. After setting this foundation, we describe the plethora of modified teleparallel theories of gravity that have been proposed in the literature. We attempt to connect them together into general classes of covariant gravitational theories. Of particular interest, we highlight the recent proposal of a teleparallel analogue of Horndeski gravity which offers the possibility of reviving all of the regular Horndeski contributions. In the second part of the Review, we first survey works in teleparallel astrophysics literature where we focus on the open questions in this regime of physics. We then discuss the cosmological consequences for the various formulations of teleparallel gravity. We do this at background level by exploring works using various approaches ranging from dynamical systems to Noether symmetries, and more. Naturally, we then discuss perturbation theory, firstly by giving a concise approach in which this can be applied in teleparallel gravity theories and then apply it to a number of important theories in the literature. Finally, we examine works in observational and precision cosmology across the plethora of proposal theories. This is done using some of the latest observations and is used to tackle cosmological tensions which may be alleviated in teleparallel cosmology. We also introduce a number of recent works in the application of machine learning to gravity, we do this through deep learning and Gaussian processes, together with discussions about other approaches in the literature.

Revisiting the electronic properties of dislocated graphene sheets

“…We now can turn to the study of the electronic degrees of freedom in an elastic medium with a wedge dislocations from the perspective of the Weitzenböck geometry. In this perspective we introduce the Dirac equation in the farmework of the teleparallel Fock-Ivanenko covariant derivative [28,29] . The equation we want to study is:…”

“…In this case, the Riemann-Cartan geometry is reduced to a space of absolute parallelism giving an analogue of teleparallel gravity. In this theory, the Riemann curvature is determined by the torsion tensor which depends on the Weitzenböck connection (see [28,29] and references therein for more details). In the rest of the paper we use the notation of [28], where • , • denote the quantities given by the Levi-Civita and Weitzenböck connection, respectively.…”

Kantowski-Sachs cosmology in scalar-torsion theory

“…In this piece of study we are interested on the evolution of the cosmological parameters in a Kantowski-Sachs background geometry in scalar-torsion theory [52]. Scalar-torsion is the analogue of scalar-tensor theory [53] in teleparallelism [54], in which a non-minimally coupled scalar field is introduced into the gravitational Action Integral and it is coupled to the torsion scalar T . Scalar-torsion is a theory of special interest in cosmological studies because it provides a geometric mechanism for the explanation of the acceleration phases of the universe [55,56], as a unification in the dark sector components of the universe [57].…”

f(T,B) gravity in a Friedmann–Lemaître–Robertson–Walker universe with nonzero spatial curvature

“…There is a plethora of modified theories inspired by teleparallelism with many interesting results in cosmology and astrophysics 16–24 . For reviews in teleparallelism, we refer the reader to references 25,26 …”