Teleparallel Gravity: From Theory to Cosmology (2106.13793v4)

Abstract: Teleparallel gravity has significantly increased in popularity in recent decades, bringing attention to Einstein's other theory of gravity. In this Review, we relate 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. 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.

• The paper establishes the equivalence between TEGR and general relativity by replacing curvature with torsion and a boundary term.
• The paper presents a covariant formulation that preserves local Lorentz invariance using tetrads as the principal dynamical variables.
• The paper extends teleparallel frameworks to modified theories like f(T) models, offering insights into dark energy dynamics and gravitational wave phenomenology.

Teleparallel Gravity: From Theory to Cosmology

The paper under discussion provides a comprehensive overview of teleparallel gravity (TG) and its various theoretical and cosmological implications. Instead of relying on the notion of spacetime curvature, teleparallel gravity offers an alternative by utilizing torsion to describe gravitational interactions. This is achieved by employing a flat connection with non-zero torsion, known as the teleparallel connection. The paper elaborates on how this framework can be perceived as a gauge theory of translations, with the torsion tensor serving as the gravitational field strength.

Formulation of Teleparallel Gravity

One of the significant aspects discussed is the distinction between teleparallel gravity and General Relativity (GR). Teleparallel Equivalent of General Relativity (TEGR) is highlighted, with emphasis on its equivalence to GR in classical predictions while differing in the underlying framework by utilizing torsion instead of curvature. This equivalence is established through the replacement of the Ricci scalar with the torsion scalar, supplemented by a boundary term to account for their equivalence in the action formulation.

Theoretical Insights and Symmetric Formulations

Delving deeper into the theoretical structure, the paper outlines how TG maintains the local Lorentz invariance. It identifies that while the tetrads and spin connection are pivotal elements, the gauge freedom associated with the latter allows for different representations in TG formulations. Importantly, the covariant formulation treats the tetrad as the primary dynamical variable, contributing to the metric while the metric-affine approach is leveraged for a broader geometric understanding.

Furthermore, the paper discusses the particular choices of the teleparallel connection and the systematic way to construct consistent teleparallel theories adhering to physical principles like local Lorentz invariance. This includes understanding the properties and applying them to cosmological scenarios like Friedmann–Lemaître–Robertson–Walker (FLRW) cosmologies, where it is pivotal to maintain cosmological symmetries to derive meaningful cosmological dynamics.

Modified Theories and Cosmological Applications

The review highlights various modified teleparallel theories of gravity that extend beyond TEGR. Of particular interest is the development of $f(T)$ theories and teleparallel equivalents of Horndeski gravity, which aim to address scenarios where standard GR may falter, such as quantum gravity considerations or dark energy dynamics. The construction of such theories typically involves generalizing the form of the torsion scalar or its modifications.

The paper surveys advancements in teleparallel astrophysics, focusing on teleparallel gravity's potential implications regarding gravitational wave phenomenology. In the observational field, constraints on TG models are placed using emerging data from precision cosmology, investigating aspects like dark energy models and cosmological tensions potentially alleviated within the teleparallel cosmological framework.

Implications and Future Developments

The theoretical framework laid out for TG provides a platform that holds potential for extensions into the quantum regime, expanding its applicability and testing against increasingly precise astrophysical and cosmological observations. The integration of machine learning techniques in TG research is particularly noted as a novel approach to refining model parameters and enhancing predictive power.

Conclusion

This paper encapsulates a detailed exploration of teleparallel gravity, emphasizing its geometric underpinnings, theoretical constructs, and potent cosmological applications. The theoretical implications are profound, providing an alternative and potentially insightful approach to gravitation contrary to the curvature-centric paradigm of GR. As astrophysical observations continue to challenge our understanding of fundamental physics, teleparallel frameworks offer promising avenues for addressing outstanding puzzles in modern cosmology. Future penetrations into quantum gravity could further delineate TG's role in the continuum of gravitational physics.