Alternative and Averaged Theories of Gravity

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General Relativity (GR) has been very successful in explaining a wide range of astrophysical observations and laboratory experiments to high precision. However, GR offers no explanation for the currently observed acceleration of the universe without the introduction of an additional "exotic" field, called Dark Energy. A different explanation for the equivalent effects of dark energy can perhaps be explained within an alternative theory of gravity possibly without the need to add an exotic field. There are two primary objectives over the next five years. The first objective of the research program attempts to define an averaged theory of gravity. Since one commonly assumes the universe is isotropic and spatially homogeneous on cosmological scales, there may be nontrivial dynamical effects that are lost due to these assumptions. On cosmological scales, one should incorporate the effects of small scale inhomogeneities on the overall macroscopic dynamics. One proposal to address the issue is to develop an averaged theory of gravity. An averaged theory of gravity based on GR is challenging due to the difficulty in defining a covariant averaging procedure and the nonlinear nature of the gravitational field equations. The Teleparallel Equivalent to General Relativity (TEGR) is dynamically equivalent to GR and therefore offers us an opportunity to develop an averaged theory of gravity. Assuming that TEGR describes gravity on small to intermediate length scales, one can then use a covariant averaging procedure already developed to determine the field equations describing the dynamics in the averaged or macroscopic space-time. As we move to precision observational cosmology, analysis of models based on averaged theories of gravity becomes much more critical. These averaged theories could yield small but important contributions in the understanding of the energy budget of the universe. Moreover, these averaged theories of gravity show promise in resolving the Hubble tension - why two independent sets of cosmological observations yield different values for the Hubble parameter. The second objective within my research program grew out of my preliminary development of an averaged theory of gravity. Teleparallel gravity is a promising class of alternative theories of gravity that assumes that the dynamics of the gravitational field are no longer encoded in the curvature via the metric as in GR, but are encoded in the torsion of a zero curvature connection via the frame field. One class of teleparallel gravity theories are based on a function of the torsion scalar T, f(T). One of the current challenges in f(T) teleparallel gravity is in the selection of a viable ansatz for the inertial frame (or equivalently for the frame and connection), in particular, when symmetry constraints are imposed on the geometry. I will develop a clear and concise procedure that can be used to impose consistent geometrical symmetries in teleparallel gravity.