Alternative and Averaged Theories of Gravity

广义相对论（GR）已经非常成功地以高精度解释了大量的天体物理观测和实验室实验。然而，如果没有引入额外的“外来”场，即暗能量，广义相对论无法解释目前观测到的宇宙加速现象。对暗能量等效效应的另一种解释也许可以用另一种引力理论来解释，而不需要添加一个奇异场。未来五年有两个主要目标。研究计划的第一个目标是试图定义重力的平均理论。由于人们通常假设宇宙在宇宙尺度上是各向同性和空间均匀的，因此由于这些假设，可能会丢失一些重要的动力学效应。在宇宙尺度上，我们应该考虑小尺度的不均匀性对整体宏观动力学的影响。解决这个问题的一个建议是建立一个引力平均理论。由于难以定义协变平均过程和引力场方程的非线性性质，基于GR的引力平均理论具有挑战性。广义相对论的遥平行等效理论（TEGR）在动力学上与广义相对论等效，因此为我们提供了一个发展平均引力理论的机会。假设TEGR在小到中等长度尺度上描述重力，那么人们可以使用已经开发的协变平均程序来确定描述平均或宏观时空中的动力学的场方程。随着我们转向精确观测宇宙学，基于平均引力理论的模型分析变得更加关键。这些平均理论可能对理解宇宙的能量收支做出微小但重要的贡献。此外，这些引力平均理论显示了解决哈勃张力的希望——为什么两组独立的宇宙观测结果产生了不同的哈勃参数值。我研究计划中的第二个目标源于我对引力平均理论的初步发展。遥平行引力是一种很有前途的替代引力理论，它假设引力场的动力学不再像在广义相对论中那样通过度规编码在曲率中，而是通过坐标系场编码在零曲率连接的扭转中。一类遥平行引力理论是建立在扭转标量T的函数f(T)的基础上的。当前f(T)遥平行引力的挑战之一是选择一个可行的惯性系（或等效的框架和连接）的ansatz，特别是当对称约束施加在几何结构上时。我将开发一个清晰而简洁的程序，可以用来在遥平行重力中施加一致的几何对称性。

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.