Alternative and Averaged Theories of Gravity

广义相对论(GR)已经非常成功地高精度地解释了广泛的天体物理观测和实验室实验。然而，如果没有引入另一个叫做暗能量的“奇异”磁场，GR就无法解释目前观测到的宇宙加速现象。对于暗能量的等效效应，另一种解释或许可以用另一种引力理论来解释，而不需要添加一个奇异的磁场。未来五年有两个主要目标。该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质的，因此可能会因为这些假设而失去一些不平凡的动力学效应。在宇宙尺度上，人们应该考虑小尺度不均匀对整体宏观动力学的影响。解决这个问题的一个建议是发展一个平均引力理论。由于定义协变平均过程的困难和引力场方程的非线性性质，基于GR的平均引力理论是具有挑战性的。相当于广义相对论(TEGR)的遥平行在动态上等同于广义相对论，因此为我们提供了一个发展平均引力理论的机会。假设TEGR在中小尺度上描述重力，那么人们可以使用已经开发的协变平均过程来确定描述平均或宏观时空中的动力学的场方程。随着我们转向精密观测宇宙学，基于平均引力理论的模型分析变得更加关键。这些平均理论可以在理解宇宙的能量平衡方面做出微小但重要的贡献。此外，这些平均引力理论在解决哈勃张力方面显示出了希望--为什么两组独立的宇宙学观测为哈勃参数产生了不同的值。我的研究计划中的第二个目标源于我初步发展的平均引力理论。遥平行引力是一类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GR中那样通过度规编码在曲率中，而是编码在通过框架场的零曲率连接的扭转中。一类远程平行引力理论是基于扭标量T，f(T)的函数。F(T)遥平行重力中当前的挑战之一是为惯性系(或等效地为系和连接系)选择一个可行的ansatz，特别是当几何上施加对称约束时。我将开发一个清晰而简明的程序，可以用来在远程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.