📝 SummarXiv

Abstract

On the largest scales, the universe appears to be almost homogeneous and isotropic, adhering to the cosmological principle. In contrast, on smaller scales inhomogeneities and anisotropy become increasingly prominent, reflecting the origin, emergence, and formation of structure in the universe. Moreover, a range of tensions between various cosmological observations may suggest it necessary to explore the consequences of departure from the ideal, uniform universe on the fundamental level. Thus, in this work, the foundation of spatially homogeneous yet anisotropic universes is studied. Specifically, when given a 3D Lie algebra of \emph{desired} Killing vector fields (as would be the case for a homogeneous yet anisotropic universe), we provide an explicit construction for the metric that has exactly those as its Killing vector fields. This construction is presented accessibly, in a directly-usable, algorithmic fashion. Some examples demonstrating the construction are worked out, including a constructive method to separate out (cosmic) time dependence in spatially homogeneous, anisotropic cosmologies.