Large-scale geometry of the Universe

The large scale geometry of the late Universe can be decomposed as R$\times {\Sigma }_3$, where R stands for cosmic time and ${\Sigma }_3$ is the three dimensional spatial manifold. We conjecture that the spatial geometry of the Universe's spatial section ${\Sigma }_3$ conforms with the Thurston-Perelman theorem, according to which the geometry of ${\Sigma }_3$ is either one of the eight geometries from the Thurston geometrization conjecture, or a combination of Thurston geometries smoothly sewn together. We assume that topology of individual geometries plays no observational role, i.e. the size of individual geometries is much larger than the Hubble radius today. We investigate the dynamics of each of the individual geometries by making use of the simplifying assumption that our local Hubble patch consists of only one such geometry, which is approximately homogeneous on very large scales, but spatial isotropy is generally violated. Spatial anisotropies grow in time in decelerating universes, but they decay in accelerating universes. The thus-created anisotropy problem can be solved by a period of primordial inflation, akin to how the flatness problem is solved. Therefore, as regards Universe's large scale geometry, any of the Thurston's geometries should be considered on a par with Friedmann's geometries. We consider two observational methods that can be used to test our conjecture: one based on luminosity distance and one on angular diameter distance measurements, but leave for the future their detailed forecasting implementations.