Research

Probing topology of the universe

Because the Einstein equation describes the local property of the spacetime, we can consider cosmological models that are globally anisotropic or inhomogeneous although they are locally isotropic and homogeneous and having a constant spatial curvature and density. In general, the spatial topology of these models are non-trivial. In other words, these spaces have a particular "shape of space". For instance, one can put a loop on these spaces that cannot be contracted to a point by continuous deformation. A surface of a donut (2-torus) is a simple example. In this surface, one can consider two types of loops that cannot be contracted to a point. Because the light goes around the loops many times, an observer would think that the world extends infinitely. To be precise, it would seem to be a miracle world in which you can see old images of yourself consecutively in distant place as if one is looking into a kaleidoscope. In most literature, cosmological models whose spaces have zero or negative curvature are categorized as "open" while those with positive curvature are categorized as "closed" ones. However, this is not true. If one permits non-trivial topology, these open spaces can be "closed". In other words, one can consider various finite universes without boundary for spaces with vanishing or negative curvature. Because these spaces have a non-trivial "shape of space", the spatial global isotropy or homogeneity is broken. Therefore, if one can observe a broken symmetry on large scales, the spatial finiteness of the universe can be proved in a scientific way.