Detectability of Torus Topology

Abstract

AbstractThe global shape, or topology, of the universe is not constrained by the equations of General Relativity, which only describe the local universe. As a consequence, the boundaries of space are not fixed and topologies different from the trivial infinite Euclidean space are possible. The cosmic microwave background (CMB) is the most efficient tool to study topology and test alternative models. Multi-connected topologies, such as the 3-torus, are of great interest because they are anisotropic and allow us to test a possible violation of isotropy in CMB data. We show that the correlation function of the coefficients of the expansion of the temperature and polarization anisotropies in spherical harmonics encodes a topological signature. This signature can be used to distinguish an infinite space from a multi-connected space on sizes larger than the diameter of the last scattering surface (DLSS). With the help of the Kullback-Leibler divergence, we set the size of the edge of the biggest distinguishable torus with CMB temperature fluctuations and E-modes of polarization to 1.15 DLSS. CMB temperature fluctuations allow us to detect universes bigger than the observable universe, whereas E-modes are efficient to detect universes smaller than the observable universe.