Detectability of Cosmic Topology in Almost Flat Universes |
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3-manifold 3-sphere astronomical survey bounds on cosmological catalogues closed orientable hyperbolic CMBR compact manifold compact topologies concrete examples corresponding solution curve cosmic topology cosmological constant covering group covering space critical density curvature radius cyclic groups density parameters depth Zmar detectability of cosmic dhor exclude certain families families of possible families of topologies Figure FLRW hyperbolic universes given manifold globally homogeneous horizon radius infinite number injectivity radius inradius rmax isometry lens space topologies lens spaces L(p,q manifolds that remain maximal inradius multiple images multiply connected nearly flat hyperbolic non-trivial topologies non-zero curvature non-zero lower bound orientable hyperbolic manifolds possible topologies quasars question of detectability recent observations remain undetectable Rio de Janeiro rmar shape smallest closed geodesic SnapPea spherical manifolds subset of lens Table Tinj topologies in universes topology is undetectable topology of nearly total density undetectable and families undetectable topologies vacuum energy values of Xobs Xhor Zmax Ωο