Detectability of Cosmic Topology in Flat Universes

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  • 27 February 2002
Abstract
A combination of recent astrophysical and cosmological observations seems to indicate that we live in an accelerating Friedmann-Lemaitre-Robertson-Walker (FLRW) universe whose spatial sections are nearly or exactly flat ($\Omega_0 \simeq 1$). Motivated by this, and in order to complete our previous investigations on the detectability of nearly flat hyperbolic and spherical universes, we study here the problem of observational detection of the topology of FLRW universes with (exactly) flat spatial sections. To this end, we first give a complete description of the diffeomorphic classification of compact flat 3-manifolds, and determine the expressions for the injectivity radii ($r_{inj}$), and for the volume of each class of Euclidean 3-manifolds. There emerges from our calculations the undetectability conditions for each (topological) class of flat universes. We also study how the bounds provided by recent cosmological observations can be used to identify flat models having undetectable topologies. To materialize and quantify the study of the detectability of flat topologies we use the undetectability conditions and an assumption by Bernshtein and Shvartsman which permits to establish a relation between topological typical lengths to the dynamics of flat models. As a particular result we show that none of the models of two specific classes of flat universes which satisfy the Bernshtein and Shvartsman condition has an undetectable topology, even if current catalogues of clusters of galaxies are used. A modified version of Bernshtein-Shvartsman assumption is also suggested and used to construct a great number of flat universes with undetectable topology, even if cosmic microwave background radiations is used.

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