The shape of space after WMAP data

Abstract

What is the shape of space is a long-standing question in cosmology. I review recent advances in cosmic topology since it has entered a new era of experimental tests. High redshift surveys of astronomical sources and accurate maps of the Cosmic Microwave Background radiation (CMB) are beginning to hint at the shape of the universe, or at least to limit the wide range of possibilities. Among those possibilites are surprising "wrap around" universe models in which space, whatever its curvature, may be smaller than the observable universe and generate topological lensing effects on a detectable cosmic scale. In particular, the recent analysis of CMB data provided by the WMAP satellite suggest a finite universe with the topology of the Poincaré dodecahedral spherical space. Such a model of a "small universe", the volume of which would represent only about 80 % the volume of the observable universe, offers an observational signature in the form of a predictable topological lens effect on one hand, and rises new issues on the early universe physics on the other hand.Keywords: Cosmology; Models of universe I. THE SHAPE OF THE UNIVERSEThe problem of the global shape of the universe can be decomposed into three intertwined questions.First, what is the space curvature? In homogeneous isotropic models of relativistic cosmology, there are only three possible answers. Three-dimensional space sections of spacetime may have zero curvature on the average -in such a case, two parallel lines keep a constant space separation and never meet, as in usual Euclidean space, sometimes called "flat space". Or space sections can be negatively curved, such as two any parallels diverge and never meet (such a space is the three-dimensional analogue of the Lobachevsky hyperbolic plane). Eventually, they can be positively curved, in which case all parallels reconverge and cross again (like on the twodimensional surface of a sphere).The property for physical space to correspond to one of these three possibilities depends on the way the total energy density of the Universe may counterbalance the kinetic energy of the expanding space. The normalized density parameter Ω 0 , defined as the ratio of the actual density to the critical value that an Euclidean space would require, characterizes the present-day contents (matter and all forms of energy) of the Universe. If Ω 0 is greater than 1, then space curvature is positive and geometry is spherical; if Ω 0 is smaller than 1 the curvature is negative and geometry is hyperbolic; eventually Ω 0 is strictly equal to 1 and space is Euclidean.The second question about the shape of the Universe is to know whether space is finite or infinite -equivalent to know whether space contains a finite or an infinite amount of matter-energy, since the usual assumption of homogeneity implies a uniform distribution of matter and energy through space. From a purely geometrical point of view, all positively curved spaces (called spherical spaces whatever their topology) are finite, but the converse is not true : flat (Eu...

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