What is the shape of space? Is it finite or infinite? Is it connected, has it "edges", "holes" or "handles"? This cosmic mystery, which has puzzled cosmologists for more than two thousands years, has recently been enlightened by a breakthrough in a new field of research: cosmic topology.
An international team involving researchers from France, the United States and Brazil recently filled a major gap in the field. They propose surprising universe models in which space, spherical yet much smaller than the observable universe, generates an optical illusion on a cosmic scale (topological lens effet).
Einstein's general relativity theory teaches us that space can have a positive, zero or negative constant curvature on the large scale, the sign of the curvature depending on the total density of matter and energy. The celebrated big bang models follow, depicting a universe starting from an initial singularity and expanding forever or not. However, Einstein's theory does not tell us whether the volume of space is finite or infinite, or what its overall topology is.
Fortunately, high redshift surveys of astronomical sources and accurate maps of the cosmic microwave background radiation are beginning to hint at the shape of the spatial universe, or at least limit the wide range of possibilities.
As a consequence, cosmic topology has gained an increased interest, as evidenced by the special session "Geometry and Topology of the Universe" organized by the American Mathematical Society during its 2001 meeting held last October in Williamstown, Mass.
Three French cosmologists were invited to present to an audience of mathematicians, physicists and astronomers the statistical method they recently devised for detecting space topology: cosmic crystallography.
The two pictures below visualize the "topological lens effect" generated by a multi-connected shape of space, and the way the topology can be determined by the pair separation histogram method.
In their latest work, to be published in Classical and Quantum Gravity, the authors and their Brazilian and American collaborators fill a gap in the cosmic topology literature by investigating the full properties of spherical universes. The simplest case is the celebrated hypersphere, which is finite yet with no boundary.
Actually there are an infinite number of spherical spaceforms, including the lens spaces and the fascinating Poincaré space. The Poincaré space is represented by a dodecahedron whose opposite faces are pairwise identified, and has volume 120 times smaller than the hypersphere. If cosmic space has such a shape, an extraordinary "spherical lens" is generated, with images of cosmic souces repeating according to the Poincaré space's 120-fold "crystal structure".
The authors give the construction and complete classification of all 3-dimensional spherical spaces, and discuss which topologies are likely to be detectable by crystallographic methods. They predict the shape of the pair separation histogram and they check their prediction by computer simulations.
The Future of Cosmic Topology
The authors are Jean-Pierre Luminet (DARC/LUTH, Observatoire de Paris, France), Roland Lehoucq (Service d’Astrophysique, CEA Saclay, France), Jean-Philippe Uzan (Laboratoire de Physique Théorique, Orsay, France), Evelise Gausmann (Université de Sao Paulo, Brésil) et Jeffrey Weeks (Canton, USA).
Observatoire de Paris
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