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.

Cosmic Crystallography
Cosmic crystallography looks at the 3-dimensional observed distribution of high redshift sources (e.g. galaxy clusters, quasars) in order to discover repeating patterns in their distribution, much like the repeating patterns of atoms observed in crystals. They showed that "pair separation histograms" are in most cases able to detect a multi- connected topology of space, in the form of spikes clearly standing out above the noise distribution as expected in the simply-connected case. The researchers have particularly studied small universe models, which explain the billions of visible galaxies are repeating images of a smaller number of actual galaxies.

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.

Spherical Lensing
Until recently, the search for the shape of space had focused on big bang models with flat or negatively curved spatial sections. Recently however, a combination of astronomical (type I supernovae) and cosmological (temperature anisotropies of the cosmic background radiation) observations seem to indicate that the expansion of the universe is accelerating, and constrain the value of space curvature in a range which marginally favors a positively curved (i.e. spherical) model. As a consequence, spherical spaceforms have come back to the forefront of cosmology.

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 Poincar� space. The Poincar� 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 Poincar� 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
Experimental projects related to cosmic crystallographic methods and to the detection of correlated pairs of circles in the cosmic background radiation are currently underway. Presently, the data are not good enough to provide firm conclusions about the topology of the Universe. Fortunately breakthroughs are expected in the coming decade: high redshift surveys of galaxies will be completed, and high angular resolution maps of the cosmic radiation temperature will be provided by the MAP and Planck Surveyor satellite missions. The new data will provide clues to the shape of the Universe we live in, a question that puzzles not only cosmologists, but also philosophers and artists.

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 Th�orique, Orsay, France), Evelise Gausmann (Universit� de Sao Paulo, Br�sil) et Jeffrey Weeks (Canton, USA).

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