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

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 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

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 Théorique, Orsay, France),
Evelise Gausmann (Université de Sao Paulo, Brésil) et Jeffrey Weeks
(Canton, USA).


E. Gausmann, R. Lehoucq, J.-P. Luminet, J.-P. Uzan and J. Weeks:
"Topological lensing in spherical spaces", Class. Quant. Grav.,
(2001) 18, 1-32 (http://arXiv.org/abs/gr-qc/0106033)

R. Lehoucq, J.-P. Uzan and J.-P. Luminet: "Limits of crystallographic
methods for detecting space topology", Astronomy and Astrophysics
(2000), 363, 1 (http://arXiv.org/abs/astro-ph/0005515)

J.-P. Luminet: L’Univers chiffonné, Fayard, Paris, 2001, 369 p.

[Image 1: http://www.obspm.fr/actual/nouvelle/dec01/Image1.gif (100KB)]
In a multi-connected Universe, the physical space is identified to
a fundamental polyhedron, the duplicate images of which form the
observable universe. Representing the structure of apparent space is
equivalent to representing its " crystalline " structure, each cell of
which is a duplicate of the fundamental polyhedron. Here is depicted
the closed hyperbolic Weeks space. As viewed from inside, it gives
the illusion of a cellular space, tiled par polyhedra distorted with
optical illusions (here only one celestial object is depicted, namely
the Earth) © Jeffrey Weeks

[Image 2:
Left, http://www.obspm.fr/actual/nouvelle/dec01/Image2.gif (10KB)
Right, http://www.obspm.fr/actual/nouvelle/dec01/Image3.gif (5KB)]
Sky map simulation in hypertorus flat space (left). The fundamental
polyhedron is a cube with length = 60% the horizon size and contains
100 "original" sources (red dots). One observes 1939 topological images
(blue dots). The Pair Separation Histogram (right) .exhibits spikes
which stand out at values and with amplitudes depending on the
topological properties of space.


Jean-Pierre Luminet

DARC-LUTH, CNRS-Observatoire de Meudon

Tél: 33 1 45 07 74 23

Fax: 33 1 45 07 79 71

E-mail: Jean-Pierre.Luminet@obspm.fr