A finite dodecahedral Universe

Cosmologists study the topology of space by analyzing in great details the temperature fluctuations of the fossil Cosmic Microwave Background [note a]. The standard cosmological model describes the Universe
as a flat infinite space in eternal, accelerated expansion under the effect of a repulsive "dark energy". The data collected by the NASA satellite WMAP (Wilkinson
Microwave Anisotropy Probe), which has recently produced a high resolution map of the CMB, allowed to check the validity of such an expansion model. Temperature fluctuations on small and mean scales (i.e.
concerning regions of the sky of relatively modest size) are compatible with the infinite flat space hypothesis. However, on angular scales larger
than 60°, the observed correlations are notably weaker that those predicted by the standard model. Thus the scientists are looking for an alternative.

CMB temperature anisotropies essentially result from density fluctuations of the primordial Universe : a photon coming from a denser region will
loose a fraction of its energy to compete against gravity, and will reach us cooler. On the contrary, photons emitted from less dense regions will be received hotter. The density fluctuations result from the superposition of
acoustic waves which propagated in the primordial plasma - see previous AlphaGalileo press releases "Vibrations of the cosmic drumhead" (6 jan 03) and "A small spherical Universe?" (12 nov 01).

The franco-american team of scientists has recently developed complex theoretical models to reproduce the amplitude of such fluctuations, which can be
considered as vibrations of the Universe itself.
In particular, they simulated high resolution CMB maps for various space topologies [2] and were able to compare their results with real WMAP
data. Depending on the underlying topology, the distribution of the fluctuations differs. For instance, in an infinite "flat" (Euclidean) space, all
wavelengths are allowed, and fluctuations must be present at all scales.

Cosmic Harmonics

Like acoustic waves, the CMB temperature fluctuations
can be decomposed into a sum of spherical harmonics [note b]. The first observable harmonics is the quadrupole (whose wavenumer is l=2). WMAP
has observed a value of the quadrupole 7 times weaker
than expected in a flat infinite Universe. The probability that such a
discrepancy occurs by chance has been estimated to 0.2% only. The octopole (whose wavenumber is l=3) is also weaker (72%) than the expected value. For larger wavenumbers up to l=900 (which
correspond to temperature fluctuations at small angular scales), observations
are remarkably consistent with the standard cosmological model.

The unusually low quadrupole value means that
long wavelengths are missing, may be because space is not big enough to sustain them. Such a situation may be compared to a vibraring string fixed at its two extremities, for which the maximum wavelength of an oscillation is twice the string length.
A natural explanation of such a phenomenon relies on a model of finite space whose size constrains the wavelengths below a maximum value. The
proposed space is the Poincaré dodecahedral space [note c].

The associated power spectrum, namely the repartition of fluctuations as a function of their wavelengths corresponding to the Poincaré dodecahedral space, strongly depends on the value of the mass-energy density parameter [note d]. There is a small interval of values within which the spectral fit is good, and in agreement with
the value of the density parameter deduced from WMAP data (1.02 plus or minus 0.02). The result is quite remarkable because the Poincaré
space has no degree of freedom. By contrast, a 3-dimensional torus, constructed by gluing together the opposite faces of a cube and which constitutes a possible topology for a finite Euclidean space, may be deformed into any parallelepiped : therefore its geometrical construction depends on 6 degrees of freedom.

The Poincaré Dodecahedral Space accounts for the low value of the quadrupole as observed by WMAP in the fluctuation spectrum, and provides a
good value of the octopole. To be confirmed, such a "soccer-ball" model of space must satisfy two experimental tests :
1) A finer analysis of WMAP data, or new data from the future European satellite "Planck Surveyor" (scheduled 2007), will be able to determine the
value of the energy density parameter with a precision of 1%. A value lower than 1.01 will discard the Poincaré space as a model for cosmic
space, whereas a value greater than 1.01 will confirm its cosmological pertinence.
2) If space has a non trivial topology, there must be particular
correlations in the CMB, namely pairs of correlated circles along which temperature fluctuations should be the same [3]. The Poincaré Dodecahedral
Space model predicts 6 pairs of circles with an angular radius of 35°. The model is therefore an ideal candidate to test the method of "matched
circles" originally devised by the American astrophysicists N. Cornish, D. Spergel and G. Starkman [4]

Notes
[a] The Cosmic Background Radiation, also called Fossil Radiation, is the relics of the radiation emitted soon after the Big Bang (about 400 000 years later), when matter and radiation decoupled.
The tiny temperature irregularities of such a radiation allow to measure the density fluctuations of the matter present at this epoch, such fluctuations being the seeds of all galaxies and galaxy clusters.
[b] The temperature fluctuations of the Cosmic Background Radiation may be decomposed into a sum of spherical harmonics , much like
the sound produced by a music instrument may be decomposed into ordinary harmonics.
The "fundamental" fixes the height of the note (as for instance a 440 hertz acoustic frequency fixes the "A" of the pitch), whereas the relative amplitudes of
each harmonics determine the tone quality (such as the A played by a piano differs from the A
played by a harpsichord). Concerning the relic radiation, the relative amplitudes of each spherical harmonics determine the power spectrum, which is a
signature of the geometry of space and of the physical conditions which prevailed at the time of CMB emission.
[c] Poincaré space may be represented by a dodecahedron (a regular polyhedron with 12 pentagonal faces) whose opposite faces are
glued after a 36° twist ; such a space is positively curved, and is a multiply
connected variant of the hypersphere, with a volume 120 times smaller.
A rocket going out of the dodecahedron by crossing a given face immediately
re-enters by the opposite face. Propagation of light rays is such that any observer whose line-of-sight intercepts one face has the illusion to see inside a copy of his own dodecahedron (since his line-of-sight re-enters the
original
dodecahedron from the opposite side).
[d] The mass-energy density parameter characterizes the contents (matter and all forms of energy) of the Universe. The curvature of space depends on the value of this parameter, usually denoted Omega. If Omega is greater than 1, then space curvature is positive and geometry
is spherical; if Omega is smaller than 1 the curvature is negative and the geometry is hyperbolic; eventually Omega is strictly equal to 1 and space is "flat" (Euclidean geometry).

Observatoire de Paris and CNRS

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A finite dodecahedral Universe

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