We humans have been trying to figure out the shape of the universe since Antiquity. In ancient Greece, using bare-hands science and the power of their imaginations, philosophers Leucippus and Democritus envisaged an infinite universe. Aristotle fathomed it as a finite ball, with the Earth at the centre. Aristotle’s view prevailed and went mostly unchallenged in Western society for almost 2,000 years, until the invention of the telescope by Galileo in 1608. In 1917, when Einstein applied his geometrical explanation of gravity (his famous theory of relativity) to the questions of cosmology, he recycled a three-sphere scenario previously posited by Bernhard Riemann. All hypotheses, dating from ancient times to today, remain contentious. Technological advances over the last decade, however, have increased our chances of actually finding an answer to this age-old question.
Currently, there are three classes of shapes considered to be contenders, each based on a posited curvature of the universe: the spherical, the hyperbolic (saddle-shaped), and, the favourite of the moment, the “standard model,” which predicts an infinite and flat universe, forever expanding under the pressure of an (as yet) inexplicable “dark energy.” Now, Weeks and a team of Parisian cosmologists have found a new plausible model: the dodecahedron.
Five years ago, out of the blue, Weeks got an email from a cosmologist asking a technical question about the vibrational modes of a particular spatial manifold. Weeks did not have an answer to the query, but he offered to find one and asked, by the way, why the interest? When he discovered cosmologists were expecting hard data from outer space that would allow them to test the shape of the real universe, his already wide eyes widened some more. “This was a dream come true for a theoretical mathematician,” he says, “to finally have some data on the way.” Before he knew it, he was collaborating with the cosmologists.
Weeks’s task, through a continuous process of back-and-forth with his cohorts, was to provide the raw materials. First he determined which geometrical structures were plausible shapes for the universe by playing around within existing classes. Second, he calculated how each would behave in space. Both tasks involved a lot of sitting around and pondering, pencil and paper in hand, fiddling with the mathematics. Then he had to work up formulas to prove his hypotheses. Finally, he devised computer programs to run his formulas. After that, it was over to the cosmologists – Jean-Pierre Luminet, Alain Riazuelo, Roland Lehoucq, and Jean-Philippe Uzan. They plugged Weeks’s geometric formulas into simulations based on assumptions about the physics of the universe. The results would be compared to reality – that is, the data soon to arrive from NASA’s Wilkinson Microwave Anisotropy Probe (WMAP).
The WMAP probe was sent up to map cosmic microwave background radiation, the echo of the origin of the universe – the assumed big bang – and provide data about its early history and scale. One particularly useful indicator of universe topology is the temperature fluctuations of radiation emanating from the big bang. In a recent article in Nature magazine, Weeks and the cosmologists explained these fluctuations by comparing them with the sound waves of musical harmonics:
“A musical note is the sum of a fundamental, a second harmonic, a third harmonic, and so on. The relative strengths of the harmonics – the note’s spectrum – determines the tone quality, distinguishing, say, a sustained middle C played on a flute from the same note played on a clarinet. Analogously, the temperature map on the microwave sky is the sum of spherical harmonics. The relative strength of the harmonics – the power spectrum – is a signature of the physics and geometry of the universe.”
When the WMAP data arrived in February, 2003, it confirmed the prevailing infinite-flat model, but only partially. All the small and medium-sized temperature waves were present as predicted. But the model failed when it came to finding the broad wavelengths that should exist in such a large and infinite universe. The probe found none. One explanation, says Weeks, is that space simply isn’t that big and thus could never produce such large waves in the first place. “A violin is never going to play the low notes of a cello because a violin’s strings aren’t long enough to support such a long sound wave,” he says. “It’s the same with the universe. Its waves cannot be larger than space itself.”
Enter the finite-dodecahedron model. The behaviour Weeks predicted for a dodecahedral universe matched all the WMAP data. “It was a very pleasant surprise,” he says. “Our model fit even better than we expected.” The future of dodecahedral space, however, still faces two major challenges. First, the model’s calculations of spatial curvature must be compared to forthcoming, more precise data from the Planck Probe (launching in 2007). The results of the probe will either fine-tune Weeks’s model or refute it entirely. Second, it must pass the “circles-in-the-sky” test. If the model is correct, a computer-coded search should be able to detect six pairs of matching circles across the cosmic horizon – echoes from the big bang vibrating against the twelve faces of the dodecahedron universe.
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