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Not wrapped up yet

02 Feb 2009

The Wraparound Universe
Jean-Pierre Luminet (translated by Eric Novak)
2008 A K Peters
£24.50/$39.00 hb 400pp

Ring of truth

Cosmologists ask questions about the history and evolution of the universe on the largest spatial and temporal scales. How fast is the universe expanding? What are the densities of the various sorts of mass–energy therein? What is its future? And how and when did it all begin? These cosmic questions may at first seem far removed from the branch of mathematics known as topology, which is the study of shapes at their most basic. It is not about the angles, corners and planes of geometry, but of pliable shapes and the handles and holes that cannot be changed by bending and stretching. Topologically, a ball is the same as a glass and a single-handled coffee mug is the same as a ring, but clearly their geometries are different. Similarly, we must separate questions about the geometry of the universe from those about its topology.

Both cosmology and topology reach back to the ancient Greeks and, likely, to the first humans who had any time to think at all. However, it is only in the last couple of centuries that the two have become proper sciences. Each relies on what has come to be known as non-Euclidean geometry, a branch of mathematics that forms a cornerstone of Einstein’s general theory of relativity and is also required to enumerate the possible topologies that could describe the universe.

The Wraparound Universe by the French cosmologist Jean-Pierre Luminet is not just a twofold popular overview of the union of these two sciences, but also a none-too-subtle plug for the author’s idea that the universe might have the large-scale equivalent of handles or holes. The universe, he argues, could be multiply connected, just like a computer-game “world” where moving off the right edge of the screen brings you back onto the left, and moving off the top brings you back to the bottom (see “A cosmic hall of mirrors”).

Luminet’s argument builds on the fact that the possible topologies for a particular surface (a computer screen, say, or the fabric of space–time) are related to the ways in which you can tile, or tessellate, the surface using repeated patterns — like the ones in M C Escher prints hanging in college dorms worldwide. The individual repeating pattern, known as the fundamental domain, determines the underlying topology.

As an example, let us return to the computer-game “world”, for which the fundamental domain is a rectangle. The topology of the computer game “world” is a torus, like the surface of a doughnut. To visualize this, take a rectangular rubber sheet (representing the computer screen) and wrap the left edge against the right, making a cylinder. Now join the two ends. Note that the computer game and a doughnut are topologically equivalent, but they have different geometries: the game is flat, while the doughnut is curved.

We can expand this notion of tiling a surface to higher dimensions easily enough. One possible 3D fundamental domain that tiles Euclidean 3D space, for example, is a cube. Crucially, the same process can be extended to curved surfaces and multidimensional spaces as well — which is just what is needed to make the link to the 4D curved manifold that cosmologists are starting to describe via high-precision measurements of the universe.

Luminet’s book covers these two disciplines, cosmology and topology, and the history of their overlap up to and including these high-precision measurements. He concentrates on observations of the Cosmic Microwave Background (CMB), repeated patterns on which could be a mark of primordial topology. He closes the book proper with the first hints from the background-measuring COBE satellite, together with the MAXIMA and BOOMERANG balloon-based experiments that such data might be compatible with his topological ideas.

In the new English-language edition being reviewed here, Luminet updates the original text (written in 2001) by adding an appendix that discusses the analysis he and colleagues have performed on the more recent data. Their analysis, he claims, reveals a strong preference for a fundamental domain in the shape of a dodecahedron — albeit one inhabiting a curved hyper-spherical universe, rather than the flat 3D Euclidean geometry we can more easily picture.

Alas, a more detailed look at the relevant data obtained using NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) shows that the conclusion seems unwarranted. Among other groups, I, along with my colleagues Anastasia Niarchou and Levon Pogosian, concluded that the “power spectrum” data used in Luminet’s original work does not statistically warrant any explanation beyond the plain-vanilla standard cosmological model: a simply connected, spatially flat universe. Examining the detailed patterns of CMB fluctuations as Niarchou and I have done more recently — with a method that uses all available information — shows that Luminet’s preferred model is actually very strongly disfavoured.

Moreover, Luminet’s explanation ignores a crucial a priori weakness of his proposal: it requires the universe to have a very slightly curved geometry. This, in turn, requires the radius of the hyper-spherical universe (the curvature scale) to be comparable to the so-called Hubble distance (the distance a beam of light could have travelled since the Big Bang). Why should these numbers be nearly equal?

In fact, an interesting topology, especially paired with a curved geometry, is neither required nor particularly supported by the data. In contrast, one of the crucial features of cosmic inflation — the fact that inflation flattens the geometry of the universe so that the curvature scale becomes immeasurably large — is supported by the vast majority of cosmological data, and therefore by cosmologists (lightly mocked as the “inflation lobby” by Luminet). Nonetheless, the paired questions of the correctness of inflation and of the topology of the universe are by no means closed. We await further data, especially from European Space Agency’s Planck Satellite, which is due to be launched in April.

Luminet otherwise presents a workmanlike introduction to modern cosmology and to topology. For a popular-science book, both topics are presented at a sufficiently high level as to confuse the completely uninitiated reader, but I suspect that the audience for this book is one that has already digested some subset of other recent books, such as those by Stephen Hawking, Brian Greene, Janna Levin and João Magueijo. To differentiate this book, Luminet attempts to have its form reflect its content: the reader is presented with arrows and page numbers in the margins, supposedly giving the book multiple connections and a “tree-like structure”. I admire the attempt, and it usefully supplements the index, but I found it easier to read the book straight through (although a Web version might be more successful).

Nowadays, science (or at least physics) progresses not by sustained argument in books but by short snippets. Books serve to consolidate knowledge and present it to students or to the public. So it is heartening to read a book like The Wraparound Universe that not only summarizes the state of the art of a field but also argues for an idea, even if that idea is, in this case, likely to be incorrect.

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