Peer before you through the cosmos and you'll see the back of your head. Well, as Dan Falk reports, that's one model of the universe modern Magellans are suggesting
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The Independent Culture
IN 1519, Ferdinand Magellan set sail from Spain in search of a westward route to the Far East. Although Magellan himself never made it back, one of his captains eventually completed a full circuit of the globe - proving once and for all that the Earth was round. Sail far enough to the west, and you return from the east.

Today, nearly 500 years later, a similar quest is under way. The body in question, however, is no longer our lonely blue planet; rather, the shape of the universe itself - or, more precisely, its topology - is now at issue. This is a subject cosmologists have long wondered about, though the arguments, until now, have been largely theoretical. That may change, however, with the launch of two satellite- based experiments within the next few years.

Topology is the branch of mathematics that asks how something is connected. A simple object like a sheet of typing paper is said to be "singly connected" - an ant that strays too close to the edge will fall off. But the surface of the Earth, as Magellan's crew discovered, is "multi- connected" - you can reach India from Spain by sailing either east or west. The earth's surface, though two- dimensional, is curved; what seems like a straight line to the navigator is actually a circle on a three-dimensional globe. Take a rocket ship into space, and your path will again seem like a straight line - but that, too, may be curved. (Unfortunately, we can't view this curvature from the "outside" as we could with the Earth. To do so we would have to be able to perceive four-dimensional space.)

"Many people would say that the universe is connected in a trivial way," explains physicist Glenn Starkman, "which means that no matter how far east you travel, you'll never come back from the west, or from any other direction." But, he goes on, there are many other plausible alternatives. The simplest of these, he says, is a universe in which travelling to the east brings you back from the west; heading north brings you back from the south; and going up means that you eventually return from beneath your starting point. But there are more bizarre variations: a traveller heading east, for example, might come back from the west, but find himself "upside-down" relative to his starting position. In other words, the path from A to B may not only be curved, but given a topological "twist" along the way.

Last autumn, Starkman organised a workshop at his home base of Case Western Reserve Uni- versity in Cleveland, dedicated to the interplay between cosmology and topology. The ultimate goal - finding the precise topology of the universe - is a grand quest, but one that has received relatively little attention outside this rarefied world on the boundary between physics and mathematics.

Part of the problem is a confusion over the exact distinction between topology and geometry. Geometry refers to the curvature of space. That's a relatively familiar idea; after all, Einstein showed more than 80 years ago that matter can warp the space around it - this was, in fact, the cornerstone of his general theory of relativity. Since then, we've come to realise that the universe itself has a curvature, linked to its total mass: the more matter there is in the universe, the greater the curvature. We also know that this value is linked to the fate of our expanding cosmos. Astronomers believe the universe began with a colossal explosion known as the Big Bang, some 15 billion years ago. Since that time, it's been steadily expanding, all of the galaxies flying away from one another. But if the universe is massive enough, this expansion will eventually cease; at some remote time in the future, the galaxies will stop - and begin to move inward. Ultimately, the cosmos will contract down in a "Big Crunch", essentially the reverse of the Big Bang.

To the mathematician, such a universe is considered "positively curved" and is analogous to a sphere (with an extra dimension thrown in). However, if the universe has less than a critical mass, it will continue to expand indefinitely; the galaxies will continue to recede, without end. Such a cosmos is described as "negatively curved". There's also a possible model where the universe has exactly the critical mass, and expands forever - but just barely. This is called a geometrically "flat" universe. Although these quantities sound rather abstract, astronomers are, in fact, able to come up with some concrete numbers. By studying the motion of galaxies and clusters of galaxies, they have been able to estimate both the total mass of the universe and the expansion rate.

The total mass of the universe, therefore, determines its overall curvature or geometry, but - and this is the tricky part - the total mass does not determine its topology. It does put constraints on the topology, but does not pin it down. "Einstein's theory doesn't really tell you about the larger-scale property of space," explains Berkeley physicist Janna Levin. "It doesn't tell you whether it's connected in some way, or if maybe it folds back on itself. At least, it doesn't specify it exactly." Princeton physicist David Spergel adds, "I think this is something mathematicians have known for a long time, but if you pick up a textbook on cosmology, there's been confusion on this issue."

Some physicists and astronomers also worry that a multi-connected universe is unnecessarily complicated. They cite "Occam's Razor", a principle that says that the simpler of competing models is usually the correct one. But, say some cosmologists and topologists, Occam's Razor cannot provide an answer here. "It all depends on what you mean by 'simple'," says Dmitri Pogosyan of the Canadian Institute for Theoretical Astrophysics (CITA) in Toronto. Is a singly connected universe - necessarily infinite - really simpler than a finite universe which folds in on itself? Perhaps, Pogosyan suggests, the simplest model of all would be a positively curved universe: such a cosmos would have a finite volume and the topology would be simple - yet recent observations have just about ruled out such a universe. Simplicity, it seems, may not be much of a guide.

The latest evidence seems to point to a universe of negative curvature - in other words, one in which gravity will never halt the on-going expansion. Although a negative curvature rules out certain topologies, the number of choices that remains is still pretty large. In fact, the list is endless. "There are an infinite number of topologies for negatively curved space," explains Cambridge physicist Neil Cornish. And each of these possible universes "has some number of loops that you can go around, and come back to where you started from".

Cornish points out that a negatively curved universe is not necessarily infinite. Most of these possible topologies, he says, are finite. One can imagine cases analogous to the surface of an infinitely long cylinder, closed in one direction but infinite in another. But the most likely topologies, he says, are "typically wrapped up in every direction". Cornish worked with Glen Starkman at the University of Toronto, where Cornish did his PhD; within weeks of graduation, he took a position with Stephen Hawking's relativity group in Cambridge.

All this talk of topology would just be philosophy if there were no way to measure it. But how do you measure something so intangible? The key to making observational tests of topology involves those "multiple paths" that you - or a beam of light - could take to get from A to B in a multi- connected universe. In the simplest of these universes, an observer who peers directly ahead would see - in theory at least - the back of his own head. In other words, some of the light that reflects off your head travels one "full circuit" of the universe and hits you front-on. Because you could also see light that has made two or more circuits, you in fact see an infinite number of images of yourself, receding into the distance in each direction - rather like the view in a barber's shop with mirrored walls. (This analogy, however, isn't perfect; for one thing, the images you see in a multi-connected universe would not be mirror-reversed.)

Because the universe is so large, of course, it is foolish to talk about seeing multiple images of a single person; instead, we should be looking for multiple images of the Milky Way galaxy - or perhaps of the giant galactic grouping known as the Virgo Cluster. But even if we live in a multi- connected universe, seeing these images won't be easy. For one thing, it would take billions of years for the light to make a circuit of the universe. Even if we could see the Milky Way "out there", we'd be seeing it as it looked billions of years ago. As one scientist put it, it would be like trying to recognise a galaxy by looking at its baby pictures.

So instead of trying to view these multiple images directly, most cosmologists are pinning their hopes on the Cosmic Microwave Background radiation - the faint "echo" of the Big Bang. By a careful analysis of this radiation, the scientists believe, it should be possible to decode the "signature" of the particular topology that governs our universe.

The challenge that Cornish and his colleagues face, then, is one of pattern recognition. As a starting point, they'll be using a high-resolution map of the microwave sky - essentially a photograph of the heavens showing which areas are hotter (emitting more radiation) and which are cooler (emitting less radiation) than the average. Then, scanning different sections of this image, they'll search for patches of radiation that have the same temperature profile. (For rather technical reasons, these patches will be circular in shape.) It's called a "pixelated search" - not a simple task, but within the reach of modern high-speed computers.

"What we're planning to do is to take these data - given to us as hot and cold spots around a balloon - and [extract] all of the possible circles you can draw on it," says Cornish. If they find any pairs of matched circles, it will be "a clear signature that we're living in one of these multi- connected universes". In fact, Cornish says, it would only take three such pairs for cosmologists - with the help of the mathematicians, of course - to determine the precise topology of the universe.

One big unknown is the size of the universe; and that could be a problem. The universe may be topologically curved or twisted, but on a scale that's simply too big to see. Remember, the light from the most distant objects astronomers can see has taken billion of years to reach the earth. The light from more remote objects - if they're out there - has not yet reached us. For this reason, astronomers sometimes speak of the "observable universe". But if we do live in a multi-connected universe, and it's curved on a relatively small scale - that is, if the topology shows up on a scale that's smaller than the size of the visible universe - then those matched circles would be awaiting discovery in the microwave background radiation.

At the moment, however, we don't have a map of the microwave sky with adequate resolution. The best images so far come from the Cosmic Background Explorer (COBE), a highly success- ful Nasa satellite launched in 1989. Pogosyan, together with Tarun Souradeep and CITA director Richard Bond, has been using computers to model various topologies in order to predict what patterns will be visible in the microwave sky. The models they have tested so far are not reflected in the COBE data - but there are many more models, still untested, that the COBE data do not rule out.

What is really needed, however, is a clearer picture of the microwave background radiation, and this may come in just a few years' time with the launch of two new satellites. The first is MAP, the Microwave Anisotropy Probe, a NASA mission set for launch in 2000. The other is the European Space Agency's Planck satellite, due to get under way in 2005. The MAP satellite should be able to measure the microwave background radiation with a resolution of just 0.2 degrees (about half the angular size of the full moon), and Planck's resolution may be two or thee times better still (by comparison, COBE has a resolution of just seven degrees).

If their quest is successful, these cosmologists and topologists will bring about a profound change in the way we view our universe. Granted, there may not be the sort of practical implications that Magellan's discovery had for the 16th-century world; space travel over cosmological distances is almost certainly still centuries away. On the other hand, just knowing that the Earth was round brought about a pervasive change in thinking. Finding evidence for topology - especially any evidence for a multi-connected universe - could bring just such a change in our world-view.

"It's the end of a process that started with Copernicus," muses Starkman. "We gradually developed this Copernican notion that there is no special place in the universe - that every place is equivalent. And if we find this topology, we'll discover that, no, every place actually isn't equivalent." Instead, he says, we'll have found a kind of "You Are Here" sign for the cosmos. "It will be the ultimate end of the Copernican revolution. We'll have found our place in the universe, and in a sense there may be no more universe to find." Of course, the universe could be too large to display topology, or it could turn out to be just singly connected. But even that discovery would be a leap forward in our understanding of the cosmos. One way or another, our picture of the universe is likely to change dramatically in the next few years. !