杏吧原创

As big questions go, it鈥檚 hard to get much bigger. Does space go on forever? Some cosmologists suspect not. Rather than stretching off into infinity, space might be a much smaller, more manageable place. If we could cross the cosmos in a spaceship, then like sailors circumnavigating the globe, we might find ourselves back where we started.

Unlike the Earth, though, the universe does not have to be round. Its true shape is still a mystery. Is it flat like a sheet of paper? Is it curved? Is it tied in a knot, tangled by gravity at the very beginning of the big bang? Does it even make sense to talk about the shape of something as complicated as the universe?

Close to an answer

After years of detective work, cosmologists are at last beginning to get a little closer to some answers. In their quest to discover the shape of things, they have already explored to the edge of the observable universe 鈥� and they are even hoping to go beyond.

It is easy to visualise the shape of ordinary objects because you can look at them from the outside. You can pick up a cup, view it at different angles, feel its curves and surfaces, poke a finger through the handle. But we can鈥檛 do this with the universe because we鈥檙e stuck inside it 鈥� in fact, according to some definitions, it is all that exists. There are, however, still ways to feel its shape from the inside.

Our best bet for studying the universe is in the form of ancient microwave radiation that survives from when the universe was 380,000 years old and the universe became less dense. The first complete map of this cosmic microwave background (CMB) was made by the COBE satellite in 1993. Then in 2001, NASA launched the Wilkinson Microwave Anisotropy Probe (WMAP) to produce much more sensitive and detailed maps of the CMB.

Both COBE and WMAP have revealed that the microwave sky is mottled with warmer and cooler splotches that reflect variations in the density of the youthful universe. The pattern fits modern cosmological models neatly, except for one problem: there are no fluctuations at the very largest scales. If you blur the microwave map into big enough pixels, around 60 degrees across or more, then it becomes an eerie blankness 鈥� there is no large-scale variation at all.

Statistically solid

This deficiency was originally seen by the COBE satellite. Many cosmologists thought it would go away when we had more precise measurements, but it hasn鈥檛. It is as clear as ever in the latest results from the WMAP probe, and it seems statistically solid. 鈥淚t鈥檚 99.9 per cent or better,鈥� says Glenn Starkman of Case Western Reserve University in Cleveland, Ohio.

One way to explain the lack of any large pattern is if the universe is small. If space wraps round on itself somehow, then in that map of the microwave sky we could be seeing the same bits of early universe several times over as if we were living in a hall of mirrors. 鈥淚f you look over here, you鈥檙e also looking over there,鈥� says Andrew Jaffe, a cosmologist at Imperial College London. That might blur away the large-scale microwave patterns.

There are good theoretical reasons for thinking space might be small. Some theories that attempt to unite gravity and quantum mechanics propose extra dimensions wrapped around far too tightly for us to see. Perhaps the three dimensions we experience are wrapped up too, albeit so loosely that they appear infinite to our limited view. 鈥淲hy should three dimensions be infinite while others are finite?鈥� asks Janna Levin of Columbia University. She suggests that dimensions might all have been born the same size and it so happened that three grew bigger than others.

In the process, they might have got knotted. In the very earliest moments of the universe, physicists suspect that quantum fluctuations were powerful enough to churn up space into a 鈥渜uantum foam鈥�, riddled with holes and constantly changing. 鈥淲e believe the shape of the early universe can change in some quantum way,鈥� says Jaffe.

Tangled space

Today鈥檚 universe might have grown from this tangled space, or from just a piece of it 鈥� perhaps a doughnut-shaped piece, or something more exotic.

Jaffe is among the small band of cosmologists trying to find out whether a small universe really could fit WMAP鈥檚 observations. They are playing with different topologies 鈥� ways to wrap up space-time.

Topology is a branch of mathematics concerned with the overall properties of shape that don鈥檛 depend on geometric details. A sphere is the same thing as a potato in topological terms, but a ring doughnut is different, because there are two fundamentally different ways to trace a loop around its surface: one passing through the hole, the other encircling it.

Another way to conceptualise the doughnut topology is by curling a sheet of paper into a cylinder, and then bending the ends through and gluing them together. What鈥檚 cunning is that this shows how you can create the same topology simply by defining opposite edges of the flat sheet of paper as being the same edge. An occupant of the papery flatland can then wander off one side and back on at the other, just as if the sheet were rolled up; or it can head off the top of the sheet and reappear at the bottom. It鈥檚 not curved like an ordinary ring doughnut 鈥� its geometry is different 鈥� but it is connected in the same way, so its topology is identical.

Cubic universe

And if you can do this in two dimensions, why not in three? Draw a cube in space, and simply declare that opposite faces of the cube are actually in the same place, as if you had glued them together. To an occupant of the cube, flying out through one face would mean flying straight back in through the opposite face. And there would be no apparent edges: if you lived in a cubic universe and could see far enough, you would just see the back of your own head, repeated ad infinitum.

You needn鈥檛 even start with a cube. You can take certain other shapes 鈥� a dodecahedron, say 鈥� and declare opposite faces to be identical. You can add twists, so that when you look at the back of your own head, it鈥檚 tilted sideways, or at some other angle. Some dimensions could be infinite while others are finite. The possibilities are endless.

There is one way to trim this embarrassing choice: find out how space is curved. The very possibility that space can curve only emerged with Einstein鈥檚 general theory of relativity, which describes gravity as a curvature of space-time. Earth certainly puts a dimple in space, or we would fall off; and other planets, stars and galaxies do the same. On large enough scales, those little dimples become irrelevant and the grand sweep of space could curve in one of three ways.

Pringle shaped

It could be flat, with zero curvature. In our expanding universe, however, space is only flat if it holds a precise, critical density of matter and energy. Any more stuff and the extra gravity pulls space into a dome, giving it a positive curvature. Any less than the critical density and space will warp in a more complex way, like the curve of a saddle or Pringles potato chip 鈥� down in one direction and up in another. This is negative curvature.

Astronomers have measured the curvature of our universe using the CMB. The most common size for the microwave hot and cold spots is known from theory to be a little less than 500,000 light years across. This means that they can be used as cosmic yardsticks to reveal how space bends.

Curved space acts like a lens; positively curved space would magnify these standard spots in the microwave sky, whereas negatively curved space would shrink them. In fact, the spots appear about a degree across, which suggests little magnification or shrinkage 鈥� it is close to the size that would be expected in flat space.

Kicked into touch

There could be just the slightest bias in one direction, however. According to the latest chart released in March this year, the spots appear just fractionally larger than they would in flat space, implying that the universe has positive curvature like the surface of a sphere. If that鈥檚 true, the ancient question would be all but answered: the universe does not stretch on forever. Instead, it curves around to form a finite universe. It would still be pretty big, of course, but limited, like the surface of the Earth; a closed but borderless cosmos.

One way to wrap up space in such a cosmos was suggested in 2003 by a team of cosmologists led by Jean-Pierre Luminet of the Paris Meudon Observatory in France. They showed that the first WMAP chart squared with a peculiar topology that became known, rather inaccurately, as the soccer ball universe. The idea was that our universe could be a slightly warped dodecahedron. Fly through one face in a spaceship and you would find yourself flying back in through another face. If there were no limit on how far we could see, we would see the pattern of hot and cold spots repeated exactly 120 times, smearing out those patterns at large angles.

However the soccer ball universe has now been kicked firmly into touch. At Imperial College, Jaffe and Anastasia Niarchou have been testing a few different topologies in a curved universe. So far they have looked at four relatively simple shapes: the dodecahedron, plus the prism, octahedron and truncated cube (see Diagram). Niarchou鈥檚 PhD project was to simulate how the microwave background would look in each of these topologies. Some of them seemed promising. Niarchou found that the dodecahedral space suppressed the large-scale patterns in the right way, and the truncated cube was an even better fit for the observations. Could it be that we live in a universe with such an odd shape, an 18-cornered cosmos? 鈥淚 was hoping it would turn out like that,鈥� says Niarchou.

18-cornered cosmos?

Her hopes were dashed when she looked in more detail, however. It turns out that the four simulated topologies all produce far too much splotchiness to fit the data. 鈥淪adly, they are all ruled out,鈥� says Jaffe.

It鈥檚 a start, but it certainly doesn鈥檛 end the search. There are an infinite number of weird topologies possible in the positively curved universe that might fit the measurements better. They include 鈥渓ens鈥� topologies in which the universe resembles a slice of a sphere, and even more exotic ones called double-action and linked-action manifolds that look like badly grown crystals with skewed facets. Their microwave signatures are too hard to calculate, however, so for now cosmologists have swept them under the rug.

There is another problem. WMAP鈥檚 curvature measurement is still very hazy. Statistical uncertainty means it could also be consistent with flat or negatively curved space. 鈥淚 don鈥檛 even consider it suggestive,鈥� says Starkman.

That leaves the door open for some distinctly bizarre possibilities. In 2004 Frank Steiner at the University of Ulm in Germany looked at the strange topologies that can exist in a negatively curved saddle-type space (New 杏吧原创, 17 April 2004, p12).

Some of these are particularly strange, with long narrow spikes. The simplest is the Picard topology, a single flared spike or horn shape. If we lived at the narrow end of the horn it would be very clear that something was up, because two dimensions would be rolled up very small. Seeing the back of your own head would be relatively easy 鈥� or, more realistically, we might see images of our galaxy and its companions only a few million light years away. That would be a dead giveaway.

Spiky space

From the more spacious flared end of the trumpet, however, the view would be much more conventional, so it is quite possible that we live in such a spiky space and just haven鈥檛 noticed. And that鈥檚 not the only possibility. Mathematicians haven鈥檛 even been able to classify all the possible saddle topologies, so it鈥檚 not obvious what signatures they might have. 鈥淩uling them all out becomes difficult,鈥� says Levin.

But there is one powerful approach that should catch any kind of wraparound topology: searching for distinctive repeating circles in the microwave background. These will reveal any universal shape, as long as it isn鈥檛 so big that it repeats beyond our 鈥渃osmic horizon鈥�. No light or signal can reach us from beyond this boundary because of the expansion of the intervening space. It currently lies at more than 40 billion light years away.

Along with David Spergel and Eiichiro Komatsu at Princeton University and Neil Cornish at Montana State University, Starkman is part of one team that has looked hard for these circles. They have found none. So far they have only published a limited search, looking for matching circles that are nearly opposite one another in the sky (Physical Review Letters, vol 92, p 201302).

Many topologies would slip though that net, but none should escape the full analysis, which should be finished soon. Early indications are that there will be no matching circles. The team expect to set an absolute lower limit on the size of the universe, about 94 per cent of the distance to the cosmic horizon.

Long shot

Perhaps the size of the universe is right in the narrow range between Starkman鈥檚 lower limit and the horizon. That would be some cosmic coincidence. 鈥淚t鈥檚 really a long shot,鈥� says Richard Bond of the University of Toronto. If the universe is bigger than the horizon, we seem doomed never to know if it wraps around. Space might be finite, but so big that any signs are out of sight. We might never know its shape.

That is, unless astronomy can somehow reach beyond the horizon. 鈥淚t鈥檚 not impossible,鈥� says Starkman. 鈥淲e would really like to see five or six times further.鈥� He likens the universe to a giant drum beaten in the first instants of the big bang. Its vibrations seeded the pattern of hot and cold spots in the microwave background, so by mapping them cosmologists can effectively listen to the drum. If the drum was finite, its shape might have favoured certain wavelengths, and although its sound seems to be something close to white noise, there might yet be some subtle notes hidden in there.

Teasing that signal out of the microwave data could be difficult, however, or even impossible. 鈥淭he question is can you tell the shape of the drum by listening to it?鈥� asks Starkman. 鈥淣o one has an answer yet.鈥� If the answer is yes, then a universe several times bigger than our horizon might reveal its shape.

Another shred of hope comes from getting a better idea of the curvature of space, and finding out if it really is positively curved. If it is, space must be finite whatever the topology. A European Space Agency probe called Planck should improve on WMAP鈥檚 measurement once it is launched early next year. But even this is unlikely to be conclusive, so it might be a long time before we know for sure.

For now we鈥檒l just have to wait and see 鈥� never have the words 鈥渨atch this space鈥� been more apt.