‘We’re trying to find the shape of space’: scientists wonder if the universe is like a doughnut

We may be living in a doughnut. It sounds like Homer Simpson’s fever dream, but that could be the shape of the entire universe – to be exact, a hyperdimensional doughnut that mathematicians call a 3-torus.

This is just one of the many possibilities for the topology of the cosmos. “We’re trying to find the shape of space,” says Yashar Akrami of the Institute for Theoretical Physics in Madrid, a member of an international partnership called Compact (Collaboration for Observations, Models and Predictions of Anomalies and Cosmic Topology). In May, the Compact team explained that the question of the shape of the universe remains wide open and surveyed the future prospects for pinning it down.

“It’s high-risk, high-reward cosmology,” says team member Andrew Jaffe, a cosmologist at Imperial College London. “I would be very surprised if we find anything, but I’ll be extremely happy if we do.”

The topology of an object specifies how its parts are connected. A doughnut has the same topology as a teacup, the hole being equivalent to the handle: you can remould a clay doughnut into a cup shape without tearing it. Similarly, a sphere, cube and banana all have the same topology, with no holes.

The idea that the whole universe can have a shape is hard to picture. In addition to the topology there is another aspect: the curvature. In his theory of general relativity in 1916, Albert Einstein showed that space can be curved by massive objects, creating the force of gravity.

Imagine space as two-dimensional, like a sheet, rather than having all three spatial dimensions. Flat space is like a flat sheet of paper, while curved space could be like the surface of a sphere (positive curvature) or a saddle (negative curvature).

These possibilities can be distinguished by simple geometry. On a flat sheet, the angles of a triangle must add up to 180 degrees. But on a curved surface, that’s no longer so. By comparing the real and apparent size of distant objects such as galaxies, astronomers can see that our universe as a whole seems to be as close to flat as we can measure: it’s like a flat sheet pocked with little dimples where each star deforms the space around it.

“Knowing what the curvature is, you know what kinds of topologies are possible,” says Akrami. Flat space could just go on for ever, like an infinite sheet of paper. That’s the most boring, trivial possibility. But a flat geometry also fits with some topologies that cosmologists euphemistically call “nontrivial”, meaning that they’re far more interesting and can get pretty mind-boggling.

There are, for mathematical reasons, precisely 18 possibilities. In general, they correspond to the universe having a finite volume but no edges: if you travel farther than the scale of the universe, you end up back where you started. It’s like the screen of a video game in which a character exiting on the far right reappears on the far left – as though the screen is twisted into a loop. In three dimensions, the simplest of these topologies is the 3-torus: like a box from which, exiting through any face, you re-enter through the opposite face.

Such a topology has a bizarre implication. If you could look out across all the universe – which would require the speed of light to be infinite – you would see endless copies of yourself in all directions, like a 3D hall of mirrors. Other, more complex topologies are variations on the same theme, where, for example, the images would appear slightly shifted – you re-enter the box in a different place, or perhaps twisted so that right becomes left.

If the universe’s volume is not too big, we may then be able to see such duplicate images – an exact copy, say, of our own galaxy. “People started looking for topology on very small scales by looking for images of the Milky Way,” says Jaffe. But it’s not entirely straightforward because of the finite speed of light – “you have to look for them as they were a long time ago” – and so you may not recognise the duplicate. Also, our galaxy is moving, so the copy won’t be in the same place as we are now. And some of the more exotic topologies would also shift it. In any event, astronomers have seen no such cosmic duplication.

If, on the other hand, the universe is really immense yet not infinite, we may never be able to distinguish between the two, says Akrami. But if the universe is finite, at least along some directions, and not much larger than the farthest we can see, then we should be able to detect its shape.

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One of the best ways to do that is to look at the cosmic microwave background (CMB): the very faint glow of heat left over from the big bang itself, which fills the cosmos with microwave radiation. First detected in 1965, the CMB is one of the key pieces of evidence that the big bang happened at all. It is very nearly uniform throughout the cosmos. But as astronomers have developed ever more precise telescopes to detect and map it across the sky, they have found tiny variations in the “temperature” of this microwave sea from place to place. These variations are remnants of random temperature differences in the nascent universe – differences that helped to seed the emergence of structure, so that matter in the universe is not spread evenly throughout the cosmos like butter on bread.

Thus the CMB is a sort of map of what the universe looked like at the earliest stage we can still observe today (about 10bn years ago), imprinted on the sky all around us. If the universe has a nontrivial topology that produces copies in some or all directions, and if its volume is not significantly larger than the sphere on which we see the projection of the CMB, then these copies should leave traces in the temperature variations. Two or more patches will match, like duplicates of fingerprints. But that’s not easy to detect, given that these variations are random and faint and that some topologies would shift the duplicates around. Nonetheless, we can search among the statistics of the tiny temperature variations and see if they are random or not. It’s pattern-seeking, like traders looking for nonrandomness in fluctuations of the stock market.

The Compact team has taken a close look at the chances of finding anything. It showed that, even though no nonrandom patterns have yet been seen in the CMB map, neither have they been ruled out. In other words, many weird cosmic topologies are still entirely consistent with the observed data. “We haven’t ruled out as many interesting topologies as some previously thought,” says Akrami.

Others outside the group agree. “Previous analyses do not rule out there being possibly observable effects due to the universe having a nontrivial topology,” says astrophysicist Neil Cornish of Montana State University in Bozeman, who devised one such analysis 20 years ago. Ralf Aurich, an astronomer at Ulm University in Baden-Württemberg, Germany, also says:“I think that nontrivial topologies are still very much a possibility.”

Isn’t it, though, a little perverse to imagine that the universe may have some twisted-doughnut shape rather than having the simplest possible topology of infinite size? Not necessarily. Going from nothing to infinity in the big bang is quite a step. “It’s easier to create small things than big things,” says Jaffe. “So it’s easier to create a universe that is compact in some way – and a nontrivial topology does that.”

Besides, there are theoretical reasons to suspect that the universe is finite. There is no agreed theory of how the universe originated, but one of the most popular frameworks for thinking about it is string theory. But current versions of string theory predict that the universe shouldn’t have just four dimensions (three of space, plus time) but at least 10.

String theorists argue that maybe all the other dimensions became highly “compactified”: they are so small that we don’t experience them at all. But then why would only six or so have become finite while the others remained infinite? “I would say it is more natural to have a compact universe, rather than four infinite dimensions and the others compact,” says Akrami.

And if the search for cosmic topology showed that at least three of the dimensions are indeed finite, says Aurich, that would rule out many of the possible versions of string theory.

“A detection of a compact universe would be one of the most staggering discoveries in human history,” says cosmologist Janna Levin of Barnard College in New York. That’s why searches like this, “though they threaten to disappoint, are worthwhile.” But if she had to place a bet, she adds: “I would wager against a small universe.”

Will we ever know the answer? “It is quite likely that the universe is finite, but with the topology scale larger than what we can probe with observations,”says Cornish. But he adds that some odd features in the CMB pattern “are exactly the kind you would expect in a finite universe, so it is worth probing further”.

The problem with seeking patterns in the CMB, Cornish says, is given how each of the 18 flat topologies can be varied, “there are an infinite number of possibilities to consider, each with its own unique predictions, so it is impossible to try them all out.” Maybe the best we can do, then, is decide which possibilities seem most probable and see if the data fits those.

Aurich says that a planned improvement of the CMB map in an international project called CMB stage 4, using a dozen telescopes in Chile and Antarctica, should help the hunt. But the Compact researchers suspect that, unless we get lucky, the CMB alone may not allow us to answer the topology question definitively.

However, they say there is plenty of other astronomical data we can use too: not just what’s on the “sphere” of the CMB map but what’s inside it, in the rest of space. “Everything in the universe is affected by the topology,” says Akrami. “The ideal case will be to combine everything that is observable and hopefully that will give us a large signal of the topology.” The team wants either to detect that signal, he says, or show that it’s impossible.

There are several instruments now in use or in construction that will fill in more details of what is inside the volume of observable space, such as the European Space Agency’s Euclid space telescope, launched last year, and the SKA Observatory (formerly the Square Kilometre Array), a system of radio telescopes being built in Australia and South Africa. “We want a census of all the matter in the universe,” says Jaffe, “which will enable us to understand the global structure of space and time.”

If we manage that – and if it turns out that the cosmic topology makes the universe finite – Akrami imagines a day when we have a kind of Google Earth for the entire cosmos: a map of everything.