Scientists are interested in whether the Universe is like a donut

Instead of extending to infinity and beyond, the topology of the universe may be such that it can eventually be mapped

We may be living in a doughnut. It sounds like Homer Simpson's sick dream, but this could be the shape of the entire universe—more precisely, the hyperdimensional donut that mathematicians call 3rd.

This is just one of many possible topologies of space. “We are trying to understand the shape of the cosmos,” says Yashar Akrami of the Institute of Theoretical Physics in Madrid, part of an international partnership called Compact (Collaboration for Observations, Models and Predictions of Anomalies and Cosmic Topology). In May the Compact team explainedthat the question of the shape of the Universe remains wide open, and considered future prospects for its solution.

“This cosmology has high risks, but also high rewards,” says team member Andrew Jaffe, a cosmologist at Imperial College London. “I'll be very surprised if we find anything, but I'll be very happy if we do.”

The topology of an object determines how its parts are connected. The donut has the same topology as a teacup: the hole is equivalent to a handle: you can reshape a plasticine donut into a cup shape without tearing it. Likewise, a sphere, cube and banana have the same topology, without holes.

The idea that the entire universe could have a form is difficult to imagine. In addition to topology, there is another aspect: curvature. In his theory of general relativity in 1916, Albert Einstein showed that space can be curved by massive objects, generating gravity.

Imagine that space is two-dimensional, like a leaf, and does not have all three spatial dimensions. Flat space is like a flat piece of paper, while curved space can be like the surface of a sphere (positive curvature) or a saddle (negative curvature).

These possibilities can be distinguished using simple geometry. On a flat sheet, the angles of the triangle should be 180 degrees. But on a curved surface this is no longer the case. By comparing the real and apparent sizes of distant objects such as galaxies, astronomers see that our Universe as a whole appears to be as close to flat as we can measure: it looks like a flat sheet dotted with little dimples where each star warps the space around it.

In the three geometric scenarios, spacetime can bend on itself, be flat, or bend outward, and the sum of the triangle's angles either exceeds, equals, or falls short of 180°.

“By knowing what the curvature is, you know what kinds of topologies are possible,” says Akrami. Flat space can simply go on forever, like an endless sheet of paper. This is the most boring and trivial possibility. But flat geometry also corresponds to some topologies that cosmologists euphemistically call “non-trivial,” which means they are much more interesting and can be quite mind-blowing.

For mathematical reasons, there are exactly 18 possibilities. In general, they correspond to the fact that the Universe has a finite volume, but has no edges: if you fly a distance greater than the scale of the Universe, you will end up where you started. It's like a video game screen in which a character who exits on the far right reappears on the far left – as if the screen is in a loop. In three dimensions, the simplest of these topologies is a 3-torus: like a box from which, when you exit through any edge, you re-enter through the opposite edge.

This topology has a bizarre consequence. If you could look around the entire Universe—for this to happen, the speed of light would have to be infinite—you would see endless copies of yourself in every direction, like in a three-dimensional hall of mirrors. Other, more complex topologies are variations on the same theme, where, for example, the images appear slightly shifted—you enter the box in a different place—or perhaps rotated so that right becomes left.

If the volume of the Universe is not too large, then we will be able to see such duplicates – an exact copy of, say, our own galaxy. “People have started looking for topology at very small scales by looking at images of the Milky Way,” Jaffe says. But it's not entirely easy because of the finite speed of light – “you have to look for them as they were a long time ago” – and so you You may not recognize the duplicate. In addition, our galaxy is moving, so the copy will not be in the same place as we are now. And some more exotic topologies will shift it even more. In any case, astronomers have not yet seen such a cosmic duplication.

One of the topologies possible for the Universe is the Euclidean 3-torus. Although it's hard to imagine in three dimensions, the equivalent in two dimensions is a 2-torus: a simple donut. Imagine a two-dimensional sheet. The two edges may curve and meet to form a cylinder, and the ends will curl to form a ring. In this case, the light will spread from one galaxy to another in different directions and will be visible in several copies. Although limited, an infinite line of sight will create the impression of infinite space.

On the other hand, if the universe is truly huge but not infinite, we may never be able to tell one from the other, Akrami says. But if the universe is finite, at least in some directions, and not much larger than the furthest thing we can see, then we should be able to determine its shape.”

One of the best ways to do this is to look at the cosmic microwave background radiation (CMB): a very faint glow of heat left over from the Big Bang that fills space with microwave radiation. First discovered in 1965, CMB is one of the key pieces of evidence that the Big Bang even happened. It is almost uniform throughout space. But as astronomers have built increasingly precise telescopes to detect it and map it across the sky, they have discovered tiny variations in the “temperature” of this microwave sea from place to place. These variations are the remnants of random temperature differences in the nascent Universe – differences that contributed to the emergence of structure, so that the matter in the Universe is not distributed evenly throughout the cosmos, like butter on bread.

Thus, the CMB is a kind of map of what the Universe looked like at the earliest stage that we can observe today (about 10 billion years ago), imprinted on the sky around us. If the Universe has a non-trivial topology that creates copies in some or all directions, and if its volume is not much larger than the sphere on which we see the CMB projection, then these copies should leave traces in temperature fluctuations. Two or more spots will match, like duplicate fingerprints. But this is not easy to detect, given that these variations are random and weak, and some topologies will bias duplicates. However, we can look up the statistics of tiny temperature fluctuations and understand whether they are random or not. This is a search for patterns, just as traders look for non-randomness in stock market fluctuations.

The Compact team carefully studied the chances of finding at least something. She showed that, although no non-random patterns have yet been discovered on the RI map, they are still not excluded. In other words, many strange cosmic topologies are still perfectly consistent with observed data. “We believe that we can exclude as many non-trivial topologies as other scientists have previously excluded,” says Akrami.

Other experts outside the group agree with this. “Previous analyzes do not exclude the possibility that there may be observed effects associated with a nontrivial topology of the universe,” says astrophysicist Neil Cornish of Montana State University in Bozeman, who developed one of these analyzes 20 years ago. Ralf Aurich, an astronomer at the University of Ulm in Baden-Württemberg, Germany, also says: “I think a non-trivial topology is still very likely.”

Isn't it too perverse to imagine that the Universe could have the shape of a twisted donut, rather than the simplest topology of infinite size? Not necessary. The transition from nothing to infinity as a result of the big bang is a pretty serious 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 sense—and the nontrivial topology allows us to do just that.”

Moreover, there are theoretical reasons to suspect that the universe is finite. There is no agreed upon theory for the origin of the universe, but one of the most popular frameworks for thinking about it is string theory. But modern versions of string theory predict that the universe should not have four dimensions (three spaces plus time), but at least ten.

String theorists argue that perhaps all the other dimensions have been greatly “densified”: they are so small that we cannot feel them at all. But then why did only six or so dimensions become finite while the rest remained infinite? “I would say that it is more natural to have a compact universe, rather than four infinite dimensions and the rest being compact,” says Akrami.

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

“Discovering a compact universe would be one of the most stunning discoveries in human history,” says cosmologist Gianna Levin of Barnard College in New York. That is why such searches, “although they threaten to disappoint, make sense.” But if she had to bet, she adds, “I'd bet against the little universe.”

Will we ever know the answer? “It is likely that the Universe is finite, but its topological scale is larger than we can probe through observations,” says Cornish. But he adds that some of the strange features in the CMB pattern are “exactly what one would expect in a finite universe, so it's worth exploring further.”

The problem with finding patterns in RI, Cornish says, is that given how each of the 18 planar topologies can be varied, “there are an infinite number of possibilities to consider, each with its own unique predictions, so it's impossible to try them all.” Perhaps the best we can do is decide which possibilities seem most likely and see if the data matches them.

Aurich says that the planned improvement of the RI map is part of an international project called “CMB stage 4” using a dozen telescopes in Chile and Antarctica should help in the search. But the Compact researchers suspect that, unless we are lucky, RI alone will not allow us to definitively answer the topology question.

However, according to them, there is a lot of other astronomical data that we can use: not only what is on the “sphere” of the RI map, but also what is inside it, in the rest of space. “Everything in the universe is affected by topology,” says Akrami. “The ideal would be to combine everything that can be observed, and hopefully that will give us a big signal about the topology.” The team wants to either detect this signal, he says, or show that it is impossible.

Several instruments are currently in use or under construction that will provide more detailed information about what lies within the observed space, such as the European Space Agency's Euclid Space Telescope, launched last year, and the SKA (formerly Square Kilometer Array), a radio telescope system under construction in Australia and South Africa. “We want to take a census of all the matter in the universe,” says Jaffe, “which will allow us to understand the global structure of space and time.”

If we succeed—and if it turns out that, because of cosmic topology, our universe is finite—then Akrami envisions a day when we will have a kind of Google Earth for the entire cosmos: a map of everything.