Maybe we live in a doughnut. It sounds like Homer Simpson’s fever dream, but that could be the shape of the entire universe—specifically, a hyperdimensional donut that mathematicians call a 3-torus.
This is just one of many possibilities for the topology of the cosmos. “We are trying to find the shape of space,” says Yashar Akrami of the Institute of Theoretical Physics in Madrid, 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 explored the future prospects for determining this.
“It’s a high-risk, high-reward cosmology,” says team member Andrew Jaffe, a cosmologist at Imperial College London. “I would be very surprised if we found anything, but I would be very happy if we did.”
The topology of an object specifies how its parts are connected to each other. A donut has the same topology as a teacup, with the hole being the same as the handle: you can shape a clay donut into a cup without tearing it. Likewise, a sphere, cube, and banana all have the same topology, with no holes.
The idea that the entire universe can have a shape is difficult to imagine. In addition to the topology, there is another aspect: the curvature. In his 1916 general theory of relativity, Albert Einstein showed that space can be curved by massive objects, creating gravity.
Imagine space as two-dimensional, like a sheet, instead of having all three spatial dimensions. The flat space resembles a flat sheet of paper, while the curved space can resemble the surface of a sphere (positive curvature) or a saddle (negative curvature).
These options are distinguished by a simple geometry. On a flat plate, the angles of a triangle must add up to 180 degrees. But on a curved surface that is no longer the case. By comparing the real and apparent sizes of distant objects such as galaxies, astronomers can see that our universe as a whole appears as flat as we can measure: it is like a flat plate full of tiny dimples in which each star distorts space. around it.
“If you know what the curvature is, you know what kinds of topologies are possible,” says Akrami. Flat space could last forever, like an infinite sheet of paper. That’s the most boring, trivial possibility. But flat geometry also fits some topologies that cosmologists euphemistically call “non-trivial,” meaning they are much more interesting and can become quite mind-boggling.
For mathematical reasons there are exactly 18 possibilities. In general, they correspond to the universe having a finite volume but no edges: if you travel beyond the scale of the universe, you end up back where you started. It looks like a video game screen where a character who ends up on the far right reappears on the far left – as if the screen has been rotated in a loop. In three dimensions, the simplest of these topologies is the 3-torus: like a box from which you enter via any plane via the opposite plane.
If you could look across the universe, you would see endless copies of yourself in every direction, like a 3D hall of mirrors
Such a topology has a bizarre implication. If you could look across the entire universe – which would require the speed of light to be infinite – you would see endless copies of yourself in every direction, like a 3D hall of mirrors. Other, more complex topologies are variations on the same theme, where, for example, the images appear slightly shifted – you re-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, we might be able to see such double images – an exact copy of our own galaxy, for example. “People started looking for topology on a very small scale by looking at images of the Milky Way,” says Jaffe. But it’s not entirely simple because of the finite speed of light – “you have to look for them as they were there a long time ago” – and so you may not recognize the duplicate. Plus, our galaxy is moving, so the copy won’t be in the same place we are now. And some of the more exotic topologies would change this too. Be that as it may, astronomers have not seen such a cosmic duplication.
On the other hand, if the universe is truly immense, but not infinite, we may never be able to distinguish between the two, says Akrami. But if the universe is finite, at least in some directions, and not much larger than the furthest 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 that fills the cosmos with microwave radiation. The CMB was first discovered in 1965 and is one of the most important pieces of evidence that the Big Bang happened at all. It is virtually uniform throughout the cosmos. But as astronomers have developed increasingly precise telescopes to detect and map it in the sky, they have discovered small variations in the “temperature” of this microwave sea from place to place. These variations are remnants of random temperature differences in the emerging universe – differences that helped create structure, so that matter in the universe is not spread evenly through the cosmos like butter on bread.
So the CMB is a kind of map of what the universe looked like at the earliest stage we can see today (about 10 billion years ago), imprinted on the sky all around us. If the universe has a non-trivial 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 match, like fingerprint duplicates. But that is not easy to detect as these variations are random and vague and some topologies would shift the duplicates. Nevertheless, we can search the statistics of the small temperature variations and see if they are random or not. It’s looking for patterns, like traders looking for non-random swings in the stock market.
The Compact team has taken a closer look at the chances of finding something. This showed that, although no non-random patterns can yet be seen in the CMB map, these have not been ruled out either. In other words, many strange cosmic topologies are still completely 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 analyzes do not rule out the possibility that there may be observable effects because the universe has a non-trivial topology,” says astrophysicist Neil Cornish of Montana State University in Bozeman, who devised such an analysis twenty years ago. Ralf Aurich, an astronomer at Ulm University in Baden-Württemberg, Germany, also says: “I think non-trivial topologies are still very possible.”
However, isn’t it a bit perverse to imagine that the universe could be shaped like a twisted donut instead of having the simplest possible topology of infinite size? Not necessary. 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 non-trivial topology does that.”
Moreover, there are theoretical reasons to suspect that the universe is finite. There is no universally accepted theory of how the universe came into existence, but one of the most popular frameworks for thinking about it is string theory. But current versions of string theory predict that the universe should have not just four dimensions (three of space, plus time), but at least ten.
String theorists argue that perhaps all other dimensions have become very “compacted”: they are so small that we no longer experience them at all. But why then should only six have become finite, while the others remained infinite? “I would say it is more natural to have a compact universe, than four infinite dimensions and the other compact,” says Akrami.
The ideal case will be to combine everything that is observable and hopefully that will give us a big signal of the topology
Yashar Akrami, cosmologist
And if the search for cosmic topology shows that at least three of the dimensions are indeed finite, Aurich says, that would rule out many of the possible versions of string theory.
“The detection of a compact universe would be one of the most astonishing discoveries in human history,” says cosmologist Janna Levin of Barnard College in New York. That’s why searches like these, “although they threaten to disappoint, are worth it.” But if she had to place a bet, she adds, “I would bet against a small universe.”
Will we ever know the answer? “It is very likely that the universe is finite, but with a topology scale that is larger than what we can investigate with observations,” says Cornish. But he adds that some strange features in the CMB pattern are “exactly the kind you would expect in a finite universe, so it’s worth investigating further.”
The problem with looking for patterns in the CMB, says Cornish, 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’s impossible to trying them all out.” Perhaps the best we can do then is decide which possibilities seem most likely and see if the data fits them.
Aurich says a planned improvement to the CMB map in an international project called CMB Phase 4, using a dozen telescopes in Chile and Antarctica, should help the hunt. But the Compact researchers suspect that unless we’re lucky, the CMB alone may not allow us to definitively answer the topology question.
However, they say there is plenty of other astronomical data we can use: 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 topology,” says Akrami. “The ideal case will be to combine everything that is observable and hopefully that will give us a big signal of the topology.” The team wants to detect that signal, he says, or show that it is impossible.
There are several instruments now in use or under construction that will fill in more detail about what lies within the volume of observable space, such as the European Space Agency’s Euclid Space Telescope, which launched last year, and the SKA Observatory (formerly the Square Kilometer Array). ), a system of radio telescopes being built in Australia and South Africa. “We want a count 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 cosmic topology makes the universe finite – Akrami envisions a day when we have a kind of Google Earth for the entire cosmos: a map of everything.