Around 2,500 years ago humankind began to recognize that our planet is round—although a few “flat Earth” supporters still deny this today. The shape of our *universe* is not clear, however. Previous studies have suggested that the cosmos probably has a fairly simple shape, such as the three-dimensional equivalent of a spherical surface or a plane. But this view may be wrong, according to cosmologists from the Collaboration for Observations, Models and Predictions of Anomalies and Cosmic Topology (COMPACT). In a paper published in April 2024, they found that the shape of the universe could be significantly more complex than previously assumed.

Earth also appears flat at first glance. This is because its radius is so large that the surface’s curvature is barely perceptible. To prove it’s a sphere, you would just have to start walking. If you were to move straight ahead without ever turning—crossing over mountains and oceans—you would inevitably return to the start at some point.

Cosmologists collect similar clues to determine the shape of the universe. They could send a spaceship out to traverse the cosmos. Instead they peer into the night sky and examine traces of the oldest light that reaches us from the depths of the cosmos. This radiation was created around 300,000 years after the big bang. Although photons had existed even before that, matter was so densely packed into what was then a small universe that the light quanta had no chance of traveling freely. Eventually, however, the universe cooled down to such an extent that it became transparent. The photons were able to spread freely through space—and still do today.

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Photons from this early time in the universe reach us as cosmic background radiation. Approximately the same pattern of old light comes from every corner of the universe. This radiation provides clues to the shape of the universe. If, for example, the curvature of the universe were to change at one point, the cosmic background radiation would not be as homogeneous as we observe it. Experts therefore assume that the universe is either curved in the same way everywhere—or is completely flat.

## A Wealth of Possibilities

It is difficult to imagine the universe having no curvature or constant curvature. That’s why it helps to use two-dimensional examples. In this analogy, to fit the homogeneous patterns of cosmic background radiation, the universe can be actually flat like a flat sheet of paper (no curvature), correspond to the three-dimensional counterpart of a spherical surface (positive curvature) or form a kind of three-dimensional saddle surface (negative curvature). In all three cases, the curvature is constant everywhere.

But this still does not reveal what the universe as a whole looks like. For example, space could be curved in the same way everywhere and still have a hole. The mathematical discipline of topology can be used to differentiate between such cases. Topologists classify geometric figures according to rough categories, such as the number of holes in them, in order to create a catalog of shapes. Cosmologists aim to assign one of these shapes to the universe.

As mathematicians have been able to prove, there are an infinite number of different categories (topologies) of curved three-dimensional surfaces. So if spacetime really is saddle-shaped or spherical, then there are theoretically any number of shapes that the universe can take on, each of which is very different from the other. Cosmological observations to date, however, point to a flat universe without curvature. If true, this insight narrows the search for the topology of the universe: as mathematician Grigori Perelman deduced in 2003, the catalog of flat three-dimensional surfaces is very small, with only 18 different categories.

## One of 18 Shapes

The shape of the universe therefore presumably corresponds to one of these 18 shapes. One is the three-dimensional analog of a sheet of paper. Our cosmos could also have holes, however. Among the 18 topologies, the three-dimensional equivalent of a donut surface is a so-called torus.

Although a donut appears curved at first glance, it turns out that this shape is actually flat. This is because a donut shape can be created using a (very flexible) sheet of paper. To do this, you first stick the opposite long sides of the sheet together, creating an elongated tube. Then you bend the tube and bring the two openings together.

This construction has many advantages. For example, if you want to describe the movement of an ant on a donut, it is usually easier to imagine the rectangular sheet of paper. The insect walks around on the piece of paper and as soon as it crosses an edge, it reenters the opposite side of the sheet—as in the popular mobile game Snake from the 2000s. If the ant moves in a straight line, it will inevitably end up back at the start at some point.

You can create a three-dimensional torus in the same way by using a three-dimensional cuboid instead of a rectangular sheet of paper. In this case, you also deform it so that opposite sides of the cuboid are glued together.

## A Donut Universe with Loops

If our universe were torus-shaped, this would have tangible consequences. If we were to shine a flashlight into the sky, for example, this light would theoretically reach us again at some point—just as the ant on the sheet of paper always returns to its starting point. Such loops exist in most of the 18 different topologies. These are trajectories that return to their starting point. At first glance, however, the loops do not seem to correspond to our observations. If light were to periodically pass through space, we would have to see several copies of the universe in the night sky.

Still, this fact does not exclude all topologies with loops. We could live in a universe so large that light has not yet had enough time to traverse it and return. Perhaps in several billion years, humans (or other beings) will have the pleasure of seeing a night sky that contains numerous images of the universe.

But even such huge loops would leave traces today. For example, the shape of the cosmos would have influenced how matter and light interacted with each other in the early universe. If the universe had holes—and therefore also loops—this fact should be reflected in the data of the cosmic background radiation.

Yet in searches from the 2000s and 2010s, experts found nothing. Most cosmologists have therefore assumed that the universe had a fairly simple structure. Research into the shape of the universe stalled—at least until COMPACT launched in 2022.

## Significantly More Possibilities Than Expected

The COMPACT team compared the latest data on cosmological background radiation with various topologies of the universe—and came up with surprising results in its first publication. Among other things, the researchers discovered that previous work had ignored many variants of the individual topologies. In other words, researchers may have previously missed other forms that could describe our cosmos. The collaboration was also able to show that the missing evidence for loops in the cosmic background radiation is far less restrictive than previously assumed. In addition to loops, there could be other traces in the cosmic background radiation that point to complicated topologies. Future work will explore what exactly these shapes could be.

The researchers backed up their arguments using three concrete example topologies of flat space: an ordinary torus and two variations of it in which the faces of the cuboid are twisted before being glued together. All three have holes. For the ordinary torus, COMPACT confirmed earlier conclusions: if our universe is torus-shaped, the loops must be so large that light has not yet been able to reach us. But the surprise was that this same idea may not apply to the modified shapes. The collaboration’s calculations show that loops of these shapes could also be significantly shorter than what was assumed from the previous data. Because of the many twists, the universe could contain copies of itself that might look different from the original, making them less easy to spot in maps of the cosmic microwave background.

Taken together, the COMPACT team’s findings show that the universe could look far more complicated than previously assumed. And that would have far-reaching consequences. The shape of our cosmos is not just an academic question. The topology of spacetime was presumably determined by the quantum processes that took place shortly after the big bang. If we knew more about the shape of the universe, we could—potentially—learn more about the complex processes that took place at its beginning.

*This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.*

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