Cosmologists Try a New Way to Measure the Shape of the Universe
Introduction
Is the universe infinite, Aristotle asked in 350 BCE, “or is this an impossibility? The decision … is … all-important to our search for the truth.” The Greek philosopher opined that “the heavens” (meaning the moon, planets, sun and other stars) travel around the Earth in circles, and that a “body which moves in a circle is not endless or infinite, but has its limit.” Moreover, he assumed that Earth sits at the center of the universe. In that case, the universe must be finite, Aristotle reasoned, for otherwise it could not have a center. With that, he’d apparently resolved an issue that had confounded both his predecessors and his contemporaries.
Circular as Aristotle’s logic was, his conclusion could still be correct. More than two millennia later, we still can’t be sure if the universe is finite or infinite. The universe could be boundless, continuing in all directions without end, or it could be sealed up in a compact shape such as a sphere or doughnut.
Of course, modern scientists have wondered about this issue as well. They’ve devised strategies to investigate the universe’s overall topology, using methods more rigorous than Aristotle’s. The first tests, conducted about two decades ago, linked a range of possible topologies to signals that might have been spotted in astronomical data. Efforts to find those signals have come up short, but hope may be on the horizon.
Recently, a group of about 15 scientists from seven countries known as the Compact collaboration has devised a new way of finding topological clues. They’re taking advantage of computational capabilities that weren’t available a decade ago, buoyed by the conviction — as they wrote in Physical Review Letters in April 2024 — that “prior searches for topology have far from exhausted the potentially significant possibilities. Much more can be done to discover, or constrain, the topology of space.”
“The size and shape of the universe is absolutely one of the most basic and important questions we could ask,” said Neil Cornish, a Montana State University astrophysicist who is not part of Compact. Given that a substantial amount of relevant data is already available, he said, “it makes sense to expend the effort to do the most complete analysis possible.”
Circles in the Sky
The Compact collaboration builds on work from more than 25 years ago. In 1998, Cornish, Glenn Starkman, a theoretical physicist at Case Western Reserve University who unofficially leads Compact, and David Spergel, who at the time was at Princeton University1, published “Circles in the Sky,” a road map for probing our cosmic topology.
Glen Starkman, a leader of the Compact collaboration, is mapping the signals that a universe with an interesting shape would produce.
David Hintz/Case Western Reserve University
The technique that the three researchers introduced would work if a few assumptions fell into place. Most important, the universe’s topology would have to allow light, traveling for almost the entire age of the universe, to take two totally different routes to get to us, in much the same way that an airplane traveling from Spain to New Zealand can fly either east across Asia or west over the Americas.
The surface of the Earth is like a sphere, but other shapes are possible for the universe as a whole. Consider, for instance, a doughnut-like torus. In this case, there are multiple ways for a light ray to travel around the surface of the torus and return to the same point. The light can loop around the outside of the doughnut, or loop through the central hole. Either way, it makes it back to where it started from.
It’s much harder to picture a torus with a three-dimensional surface (rather than our two-dimensional example), but it can be modeled by a cube — albeit one with some unusual properties. Imagine living inside a special kind of cube where each face is somehow connected to (or “identified with”) its opposite side. If you were to walk out the left-hand face of the cube, you’d emerge on the right. Similarly, you would pass from top to bottom and from front to back.
In “Circles in the Sky,” Cornish, Spergel and Starkman explained how cosmological data might reveal that our universe has topology like that of a 3D torus (one of many shapes they considered). They proposed looking for this evidence in the cosmic microwave background (CMB), a steady stream of photons from the early universe that reaches us from all directions. The CMB tells us how the universe looked just 380,000 years after the Big Bang, when light was first able to travel through the cosmos unimpeded. By observing these photons today, we can map a spherical surface called the last scattering surface (LSS) — a snapshot of the universe at that early time. The brightness and temperature throughout the surface appear to be remarkably uniform, with variations of just one part in 100,000 from one spot to another.
That sphere, the LSS, is essentially the farthest thing we can see. Cornish, Spergel and Starkman imagined our universe as a 3D torus, depicted (figuratively) as a rectangular box. Now imagine: What if we put the LSS sphere in the middle of the box, but it didn’t quite fit? It would be like squeezing a basketball into a shoebox.
In that case, the sphere would pop out of the sides of the box. If we look at the places where the sphere intersects the box, we’ll find two circles on opposite sides. And since opposite sides of the box are identical — remember, the box is our 3D torus stand-in — those two circles will be identical as well.
With this in mind, you can search for features on opposite ends of the CMB sky that appear to be identical circles.
The researchers carried out an exhaustive search for such circles, relying upon CMB data from the Wilkinson Microwave Anisotropy Probe (WMAP). Yet they didn’t uncover any, as they reported in a 2004 paper. “Our initial searches using WMAP data and later searches by the Planck collaboration” — carried out roughly a decade later — “did not find any evidence for light wrapping around the universe in this way,” Cornish said.
The absence of circles could mean that the universe is infinite and unbounded. But another possibility is that the universe is finite but much larger than the LSS. In that case, the sphere of the LSS never intersects the much larger box of the universe.
To push forward, Starkman and his colleagues needed to find another technique that could work in a universe that’s larger than the portion we can see.
Hearing the Shape of a Drum
Compact’s new approach is based on an old idea raised in a 1966 paper by the mathematician Mark Kac. Starkman had known about this paper for many years, and he recognized that it could provide an alternative method for probing cosmic topology if the circle test did not pan out.
The premise of Kac’s work, explained Compact collaborator Yashar Akrami, a physicist at the Institute for Theoretical Physics in Madrid, is this: “If I close my eyes and hear the sound coming from a drum, can I actually analyze that sound, [determining] which frequencies exist and the amplitudes of the different modes? Can I go from there back to inferring the shape of the drum?” The Compact team plans to do the same thing with the universe by analyzing the sound (or acoustic) waves that left their imprint in the CMB and other cosmological data.
Yashar Akrami is calculating the signatures that particular topologies would imprint on cosmological data.
Laura Marcos Mateos
There are slight variations in the CMB: Some spots on the last scattering surface (where the CMB photons originated) are a bit warmer than average, while others are a bit cooler. These patterns were created by sound waves propagating through the plasma of the early universe. The waves themselves were triggered by minute quantum fluctuations in the fabric of space as it rapidly expanded during the first moments of the Big Bang, not unlike the waves that might be created if rocks of various sizes were dropped randomly into a pond. The crests of the waves would correspond to places of slightly elevated temperature or density, whereas troughs would signify places of depressed temperature or density.
This pattern is imprinted in the CMB. However, you can’t just look at a CMB map and see waves leaping off the page. You need to make a detailed examination of statistical correlations in order to measure the size distribution of troughs and crests. It’s like analyzing a scratchy recording of Beethoven’s Ninth Symphony to reconstruct the original sheet music and see what notes the universe was playing at its birth. “A piccolo and a tuba are both wind instruments, but you can easily hear the difference because of the notes they produce,” said Craig Copi, a Compact team member at Case Western.
How might identifying those notes help? A universe with a particular topology may amplify some notes and muffle others. Consider, for instance, a puzzling feature of the CMB. Imagine you have two telescopes. One points straight up. The other is 10 degrees off. Statistically, if you measure the temperature of the CMB at both locations, the results will be correlated. If one spot is warmer than average, the other is also likely to be warmer than average. Other angles will show an anticorrelation — one spot warm, the other cold.
These relationships hold for all angles between zero and 60 degrees. “But above 60 [degrees], there is no correlation, and we have no explanation for that,” Copi said.
Topology might be the answer. If you have a taut string of a certain length and you pluck it, the notes it can make will have a maximum wavelength, meaning there are low notes it simply cannot produce. Correlations in the CMB above 60 degrees would likely be caused by long-wavelength fluctuations, but large-angle correlations of that magnitude have not been seen. “Maybe our drum [universe] doesn’t produce those notes [because] topology would naturally cut off that scale,” Copi suggested. In other words, maybe we live in a piccolo rather than a tuba.
So how do we find out? The first step is to map out the sounds that we expect to be produced by the various topologies.
Possible Topologies
Compact researchers have started with the easiest topologies — the 17 different flat spaces, starting with a simple 3D torus, labeled E1, and proceeding through more complicated configurations up to E17.
A Compact collaboration paper, published last year, presented the templates for the nine flat topologies that are called “orientable,” which means that if you are on such a surface and you’re pointing upward, and you travel around in a loop, you’ll still be pointing up when you return to your starting point. A paper with templates for the remaining eight nonorientable flat topologies should become available early this year. A nonorientable surface (such as a Möbius strip) has a built-in twist; upon completing a loop around such a space, you’ll change your orientation from right side up to upside down (or vice versa).
Akrami and his doctoral student are starting the next phase, working on the signatures of topologies with positive curvature, like a sphere. There are five general classes of these topologies.
The Compact team is also exploring how to study the topology of the universe not just with the CMB, but also by using the distribution of galaxies across the universe. Whereas the CMB provides just 2D data — a sort of cross section of the universe — galaxies fill the entire 3D volume of space, providing far more data points to analyze. Compact members are hoping that improved maps of the galaxy distribution due to come in over the next several years from the Euclid, Roman and Spherex space telescopes can enhance the search for cosmic topology.
Cornish views Compact as a “low-probability, high-reward” proposition. “If I had to bet, I don’t think they’re going to find anything,” he said. “But the question is so important,” he added, that it ought to be explored “to the fullest extent.”
Starkman doesn’t believe that anyone can assess the probability of success or failure just yet. “When it comes to the topology of the universe, we simply have no idea what to expect,” he said. He is motivated to carry on, in part, because topology can potentially explain the anomalies in the CMB — not only the apparent 60-degree cutoff in its statistical correlations, but also the perplexing differences in the patterns observed above and below the orbital plane of the solar system (called the North-South asymmetry). Starkman can’t say for sure that these anomalies are due to topology, but he has yet to hear any other convincing explanation.
These confounding patterns, Cornish acknowledged, could be “caused by the universe having a particular shape,” or they could be due to “a random fluke.”
Compact investigators have found that the data that’s accumulated over recent decades has strengthened the case for the physical reality of these anomalies rather than weakening it, and time will tell whether that trend continues. The group intends to carry out its work over the next five to 10 years, Akrami said. “We’ll either find the topology of the universe,” he said, or determine that “the universe is so big that its topology cannot be detected” — at least with the tools available to us now.
Some may find the latter outcome disappointing, if it comes to pass. After all that effort, we’d surely know a lot more than Aristotle ever did. But we’d still lack an answer to the question he deemed “all-important to our search for the truth.”
Notes
1: David Spergel is now the president of the Simons Foundation, which funds this editorially independent magazine. Simons Foundation affiliations have no influence on our coverage. For more information about our editorial independence and our relationship to the Simons Foundation, please see our About page. (Return to article.)
Correction: January 27, 2025
An earlier version of this article misspelled Mark Kac’s name.
