“As long as they live for long enough, they will always become large cosmological beasts,” says Ricardo Ferreira, a cosmologist at the University of Coimbra in Portugal. He’s not talking about actual beasts but rather about hypothetical humongous sheets of spacetime that could divide one region of the universe from another. Such so-called domain walls are the natural outcome of theories that try to solve some of the deepest mysteries in physics, such as the origins of gravity. As Ferreira says, however, had they formed after the big bang, by today they’d be the dominant source of energy in our universe, and there’s no evidence that’s the case. So any theory invoking their existence has been considered suspect—until now, perhaps.
In a theoretical study that was recently posted on the preprint server arXiv.org, Ferreira and his colleagues have shown that if these domain walls formed, grew and then mostly annihilated in fractions of a second after the big bang—thus accounting for their absence in today’s universe—they would have created a random, cosmos-suffusing background of ripples in spacetime. And given tentative claims from multiple collaborating groups of astronomers, such a background “hum” of stochastic gravitational waves may have already been detected. According to the theoretical study, the few domain walls that did not annihilate would now appear as black holes of about one solar mass. In some scenarios, these black holes would be even smaller and would be numerous enough to be dark matter—the unseen stuff that’s thought to make up about a quarter of the universe.
“The authors have done quite a thorough analysis,” says theoretical physicist Tanmay Vachaspati of Arizona State University, who was not involved in the study. “The idea is fascinating in that it simultaneously addresses the stochastic gravitational-wave background observations and raises the possibility that there is an abundance of small black holes.”
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The rationale for the existence of domain walls comes from extensions to the Standard Model of particle physics, which in essence describes all known fundamental particles and forces of nature (all, that is, except for gravity). But even leaving gravity aside, the Standard Model is still incomplete. It considers neutrinos to be massless, for example, whereas observations tell us otherwise, and it cannot explain enigmas such as the cosmic abundance of matter over antimatter. Most modifications to the Standard Model that try to solve such problems rely on various sorts of “symmetry-breaking” events, in which nearly uniform conditions that exist at high energies break down as the universe expands and cools, a bit like how mud transforms from smooth to cracked as it dries.
When symmetry breaks, domain walls can be born; understanding the process, however, requires grappling with a concept called a scalar field—something like an invisible field with a value at every point in spacetime but no direction. A scalar field can be used to describe, for instance, how the temperature varies from place to place in a room. Domain walls can arise from scalar fields that exhibit “discrete” symmetry, meaning that the fields can have two (or more) different states with the same energy. Imagine a mass of mud atop which floats a greasy plastic ball. Let’s say that as the mud cools and solidifies, two shallow depressions of equal depth form on either side of the ball. Eventually—assuming it doesn’t get stuck in the mud—the ball will randomly roll down to the bottom of one or the other depression. Before this occurs, the mud and ball together can be thought of as possessing a discrete symmetry, which breaks when the mud cools and the ball rolls down. A domain wall is akin to the mound of dried mud between the depressions—which the ball would have to go over if it had to move from one dip to the other. The domain wall is an energy barrier.
Some extensions of the Standard Model exhibit similar discrete symmetry breaking, wherein the scalar field representing the minimal energy state, or vacuum state, of spacetime itself takes on either a positive or a corresponding negative value as the universe expands and cools, giving rise to two distinct vacuum states.
As far as the cosmos is concerned, “there is no preference between one or the other [vacuum state], so different places in the universe will choose randomly between the two,” Ferreira says. Traversing from one to the other requires scaling an energy barrier. Such high-energy barriers between the two vacuum states—not unlike the mound of mud—constitute the topological spacetime structures called domain walls.
“Since discrete symmetries, which are the basis of domain walls, are common in high-energy physics..., it is quite possible that domain walls exist,” says Fuminobu Takahashi of Tohoku University in Japan, who studies domain walls and physics beyond the Standard Model and was also not associated with the new study.
Such a possibility was borne out in the lab in 2021 when a team led by Cheng Chin, an experimental physicist at the University of Chicago, demonstrated the formation of domain walls in a collection of about 40,000 ultracold atoms in what is called a Bose-Einstein condensate (BEC). Initially, all atoms in a BEC are in the same quantum ground state. The team then disturbed the condensate ever so slightly. This caused some atoms on one side of it to transition to one minimal momentum state, whereas some on the other side entered another minimal state with the exact opposite momentum, like a mirror image. The physical border between the two regions, comprised of the remaining atoms, formed a domain wall. “There’s this kind of a barrier that separates the two minima,” Chin says.
In cosmology, however, such domain walls pose a problem. Because the regions of spacetime they separate would have the same energy, these walls have no incentive to prefer one region over the other and hence would be stable over time. “If the discrete symmetry was exact, [the domain walls] would live forever,” Ferreira says. “And if they live until today, they will dominate the energy density of the universe. And we know we don’t live in a universe which is dominated by the domain walls.”
David Spergel, an astrophysicist at Princeton University and president of the Simons Foundation, agrees. “Models where the two minima are equal and where stable domain walls formed are ruled out as domain walls grow to dominate the universe,” he says.
But domain walls haven’t done so. A wealth of observational and theoretical evidence shows with high confidence that our universe is dominated by dark energy and dark matter and not by the energy density of domain walls.
There’s a way out of this impasse. Researchers have known that domain walls can dissipate if the discrete symmetry is not exact and instead breaks with a small bias such that one region of spacetime has a smidgen more energy than the neighboring region. Because of the small difference in the energies, the spacetime regions will exert some net pressure on the intervening domain wall, which itself has an inherent tension. As the universe expands, this differential pressure will eventually equal the tension of the domain wall, causing it to spontaneously collapse and become part of one region or the other. This could explain why we don’t see domain walls today.
In the latest study, Ferreira and his colleagues focused on one version of biased symmetry breaking that could have unfolded in the early universe. They used computational simulations and other techniques to examine how a network of domain walls would then rise and fall from such an off-kilter start. They found that these topological spacetime structures would generate gravitational waves all the way up to and through their ultimate collapse. “Almost independently of the initial details, as long as they live long enough, they become very loud cosmological relics,” Ferreira says. “They can naturally give you big signals.”
Across the eons, cosmic expansion would have stretched out the domain-wall-generated ripples in spacetime to very long wavelengths and low frequencies. Today, these gravitational waves would mainly be in the nanohertz frequency range—which is exactly the range in which an international consortium of astronomers says it has detected a potential gravitational-wave background signal.
Last year multiple groups that had been monitoring millisecond pulsars—neutron stars that are rotating hundreds of times per second—found changes in the periodicity of these cosmic beacons over a 15-year period. These changes are consistent with a stochastic background of nanohertz gravitational waves. The idea is that the passing waves can ever so slightly alter the observed timing of the pulsars’ otherwise clockworklike cosmic pings. Such a signal would be very subtle, only approaching detectability via correlating timing shifts across multiple pulsars.
The favored, but not conclusive, explanation is that these putative pulsar-jostling waves were generated by the mergers of supermassive black holes. But this explanation suffers from a major problem, Ferreira says. Based on our current understanding of supermassive black holes, “the time it takes for them to merge is bigger than the age of universe,” he says.
This leaves the door open for other explanations. Ferreira and his colleagues argue that if most of the domain walls that formed in the early universe annihilated when the cosmos had a temperature of about two trillion degrees Celsius, the result would be a similar nanohertz gravitational-wave signal. It’s “quite suggestive,” Ferreira says, that this temperature is also approximately at which the infant universe abruptly transitioned from being a hot, dense plasma of quarks and gluons to being filled with larger clumps of quarks—particles called hadrons that are the subatomic building blocks of matter.
The new study is “an important step toward understanding the gravitational waves produced by domain-wall decay,” Takahashi says.
Not all domain walls would have collapsed at that temperature, however. According to the analysis of Ferreira and his colleagues, some walls could survive if they enclosed bubbles of spacetime bigger than the radius of the observable universe at that time. As cosmic history unfolded and spacetime expanded, these bubbles and their domain walls would eventually enter the observable universe. Such structures would appear as overdense regions of energy. And “an overdensity ... will collapse to a black hole,” Ferreira says.
Such “primordial” black holes would have about the mass of our sun, but according to the estimates of Ferreira and his co-authors, these objects wouldn’t be abundant enough to constitute the universe’s dark matter.
If, however, the domain-wall annihilation happened even earlier in the evolution of the universe, at even higher temperatures of about 1020 kelvins, this would lead to a modern-day surfeit of asteroid-mass primordial black holes sufficient to account for dark matter. Importantly, this earlier, higher-temperature domain-wall decay would create a very different gravitational-wave signature. Such “primordial black holes would have a complimentary signal ... around the hertz frequency,” Ferreira says. “So it’ll be testable.” Of course, this would mean that domain-wall collapse wouldn’t explain the nanohertz signal that astronomers claim to have seen; that signal might indeed have a more mundane explanation.
Future gravitational-wave detectors such as Europe’s proposed Einstein Telescope, as well as augmented versions of the current detectors spread around the world, called the Laser Interferometer Gravitational-Wave Observatory (LIGO), Virgo and the Kamioka Gravitational-Wave Detector (KAGRA), will be sensitive to such signals, Ferreira says. “If they measure the signals, and the spectrum has ... the typical shape of a gravitational wave generated from domain walls..., it’s a strong hint for primordial black holes [as] dark matter.”
“It will be interesting to see if future gravitational-wave observatories find the asteroid-mass black holes that the model predicts,” Vachaspati agrees.
Such empirical observations would also have implications for building extensions to the Standard Model because domain walls depend on the presence of scalar fields in the early universe. “If the model is further validated by cosmological observations, it would imply the existence of the scalar sector,” Vachaspati says.
If so, we’d have found yet another new window on the early universe and its evolution. “Domain walls are like time capsules that have the potential to give us information about very high-energy physics,” Takahashi says.