In physics, it occasionally happens that an idea that scientists long ago declared dead suddenly gets a new lease on life. This is exactly what happened with a 19th-century hypothesis, when the Scottish physicist Lord Kelvin imagined atoms as tiny knots in an infinite "aether". This picture quickly fell apart under the onslaught of modern atomic theory – but today, nearly a century and a half later, the mathematical idea of knots returns to the stage in a completely different context: as a possible key to answering the question of why the universe as we know it even exists.

A new theoretical work by a team of Japanese physicists shows that "cosmic knots"—exotic, topologically protected structures in the fields that describe particles and forces—can naturally form in the early universe. These knots, if they indeed existed, could explain why a small excess of ordinary matter relative to antimatter remained after the Big Bang. Without this microscopic excess, the universe today would be filled only with radiation, and not with galaxies, stars, and ourselves.

The study was published in the prestigious journal Physical Review Letters and comes from an international research environment centered around Hiroshima University. In this work, the physicists connect several of the biggest puzzles of modern cosmology – the origin of the matter excess, dark matter, the masses of neutrino particles, and the so-called strong CP problem – into a single common framework based on the idea of knots in fields.

Lord Kelvin, the Aether, and Knots: From a Failed Picture of Atoms to Modern Cosmology

When Lord Kelvin proposed in 1867 that atoms were actually stable knots in a hypothetical aether, physics did not know about the electron, the proton, or quantum mechanics. It sought refuge in continuous media and geometric images. As experiments in the early 20th century showed that the aether does not exist, and atomic structure came down to nuclei and electrons, Kelvin was quickly cited by everyone as an example of a beautiful, but flawed concept.

However, the mathematical theory of knots itself—the way closed loops can be tangled, intertwined, and deformed without breaking—continued to develop in pure mathematics and later in condensed matter physics, and even in descriptions of fields in particle physics. In topology, a branch of mathematics that studies properties that do not vanish when an object is stretched or twisted, knots represent particularly stable configurations: to untie them, you must cut the thread.

In the new work by the Japanese team, this old topological idea does not return in the form of knotted atoms, but as knots in the invisible fields that permeate the universe after the Big Bang. These knots are not objects that we could ever "see" with a telescope like a rope or a cable, but abstract configurations in the equations that describe fundamental forces and particles. Yet, their presence could leave very real traces in the structure of the universe.

The Problem of Missing Antimatter and Baryogenesis

According to the standard Big Bang scenario, the universe should have created almost equal amounts of matter and antimatter at the very beginning. Every particle has its antiparticle twin—the electron has its positron, the proton its antiproton. When they meet, they mutually annihilate and turn into pure radiation. If the initial ratio was indeed one-to-one, it is natural to expect that after a short time, all the content of the universe would have disappeared in a flash of gamma-rays.

Instead, observations show that the visible universe is almost entirely built of matter, while antimatter is extremely rare. Theoretical calculations suggest that for every billion particle-antiparticle pairs, only one "excess" piece of matter survived. It is precisely this miniature excess that enabled the formation of atoms, stars, galaxies, and life. The process that produced this excess is called baryogenesis, but its exact mechanism remains one of the deepest open questions in cosmology.

The Standard Model of particle physics—the theoretical framework that brilliantly describes almost all known particles and forces, except gravity—fails to explain such a large asymmetry between matter and antimatter. Additional ingredients are needed, new symmetries and particles that go beyond what has already been discovered in accelerator experiments. The new model of cosmic knots attempts to fit in precisely here, offering a natural way to create the matter excess without introducing too many ad hoc elements.

The Hiroshima Team and the Institute for "Knot" States of Matter

The research is signed by physicists gathered around the International Institute for Sustainability with Intertwined Chiral Metamatter (WPI-SKCM²) at Hiroshima University, along with collaborators from Keio University in Japan and the German laboratory Deutsches Elektronen-Synchrotron (DESY). This is an interdisciplinary center that deals with phenomena where "knots" and related topological structures play a role in various fields—from a new generation of materials to fundamental physics.

The authors of the study combine knowledge from theoretical particle physics, cosmology, and mathematical topology. In the work, they show that in a realistic extension of the Standard Model, the kind usually studied for neutrino masses and dark matter, topological knots spontaneously form in the fields. These knots, which they call knot solitons, are not just an exotic mathematical curiosity, but can play a key role in baryogenesis.

Two Old Symmetries in a New Combination: B–L and Peccei–Quinn

At the heart of the new model are two symmetries that physicists have been studying for decades, but which no one has systematically combined into a unified scenario for the early universe until now. The first is the so-called B–L symmetry, which refers to the difference between the number of baryons (particles like protons and neutrons) and leptons (like electrons and neutrino particles). In this theory, B–L is not just a practical accounting quantity, but a gauge symmetry, meaning it requires a new force with corresponding interaction "carriers".

The second key component is the Peccei–Quinn (PQ) symmetry, introduced to solve the so-called strong CP problem. This problem arises from the fact that the theory of the strong nuclear force should, in principle, allow a small violation of the symmetry between matter and antimatter, but experiments have not found such an effect in the properties of the neutron for decades. The Peccei–Quinn symmetry elegantly removes this unwanted term, and as a byproduct introduces a new hypothetical particle—the axion—one of the main candidates for dark matter.

The very fact that PQ symmetry solves the strong CP problem and provides a dark matter candidate makes it extremely attractive. But the authors of the new paper decided to "pair" it with B–L symmetry. In doing so, they carefully choose for PQ to remain a global symmetry (i.e., it does not become a new force), in order to preserve the fine balance needed for the axion to retain its desired properties. B–L, on the other hand, is introduced as a local (gauge) symmetry, which naturally leads to the existence of heavy right-handed neutrinos—particles that are imposed anyway in most baryogenesis scenarios.

From Cosmic Strings to Knots in the Field

In the very early universe, immediately after the Big Bang, temperatures were so high that forces and particles behaved differently than they do today. As the universe expanded and cooled, it went through a series of phase transitions—sudden changes in the state of the fields, similar to water freezing into ice, but at the level of fundamental forces. These phase transitions could have left "scars" in the structure of space, known as topological defects.

One type of such defects are cosmic strings—extremely thin, but massive "cracks" in the fields, stretching through the universe like taut threads. Although still hypothetical, cosmic strings often appear in theories of grand unification and other extensions of the Standard Model. In the new work, the breaking of B–L symmetry creates magnetic flux tubes (strings), while PQ symmetry gives birth to vortex structures similar to superfluid vortices.

The key idea is that these two types of defects can "lock" together into a more stable configuration. The B–L string carries a magnetic flux, while the PQ vortex has no flux of its own, but through the so-called Chern–Simons coupling, it can "pump" charge into the magnetic tube. In this way, the tension that would otherwise cut the string ring is balanced by the additional energy of the connected structure, and the whole combination becomes a metastable knot—a topological soliton that cannot simply be stretched into a flat configuration.

Such knots are not just a mathematical drawing on a board. Their mass and energy can be enormous on the early-universe scale, and due to topological protection, they can live long enough to influence the dynamics of the universe's expansion. It is precisely this longevity that opens up space for a scenario in which knots take temporary dominance over the total energy density of the universe.

The "Knot-Dominated Era" – A Period When Knots Ruled the Universe

The authors introduce the term "knot-dominated era," a brief period after the Big Bang during which the energy stored in the knots exceeded the energy of radiation and ordinary matter. While photons are diluted and lose energy as the universe grows, knots behave more like cold matter: their density falls slower, so they can become the dominant component of the total energy budget.

But this dominance does not last forever. Although topologically protected, knots can decay through quantum tunneling—a process in which a system passes over an energy barrier that would classically be inaccessible to it. Through tunneling, the knot can "untie" and release its energy in the form of particles. In this model, it is precisely this collapse that triggers the chain of events that creates the matter excess.

When the knot decays, it releases a large number of heavy right-handed neutrinos, scalars, and new bosons associated with B–L symmetry. These heavy neutrinos then decay into lighter particles with a slight bias in favor of matter relative to antimatter. This small, but systematic "tilt" is sufficient to create the initial asymmetry. In the next step, known electroweak processes in the hot universe convert this asymmetry into a permanent excess of baryons—the protons and neutrons from which we are built.

By calculating how efficient the knots are in creating these heavy neutrinos, what mass these particles have, and how they reheat the universe upon decay, the authors show that the model naturally leads to a reheating temperature of about 100 giga-electronvolts (GeV). Interestingly, it is precisely around this energy scale that the window for processes that can convert leptonic asymmetry into baryonic asymmetry closes. In other words, the model "hits" a physically meaningful moment in the history of the universe when the asymmetry had to solidify if it wanted to survive until today.

Potential Gravitational Wave Signature of Knots

One of the biggest advantages of the cosmic knot model is that it offers a concrete, physically measurable trace that future experiments could look for: a background of gravitational waves. Knots and the string networks that form them are expected to emit gravitational waves—ripples in the very fabric of spacetime—throughout their life, whenever the tangled threads twitch, join, or decay.

Such a signal would not look like the brief flash that detectors occasionally record today during black hole collisions. Instead, it would be a continuous "sound background"—a noise of gravitational waves of different frequencies, the spectrum of which could differ from other theoretical models of the early universe. According to the authors' calculation, the dominance of knots and their decay could shift the gravitational spectrum towards higher frequencies, in the range where future missions will have the best sensitivity.

The European Space Agency is planning the LISA (Laser Interferometer Space Antenna) mission, a space interferometer sensitive to medium-frequency gravitational waves, while the Cosmic Explorer and DECIGO projects are in preparation in the US and Japan. If these detectors record a background of gravitational waves in the coming decades that corresponds to the predicted "signature" of knots, it would be a strong—though not necessarily final—argument in favor of this scenario.

An additional advantage is that the model does not stand alone: the same framework predicts the axion as a candidate for dark matter and heavy right-handed neutrinos that participate in the creation of the matter excess. Axion and neutrino research are already separate experimental fields. If traces of the axion, specific properties of neutrino particles, and the corresponding gravitational background appear in laboratories and cosmic observations in parallel, the puzzle could be put together into a coherent picture.

What the New Theory Means for Our Picture of the Universe

It is important to emphasize that this is a purely theoretical model. No one has yet "seen" a cosmic knot, nor is there an experiment that could directly record it. Researchers work with field equations, symmetries, and topological arguments, building a scenario that is mathematically consistent and in line with known physical constraints. Their work shows that such knots are not forbidden by existing knowledge and that they can arise naturally within the framework of a realistic extension of the Standard Model.

For particle physics, the scenario is interesting because it bundles multiple problems into one package instead of introducing a separate solution for each. The B–L symmetry introduces heavy neutrino particles that are needed anyway to explain the masses of ordinary neutrinos. Peccei–Quinn symmetry solves the strong CP problem and opens the door to the axion as a candidate for dark matter. Cosmic knots arise as a consequence of these very symmetries and thereby take care of baryogenesis, i.e., the matter excess. This gives the theory a degree of "economy" that physicists appreciate.

For cosmology, the model presents another possible story about what happened in the first fractions of a second after the Big Bang. In the last few decades, dozens of different baryogenesis mechanisms have been proposed, from leptogenesis through phase transitions to exotic short-lived particles. Many of them are difficult to test, precisely because they take place at energies and times that we cannot directly reach. Cosmic knots stand out because their "echo" in gravitational waves could be within reach of future observations.

Of course, the same gravitational background "singing" could also be explained by some other model—for example, a network of ordinary cosmic strings created during grand unification or other exotic processes in the early universe. Even if future observations show a signal, a whole series of additional tests and comparisons will be needed to narrow down the space of possible explanations. But the fact that the new theory is even falsifiable, i.e., that it offers clear predictions, makes it a serious candidate in the game.

The authors of the paper emphasize that the next step is a more precise modeling of the formation and decay of knots, as well as a detailed simulation of their gravitational wave "score." It is necessary to investigate for which values of parameters—particle masses, coupling strengths, and phase transition times—knots arise often enough and live long enough to truly have an observable effect. Only then will it be possible to analyze in parallel what is expected from future detectors and other experiments.

Topology as the Thread Connecting the Micro and Macro Universe

Perhaps the most interesting philosophical aspect of this story is that it re-emphasizes the power of topology in understanding nature. Kelvin intuitively suspected in the 19th century that knots could play a role in the structure of matter, although he did not have the right mathematical or experimental tools to prove it. Today, when field physics and numerical simulations enable the description of extremely complex configurations, the old idea gets a new, much more sophisticated form.

Knots in the field—topological solitons—appear in various branches of physics, for example, in magnetic materials, superfluids, and quantum liquids. If it is confirmed that similar structures also played a role in early cosmology, it would mean that the same mathematical principles connect the phenomenology on completely different scales: from laboratory material samples to the entire universe. Such "universality" of topological ideas is one of the reasons why more and more scientific attention is being invested in them.

For now, cosmic knots remain an elegant, but yet unproven scenario. But the mere possibility that the problem of the "missing" antimatter can be solved by recycling one of the most unusual ideas of the 19th century reminds us how non-linear science sometimes is. Ideas that one day seem forever abandoned can, in a completely different context and with much richer apparatus, return as serious candidates for explaining the deepest puzzles of the universe.