Disordered hyperuniform modulated phases and the cosmic web from one free energy

The universe is filled with patterns that seem to defy simple explanation. On the smallest scales, inside materials like glass or the cells of a living leaf, matter often arranges itself in ways that look random but possess a hidden, deep order. Scientists call this "disordered hyperuniformity," a state where the material is jumbled like a liquid but suppresses the large-scale ripples and clumps that usually appear in such messes. It is a property found in everything from the way sand piles on Mars to the arrangement of light-sensitive cells in bird eyes. On the largest scales, the cosmos itself displays a similar, vast architecture: a cosmic web of filaments and dense knots of galaxies separated by enormous, empty voids. For decades, physicists have treated these two worlds—the microscopic world of soft materials and the macroscopic world of gravity—as entirely separate realms, governed by different rules and studied by different communities. One is driven by the thermal jiggling of atoms, while the other is shaped by the relentless pull of gravity over billions of years.

A new study challenges this separation by proposing that these two vastly different structures might emerge from a single, unified source. The researchers, working with a mathematical model of how matter organizes itself, asked a simple question: could the same underlying energy rules that create the intricate labyrinths of soft materials also generate the cosmic web of the universe, depending only on how the system moves? They constructed a theoretical framework that describes matter not by tracking individual particles, but by looking at the density of the material as a whole. This framework includes two competing forces: one that encourages the material to form a specific, repeating pattern at a certain size, and another that mimics the long-range pull of gravity, which tries to pull matter together over vast distances. By running computer simulations that push this model to its extremes, the team discovered that the same set of rules can produce both the microscopic labyrinth and the cosmic web, simply by changing the speed at which the material responds to these forces.

In the first scenario, the researchers simulated a system where the material moves slowly and sluggishly, like a thick fluid where friction dominates. In this "overdamped" state, the material cannot build up speed; it simply settles into the nearest stable arrangement. Under these conditions, the competition between the pattern-forming force and the gravitational pull results in a frozen, bicontinuous maze. This structure looks like a sponge or a labyrinth, with two interwoven networks of material and empty space that are perfectly balanced. The researchers found that this state is a specific type of disordered hyperuniformity, known as a Class III state, which is exactly the kind of order seen in confined fluids and block copolymers. Crucially, the large-scale smoothness of this maze was not created by the movement of the material itself. Instead, the simulation showed that the system simply preserved the smoothness it started with. If the initial material was smooth on large scales, the final frozen maze remained smooth, regardless of how the small-scale patterns formed. The dynamics of the system acted like a filter, freezing the large-scale features in place while allowing the small-scale structures to evolve.

In the second scenario, the researchers switched the rules to simulate a system where inertia matters, like the early universe where matter flies freely under the influence of gravity. Here, the material is not slowed by friction; it accelerates and builds momentum. When the team applied this "inertial" dynamic to the same underlying energy rules, the result was a cosmic web. The simulation produced a network of filaments and dense nodes separated by vast voids, strikingly similar to the distribution of galaxies in our own universe. The researchers tested this by comparing their simulation to a universe governed only by gravity. They found that the tidal skeleton of the web—the way the matter collapses into sheets, filaments, and nodes—matched the gravity-only prediction with an accuracy of within 0.03 in every morphological category. The only difference was that the ordering force in their model prevented the matter from compressing into infinitely dense points, capping the density contrast at a specific limit. This suggests that the cosmic web's large-scale structure is robust and can be generated by a broader set of physical rules than just gravity alone.

Perhaps the most surprising discovery lies in how these two worlds connect. The researchers showed that the transition from the microscopic labyrinth to the cosmic web is continuous. By adjusting a single parameter that represents the "stiffness" or pressure of the material, they could slide the system from one state to the other. In the cosmic web simulation, they found that the ordering force, which creates the small-scale texture, leaves a subtle but measurable imprint on the largest scales. As the material clumps together, the ordering term generates a kind of residual pressure that slightly suppresses the growth of the very largest structures. This suppression is smooth and predictable, acting like a gentle brake on the largest scales without distorting the overall shape of the spectrum. However, as the researchers pushed the model toward the limit of "cold matter"—where the material has no internal pressure at all—this residual effect vanished. In this limit, the simulation showed that the large-scale structure of the universe tracks the growth of gravity with an accuracy better than one percent over a growth factor of 1363. This means that the universe's large-scale smoothness is not a fragile accident but a robust inheritance from the initial conditions, preserved even as the universe expands and collapses.

The study also clarified a fundamental question about the origin of order in the universe. For a long time, it was debated whether the smoothness of the cosmic web was a result of the complex, non-linear dance of gravity and matter, or if it was simply a memory of the smooth conditions that existed at the very beginning. The simulations provided a definitive answer: the large-scale smoothness is inherited. The researchers ran two sets of simulations starting with identical small-scale structures but different large-scale smoothness. In the slow, friction-dominated limit, the system froze the large-scale smoothness of the starting point. In the fast, gravity-dominated limit, the system amplified that starting smoothness without changing its character. The dynamics of the universe did not create the smoothness; it merely preserved and magnified what was already there. This finding suggests that the anomalous suppression of long-wavelength fluctuations, which defines disordered hyperuniformity, is not necessarily a sign of a system in perfect equilibrium or a specific ordering mechanism. Instead, it can emerge naturally as a consequence of how a system evolves from its initial state, whether that system is a soft material in a lab or the entire cosmos.

The implications of this work extend beyond just explaining the shape of the universe. By showing that a single free energy functional can describe both the microscopic world of soft matter and the macroscopic world of cosmology, the researchers have provided a common language for two fields that have rarely spoken to each other. The model demonstrates that the same mathematical principles governing the formation of a labyrinth in a polymer melt can also govern the formation of the cosmic web. This unification suggests that the deep order found in disordered systems is a universal feature of nature, arising from the interplay between local ordering forces and long-range interactions. The study does not claim to have solved every mystery of the cosmos, but it offers a powerful new perspective: that the universe and the materials we touch may be different expressions of the same underlying physical laws, separated only by the speed at which they move and the scale at which we observe them. The results, derived entirely from computer simulations based on these theoretical rules, provide a concrete bridge between the microscopic and the cosmic, showing that the universe's grand architecture is built on the same foundation as the smallest patterns of matter.