The emergence of cosmological structures and the fundamental expansion of the universe have long been attributed to the intricate interplay of general relativity and quantum field theory. However, a groundbreaking paper submitted to the arXiv preprint server on August 7, 2026, by physicist Abhishodh Prakash, proposes a radical alternative. The research, titled "Classical fractons as a model for cosmological dynamics," suggests that the large-scale evolution of a flat, matter-dominated universe can be modeled using classical fractons—Hamiltonian systems characterized by constrained mobility and the conservation of multipole moments. By analyzing a specific family of fracton Hamiltonians, Prakash demonstrates that the salient features of our universe, including its expansion rate and the formation of large-scale structures, emerge naturally as attractor properties rather than as the result of highly specific initial conditions or "fine-tuning."
The Fracton Framework and Dipole Conservation
Fractons are a relatively new class of quasiparticles that first gained prominence in condensed matter physics due to their unusual mobility constraints. Unlike traditional particles, a single fracton is immobile; however, pairs or clusters can move under specific conditions, often related to the conservation of dipole moments. While most research into fractons has focused on quantum many-body systems and lattice models, Prakash’s work shifts the focus to classical Hamiltonian systems.
The core of the study revolves around a scale-invariant, dipole-conserving two-parameter family of fracton Hamiltonians, denoted as $H_alpha,beta$. In these systems, the total momentum and the total dipole moment are conserved quantities. This conservation significantly restricts the phase space of the system, leading to unique dynamical behaviors. The research identifies that while the full phase space of these classical fractons admits no attractors, a projection onto configuration or "shape" variables reveals the development of robust attractors. This means that regardless of the initial positions and velocities of the particles, the system tends to evolve toward specific geometric configurations over time.
Mathematical Separation: Scale vs. Shape Dynamics
A pivotal aspect of Prakash’s analysis is the separation of coordinates into two distinct components: "scale" and "shape." Scale refers to the overall size or expansion factor of the system, represented as $R(t)$, while shape refers to the relative distribution and arrangement of the particles within that scale.
The study finds that the shape dynamics are autonomous, meaning they evolve independently of the overall scale. These dynamics admit fixed points, which are essentially stable arrangements that the particles settle into. Once the shape reaches a fixed point, the evolution of the system becomes purely a matter of scale. The mathematical derivation shows that the scale evolution follows a power-law form: $R(t) propto |t|^alpha/(alpha-beta)$.
This result is significant because it provides a mechanical basis for universal expansion. The fixed points that determine the distribution of the expanding particles are identified as "central configurations" of power-law Riesz potentials. Riesz potentials are mathematical generalizations of the more familiar Newtonian or Coulomb potentials, and their central configurations represent states where the force on each particle is directed toward the center of mass and is proportional to its distance from that center.
The "Distinguished Model" and the Einstein-de Sitter Connection
Among the family of Hamiltonians explored, Prakash identifies a "distinguished model" where the parameters are set to $(alpha,beta) = (-2,1)$. This specific configuration yields results that are strikingly similar to the observed universe.
First, the scale evolution for this model takes the form $R(t) propto |t|^2/3$. This is identical to the Einstein-de Sitter model, which describes a flat, matter-dominated universe in standard Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology. In the Einstein-de Sitter framework, this $2/3$ power-law expansion is a fundamental result of the Friedmann equations; in Prakash’s model, it emerges as a natural consequence of fracton dynamics.
Furthermore, the fixed-point equation for the $(-2,1)$ model corresponds to the equal-mass Newtonian central configuration. In simulations involving a large number of particles ($N$), the distribution of these particles approaches a homogeneous ball. This mirrors the large-scale homogeneity and isotropy observed in our universe, as described by the Cosmological Principle. Perhaps most intriguing is the discovery that the homothetic trajectories of this fracton model admit a zero-energy Newtonian gravitational dual, suggesting a deep underlying link between fracton constraints and gravitational physics.
Simulation Results: Small N vs. Large N Dynamics
To validate the theoretical findings, the research employed numerical simulations at various scales. At moderate values of $N$, the systems were shown to approach the predicted shape fixed points even when starting from entirely random initial data. This stability confirms that the cosmological structures are indeed attractors within the shape space.
However, as the number of particles increases (large $N$), the simulations reveal a much richer and more complex structure. Instead of all particles settling into a single static arrangement, they form "bound clusters." These clusters retain internal motion and have an approximately fixed physical size, much like galaxies or star clusters in the real universe.
The centers of these clusters then approach "unequal-mass Newtonian central configurations." Despite the internal complexity and motion of the clusters, the system as a whole preserves large-scale homogeneity. Prakash introduces a "scale-separation conjecture" to explain this, which yields an effective unequal-mass fracton dynamics for the centers of the clusters. This effective dynamics also possesses a zero-energy Newtonian gravitational dual, reinforcing the model’s consistency across different scales of observation.
The Janus Point and the Arrow of Time
One of the most profound implications of the fracton model is its treatment of time and entropy. The trajectories within the $H_alpha,beta$ family generically exhibit what is known as a "bidirectional arrow of time." This is centered around a "Janus point"—a point of minimum size and maximum density in the system’s history.
As the system moves away from the Janus point in either direction (into the past or the future), both the scale of the system and its "shape complexity" grow. This provides a theoretical explanation for why we perceive time as moving in a specific direction: we are moving away from a point of lower complexity toward a state of higher complexity.
The research also tracks the Boltzmann entropy of the system. In standard thermodynamics, entropy is expected to increase over time. In this fracton model, Prakash demonstrates that Boltzmann entropy grows logarithmically as the system expands away from the Janus point. This growth in entropy, coupled with the growth in shape complexity, reproduces the thermodynamic structure of a flat, matter-dominated cosmology without requiring a singular "Big Bang" event in the traditional, non-mathematical sense.
Scientific Context and Historical Timeline
The proposal of using fractons to model cosmology sits at the intersection of several decades of theoretical development:
- 1930s-1960s: Development of the Einstein-de Sitter model and FLRW cosmology, establishing the $t^2/3$ expansion for matter-dominated universes.
- 2011-2015: The discovery of fracton topological phases in condensed matter physics, primarily through the work of researchers like Haah, Vijay, and Fu.
- 2017-2022: Theoretical physicists began exploring the "fracton-elasticity duality" and the potential for fracton physics to explain aspects of linearized gravity and emergent geometry.
- 2024-2025: Increasing interest in the "Janus point" theory (pioneered by Julian Barbour and others) as a solution to the problem of time in physics.
- August 7, 2026: Submission of "Classical fractons as a model for cosmological dynamics," which synthesizes these fields into a cohesive toy model for the universe.
Analysis of Implications: Solving the Fine-Tuning Problem
The primary advantage of the fracton model, as highlighted by Prakash, is the elimination of fine-tuning. In standard cosmological models, the universe must begin with extremely specific density and expansion parameters to avoid collapsing immediately or expanding too quickly for structures to form. This is often addressed through the theory of cosmic inflation, which requires its own set of specific conditions.
In the fracton model, the "Einstein-de Sitter" behavior is an attractor. This means that a wide range of initial conditions will naturally lead to the same cosmological outcome. If our universe behaves like a classical fracton system, its current state is not a statistical fluke but an inevitable destination.
While the scientific community has yet to provide a peer-reviewed consensus on the paper, the implications are vast. If gravity can be effectively modeled as a consequence of dipole conservation and fracton-like constraints, it could lead to a new understanding of dark matter. The "bound clusters" observed in the large-$N$ simulations provide a natural mechanism for the formation of dark matter halos without the need for additional particles outside the standard model.
Potential Challenges and Future Research
Despite the elegance of the model, several questions remain. The current paper presents a "toy model," meaning it simplifies certain aspects of reality to make the mathematics tractable. For instance, the model focuses on a matter-dominated universe and does not yet account for the current era of dark energy-dominated accelerated expansion (the Lambda-CDM model).
Future research will likely focus on:
- Incorporating Dark Energy: Modifying the Hamiltonian $H_alpha,beta$ to account for a positive cosmological constant.
- Quantum Transitions: Investigating how these classical fracton attractors behave when quantum mechanical effects are introduced, particularly near the Janus point.
- Observational Tests: Determining if the logarithmic entropy growth or the specific cluster distributions predicted by the model can be detected in cosmic microwave background (CMB) data or large-scale galaxy surveys.
The work of Abhishodh Prakash represents a significant shift in how theoretical physicists might approach the "Big Questions" of the universe’s origin and evolution. By demonstrating that the structure of the cosmos can emerge from the simple, constrained dynamics of classical fractons, the paper opens a new window into the fundamental nature of space, time, and gravity.
