Rebuilding Dark Energy From Scratch: Gravity Without Energy Conservation Gets a New Mathematical Toolkit

One of the deepest puzzles in modern physics may have just acquired a powerful new set of mathematical tools. In a study published in The European Physical Journal C, Gabriel Gómez of Universidad Mayor, together with Guillermo Palma and Norman Cruz of Universidad de Santiago de Chile, has developed a systematic framework for reconstructing an entire class of cosmological models built on unimodular gravity — a modified version of Einstein’s theory in which energy is not strictly conserved. Instead of guessing what form the mysterious energy exchange should take, the team shows how it can be extracted directly from the structure of the universe’s own dynamical evolution, opening a path toward dark energy models that emerge from the mathematics rather than from ad hoc assumptions.

Unimodular gravity has long intrigued theorists precisely because of how it handles the cosmological constant problem, arguably the most notorious discrepancy between theory and observation in physics. Quantum field theory predicts a vacuum energy vastly larger than what astronomers measure, yet in unimodular gravity the situation changes fundamentally. By restricting the symmetry of Einstein’s theory so that transformations must preserve the four-dimensional volume element, the theory yields only the trace-free part of Einstein’s field equations. The cosmological constant then ceases to be a parameter fixed by the microscopic physics and instead appears as a constant of integration, much like the energy of a pendulum, whose value is determined by initial conditions rather than by quantum loops.

The price of this elegant restructuring is a modified conservation law. In standard general relativity, the energy-momentum tensor of matter is covariantly conserved, which forces the cosmological constant to be truly constant. In unimodular gravity, that constraint is relaxed, and the divergence of the energy-momentum tensor is balanced by the gradient of the Lagrange multiplier that enforces the volume restriction. Solving the resulting equation reveals that the cosmological term splits into two pieces: a fixed integration constant and an arbitrary function of time, which the authors call the energy diffusion function. This function quantifies exactly how much local energy conservation fails, and it acts as a continuous source or sink of energy for the cosmic fluids filling the universe.

Physically motivated guesses for this diffusion function have been proposed before. One striking example draws on the continuous spontaneous localization model of quantum collapse, in which energy is genuinely created during wavefunction collapse, suggesting a diffusion term proportional to the energy density of the fluid itself. Earlier work showed that when only dark matter diffuses, the resulting cosmology closely reproduces the standard Lambda-CDM model, and related diffusion scenarios have even been explored as a way to ease the Hubble tension, the persistent disagreement between different measurements of the universe’s expansion rate. The trouble, as the Chilean team emphasizes, is that no widely accepted diffusion function follows from an established physical process, so most proposals rest on phenomenological convenience or mathematical simplicity rather than principle.

The new study attacks this problem from the opposite direction. Rather than postulating a diffusion law and studying its consequences, the authors recast the cosmological equations as an autonomous dynamical system and ask what diffusion functions the phase-space structure itself demands. The key move is to introduce a dimensionless variable called the diffusion slope, defined as minus the logarithmic derivative of the diffusion function with respect to the e-fold number, a natural clock for cosmic expansion. For the system to close, this slope must be invertible along the trajectories — a condition satisfied whenever it evolves monotonically. Remarkably, when the slope is held constant, the formalism automatically spits out the power-law diffusion functions that had previously been introduced by hand, suggesting that the dynamical systems perspective can rediscover and organize known models rather than merely accommodate them.

With the autonomous system in hand, the team mapped out the fixed points of cosmic evolution and their stability. The familiar matter-dominated era appears as a saddle point, not an attractor, meaning the universe inevitably passes through it rather than lingering. The de Sitter solution, where expansion accelerates at a constant rate under the influence of the integration constant, emerges as a genuine late-time attractor for positive diffusion slopes. More intriguingly, the analysis uncovered a novel matter-diffusion scaling solution in which the diffusion term tracks the dark matter density, maintaining a constant fractional share of the cosmic energy budget. Within a specific range of parameters, this scaling regime itself drives accelerated expansion — entirely without a cosmological constant — although it represents a transient stage rather than a final destiny.

Perhaps the most provocative result concerns the purely diffusion-dominated configuration. When the diffusion function becomes constant, it behaves exactly like a cosmological constant, producing exponential expansion with an effective equation-of-state parameter of minus one. In that regime, the diffusion sector alone can power late-time acceleration, with no fundamental vacuum energy required at all. The stability analysis revealed subtlety here: linear theory alone cannot settle the fate of this point because one eigenvalue vanishes, so the researchers supplemented their analytic work with numerical integration of the phase-space flow, showing that trajectories are attracted along one direction but repelled along another. The diffusion-dominated solution is therefore a saddle, a waystation the cosmos may visit but not a permanent home — the true endpoint remains the de Sitter state governed by the integration constant.

Beyond the asymptotic regimes, the authors built a full reconstruction machinery analogous to potential reconstruction in scalar-field cosmology. By specifying a curvature function that controls how the diffusion slope bends in logarithmic space, one can integrate the slope evolution and then reconstruct the diffusion function itself through a simple exponential integral. The constant-curvature family already displays rich behavior: curvature equal to one recovers power-law diffusion, curvature greater than one drives the diffusion term smoothly to zero as the universe expands, and curvature below one produces a finite-time singularity in the slope, signaling a breakdown of the description. More elaborate curvature functions that cross unity allow the slope to station at multiple values, enabling trajectories that interpolate between a rapidly decaying diffusion contribution in the early universe and an asymptotically constant component at late times — precisely the kind of transition a viable dark energy candidate might need.

The framework also clarifies what thermodynamics demands. Using the Gibbs relation, the team showed that the second law of thermodynamics requires the diffusion function to be non-increasing as the universe expands, guaranteeing positive entropy production. Notably, this condition constrains only the trend, not the sign, of the diffusion term, which can push the effective cosmological constant either up or down. The reconstruction formalism respects this automatically: because the diffusion function is built from a strictly positive exponential factor, it cannot change sign dynamically, keeping every reconstructed model consistent with the thermodynamic constraint from the outset.

The implications reach well beyond formal elegance. Because the diffusion sector exchanges energy with dark matter, it should alter the growth of cosmic structure, leaving fingerprints in the matter power spectrum and in the growth rate of galaxies that could distinguish diffusion cosmologies from Lambda-CDM. The authors identify this perturbative analysis, together with a direct confrontation of the reconstructed models with observational data, as the natural next step of their program. If the coming generation of surveys continues to hint that dark energy is not perfectly constant — as recent measurements have suggested — then a framework that generates and classifies diffusion models from first principles, rather than by trial and error, may prove exactly what cosmologists need to make sense of a universe that refuses to sit still.

Subject of Research: Cosmological diffusion models and energy non-conservation in unimodular gravity, analyzed through dynamical systems reconstruction

Article Title: Unimodular gravity with arbitrary diffusion function: a dynamical system reconstruction approach

Article References: Unimodular gravity with arbitrary diffusion function: a dynamical system reconstruction approach. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16363-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16363-y

Keywords: unimodular gravity, cosmological constant problem, dark energy, energy diffusion function, dynamical systems, phase-space fixed points, de Sitter expansion, scaling solutions, entropy production, Lambda-CDM, Hubble tension, theoretical cosmology