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

The research, carried out by B. Yildirim and A. A. Coley of the Department of Mathematics and Statistics at Dalhousie University in Halifax, Canada, approaches the problem through the mathematics of scalar-tensor gravity. This class of theories extends Einstein’s general relativity by adding a scalar field that dynamically couples to the curvature of spacetime. Scalar-tensor gravity has long attracted attention in cosmology because it provides a natural mathematical setting in which nonsingular bouncing universes can be realized. In such models, the scalar field’s evolution can drive a contraction to a halt and trigger a rebound, replacing the dreaded big bang singularity with a smooth, finite transition. But while bouncing cosmologies have been studied extensively in homogeneous settings, the inclusion of localized objects such as black holes makes the field equations dramatically harder to solve.

To make progress, the authors adopted a perturbative strategy, treating the black hole as a small deviation from an otherwise perfectly uniform cosmos. At leading order in their perturbative scheme, controlled by a small parameter epsilon, the solution is a spatially flat Friedmann-Lemaître-Robertson-Walker, or FLRW, spacetime undergoing a bounce, sourced by a perfect fluid of radiation. This background captures the essence of a bouncing cosmology in the simplest possible form: the universe contracts, reaches a minimum size at a moment identified with a conformal time coordinate eta equal to zero, and then re-expands. This leading-order solution respects what the authors call the parabolic structure of the bounce, a smoothness condition on how the scale factor behaves at the turning point.

At the next order in the expansion, the team embedded a central inhomogeneity into the bouncing background using a generalized McVittie geometry. The original McVittie solution, constructed in 1933, is a celebrated exact solution of general relativity that describes a mass concentrated at the center of an expanding universe, providing a mathematically tractable bridge between black hole physics and cosmology. By generalizing this construction to the scalar-tensor setting and treating it perturbatively, Yildirim and Coley encoded the gravitational imprint of a localized compact object within the contracting and rebounding cosmos. The perturbations appear as first-order corrections to the metric and to the scalar field, and the coupled field equations were solved as a series expansion carried up to fourth order in the parameter eta near the bounce.

A central technical challenge arose from the nature of the matter content near the inhomogeneity. In the vicinity of a concentrated mass, the stress-energy generically becomes anisotropic, meaning that pressure differs along the radial direction compared with the tangential directions. The authors therefore first allowed an anisotropic fluid with separate radial and tangential pressures, whose diagonal components suffice to solve the diagonal components of the field equations. They then imposed the physically motivated condition that the stress-energy reduce to a perfect fluid, one with a single isotropic pressure, far from the center. The resulting perfect fluid solution contains three arbitrary functions, which are constrained by demanding that the spacetime smoothly asymptote to the homogeneous FLRW background as the radial coordinate tends to infinity, ensuring that the black hole’s influence fades with distance as it must.

With suitable initial conditions chosen to preserve the parabolic structure of the bounce, a remarkable simplification emerged. The solution’s integration constants consolidate into a single quantity, denoted d0, which the authors identify as the true perturbative parameter of the problem. When d0 is set to zero, every perturbation vanishes and the spacetime reverts exactly to the homogeneous bouncing FLRW universe. When d0 is small but nonzero, a localized inhomogeneity, and with it a small evolving horizon, appears in the geometry. This clean parametrization means the entire structure of the black hole embedding is controlled by one number, allowing the authors to track precisely how the compact object’s gravitational field responds to the cosmic contraction and rebound.

The key result of the analysis concerns the fate of that horizon. Yildirim and Coley find a small evolving horizon whose radius scales linearly with the perturbative parameter, roughly as d0, and which they interpret as the horizon of the central inhomogeneity. Crucially, this horizon persists through the bounce at eta equals zero, supporting the interpretation that a black hole present before the cosmological transition survives it and continues to exist in the expanding universe on the other side. Intriguingly, the evolution is not symmetric about the bounce: the horizon’s behavior on the contracting side differs from its behavior on the expanding side, suggesting that the cosmic reversal leaves a subtle imprint on the object even as it survives. In a cosmological context, such findings resonate with long-standing speculations about black holes from a previous cosmic epoch, sometimes discussed in connection with ideas about the origin of supermassive black holes and the possible relics of a pre-bounce universe.

The authors are careful about what their construction does and does not establish. The small horizon they track is a future outer trapping horizon, a locally defined surface characterized by the convergence properties of outgoing and ingoing light rays, and it behaves as such in a two-sided neighborhood of the bounce. Because the entire construction is local and perturbative, it does not demonstrate the existence of a global event horizon, the teleological boundary beyond which nothing can escape to infinity. The distinction matters in a dynamical spacetime, where global horizons are notoriously difficult to define and can depend on the entire future evolution of the cosmos. Still, the persistence of a local trapping horizon through a nonsingular bounce is a nontrivial and suggestive result, indicating that the mechanisms of black hole formation and survival may be more robust under extreme cosmological conditions than simpler arguments had implied.

The work also fits into a broader effort to understand how inhomogeneities behave in bouncing scenarios. Previous studies have explored whether structure formation is possible through a bounce, whether primordial black holes can survive ekpyrotic contractions, and how numerical simulations of nonsingular bouncing spacetimes handle black holes and their horizons. By providing an analytic, perturbative solution in scalar-tensor gravity, the Dalhousie team adds a complementary tool to this mostly numerical and heuristic literature. The mathematical framework, combining generalized McVittie geometries with a controlled expansion around a radiation-dominated bounce, offers a concrete laboratory in which questions about horizons, matter anisotropies, and scalar-field dynamics can be addressed with precision. While the analysis is idealized, assuming spherical symmetry, a radiation fluid, and a small perturbation strength, it demonstrates that black hole persistence through a bounce is a mathematically consistent possibility in a well-motivated class of gravitational theories, bringing the speculative picture of black holes bridging cosmic epochs one step closer to rigorous footing.

Subject of Research: Perturbative scalar-tensor cosmology modeling black hole survival through a nonsingular bouncing universe

Article Title: Black hole persistence in scalar-tensor theories

Article References: Yildirim, B., & Coley, A. A. (2026). Black hole persistence in scalar-tensor theories. General Relativity and Gravitation, 58(9), Article 111. https://doi.org/10.1007/s10714-026-03610-6

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03610-6

Keywords: black holes, scalar-tensor gravity, bouncing cosmology, general relativity, McVittie metric, trapping horizon, FLRW spacetime, cosmic bounce, perturbation theory, theoretical cosmology, nonsingular cosmology, gravitational physics