Unimodular Gravity with Arbitrary Diffusion Function: A Dynamical System Reconstruction Approach

Abstract

We investigate cosmological diffusion models in unimodular gravity within a dynamical systems reconstruction framework. By treating the logarithmic slope of the diffusion sector as an invertible dynamical variable, the diffusion function can be systematically reconstructed from the phase-space structure of the cosmological evolution. Under these conditions, we determine the physically admissible fixed points of the system, identifying novel matter--diffusion scaling solutions associated with power-law diffusion sectors, as well as purely diffusion-dominated configurations capable of driving late-time accelerated expansion without requiring an explicit cosmological constant term. The local behavior around the fixed points is then extended to the full cosmological evolution, providing a framework to explore the global implications of diffusion cosmologies. Beyond the asymptotic fixed-point structure, we further develop a reconstruction formalism based on the dynamical evolution of the diffusion slope, allowing for trajectories interpolating between different diffusion regimes during the cosmic history. Our results establish a systematic framework for constructing and classifying viable diffusion cosmologies in unimodular gravity directly from the phase-space dynamics.