Dissipative Unimodular Gravity: Linking Energy Diffusion to Bulk Viscosity as an Alternative to $\Lambda$CDM under DESI DR2 Data

In this paper, we explore a theoretical and observational study of the presence of viscosity in the Unimodular Gravity formalism, a pioneering approach that, to the best of our knowledge, has not been previously explored in this context. Specifically, we study a flat FLRW universe at late times, where matter experiences dissipative processes in the form of a bulk viscosity, in the framework of Eckart's theory, which is linked to the energy diffusion function $Q$ through the power law $\xi=\xi_{0}\left|Q\right|^{1/2}$, being $\xi_{0}$ a positive dimensionless parameter. By assuming the ansatz $Q=\nu H^{2}$, where $H$ is the Hubble parameter and $\nu$ is a dimensionless arbitrary constant, we find analytical solutions for the cosmological evolution. We test these models against the most recent cosmological observations, including type Ia supernovae, baryon acoustic oscillations, cosmic chronometers, gravitational lensing, and black hole shadow data. Our results show that two of the tested models provide a significantly better fit to the data ($\chi_{\text{min}}^{2}$) and remain as competitive as $\Lambda$CDM model according to the Bayesian Information Criterion. These findings, combined with the inherent ability of Unimodular Gravity to alleviate the cosmological constant problem, position dissipative UG as a robust and compelling alternative to the standard model, potentially suggesting that a very small but nontrivial energy nonconservation is compatible with the late-time observational data.