Unified dark sector approaches to cosmological tensions
We present an analytical study of minimal mass-varying dark matter models, in which a single scalar field controls the dark matter mass, focusing on their utility in simultaneously addressing multiple cosmological tensions. We show that such minimal models can naturally become relevant at two critical epochs in the history of the Universe; matter-radiation equality and the present day, generating an early dark energy-like energy injection, as motivated by the Hubble tension, and an apparent phantom crossing, as suggested by baryon acoustic oscillation data from the DESI project. Crucially, the characteristic timing of these effects is tied to the cosmological background evolution itself, rather than specific energy scales. In addition, the intermediate evolution reduces the clustering matter density and can therefore lower $S_8$, in the direction preferred by data, while avoiding new long-range interactions. We derive analytic arguments to understand the conditions for these effects to arise in the same model and assess whether they can occur without introducing additional fine-tuning. This provides a simple framework for identifying which potentials and couplings to dark matter can simultaneously affect early- and late-time cosmological tensions.