William Giarè, Jeremy Sakstein

We investigate whether early- and late-time dark energy could arise from a single scalar field. Adopting a bottom-up perspective, we first identify the sequence of dynamical regimes that any unified scenario must traverse to account for both an early dark energy phase relevant for pre-recombination solutions of the Hubble tension and the late-time acceleration of the Universe. We derive the corresponding requirements on the scalar energy density and equation of state. We then adopt a complementary top-down perspective and translate these requirements into constraints on the phase-space structure of minimally coupled scalar fields with tracking-like dynamics. We show that satisfying all requirements necessitates a potential with three distinct slopes, arranged in a steep-steeper-shallow hierarchy. This conclusion remains unchanged in the presence of conformal couplings to dark matter. These results place strong constraints on attempts to construct unified models of early- and late-time dark energy. We discuss implications for model-building.