Non-minimally coupled scalar field dark sector of the Universe: in-depth (Einstein frame) case study

Abstract In this study, motivated by recent results from DESI DR2 (Abdul Karim et al 2025 Phys. Rev. D 112 083515; Lodha et al 2025 Phys. Rev. D 112 083511) indicating a possible preference for evolving dark energy in some combined data analysis, we analyze spatially flat FLRW interacting scalar-tensor cosmological models with non-minimal coupling between the scalar field (SF) and matter/cosmological dust in the Einstein conformal frame. By using modified expansion normalized variables that account for negative values of the SF potential, we derive cosmological dynamical system equations and expressions for physical variables. Five specific SF models (axions/axion-like particles, cyclic ekpyrotic, exponential with a constant, quintessence, and SF dark matter) are examined in depth to determine how they evolve, as they serve as representative candidates used in the literature for the evolving dark energy in the late Universe. Using appropriate mathematical methods (e.g. linear stability, center manifold and Poincaré sphere), we present critical points along with their character and physical interpretation with respect to the possible evolution of the Universe. Considering both positive and negative values of the coupling parameter allows us to examine the transfer of energy from the scalar sector to dust and from matter to the SF. Considering four representative absolute values of this parameter provides a comprehensive analysis that incorporates all significant types of dynamical system evolution (from mathematical and physical perspectives). The initial conditions for the in-depth numerical analysis were constructed analytically from observationally motivated Planck 2018 (Aghanim et al 2020 Astron. Astrophys. 641 A6) and DESI DR2 (Abdul Karim et al 2025 Phys. Rev. D 112 083515) CPL parameterizations.