Late time evolution of negatively curved FLRW models

We study the late time evolution of negatively curved Friedmann–Lemaître–Robertson–Walker (FLRW) models with a perfect fluid matter source and a scalar field nonminimally coupled to matter. Since under mild assumptions on the potential V, it is already known—see e.g., Giambò and Miritzis (Class Quantum Grav 27:095003, 2010)—that equilibria corresponding to nonnegative local minima for V are asymptotically stable, we classify all cases where one of the energy components eventually dominates. In particular for nondegenerate minima with zero critical value, we rigorously prove that if γ, the parameter of the equation of state is larger than 2/3, then there is a transfer of energy from the fluid and the scalar field to the energy density of the scalar curvature. Thus, the scalar curvature, if present, has a dominant effect on the late evolution of the universe and eventually dominates over both the perfect fluid and the scalar field. The analysis in complemented with the case where V is exponential, and therefore the scalar field diverges to infinity.