Energy exchange for homogeneous and isotropic universes with a scalar field coupled to matter

Understand

We study the late time evolution of flat and negatively curved Friedmann-Robertson-Walker (FRW) models with a perfect fluid matter source and a scalar field arising in the conformal frame of $f(R)$ theories nonminimally coupled to matter.

• Under mild assumptions on the potential V we prove that equilibria corresponding to non-negative local minima for V are asymptotically stable, as well as horizontal asymptotes approached from above by V.
• We classify all cases of the flat model where one of the matter components eventually dominates.
• In particular for a nondegenerate minimum of the potential with zero critical value we prove in detail that if the parameter of the equation of state is larger than one, then there is a transfer of energy from the fluid to the scalar field and the later eventually dominates in a generic way.