Crossing the phantom boundary in f(R) modified gravity: Jordan vs. Einstein frames

We study the capability of f(R) gravity models to allow for crossing the phantom boundary in both Jordan and Einstein conformal frames. In the Einstein frame, these models are equivalent to Einstein gravity with a scalar field minimally coupled to gravity. This scalar degree of freedom appears as a quintessence field with a coupling to the matter sector. We investigate the evolution of the equation of state parameter for some cosmologically viable f(R) gravity models in both conformal frames. This investigation (beyond mere theoretical arguments) acts as an operational tool to distinguish the physical status of the two conformal frames. It shows that this status is not the same.