We rewrite the action (2.1) in the form
The conformal factor is field-dependent. From the matter action (2.32) the scalar field is directly coupled to matter in the Einstein frame. In order to see this more explicitly, we take the variation of the action (2.32) with respect to the field :
The strength of the coupling between the field and matter can be quantified by the following quantityf (R) gravity . It then follows that
Let us consider the flat FLRW spacetime with the metric (2.12) in the Jordan frame. The metric in the Einstein frame is given by
Equations (2.48) and (2.49) show that the field and matter interacts with each other, while the total energy density and the pressure satisfy the continuity equation . More generally, Eqs. (2.48) and (2.49) can be expressed in terms of the energy-momentum tensors defined in Eqs. (2.34) and (2.37): (see also ).
In the absence of a field potential (i.e., massless field) the field mediates a long-range fifth force with a large coupling (), which contradicts with experimental tests in the solar system. In f (R) gravity a field potential with gravitational origin is present, which allows the possibility of compatibility with local gravity tests through the chameleon mechanism [344, 343].
In f (R) gravity the field is coupled to non-relativistic matter (dark matter, baryons) with a universal coupling . We consider the frame in which the baryons obey the standard continuity equation , i.e., the Jordan frame, as the “physical” frame in which physical quantities are compared with observations and experiments. It is sometimes convenient to refer the Einstein frame in which a canonical scalar field is coupled to non-relativistic matter. In both frames we are treating the same physics, but using the different time and length scales gives rise to the apparent difference between the observables in two frames. Our attitude throughout the review is to discuss observables in the Jordan frame. When we transform to the Einstein frame for some convenience, we go back to the Jordan frame to discuss physical quantities.
This work is licensed under a Creative Commons License.