We rewrite the action (2.1) in the form

where Using Eq. (2.25) and the relation , the action (2.27) is transformed as We obtain the Einstein frame action (linear action in ) for the choice This choice is consistent if . We introduce a new scalar field defined by From the definition of in Eq. (2.26) we have that . Using Eq. (2.26), the integral vanishes on account of the Gauss’s theorem. Then the action in the Einstein frame is where Hence the Lagrangian density of the field is given by with the energy-momentum tensorThe 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 :

that is Using Eq. (2.24) and the relations and , the energy-momentum tensor of matter is transformed as The energy-momentum tensor of perfect fluids in the Einstein frame is given by The derivative of the Lagrangian density with respect to isThe strength of the coupling between the field and matter can be quantified by the following quantity

which is constant in f (R) gravity [28]. It then follows that where . Substituting Eq. (2.41) into Eq. (2.36), we obtain the field equation in the Einstein frame: This shows that the field is directly coupled to matter apart from radiation ().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

which leads to the following relations (for ) where Note that Eq. (2.45) comes from the integration of Eq. (2.40) for constant . The field equation (2.42) can be expressed as where Defining the energy density and the pressure , Eq. (2.46) can be written as Under the transformation (2.44) together with , , and , the continuity equation (2.17) is transformed asEquations (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):

which correspond to the same equations in coupled quintessence studied in [23] (see also [22]).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.

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