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 quantity
which is constant in f (R) gravity [28 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
):
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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