Taking the variation of the action (10.1) with respect to and , we obtain the following field equations

where is the Ricci scalar in metric f (R) gravity, and is the energy-momentum tensor of matter. In order to find the relation with f (R) theories in the metric and Palatini formalisms, we consider the following correspondence Recall that this potential (which is the gravitational origin) already appeared in Eq. (2.28). We then find that Eqs. (2.4) and (2.7) in metric f (R) gravity are equivalent to Eqs. (10.2) and (10.3) with the BD parameter . Hence f (R) theory in the metric formalism corresponds to BD theory with [467, 579, 152, 246, 112]. In fact we already showed this by rewriting the action (2.1) in the form (2.21). We also notice that Eqs. (9.4) and (9.2) in Palatini f (R) gravity are equivalent to Eqs. (2.4) and (2.7) with the BD parameter . Then f (R) theory in the Palatini formalism corresponds to BD theory with [262, 470, 551]. Recall that we also showed this by rewriting the action (2.1) in the form (9.8).One can consider more general theories called scalar-tensor theories [268] in which the Ricci scalar is coupled to a scalar field . The general 4-dimensional action for scalar-tensor theories can be written as

where and are functions of . Under the conformal transformation , we obtain the action in the Einstein frame [408, 611] where . We have introduced a new scalar field to make the kinetic term canonical:We define a quantity that characterizes the coupling between the field and non-relativistic matter in the Einstein frame:

Recall that, in metric f (R) gravity, we introduced the same quantity in Eq. (2.40), which is constant (). For theories with constant, we obtain the following relations from Eqs. (10.7) and (10.8): In this case the action (10.5) in the Jordan frame reduces to [596] In the limit that we have , so that Eq. (10.10) recovers the action of a minimally coupled scalar field in GR.Let us compare the action (10.10) with the action (10.1) in BD theory. Setting , the former is equivalent to the latter if the parameter is related to via the relation [343, 596]

This shows that the GR limit () corresponds to the vanishing coupling (). Since in metric f (R) gravity one has , as expected. The Palatini f (R) gravity corresponds to , which corresponds to the infinite coupling (). In fact, Palatini gravity can be regarded as an isolated “fixed point” of a transformation involving a special conformal rescaling of the metric [247]. In the Einstein frame of the Palatini formalism, the scalar field does not have a kinetic term and it can be integrated out. In general, this leads to a matter action which is non-linear, depending on the potential . This large coupling poses a number of problems such as the strong amplification of matter density perturbations and the conflict with the Standard Model of particle physics, as we have discussed in Section 9.Note that BD theory is one of the examples in scalar-tensor theories and there are some theories that give rise to non-constant values of . For example, the action of a nonminimally coupled scalar field with a coupling corresponds to and , which gives the field-dependent coupling . In fact the dynamics of dark energy in such a theory has been studied by a number of authors [22, 601, 151, 68, 491, 44, 505]. In the following we shall focus on the constant coupling models with the action (10.10). We stress that this is equivalent to the action (10.1) in BD theory.

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