### 2.2 Equivalence with Brans–Dicke theory

The f (R) theory in the metric formalism can be cast in the form of Brans–Dicke (BD) theory [100] with
a potential for the effective scalar-field degree of freedom (scalaron). Let us consider the following action
with a new field ,
Varying this action with respect to , we obtain
Provided it follows that . Hence the action (2.18) recovers the action (2.1) in
f (R) gravity. If we define
the action (2.18) can be expressed as
where is a field potential given by
Meanwhile the action in BD theory [100] with a potential is given by

where is the BD parameter and . Comparing Eq. (2.21) with
Eq. (2.23), it follows that f (R) theory in the metric formalism is equivalent to BD theory
with the parameter [467, 579, 152] (in the unit ). In Palatini f (R) theory
where the metric and the connection are treated as independent variables, the
Ricci scalar is different from that in metric f (R) theory. As we will see in Sections 9.1 and
10.1, f (R) theory in the Palatini formalism is equivalent to BD theory with the parameter
.