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
.