10.1 Brans–Dicke theory and the equivalence with f (R) theories
Let us start with the following 4-dimensional action in BD theory
where
is the BD parameter,
is a potential of the scalar field
, and
is a matter
action that depends on the metric
and matter fields
. In this section we use the unit
, but we recover the gravitational constant
and the reduced Planck mass
when the discussion becomes transparent. The original BD theory [100] does not possess the field
potential
.
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.