### 10.3 Local gravity constraints

We study local gravity constraints (LGC) for BD theory given by the action (10.10). In the absence of
the potential the BD parameter is constrained to be from solar-system
experiments [616, 83, 617]. This bound also applies to the case of a nearly massless field with the potential
in which the Yukawa correction is close to unity (where is a scalar-field mass and
is an interaction length). Using the bound in Eq. (10.11), we find that
This is a strong constraint under which the cosmological evolution for such theories is difficult to be
distinguished from the CDM model ().
If the field potential is present, the models with large couplings () can be consistent with
local gravity constraints as long as the mass of the field is sufficiently large in the region of high
density. For example, the potential (10.23) is designed to have a large mass in the high-density region so
that it can be compatible with experimental tests for the violation of equivalence principle through the
chameleon mechanism [596]. In the following we study conditions under which local gravity constraints can
be satisfied for the model (10.23).

As in the case of metric f (R) gravity, let us consider a configuration in which a spherically symmetric
body has a constant density inside the body with a constant density outside the
body. For the potential in the Einstein frame one has under the
condition . Then the field values at the potential minima inside and outside the body are

The field mass squared at () is approximately given by
Recall that is roughly the same order as the present cosmological density . The
baryonic/dark matter density in our galaxy corresponds to . The mean density of Sun
or Earth is about . Hence and are in general much larger than for
local gravity experiments in our environment. For the chameleon mechanism we discussed in
Section 5.2 can be directly applied to BD theory whose Einstein frame action is given by Eq. (10.6) with
.

The bound (5.56) coming from the possible violation of equivalence principle in the solar system
translates into

Let us consider the case in which the solutions finally approach the de Sitter point (e) in Table 1. At this
de Sitter point we have with given in Eq. (10.26). Then the following
relation holds
Substituting this into Eq. (10.32) we obtain
where is the Ricci scalar at the de Sitter point. Since is smaller than 1/2 from
Eq. (10.28), it follows that . Using the values and
, we get the bound for [596]:
When and we have and , respectively. Hence the model can be
compatible with local gravity experiments even for .