5.1 Linear expansions of perturbations in the spherically symmetric background
First we decompose the quantities
,
, and
into the background part and the perturbed
part:
,
, and
about the approximate Minkowski
background (
). In other words, although we consider
close to a mean-field value
, the
metric is still very close to the Minkowski case. The linear expansion of Eq. (2.7) in a time-independent
background gives [470
, 250
, 154
, 448
]
where
and
The variable
is defined in Eq. (4.67). Since
for viable f (R) models, it follows that
(recall that
).
We consider a spherically symmetric body with mass
, constant density
, radius
, and vanishing density outside the body. Since
is a function of the distance
from
the center of the body, Eq. (5.1) reduces to the following form inside the body (
):
whereas the r.h.s. vanishes outside the body (
). The solution of the perturbation
for positive
is given by
where
(
) are integration constants. The requirement that
as
gives
. The regularity condition at
requires that
. We match two solutions (5.4)
and (5.5) at
by demanding the regular behavior of
and
. Since
, this
implies that
is also continuous. If the mass
satisfies the condition
, we obtain the
following solutions
As we have seen in Section 2.3, the action (2.1) in f (R) gravity can be transformed to the Einstein
frame action by a transformation of the metric. The Einstein frame action is given by a linear action in
,
where
is a Ricci scalar in the new frame. The first-order solution for the perturbation
of the
metric
follows from the first-order linearized Einstein equations in the
Einstein frame. This leads to the solutions
and
.
Including the perturbation
to the quantity
, the actual metric
is given by [448
]
Using the solution (5.7) outside the body, the
and
components of the metric
are
where
and
are the effective gravitational coupling and the post-Newtonian parameter,
respectively, defined by
For the f (R) models whose deviation from the
CDM model is small (
), we have
and
. This gives the following estimate
where
is the gravitational potential at the surface of the body. The
approximation
used to derive Eqs. (5.6) and (5.7) corresponds to the condition
Since
, it follows that
The validity of the linear expansion requires that
, which translates into
. Since
at
, one has
under the condition (5.12). Hence
the linear analysis given above is valid for
.
For the distance
close to
the post Newtonian parameter in Eq. (5.10) is given by
(i.e., because
). The tightest experimental bound on
is given by [616
, 83
, 617
]:
which comes from the time-delay effect of the Cassini tracking for Sun. This means that
f (R) gravity models with the light scalaron mass (
) do not satisfy local gravity
constraints [469, 470
, 245, 233, 154, 448, 330
, 332
]. The mean density of Earth or Sun is of the order of
, which is much larger than the present cosmological density
. In
such an environment the condition
is violated and the field mass
becomes large such that
. The effect of the chameleon mechanism [344
, 343
] becomes important in this non-linear
regime (
) [251
, 306
, 134
, 101]. In Section 5.2 we will show that the f (R) models
can be consistent with local gravity constraints provided that the chameleon mechanism is at
work.