### 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 [470250154448]
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 [61683617]:

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 [469470245233154448330332]. 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 [344343] becomes important in this non-linear regime ([251306134101]. 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.