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 ():
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 
For the f (R) models whose deviation from the CDM model is small (), we have and . This gives the following estimate
Since , it follows that
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]: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.
This work is licensed under a Creative Commons License.