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 solutionsAs 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 byFor 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 conditionSince , 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.http://www.livingreviews.org/lrr-2010-3 |
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