The action (2.1) in f (R) gravity can be transformed to the Einstein frame action (2.32) with the coupling between the scalaron field and non-relativistic matter. Let us consider a spherically symmetric body with radius in the Einstein frame. We approximate that the background geometry is described by the Minkowski space-time. Varying the action (2.32) with respect to the field , we obtain. Recall that the field potential is given in Eq. (2.33). The energy density in the Einstein frame is related with the energy density in the Jordan frame via the relation . Since the conformal transformation gives rise to a coupling between matter and the field, is not a conserved quantity. Instead the quantity corresponds to a conserved quantity, which satisfies . Note that Eq. (5.15) is consistent with Eq. (2.42).
In the following we assume that a spherically symmetric body has a constant density inside the body () and that the energy density outside the body () is (). The mass of the body and the gravitational potential at the radius are given by and , respectively. The effective potential has minima at the field values and :
When the effective potential has a minimum for the models with , which occurs, e.g., for the inverse power-law potential . The f (R) gravity corresponds to a negative coupling (), in which case the effective potential has a minimum for . As an example, let us consider the shape of the effective potential for the models (4.83) and (4.84). In the region both models behave as
In order to solve the “dynamics” of the field in Eq. (5.15), we need to consider the inverted effective potential . See the lower panel of Figure 3 for illustration [which corresponds to the model (4.84)]. We impose the following boundary conditions:
In the region one can approximate the r.h.s. of Eq. (5.15) as around , where . Hence the solution to Eq. (5.15) is given by , where and are constants. In order to avoid the divergence of at we demand the condition , in which case the solution is
In the region the field evolves toward larger values with the increase of . In the lower panel of Figure 3 the field stays around the potential maximum for , but in the regime it moves toward the left (largely negative region). Since in this regime we have that in Eq. (5.15), where we used the condition . Hence we obtain the following solution
Since the field acquires a sufficient kinetic energy in the region , the field climbs up the potential hill toward the largely negative region outside the body (). The shape of the effective potential changes relative to that inside the body because the density drops from to . The kinetic energy of the field dominates over the potential energy, which means that the term in Eq. (5.15) can be neglected. Recall that one has under the condition [see Eq. (5.22)]. Taking into account the mass term , we have on the r.h.s. of Eq. (5.15). Hence we obtain the solution with constants and . Using the boundary condition (5.25), it follows that and hence
Three solutions (5.26), (5.27) and (5.28) should be matched at and by imposing continuous conditions for and . The coefficients , , and are determined accordingly : [344, 343] by assuming that the field is frozen at in the region .
The radius is determined by the following condition
If the field value at is away from , the field rolls down the potential for . This corresponds to taking the limit in Eq. (5.40), in which case the field profile outside the body is given by
Let us consider the case in which is close to , i.e.[344, 343]. Neglecting second-order terms with respect to and in Eq. (5.40), it follows that
Since under the conditions and , the amplitude of the effective coupling becomes much smaller than 1. In the original papers of Khoury and Weltman [344, 343] the thin-shell solution was derived by assuming that the field is frozen with the value in the region . In this case the thin-shell parameter is given by , which is different from Eq. (5.43). However, this difference is not important because the condition is satisfied for most of viable models .
We derive the bound on the thin-shell parameter from experimental tests of the post Newtonian parameter in the solar system. The spherically symmetric metric in the Einstein frame is described by 
Let us transform the metric (5.46) back to that in the Jordan frame under the inverse of the conformal transformation, . Then the metric in the Jordan frame, , is given by
The term in Eq. (5.48) is smaller than under the condition . Provided that the field reaches the value with the distance satisfying the condition , the metric does not change its sign for . The post-Newtonian parameter is given byf (R) gravity () this corresponds to .
Let us next discuss constraints on the thin-shell parameter from the possible violation of equivalence principle (EP). The tightest bound comes from the solar system tests of weak EP using the free-fall acceleration of Earth () and Moon () toward Sun . The experimental bound on the difference of two accelerations is given by [616, 83, 617]
Provided that Earth, Sun, and Moon have thin-shells, the field profiles outside the bodies are given by Eq. (5.44) with the replacement of corresponding quantities. The presence of the field with an effective coupling gives rise to an extra acceleration, . Then the accelerations and toward Sun (mass ) are [134, 596] f (R) gravity. This bound provides a tighter bound on model parameters compared to (5.51).
Since the condition is satisfied for , one has from Eq. (5.43). Then the bound (5.55) translates into
We place constraints on the f (R) models given in Eqs. (4.83) and (4.84) by using the experimental bounds discussed above. In the region of high density where is much larger than , one can use the asymptotic form (5.19) to discuss local gravity constraints. Inside and outside the spherically symmetric body the effective potential for the model (5.19) has two minima at
The bound (5.56) translates into
At the de Sitter point the model (5.19) satisfies the condition . Substituting this relation into Eq. (5.58), we find
We use the approximation that and are of the orders of the present cosmological density and the baryonic/dark matter density in our galaxy, respectively. From Eq. (5.59) we obtain the bound 
If we consider the model (4.81), it was shown in  that the bound (5.56) gives the constraint . This means that the deviation from the CDM model is very small. Meanwhile, for the models (4.83) and (4.84), the deviation from the CDM model can be large even for , while satisfying local gravity constraints. We note that the model (4.89) is also consistent with local gravity constraints.
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