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

where is a distance from the center of symmetry that is related to the distance in the Jordan frame via . The effective potential is defined by where is a conserved quantity in the Einstein frame [343]. 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 :

The former corresponds to the region of high density with a heavy mass squared , whereas the latter to a lower density region with a lighter mass squared . In the case of Sun, for example, the field value is determined by the homogeneous dark matter/baryon density in our galaxy, i.e., .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

For this functional form it follows that The r.h.s. of Eq. (5.20) is smaller than 1, so that . The limit corresponds to . In the limit one has and . This property can be seen in the upper panel of Figure 3, which shows the potential for the model (4.84) with parameters and . Because of the existence of the coupling term , the effective potential has a minimum at Since in the region of high density, the condition is in fact justified (for and of the order of unity). The field mass about the minimum of the effective potential is given by This shows that, in the regime , is much larger than the present Hubble parameter (). Cosmologically the field evolves along the instantaneous minima characterized by Eq. (5.22) and then it approaches a de Sitter point which appears as a minimum of the potential in the upper panel of Figure 3.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:

The boundary condition (5.25) can be also understood as . The field is at rest at and starts to roll down the potential when the matter-coupling term in Eq. (5.15) becomes important at a radius . If the field value at is close to , the field stays around in the region . The body has a thin-shell if is close to the radius of the body.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 fact, this satisfies the boundary condition (5.24).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

where and are constants.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 [575]:

where If the mass outside the body is small to satisfy the condition and , we can neglect the contribution of the -dependent terms in Eqs. (5.29) – (5.32). Then the field profile is given by [575] Originally a similar field profile was derived in [344, 343] by assuming that the field is frozen at in the region .The radius is determined by the following condition

This translates into where is the gravitational potential at the surface of the body. Using this relation, the field profile (5.37) outside the body reduces toIf 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

This shows that the effective coupling is of the order of and hence for local gravity constraints are not satisfied.Let us consider the case in which is close to , i.e.

This corresponds to the thin-shell regime in which the field is stuck inside the star except around its surface. If the field is sufficiently massive inside the star to satisfy the condition , Eq. (5.39) gives the following relation where is called the thin-shell parameter [344, 343]. Neglecting second-order terms with respect to and in Eq. (5.40), it follows that where is the effective coupling given bySince 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 [575].

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 [251]

where and are functions of and . In the weak gravitational background ( and ) the metric outside the spherically symmetric body with mass is given 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

Under the condition we obtain the following relations In the following we use the approximation , which is valid for . Using the thin-shell solution (5.44), it follows that where we have used the approximation and hence .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 by

The experimental bound (5.14) can be satisfied as long as the thin-shell parameter is much smaller than 1. If we take the distance , the constraint (5.14) translates into where is the thin-shell parameter for Sun. In f (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 [343]. 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 [343]

where is the thin-shell parameter of Earth, and , , are the gravitational potentials of Sun, Earth and Moon, respectively. Hence the condition (5.52) translates into [134, 596] which corresponds to in 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

where and is the Ricci scalar at the late-time de Sitter point. In the following we consider the case in which the Lagrangian density is given by (5.19) for . If we use the models (4.83) and (4.84), then there are some modifications for the estimation of . However this change should be insignificant when we place constraints on model parameters.At the de Sitter point the model (5.19) satisfies the condition . Substituting this relation into Eq. (5.58), we find

For the stability of the de Sitter point we require that , which translates into the condition . Hence the term in Eq. (5.59) is smaller than 0.25 for .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 [134]

Under this condition one can see an appreciable deviation from the CDM model cosmologically as decreases to the order of .If we consider the model (4.81), it was shown in [134] 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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