Inside the spherically symmetric body (), Eq. (11.12) gives
In the following we shall derive the analytic field profile by using the linear expansion in terms of the gravitational potential . This approximation is expected to be reliable for . From Eqs. (11.14) – (11.16) it follows that
In the region the field derivative of the effective potential around can be approximated by . The solution to Eq. (11.19) can be obtained by writing the field as , where is the solution in the Minkowski background and is the perturbation induced by . At linear order in and we obtain:
In the region the field evolves towards larger values with increasing . Since the matter coupling term dominates over in this regime, it follows that . Hence the field perturbation satisfies
In the region outside the body () the field climbs up the potential hill after it acquires sufficient kinetic energy in the regime . Provided that the field kinetic energy dominates over its potential energy, the r.h.s. of Eq. (11.13) can be neglected relative to its l.h.s. of it. Moreover the terms that include and in the square bracket on the l.h.s. of Eq. (11.13) is much smaller than the term . Using Eq. (11.17), it follows that
The coefficients are known by matching the solutions (11.21), (11.23), (11.25) and their derivatives at and . If the body has a thin-shell, then the condition is satisfied. Under the linear expansion in terms of the three parameters , , and we obtain the following field profile :
In order to derive the above field profile we have used the fact that the radius is determined by the condition , and hence
In terms of a linear expansion of , the field profile (11.28) outside the body is
From Eq. (11.26) the field value and its derivative around the center of the body with radius are given by[349, 350] for the f (R) model (4.84). However there exists a thin-shell field profile even for (and ) around the center of the body. In fact, the derivative can change its sign in the regime for thin-shell solutions, so that the field does not reach the curvature singularity at .
For the inverse power-law potential , the existence of thin-shell solutions was numerically confirmed in  for . Note that the analytic field profile (11.26) was used to set boundary conditions around the center of the body. In Figure 10 we show the normalized field versus for the model with , , , and . While we have neglected the term relative to to estimate the solution in the region analytically, we find that this leads to some overestimation for the field value outside the body (). In order to obtain a numerical field profile similar to the analytic one in the region , we need to choose a field value slightly larger than the analytic value around the center of the body. The numerical simulation in Figure 10 corresponds to the choice of such a boundary condition, which explicitly shows the presence of thin-shell solutions even for a strong gravitational background.
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