### 11.2 Constant density star

Let us consider a constant density star with . We also assume that the density outside the star is constant, . We caution that the conserved density in the Einstein frame is given by  [343]. However, since the condition holds in most cases of our interest, we do not distinguish between and in the following discussion.

Inside the spherically symmetric body (), Eq. (11.12) gives

Neglecting the field contributions in Eqs. (11.8) – (11.11), the gravitational background for is characterized by the Schwarzschild interior solution. Then the pressure inside the body relative to the density can be analytically expressed as
where is the gravitational potential at the surface of the body:
Here is the mass of the spherically symmetric body. The density is much smaller than , so that the metric outside the body can be approximated by the Schwarzschild exterior solution

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

Substituting these relations into Eq. (11.13), the field equation inside the body is approximately given by
If is close to at , 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 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

where satisfies the equation . The solution of with the boundary conditions at is given by , where is a constant. Plugging this into Eq. (11.20), we get the following solution for  [594]:

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

where obeys the equation . Hence we obtain the solution
where and are constants.

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

whose solution satisfying the boundary condition is
where is a constant.

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 [594]:

where is the thin-shell parameter, and
where
As long as , the parameter is much smaller than 1.

In order to derive the above field profile we have used the fact that the radius is determined by the condition , and hence

where is defined by
which is much smaller than 1. Using Eq. (11.32) we obtain the thin-shell parameter

In terms of a linear expansion of , the field profile (11.28) outside the body is

where the effective coupling is
To leading-order this gives , which agrees with the result (5.45) in the Minkowski background. As long as and , the effective coupling can be much smaller than the bare coupling , even in a strong gravitational background.

From Eq. (11.26) the field value and its derivative around the center of the body with radius are given by

In the Minkowski background (), Eq. (11.38) gives for (or for ). In the strong gravitational background () the second term in the square bracket of Eq. (11.38) can lead to negative for (or positive for ), which leads to the evolution of toward 0. This effects comes from the presence of the strong relativistic pressure around the center of the body. Unless the boundary conditions at are appropriately chosen the field tends to evolve toward , as seen in numerical simulations in [349350] 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  [594].

For the inverse power-law potential , the existence of thin-shell solutions was numerically confirmed in  [594] 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.