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 [594
]:
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
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.
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