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
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 [575
]:
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.
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 [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]
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 by
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
]
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 [134
]
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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