In the context of dark energy it is possible to construct viable single-field models based on BD theory. In
what follows we discuss cosmological dynamics of dark energy models based on the action (10.10) in the flat
FLRW background given by (2.12
) (see, e.g., [596
, 22
, 85, 289, 5, 327, 139, 168] for dynamical analysis in
scalar-tensor theories). Our interest is to find conditions under which a sequence of radiation, matter, and
accelerated epochs can be realized. This depends upon the form of the field potential
. We first carry
out general analysis without specifying the forms of the potential. We take into account non-relativistic
matter with energy density
and radiation with energy density
. The Jordan frame is
regarded as a physical frame due to the usual conservation of non-relativistic matter (
).
Varying the action (10.10
) with respect to
and
, we obtain the following equations
We introduce the following dimensionless variables
and also the density parameters These satisfy the relationName | ![]() |
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(a) ![]() |
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0 | ![]() |
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(b1) Kinetic 1 | ![]() |
0 | 0 | ![]() |
(b2) Kinetic 2 | ![]() |
0 | 0 | ![]() |
(c) Field dominated | ![]() |
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0 | ![]() |
(d) Scaling solution | ![]() |
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(e) de Sitter | 0 | 1 | 0 | ![]() |
If is a constant, i.e., for the exponential potential
, one can derive fixed points for
Eqs. (10.18
) – (10.20
) by setting
(
). In Table 1 we list the fixed points of the
system in the absence of radiation (
). Note that the radiation point corresponds to
. The point (a) is the
-matter-dominated epoch (
MDE) during which the
density of non-relativistic matter is a non-zero constant. Provided that
this can be used
for the matter-dominated epoch. The kinetic points (b1) and (b2) are responsible neither for
the matter era nor for the accelerated epoch (for
). The point (c) is the scalar-field
dominated solution, which can be used for the late-time acceleration for
. When
this point yields the cosmic acceleration for
. The scaling
solution (d) can be responsible for the matter era for
, but in this case the condition
for the point (c) leads to
. Then the energy fraction of the pressureless
matter for the point (d) does not satisfy the condition
. The point (e) gives rise to
the de Sitter expansion, which exists for the special case with
[which can be also
regarded as the special case of the point (c)]. From the above discussion the viable cosmological
trajectory for constant
is the sequence from the point (a) to the scalar-field dominated point (c)
under the conditions
and
. The analysis based on the
Einstein frame action (10.6
) also gives rise to the
MDE followed by the scalar-field dominated
solution [23, 22].
Let us consider the case of non-constant . The fixed points derived above may be regarded as “instantaneous”
points7 [195, 454]
varying with the time-scale smaller than
. As in metric f (R) gravity (
) we are
interested in large coupling models with
of the order of unity. In order for the potential
to
satisfy local gravity constraints, the field needs to be heavy in the region
such that
. Then it is possible to realize the matter era by the point (d) with
. Moreover the
solutions can finally approach the de Sitter solution (e) with
or the field-dominated
solution (c). The stability of the point (e) was analyzed in [596
, 250, 242] by considering
linear perturbations
,
and
. One can easily show that the point (e) is stable for
For the f (R) model (5.19) the field
is related to the Ricci scalar
via the relation
. Then the potential
in the Jordan frame can be
expressed as
During the radiation and deep matter eras one has from Eqs. (10.12
) – (10.13
)
by noting that
is negligibly small relative to the background fluid density. From Eq. (10.14
) the field is
nearly frozen at a value satisfying the condition
. Then the field
evolves along the
instantaneous minima given by
Since around
, the instantaneous fixed point (d) can be responsible for the
matter-dominated epoch provided that
. The variable
decreases in time irrespective
of the sign of the coupling
and hence
. The de Sitter point is characterized by
,
i.e.,
The above discussion shows that for the model (10.23) the matter point (d) can be followed by the
stable de Sitter solution (e) for
. In fact numerical simulations in [596
] show that the sequence
of radiation, matter and de Sitter epochs can be in fact realized. Introducing the energy density
and
the pressure
of dark energy as we have done for metric f (R) gravity, the dark energy
equation of state
is given by the same form as Eq. (4.97
). Since for the
model (10.23
)
increases toward the past, the phantom equation of state (
) as well as
the cosmological constant boundary crossing (
) occurs as in the case of metric
f (R) gravity [596
].
As we will see in Section 10.3, for a light scalar field, it is possible to satisfy local gravity constraints
for . In those cases the potential does not need to be steep such that
in
the region
. The cosmological dynamics for such nearly flat potentials have been
discussed by a number of authors in several classes of scalar-tensor theories [489, 451, 416
, 271
].
It is also possible to realize the condition
, while avoiding the appearance of a
ghost [416, 271].
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