4.1 Dynamical equations
We introduce the following variables
together with the density parameters
It is straightforward to derive the following equations
where
is the number of e-foldings, and
From Eq. (4.68) the Ricci scalar
can be expressed by
. Since
depends on
, this means
that
is a function of
, that is,
. The
CDM model,
, corresponds to
. Hence the quantity
characterizes the deviation of the background dynamics
from the
CDM model. A number of authors studied cosmological dynamics for specific
f (R) models [160
, 382
, 488, 252
, 31
, 198, 280, 72, 41, 159, 235, 1, 279, 483, 321, 432].
The effective equation of state of the system is defined by
which is equivalent to
. In the absence of radiation (
) the fixed points for the
above dynamical system are
The points
and
are on the line
in the
plane.
The matter-dominated epoch (
and
) can be realized only by the point
for
close to 0. In the (
) plane this point exists around
. Either the point
or
can be responsible for the late-time cosmic acceleration. The former is a de Sitter point (
) with
, in which case the condition (2.11) is satisfied. The point
can give rise to the
accelerated expansion (
) provided that
, or
, or
.
In order to analyze the stability of the above fixed points it is sufficient to consider only time-dependent
linear perturbations
(
) around them (see [170, 171
] for the detail of such
analysis). For the point
the eigenvalues for the
Jacobian matrix of perturbations are
where
and
with
. In the limit that
the latter two
eigenvalues reduce to
. For the models with
, the solutions cannot remain for a
long time around the point
because of the divergent behavior of the eigenvalues as
.
The model
(
) falls into this category. On the other hand, if
, the latter two eigenvalues in Eq. (4.77) are complex with negative real parts.
Then, provided that
, the point
corresponds to a saddle point with a damped
oscillation. Hence the solutions can stay around this point for some time and finally leave for
the late-time acceleration. Then the condition for the existence of the saddle matter era is
The first condition implies that viable f (R) models need to be close to the
CDM model during the
matter domination. This is also required for consistency with local gravity constraints, as we will see in
Section 5.
The eigenvalues for the Jacobian matrix of perturbations about the point
are
where
. This shows that the condition for the stability of the de Sitter point
is [440, 243, 250
, 26
]
The trajectories that start from the saddle matter point
satisfying the condition (4.78) and then
approach the stable de Sitter point
satisfying the condition (4.80) are, in general, cosmologically
viable.
One can also show that
is stable and accelerated for (a)
,
, (b)
,
, (c)
,
, (d)
,
. Since
both
and
are on the line
, only the trajectories from
to
are allowed (see Figure 2). This means that only the case (a) is viable as a stable and
accelerated fixed point
. In this case the effective equation of state satisfies the condition
.
From the above discussion the following two classes of models are cosmologically viable.
- Class A: Models that connect
(
,
) to
(
)
- Class B: Models that connect
(
,
) to
(
)
From Eq. (4.56) the viable f (R) dark energy models need to satisfy the condition
, which is consistent
with the above argument.