In order to discuss cosmological dynamics it is convenient to introduce the dimensionless variables:
by which Eq. (9.12 Differentiating Eq. (9.11) with respect to
, it follows that
The fixed points of Eqs. (9.16) and (9.17
) can be found by setting
and
.
Even when
is not constant, except for the cases
and
, we obtain the
following fixed points [253
]:
The stability of the fixed points can be analyzed by considering linear perturbations about them. As long as
and
are bounded, the eigenvalues
and
of the Jacobian matrix of linear
perturbations are given by
In the CDM model (
) one has
and
.
Then the points
,
, and
correspond to
,
(radiation
domination, unstable),
,
(matter domination, saddle), and
,
(de Sitter epoch, stable), respectively. Hence the sequence of radiation, matter, and
de Sitter epochs is in fact realized.
Let us next consider the model with
and
. In this case the
quantity
is
In Figure 8 we plot the evolution of
as well as
and
for the model
with
. This shows that the sequence of (
) radiation domination (
), (
)
matter domination (
), and de Sitter acceleration (
) is realized. Recall that in
metric f (R) gravity the model
(
,
) is not viable because
is negative. In Palatini f (R) gravity the sign of
does not matter because there
is no propagating degree of freedom with a mass
associated with the second derivative
[554].
In [21, 253] the dark energy model
was constrained by the combined analysis of
independent observational data. From the joint analysis of Super-Nova Legacy Survey [39], BAO [227] and
the CMB shift parameter [561], the constraints on two parameters
and
are
and
at the 95% confidence level (in the unit of
) [253]. Since the allowed values of
are close to 0, the above model is not particularly favored over the
CDM model. See
also [116, 148, 522, 46, 47] for observational constraints on f (R) dark energy models based on the
Palatini formalism.
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