In [26] it was shown that
needs to be close to 0 during the radiation domination as well as the
matter domination. Hence the viable f (R) models are close to the
CDM model in the region
.
The Ricci scalar remains positive from the radiation era up to the present epoch, as long as it does not
oscillate around
. The model
(
,
) is not viable because the
condition
is violated.
As we will see in Section 5, the local gravity constraints provide tight bounds on the deviation
parameter in the region of high density (
), e.g.,
for
[134
, 596
].
In order to realize a large deviation from the
CDM model such as
today (
)
we require that the variable
changes rapidly from the past to the present. The f (R) model
given in Eq. (4.81
), for example, does not allow such a rapid variation, because
evolves as
in the region
. Instead, if the deviation parameter has the dependence
In the model (A) the following relation holds at the de Sitter point:
where Similarly the model (B) satisfies [568]
Another model that leads to an even faster evolution of is given by [587
]
The models (A), (B) and (C) are close to the CDM model for
, but the deviation from it
appears when
decreases to the order of
. This leaves a number of observational signatures such as
the phantom-like equation of state of dark energy and the modified evolution of matter density
perturbations. In the following we discuss the dark energy equation of state in f (R) models. In Section 8
we study the evolution of density perturbations and resulting observational consequences in
detail.
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