The effective equation of state of the system is defined by
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 aref (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[440, 243, 250, 26]
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.
From Eq. (4.56) the viable f (R) dark energy models need to satisfy the condition , which is consistent with the above argument.
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