In order to discuss cosmological dynamics it is convenient to introduce the dimensionless variables:

by which Eq. (9.12) can be written as Differentiating and with respect to , we obtain [253] where The following constraint equation also holds Hence the Ricci scalar can be expressed in terms of and .Differentiating Eq. (9.11) with respect to , it follows that

from which we get the effective equation of state: The cosmological dynamics is known by solving Eqs. (9.16) and (9.17) with Eq. (9.18). If is not constant, then one can use Eq. (9.19) to express and in terms of and .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]:

- 1.
- : ,
- 2.
- : ,
- 3.
- : .

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

- 1.
- : ,
- 2.
- : ,
- 3.
- : .

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

The constraint equation (9.19) gives The late-time de Sitter point corresponds to , which exists for . Since in this case, the de Sitter point is stable with the eigenvalues . During the radiation and matter domination we have (i.e., ) and hence . corresponds to the radiation point () with the eigenvalues , whereas to the matter point () with the eigenvalues . Provided that , and correspond to unstable and saddle points respectively, in which case the sequence of radiation, matter, and de Sitter eras can be realized. For the models , it was shown in [253] that unified models of inflation and dark energy with radiation and matter eras are difficult to be realized.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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