1  Note that we do not take into account a direct coupling between the Ricci scalar and matter (such as ) considered in [439, 80, 81, 82, 248].  
2  This result is a consequence of the action principle, but it can be derived also by a direct calculation, using the Bianchi identities.  
3  There are some other works about theoretical constructions of f (R) models based on quantum gravity, supergravity and extra dimensional theories [341, 345, 537, 406, 163, 287, 288, 518, 519].  
4  In [426] the reheating temperature is estimated by taking the maximum value of reached around the ten oscillations of . Meanwhile we estimate at the epoch where becomes a dominant contribution to the total energy density (as in [364]).  
5  The cosmological dynamics for the model was studied in [396].  
6  If we define and plugging it into Eq. (7.80), we obtain the perturbed action for the field after the
partial integration: 

7  In strict mathematical sense the instantaneous fixed point is not formally defined because it varies with time.  
8  There are several works about the background cosmological dynamics for some models [15, 14, 229]. 
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