List of Footnotes

1 Note that we do not take into account a direct coupling between the Ricci scalar and matter (such as f1(R )ℒM) 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 ρM reached around the ten oscillations of R. Meanwhile we estimate Tr at the epoch where ρM becomes a dominant contribution to the total energy density (as in [364]).
5 The cosmological dynamics for the model f(R) = R − μRc [1 − exp (− 2R∕Rc)] was studied in [396].
6 If we define X = √Qs-ℛ and plugging it into Eq. (7.80View Equation), we obtain the perturbed action for the field X after the partial integration:
∫ [ ] (2) 3 ∘ --(0) 1 ˙2 1 1- 2 1 2 2 δS = dtd x − g 2X − 2 a2(∇X ) − 2M sX ,
where ∘ −-g(0) = a3 and M s is defined in Eq. (7.82View Equation). Then, for the field X, we obtain the Klein–Gordon equation □X = M 2X s in the large-scale limit (k → 0), which defines the mass M s in an invariant way in the FLRW background.
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 f(R,𝒢) models [15, 14, 229].