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. (7.80 ), we obtain the perturbed action for the field $X$ after the partial integration: where $∘ −-g(0) = a3$ and $M s$ is defined in Eq. (7.82 ). 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 ].