Cosmographic nature of the early universe from extra dimensional perspective

In this study, making use of the $f\left(R\right)$ theory of gravity in a D-dimensional Kaluza-Klein type framework, we discuss whether we need to assume additional dimensions in order to explain two stages (inflation and deceleration phases) of the universe. To reach this goal, we start with a higher dimensional form of the Friedmann equation written in the $f\left(R\right)$ gravity. For the inflation era occurring after a beginning vacuum phase of the universe, we conclude that a 10-dimensional spacetime model is needed as predicted in the string theory. However, after using a fixed field condition, it is observed that additional dimensions are compactified on the standard four spacetime dimensions in the deceleration phase. Moreover, the additional dimensions may be responsible for the dynamics of the dark energy dominated era, that's why we can state that the source of dark energy may be the extra dimensions.