Emergence of space and the general dynamic equation of the Friedmann-Robertson-Walker universe

A novel idea---that the cosmic acceleration can be understood from the perspective that spacetime dynamics is an emergent phenomenon---was proposed by T. Padmanabhan. The Friedmann equations of a Friedmann-Robertson-Walker universe can be derived from different gravity theories with different modifications of Padmanabhan's proposal. In this paper, a unified formula is proposed, in which those modifications can be treated as special cases when the formula is applied to different universe radii. Furthermore, the dynamic equations of a Friedmann-Robertson-Walker universe in the $f(R)$ theory and deformed Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz theory are investigated. The results show the validity of our proposed formula.