Emergence of spaces and the dynamic equations of FRW universes in the $f(R)$ theory and deformed Ho\v{r}ava-Lifshitz theory

TL;DR

This paper derives dynamic equations for FRW universes within $f(R)$ and deformed Hořava-Lifshitz theories by applying the emergent space paradigm, linking horizon entropy and degrees of freedom to cosmic evolution.

Contribution

It extends the emergent space framework to $f(R)$ and deformed HL theories, establishing a general relation between horizon entropy and degrees of freedom.

Findings

Derived dynamic equations for $f(R)$ and deformed HL theories.

Established a general relation between horizon entropy and surface degrees of freedom.

Connected emergent space ideas with modified gravity theories.

Abstract

It has been shown that Friedmann equation of FRW universe can be derived from the idea which says cosmic space is emergent as cosmic time progresses and our universe is expanding towards the state with the holographic equipartition by Padmanabhan. In this note, we give a general relationship between the horizon entropy and the number of the degrees of freedom on the surface, we also obtain the corresponding dynamic equations by using the idea of emergence of space in the $f(R)$ theory and deformed Ho\v{r}ava-Lifshitz (HL) theory.