Minimal Length, Friedmann Equations and Maximum Density

TL;DR

This paper extends thermodynamic derivations of Friedmann equations to include generalized entropy laws inspired by quantum gravity, revealing a maximum energy density that prevents singularities and leads to nonsingular cosmological evolution.

Contribution

It generalizes the derivation of Friedmann equations to arbitrary entropy-area laws and demonstrates the existence of a maximum energy density, ensuring nonsingular cosmological models.

Findings

Existence of a maximum energy density near Planck scale.

Nonsingular evolution independent of spatial curvature.

Finite-time reachability of maximum energy density.

Abstract

Inspired by Jacobson's thermodynamic approach[gr-qc/9504004], Cai et al [hep-th/0501055,hep-th/0609128] have shown the emergence of Friedmann equations from the first law of thermodynamics. We extend Akbar--Cai derivation [hep-th/0609128] of Friedmann equations to accommodate a general entropy-area law. Studying the resulted Friedmann equations using a specific entropy-area law, which is motivated by the generalized uncertainty principle (GUP), reveals the existence of a maximum energy density closed to Planck density. Allowing for a general continuous pressure $p(\rho,a)$ leads to bounded curvature invariants and a general nonsingular evolution. In this case, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point which leaves the big bang singularity inaccessible from a spacetime prospective. The existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature $k$. As an example we study the evolution of the equation of state $p=\omega \rho$ through its phase-space diagram to show the existence of a maximum energy which is reachable in a finite time.