Holography in a Radiation-dominated Universe with a Positive Cosmological Constant

TL;DR

This paper explores the holographic principle in a radiation-dominated universe with a positive cosmological constant, demonstrating how the Friedmann equation can be expressed as the Cardy formula and identifying conditions for entropy bounds saturation.

Contribution

It introduces a cosmological D-bound on entropy and shows its saturation corresponds to the Friedmann equation matching the Cardy-Verlinde formula in a brane-world scenario.

Findings

Friedmann equation can be written as the Cardy formula under certain conditions.

The cosmological D-bound is saturated when the brane crosses the black hole horizon.

At saturation, the entropy of radiation matches the Cardy-Verlinde formula.

Abstract

We discuss the holographic principle in a radiation-dominated, closed Friedmann-Robertson-Walker (FRW) universe with a positive cosmological constant. By introducing a cosmological D-bound on the entropy of matter in the universe, we can write the Friedmann equation governing the evolution of the universe in the form of the Cardy formula. When the cosmological D-bound is saturated, the Friedmann equation coincides with the Cardy-Verlinde formula describing the entropy of radiation in the universe. As a concrete model, we consider a brane universe in the background of Schwarzschild-de Sitter black holes. It is found that the cosmological D-bound is saturated when the brane crosses the black hole horizon of the background. At that moment, the Friedmann equation coincides with the Cardy-Verlinde formula describing the entropy of radiation matter on the brane.