Holography in a radiation-dominated universe with a positive cosmological constant

We discuss the holographic principle in a radiation-dominated, closed Friedmann-Robertson-Walker 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.