Holographic interacting dark energy in the braneworld cosmology

We investigate a model of brane cosmology to find a unified description of the radiation-matter-dark energy universe. It is of the interacting holographic dark energy with a bulk-holographic matter $\chi$. This is a five-dimensional cold dark matter, which plays a role of radiation on the brane. Using the effective equations of state ${\omega }_{\mathrm{\Lambda }}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ instead of the native equations of state ${\omega }_{\mathrm{\Lambda }}$, we show that this model cannot accommodate any transition from the dark energy with $\omega^{\rm eff}{\rm \Lambda}\ge-1$tothephantomregime$\omega^{\rm eff}{\rm \Lambda}<-1$. Furthermore, the case of interaction between cold dark matter and five dimensional cold dark matter is considered for completeness. Here we find that the redshift of matter-radiation equality $z_{\mathrm{e}\mathrm{q}}$ is the same order as $z_{\mathrm{e}\mathrm{q}}^{\mathrm{o}\mathrm{b}}=2.4\times {10}^4{\mathrm{\Omega }}_{\mathrm{m}}{h}^2$. Finally, we obtain a general decay rate $\mathrm{\Gamma }$ which is suitable for describing all interactions including the interaction between holographic dark energy and cold dark matter.