Role of the Brans-Dicke scalar in the holographic description of dark energy

We study cosmological application of the holographic energy density in the Brans-Dicke theory. Considering the holographic energy density as a dynamical cosmological constant, it is more natural to study it in the Brans-Dicke theory than in general relativity. Solving the Friedmann and Brans-Dicke field equations numerically, we clarify the role of Brans-Dicke field during evolution of the universe. When the Hubble horizon is taken as the IR cutoff, the equation of state ($w_{\Lmd}$) for the holographic energy density is determined to be 5/3 when the Brans-Dicke parameter $\omg$ goes infinity. This means that the Brans-Dicke field plays a crucial role in determining the equation of state. For the particle horizon IR cutoff, the Brans-Dicke scalar mediates a transition from $w_{\Lmd} = -1/3$ (past) to $w_{\Lmd} = 1/3$ (future). If a dust matter is present, it determines future equation of state. In the case of future event horizon cutoff, the role of the Brans-Dicke scalar and dust matter are turned out to be trivial, whereas the holographic energy density plays an important role as a dark energy candidate with $w_{\Lmd} =-1$.