Holographic Dark Energy in Brans-Dicke Theory with Logarithmic Form of Scalar Field

In this paper, an interacting holographic dark energy model with Hubble horizon as an infra-red cut-off is considered in the framework of Brans-Dicke theory. We assume the Brans-Dicke scalar field as a logarithmic form ϕ = ϕ0ln(α + βa), where a is the scale factor, α and β are arbitrary constants, to interpret the physical phenomena of the Universe. The equation of state parameter wh and deceleration parameter q are obtained to discuss the dynamics of the evolution of the Universe. We present a unified model of holographic dark energy which explains the early time acceleration (inflation), medieval time deceleration and late time acceleration. It is also observed that wh may cross the phantom divide line in the late time evolution. We also discuss the cosmic coincidence problem. We obtain a time-varying density ratio of holographic dark energy to dark matter which is a constant of order one (r∼O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r\sim \mathcal {O}(1)$\end{document}) during early and late time evolution, and may evolve sufficiently slow at present time. Thus, the model successfully resolves the cosmic coincidence problem.