New agegraphic dark energy in Brans-Dicke theory with logarithmic form of scalar field

In a very recent paper, the current authors (arXiv:gr-qc/1609.01477) have proposed and analyzed in detail the logarithmic form of Brans-Dicke scalar field $\phi$ as $\phi \propto ln(\alpha+\beta a)$, where $\alpha$ and $\beta$ are positive constants, to alleviate the problems of interacting holographic dark energy models in Brans-Dicke theory. In this paper, the cosmological evolution of a new agegraphic dark energy (NADE) model within the framework of Friedmann-Robertson-Walker Universe is analyzed with the same form of scalar field in Brans-Dicke theory. We derive the equation of state parameter $w_D$ and deceleration parameter $q$ of NADE model. It is observed that $w_D\rightarrow -1$ when $a\rightarrow \infty$, i.e., the NADE mimics cosmological constant in the late time evolution. Indeed, due to the assumption of logarithmic form of Brans-Dicke scalar field the NADE in Brans-Dicke theory behaves like NADE in general relativity in the late time evolution. The NADE model shows a phase transition from matter dominated phase in early time to accelerated phase in late time. We further extend NADE model by including the interaction between dark matter and NADE. In this case, $w_D$ definitely crosses the phantom divide line ($w_D=-1$) in the late time evolution. The phase transition from matter dominated to NADE dominated phase may be achieved at early stage in interacting model. Further, we show that the interacting NADE model resolves the cosmic coincidence problem as the energy density ratio may evolve sufficiently slow at present.