Quintom models with an equation of state crossing−1

Abstract

In this paper, we investigate a kind of special quintom model, which is made of a quintessence field ${\ensuremath{\phi}}_{1}$ and a phantom field ${\ensuremath{\phi}}_{2}$, and the potential function has the form of $V({\ensuremath{\phi}}_{1}^{2}\ensuremath{-}{\ensuremath{\phi}}_{2}^{2})$. This kind of quintom field can be separated into two kinds: the hessence model, which has the state of ${\ensuremath{\phi}}_{1}^{2}>{\ensuremath{\phi}}_{2}^{2}$, and the hantom model with the state ${\ensuremath{\phi}}_{1}^{2}<{\ensuremath{\phi}}_{2}^{2}$. We discuss the evolution of these models in the $\ensuremath{\omega}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\omega}}^{\ensuremath{'}}$ plane ($\ensuremath{\omega}$ is the state equation of the dark energy, and ${\ensuremath{\omega}}^{\ensuremath{'}}$ is its time derivative in units of Hubble time), and find that according to $\ensuremath{\omega}>\ensuremath{-}1$ or $<\ensuremath{-}1$, and the potential of the quintom being climbed up or rolled down, the $\ensuremath{\omega}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\omega}}^{\ensuremath{'}}$ plane can be divided into four parts. The late time attractor solution, if existing, is always quintessencelike or $\ensuremath{\Lambda}$-like for hessence field, so the big rip does not exist. But for hantom field, its late time attractor solution can be phantomlike or $\ensuremath{\Lambda}$-like, and sometimes, the big rip is unavoidable. Then we consider two special cases: one is the hessence field with an exponential potential, and the other is with a power law potential. We investigate their evolution in the $\ensuremath{\omega}\mathrm{\text{\ensuremath{-}}}{\ensuremath{\omega}}^{\ensuremath{'}}$ plane. We also develop a theoretical method of constructing the hessence potential function directly from the effective equation-of-state function $\ensuremath{\omega}(z)$. We apply our method to five kinds of parametrizations of equation-of-state parameter, where $\ensuremath{\omega}$ crossing $\ensuremath{-}1$ can exist, and find they all can be realized. At last, we discuss the evolution of the perturbations of the quintom field, and find the perturbations of the quintom ${\ensuremath{\delta}}_{Q}$ and the metric $\ensuremath{\Phi}$ are all finite even at the state of $\ensuremath{\omega}=\ensuremath{-}1$ and ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\ne}0$.