Quintom models with an equation of state crossing -1

In this paper, we investigate a kind of special quintom model, which is made of a quintessence field $ϕ_1$ and a phantom field $ϕ_2$, and the potential function has the form of $V(ϕ_1^2-ϕ_2^2)$. This kind of quintom fields can be separated into two kinds: the hessence model, which has the state of $ϕ_1^2>ϕ_2^2$, and the hantom model with the state $ϕ_1^2<ϕ_2^2$. We discuss the evolution of these models in the $ω$-$ω'$plane ($ω$ is the state equation of the dark energy, and $ω'$ is its time derivative in unites of Hubble time), and find that according to $ω>-1$ or $<-1$, and the potential of the quintom being climbed up or rolled down, the $ω$-$ω'$ plane can be divided into four parts. The late time attractor solution, if existing, is always quintessence-like or $Λ$-like for hessence field, so the Big Rip doesn't exist. But for hantom field, its late time attractor solution can be phantom-like or $Λ$-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 $ω$-$ω'$ plane. We also develop a theoretical method of constructing the hessence potential function directly from the effective equation of state function $ω(z)$. We apply our method to five kinds of parametrizations of equation of state parameter, where $ω$ crossing -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 $δ_Q$ and the metric $Φ$ are all finite even if at the state of $ω=-1$ and $ω'\neq0$.

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