The Quintom Models With State Equation Crossing − 1

In this paper, we study a kind of special quintom models, which are made of a quintessence field φ 1 and a phantom field φ 2 , and their potential functions have the form of V (φ 2 1 − φ 2 2). These models have simple kinetic functions, so the analysis of them is simple. These models are separated into two kind: the hessence models, which have φ 2 1 > φ 2 2 , and the hantom, which have φ 2 1 < φ 2 2. We discuss the evolution of these models in the plane defined by ω (the state equation parameter of the dark energy) and ω ′ (the derivative of ω with respect to the logarithm of the scale factor), and find that the ω-ω ′ plane can be divided into four parts according to two conditions: one is that the field being quintessence-like or phantom-like; the other is that the potential being climbed up or rolled down. For the late time attractor solutions, if existing, which are always quintessence-like or Λ-like for hessence fields, so the Big Rip doesn't exist in this kind of models. But for hantom fields, their late time attractor solutions can be phantom-like or Λ-like, and the Big Rip is unavoidable for some hantom models. As two example hessence models with the exponential potential and power law potential, we study their evolution in the ω-ω ′ plane. At last, we show the way to construct the potential function from the parametrized state equations ω(z). For five kind of parametrized methods, where ω crossing −1 can exist, we build their potential functions, and find they all can be realized in hessence models. Especially, we discuss a kind of oscillating ω(z), and find its potential is also an oscillating function.