2-field model of dark energy with canonical and non-canonical kinetic terms
For some parametrizations of the dark energy equation of state that varies in time there is transition from quintessence to phantom or vice versa at a certain redshift. Quintom – the 2-field model with 2 canonical kinetic terms (one with the “+” sign for quintessence and one with the “-” sign for phantom) and a potential U(φ, ξ) in Lagrangian – is one of the most popular scalar field models allowing for such behaviour. We generalize quintom to include the tachyonic kinetic term along with the classical one. For such a model we obtain the expressions for energy density and pressure. For the spatially flat, homogeneous and isotropic Universe with Friedmann-Robertson-Walker metric of 4-space we derive the equations of motion for the fields. We discuss in detail the reconstruction of the scalar fields potential U(φ, ξ). Such a reconstruction cannot be done unambiguously, so we consider 3 simplest forms of U(φ, ξ): the product of Φ(φ) and Ξ(ξ), the sum of Φ(φ) and Ξ(ξ) and the sum of Φ(φ) and Ξ(ξ) to the κth power. The second additional assumption that should be made is about the dependence of either kinetic term Xφ or Xξ on the scale factor a. For each case we obtain the reconstructed potentials in the parametric form. If it is possible to invert dependences of the fields φ and ξ on the scale factor a and obtain the analytical expressions for a(φ) and a(ξ) then we can find the potentials U(φ, ξ) in explicit form. From the obtained explicit expressions it is clear that they are not suitable for practical use for the multicomponent cosmological models with realistic parametrizations of the dark energy equation of state crossing −1. On the other hand, the parametric dependences which define the potential U(φ, ξ) are suitable for multicomponent cosmological models and all parametrizations of the dark energy equation of state.
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