Approximate wφ - Ωφ Relations in Quintessence Models

Quintessence field is a widely-studied candidate of dark energy. There is ＂tracker solution＂ in quintessence models, in which evolution of the field φ at present times is not sensitive to its initial conditions. When the energy density of dark energy is negleetable （Ωφ 〈〈 1）, evolution of the tracker solution can be well analysed from ＂tracker equation＂. In this paper, we try to study evolution of the quintessence field from ＂full tracker equation＂, which is valid for all spans of Ωφ. We get stable fixed points of we and wφ （noted as wφ and Ωφ） from the ＂full tracker equation＂, i.e., we and ωφ will always approach ωφ and Ωφ respectively. Since wφ and Ωφ are analytic functions of φ, analytic relation of φ can be obtained, which is a good approximation for the we φ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities ωφ 〈 we and 〈ωφ are statisfied if the we （or ωφ） decreases with time. In this way, the potentiai U（φ） can be constrained directly from observations, by no need of solving the equations of motion numerically.     

Constraining a New Type of Dark Energy Model from Present Observations

We propose a new type of dark energy (DE) model, in which the equation of state of DE ω de is a simple function of the fractional energy density de instead of the redshift z. We assume three DE models of this type, and fit them with present observations to get constraints of DE, which are also compared with the CPL model. It is shown that a suitable ω de (Ωde ) model can give smaller χ2 or smaller errors of ω de than that of the CPL model. This new type of DE model can help to study the essential properties and nature of DE.