Approximate $w_ϕ\simΩ_ϕ$ 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 neglectable ($Ω_ϕ\ll1$), 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 $Ω_ϕ$. We get stable fixed points of $w_ϕ$ and $Ω_ϕ$ (noted as $\hat w_ϕ$ and $\hatΩ_ϕ$) from the "full tracker equation", i.e., $w_ϕ$ and $Ω_ϕ$ will always approach $\hat w_ϕ$ and $\hatΩ_ϕ$ respectively. Since $\hat w_ϕ$ and $\hatΩ_ϕ$ are analytic functions of $ϕ$, analytic relation of $\hat w_ϕ\sim\hatΩ_ϕ$ can be obtained, which is a good approximation for the $w_ϕ\simΩ_ϕ$ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities $\hat w_ϕ<w_ϕ$ and $\hatΩ_ϕ<Ω_ϕ$ are statisfied if the $w_ϕ$ (or $\hat w_ϕ$) is decreasing with time. In this way, the potential $U(ϕ)$ can be constrained directly from observations, by no need of solving the equations of motion numerically.

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