Approximate $w_\phi\sim\Omega_\phi$ 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 $\phi$ at present times is not sensitive to its initial conditions. When the energy density of dark energy is neglectable ($\Omega_\phi\ll1$), 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 $\Omega_\phi$. We get stable fixed points of $w_\phi$ and $\Omega_\phi$ (noted as $\hat w_\phi$ and $\hat\Omega_\phi$) from the "full tracker equation", i.e., $w_\phi$ and $\Omega_\phi$ will always approach $\hat w_\phi$ and $\hat\Omega_\phi$ respectively. Since $\hat w_\phi$ and $\hat\Omega_\phi$ are analytic functions of $\phi$, analytic relation of $\hat w_\phi\sim\hat\Omega_\phi$ can be obtained, which is a good approximation for the $w_\phi\sim\Omega_\phi$ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities $\hat w_\phi<w_\phi$ and $\hat\Omega_\phi<\Omega_\phi$ are statisfied if the $w_\phi$ (or $\hat w_\phi$) is decreasing with time. In this way, the potential $U(\phi)$ can be constrained directly from observations, by no need of solving the equations of motion numerically.