Approximate $w_phisimOmega_phi$ 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 ($Omega_phill1$), 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 $hatOmega_phi$) from the full tracker equation, i.e., $w_phi$ and $Omega_phi$ will always approach $hat w_phi$ and $hatOmega_phi$ respectively. Since $hat w_phi$ and $hatOmega_phi$ are analytic functions of $phi$, analytic relation of $hat w_phisimhatOmega_phi$ can be obtained, which is a good approximation for the $w_phisimOmega_phi$ 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 $hatOmega_phi<Omega_phi$ 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.