Early Tracking Behavior in Small-field Quintessence Models

We study several quintessence models which are singular at Q=0, and use a simple initial constraint $Q_i\ge H_{inflation}/2\pi$ to see when they enter tracking regime, disregarding the details of inflation. We find it can give strong constraints for the inverse power-law potential $V=V_0Q^{-\alpha}$, which has to enter tracking regime for ${\rm ln}z \sim 10$. While for the supergravity model $V=V_0Q^{-\alpha}{\rm exp}(kQ^2/2)$, the constraint is much weakened. For another kind inverse power-law potential $V=V_0{\rm exp}(\lambda/Q)$, it exhibits no constraints.