Early Tracking Behavior in Small-field Quintessence Models

Abstract

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