the energy density at around $t=0$ is caused by the large Hubble friction due to the spatial curvature term. Hence, the field at $t\approx 0$ rolls slowly even if the potential is steep. As the spatial curvature term decays, eventually the potential energy starts to dominate or the energy density starts to decay. The potential dominance starts at $t\approx H_*^{-1}$ while the decay of the energy density starts when $1+w_{\phi}$ becomes $O(1)$, i.e. at $t\approx \epsilon_{*}^{-1/2}H_*^{-1}$. Thus, the scenario changes depending on whether $\epsilon_{*}$ is large or small compared with unity. It is shown in Fig. \[fig:evol\] schematically how the evolution of the energy density depends on $\epsilon_{*}$, assuming that $\epsilon$ stays constant until the field falls down to a plateau of the potential.

![A schematic picture of the evolution of the spatial curvature term (the dashed-dotted line in red), the kinetic term of the field (the dotted curves in green), and the potential term of the field (the solid curves in blue). []{data-label="fig:evol"}](evol.eps){width="80mm"}

First, we consider the case with $\epsilon_{*}\ll 1$. In this case potential energy starts to dominate at $t\approx H_*^{-1}$ after a short curvature dominated stage, and there is no rapid-roll regime. Once the potential term dominates, 