Dark Energy from a Phantom Field Near a Local Potential Minimum

We examine dark energy models in which a phantom field $\varphi$ is rolling near a local minimum of its potential $V(\varphi )$.We require that $(1/V)(dV/d\phi) \ll 1$,but$(1/V)(d^2 V/d\phi^2)$ can be large. Using techniques developed in the context of hilltop quintessence, we derive a general expression for $w$ as a function of the scale factor, and as in the hilltop case, we find that the dynamics of the field depend on the value of $(1\mathrm{/}V)(d^{2}V\mathrm{/}d{\varphi }^2)$ near the mimimum. Our general result gives a value for $w$ that is within 1% of the true (numerically-derived) value for all of the particular cases examined. Our expression for $w(a)$ reduces to the previously-derived phantom slow-roll result of Sen and Scherrer in the limit where the potential is flat, $(1\mathrm{/}V)(dV\mathrm{/}d\varphi )\ll 1$.