Dark energy from a quintessence (phantom) field rolling near a potential minimum (maximum)

We examine dark-energy models in which a quintessence or a phantom field, phi, rolls near the vicinity of a local minimum or maximum, respectively, of its potential V(phi). Under the approximation that (1/V)(dV/d phi)< 1, [although (1/V)(d(2)V/d phi(2)) can be large], we derive a general expression for the equation-of-state parameter w as a function of the scale factor for these models. The dynamics of the field depends on the value of (1/V)(d(2)V/d phi(2)) near the extremum, which describes the potential curvature. For quintessence models, when (1/V)(d(2)V/d phi(2))< 3/4 at the potential minimum, the equation-of-state parameter w(a) evolves monotonically, while for (1/V)(d(2)V/d phi(2))> 3/4, w(a) has oscillatory behavior. For phantom fields, the dividing line between these two types of behavior is at (1/V)(d(2)V/d phi(2))=-3/4. Our analytical expressions agree within 1% with the exact (numerically derived) behavior, for all of the particular cases examined, for both quintessence and phantom fields. We present observational constraints on these models.