Thawing quintessence with a nearly flat potential

The thawing quintessence model with a nearly flat potential provides a natural mechanism to produce an equation of state parameter, w, close to -1 today. We examine the behavior of such models for the case in which the potential satisfies the slow-roll conditions: (1/V) (dV/d) ^21 and (1/V) (d^2V/d^2) 1, and we derive the analog of the slow-roll approximation for the case in which both matter and a scalar field contribute to the density. We show that in this limit, all such models converge to a unique relation between 1+w, _, and the initial value of (1/V) (dV/d). We derive this relation and use it to determine the corresponding expression for w (a), which depends only on the presentday values for w and _. For a variety of potentials, our limiting expression for w (a) is typically accurate to within 0. 005 for w<-0. 9. For redshift z1, w (a) is well fit by the Chevallier-Polarski-Linder parametrization, in which w (a) is a linear function of a.