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/dφ)]2<<1 and (1/V)(d2V/dφ2)<<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, Ωφ, and the initial value of (1/V)(dV/dφ). 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 Ωφ. For a variety of potentials, our limiting expression for w(a) is typically accurate to within δw < or approx. 0.005 for w<-0.9. For redshift z < or approx. 1, w(a) is well fit by the Chevallier-Polarski-Linder parametrization, in which w(a) is a linear function of a.