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)]²1 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 $1+w$, Ω_φ, 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 Ω_φ. For a variety of potentials, our limiting expression for $w(a)$ is typically accurate to within δw0.005 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.