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: $[(1/V)(dV/d\varphi ){]}^2\ll 1$ and $(1/V)(d^{2}V/d{\varphi }^2)\ll 1$, 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 $1+w$, ${\Omega }_{\varphi }$, and the initial value of $(1/V)(dV/d\varphi )$. 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 ${\Omega }_{\varphi }$. For a variety of potentials, our limiting expression for $w(a)$ is typically accurate to within $\delta w\lesssim 0.005$ for $w<-0.9$. For redshift $z\lesssim 1$, $w(a)$ is well fit by the Chevallier-Polarski-Linder parametrization, in which $w(a)$ is a linear function of $a$.

• Received 20 December 2007

DOI:https://doi.org/10.1103/PhysRevD.77.083515