Thawing quintessence with a nearly flat potential

Abstract

The thawing quintessence model with a nearly flat potential provides a natural mechanism to produce an equation of state parameter, $w$, close to $\ensuremath{-}1$ today. We examine the behavior of such models for the case in which the potential satisfies the slow-roll conditions: $[(1/V)(dV/d\ensuremath{\phi}){]}^{2}\ensuremath{\ll}1$ and $(1/V)({d}^{2}V/d{\ensuremath{\phi}}^{2})\ensuremath{\ll}1$, 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 $1+w$, ${\ensuremath{\Omega}}_{\ensuremath{\phi}}$, and the initial value of $(1/V)(dV/d\ensuremath{\phi})$. 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 ${\ensuremath{\Omega}}_{\ensuremath{\phi}}$. For a variety of potentials, our limiting expression for $w(a)$ is typically accurate to within $\ensuremath{\delta}w\ensuremath{\lesssim}0.005$ for $w<\ensuremath{-}0.9$. For redshift $z\ensuremath{\lesssim}1$, $w(a)$ is well fit by the Chevallier-Polarski-Linder parametrization, in which $w(a)$ is a linear function of $a$.