Slow-roll k-essence
We derive slow-roll conditions for thawing k-essence with a separable Lagrangian $p(X,ϕ)=F(X)V(ϕ)$. We examine the evolution of the equation of state parameter, $w$, as a function of the scale factor $a$, for the case where $w$ is close to -1. We find two distinct cases, corresponding to $X \approx 0$ and $F_X \approx 0$, respectively. For the case where $X\approx0$ the evolution of $ϕ$ and hence $w$ is described by only two parameters, and $w(a)$ is model-independent and coincides with similar behavior seen in thawing quintessence models. This result also extends to non-separable Lagrangians where $X\approx0$. For the case $F_X \approx 0$, an expression is derived for $w(a)$, but this expression depends on the potential $V(ϕ)$, so there is no model-independent limiting behavior. For the $X \approx 0$ case, we derive observational constraints on the two parameters of the model, $w_0$ (the present-day value of $w$), and the $K$, which parametrizes the curvature of the potential. We find that the observations sharply constrain $w_0$ to be close to -1, but provide very poor constraints on $K$.