Slow-roll k-essence

We derive slow-roll conditions for thawing k-essence with a separable Lagrangian $p(X,\ensuremath{\phi})=F(X)V(\ensuremath{\phi})$. 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 $\ensuremath{-}1$. We find two distinct cases, corresponding to $X\ensuremath{\approx}0$ and ${F}_{X}\ensuremath{\approx}0$, respectively. For the case where $X\ensuremath{\approx}0$ the evolution of $\ensuremath{\phi}$ 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 nonseparable Lagrangians where $X\ensuremath{\approx}0$. For the case ${F}_{X}\ensuremath{\approx}0$, an expression is derived for $w(a)$, but this expression depends on the potential $V(\ensuremath{\phi})$, so there is no model-independent limiting behavior. For the $X\ensuremath{\approx}0$ 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 $\ensuremath{-}1$, but provide very poor constraints on $K$.

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