A new generic evolution for $k$-essence dark energy with $w\approx -1$

We reexamine $k$-essence dark energy models with a scalar field $\varphi$ and a factorized Lagrangian, $\mathcal{L}=V(\varphi )F(X)$, with $X = \frac{1}{2} \nabla_\mu \phi \nabla^\mu \phi.$Avalueoftheequationofstateparameter,$w$,near$-1$requireseither$X \approx 0$or$dF/dX \approx 0$. Previous work showed that thawing models with $X\approx 0$ evolve along a set of unique trajectories for $w(a)$, while those with $dF\mathrm{/}dX\approx 0$ can result in a variety of different forms for $w(a)$. We show that if $dV\mathrm{/}d\varphi$ is small and $(1\mathrm{/}V)(dV\mathrm{/}d\varphi )$ is roughly constant, then the latter models also converge toward a single unique set of behaviors for $w(a)$, different from those with $X\approx 0$. We derive the functional form for $w(a)$ in this case, determine the conditions on $V(\varphi )$ for which it applies, and present observational constraints on this new class of models. We note that $k$-essence models with $dF\mathrm{/}dX\approx 0$ correspond to a dark energy sound speed $c_s^2\approx 0$.