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

Abstract

We reexamine $k$-essence dark energy models with a scalar field $ϕ$ and a factorized Lagrangian, $\mathcal L = V(ϕ)F(X)$, with $X = \frac{1}{2} \nabla_μϕ\nabla^μϕ.$ A value of the equation of state parameter, $w$, near $-1$ requires either $X \approx 0$ 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 $dF/dX \approx 0$ can result in a variety of different forms for $w(a)$. We show that if $dV/dϕ$ is small and $(1/V)(dV/dϕ)$ 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 $V(ϕ)$ for which it applies, and present observational constraints on this new class of models. We note that $k$-essence models with $dF/dX \approx 0$ correspond to a dark energy sound speed $c_s^2 \approx 0$.