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

## TL;DR

This paper explores a new class of $k$-essence dark energy models with specific potential conditions, deriving their behavior and observational constraints, expanding understanding of models with $w$ close to -1.

## Contribution

It introduces a novel evolution class for $k$-essence models with nearly constant $V()$, deriving $w(a)$ and observational constraints for these models.

## Key findings

- Models with small $dV/d$ converge to unique $w(a)$ behaviors.
- Derived the functional form of $w(a)$ for the new class of models.
- Provided observational constraints on the new $k$-essence models.

## Abstract

We reexamine $k$-essence dark energy models with a scalar field $\phi$ and a factorized Lagrangian, $\mathcal L = V(\phi)F(X)$, with $X = \frac{1}{2} \nabla_\mu \phi \nabla^\mu \phi.$ 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 $dV/d\phi$ is small and $(1/V)(dV/d\phi)$ 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 $V(\phi)$ 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 $c_s^2 \approx 0$.