Can the dark energy equation-of-state parameter w be less than $-1?$

Models of dark energy are conveniently characterized by the equation-of-state parameter $w=p/\rho ,$ where $\rho$ is the energy density and p is the pressure. Imposing the dominant energy condition, which guarantees stability of the theory, implies that $w>~-1.$ Nevertheless, it is conceivable that a well-defined model could (perhaps temporarily) have $w<\mathrm{}-1\hspace{0.5em},$ and indeed such models have been proposed. We study the stability of dynamical models exhibiting $w<\mathrm{}-1$ by virtue of a negative kinetic term. Although naively unstable, we explore the possibility that these models might be phenomenologically viable if thought of as effective field theories valid only up to a certain momentum cutoff. Under our most optimistic assumptions, we argue that the instability time scale can be greater than the age of the universe, but only if the cutoff is at or below ${10}^{-3}\mathrm{eV}.$ We conclude that it is difficult, although not necessarily impossible, to construct viable models of dark energy with $w<\mathrm{}-1;$ observers should keep an open mind, but the burden is on theorists to demonstrate that any proposed new models are not ruled out by rapid vacuum decay.