Gradient instability of a negative-pressure perfect fluid (${\omega }^2=c_s^2{k}^2$)

For a barotropic perfect fluid with constant $w$, its cosmological adiabatic sound speed satisfies $c_s^2=w$. If $w<0$, the short-wavelength pressure term has the wrong sign for restoring oscillations: a mode grows as $\exp (\sqrt{-w}k\tau )$ when expansion can be neglected. For $w=-1/3$ in a coasting background with constant conformal expansion rate $K$, the exponents are $-K\pm \sqrt{K^2+k^2/3}$. This is a gradient instability, not ordinary positive-pressure Jeans instability. A network with tension or elastic anisotropic stress needs different constitutive perturbation equations; a negative background equation of state alone does not establish its perturbation sound speed.