= Gradient instability of a negative-pressure perfect fluid
{title2=$\omega^2=c_s^2 k^2$}
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.
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