Weak-drag dust density waves ($s_{\pm }=\pm i\omega -ϵ\Omega ({\omega }^2+{\Omega }^2)/(2{\omega }^2)+O({ϵ}^2)$)

For fixed ${\omega }^2={\Omega }^2-2\pi G{\sigma }_0|k|+c^2{k}^2>0$, the two oscillatory roots of the dust gravitational dispersion relation with gas drag acquire negative real parts at first order in $ϵ$. Their frequencies have no first-order shift. The expansion is not uniform near ${\omega }^2=0$, where the oscillation and damping scales can become comparable.