Positive-root criterion for dust self-gravity with drag ($P(0)=ϵ\Omega ({\omega }^2-{\Omega }^2)<0$)

For any nonzero positive drag, the cubic dust gravitational dispersion relation with gas drag is negative at $s=0$ if ${\omega }^2<{\Omega }^2$ and positive at sufficiently large positive $s$. The intermediate value theorem therefore guarantees a growing real root. This existence argument is independent of the weak-drag expansion and includes the dynamically unstable ${\omega }^2<0$ regime.