Jeans instability with Yukawa gravity ($k^2<4\pi G{\rho }_0/v_s^2-{\alpha }^2$)

For the screened attraction $-Gm{e}^{-\alpha r}/r$ with $\alpha \ge 0$, $({\nabla }^2-{\alpha }^2)\varphi =4\pi G\rho$. A static homogeneous barotropic medium with sound speed $v_s>0$ has ${\omega }^2=k^2[v_s^2-4\pi G{\rho }_0/(k^2+{\alpha }^2)]$. A growing mode exists iff $4\pi G{\rho }_0>v_s^2{\alpha }^2$, with $0<k<\sqrt{4\pi G{\rho }_0/v_s^2-{\alpha }^2}$. For nonzero screening the growth rate tends to zero as $k\to 0$; the homogeneous mode is not an exponentially growing density fluctuation. Finite interaction range can therefore remove the instability entirely, while all sufficiently small nonzero wavenumbers remain unstable when the threshold is exceeded.