The Jeans Instability

Linear perturbative analyis on the self-gravitating fluid equations under the assumption of infinite, constant (v_0), static background density (\rho_0) results in the following 2nd order PDE in the density perturbation (\rho_1):

\frac{\partial^2\rho_1}{\partial t^2} - C_s^2 \triangledown^2 \rho_1 - 4\pi G \rho_0 \rho_1 = 0

where

\rho_1 \equiv density ~perturbation C_s \equiv sound ~speed

This is a wave equation, so we propose a solution of the form:

\rho_1(\vec{x},t) = C e^{i(\vec{k} \cdot \vec{x} - \omega t)}

Plugging this into the wave equation above gives the following dispersion relation:

\omega ^2 = C_s^2 k^2 - 4 \pi G \rho_0