 Solution

In the formal static-background limit, $H=0$ and $\bar{\rho }$ is constant. The gravity-dominated linear cosmological density perturbation equation reduces toThe Static-universe Jeans modes areThe unstable perturbation grows exponentially rather than as $t^{2/3}$. In the expanding case, the $2H\dot{\delta }$ term resists peculiar motion and the falling background density also lengthens the gravitational growth timescale. A globally homogeneous static self-gravitating fluid is not obtained merely by turning off expansion; this calculation is the conventional local Jeans-background idealization, with the uniform background force removed or otherwise supported.