 Solution

For $k^{(p)}\ll k_J^{(p)}$, neglect pressure support. The growing and decaying Jeans instability modes obeyIn an Einstein-de Sitter universe, $a\propto t^{2/3}$ gives $H=2/(3t)$, and the matter-density relation $8\pi G\bar{\rho }=3H^2$ gives $4\pi G\bar{\rho }=2/(3t^2)$. ThusTry a power law ${\delta }_{\mathbf{k}}\propto t^p$. Its characteristic equation isTherefore the Einstein-de Sitter density-growth modes areThis is a linear result while $|\delta |\ll 1$; a growing mode eventually leaves that regime.