Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 14D / i / Solution

Expand the density contrast in Fourier modes, $\delta (\mathbf{x},t)=\int {\delta }_{\mathbf{k}}(t)e^{i\mathbf{k}\cdot \mathbf{x}}{d}^{3}k$. In the static universe each mode satisfiesFor $k>k_J$ there are sound wavesFor $k<k_J$, Jeans instability gives growing and decaying exponentials with rate $c_s\sqrt{k_J^2-k^2}$. At the Jeans wavenumber the two solutions are constant and linear in time.