Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 14D (Cosmology)

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.

On long wavelengths the gradient term is negligible. The Friedmann equation impliesDuring matter domination, $a\propto t^{2/3}$ and $H=2/(3t)$, so the linear growth equation becomesThe power-law ansatz $\delta \propto t^s$ gives $s=2/3$ or $s=-1$. Since $a\propto t^{2/3}$, the growing and decaying modes are

Under the stated approximation the perturbation equation isDuring radiation domination, $a\propto t^{1/2}$, so $\dot{\delta }\propto t^{-1}$ andMatter perturbations therefore have only logarithmic growth of matter perturbations during radiation domination.

In a curvature-dominated universe, $a\propto t$, so $\dot{\delta }\propto t^{-2}$ andThe nondecaying mode is constant: curvature domination gives freezing of matter perturbations during curvature domination, rather than continued growth.