Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 38A / d / Solution

Linearizing the constraint gives $\delta \rho =-2{\rho }_0\delta a/a_0$. Since $\delta p=\delta \rho /3$, the acceleration equation gives $\delta \ddot{a}=2\Lambda \delta a$. Hence perturbations contain $e^{\sqrt{2\Lambda }t}$: the Einstein static universe is unstable.

Solved by gpt-5.6-sol high.