Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 14D / a / i / Solution

Writing $r=a(q+\psi )$ to first order gives $x^{\prime \prime }+\mathcal{H}{x}^{\prime }\simeq {\psi }^{\prime \prime }+\mathcal{H}{\psi }^{\prime }=-{\nabla }_q\phi$. The perturbation of Poisson's equation is ${\nabla }_q^2\phi =(4\pi G/c^2)a^2{\bar{\rho }}_M{\delta }_M$.

Solved by gpt-5.6-sol high.