Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 3 / a / Solution

For each Fourier mode, the cosmological Poisson equation givesBecause nonrelativistic matter has ${\bar{\rho }}_m\propto a^{-3}$, this implies $\Phi \propto {\Delta }_m/a$. Write $\Phi =C{\Delta }_m/a$ and use $\mathcal{H}=a^{\prime }/a$. Direct differentiation in the stated potential equation givesThe supplied Friedmann equation identity therefore yieldsSince $d/d\tau =ad/dt$, this becomes the linear growth equation