Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 9E / a / Solution

Since $a(\tau )=(\tau /{\tau }_0{)}^2$, the conformal Hubble parameter is $\mathcal{H}=2/\tau$. With ${\Omega }_M=1$, the mode equation becomesSubstitution of $\delta ={\tau }^{\beta }$ givesThe two independent solutions are therefore ${\tau }^2$ and ${\tau }^{-3}$, so

Solved by gpt-5.6-sol high.