Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 15A / a / Solution

The homogeneous part of the continuity equation gives $\dot{\bar{n}}+3H\bar{n}=0$. At first order,For the density contrast $\delta =\delta n/\bar{n}$, the background equation cancels the expansion terms, leaving

Solved by gpt-5.6-sol high.