Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 9B / a / Solution

For a comoving region, $E=\rho V$ and $V=a^3{V}_0$. The work law givesso differentiation with respect to time yieldsSince $\dot{V}/V=3\dot{a}/a$, division by $V$ gives the cosmological perfect-fluid continuity equation

Solved by gpt-5.6-sol high.