Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 1 / i / Solution

Write $H=\dot{a}/a$ for the Hubble parameter. Subtracting the Friedmann equation from the second equation gives the Friedmann acceleration equation,Since $\dot{H}=\ddot{a}/a-H^2$, eliminating $H^2$ with the first equation yieldsDifferentiating the first equation, with constant cosmological constant and curvature parameter, now givesThus the cosmological perfect-fluid continuity equation is $\dot{\rho }+3H(\rho +p/c^2)=0$. Multiplying by $a^3$ and using $d(a^3)/dt=3H{a}^3$ provesThis is local conservation of energy: expansion changes the energy in a comoving volume through pressure work.