Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 15D / i / Solution

Multiply the Friedmann equation by $a^2$ and differentiate:Insert the cosmological continuity equation to obtainOn a nonstatic branch divide by $2a\dot{a}$; continuity extends the result across isolated turning points. The resulting Friedmann acceleration equation isAn everywhere static branch needs the independent spatial Einstein field equations: the two differentiated constraints alone cannot determine the pressure when $\dot{a}\equiv 0$. This is the qualification to deriving the acceleration equation solely by division.