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

For the barotropic equation of state $P=w\rho$, the continuity equation separates asSince $a(t_0)=1$,which is the constant-equation-of-state density scaling.

When $Q=0$, the expanding branch satisfieswhere $H_0=\dot{a}(t_0)>0$. Because $w>-1$, integration givesand hence the flat constant-equation-of-state scale factorIt vanishes atas $|w|<1$ ensures $1+w>0$.