Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 9B (Cosmology)

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

Put $K=8\pi G/(3c^2)$. Differentiating $Q=K\rho a^2-{\dot{a}}^2$ givesThe continuity equation and the Friedmann acceleration equation implyandThese terms cancel, soThis is the First integral of the Friedmann acceleration equation.

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 $1+w>0$.