Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 1 / iii

Put $K=2q_0-1>0$ and parametrize the scale factor byOn the expanding branch $0<\theta <\pi$, the equation from part (ii) givesConsequently,Choose the origin of cosmic time at the Big Bang, $y=0$, $\theta =0$. Integration proves the closed matter-dominated Friedmann solution:The same parametrization continues through maximum expansion at $\theta =\pi$ to collapse at $\theta =2\pi$: $dt/d\theta$ remains nonnegative while $dy/d\theta$ changes sign. The maximum scale factor is $y_{\max }=2q_0/K$, reached at $H_0{t}_{\max }=\pi q_0/K^{3/2}$; the total lifetime is twice this time. The squared equation alone does not prescribe the sign of $\dot{y}$, so this continuation supplies the collapsing branch explicitly.

For $q_0=1/2$, the expanding Friedmann equation reduces to $\dot{y}=H_0{y}^{-1/2}$. Separating variables and placing the Big Bang at $t=0$ givesThus the flat pressureless Einstein-de Sitter universe hasIn particular its present age is $2/(3H_0)$.

Put $B=1-2q_0>0$ and introduce $\psi >0$ throughThe expanding equation becomesDividing these expressions yieldsIntegrating from the Big Bang at $\psi =0$ proves the open matter-dominated Friedmann solution:The scale factor grows indefinitely. At early times $y\propto t^{2/3}$, as in the matter-dominated flat case, whereas at late times $\dot{y}\to H_0\sqrt{B}$ and $y\propto t$ as curvature becomes dominant.