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

For pressureless matter, the Friedmann acceleration equation with vanishing cosmological constant isEvaluating at the present cosmic time and using the definition of the deceleration parameter givesThe denominator is the critical density, so the cosmological density parameter satisfies the dust relationThis uses pressureless matter and $\Lambda =0$; spatial flatness has not been assumed.

Write $y=a/a_0$. Conservation of pressureless matter gives ${\rho }_m={\rho }_{m,0}{y}^{-3}$. The Friedmann equation, retaining spatial curvature, isSince $\dot{y}=yH$ and ${\Omega }_{m,0}=2q_0$, multiplication by $y^2$ givesIn the curvature convention $H^2=8\pi G{\rho }_m/3-k{c}^2/a^2$, this also fixes $-k{c}^2/(a_0^2{H}_0^2)=1-2q_0$.

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.