Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 15D (Cosmology)

In the Newtonian dust calculation, use a comoving spherical shell of physical radius $R(t)=a(t)\chi$. Homogeneity gives its enclosed mass $M=4\pi \rho R^3/3$, and the shell theorem removes the gravitational force of exterior shells. Assume pressureless matter, conserved enclosed mass, no cosmological constant, and no shell crossing. Its specific conserved mechanical energy isHomogeneity permits the energy per ${\chi }^2$ to be a constant, written $-k{c}^2/2$. Dividing by $a^2$ givesThis is a Newtonian derivation for dust. For radiation the enclosed gravitating mass is not conserved, so this same mechanical-energy argument cannot simply be reused; general relativity supplies the same Friedmann constraint with the total energy density divided by $c^2$, together with pressure-dependent fluid conservation.

The fluid conservation equation is $\dot{\rho }+3H(\rho +P/c^2)=0$. With the radiation equation of state, it becomes $\dot{\rho }+4H\rho =0$, giving $\rho a^4={\rho }_0$. Define the critical density ${\rho }_{\mathrm{crit}}=3H^2/(8\pi G)$ and cosmological density parameter $\Omega =\rho /{\rho }_{\mathrm{crit}}$. Since $a_0=1$,Evaluating the Friedmann equation today gives $H_0^2+k{c}^2=H_0^2{\Omega }_0$, hence

For ${\Omega }_0>1$, introduce conformal time by $dt/d\tau =a$. Since $a^{\prime }=a\dot{a}$, the constraint becomesOn the initial expanding branch, separate variables and choose $a=t=\tau =0$ at the Big Bang. With $\alpha =H_0\sqrt{{\Omega }_0-1}$,The second formula follows by integrating $dt/d\tau =a$. The model reaches maximum size at $\alpha \tau =\pi /2$ and recollapses at $\alpha \tau =\pi$.

Today $a=1$ and $H_0>0$ selects the expanding branch, soSubstitution yields the requested ageIn particular the numerator is $\sqrt{{\Omega }_0}-1$, not $\sqrt{{\Omega }_0-1}$; the PDF displays the former. The flat-model limit is $1/(2H_0)$.