Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 1 / c / Solution

Normalize the present scale factor to $a(t_0)=1$. In a flat universe containing radiation in cosmology, pressureless matter, and a cosmological constant, the Friedmann equation iswhere ${\Omega }_{r,0}+{\Omega }_{m,0}+{\Omega }_{\Lambda ,0}=1$. Since $dt=da/(aH)$, the age of an FLRW universe isFor a matter-only flat Einstein-de Sitter universe,Taking the present value $H_0\simeq 70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, for which the Hubble time is $H_0^{-1}\simeq 14.0\mathrm{Gyr}$, gives