Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 1 / b

For $0<{\Omega }_M<1$, set $x=a^{3/2}$ and choose the expanding branch of the Friedmann equation. ThenSeparating variables and choosing cosmic time $t=0$ at the Big Bang givesThus the flat dust-and-vacuum expansion isSetting $a(t_0)=1$ gives the age of a flat matter-Lambda universe,The two matter-vacuum age endpoint limits follow below.

As ${\Omega }_M\downarrow 0$, use $\mathrm{arsinh}z=\log (z+\sqrt{1+z^2})\sim \log (2z)$ to obtainAt fixed Hubble constant, the age of a flat matter-Lambda universe diverges logarithmically. The limiting purely vacuum-dominated de Sitter spacetime has an exponential scale factor and no Big Bang at a finite value of cosmic time in this flat slicing.

As ${\Omega }_M\uparrow 1$, use $\mathrm{arsinh}z\sim z$ near zero. ConsequentlyThe flat dust-and-vacuum expansion tends to $a(t)=(3H_{0}t/2{)}^{2/3}$, the Einstein-de Sitter universe.