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

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.