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

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.