Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 1 / c / Solution

Dropping radiation in cosmology in the Friedmann equation givesTo evaluate this matter-vacuum conformal age integral, put $a=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}y$. The square root in the denominator becomes ${\Omega }_m^{2/3}{\Omega }_{\Lambda }^{-1/6}\sqrt{y(1+y^3)}$, whereas $da={\Omega }_m^{1/3}{\Omega }_{\Lambda }^{-1/3}dy$. ThereforeThe radiation-free approximation is not valid arbitrarily close to the lower limit. Its extension to zero nevertheless gives a good leading conformal time estimate: the early-time correction has size $O(H_0^{-1}\sqrt{{\Omega }_r}/{\Omega }_m)$, small compared with the total age when matter-radiation equality occurs sufficiently early. The apparent $y^{-1/2}$ endpoint singularity is integrable.