The <age of an FLRW universe> follows from $dt=da/(aH)$. Neglecting radiation and spatial curvature gives
$$
t_0=\frac1{H_0}\int_0^1
\frac{da}{a\sqrt{\Omega_{m,0}a^{-3}+\Omega_{\Lambda,0}}}
=\frac1{H_0}\int_0^1
\frac{a^{1/2}\,da}{\sqrt{\Omega_{m,0}+\Omega_{\Lambda,0}a^3}}.
$$
For an <Einstein-de Sitter universe>, $\Omega_{m,0}=1$ and $\Omega_{\Lambda,0}=0$, so
$$
\boxed{t_0=H_0^{-1}\int_0^1a^{1/2}\,da=\frac{2}{3}H_0^{-1}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-310/1/c/solution