 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-62/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For

0 < Ω_{M} < 1

x = a^{3/2}

and choose the expanding branch of the Friedmann equation. Then

\overset{x}{˙} = \frac{3}{2} H_{0} Ω_{M} + (1 - Ω_{M}) x^{2} .

Separating variables and choosing cosmic time

t = 0

at the Big Bang gives

\frac{arsinh ( x ( 1 - Ω _{M} ) / Ω _{M} )}{1 - Ω _{M}} = \frac{3}{2} H_{0} t .

a (t) = (\frac{Ω _{M}}{1 - Ω _{M}})^{1/3} sinh^{2/3} (\frac{3}{2} H_{0} 1 - Ω_{M} t) .

Setting

a (t_{0}) = 1

t_{0} = \frac{2}{3 H _{0} 1 - Ω _{M}} arsinh \frac{1 - Ω _{M}}{Ω _{M}} .

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