 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-67/2/ii/solution

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

H (z) = H_{0} (1 + z)^{3/2}

r_{h} (z) = \frac{c}{H _{0}} \int_{z}^{\infty} (1 + z^{'})^{- 3/2} d z^{'} = \frac{c}{H _{0}} [- 2 (1 + z^{'})^{- 1/2}]_{z}^{\infty} .

Thus

r_{h} (z) = \frac{2 c}{H _{0} 1 + z} .

It tends to zero towards the initial singularity and increases without bound as

z \to - 1

in the infinite future. At the present time it is

2 c / H_{0} = 3 c t_{0}

t_{0} = 2/ (3 H_{0})

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