 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-70/2/d/solution

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

A radial null ray obeys

c d τ = d r / 1 + r^{2} / r_{curv}^{2}

. Hence

c (τ_{0} - τ_{*}) = r_{curv} arsinh \frac{r _{*}}{r _{curv}}, r_{*} = r_{curv} sinh \frac{c ( τ _{0} - τ _{*} )}{r _{curv}} .

For the open matter solution, let

β = 1 - Ω_{m}

a^{'} = H_{0} Ω_{m} a + β^{2} a^{2}

, and the age in conformal time is

τ_{0} = \frac{1}{H _{0}} \int_{0}^{1} \frac{d a}{a ( Ω _{m} + β ^{2} a )} = \frac{2}{H _{0} β} arsinh \frac{β}{Ω _{m}} = \frac{2}{H _{0} β} artanh β .

Thus

τ_{0} = \frac{1}{H _{0} β} ln \frac{1 + β}{1 - β}, r_{*} ≃ \frac{c}{H _{0} β} sinh (2 artanh β) = \frac{2 c}{H _{0} Ω _{m}} .

Here the stated approximation

τ_{*} ≪ τ_{0}

was used only in the last distance estimate. The flat limit is smooth:

τ_{0} \to 2/ H_{0}

and

r_{*} \to 2 c / H_{0}

. Retaining

τ_{*}

in the preceding null-ray formula gives the finite-emission-time correction.

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