 Solution

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

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

a (t_{0}) = 1

a (t) = (t / t_{0})^{2/3}

. A photon emitted inward today at the current comoving particle horizon starts at

χ_{0} = 3 c t_{0}

t

it has traversed the comoving distance

\int_{t_{0}}^{t} \frac{c d t ^{'}}{a ( t ^{'} )} = 3 c t_{0} [(\frac{t}{t _{0}})^{1/3} - 1] .

At arrival at Earth this equals

χ_{0}

, hence

(t / t_{0})^{1/3} = 2

and

t_{arrival} = 8 t_{0}, t_{travel} = 7 t_{0} .

The emitting point is at the horizon location today, not a photon already arriving today from the initial singularity. The calculation of photon arrival from an Einstein-de Sitter particle horizon follows the future inward null geodesic from that event.

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