 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-55/1/b/3/solution

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

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

. The remaining future conformal time is

η_{\infty} - η_{0} = \int_{t_{0}}^{\infty} \frac{d t}{a ( t )} = \frac{t _{0}}{a _{0}} .

A light ray sent to a comoving reflector at separation

χ

and immediately returned consumes

2 χ

d_{m a x} = a_{0} χ_{m a x} = \frac{t _{0}}{2} = \frac{1}{H _{0}} .

Finite-time reception requires

d_{0} < d_{m a x}

. Indeed the total travel equation is

2 d_{0} / a_{0} = (t_{0} / a_{0}) (1 - t_{0} / t_{r})

t_{r} = t_{0} / (1 - 2 d_{0} / t_{0})

. At the limiting distance the return time tends to infinity. This round-trip radius is half the current physical cosmological event horizon radius.

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