 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2/9b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

d_{H} (t) = a (t) \int_{0}^{t} \frac{c d t ^{'}}{a ( t ^{'} )} .

a (t) \propto t^{2/3}

, so

d_{H} (t_{0}) = t_{0}^{2/3} c \int_{0}^{t_{0}} t^{' - 2/3} d t^{'} = 3 c t_{0} .

Since

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

d_{H} (t_{0}) = \frac{2 c}{H _{0}} .

The Horizon problem is that widely separated regions of the observed universe have nearly identical conditions even though, in a decelerating standard Big Bang history, their past light cones do not overlap early enough to establish causal thermal equilibrium.

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