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 Incoming links: Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3 / c

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3 / d / Solution 2026-10-25

Adiabatic photon redshifting gives

T \propto a^{- 1}

. Using the stipulated temperatures,

1 + z_{rec} ≃ 1000, z_{rec} ≃ 999.

In the flat dust approximation of part c,

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

, so

t_{rec} ≃ (2/3 H_{0}) 100 0^{- 3/2} ≃ 2.75 \times 1 0^{5}

years. This approximation omits the significant radiation contribution near the actual recombination epoch.

The elementary travel-distance estimate is

c t_{rec} ≃ 8.4 \times 1 0^{4}

pc, whose corresponding comoving scale today is about

84

Mpc. In an expanding universe, however, that is not the exact proper particle-horizon radius. Integrating the null geodesic gives

d_{hor} (t) = a (t) c \int_{0}^{t} \frac{d t ^{'}}{a ( t ^{'} )} = 3 c t

for dust. Thus the proper horizon at this epoch is about

0.25

Mpc and its present comoving size about

250

Mpc. The factor three accounts for expansion during light propagation. These scales describe the limited causal domain in the simple decelerating model. The near-uniform CMB temperature across much more widely separated last-scattering regions motivates the horizon problem; the question's rough

c t

estimate and the exact dust horizon should not be conflated.

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