 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2022/s02/3/b/solution

ρ_{m} \propto R^{- 3}

give

\dot{R} \propto R^{- 1/2}

, hence

R \propto t^{2/3}

and

H = 2/ (3 t)

. For

H_{0} = 70 km s^{- 1} Mp c^{- 1} = 2.269 \times 1 0^{- 18} s^{- 1}

t_{0} = \frac{2}{3 H _{0}} ≃ 9.31 \times 1 0^{9} yr .

The particle horizon is a proper distance on a constant-time slice, not simply

c t

r_{p} (t) = c t^{2/3} \int_{0}^{t} (t^{'})^{- 2/3} d t^{'} = 3 c t .

Therefore

r_{p} (t_{0}) = \frac{2 c}{H _{0}} ≃ 8.57 \times 1 0^{3} Mpc ≃ 27.9 \times 1 0^{9} light years .

These are estimates within a spatially flat, matter-only model, rather than the measured age and particle horizon of the actual late-time universe.

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