 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012/s2/3/4/solution

Convert both inputs to energy density. The matter contribution, including rest energy, is

u_{M, 0} = ρ_{M, 0} c^{2} = 2.25 \times 1 0^{- 10} J m^{- 3}

. The given photon value is

u_{γ, 0} = 5 \times 1 0^{5} (1.602 \times 1 0^{- 19}) = 8.01 \times 1 0^{- 14} J m^{- 3}

. Hence

\frac{u _{M, 0}}{u _{γ, 0}} ≃ 2.80 \times 1 0^{3} .

(1 + z)^{3}

(1 + z)^{4}

1 + z_{eq} = \frac{u _{M, 0}}{u _{γ, 0}}, z_{eq} ≃ 2.8 \times 1 0^{3} .

This uses the supplied CMB photon density. Additional relativistic species change equality for total radiation.

 New to topics? Read the docs here!