 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-70/2/f/solution

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

For instantaneous complete hydrogen reionization, take

n_{e 0} = Ω_{b} ρ_{crit, 0} / (m_{p} c^{2}) = 3 Ω_{b} H_{0}^{2} / (8 π G m_{p})

and

n_{e} (z) = n_{e 0} (1 + z)^{3}

. Helium or incomplete ionization would multiply this by the appropriate free-electron count per baryon. In the matter-plus-curvature model,

H (z) = H_{0} (1 + z) 1 + Ω_{m} z, ∣ d t ∣ = \frac{d z}{( 1 + z ) H ( z )} .

κ = \frac{3 Ω _{b} σ _{T} c H _{0}}{8 π G m _{p}} \int_{0}^{z_{re}} \frac{1 + z}{1 + Ω _{m} z} d z .

u = 1 + Ω_{m} z

and

β^{2} = 1 - Ω_{m}

. The integral is

2 Ω_{m}^{- 2} [u^{3} /3 - β^{2} u]_{1}^{u_{re}}

κ = \frac{Ω _{b} C _{H}}{Ω _{m}^{2}} [u_{re}^{3} - 1 - 3 (1 - Ω_{m}) (u_{re} - 1)], u_{re} = 1 + Ω_{m} z_{re}, C_{H} = \frac{σ _{T} c H _{0}}{4 π G m _{p}} .

At high redshift where matter dominates over curvature, this approaches

κ ≃ Ω_{b} C_{H} (1 + z_{re})^{3/2} / Ω_{m}

. The exact expression should be retained when

Ω_{m} z_{re}

is not large.

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