Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 1 / d / Solution

To first order, the radial and transverse comoving separations are

Δ χ_{∥} = \frac{Δ z}{H ( z )} = \frac{f ( z , Ω _{m, 0} )}{H _{0}} Δ z, Δ χ_{⊥} = χ (z) Δ θ = \frac{I ( z ; Ω _{m, 0} )}{H _{0}} Δ θ,

where

f (z, Ω_{m, 0}) = \frac{1}{Ω _{m, 0} ( 1 + z ) ^{3} + 1 - Ω _{m, 0}} .

The local spatial geometry is that of Euclidean space, so the Pythagorean theorem gives

Δ χ = \frac{f}{H _{0}} (Δ z)^{2} + (\frac{I}{f})^{2} (Δ θ)^{2} .

Consequently the requested function is

y (z, Ω_{m, 0}) = \frac{I ( z ; Ω _{m, 0} )}{f ( z , Ω _{m, 0} )} = H (z) χ (z) .

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