D_{A} ≡ \frac{S}{θ}

θ(t_{e}) = \frac{S}{ω̃a(t_{e})}

θ(z) = \frac{S(1 + z)}{ω̃(z)}

\frac{a(t_{e})}{a(now)} = \frac{1}{1 + z}

ω̃(z) = \frac{2c}{H_{0}} (\frac{1}{1 + z} ) (\frac{1}{Ω_{M}^{2}} ) [Ω_{M}z − (2 − Ω_{M}) (\sqrt{1 + Ω_{M}z} − 1)]

θ(z) = \frac{S(1 + z)}{\frac{2c}{H_{0}} (\frac{1}{1+z} ) (\frac{1}{Ω_{M}^{2}} ) [Ω_{M}z − (2 − Ω_{M}) (\sqrt{1 + Ω_{M}z} − 1)]}

= \frac{(SH_{0}/2c) (1 + z)^{2} Ω_{M}^{2}}{[Ω_{M}z − (2 − Ω_{M}) (\sqrt{1 + Ω_{M}z} − 1)]}