In the flat <Lambda-CDM model>,
$$
d_A(z)=\frac1{1+z}\frac1{H_0}
\int_0^z\frac{dz'}{\sqrt{\Omega_{m,0}(1+z')^3+1-\Omega_{m,0}}}.
$$
At low redshift, $H(z)=H_0+O(z)$, so the <angular diameter distance> satisfies $d_A(z)=z/H_0+O(z^2)$ and initially increases from zero. At high redshift, <pressureless matter> dominates and the integral converges:
$$
\chi(z)\longrightarrow\chi_\infty<\infty,
\qquad
d_A(z)\sim\frac{\chi_\infty}{1+z}\longrightarrow0.
$$
Thus the positive continuous function $d_A$ rises from zero and returns toward zero, so it attains an interior maximum. This is the <angular diameter distance turnover>; beyond it, a fixed physical size appears larger at higher redshift.

If the <variable dark-energy equation of state> differs slightly from $-1$, matter still controls the large-$z$ expansion, while the same low-$z$ limit holds. The turnover therefore persists under such a small change. More generally, it persists whenever the high-redshift comoving distance grows more slowly than $1+z$.

Solution

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

{title2=$d_A$}
{wiki}

The angular diameter distance is defined by $d_A=x/\theta$ for a small transverse physical size $x$ subtending angle $\theta$. In a spatially flat <Friedmann-Lemaitre-Robertson-Walker metric>,
$$
d_A(z)=\frac{\chi(z)}{1+z},
$$
where $\chi$ is the <comoving radial distance>.

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