= Constant-equation-of-state angular diameter distance maximum
{title2=$1+z_m=\alpha^{1/(\alpha-1)},\quad\alpha=3(1+w)/2$}
For a flat universe with a single constant <equation-of-state parameter> $w>-1$, $H(z)=H_0(1+z)^\alpha$. Its <angular diameter distance> is $H_0^{-1}[(1+z)^{-\alpha}-(1+z)^{-1}]/(1-\alpha)$, with logarithmic limit $H_0^{-1}\log(1+z)/(1+z)$ at $\alpha=1$. Differentiation gives the displayed unique maximum, interpreted as $z_m=e-1$ at $\alpha=1$. The distance vanishes at both endpoints $z=0$ and $z\to\infty$. As $w\downarrow-1$, the maximum moves to infinite <cosmological redshift>.
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