Constant-equation-of-state angular diameter distance maximum ($1+z_m={\alpha }^{1/(\alpha -1)},\alpha =3(1+w)/2$)

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.