Angular diameter distance in an open dust universe ($D_A=S_k(D_C)/(1+z)$)

Normalize the present scale factor to one and assume dust plus curvature with no cosmological constant. Then $H^2=H_0^2[{\Omega }_M{a}^{-3}+(1-{\Omega }_M)a^{-2}]$ and $D_C=(c/H_0){\int }_{1/(1+z)}^{1}da/[\sqrt{a}\sqrt{{\Omega }_M+(1-{\Omega }_M)a}]$. For positive ${\Omega }_K=1-{\Omega }_M$, the transverse comoving distance is $S_k(D_C)=c\sinh (H_0\sqrt{{\Omega }_K}{D}_C/c)/(H_0\sqrt{{\Omega }_K})$. At fixed radial distance it exceeds the flat transverse distance. At fixed redshift the radial distance must also be calculated from the expansion history. The empty limit is $D_C=(c/H_0)\ln (1+z)$, while the flat dust limit is $2c[1-(1+z{)}^{-1/2}]/H_0$.