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For instantaneous complete hydrogen reionization, take $n_{e0}={\Omega }_b{\rho }_{\mathrm{crit},0}/(m_p{c}^2)=3{\Omega }_b{H}_0^2/(8\pi G{m}_p)$ and $n_e(z)=n_{e0}(1+z{)}^3$. Helium or incomplete ionization would multiply this by the appropriate free-electron count per baryon. In the matter-plus-curvature model,The reionization optical depth in an open matter universe is thereforeSet $u=\sqrt{1+{\Omega }_{m}z}$ and ${\beta }^2=1-{\Omega }_m$. The integral is $2{\Omega }_m^{-2}[u^3/3-{\beta }^{2}u{]}_1^{u_{\mathrm{re}}}$, givingAt high redshift where matter dominates over curvature, this approaches $\kappa \simeq {\Omega }_b{C}_H(1+z_{\mathrm{re}}{)}^{3/2}/\sqrt{{\Omega }_m}$. The exact expression should be retained when ${\Omega }_m{z}_{\mathrm{re}}$ is not large.