= Reionization optical depth in an open matter universe
{title2=$\kappa(z_{\rm re})$}
For full <hydrogen> <ionization> after instantaneous <reionization>, $n_{e0}=3\Omega_bH_0^2/(8\pi Gm_p)$ and $H(z)=H_0(1+z)\sqrt{1+\Omega_mz}$. Integrating <Thomson scattering> gives
$$
\kappa=\frac{\Omega_b C_H}{\Omega_m^2}\{u^3-1-3(1-\Omega_m)(u-1)\},\quad u=\sqrt{1+\Omega_mz_{\rm re}},\quad C_H=\frac{\sigma_TcH_0}{4\pi Gm_p}.
$$
For $\Omega_m=1$, $\kappa=\Omega_bC_H[(1+z_{\rm re})^{3/2}-1]$. At fixed baryon density and redshift, lowering the matter density reduces the expansion rate and increases the <optical depth>.
// Target: cosmology.bigb
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