Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 70 / 2 / g

For ${\Omega }_m=1$ the preceding result reduces to $\kappa ={\Omega }_b{C}_H[(1+z_{\mathrm{re}}{)}^{3/2}-1]$. Using $C_H=0.032$ and ${\Omega }_b=0.05$ givesAt fixed $H_0$ and ${\Omega }_b$, a lower matter density lowers $H(z)$ relative to the flat dust value at positive redshift. The same electron density then persists for a longer scattering time, increasing the optical depth. Hence the reionization redshift giving unit depth is lower in an open universe.

In the high-redshift matter regime the scaling is $1+z_{\mathrm{re}}\propto {\Omega }_m^{1/3}$ at fixed optical depth; for ${\Omega }_m=0.3$ this suggests a redshift near fifty rather than seventy. This comparison holds the baryon density fixed, not the baryon fraction ${\Omega }_b/{\Omega }_m$, and is not an estimate of the actual observed reionization epoch.