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

Normalize $a_0=1$ and use the open dust model, with no cosmological constant and negligible present radiation. The Friedmann equation at the present epoch reads $H_0^2=H_0^2{\Omega }_m-k{c}^2$. Since $k=-r_{\mathrm{curv}}^{-2}$, the curvature radius of an open FLRW universe isIn the metric's $c=1$ convention the numerator is one. With $a_0=1$ this is also the present physical curvature radius; at scale factor $a$ the physical radius is $a{r}_{\mathrm{curv}}$. Inferring curvature from ${\Omega }_m$ alone would require this matter-only assumption; an additional dark-energy density would also enter the present Friedmann equation.