Past exam of the general relativity and cosmology course of the University of Oxford / 2012 / 3 / b / Solution

For the expanding empty model, $\dot{a}=H_0$ in the preceding normalization. If $t_B$ denotes the origin of the expanding slicing,The constant-time spatial slices are negatively curved hyperbolic spaces, with curvature radius $R_{\mathrm{curv}}=a/\sqrt{-k}=c(t-t_B)$ and three-dimensional scalar curvature $6k/a^2$. In the usual simply connected description they are copies of hyperbolic three-space. This is the Milne model: its four-dimensional spacetime is actually flat. Setting $T=(t-t_B)\cosh \chi$ and $R=c(t-t_B)\sinh \chi$ converts the metric to the interior of a future light cone in Minkowski spacetime. Its zero-scale-factor boundary is a slicing boundary, not a curvature singularity.

Add separately conserved pressureless matter, with present mass-equivalent density ${\rho }_{M0}$. Define the cosmological density parametersThenCurvature and dust have equal magnitudes at $a_K={\Omega }_M/{\Omega }_K$, givingFor the almost-empty open case $0<{\Omega }_M<1/2$, curvature domination begins at positive redshift. Equality is today for ${\Omega }_M=1/2$ and in the future for $1/2<{\Omega }_M<1$. With ${\Omega }_M=0$, curvature dominates at all times and there is no finite matter–curvature equality; with ${\Omega }_K=0$, curvature never dominates.