Past exam of the general relativity II course of the University of Oxford / 2015 / 3 / a / Solution

The scale factor $a(t)>0$ specifies the common time dependence of proper distance in cosmology between comoving observers. Their comoving coordinates remain fixed, while their separation is multiplied by $a(t)$. Thus $\dot{a}>0$ means expansion and $\dot{a}<0$ means contraction. The coordinate $t$ is cosmic time, measured by the proper time of a comoving observer.

The parameter $k$ fixes the sign of the spatial curvature of an FLRW universe. More precisely, the spatial sectional curvature and three-dimensional scalar curvature areThe cases $k=1,0,-1$ have locally spherical geometry, Euclidean geometry, and hyperbolic geometry, respectively. For $k=\pm 1$, $a$ is the curvature radius of a constant-curvature space in the displayed normalization. Local curvature alone does not fix the global topology.