= Solution
{c}
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> are
$$
\boxed{K_{\mathrm{spatial}}=\frac{k}{a^2},\qquad {}^{(3)}R=\frac{6k}{a^2}.}
$$
The cases $k=1,0,-1$ have locally <spherical geometry>, <Euclidean geometry>, and <hyperbolic geometry>, respectively. For $k=\pm1$, $a$ is the <curvature radius of a constant-curvature space> in the displayed normalization. Local <curvature> alone does not fix the global <topology>.
Back to article page
