We first consider Robertson-Walker cosmologies that are also vacuum solutions, that is, for which 1) \begin{equation} \rho = 0 = p \end{equation} Friedmann's equation reduces to \begin{equation} \dot{a}^2 = \frac{\Lambda a^2}{3} - k \label{frwvac} \end{equation}

If $k=0$, then either $\Lambda=0$, in which case $a$ is constant and we have Minkowski space, or $\Lambda>0$, in which case \begin{equation} a = e^{\pm qt} \end{equation} where \begin{equation} q = \sqrt{\frac{|\Lambda|}{3}} \end{equation} We choose the exponent to be positive, so that the universe is expanding, to match current observations; the resulting spacetime is called de Sitter space.

Similarly, if $\Lambda=0$, then ($\ref{frwvac}$) implies either that $k=0$, which we have already considered, or that $k=-1$ and \begin{equation} a = \pm t \end{equation} However, the resulting spacetime turns out to be Minkowski space in disguise, as can be seen by making the coordinate transformation \begin{align} R &= t \sinh\psi = tr \\ T &= t \cosh\psi = t\sqrt{1+r^2} \end{align} after which the line element becomes \begin{equation} ds^2 = -dT^2 + dR^2 + R^2 \left( d\theta^2 + \sin^2\theta\,d\phi^2 \right) \end{equation}

The remaining cases can be handled similarly; the result is shown in Table 2 below. Remarkably, even though there are in principle nine possible combinations of $k$ and $\Lambda$, there are only three vacuum Friedmann solutions; different entries in the table for the same geometry correspond to different ways of slicing up that geometry.

Table 2: Classification of vacuum Friedmann models.