Non-vacuum Friedmann solutions can be classified using similar methods to those used in the vacuum case. We restrict the discussion here to some special cases. In particular, we assume throughout this section that $\Lambda=0$, which brings Friedmann's equation to the form \begin{equation} \dot{a}^2 = \frac{8\pi\rho a^2}{3} - k \end{equation}

Solving for $k$ results in \begin{equation} k = \frac{8\pi\rho a^2}{3} - \dot{a}^2 \end{equation} If we define the critical density by \begin{equation} \rho_c = \frac{3}{8\pi} \frac{\dot{a}^2}{a^2}\bigg|_{\hbox{now}} \end{equation} then \begin{equation} k = \frac{8\pi a^2}{3} \left( \rho - \rho_c \right) \end{equation} Thus, in these models, the shape of the universe can be determined by comparing the observed value $\rho$ of the energy density to the critical value $\rho_c$: If $\rho>\rho_c$, then $k=1$; if $\rho=\rho_c$, then $k=0$; and if $\rho<\rho_c$, then $k=-1$.

As discussed later in this chapter, the value of $\rho_c$ is related to the cosmological redshift; the current value is approximately \begin{equation} \rho_c \approx 2\times10^{-29} \frac{\hbox{g}}{\hbox{cm}^3} \end{equation} Meanwhile, the estimated average energy density of the galaxies we can see is approximately \begin{equation} \rho_{\hbox{galaxies}} \approx 1\times10^{-30} \frac{\hbox{g}}{\hbox{cm}^3} \end{equation} These numbers are astonishingly close! It is currently an open question whether there is enough matter in the universe in other forms to lead to a closed universe ($k=1$), a mystery commonly referred to as the problem of “missing matter.”