= Solution
Let $\mathcal H=a'/a=\dot a$, where a prime denotes differentiation with respect to <conformal time> $d\tau=dt/a$. Multiplying the <Friedmann equation> by $a^2$ gives
$$
\mathcal H^2+kc^2=\frac{8\pi G}{3}\rho a^2.
$$
The <critical density> is the value obtained when $k=0$, so
$$
\Omega=\frac{\rho}{\rho_{\rm crit}}
=\frac{8\pi G\rho a^2}{3\mathcal H^2},
\qquad
\boxed{\ \frac{kc^2}{\mathcal H^2}=\Omega-1\ }.
$$
Since $\mathcal H'=a\ddot a$, the pressure-free <Raychaudhuri equation> gives
$$
2\mathcal H'=-\frac{8\pi G}{3}\rho a^2
=-(\mathcal H^2+kc^2),
$$
or
$$
2\mathcal H'+\mathcal H^2+kc^2=0.
$$
Differentiating $\Omega=1+kc^2/\mathcal H^2$ and substituting this equation yields
$$
\boxed{\ \Omega'=\mathcal H\Omega(\Omega-1)\ }.
$$
For an expanding universe $\mathcal H>0$. The <Einstein-de Sitter universe> $\Omega=1$ is an unstable constant solution: curves starting with $0<\Omega<1$ decrease towards $0$, while curves starting with $\Omega>1$ rise away from $1$ and diverge as the closed model reaches maximum expansion. This is the <flatness problem>.
Back to article page
