= Solution
With a prime denoting conformal-time differentiation, $a'=a\dot a$ and $\mathcal H=a'/a=\dot a$. Multiplying the <Friedmann equation> by $a^2$ gives $\mathcal H^2+kc^2=8\pi G\rho a^2/3$, and hence
$$
\boxed{kc^2/\mathcal H^2=\Omega-1.}
$$
The <Friedmann acceleration equation> and $P=w\rho c^2$ give
$$
\mathcal H'=a\ddot a=-\frac{4\pi G}{3}\rho a^2(1+3w)
=-\frac{1+3w}{2}(\mathcal H^2+kc^2).
$$
Differentiate $\Omega-1=kc^2/\mathcal H^2$ and substitute this expression:
$$
\boxed{\Omega'=(1+3w)\mathcal H\Omega(\Omega-1).}
$$
For $k=0$, $\Omega=1$ identically and the same equation holds directly. The relations assume $\mathcal H\ne0$, as is true on the expanding branch in the question.
For pressureless matter, $w=0$, the <cosmological density parameter flow> points away from $\Omega=1$: initial $\Omega>1$ increases, while $0<\Omega<1$ decreases. In the closed case the expanding branch can end at a turnaround, where $\mathcal H=0$ and this density parameter diverges. For vacuum domination, $w=-1$, the sign reverses: values above one decrease and positive values below one increase toward one. The plotted curves are representative conformal-time solutions with $c=1$ and separately chosen origins; inflationary future infinity occurs at a finite conformal time.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4-density.png]
{title=Conformal-time evolution of the cosmological density parameter during matter domination and inflation}
In a decelerating matter or radiation era, small deviations from flatness grow: near one, $d\log|\Omega-1|/d\log a\simeq1+3w>0$. Observed near-flatness therefore requires extremely small earlier deviations, the <flatness problem>. During accelerated expansion with $w<-1/3$, this exponent is negative and curvature is driven toward negligible relative size. For $w=-1$, $|\Omega-1|$ decays approximately as $a^{-2}$. Sufficiently long <cosmic inflation> supplies the <inflationary solution of the flatness problem> before ordinary decelerating evolution resumes.
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