= Density-parameter divergence at a cosmological turnaround
{title2=$\Omega=\rho/\rho_c,\quad\rho_c=3H^2/(8\pi G)$}
At maximum expansion of a <closed matter-dominated Friedmann solution>, the <Hubble parameter> vanishes while the physical <mass density> stays positive and finite. Its <critical density> therefore vanishes and the <cosmological density parameter> diverges. This is a singularity of that ratio rather than a <curvature singularity>. In the cycloid variable $\theta$, $\Omega_M=\sec^2(\theta/2)$, diverging at $\theta=\pi$ and approaching one at the <Big Bang> and <Big Crunch> ends.
\Image[/past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2013-closed-dust-hubble-and-density.png]
{title=A closed dust universe: the Hubble parameter changes sign while its density parameter diverges at maximum expansion}
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