= Closed matter-dominated Friedmann solution
{title2=$a=A(1-\cos u),\quad t=A(u-\sin u)/\sqrt{k}$}
A positively curved <pressureless matter> universe with zero <cosmological constant> has the <cycloid> parametrization displayed above. In <natural units>, if the <Friedmann equation> is $\dot a^2=C/a-k$ with $C,k>0$, then $A=C/(2k)$ and $u=\sqrt{k}\tau$ for <conformal time> measured from the <Big Bang>. Substituting in $a'^2=Ca-ka^2$ verifies the <scale factor>; integrating $dt=a\,d\tau$ gives <cosmic time>. The universe reaches maximum size at $u=\pi$ and recollapses into a <Big Crunch> at $u=2\pi$.
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