= Flat dust universe with a cosmological constant
{title2=$a\propto\sinh^{2/3}(\sqrt{3\Lambda}\,t/2)$}
For <pressureless matter>, put $C=8\pi G\rho_m a^3/3>0$ and use units $c=1$. The flat <Friedmann equation> is $\dot a^2=C/a+\Lambda a^2/3$. Substitution $u=a^{3/2}$ gives $\dot u^2=(9/4)(C+\Lambda u^2/3)$. With a <Big Bang> at $t=0$, positive <cosmological constant> gives $a=(3C/\Lambda)^{1/3}\sinh^{2/3}(\sqrt{3\Lambda}\,t/2)$. Negative <cosmological constant> replaces the hyperbolic sine by $\sin(\sqrt{3|\Lambda|}\,t/2)$ and produces recollapse at $2\pi/\sqrt{3|\Lambda|}$. Zero <cosmological constant> gives $a=(9C/4)^{1/3}t^{2/3}$.
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