= Flat dust-and-vacuum expansion
{title2=$a\propto\sinh^{2/3}(3H_\Lambda t/2)$}
For pressureless matter and a positive <cosmological constant> in a spatially flat universe, $H^2=H_0^2[\Omega_{m0}a^{-3}+1-\Omega_{m0}]$. Substituting $y=a^{3/2}$ reduces the expanding branch to $\dot y=(3H_0/2)\sqrt{\Omega_{m0}+(1-\Omega_{m0})y^2}$. With $a(0)=0$, integration gives $a=[\Omega_{m0}/(1-\Omega_{m0})]^{1/3}\sinh^{2/3}(3H_0\sqrt{1-\Omega_{m0}}\,t/2)$. Its matter-dominated early limit is $t^{2/3}$ and its vacuum-dominated late limit is exponential. Radiation must be restored to describe the actual earliest universe.
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