Flat dust-and-vacuum expansion ($a\propto {\sinh }^{2/3}(3H_{\Lambda }t/2)$)

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.