= Minimum expansion speed in a matter-Lambda universe
{title2=$a_{\min}=a_0(\Omega_{m,0}/2\Omega_{\Lambda,0})^{1/3}$}
For pressureless matter and a positive <cosmological constant>, the <Friedmann equation> gives $\dot a^2=H_0^2[\Omega_{m,0}a_0^3/a+\Omega_{k,0}a_0^2+\Omega_{\Lambda,0}a^2]$. Differentiating with respect to $a$ gives the displayed unique minimum of this expression for positive matter density. It is a minimum of the actual expanding solution's speed only if that branch reaches it; sufficient positive spatial curvature can cause recollapse first. At the minimum, $\rho_m=2\rho_\Lambda$ and the <Friedmann acceleration equation> changes sign.
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