Minimum expansion speed in a matter-Lambda universe ($a_{\min }=a_0({\Omega }_{m,0}/2{\Omega }_{\Lambda ,0}{)}^{1/3}$)

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.