Matter density flow with a cosmological constant (${\Omega }_m^{\prime }=\mathcal{H}{\Omega }_m({\Omega }_m-1-2{\Omega }_{\Lambda })$)

For separately conserved pressureless matter, ${\rho }_m\propto a^{-3}$. The deceleration parameter identity $\dot{H}/H^2=-(1+q)$ and ${\Omega }_m\propto {\rho }_m/H^2$ give ${\dot{\Omega }}_m=H{\Omega }_m(2q-1)$. A dust-plus-vacuum universe has $q={\Omega }_m/2-{\Omega }_{\Lambda }$; conversion to conformal time yields the displayed flow. The familiar matter-only equation requires negligible cosmological constant. In a spatially flat dust-plus-vacuum universe, ${\Omega }_m+{\Omega }_{\Lambda }=1$ and ${\Omega }_m^{\prime }=-3\mathcal{H}{\Omega }_m{\Omega }_{\Lambda }$, so flatness does not require ${\Omega }_m=1$.