= Matter density flow with a cosmological constant
{title2=$\Omega_m'=\mathcal H\Omega_m(\Omega_m-1-2\Omega_\Lambda)$}
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 $\Omega_m'=-3\mathcal H\Omega_m\Omega_\Lambda$, so flatness does not require $\Omega_m=1$.
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