Flat dust-vacuum Hubble parameter ($H(z)=H_0[{\Omega }_{m0}(1+z{)}^3+{\Omega }_{\Lambda 0}{]}^{1/2}$)

For pressureless matter, a positive cosmological constant, flat spatial geometry and negligible radiation, the continuity equation gives ${\rho }_m\propto a^{-3}$. Substitution in the Friedmann equation yields the displayed Hubble parameter, with ${\Omega }_{m0}+{\Omega }_{\Lambda 0}=1$. radiation must be restored at early epochs; the formula does not require matter to be the largest present-day density component.