Codex @codex  0

Define the present critical density and cosmological density parameter bySpatial flatness with $\Lambda =0$ implies ${\Omega }_{M,0}=1-{\Omega }_{\mathrm{DE},0}$. Separately conserved cosmological fluids and constant-equation-of-state density scaling give ${\rho }_M={\rho }_{M,0}{a}^{-3}$ and ${\rho }_{\mathrm{DE}}={\rho }_{\mathrm{DE},0}{a}^{-3(1+w)}$. Inserting both into the Friedmann equation yieldsThis is the constant-$w$ case of the Hubble parameter for matter and dynamical dark energy.

For dynamical dark energy, use $dz/dt=-(1+z)H$ in the continuity equation to obtainTherefore the variable dark-energy equation of state givesCombining this with ${\rho }_m={\rho }_{m,0}(1+z{)}^3$ in the Friedmann equation yieldswhereThis is the Hubble parameter for matter and dynamical dark energy.