Hubble parameter for matter and dynamical dark energy

Assume spatial flatness, negligible radiation, separately conserved pressureless matter and dark energy, and $a_0=1$. Let $w(z)=P_{\mathrm{DE}}/{\rho }_{\mathrm{DE}}$ be the dark-energy equation-of-state parameter. The cosmological continuity equation givesCombining this with ${\rho }_m={\rho }_{m,0}(1+z{)}^3$ in the Friedmann equation yieldswith ${\Omega }_{m,0}+{\Omega }_{\mathrm{DE},0}=1$. For $w=-1$, the dark-energy factor is constant and this reduces to the matter-plus-cosmological constant model. Nonzero cosmological spatial curvature, radiation or energy exchange between the components requires additional terms or modified continuity equations.