Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 1 / iv / Solution

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.