Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 1 / b / Solution

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.