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

Assume both densities are nonnegative and ${\Omega }_{\mathrm{DE},0}<1$, so a nonzero pressureless matter component exists. Their ratio obeysThus matter domination givesAt fixed $H_0$, the early expansion rate does depend on the present dark energy fraction: increasing it reduces ${\rho }_{M,0}=(1-{\Omega }_{\mathrm{DE},0})3H_0^2/(8\pi G)$. There is no contradiction with negligible early dark energy, since the leading expression is simply $H^2\sim 8\pi G{\rho }_M/3$. At fixed physical matter density, that leading rate is independent of the negligible component.

The endpoint ${\Omega }_{\mathrm{DE},0}=1$ has no matter, so the ratio argument does not apply. Its Hubble parameter is $H=H_0{a}^{-3(1+w)/2}$: it diverges for $-1<w<0$, is constant for $w=-1$, and tends to zero for $w<-1$.