Matter-vacuum density equality ($1+z_{m\Lambda }=({\Omega }_{\Lambda 0}/{\Omega }_{m0}{)}^{1/3}$)

For conserved pressureless matter and a true cosmological constant, equality follows by comparing the matter density proportional to $(1+z{)}^3$ with the constant vacuum density. This is not the onset of accelerated expansion: in a matter-plus-vacuum model acceleration starts at ${\rho }_m=2{\rho }_{\Lambda }$.