Dust relation between deceleration and density ($q={\Omega }_m/2$)

For pressureless matter with zero cosmological constant, the Friedmann acceleration equation gives $q=4\pi G\rho /(3H^2)={\Omega }_m/2$. Spatial curvature need not vanish. Consequently $k{c}^2/a^2=(2q-1)H^2$; this relation does not hold for a general fluid mixture.