 Solution

For dust, $P=0$ and $w=0$. The continuity equation gives $\rho \propto R^{-3}$, and the flat, zero-$\Lambda$ Friedmann equation integrates to $R\propto t^{2/3}$. ThusDirectly, a power-law scale factor $R\propto t^p$ has $q=(1-p)/p$, which gives the same constant. This is the usual pressureless Einstein-de Sitter model.