Scale factor in a conformal-time density parameter (${\Omega }_i=8\pi G{a}^2{\bar{\rho }}_i/(3{\mathcal{H}}^2)$)

For physical density ${\bar{\rho }}_i$ and conformal time, $\mathcal{H}=a^{\prime }/a=aH$. Substitution into the definition ${\Omega }_i=8\pi G{\bar{\rho }}_i/(3H^2)$ gives the displayed formula. Omitting $a^2$ gives ${\Omega }_i/a^2$, which need not equal one even in a spatially flat matter-only universe. In particular, the conformal growth source $\frac{3}{2}{\mathcal{H}}^2{\Omega }_m$ equals $4\pi G{a}^2{\bar{\rho }}_m$ with the physical density parameter.