Flat matter-radiation conformal solution ($a(\eta )=H_0\sqrt{{\Omega }_{R,0}}\eta +{\Omega }_{M,0}{H}_0^2{\eta }^2/4$)

For a spatially flat FLRW metric containing conserved pressureless matter and radiation in cosmology, with $a_0=1$ and conformal time zero at the Big Bang, the Friedmann equation gives $a^{\prime 2}=H_0^2({\Omega }_{M,0}a+{\Omega }_{R,0})$. Separating the positive square root gives the displayed scale factor. At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{R,0}/{\Omega }_{M,0}$ and ${\eta }_{\mathrm{eq}}=2(\sqrt{2}-1)\sqrt{{\Omega }_{R,0}}/(H_0{\Omega }_{M,0})$. Here the prime denotes differentiation with respect to $\eta$, not $c\eta$.