= Radiation-matter scale factor in conformal time
{title2=$a(\tau)=H_0\sqrt{\Omega_{r,0}}\tau+\tfrac14H_0^2\Omega_{m,0}\tau^2$}
In a flat universe containing separately conserved <radiation in cosmology> and <pressureless matter>, the <Friedmann equation> becomes $(a')^2=H_0^2(\Omega_{r,0}+\Omega_{m,0}a)$ in <conformal time>. Setting $a=0$ at $\tau=0$ gives the stated quadratic-plus-linear <scale factor>. The linear term controls <radiation domination> and the quadratic term controls <matter domination>. The formula presumes $a_0=1$, $\Omega_{r,0}+\Omega_{m,0}=1$ and no <cosmological constant>.
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