= Solution
Normalize the <scale factor> so that $a_0=1$. With separately conserved <pressureless matter> and <radiation in cosmology>, their <energy densities> scale as $a^{-3}$ and $a^{-4}$. Hence the flat <Friedmann equation> is
$$
H^2=H_0^2\bigl(\Omega_{m,0}a^{-3}+\Omega_{r,0}a^{-4}\bigr),\qquad
\mathcal H^2=H_0^2\bigl(\Omega_{m,0}a^{-1}+\Omega_{r,0}a^{-2}\bigr).
$$
Equality of the two densities defines <matter-radiation equality>, with $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}$. Define the dimensionless variable
$$
\boxed{y=\frac a{a_{\rm eq}}=a\frac{\Omega_{m,0}}{\Omega_{r,0}}=\frac{\bar\rho_m}{\bar\rho_r}.}
$$
Substituting $a=a_{\rm eq}y$ gives
$$
\boxed{\mathcal H^2=C\left(\frac1y+\frac1{y^2}\right),\qquad C=\frac{H_0^2\Omega_{m,0}^2}{\Omega_{r,0}}.}
$$
Both reference density fractions are positive. In the two-component flat model their sum at the reference epoch is one; no dark-energy component is being included in this background calculation.
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