Solution
= Solution
At <matter-radiation equality>, $\rho_{M,0}a^{-3}=\rho_{R,0}a^{-4}$, so $a_{\rm eq}=\Omega_{R,0}/\Omega_{M,0}$. Evaluating the integrated square-root relation gives
$$
\boxed{\eta_{\rm eq}=\frac{2(\sqrt2-1)\sqrt{\Omega_{R,0}}}{\Omega_{M,0}H_0}.}
$$
This assumes both present densities are positive and uses the same <Big Bang> origin for <conformal time>.