= Solution
Adiabatic <photon> cooling gives $a_*\simeq1/1200$. During <matter domination>, $H(a)\simeq H_0\sqrt{\Omega_m}a^{-3/2}$, so the comoving sound distance traveled before last scattering is
$$
s_*\simeq\frac{c}{\sqrt3}\int_0^{a_*}\frac{da}{a^2H(a)}
=\frac{2c\sqrt{a_*}}{\sqrt3H_0\sqrt{\Omega_m}}.
$$
Multiplication by $a_*$ gives the physical <sound horizon> radius:
$$
\boxed{R_{s,*}\simeq\frac{2c}{\sqrt3H_0\sqrt{\Omega_m}}(1200)^{-3/2}
\simeq\frac{0.119\,\mathrm{Mpc}}{\sqrt{\Omega_m}}
\left(\frac{70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}}{H_0}\right).}
$$
This is a radius, or one-way sound-propagation length, rather than its diameter. The integral treats the early radiation interval as negligible under the specified late-equality approximation; it also neglects curvature at last scattering, requiring $\Omega_ma_*^{-3}\gg(1-\Omega_m)a_*^{-2}$. A very small matter fraction would violate those approximations.
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