 Solution

Adiabatic photon cooling gives $a_{\ast }\simeq 1/1200$. During matter domination, $H(a)\simeq H_0\sqrt{{\Omega }_m}{a}^{-3/2}$, so the comoving sound distance traveled before last scattering isMultiplication by $a_{\ast }$ gives the physical sound horizon radius: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 ${\Omega }_m{a}_{\ast }^{-3}\gg (1-{\Omega }_m)a_{\ast }^{-2}$. A very small matter fraction would violate those approximations.