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In standard cosmology, the recombination time is estimated to be $t_{\text{rec}} \approx 380~000~\mathrm{years}$ after the Big Bang, when matter and electromagnetic radiation becomes decoupled. It's the time the CMB was created (or "liberated" from matter).

Since $\rho = \rho_{\text{mat}} + \rho_{\text{rad}}$ and $p = \frac{1}{3} \, \rho_{\text{rad}}$, the sound velocity in the cosmic fluid is (I'm using $c = 1$ here): \begin{equation}\tag{1} v_{\text{sound}} = \sqrt{\frac{dp}{d\rho}} = \frac{1}{\sqrt{3}} \, \frac{1}{\sqrt{1 + \frac{3 \rho_{\text{mat}}}{4 \rho_{\text{rad}}}}}. \end{equation} At time $t_{\text{rec}}$, it is often said (written) that $\frac{3 \rho_{\text{mat}}}{4 \rho_{\text{rad}}} \approx 1$ (so the sound velocity is $v_{\text{sound}} \approx 1/\sqrt{6}$ at recombination time). How can we justify this?

In other words, how do we calculate (or predict) the recombination time? I suspect that it's from the Saha equation (which I don't know much), and not from the energy density relation which feels a bit arbitrary: \begin{equation}\tag{2} 3 \rho_{\text{mat}} \approx 4 \rho_{\text{rad}}. \end{equation}

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The matter density scales as $a^{-3}$ whilst the radiation density scales as $a^{-4}$.

At the present epoch $\rho_{{\rm rad},0} \sim 6\times 10^{-5}$ of the critical density, whereas $\rho_{{\rm mat},0} \sim 0.0486$ of the critical density.

At the epoch of recombination $z_{\rm rec} \sim 1100$. This does indeed come from the Saha equation and is a standard piece of bookwork which calculates the redshift at which some fraction of the hydrogen becomes atomic, though getting the exact answer is a bit more complex than that (see for example here). The scale factor $a$ changes as $(1+z)^{-1}$, so, if we go back to the epoch of recombination $$ \frac{\rho_{\rm mat}}{\rho_{\rm rad}} = \frac{\rho_{{\rm mat},0}}{ \rho_{{\rm rad},0}} \left(1+z_{\rm rec}\right)^{-1} \simeq 0.74$$

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