# How to justifiy that $\rho_{\text{rad}} \approx \rho_{\text{mat}}$ at recombination time?

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): $$$$\tag{1} v_{\text{sound}} = \sqrt{\frac{dp}{d\rho}} = \frac{1}{\sqrt{3}} \, \frac{1}{\sqrt{1 + \frac{3 \rho_{\text{mat}}}{4 \rho_{\text{rad}}}}}.$$$$ At time $$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: $$$$\tag{2} 3 \rho_{\text{mat}} \approx 4 \rho_{\text{rad}}.$$$$

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 $$\frac{\rho_{\rm mat}}{\rho_{\rm rad}} = \frac{\rho_{{\rm mat},0}}{ \rho_{{\rm rad},0}} \left(1+z_{\rm rec}\right)^{-1} \simeq 0.74$$