Is there an exact theoretical scale factor associated with the CMB? More specifically, since the CMB is associated with a temperature of around 3000K (as I understand), is there an exact thermodynamic relationship that says what the scale factor must be for the Universe to have that temperature at that time?


In terms of temperature and scale factor then $$T_0 = (1+z)^{-1} T_{rec},$$ where $T_{rec}$ is the recombination temperature and $T_0$ is the temperature of the CMB now. To put this interms of scale factor, we note that $a = (1+z)^{-1}$. So $$ a_{rec} = \frac{T_0}{T_{rec}}$$

The universe is "matter-dominated" at the epoch of recombination and in this case $a(t) \propto t^{2/3}$. Thus if we define now as $t=t_0$, then $$ a_{rec} = \left(\frac{t_{rec}}{t_0}\right)^{2/3}$$ and so in terms of time $$t_{rec} = t_0 \left(\frac{T_0}{T_{rec}}\right)^{3/2}\ .$$

Note that "the" surface of last scattering doesn't exist uniquely. Photons in the CMB were emitted at a range of redshifts from about 900-1200.

The value of $T_{rec}$ at which recombination takes place is calculated using the Saha equation for the ionisation equilibrium of hydrogen and can be refined to take account of various other corrections as I explain in this answer. It also depends (weakly) on the baryonic density of the universe.

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