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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?

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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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