Scale factor at surface of last scattering 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?
 A: 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.
