How is the Redshift of Recombination Calculated? In all the text, they do a hand waving exercise where they say the re-ionizing temperature of Hydrogen in the plasma is 3,000K, the current temperature of space is 2.75K, so you divide 3000/2.75 and get 1090.  But it turns out the real calculation is considerably more difficult.  I've seen two versions:


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*The redshift is calculated as the point at which the Optical Depth is 1.  Meaning that it's very likely the photon has collided just once on it's trip from the Surface of Last Scattering to the observer.





*The redshift is calculated as the point where the Visibility Function reaches its maximum.


 
Which is it?
 A: $z=1090$ (or $z\simeq 1100$) is often quoted and is approximately where the integrated optical depth of a photon to Thomson scattering reaches 1. The exact value depends precisely how much physics you are able or willing to put into the calculation.
An optical depth of 1 corresponds to the mean free path of a photon (i.e. when a photon travels through a scattering medium, the average path length it travels before scattering corresponds to an optical depth of 1), so this represents how far back a typical CMB photon has travelled from.
You can use other definitions if you like. For example the visibility function represents the probability distribution that any particular photon in the CMB originated at redshift $z$. So you could choose the peak of this probability function and it is obviously closely related to how the optical depth ramps up with increasing $z$. But it is not exactly the same (although you would be hard pressed to see any difference in the x-axis values in the plots shown in the question), in the same sense that the peak of a probability distribution is not the mean value unless the distribution is perfectly symmetric. This is shown more clearly in the plot below (upper panel). This shows the asymmetry in the visibility function versus redshift and indicates that the peak of the probability distribution is at a slightly higher $z$ than the mean.

There is no one definition. The CMB formed over tens of thousands of years and at a range of redshifts.
See Understanding the recombination and decoupling periods in the early universe and For how long CMB was being emitted?
