How does this number get calculated?
About 380,000 years after the Big Bang the temperature of the universe fell to the point where nuclei could combine with electrons to create neutral atoms.
http://en.wikipedia.org/wiki/Photon_epoch
I've seen it in many places (or something close to that), but there is never a citation or any explanation.
The calculation is described in detail in the Wikipedia article on recombination.
If you consider the ionisation of hydrogen as a reaction:
$$ p + e \rightarrow H + \gamma $$
Then you can write down an expression for the equilibrium constant as a function of temperature using the Saha equation:
$$ \frac{n_pn_e}{n_H} = \left( \frac{m_ek_BT}{2\pi\hbar^2} \right)^{3/2} \exp \left( \frac{-E_I}{k_BT} \right) $$
If you take 50% ionisation you can work out the corresponding temperature and it turns out to be about 4,000K. So now it's just a matter of relating the temperature of the universe to the time after the Big Bang. Once we're past the various phase transitions that happened in the first few instants after the Big Bang the temperature is inversely proportional to the scale factor. Sadly there isn't a simple equation to give the scale factor as a function of time, however it's a straightforward numerical calculation, and the result is that the temperature was 4,000K about 380,000 years after the Big Bang.
That's how the figure of 380,000 years is calculated.
Scientists calculated this knowing that in the time between the Big Bang and the Era of Recombination there was a large 'soup' of superheated particles, which cooled down as the universe was expanding. Atoms could not form because every time an electron tried to 'orbit' a proton it was knocked out of orbit by a high energy photon. These high energy photons are crucial and they got their high energy from the immense heat at the beginning: at Big Bang. These photons are thermal radiation. Thermal Radiation is what we observe in the CMBR. Recently the Planck Telescope has taken some measurements of the levels of radiation across the universe, and it was not the first telescope to do this. Scientists, after collecting information about the radiation across the universe, can subtract the amount of radiation from different sources and then they look at CMBR, which dates back to the Era of Recombination. From the amount of radiation they can then calculate the temperature at that time and can deduce that atoms could form because the photons would be less energetic and would not 'wreck' every new-formed atom. ~More Information: https://m.youtube.com/watch?v=1loJTy6bOu8 http://ned.ipac.caltech.edu/level5/March02/Plionis/Plionis1_2.html ~
