The Last Scattering Surface

From the evolution of the temperature, T oc (1 + z), and the number density of non-relativistic particles, v x (I z y we can infer that the universe was once sufficiently hot and dense to ionize the hydrogen that today makes up most of the baryon density of the Universe. Naively, we might expect this to occur when T i Ry = 13.6 eV, the binding energy of Hydrogen. 

To do the calculation in more detail, for the baryons (nuclei) and electrons, we define the ionization fraction X (t), the ratio of the density of ions to neutrals if we assume overall charge-neutrality, this is equal to the ratio of the free electron number density to that of neutrals. We also assume that the number density of any massive species (i) is large enough that it can be described by a Boltzmann distribution, 

We show the Hydrogen ionization fraction in figure 10.1. We find a very rapid transition from X 1 to X 0 at z 1,100, corresponding to T(z) = 0.3eV. This is considerably lower than our naive expectation of 13.6eV, due to the small prefactors on the right hand side of Eq. 10.7, themselves due to the very small value of nB/n-y = 2.7 x 10 8(flBh2) there are many more photons than baryons, and so even the small fraction in the high-energy tail of the Boltzmann distribution are sufficient to keep the Universe ionized. 

If we follow the evolution of the ionization fraction yet further in time, we would find that the universe never completely recombines rather, we are left with a residual ionization fraction X(0) 10-4. As the universe expands and 

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