# Why is the temperature of electron-proton recombination obtained from Saha equation so small?

I read a blog about the temperature of the Universe at recombination phase.

The ionization potential of a hydrogen atom is $$13.6\, eV$$, throw it into the thermal energy equation $$E=\frac{3}{2}kT$$, will yield a temperature about $$105000 \, K$$, which is incorrect.

By manipulating the Saha equation, we obtain $$\frac{x_e^2}{(1-x_e)}=\frac{1}{1.6(1+z)^3}(\frac{m_ek_B}{2\pi h^2})^{3/2}T^{3/2}\exp(-\frac{E_1}{k_BT})$$, where $$x_e$$ is the fraction of free electron.

Solving it numerically, the temperature at which recombination is nearly complete ($$x_e \approx 0$$) is around $$3000 \, K$$.

This is way smaller than $$105000\, K$$. Not only that, but if we slightly increase the temperature, say $$4000 \, K$$, the free electron abundance surges to about 60%.

Mathematically, I think it's because the exponential term in the Saha equation dominates and make the result very sensitive to even small changes.

However, I am having trouble to physically interpret this phenomenon, why is $$x_e$$, the fraction of free electron, can be so high at relative low temperatures such as $$4000\, K$$, which is far below $$105000 \, K$$.