= Lyman-alpha forest photoionization scaling
{c}
{title2=$n_{\rm HI}\simeq\alpha(T)n_H^2/\Gamma$}
In optically thin, almost fully ionized <hydrogen> in <photoionization equilibrium>, $n_{\rm HI}\Gamma=n_en_p\alpha(T)$ and $n_e\simeq n_p\simeq n_H$. Thus the neutral fraction is approximately $\alpha(T)n_H/\Gamma$, while the neutral density is proportional to $n_H^2$. At fixed overdensity, cosmic dilution gives $n_H\propto(1+z)^3$, hence $n_{\rm HI}\propto\alpha(T)(1+z)^6/\Gamma$. Gas density, temperature, ionizing radiation and path length are all required to infer a baryon density from <Lyman-alpha absorption>.
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