 Lyman-alpha forest photoionization scaling

In optically thin, almost fully ionized hydrogen in photoionization equilibrium, $n_{\mathrm{HI}}\Gamma =n_e{n}_p\alpha (T)$ 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 $n_{\mathrm{HI}}\propto \alpha (T)(1+z{)}^6/\Gamma$. Gas density, temperature, ionizing radiation and path length are all required to infer a baryon density from Lyman-alpha absorption.