Lyman-alpha forest photoionization scaling ($n_{\mathrm{HI}}\simeq \alpha (T)n_H^2/\Gamma$)

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.

Assume optically thin photoionization equilibrium and an absorber depth of order its local Jeans length, $L\propto (f_{g}T/n_H{)}^{1/2}$, where $f_g$ is the local gas fraction. Then $N_{\mathrm{HI}}\simeq \alpha (T)n_H^{2}L/\Gamma \propto [\alpha (T)/\Gamma ](f_{g}T{)}^{1/2}{n}_H^{3/2}$. Solving for $n_H$ and substituting in $N_H=n_{H}L$ gives the displayed correction. Thus the ionized mass integral weights the neutral-hydrogen column-density distribution by $N_{\mathrm{HI}}^{1/3}$ rather than $N_{\mathrm{HI}}$. This model does not apply without modification to optically thick systems or arbitrary unconfined gas.