Ionized gas density from an absorption distribution (${\Omega }_g=\frac{H_0{\mu }_H{m}_H}{c{\rho }_{\mathrm{crit},0}}\int \frac{N}{x_{\mathrm{HI}}}{f}_X(N,z)dN$)

A neutral-hydrogen column density $N$ traces total hydrogen column $N/x_{\mathrm{HI}}$ only after specifying its neutral fraction $x_{\mathrm{HI}}$. Integrating the latter against the distribution per absorption distance gives the comoving gas mass density in the displayed expression. The factor ${\mu }_H$ includes helium if desired. In photoionization equilibrium for optically thin, almost fully ionized hydrogen, $x_{\mathrm{HI}}\simeq \alpha (T)n_H/\Gamma$, so gas density, temperature, photoionization rate and absorber size are required. An observed distribution alone does not fix the total baryon inventory.