= Neutral gas density from an absorption distribution
{title2=$\Omega_{\rm HI}=\frac{H_0m_H}{c\rho_{\rm crit,0}}\int Nf_X(N,z)\,dN$}
Let $f_X=\partial^2\mathcal N/(\partial N\partial X)$ count neutral <hydrogen> absorbers per <column density> and <absorption distance>. The summed column per unit proper light-path length is the mean neutral <number density>. Since $d\ell=c\,dz/[(1+z)H(z)]$ and $dX/dz=H_0(1+z)^2/H(z)$, that density is $(H_0/c)(1+z)^3\int Nf_X\,dN$. Dividing the proper <mass density> by $(1+z)^3$ gives the comoving density in the displayed formula. The denominator is today's <critical density>; including neutral helium requires a stated mass correction. For a distribution per redshift, replace $H_0f_X$ by $H(z)f_z/(1+z)^2$.
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