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Use absorption distance $X$ and its distribution $f_X$. Summing hydrogen columns per unit $X$ gives $\int N{f}_{X}dN$. The physical light-path length per $X$ isThe mean neutral-hydrogen number density is therefore $(H_0/c)(1+z{)}^3\int N{f}_{X}dN$. Divide by $(1+z{)}^3$ to obtain its comoving density, multiply by hydrogen atomic mass, and divide by today's mass critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$. This derives the neutral gas density from an absorption distribution:If “neutral gas” includes helium, replace $m_H$ by ${\mu }_H{m}_H$, with a stated abundance correction approximately ${\mu }_H\simeq 1.3$. Equivalently, for the distribution per redshift the prefactor inside the integral is $m_{H}H(z)/[c{\rho }_{\mathrm{crit},0}(1+z{)}^2]$ multiplying $N{f}_z$. In the PDF's energy-density convention, both the mass density and critical density are multiplied by $c^2$, leaving their ratio unchanged. The usual quoted abundance uses comoving mass density relative to today's critical density; a ratio of physical density to the epoch's critical density instead multiplies this answer by $(1+z{)}^3{H}_0^2/H(z{)}^2$.

To infer total gas, one needs the neutral fraction $x_{\mathrm{HI}}=N_{\mathrm{HI}}/N_H$ for each absorber. The ionized gas density from an absorption distribution isThis is an ionization-corrected integral, not an inference available from the observed columns alone. For optically thin, almost fully ionized hydrogen, photoionization equilibrium gives $\Gamma n_{\mathrm{HI}}=\alpha (T)n_e{n}_{\mathrm{HII}}\simeq \alpha (T)n_H^2$. If the absorbing depth $L$ is specified, $N=\alpha (T)n_H^{2}L/\Gamma$ andThus photoionization rate, temperature and depth are all needed.

One useful physical closure sets $L$ to a local Jeans length, $L\propto (f_{g}T/n_H{)}^{1/2}$, with local gas fraction $f_g$. Solving $N\propto [\alpha (T)/\Gamma ](f_{g}T{)}^{1/2}{n}_H^{3/2}$ then gives the Jeans-length ionization correction for the Lyman-alpha forest:This illustrates why weak, highly ionized forest lines can dominate the total gas inventory even though damped Lyman-alpha systems dominate the neutral-column-weighted inventory. It is a model for locally confined optically thin gas; optically thick Lyman limit systems and damped Lyman-alpha systems require a separate radiative transfer treatment. The physical basis and observational application of this closure are developed in Schaye's absorption-based matter-density calculation.

No measured normalization of the neutral-hydrogen column-density distribution, ionizing field or gas temperature is supplied, so the paper's formulas do not determine numerical densities on their own. With observational data and suitable ionization corrections, the high-redshift Lyman-alpha forest gives a total gas density of order a few $10^{-2}$ relative to today's critical density, comparable to the cosmological baryon density ${\Omega }_b\sim 0.04$–$0.05$ for $H_0\sim 70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. The conclusion is that a large fraction of baryons at $z\simeq 3$ reside in diffuse ionized gas, rather than that every baryon has been individually accounted for. Schaye's analysis and Kim and collaborators' survey support that interpretation under their stated gas and ionizing-background assumptions.

The neutral-gas reservoir traced predominantly by damped Lyman-alpha systems is much smaller, of order $10^{-3}$ in the same convention. It is broadly comparable in order of magnitude to today's luminous stellar matter, whose cosmic stellar mass density is a few $10^{-3}$, but exact comparisons depend on cosmology, helium correction and which stellar components are counted. Storrie-Lombardi, McMahon and Irwin's neutral-gas census finds especially close comparison with present stellar disks. Most high-redshift gas is ionized; the smaller neutral reservoir has a mass of roughly the same order as today's visible stellar population. This comparison does not imply that the measured neutral reservoir alone simply becomes all present-day stars: gas accretion, recycling, ionization changes and outflows continue during galaxy evolution.

The neutral gas density from an absorption distribution is comparatively robust for high columns where the hydrogen is mostly neutral, but saturated lines, dust-obscured quasars, incomplete surveys and small numbers of rare high-column systems affect the integral. The ionized gas density from an absorption distribution has additional systematic uncertainty in the cosmic ionizing background, temperature, absorber depth, helium treatment and small-scale density inhomogeneity. At low columns, line blending and incompleteness matter; at high columns, attenuation makes an optically thin ionization correction invalid. A single power law extrapolated outside the measured column range can also produce a misleading mass integral. Hot gas, molecular gas and gas missed by the selected absorption population are not included automatically. These are observational and modelling qualifications to a baryon census, not failures of the column-mass accounting itself.

Specify the path convention before integrating the neutral-hydrogen column-density distribution. Write $f_z={\partial }^2\mathcal{N}/(\partial N\partial z)$ and $f_X={\partial }^2\mathcal{N}/(\partial N\partial X)$, with absorption distance satisfyingThe lower panel of the preceding sketch shows the approximate decrease of $f$ with $N=N_{\mathrm{HI}}$ near $z=3$. Ordinary forest columns are roughly described by $f\propto N^{-\beta }$, $\beta \sim 1.5$, but the slope changes through partially shielded Lyman limit systems and high-column damped Lyman-alpha systems. No single slope should be extrapolated through all these regimes. In logarithmic bins the number is $N\ln (10)f$, whereas their neutral mass contribution is proportional to $N^{2}f$; confusing these two weightings would give a wrong baryon inventory.

To derive the neutral gas density from an absorption distribution, a redshift interval $dz$ has proper light-path length $d\ell =cdz/[(1+z)H]$. The sum of neutral hydrogen columns per path length estimates the mean proper number density:The proper mass density is $m_H{\overline{n}}_{\mathrm{HI}}$; division by $(1+z{)}^3$ converts it to a comoving mass density. Thus, with today's critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$,A representative finite-sample estimator replaces the last integral by ${\sum }_j{N}_j/\Delta X$, with completeness and selection corrections. This defines a comoving inventory divided by today's critical density; a fraction of the instantaneous critical density has a different normalization. Neutral gas including accompanying helium is obtained by multiplying by a specified mass correction, approximately $1.3$; hydrogen-only ${\Omega }_{\mathrm{HI}}$ must not silently include it.

At $z\sim 3$ the observed neutral inventory is of order $10^{-3}$, far below the cosmological baryon density of order $0.04$–$0.05$ for $h\sim 0.7$. High-column damped Lyman-alpha systems contribute most of this neutral mass although weak forest lines dominate the counts; for $\beta =1.5$ the mass per logarithmic column interval increases as $N^{1/2}$ in the forest regime. The small neutral inventory therefore does not imply a small total gas inventory: the intergalactic medium contains a large ionized baryon reservoir invisible to a neutral-only count. Recovering that reservoir requires an ionization correction and a gas-density model. The survey normalization and high-column contribution are described in the neutral-gas inventory analysis.