= Solution
The neutral-hydrogen <column density> is $N_{\rm HI}=\int n_{\rm HI}\,d\ell$, with units $\mathrm{cm}^{-2}$. Define the <neutral-hydrogen column-density distribution> by
$$
f(N,z)=\frac{\partial^2\mathcal N}{\partial N\,\partial z},
\qquad d\mathcal N=f(N,z)\,dN\,dz.
$$
Here $\mathcal N$ counts absorption systems along a <quasar> sightline. A histogram per logarithmic column interval instead measures $N\ln(10)f(N,z)$; forgetting this factor changes the inferred slope. Another common convention uses the <absorption distance> $X$, with $dX/dz=H_0(1+z)^2/H(z)$, and $f_X=f/(dX/dz)$.
At a fixed epoch, weak <Lyman-alpha forest> lines have an approximately falling power-law distribution, $f(N,z)\simeq A(z)N^{-\beta}$ with $\beta\simeq1.4$–$1.5$ over representative low-column ranges. Stronger absorbers deviate from one universal slope, and the normalization and fitted slopes depend on epoch and sample. The low-column fit is measured, for example, in https://arxiv.org/abs/astro-ph/0101005[Kim and collaborators' forest survey]. The sketch distinguishes that empirical low-column trend from an illustrative higher-column continuation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-73-column-distribution.png]
{title=Neutral-hydrogen column-density distribution with forest, Lyman-limit and damped-system thresholds}
Ordinary <Lyman-alpha forest> lines cover roughly $10^{12}\lesssim N_{\rm HI}\lesssim10^{17.2}\,\mathrm{cm}^{-2}$. They are numerous resonant <Lyman-alpha absorption> features from intervening gas at different <cosmological redshifts>, collectively resembling a forest in a <quasar> spectrum. Many are optically thin at the ionization edge even when their line centers are saturated.
A <Lyman limit system> becomes optically thick at the $91.2\,\mathrm{nm}$ hydrogen ionization edge. With $\sigma_{912}\simeq6.3\times10^{-18}\,\mathrm{cm}^2$, $\tau_{912}=N_{\rm HI}\sigma_{912}\gtrsim1$ requires
$$
N_{\rm HI}\gtrsim1.6\times10^{17}\,\mathrm{cm}^{-2}\simeq10^{17.2}\,\mathrm{cm}^{-2}.
$$
Its spectrum consequently has a strong continuum decrement below the redshifted Lyman edge. In a classification separating the highest columns, <Lyman limit systems> extend up to the damped-system boundary.
A <damped Lyman-alpha system> conventionally has $N_{\rm HI}\gtrsim2\times10^{20}\,\mathrm{cm}^{-2}$. Its <Lyman-alpha absorption> is sufficiently strong that the Lorentzian wings of the <Voigt profile> are prominent. “Damped” refers to the finite radiative lifetime of the atomic excited state, not an inference that the gas has a high <temperature>. These systems trace substantial neutral reservoirs, usually associated with <galaxies>.
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