Annu. Rev. Astron. Astrophys. 1998. 36: 267-316
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3.4. The Column Density Distribution

In absorption line studies, the column density distribution function (CDDF), i.e., the number of absorbers in a given column density bin occupies a similarly, central place (and provides similarly vague information) as the luminosity function in the study of galaxies. The observational determination of the CDDF relies on a patchwork of techniques, owing to the large dynamic range of observable HI absorption. The CDDF can be measured relatively unambigously from Voigt profile fitting for column densities between N(HI) greater than a few times 1012 cm-2, the detection limit for typical Keck spectra, to about a few times 1014cm-2, where the linear part of the curve of growth ends. The weakest lines (N(HI) ltapprox 1013 cm-2) are so numerous that they begin to overlap, requiring application of confusion corrections (Hu et al. 1995). Above ~ 1014 cm-2, once a line is saturated, there is a certain degeneracy between a small change in apparent line width and a large change in column density. Then the column densities are relatively difficult to measure exactly by any means. The situation is complicated further by the appearance of noticeable multi-component structure in absorption systems with column densities above 1015 cm-2 (e.g., Cowie et al 1995). Blending among these components due to the large thermal width of the (already saturated) Lyalpha can mimick large Doppler parameter/ high column density lines. Simultaneous fits to the higher order Lyman lines (which are less saturated owing to their lower oscillator strengths) can only help to some degree. Eventually, for systems with column density N(HI) gtapprox 1017 cm-2 the discontinuity at the Lyman limit (LL) can be observed (1017.3 cm-2 corresponds to tauLL ~ 1) giving again a relatively precise measure of the HI column (Lanzetta 1988; Sargent et al 1989). From N gtapprox 1018.5 on, the damping wings of Lyalpha become detectable. The line width is now entirely a measure of the column density and can again be read off by Voigt profile fitting, or, in lower resolution data, directly from the equivalent width (Wolfe et al 1986).

Carswell et al. (1984) found that the number of absorbers calN per unit HI column density interval can be parametrized as

Equation 10     (10)

Tytler (1987a) has drawn attention to the remarkable fact that when results from higher column density surveys are included a single power law with slope beta = 1.5 fits the whole range of observable column densities well. Keck spectra (Hu et al. 1995, Lu et al. 1996, Kirkman & Tytler 1997, Kim et al. 1997) appear to show that the the power law extends over ten orders of magnitude in column density from 1012 to 1022 cm-2 (if the confusion correction made at the low column density end is justified). A customary working definition of the CDDF (Tytler 1987a) is

Equation 11     (11)

which gives the total number calN of absorbers in HI column density bin [N, N + DeltaN], found over the total surveyed redshift distance sumi DeltaiX, where X has been defined earlier.

One of the more recent measurements (Hu et al 1995) gives

Equation 12     (12)

Various authors (Bechtold 1987; Carswell et al 1987, Petitjean et al. 1993; Meiksin & Madau 1993; Giallongo et al 1993) have presented evidence for departures from a single power law which seem to be borne out by the new Keck data. A steepening for log N gtapprox 14 explains why individual high resolution spectra tend to yield beta ~ 1.7 - 1.8 for the regions (log N ~ 13-15) for which they are most sensitive (e.g. Carswell et al. 1984, 1987; Atwood et al 1985; Rauch et al 1992). In the high column density range (beyond log N ~ 16) f (N) flattens and damped systems are more abundant than they should be judging from an extrapolation of the lower column density power law. Weak evolution of f (N) may occur in the sense that this turnover moves slightly with redshift (Carswell et al 1987, Kim et al. 1997) but the evidence is currently not overwhelming, given that the dip in f (N) occurs in a column density range, where the determination of N(HI) is least certain (see above).

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