Annu. Rev. Astron. Astrophys. 1998. 36: 267-316
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2.4. High Resolution Spectroscopy: Voigt Profile Decomposition

If the Lyalpha forest is seen as an assemblage of redshifted lines the standard arsenal of notions and techniques from stellar spectroscopy becomes applicable. For lower resolution data, the equivalent width provides a combined measure of line width and strength. In high resolution spectra (FWHM < 25 kms-1) where the typical Lyalpha line is resolved, the line shapes are found to be reasonably well approximated by Voigt profiles (Carswell et al 1984). Then line width (Doppler parameter b (= sqrt2 sigma)), column density N(HI), and redshift z of an absorption line are the basic observables. The statistics of the Lyalpha forest from high resolution studies largely have been cast in terms of the distribution functions of these three quantities and their correlations. The main advantage of the high resolution approach is the opportunity of determining the shape of these distribution functions without parametric prejudices, by directly counting lines with parameters in a certain range.

The standard approach to Voigt profile fitting (Webb 1987; Carswell et al. 1987) relies on chi2 minimization to achieve a complete decomposition of the spectrum into as many independent Voigt profile components as necessary to make the chi2 probability consistent with random fluctuations. For stronger Lyalpha lines the higher order Lyman lines can provide additional constraints when fitted simultaneously. The absorption lines are measured against a QSO continuum estimated locally from polynomial fits to spectral regions deemed free of absorption. A local high order continuum fit (as compared to a global extrapolation with a physical model for the QSO continuum) is necessary because the spectra are patched together from many individual echelle orders with strong variations in sensitivity. These variations do not divide out completely when dividing by the flux of a standard star because the light going through a slit narrow enough to ensure slit-width limited resolution varies with the seeing conditions and with the position of the object on the slit. When applying a local fit to the continuum the zeroth order contribution tends to be underestimated, i.e., the continuum is drawn too low, which is the main drawback of this method.

Given sufficient spectral resolution, and assuming that Lyalpha clouds are discrete entities (in the sense of some of the models to be discussed below) the profile fitting approach is the most physically meaningful way of extracting information from the Lyalpha forest. If the absorber is a gas cloud with a purely Gaussian velocity dispersion (a thermal Maxwell-Boltzmann distribution, plus any Gaussian contributions from turbulence) a Voigt profile provides an exact description of the absorption line shape. The Doppler parameter can then be written as the quadratic sum of its individual contributions:

Equation 5     (5)

Unfortunately, in most more realistic models of the absorbing gas finite velocity and density gradients invalidate the assumptions underlying Voigt profile fitting, and the line parameters may have less immediate physical meaning. Departures of the absorption line shape from a Voigt profile may contain valuable information about the underlying nature of the absorption systems, and different scenarios may have quite different observational signatures. Rotational motion (Weisheit 1978; Prochaska & Wolfe 1997), gravitational collapse (McGill 1990; Meiksin 1994; Rauch 1996) and galactic outflows (Fransson & Epstein 1982; Wang 1995) have been discussed in terms of the likely absorption line shapes they produce. As yet, the quantitative application of these results has proven difficult, because of the lack of realistic prototypical models for the actual line formation, the rather subtle departures from Voigt profiles expected, and the wide variety of profiles actually encountered.

Non-Voigt profiles can still be fitted as blends of several Voigt profiles, but the information about the non-thermal motion is encoded in the spatial correlations between the individual profiles (Rauch 1996). Also, there is no guarantee that the number of components necessary for a good fit converges with increasing signal-to-noise ratio. Clearly, for more general line formation models, global techniques of extraction the velocity information may be more appropriate.

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