The b Distribution and the Velocity Structure of Absorption Peaks in the Lyman-Alpha Forest

A theory is developed which relates the observed b-parameter of a Lyman-alpha absorption line to the velocity-curvature of the corresponding peak in the optical depth fluctuation. Its relation to the traditional interpretation of b as the thermal broadening width is discussed. It is demonstrated that, independent of the details of the cosmological model, the differential b distribution has a high b asymptote of $dN/db \propto b^{-m}$, where $m \geq 5$, when we make the reasonable assumption that low-curvature fluctuations are statistically favored over high-curvature ones. There in general always exist absorption lines much broader than the thermal width. We then develop a linear perturbative analysis of the optical depth fluctuation, which yields a single-parameter prediction for the full b distribution: in addition to exhibiting the general high velocity tail, it explains the observed sharp low b cut-off. The dependence of the b distribution on cosmological parameters, such as $\Omega$ and the power spectrum, and reionization history as well as observation/simulation resolution is derived and discussed.