The b distribution and the velocity structure of absorption peaks in the Ly alpha forest

A theory is developed that relates the observed b parameter of a Ly alpha a bsorption line to the velocity curvature of the corresponding peak in the o ptical depth fluctuation. Its relation to the traditional interpretation of b as the thermal broadening width is discussed. It is demonstrated that, i ndependent of the details of the cosmological model, the differential b dis tribution has a high-b asymptote of dN/db proportional to b(-m), where m gr eater than or equal to 5, when we make the reasonable assumption that low-c urvature fluctuations are statistically favored over high-curvature ones. T here in general always exist lines much broader than the thermal width. We develop a linear perturbative analysis of the optical depth fluctuation, wh ich yields a single-parameter prediction for the full b distribution. In ad dition to exhibiting the high-velocity tail, it qualitatively explains the observed sharp low-b cutoff-a simple reflection of the fact that high-curva ture fluctuations are relatively rare. Although the existence of the high-b asymptote, which is independent of the validity of the linear expansion, i s consistent with the observed b distribution, a detailed comparison of the linear prediction with six observational data sets indicates that higher o rder corrections are not negligible. The perturbative analysis nonetheless offers valuable insights into the dependence of the b distribution on cosmo logical parameters such as Omega and the power spectrum. A key parameter is the effective smoothing scale of the optical depth fluctuation, which is i n turn determined by three scales: the thermal broadening width, the baryon smoothing scale (approximately the Jeans scale), and the observation/simul ation resolution. The first two are determined by reionization history, but are comparable in general, whereas the third varies by about an order of m agnitude in current hydrodynamic simulations. Studies with non-resolution-d ominated b distributions can be used to probe the reionization history of t he universe.