Properties of High-Redshift Lyman Alpha Clouds II. Statistical Properties of the Clouds

Curve of growth analysis, applied to the Lyman series absorption ratios deduced in our previous paper, yields a measurement of the logarithmic slope of distribution of \Lya\ clouds in column density $N$. The observed exponential distribution of the clouds' equivalent widths $W$ is then shown to require a broad distribution of velocity parameters $b$, extending up to 80 km s$^{-1}$. We show how the exponential itself emerges in a natural way. An absolute normalization for the differential distribution of cloud numbers in $z$, $N$, and $b$ is obtained. By detailed analysis of absorption fluctuations along the line of sight we are able to put upper limits on the cloud-cloud correlation function $\xi$ on several megaparsec length scales. We show that observed $b$ values, if thermal, are incompatible, in several different ways, with the hypothesis of equilibrium heating and ionization by a background UV flux. Either a significant component of $b$ is due to bulk motion (which we argue against on several grounds), or else the clouds are out of equilibrium, and hotter than is implied by their ionization state, a situation which could be indicative of recent adiabatic collapse.