2.1. Distribution of Column Densities and Evolution

The bivariate distribution f(N_HI, z) of H I column densities and redshifts is defined by the probability dP that a line-of-sight intersects a cloud with column density N_HI in the range dN_HI, at redshift z in the range dz,

(15)

As a function of column, a single power-law with slope -1.5 appears to provide at high redshift a surprisingly good description over 9 decades in N_HI, i.e. from 10¹² to 10²¹ cm^-2. It is a reasonable approximation to use for the distribution of absorbers along the line-of-sight:

(16)

Ly forest clouds and Lyman-limit systems appear to evolve at slightly different rates, with = 1.5 ± 0.4 for the LLS and = 2.8 ± 0.7 for the forest lines. Let us assume, for simplicity, a single redshift exponent, = 2, for the entire range in column densities. In the power-law model (16) the number N of absorbers with columns greater than N_HI per unit increment of redshift is

(17)

A normalization value of A = 4.0 x 10⁷ produces then ~ 3 LLS per unit redshift at z = 3, and, at the same epoch, ~ 150 forest lines above N_HI = 10^13.8 cm^-2, in reasonable agreement with the observations.

If absorbers at a given surface density are conserved, with fixed comoving space number density n = n₀ (1 + z)³ and geometric cross-section , then the intersection probability per unit redshift interval is

(18)

If the Universe is cosmologically flat, the expansion rate at early epochs is close to the Einstein-de Sitter limit, and the redshift distribution for conserved clouds is predicted to be

(19)

The rate of increase of f(N_HI, z) with z in both the Ly forest and LLS is considerably faster than this, indicating rapid evolution. The mean proper distance between absorbers along the line-of-sight with columns greater than N_HI is

(20)

For clouds with N_HI > 10¹⁴ cm^-2, this amounts to L ~ 0.7 h^-1 _M^-1/2 Mpc at z = 3. At the same epoch, the mean proper distance between LLS is L ~ 30 h^-1 _M^-1/2 Mpc.