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2.1. Distribution of Column Densities and Evolution

The bivariate distribution f(NHI, 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 NHI in the range dNHI, at redshift z in the range dz,

Equation 15 (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 NHI, 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:

Equation 16 (16)

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

Equation 17 (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 NHI = 1013.8 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 Sigma, then the intersection probability per unit redshift interval is

Equation 18 (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

Equation 19 (19)

The rate of increase of f(NHI, z) with z in both the Lyalpha 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 NHI is

Equation 20 (20)

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

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