**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*,

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 10^{12} to
10^{21} cm^{-2}.
It is a reasonable approximation to use for the distribution
of absorbers along the line-of-sight:

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

A normalization value of *A* = 4.0 x 10^{7} 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* = *n*_{0} (1 +
*z*)^{3} and geometric cross-section
, then
the intersection probability per unit redshift interval is

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

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

For clouds with *N*_{HI} > 10^{14} 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.