  Analytic derivations of the correlation function and the column density
distribution for neutral hydrogen in the IGM are presented, assuming that the
non-linear baryonic mass density distribution in the IGM is lognormal. This
ansatz was used earlier by Bi & Davidsen (1997) to perform 1D simulations of
lines-of-sight and analyse the properties of absorption systems. Our approach
is completely analytic, which allows us to explore a wide region of the
parameter space for our model. The analytic results have been compared with
observations, whenever possible. Two kinds of correlation functions are
defined: along the line-of-sight (LOS) and across the transverse direction. We
find that the effects on the LOS correlation due to change in cosmology and the
slope of the equation of state of the IGM, \gamma are of the same order, which
means that we cannot constrain both the parameters simultaneously. However, it
is possible to constrain \gamma and its evolution using the observed LOS
correlation function at different epochs, provided one knows the background
cosmology. We suggest that the constraints on the evolution of \gamma obtained
using the LOS correlation can be used as an independent tool to probe the
reionisation history of the universe. From the transverse correlation function,
we find that the excess probability, over random, of finding two neutral
hydrogen overdense regions separated by an angle \theta, is always less than 1
per cent for redshifts greater than 2. Our models also reproduce the observed
column density distribution for neutral hydrogen and the shape of the
distribution depends on \gamma. Our calculations suggest that one can rule out
\gamma > 1.6 for z \simeq 2.31 using the column density distribution. However,
one cannot rule higher values of \gamma at higher redshifts.

Astrophysics