  We have performed a detailed statistical study of the evolution of structure
in a photoionized intergalactic medium (IGM) using analytical simulations to
extend the calculation into the mildly non-linear density regime found to
prevail at z = 3. Our work is based on a simple fundamental conjecture: that
the probability distribution function of the density of baryonic diffuse matter
in the universe is described by a lognormal (LN) random field. The LN field has
several attractive features and follows plausibly from the assumption of
initial linear Gaussian density and velocity fluctuations at arbitrarily early
times. Starting with a suitably normalized power spectrum of primordial fluc-
tuations in a universe dominated by cold dark matter (CDM), we compute the
behavior of the baryonic matter, which moves slowly toward minima in the dark
matter potential on scales larger than the Jeans length. We have computed two
models that succeed in matching observations. One is a non-standard CDM model
with Omega=1, h=0.5 and \Gamma=0.3, and the other is a low density flat model
with a cosmological constant(LCDM), with Omega=0.4, Omega_Lambda=0.6 and h=.65.
In both models, the variance of the density distribution function grows with
time, reaching unity at about z=4, where the simulation yields spectra that
closely resemble the Ly-alpha forest absorption seen in the spectra of high z
quasars. The calculations also successfully predict the observed properties of
the Ly-alpha forest clouds and their evolution from z=4 down to at least z=2,
assuming a constant intensity for the metagalactic UV background over this
redshift range. However, in our model the forest is not due to discrete clouds,
but rather to fluctuations in a continuous intergalactic medium. (This is an
abreviated abstract; the complete abstract is included with the manuscript.)
