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 distribution 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 fluctuations 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 Ω = 1, h = 0.5 and Γ = 0.3, and the other is a low density flat model with a cosmological constant (LCDM), with Ω = 0.4, Ω Λ = 0.6 and h = 0.65. 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 Lyα forest absorption seen in the spectra of high z quasars. The calculations also successfully predict the observed properties of the Lyα 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. At z = 3, typical clouds with measured neutral hydrogen column densities N HI = 10 15.3 , 10 13.5 , and 10 11.5 cm −2 correspond to fluctuations with mean total densities approximately 10, 1, and 0.1 times the universal mean baryon density. Perhaps surprisingly, fluctuations whose amplitudes are less than or equal to the mean density still appear as "clouds" because in our model more than 70% of the volume of the IGM at z = 3 is filled with gas at densities below the mean value.We find that the column density distribution of Lyα forest lines can be fit to f (N HI ) ∝ N −β HI , with β = 1.46 in the range 12.5 < log N HI < 14.5, matching recent -2 -Keck results. At somewhat higher column densities the distribution steepens, giving β = 1.80 over the range 14.0 < log N HI < 15.5, and matching earlier observations for these stronger lines. The normalization of the line numbers in our model also agrees with observations if the total baryon density is Ω b = 0.015h −2 and the ionizing background intensity is J 21 = 0.18. Alternatively, if J 21 = 0.5 as recently estimated for the background due to observed quasars at z = 2.5, then Ω b = 0.025h −2 yields the observed number of Lyα lines and the observed mean opacity. The model predicts that about 80% of the baryons in the universe are associate...

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