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We have performed a detailed statistical study of the evolution of structurein a photoionized intergalactic medium (IGM) using analytical simulations toextend the calculation into the mildly non-linear density regime found toprevail at z = 3. Our work is based on a simple fundamental conjecture: thatthe probability distribution function of the density of baryonic diffuse matterin the universe is described by a lognormal (LN) random field. The LN field hasseveral attractive features and follows plausibly from the assumption ofinitial linear Gaussian density and velocity fluctuations at arbitrarily earlytimes. Starting with a suitably normalized power spectrum of primordial fluc-tuations in a universe dominated by cold dark matter (CDM), we compute thebehavior of the baryonic matter, which moves slowly toward minima in the darkmatter potential on scales larger than the Jeans length. We have computed twomodels that succeed in matching observations. One is a non-standard CDM modelwith Omega=1, h=0.5 and \Gamma=0.3, and the other is a low density flat modelwith 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 withtime, reaching unity at about z=4, where the simulation yields spectra thatclosely resemble the Ly-alpha forest absorption seen in the spectra of high zquasars. The calculations also successfully predict the observed properties ofthe 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 thisredshift range. However, in our model the forest is not due to discrete clouds,but rather to fluctuations in a continuous intergalactic medium. (This is anabreviated abstract; the complete abstract is included with the manuscript.)Comment: Wrong Fig. 10 is corrected. Our custom made postscript is available  at ftp://hut4.pha.jhu.edu/incoming/igm, or contact Arthur Davidsen  (afd@pha.jhu.edu) for nice hardcopies; accepted for publication in Ap

Evolution of Structure in the Intergalactic Medium and the Nature of the Lyα Forest