We use an Eulerian hydrodynamic cosmological simulation to model the Ly alpha forest in a spatially flat, COBE-normalized, cold dark matter model with Omega = 0.4. We find that the intergalactic, photoionized gas is predicted to collapse into sheetlike and filamentary structures which give rise to absorption lines having characteristics similar to the observed Ly alpha forest. A typical filament is similar to 500 h(-1) kpe long with thickness similar to 50 h(-1) kpc (in proper units), and baryonic mass similar to 10(10) h(-1) M(.). In comparison our cell size is (2.5, 9) h(-1) kpc in the two simulations we perform, with true resolution perhaps a factor of 2.5 worse than this. The gas temperature is in the range 10(4)-10(5) K, and it increases with time as structures with larger velocities collapse gravitationally. We show that the predicted distributions of column densities, b-parameters, and equivalent widths of the Ly alpha forest clouds agree reasonably with observations, and that their evolution is consistent with the observed evolution, if the ionizing background has an approximately constant intensity between z = 2 and z = 4. A new method of identifying lines as contiguous regions in the spectrum below a fixed flux threshold is suggested to analyze the absorption lines, given that the Ly alpha spectra arise from a continuous density held of neutral hydrogen rather than discrete clouds. We also predict the distribution of transmitted flux and its correlation along a spectrum and on parallel spectra, and the He II flux decrement as a function of redshift. We predict a correlation length of similar to 80 h(-1) kpc perpendicular to the line of sight for features in the Ly alpha forest. In order to reproduce the observed number of lines and average flux transmission, the baryon content of the clouds may need to be significantly higher than in previous models because of the low densities and large volume-filling factors we predict. If the background intensity J(Hl) is at least that predicted from the observed quasars, Omega(b) needs to be as high as similar to 0.25 h(-2). The model also predicts that most of the baryons at z > 2 are in Ly alpha clouds, and that the rate at which the baryons move to more overdense regions is slow. A large fraction of the baryons which are not observed at present in galaxies might be intergalactic gas in the currently collapsing structures, with T similar to 10(5)-10(6) K. All our results on the statistical properties of the simulated spectra are predictions that can be directly tested by applying the same methods to observed spectra. We are making the simulated spectra electronically available.
