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We outline the physical picture of the high-redshift Ly-alpha forest that has emerged from cosmological simulations, describe statistical characteristics of the forest that can be used to test theories of structure formation, present a preliminary comparison between simulation results and measurements from Keck HIRES spectra, and discuss a recent determination of the slope and amplitude of the linear mass power spectrum P(k) at z=2.5 from moderate resolution spectra. The physical picture is simple if each QSO spectrum is viewed as a continuous non-linear map of the line-of-sight density field rather than a collection of discrete absorption lines. The distribution of flux decrements depends mainly on the amplitude and PDF (Gaussian vs. non-Gaussian) of the primordial density fluctuations. The threshold crossing frequency, analogous to the 3-d genus curve, responds to the shape and amplitude of P(k) and to the values of Omega and Lambda. Open and Lambda-CDM models agree well with the measured flux decrement distribution at smoothing lengths of 25 km/s and 100 km/s and with the threshold crossing frequency at 100 km/s. Discrepancy with the observed threshold crossing frequency at 25 km/s may reflect the combined effects of noise in the data and limited mass resolution of the simulations. The slope of the measured P(k) agrees with the predictions of inflation+CDM models. Combining the amplitude with COBE normalization imposes a constraint on these models of the form Omega h^x n^y Omega_b^z = constant. Assuming Gaussian primordial fluctuations and a power spectrum shape parameter Gamma~0.2, consistency of the measured P(k) with the observed cluster mass function at z=0 requires Omega=0.46^{+0.12}_{-0.10} for an open universe and Omega=0.34^{+0.13}_{-0.09} for a flat universe (1-sigma errors). (Shortened)


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