Many complex multi-component numerical simulations have been performed of the Lyman forest, which include the effects of dark matter (N-body), baryons (hydrodynamics), chemical composition (reactive networks), and microphysical response (radiative cooling and heating). See, for example, [67, 118, 166], which represent some of the earliest comprehensive simulations. For the most part, all these calculations have been able to fit the observations reasonably well, including the column density and Doppler width distributions, the size of absorbers [62], and the line number evolution. Despite the fact that the cosmological models and parameters are different in each case, the simulations give roughly similar results provided that the proper ionization bias is used, , where is the baryonic density parameter, is the Hubble parameter and is the photoionization rate at the hydrogen Lyman edge. (However, see [50] for a discussion of the sensitivity of statistical properties on numerical resolution.) A theoretical paradigm has thus emerged from these calculations in which Ly absorption lines originate from the relatively smaller scale structure in pregalactic or intergalactic gas through the bottom-up hierarchical formation picture in CDM-like Universes. The absorption features originate in structures exhibiting a variety of morphologies commonly found in numerical simulations (see Figure 11), including fluctuations in underdense regions, spheroidal minihalos, and filaments extending over scales of a few Mpc.

UpdateMachacek et al. [115] expanded on earlier work to compare several Ly statistical measures from five different background cosmological models, including standard critical density Cold Dark Matter (CDM), open CDM, flat CDM with a cosmological constant, standard CDM with a tilted density spectrum, and a flat model with mixed hot and cold dark matter. All models were chosen to match local or low redshift observations, and most were also consistent with COBE measurements of the CMBR. The calculations were designed to establish which statistics are sensitive to different cosmological models. In particular, they find that the line number count above a given column density threshold is relatively insensitive to background models. On the other hand, the shape of the optical depth probability distribution function is strongly correlated to the amount of small scale power in density fluctuations, and is thus a good discriminator among models on scales of a few hundred kpc.

Meiksin et al. [117] followed up with more detailed comparisons of Ly systems in several cosmologies
with observed high resolution QSO spectra. Although all models are consistent with previous studies in that
they give reasonably good statistical agreement with observed Ly properties, under closer scrutiny
none of the numerical models they considered passed all the tests, which included spectral flux,
wavelet decomposed amplitude, and absorption line profile distributions. They suggest that
comparisons might be improved, particularly in optically thin systems, by more energy injection
into the IGM from late He^{+} reionization or supernovae-driven winds, or by a larger baryon
fraction.

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