Predictions for the clustering properties of the Lyman-alpha Forest - I.   One-point statistics
  We present predictions for the one-point probability distribution and cumulants of the transmitted QSO flux in the high redshift Lyman-$\alpha$ Forest. We make use of the correlation between the Lyman-$\alpha$ optical depth and the underlying matter density predicted by gravitational instability theory and seen in numerical hydrodynamic simulations. We have modelled the growth of matter fluctuations using the non-linear shear-free dynamics, an approximation which reproduces well the results of perturbation theory for the cumulants in the linear and weakly non-linear clustering regime. As high matter overdensities tend to saturate in spectra, the statistics of the flux distribution are dominated by weakly non-linear overdensities. As a result, our analytic approach can produce accurate predictions, when tested against N-body simulation results, even when the underlying matter field has rms fluctuations larger than unity. Our treatment can be applied to either Gaussian or non-Gaussian initial conditions. Here we concentrate on the former case, but also include a study of a specific non-Gaussian model. We discuss how the methods and predictions we present can be used as a tool to study the generic clustering properties of the \lya forest at high-redshift.   With such an approach, rather than concentrating on simulating specific cosmological models, we may be in the position to directly test our assumptions for the Gaussian nature of the initial conditions, and the gravitational instability origin of structure itself. In a separate paper we present results for two-point statistics. 
Enrique Gaztanaga and Rupert A.C. Croft