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 root-mean-square 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 Lyman-ar forest at high redshift. With such an approach, rather than concentrating on simulating specific cosmological models, we may be in a 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.