A first step towards a direct inversion of the Lyman forest in QSO   spectra
  A method for the recovery of the real space line-of-sight mass density field from Lyman absorption in QSO spectra is presented. The method makes use of a Lucy-type algorithm for the recovery of the HI density. The matter density is inferred from the HI density assuming that the absorption is due to a photoionized intergalactic medium which traces the mass distribution as suggested by recent numerical simulations. Redshift distortions are corrected iteratively from a simultaneous estimate of the peculiar velocity. The method is tested with mock spectra obtained from N-body simulations. The density field is recovered reasonably well up to densities where the absorption features become strongly saturated. The method is an excellent tool to study the density probability distribution and clustering properties of the mass density in the (mildly) non-linear regime. Combined with redshift surveys along QSO sightlines the method will make it possible to relate the clustering of high-redshift galaxies to the clustering of the underlying mass density. We further show that accurate estimates for \Omega_{bar}h^2)^2 J^{-1} H(z)^{-1} and higher order moments of the density probability function can be obtained despite the missing high density tail of the density distribution if a parametric form for the probability distribution of the mass density is assumed. 
Adi Nusser and Martin Haehnelt