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We revisit the proximity effect produced by QSOs at redshifts 2.1-3.3 applying the FLO approach (Saitta et al. 2008) to a sample of ~6300 Ly-alpha lines fitted in 21 high resolution, high signal-to-noise spectra. This new technique allows to recover the hydrogen density field from the HI column densities of the lines in the Ly-alpha forest, on the basis of simple assumptions on the physical state of the gas. To minimize the systematic uncertainties that could affect the density recovering in the QSO vicinity, we carefully determined the redshifts of the QSOs in our sample and modelled in detail their spectra to compute the corresponding ionising fluxes. The mean density field obtained from the observed spectra shows a significant over-density in the region within 4 proper Mpc from the QSO position, confirming that QSOs are hosted in high density peaks. The absolute value of rho/<rho> for the peak is uncertain by a factor of ~3, depending on the assumed QSO spectral slope and the minimum HI column density detectable in the spectra. We do not confirm the presence of a significant over-density extending to separations of ~15 proper Mpc from the QSO, claimed in previous works at redshifts <z>=2.5 and 3.8. Our best guess for the UV background ionisation rate based on the IGM mean density recovered by FLO is Gamma_UVB ~ 10^{-12} s^{-1}. However, values of Gamma_UVB ~ 3x10^{-12} s^{-1} could be viable if an inverted temperature-density relation with index alpha=-0.5 is adopted.
[Abridged] We present FLO (From Lines to Over-densities), a new technique to reconstruct the hydrogen density field for the Lya forest lines observed in high resolution QSO spectra. The method is based on the hypothesis that the Lya lines arise in th e low to intermediate density intergalactic gas and that the Jeans length is the typical size of the Lya absorbers. The reliability of FLO is tested against mock spectra obtained from cosmological simulations. The recovering algorithm gives satisfactory results in the range from the mean density to over-densities of ~30 and reproduces correctly the correlation function of the density field and the 1D power spectrum on scales between ~20 and 60 comoving Mpc. A sample of Lya forests from 22 high resolution QSO spectra is analysed, covering the redshift range 1.7<z<3.5. For each line of sight, we fit Voigt profiles to the lines of the Lya forest, providing the largest, homogeneous sample of fitted Lya lines ever studied. The line number density evolution with redshift follows a power-law relation: dn/dz=(166 +/- 4) [(1+z)/3.5]^{(2.8 +/- 0.2)} (1 sigma errors). The two-point correlation function of lines shows a signal up to separations of ~2 comoving Mpc; weak lines (log N(HI)<13.8) also show a significant clustering but on smaller scales (r<1.5 comoving Mpc). We estimate with FLO the hydrogen density field toward the 22 observed lines of sight. The redshift distribution of the average densities computed for each QSO is consistent with the cosmic mean hydrogen density in the analysed redshift range. The two-point correlation function and the 1D power spectrum of the delta field are estimated. The correlation function shows clustering signal up to ~4 comoving Mpc.
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