Using high resolution ( R \sim 45 000 ) , high S/N ( \sim 20–50 ) VLT/UVES data , we have analyzed the Ly \alpha forest of 3 QSOs in the neutral hydrogen ( H i ) column density range N _ { H \textsc { i } } = 10 ^ { 12.5 - 16 } \mathrm { cm } ^ { -2 } at 1.5 < z < 2.4 .
We combined our results with similar high-resolution , high S/N data in the literature at z > 2.4 to study the redshift evolution of the Ly \alpha forest at 1.5 < z < 4 .
We have applied two types of analysis : the traditional Voigt profile fitting and statistics on the transmitted flux .
The results from both analyses are in good agreement :

1 . The differential column density distribution function , f ( N _ { H \textsc { i } } ) , of the Ly \alpha forest shows little evolution in the column density range N _ { H \textsc { i } } = 10 ^ { 12.5 - 14 } \mathrm { cm } ^ { -2 } , f ( N _ { H \textsc { i } } ) \propto N _ { H \textsc { i } } ^ { - \beta } , with \beta \sim 1.4 –1.5 at 1.5 < z < 4 and with a possible increase of \beta to \beta \sim 1.7 at z < 1.8 .
A flattening of the power law slope at lower column densities at higher z can be attributed to more severe line blending .
A deficiency of lines with N _ { H \textsc { i } } > 10 ^ { 14 } \mathrm { cm } ^ { -2 } is more noticeable at lower z than at higher z .
The one-point function and the two-point function of the flux confirm that strong lines do evolve faster than weak lines .

2 . The line number density per unit redshift , dn / dz , at N _ { H \textsc { i } } = 10 ^ { 13.64 - 16 } \mathrm { cm } ^ { -2 } is well fitted by a single power law , dn / dz \propto ( 1 + z ) ^ { 2.19 \pm 0.27 } , at 1.5 < z < 4 .
In combination with the HST results from the HST QSO absorption line key project , the present data indicate that a flattening in the number density evolution occurs at z \sim 1.2 .
The line counts as a function of the filling factor at the transmitted flux F in the range 0 < F < 0.9 are constant in the interval 1.5 < z < 4 .
This suggests that the Hubble expansion is the main drive governing the forest evolution at z > 1.5 and that the metagalactic UV background changes more slowly than a QSO-dominated background at z < 2 .

3 . The observed cutoff Doppler parameter at the fixed column density N _ { H \textsc { i } } = 10 ^ { 13.5 } \mathrm { cm } ^ { -2 } , b _ { c, \mathrm { 13.5 } } , shows a weak increase with decreasing z , with a possible local b _ { c, \mathrm { 13.5 } } maximum at z \sim 2.9 .

4 . The two-point velocity correlation function and the step optical depth correlation function show that the clustering strength increases as z decreases .

5 . The evolution of the mean H i opacity , \overline { \tau } _ { H \textsc { i } } , is well approximated by an empirical power law , \overline { \tau } _ { H \textsc { i } } \propto ( 1 + z ) ^ { 3.34 \pm 0.17 } , at 1.5 < z < 4 .

6 . The baryon density , \Omega _ { \mathrm { b } } , derived both from the mean H i opacity and from the one-point function of the flux is consistent with the hypothesis that most baryons ( over 90 % ) reside in the forest at 1.5 < z < 4 , with little change in the contribution to the density , \Omega , as a function of z .