5 Conclusions

Using the new, large dataset from high S/N, high resolution VLT/UVES data combined with one Keck/HIRES QSO in the literature, the minimum cutoff Doppler parameter as a function of $N_{\mbox{H~{\sc i}}}$, $b_{\rm c}(N_{\mbox{H~{\sc i}}})$, of the Ly$\alpha$ forest has been derived at $<z> \,=$ 2.1, 3.3 and 3.8. We have found:

1.

When incompleteness is accounted for, the derived $(\Gamma-1)$ is consistent with no-z evolution with a suggestion of lower $(\Gamma-1)$at $z \sim 3.1$, while $b_{\rm c}(13.6)$ increases as z decreases. These results suggest that $(\gamma-1)$ shows no z-evolution within uncertainties and that T₀ decreases as z decreases, assuming the QSO-dominated UV background from Haardt & Madau (1996).

2.

The $(\Gamma-1)$ and $b_{\rm c}(13.6)$ values show a large fluctuation when derived using a subsample even at the similar redshifts. The fluctuation is larger at z > 3 than at z < 3. Although smaller number of lines in the fit could result in this fluctuation, we use a similar number of lines to derive $(\Gamma-1)$ and $b_{\rm c}(13.6)$ for each subsample. This result suggests that there might be a large fluctuation in the IGM temperature along different sightlines even at similar z.

3.

There is a suggestion of a flatter $(\Gamma-1)$ and higher $b_{\rm c}(13.6)$along each line of sight as z decreases at z > 3 more than 3$\sigma$, probably due to the He II reionization. At $z \sim 2.1$, there is no such a significant trend along each line of sight. This result occurs due to the lack of lower- $N_{\mbox{H~{\sc i}}}$and smaller-b lines in the lower-z part of the spectra at z > 3. Our result implies that the impact of the He II reionization on the Ly$\alpha$ forest might be mainly on the lower- $N_{\mbox{H~{\sc i}}}$ forest and that its significance might be smaller than previously suggested.

4.

The lack of strong discontinuity of $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ at $N_{\mbox{H~{\sc i}}} = 10^{14-17}~{\rm cm}^{-2}$ at $z \sim 3$ suggests that $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ might not be a good observational tool to probe the He II reionization and that the UV background might be strongly contributed by local, high-z galaxies at z < 3.

Acknowledgements

We are indebted to M. Dessauges-Zavadsky for support during the observations of Q0055-269. We thank Martin Haehnelt, Joop Schaye, Tom Theuns and Saleem Zaroubi for insightful discussions. TSK thanks Dave Jewitt, Bob Carswell and Glenn Morris for their careful reading of the manuscript. This work has been conducted with partial support by the Research Training Network "The Physics of the Intergalactic Medium'' set up by the European Community under the contract HPRN-CT2000-00126 RG29185 and by ASI through contract ARS-98-226.