3 The physical state of the IGM

3.1 The z-evolution of the Doppler cutoff

To derive $b_{\rm c}(N_{\mbox{H~{\sc i}}})$, we adopted two methods: the iterative power-law fit (Schaye et al. 1999, 2000) and the smoothed b power-law fit (Bryan & Machacek 2000). From these two power-law fit methods, we derived $\log(b_{0,k})$ and $(\Gamma-1)_{\rm k}$ of Eq. (1) for k = i or s, where subscripts "i'' and "s'' indicate the iterative power-law fit and the smoothed b power-law fit, respectively. For the iterative power-law fit, 100 bootstrap realizations were averaged with an exclusion threshold of 0.5 absolute mean deviation (AMD). For the smoothed b power-law fit, a smoothing constant of 3 km s^-1 was used for each subsample having 30 lines and the robust power-law fit was applied.

Schaye et al. (2000) used the iterative power-law fit for $N_{\mbox{H~{\sc i}}}= 10^{12.5-14.5} \, {\rm cm}^{-2}$at 2 < z < 3.7 and for $N_{\mbox{H~{\sc i}}} = 10^{12.5-14.8} \ {\rm cm}^{-2}$ at 3.7 < z < 4.4. Lines with $N_{\mbox{H~{\sc i}}} \le 10^{13} \ {\rm cm}^{-2}$, however, suffers from incompleteness due to line blending. For example, at $z \sim 2.1$, lines with $N_{\mbox{H~{\sc i}}} \le 10^{13} \ {\rm cm}^{-2}$are affected by incompleteness and this threshold increases with z (cf. Hu et al. 1995; Kim et al. 1997, 2001a, 2001b). Incompleteness causes a bias in the measurements of $(\Gamma-1)$. Due to the lack of lines with $N_{\mbox{H~{\sc i}}}$ below this threshold and $b \le 15$ km s^-1 (cf. Kim et al. 2001a), an observationally measured $(\Gamma-1)$ value becomes flatter than its true value. We have defined different lower $N_{\mbox{H~{\sc i}}}$ fitting thresholds at the various redshifts in order to avoid this bias (see Fig. 1) and to obtain a stable estimate of $(\Gamma-1)$ and $b_{\rm c}$ at the fixed column density $N_{\mbox{H~{\sc i}}} = 10^{13.6} \ {\rm cm}^{-2}$, $b_{\rm c}(13.6)$.

Figure 1 shows the $N_{\mbox{H~{\sc i}}}$-b diagram at $<z> \, = 2.1$, 3.3 and 3.8 for Sample A. The fitted parameters are listed in Table 2, as well as the $N_{\mbox{H~{\sc i}}}$ ranges used in the fit. At higher z, several lines show a b value smaller than $b_{\rm c}(N_{\mbox{H~{\sc i}}})$ at $<z> \, = 2.1$ (shaded area). In fact, $b_{\rm c}(13.6)$ from the both power-law fits increases as z decreases. The slopes $(\Gamma -1)_{\rm i}$ at $<z> \, = 2.1$ and 3.8 are similar within errors, while the slope $(\Gamma-1)_{\rm s}$ at $<z> \, = 2.1$ is flatter than the one at $<z>\,=3.8$ more than $3\sigma$. Both slopes shows the lowest value at $<z>\,=3.3$ (see Table 2). Note that, however, the errors from the bootstrap method are likely to be underestimated (cf. Schaye et al. 2000). The close examination of Fig. 1 suggests that $(\Gamma-1)_{\rm s}$at $<z>\,=3.8$ might well be overestimated and that the real $(\Gamma-1)$ might be inbetween $(\Gamma -1)_{\rm i}$and $(\Gamma-1)_{\rm s}$.

We visually compared our $b_{\rm c,i}(13.6)$ with the results of Schaye et al. (2000; their Fig. 1), although their sample corresponds to our Sample B. As shown in Table 2, their $b_{\rm c}(13.6)$ values are larger than our $b_{\rm c}(13.6)$ values at similar z with the differences larger at smaller z. Their $(\Gamma -1)_{\rm i}$ values appear to be flatter than our values at all z, likely caused by their lack of accounting for incompleteness of observed lines.

3.2 Fluctuations of the Doppler cutoff

Figure 2: Slopes $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$as a function of a lower $N_{\mbox{H~{\sc i}}}$ threshold $N_{\rm {\mbox{H~{\sc i}}}, th}$ for Sample A. Open circles, open squares and open diamonds indicate $<z> \, = 2.1$, 3.3 and 3.8, respectively.

Figure 3: Slopes $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$as a function of z. Open symbols (circles, squares and diamonds at $<z> \, = 2.1$, 3.3 and 3.8, respectively), filled symbols (same as in open symbols) and filled triangles represent the individual members of Sample B, Sample A and Sample B averaged at each z, respectively.

Figure 2 shows $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$as a function of the lower $N_{\mbox{H~{\sc i}}}$ threshold $N_{\rm {\mbox{H~{\sc i}}}, th}$ for Sample A. The lack of lines due to incompleteness results in flatter $(\Gamma -1)_{\rm i}$ at lower $N_{\rm {\mbox{H~{\sc i}}}, th}$. When the lower $N_{\mbox{H~{\sc i}}}$ threshold increases so that incompleteness does not affect the $N_{\mbox{H~{\sc i}}}$-b distribution severely, $(\Gamma -1)_{\rm i}$ becomes stabilized (the lower $N_{\rm {\mbox{H~{\sc i}}}, th} \sim 10^{13} \ {\rm cm}^{-2}$ at $<z> \ = 2.1$ and 3.3, and the lower $N_{\rm {\mbox{H~{\sc i}}}, th} \sim 10^{13.3} \ {\rm cm}^{-2}$ at $<z> \ = 3.8$). If the available lines in the fit, however, become too small for higher $N_{\rm {\mbox{H~{\sc i}}}, th}$, $(\Gamma -1)_{\rm i}$becomes rather ill-defined. On the other hand, the lower $N_{\rm {\mbox{H~{\sc i}}}, th}$does not affect the $b_{\rm c,i}(13.6)$ values since $b_{\rm c,i}$ at $N_{\mbox{H~{\sc i}}} \sim 10^{13.6} \ {\rm cm}^{-2}$behaves more like a pivotal point. The slopes $(\Gamma-1)_{\rm s}$and $b_{\rm c,s}(13.6)$ from the smoothed b power-law fit also show similar behaviors.

Figure 3 shows $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$ as a function of z for the individual members of Sample B (open symbols), the averaged values from Sample B (filled triangles) and Sample A (filled symbols). The members of Sample B show the large fluctuation of $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$even at the similar z (see Table 2). With more lines available to sharpen the lower cutoff envelope, $(\Gamma -1)_{\rm i}$ becomes steeper than the ones derived from the individual members using a smaller number of lines, possibly approaching asymptotic values (Sample A: filled circles, filled squares and filled diamonds). For both Sample A and Sample B averaged (filled triangles), $(\Gamma -1)_{\rm i}$ shows the lowest value at $<z> \ = 3.3$, although uncertainties for Sample B averaged are rather large at $<z> \ = 3.3$ and 3.8. While the 1$\sigma$ is 0.037 at $<z> \ = 2.1$, it becomes twice as large as that at $<z> \ = 3.3$ and 3.8. This larger fluctuation at z > 3 is caused in part by a smaller data size and in part by the cosmic variance. The fluctuation at $<z> \ = 2.1$ is mostly caused by different sightlines. A single, long sightline does not show any significant difference between the lower-zpart and the higher-z part, such as HE1122-1648 and HE2217-2818. On the other hand, at $<z> \ = 3.3$ and 3.8, even a single, long sightline shows a difference between the lower-z part and the higher-z part more than 3$\sigma$ (see Sect. 3.3 for more discussions).

Similarly, $b_{\rm c,i}(13.6)$ derived from a larger number of lines is smaller, possibly approaching asymptotic values (Sample A: filled circles, filled squares and filled diamonds). Both the $b_{\rm c,i}(13.6)$ values from Sample A and Sample B averaged (filled triangles) increase as z decreases, although the uncertainties for Sample B averaged are rather large. Note again that the larger fluctuation of $b_{\rm c,i}(13.6)$ at $<z> \ = 3.3$ and 3.8 (keep in mind the smaller sample sizes at z > 3 at the same time). The smoothed b power-law fit also produces the large fluctuations of $(\Gamma-1)_{\rm s}$ and $b_{\rm c,s}(13.6)$ for the individual members of Sample B (see Table 2).

Our $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$ for Sample A are not consistent with the Schaye et al. values by more than $3\sigma$. Our $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$for Sample B, however, are completely in agreement with theirs. This result shows that deriving $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$ depends largely on the number of lines in the fit and is subject to the cosmic variance. Experiments with the fitted line parameters show that in general more than 200 lines in the fit stabilizes the results (cf. Schaye et al. 1999). The large fluctuation found from sightline to sightline in similar redshift ranges using the number of lines smaller than 200 could explain in part the previous contradictory conclusions on the evolution of $(\Gamma-1)$ and $b_{\rm c}(13.6)$, which were usually based on a small number of lines and of sightlines.

There are several known voids (or regions devoid of absorption lines with $N_{\mbox{H~{\sc i}}} \ge 10^{13.5} \ {\rm cm}^{-2}$) in the present data. One void ( $z \sim 3.2$) towards Q0302-003, two ( $z \sim 3.1$ and 3.3) towards Q0055-269 and three ( $z \sim 1.9$, 2.2 and 2.3) towards HE2217-2818 have been identified. The regions containing voids of Q0302-003 and Q0055-269 show a flatter $(\Gamma -1)_{\rm i}$ by more than 7$\sigma$ (two lower squares at $z \sim 3.1$in the upper panel of Fig. 3) and a higher $b_{\rm c,i}(13.6)$ by more than $3\sigma$ (two higher squares at $z \sim 3.1$ in the lower panel of Fig. 3) compared to the ones derived from Q0055-269 without the voids. The forest of HE2217-2818, however, does not show any significant difference between the regions with the voids and without the voids. Similarly, the HE1122-1648 forest without any recognizable voids does not show any significant difference between the lower-z part and the higher-z part of the spectrum.

Even though voids are produced by the enhanced radiations from local sources, the increase of the H I photoionizing photons in voids does not increase significantly the temperature of the forest as whole (Haehnelt & Steinmetz 1998). A fixed $N_{\mbox{H~{\sc i}}}$, however, corresponds to a higher overdensity due to the increased photoionization. From the equation of state, a higher overdensity means a higher temperature. Therefore, if voids are produced by enhanced ionizations, there should be an increase of $b_{\rm c,i}(13.6)$and a decrease of $(\Gamma -1)_{\rm i}$ compared to the forest without any extra heating sources at similar z (Haehnelt & Steinmetz 1998; Schaye et al. 1999).

The lack of a significant difference in $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$from the HE2217-2818 forest at $z \sim 2.1$ suggests that the origin of the voids towards HE2217-2818 is different. This could be due to a density fluctuation (Heap et al. 2000) and/or due to a shock heating by galactic infalls or by galactic winds (Theuns et al. 2001a).

Simulations often show that some Ly$\alpha$ lines are broadened by shock heating (Bryan & Machacek 2000; Theuns et al. 2001a). They also show that shock heating is not very important at z > 2 for the lower column density forest considered here. In addition, the cutoff b method is less sensitive than other methods to shock-heated gas (cf. Theuns et al. 2001b).

One of the other main candidates for the fluctuations of $(\Gamma -1)_{\rm i}$ and $b_{\rm c,i}(13.6)$ is the He II reionization at $z \sim 3$(Haehnelt & Steinmetz 1998; Songaila 1998; Schaye et al. 1999).

3.3 The HeII reionization

There have been various claims on a possible He II reionization at $z \sim 3$from observations (Reimers et al. 1997; Songaila 1998; Heap et al. 2000; Kriss et al. 2001). One of the consequences of the He II reionization is an increase of the temperature of the Ly$\alpha$ forest (Haehnelt & Steinmetz 1998). In fact, various theoretical models have predicted a $(\gamma-1) \sim 0$ (thus a flatter $(\Gamma-1)$) and a higher $T_{\rm0}$(thus a higher $b_{\rm c}(13.6)$) when the He III bubbles surrounding the ionizing sources overlap and increase the radiations in the IGM (Haehnelt & Steinmetz 1998; Ricotti et al. 2000; Schaye et al. 2000).

To study the effect on the Ly$\alpha$ forest caused by the He II reionization, we re-grouped the individual members of Sample B. For each z bin, individual members were divided into two subgroups according to their redshift ranges, i.e. a lower-z part and a higher-z part. This method of grouping has been taken since the absorption lines from the same sightline might be correlated. The number of lines at each subgroup is sampled to be similar (see Table 2).

There is no significant difference in $(\Gamma -1)_{\rm i}$and $b_{\rm c,i}(13.6)$ between $z \sim 2.0$and $z \sim 2.1$. There is, however, a significant difference by more than $\sim$3$\sigma$between the lower-z and the higher-z parts at $z \sim 3.3$ and at $z \sim 3.8$. The slope $(\Gamma -1)_{\rm i}$decreases at lower z for each bin at z > 3, while the $b_{\rm c,i}(13.6)$increases. In short, along each line of sight at z > 3, $(\Gamma -1)_{\rm i}$ decreases and $b_{\rm c,i}(13.6)$ increases at z decreases, as expected from the He II reionization at $z \sim 3$. This trend, however, does not continue at z < 3, i.e. the HE1122-1648 forest and the HE2217-2818 forest do not show the similar behavior. Theuns et al. (2001b) find a similar behavior of b values from the wavelet analysis using the same data presented here. This is interpreted as a result of HeII reionization at $z \sim 3.3$. They do find, however, a cold region, i.e. a region with a lower average b values than the adjacent regions along HE2217-2818.

Deriving the cutoff Doppler parameters using a small number of lines introduces a large scatter (Fig. 3). Therefore, a flatter $(\Gamma -1)_{\rm i}$ and a higher $b_{\rm c,i}(13.6)$ shown at the lower-z part of the bins at z > 3 ($z \sim$ 3.1 and 3.6) might be an artifact of the small number of lines in the fit. More available lines in the fit, however, tends to increase $(\Gamma -1)_{\rm i}$ and to decrease $b_{\rm c,i}(13.6)$as seen in Sect. 3.2, contrary to this result.

The $N_{\mbox{H~{\sc i}}}$-b distributions at $z \sim 3.1$and $z \sim 3.6$ show a lack of lines with $N_{\mbox{H~{\sc i}}} \le 10^{13.7} \ {\rm cm}^{-2}$ and $b \le 20$ km s^-1with respect to the ones at $z \sim 3.4$ and 3.9, while the $N_{\mbox{H~{\sc i}}}$-b distributions for higher- $N_{\mbox{H~{\sc i}}}$ lines are similar (the figures not shown). This lack of lower- $N_{\mbox{H~{\sc i}}}$ and lower-b lines results in a flatter $(\Gamma -1)_{\rm i}$and a higher $b_{\rm c,i}(13.6)$ at $z \sim 3.1$ and 3.6. This, however, can not be caused by incompleteness due to line blending. The number of lower- $N_{\mbox{H~{\sc i}}}$ lines are similar at $z \sim 3.1$ and $z \sim 3.4$, and at $z \sim 3.6$ and $z \sim 3.9$, i.e. b values corresponding to lower- $N_{\mbox{H~{\sc i}}}$ lines are higher at $z \sim 3.1$ and $z \sim 3.6$ than those at $z \sim 3.4$ and $z \sim 3.9$. In addition, line blending is more severe at higher z and should have resulted in a lack of lower- $N_{\mbox{H~{\sc i}}}$ and lower-b lines at $z \sim 3.4$ and $z \sim 3.9$ rather than at $z \sim 3.1$ and $z \sim 3.6$ as observed. What is observed at $z \sim 3.1$ and $z \sim 3.6$ shows the opposite behavior expected from line blending. In fact, the lower- $N_{\mbox{H~{\sc i}}}$ and higher-b lines seen at $z \sim 3.1$ and at $z \sim 3.6$are expected from the He II reionization since it has a greater effect on the lower- $N_{\mbox{H~{\sc i}}}$ forest than on the higher- $N_{\mbox{H~{\sc i}}}$ forest (Theuns 2001, private communication).

Songaila (1998) found that there is an abrupt, sharp discontinuity in the ratio of Si IV column density to C IV column density, $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$, at $z \sim 3$, below which $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ is always less than $\sim$0.07. The observed $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ by Songaila (1998) suggests that the UV background is softer than a QSO-dominated background at z > 3 and becomes harder as expected from a QSO-dominated background at z < 3. This observation has been interpreted as the complete He II reionization by $z \sim 3$, i.e. the overlap of the He III bubbles.

Figure 4 shows $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ of the Ly$\alpha$ forest at $N_{\mbox{H~{\sc i}}} = 10^{14-17}~{\rm cm}^{-2}$ as a function of z from the QSOs in Table 1 except from Q0000-263 (Kim et al. 2001, in preparation). We only include the H I systems having higher Lyman lines other than Ly$\alpha$, such as Ly$\beta$, Ly$\gamma$, etc. This selection enables us to estimate an accurate H I column density and to assign Si IV and C IV to a H I line more reliably. Note that there are no data points at 2.6 < z < 2.9 at which the discontinuity of $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$has been reported to be the largest. Although the bulk of the forest shows lower $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ at z < 2.6, there are the forest lines with $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ larger than 0.07. The similar results have been reported at z > 2 by Boksenberg et al. (1998). Bear in mind that no specific H I column density range used in their Fig. 1 is given in their paper. In addition, note that their results are based on the individual, fitted components, while ours are based on the integrated absorption lines. Using the component-by-component analysis introduces an additional scatter in the diagram due to the different velocity structures in Si IV and C IV. The number of components for one integrated system in our data is in general from 1 to 3. The component-by-component analysis on our data also shows the similar trend found in Fig. 4 from the integrated profiles.

Our observations do not suggest any abrupt change in $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ at 1.6 < z < 3.6, i.e. no abrupt change in the softness parameter in general. This result suggests that some forest clouds are exposed to a soft UV background at z < 2.6. The lack of a sharp change in $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$at $z \sim 3$ does not mean, however, that the He II reionization did not occur at $z \sim 3$. Rather, our $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$indicates that $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$might not be a good observational tool to probe the He II reionization. What it probes is a softness parameter of the UV background. Increasing evidences of the contribution to the UV background from local, high-z galaxies would explain our high $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ at z < 2.6 without any difficulty (Giroux & Shull 1997; Shull et al. 1999; Bianchi et al. 2001; Steidel et al. 2001). In addition, the lack of strong z-evolution of $(\Gamma-1)$ found for $N_{\mbox{H~{\sc i}}} = 10^{13.7-14.5} \ {\rm cm}^{-2}$ at z > 3is in agreement with the lack of strong evolution of $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ at $N_{\mbox{H~{\sc i}}} = 10^{14-17}~{\rm cm}^{-2}$. In short, the He II reionization at $z \sim 3$(Reimers et al. 1997; Heap et al. 2000; Smette et al. 2002) shows its impact on the Ly$\alpha$ forest mainly at $N_{\mbox{H~{\sc i}}} \le 10^{13.7}~{\rm cm}^{-2}$and might be very inhomogeneous. Its strength might not be as strong as previously suggested and $N_{\mbox{Si~{\sc iv}}}$/ $N_{\mbox{C~{\sc iv}}}$ might not be a best observational tool to probe the He II reionization.

Figure 4: The ratio of $N_{\mbox{Si~{\sc iv}}} /N_{\mbox{C~{\sc iv}}}$ as a function of z. Only the Ly$\alpha$ forest with $N_{\mbox{H~{\sc i}}} \le 10^{17}~{\rm cm}^{-2}$ is included. Filled circles represent a forest with $N_{\mbox{C~{\sc iv}}} = 10^{12.7 - 14}~{\rm cm}^{-2}$, while open circles represent a forest with $N_{\mbox{C~{\sc iv}}} > 10^{14}~{\rm cm}^{-2}$. Open squares represent upper limits or lower limits.