4 The Voigt profile analysis of the Ly$\alpha$ forest

In addition to the three QSOs observed with UVES, we have used the published line lists of three QSOs at higher redshift observed with similar resolution and S/N. Table 2 lists all the analyzed QSOs with their properties and the relevant references. We have avoided a region close to a damped Ly$\alpha$ system in the spectrum of Q0000-263 and a lower S/N region in the spectrum of HS 1946+7658. The spectrum of Q0302-003 does not include the region of the known void at $z \sim 3.17$(cf. Dobrzycki & Bechtold 1991). The fitted line parameters with the associated errors of HS 1946+7658 and Q0000-263 were generated by VPFIT with a $\chi^{2}$ threshold of $\chi^2 \le 1.2$ and $\chi^{2} \le 1.1$, respectively. The line list of Q0302-003 was generated by an automatized version of the Voigt profile fitting program by Hu et al. (1995) with $\chi^2 \sim 1$ and the errors associated with the fitted parameters are not published.

The results of profile fitting are known to be sensitive to the data quality as well as to the characteristics of the fitting program. As a consequence, comparing line lists obtained with different criteria is not usually straightforward. Due to the use of a different fitting program, the line list of Q0302-003 at $z \sim 2.9$ should be treated with caution when combined with other line lists. A systematic difference in b and $N_H{\sc i}$ from VPFIT can introduce a slightly different behavior of the Ly$\alpha$ forest at $z \sim 2.9$. While the difference would not change the study of the line number density or the correlation function significantly, it can affect the determination of a lower cutoff b envelope in the $N_H{\sc i}$-b diagrams. Furthermore, the six QSOs in Table 2 cover the Ly$\alpha$ forest at $1.54 < z_{\rm Ly\alpha} < 4.0$ with a fairly regular spacing. There is very little overlap between the Ly$\alpha$ forests of the different QSOs and the effects of cosmic variance in the individual lines of sight might be important.

 

 

Table 2: Analyzed QSOs.

QSO	$z_{{\rm em}}$	$\lambda\lambda$	$z_{\rm Ly\alpha}$	d $X^{{\rm a}}$
HE 0515-4414	1.719	3090-3260	1.54-1.68	0.365
J2233-606 $^{{\rm b}}$	2.238	3400-3850	1.80-2.17	1.104
HE 2217-2818	2.413	3550-4050	1.92-2.33	1.286
HS 1946+7658 $^{{\rm c}}$	3.051	4252-4635	2.50-2.81	1.157
Q0302-003 $^{{\rm d}}$	3.290	4410-5000	2.63-3.11	1.878
Q0000-263 $^{{\rm e}}$	4.127	5450-6100	3.48-4.02	2.540

$^{{\rm a}}$

For $q_{\rm0}=0$.

$^{{\rm b}}$

See also Cristiani & D'Odorico (2000).

$^{{\rm c}}$

Kirkman & Tytler (1997).

$^{{\rm d}}$

Hu et al. (1995).

$^{{\rm e}}$

Lu et al. (1996).

4.1 The differential density distribution function

The differential density distribution function, $f(N_H{\sc i})$, is defined as the number of absorption lines per unit absorption distance path and per unit column density as a function of $N_H{\sc i}$(equivalent to the luminosity function of galaxies). The absorption distance path X(z) is defined as $X(z) \equiv {1\over 2} [(1+z)^{2} -1]$ for $q_{\rm0}=0$ or as $X(z) \equiv {2\over 3} [(1+z)^{3/2} -1]$ for $q_{\rm0} = 0.5$. We used $q_{\rm0}=0$ for dX to compare our $f(N_H{\sc i})$ with the published $f(N_H{\sc i})$ from the literature (Table 2 lists the values of dX for $q_{\rm0}=0$). Empirically, $f(N_H{\sc i})$ is fitted by a power law: $f(N_H{\sc i}) = A\,N_H{\sc i}^{-\beta }$.

Figure 4 shows the observed $\log f(N_H{\sc i})$as a function of $\log N_H{\sc i}$ for different redshifts. The dotted line represents the incompleteness-corrected $f(N_H{\sc i})$ at $z \sim 2.8$ from Hu et al. (1995), i.e. $f(N_H{\sc i}) =4.9 \times 10^{7} \, N_H{\sc i}^{-1.46}$. Note that the apparent flattening of the slope towards lower column densities in the observed $\log N_H{\sc i}$- $\log f(N_H{\sc i})$diagram is caused by line blending and limited S/N, i.e. incompleteness, which becomes more severe at higher z. For incompleteness-corrected $f(N_H{\sc i})$at z > 2.5 (Hu et al. 1995; Lu et al. 1996; Kim et al. 1997), this apparent flattening disappears. The incompleteness-corrected $f(N_H{\sc i})$ at $z \sim 2.8$ over $N_H{\sc i}= 10^{12.5-16} \ {\rm cm}^{-2}$is similar to the incompleteness-corrected $f(N_H{\sc i})$ at $z \sim 3.7$ (Lu et al. 1996) and at $z \sim 3.2$ (Kim et al. 1997) over the same column density range. The amount of incompleteness extrapolated from at z > 2.5 (Hu et al. 1995; Lu et al. 1996; Kim et al. 1997; Kirkman & Tytler 1997) becomes negligible at z < 2.4 and we assume the observed $f(N_H{\sc i})$ as representative of the actual $f(N_H{\sc i})$ at z < 2.4.

In the column density range $N_H{\sc i} = 10^{12.5-14} \ {\rm cm}^{-2}$, the observed $f(N_H{\sc i})$ at z < 2.4 is in good agreement for the different QSOs and also agrees with the incompleteness-corrected $f(N_H{\sc i})$ at 2.6< z < 4.0. This suggests that there is very little evolution in $f(N_H{\sc i})$ in the interval 1.5 <z < 4for forest lines with $N_H{\sc i} = 10^{12.5-14} \ {\rm cm}^{-2}$. At $N_H{\sc i} = 10^{14-16} \ {\rm cm}^{-2}$, $f(N_H{\sc i})$shows differences at different z. Kim et al. (1997) noted that at lower z, $f(N_H{\sc i})$ starts to deviate from a single power law for $N_H{\sc i} > 10^{14} \ {\rm cm}^{-2}$ and that the column density at which the deviation from a single power-law starts decreases as z decreases. The deviation from the single power-law in $f(N_H{\sc i})$ is evident in Fig. 4. While the forest at $z \sim 3.7$ is still well approximated by a single power-law over $N_H{\sc i} = 10^{14-16} \ {\rm cm}^{-2}$, the forest at z<2.4 starts to deviate from the power law at $N_H{\sc i} \ge 10^{14.1} \ {\rm cm}^{-2}$ with a decreasing number of lines at $N_H{\sc i} = 10^{14-16} \ {\rm cm}^{-2}$.

Table 3 lists the parameters of a maximum-likelihood power-law fit to various column density ranges. These column density ranges are selected for comparison with the previous observational results of Kim et al. (1997) and Penton et al. (2000) and with simulations of Zhang et al. (1998) and Machacek et al. (2000). At $z \sim 2.1$, the slope $\beta$ is approximately 1.4 in the interval $N_H{\sc i} = 10^{12.5-14} \ {\rm cm}^{-2}$ and 1.68 in the interval $N_H{\sc i} = 10^{14-16} \ {\rm cm}^{-2}$, i.e. the slope is steeper for higher column density clouds. At $z \sim 1.61$, the slopes $\beta \sim 1.70$-1.72 are steeper for both column density ranges. This indicates that the slope of $f(N_H{\sc i})$ increases from $z \sim 2.1$ to $z \sim 1.6$. Assuming a curve of growth with $b = 25 \pm 5$ km s^-1, Penton et al. (2000) found that the slope of $f(N_H{\sc i})$ at $z \sim 0.036$ over $N_H{\sc i} = 10^{12.5-14} \ {\rm cm}^{-2}$and over $N_H{\sc i} = 10^{14-16} \ {\rm cm}^{-2}$ is $\beta = 1.72 \pm 0.06$and $\beta = 1.43 \pm 0.35$, respectively. The slopes over $N_H{\sc i} = 10^{12.5-14} \ {\rm cm}^{-2}$are steeper at $z \sim 0.036$ and at $z \sim 1.6$ than at z > 1.8, and suggest that the incompleteness correction at z > 1.8 might be underestimated or that the slope becomes intrinsically steeper at z < 1.8. The slopes over $N_H{\sc i} = 10^{12.8-14} \ {\rm cm}^{-2}$and $N_H{\sc i} = 10^{14-16} \ {\rm cm}^{-2}$ are in agreement with the ones found by Kim et al. (1997) at $z \sim 2.3$. However, our measurement of $\beta = 1.48 \pm 0.15$ at $z \sim 2.1$ (only from HE 2217-2818 and J2233-606) over $N_H{\sc i} = 10^{13.1-14.5} \ {\rm cm}^{-2}$ is lower than the previous determination of $\beta = 1.79 \pm 0.10$ over the same column density range at $z \sim 1.85$ by Kulkarni et al. (1996).

While these observed $\beta$ values at 1.5 < z < 2.4 can be obtained with semi-analytic models by Hui et al. (1997), they are lower than the values predicted from numerical simulations (Zhang et al. 1998; Machacek et al. 2000), by more than $2 \sigma$. The slope depends on the amplitude of the power spectrum and models with less power produce steeper slopes. Thus, the steeper slopes from the simulations by Zhang et al. (1998) and Machacek et al. (2000) suggest that their index for the power spectrum, $n_{\rm p}=1$, might be smaller than the actual index of the power spectrum.

Figure 4: The differential density distribution function as a function of $\log N_H{\sc i}$ without the incompleteness correction due to line blending and limited S/N. The vertical error bars represent the $1\sigma$ Poisson errors.

 

 

Table 3: The power-law fit of the distribution functions, $f(N_H{\sc i}) = A\,N_H{\sc i}^{-\beta }$.

	$N_H{\sc i} = 10^{12.5-14.0} \ {\rm cm}^{-2}$			$N_H{\sc i} = 10^{14.0-16.0} \ {\rm cm}^{-2}$			$N_H{\sc i} = 10^{12.8-14.3} \ {\rm cm}^{-2}$			$N_H{\sc i} = 10^{12.5-16.0} \ {\rm cm}^{-2}$
z	$\log A$	$\beta$		$\log A$	$\beta$		$\log A$	$\beta$		$\log A$	$\beta$
1.61	$10.65 \pm 0.86$	$1.72 \pm 0.16$		$10.38 \pm 0.35$	$1.70 \pm 0.42$		$10.85 \pm 0.74$	$1.74 \pm 0.21$		$10.47 \pm 0.88$	$1.71 \pm 0.10$
1.98	$6.19 \pm 1.07$	$1.35 \pm 0.09$		$6.49 \pm 0.59$	$1.40 \pm 0.21$		$8.02 \pm 1.00$	$1.49 \pm 0.11$		$8.66 \pm 1.09$	$1.54 \pm 0.05$
2.13	$6.66 \pm 1.11$	$1.38 \pm 0.08$		$14.48 \pm 0.65$	$1.94 \pm 0.24$		$7.80 \pm 1.05$	$1.46 \pm 0.10$		$9.05 \pm 1.14$	$1.56 \pm 0.05$
$1.94^{{\rm a}}$	$6.97 \pm 1.28$	$1.41 \pm 0.06$		$10.43 \pm 0.80$	$1.68 \pm 0.15$		$8.16 \pm 1.21$	$1.50 \pm 0.07$		$9.02 \pm 1.30$	$1.57 \pm 0.03$

$^{{\rm a}}$

$1.54 \le z \le 2.33$.

4.2 The evolution of the line number density

The line number density per unit redshift is defined as dn/dz = (dn/d $z)_{\rm0} (1+z)^{\gamma}$, where (dn/d$z)_{\rm0}$ is the local comoving number density of the forest. For a non-evolving population in the standard Friedmann universe with the cosmological constant $\Lambda = 0$, $\gamma = 1$ and 0.5 for $q_{\rm0}=0$ and 0.5, respectively. In practice, the measured $\gamma$ is dependent on the chosen column density interval, the redshift and the spectral resolution. Therefore, comparisons between individual studies are complicated (Kim et al. 1997).

Figure 5 shows the number density evolution of the Ly$\alpha$ forest in the interval $N_H{\sc i} = 10^{13.64 - 16} \ {\rm cm}^{-2}$. This range has been chosen to allow a comparison with the HST results from the HST QSO absorption line Key Project at z < 1.5 from Weymann et al. (1998), for which a threshold in the equivalent width of 0.24 Å was adopted. We assumed the conversion between the equivalent width and the column density to be $N_H{\sc i} = 1.33 \times 10^{20}\,W/\lambda_{\rm0}^{2}f$, where W is the equivalent width in angstrom, $\lambda_{\rm0}$ is the wavelength of Ly$\alpha$ in angstrom, f is the oscillator strength of Ly$\alpha$(Cowie & Songaila 1986). The value of the square (Penton et al. 2000) was estimated under the assumption of b=25 km s^-1 from the equivalent widths (corresponding to the column density range $N_H{\sc i} \ge 10^{14} \ {\rm cm}^{-2}$) and is lower than the extrapolated dn/dz at $z \sim 0.04$ from the Weymann et al. results, but within the error bar. Pentagons (Savaglio et al. 1999 from the line fitting analysis) also correspond to the column density range $N_H{\sc i} \ge 10^{14} \ {\rm cm}^{-2}$. Note that $N_H{\sc i}$ from W depends on an assumed b parameter, resolution and S/N. Also note that including lines with $N_H{\sc i} \ge 10^{16}\ {\rm cm}^{-2}$from line fitting analyses introduces a further uncertainty on the line counting since different programs deblend completely saturated lines differently, resulting in different numbers of lines for the same saturated lines.

The long-dashed line is the maximum-likelihood fit to the UVES and the HIRES data at z > 1.5: dn/d $z = (9.06 \pm 0.40) \,(1+z)^{2.19 \pm 0.27}$. This $\gamma$ is lower than previously reported $\gamma \sim 2.75 \pm 0.30$ (Lu et al. 1991; Kim et al. 1997). This slope is steeper than the expected values for the non-evolving forest for a universe with $\Omega_\Lambda = 0.7$, $\Omega_{\rm m} = 0.3$ and $\Omega = 1$. These results suggest that the Ly$\alpha$ forest at $N_H{\sc i} = 10^{13.64-16}~ {\rm cm}^{-2}$ evolves and that its evolution slows down as z decreases. Interestingly, the HST data point at $<\!z\!> \ =1.6$ (the open triangle at the boundary of the shaded area), which has been measured in the line-of-sight to the QSO UM 18 and was suggested to be an outlier by Weymann et al. (1998), is now in good agreement with the extrapolated fit from higher z.

Despite the different line counting methods between the HST observations (based on the equivalent width) and the high-resolution observations (based on the profile fitting), a change of the slope in the Ly$\alpha$number density does seem to be real. The UVES observations suggest that the slow-down in the evolution does occur at $z \sim 1.2$, rather than at $z \sim 1.7$ as previously suggested (Impey et al. 1996; Weymann et al. 1998), although the different methods of line counting at higher and lower z make it a little uncertain. At least, down to $z \sim 1.5$, the number density of the forest evolves as at higher z, which suggests that any major drive governing the forest evolution at z > 2 continues to dominate the forest evolution down to $z \sim 1.5$. Since the Hubble expansion is the main drive governing the forest evolution at z > 2 (Miralda-Escudé et al. 1996), the continuously decreasing number density of the forest down to $z \sim 1.5$ implies that the Hubble expansion continues to dominate the forest evolution down to $z \sim 1.5$.

Figure 5: The number density evolution of the Ly$\alpha$ forest. The column density range $N_H{\sc i} = 10^{13.64 - 16} \ {\rm cm}^{-2}$ has been chosen to allow a comparison with the HST results from Weymann et al. (1998), which are shown as open triangles. Filled symbols are estimated from HE 0515-4414 at $<\!z\!> \ = 1.61$, from J2233-606 at $<\!z\!> \ = 1.98$ and from HE 2217-2818 at $<\!z\!> \ = 2.13$, respectively. The star, open circles, and the diamond are taken from the HIRES data at similar resolutions by Lu et al. (1996), Kim et al. (1997), and Kirkman & Tytler (1997), respectively. The square and pentagons are taken from Penton et al. (2000) and Savaglio et al. (1999), respectively, over $N_H{\sc i} \ge 10^{14} \ {\rm cm}^{-2}$. Horizontal solid lines represent the z interval over which the number density was estimated. Vertical solid lines represent the $1\sigma$ Poisson errors. The shaded area is the z range where UVES is extremely sensitive. The long-dashed line is the maximum likelihood fit to the UVES and the HIRES data at z > 1.5. The UVES observations indicate that the number density evolution of the Ly$\alpha$ forest at z > 2.4 continues at least down to $z \sim 1.5$ and that a slope change occurs at $z \sim 1.2$.

Figure 6 is similar to Fig. 5, except for the $N_H{\sc i}$ range: $N_H{\sc i} = 10^{13.1-14} \ {\rm cm}^{-2}$. The correction for incompleteness due to line blending is still negligible in this column density range (Fig. 4 shows that the number of lines per unit column density over $N_H{\sc i} = 10^{13.1-14} \ {\rm cm}^{-2}$is still well represented by a single power-law). Again, the square from Penton et al. (2000) is estimated from the equivalent widths with the assumed b=25 km s^-1. The dot-dashed line is the maximum-likelihood fit to the lower column density forest of the UVES and the HIRES data: dn/d $z = (55.91 \pm 2.00) \, (1+z)^{1.10 \pm 0.21}$. At 2.4 < z < 4 and at 2.1 < z < 4, $\gamma = 0.90 \pm 0.29$ and $\gamma = 1.00 \pm 0.22$, respectively. For the column density range $N_H{\sc i} = 10^{13.1-14} \ {\rm cm}^{-2}$, the forest does not show any strong evolution.

Note that the point at $<\!z\!> \ = 2.66$ (diamond) from Kirkman & Tytler (1997) indicates a number density twice as large as than at $<\!z\!> \ = 2.87$in the interval $N_H{\sc i} = 10^{13.1-14.0} \ {\rm cm}^{-2}$(excluding the $<\!z\!> \ = 2.66$ forest, the maximum-likelihood fit becomes dn/d $z = (47.77 \pm 1.84) \, (1+z)^{1.18 \pm 0.22}$). Although this discrepancy could result from a real cosmic variance of the number density from sightline to sightline, the number density in the interval $N_H{\sc i} = 10^{13.64-16.0} \ {\rm cm}^{-2}$ from the same line of sight is in good agreement with other HIRES data. The differential density distribution function (Fig. 4) and the mean H I opacity (Fig. 15) towards this line of sight suggest that the discrepancy at $N_H{\sc i} = 10^{13.1-14.0} \ {\rm cm}^{-2}$is due to overfitting, which, as discussed in Sect. 3, may occur especially in high S/N data.

As previously noticed (Kim et al. 1997), the lower column density forest evolves at a slower rate than the higher column density forest. The evolutionary rate $\propto (1+z)^{1.10 \pm 0.21}$would be consistent with no evolution for $q_{\rm0} = 1$ or mild evolution for $q_{\rm0} = 0.5$. For $\Omega_\Lambda = 0.7$, $\Omega_{\rm m} = 0.3$ and $\Omega = 1$, the Ly$\alpha$ forest with $N_H{\sc i} = 10^{13.1-14} \ {\rm cm}^{-2}$ is mildly evolving at z > 1.5. The Ly$\alpha$ forest with $N_H{\sc i} = 10^{13.1-14} \ {\rm cm}^{-2}$ appears more numerous at $z \sim 0$ than expected when extrapolating from the z > 1.5 range.

Figure 6: The number density evolution of the Ly$\alpha$ forest at the $N_H{\sc i}$ range $N_H{\sc i} = 10^{13.1-14.0} ~ {\rm cm}^{-2}$. Symbols are the same as in Fig. 5. The dot-dashed line (solid line) is the maximum likelihood fit to the lower column density forest including the $<\!z\!>$ = 2.66 forest (excluding the $<\!z\!>$ = 2.66 forest), while the dashed line represents the maximum likelihood fit to the forest at $N_H{\sc i} = 10^{13.64-16}~ {\rm cm}^{-2}$ from Fig. 5.

4.3 The lower cutoff b in the $\mathsfsl{N_{{\mathsf{H}}\,{\mbox{\fontsize{6}{8}\selectfont\sffamily\bfseries I}}}-b}$ distribution

4.3.1 The lower cutoff $\mathsfsl{b}$ parameter

For a photoionized gas, a temperature-density relation exists, i.e. the equation of state: $T=T_{\rm0} \, (1+\delta)^{\gamma_{T}-1}$, where T is the gas temperature, T₀ is the gas temperature at the mean gas density, $\delta$ is the baryon overdensity, $(\rho_{\rm b}-\overline{\rho}_{\rm b})/\overline{\rho}_{\rm b}$ ( $\overline{\rho}_{\rm b}$ is the mean baryon density), and $\gamma_{\rm T}$is a constant which depends on the reionization history (Hui & Gnedin 1997). For an abrupt reionization at $z \gg 5$, the temperature of the mean gas density decreases as z decreases after the reionization, eventually approaching an asymptotic $T_{\rm0}$. For a generally assumed QSO-dominated UV background with a sudden turn-on of QSOs at 5 < z < 10, T₀ decreases as z decreases at 2 < z < 4(Hui & Gnedin 1997).

Under the assumption that there are some lines which are broadened primarily by the thermal motion at any given column density, this equation of state translates into a lower cutoff $b(N_H{\sc i})$envelope in the $N_H{\sc i}$-b distribution: T and $\delta$ can be derived from b and $N_H{\sc i}$. For the equation of state $T=T_{0}\,(1+\delta)^{\gamma_{T}-1}$, $b_{\rm c}(N_H{\sc i})$ becomes

$$\log(b_{\rm c}) = \log(b_{0}) + (\Gamma_{\rm T}-1) \, \log(N_H{\sc i}),$$ (1)

where $\log(b_{0})$ is the intercept of the cutoff in the $\log (N_H{\sc i})$-$\log b$ diagram and $(\Gamma _{\rm T}-1)$is the slope of the cutoff (Schaye et al. 1999). The cutoff slope $(\Gamma _{\rm T}-1)$ is proportional to $(\gamma_{\rm T}-1)$. This cutoff envelope provides a probe of the gas temperature of the IGM at a given z, thus giving a powerful constraint on the thermal history of the IGM (Hu et al. 1995; Lu et al. 1996; Kim et al. 1997; Kirkman & Tytler 1997; Zhang et al. 1997; Schaye et al. 1999; Bryan & Machacek 2000; McDonald et al. 2000; Ricotti et al. 2000; Schaye et al. 2000).

Figure 7: The $\log N_H{\sc i}$-b diagrams with the power law fits over $N_H{\sc i} = 10^{12.5-14.5} ~ {\rm cm}^{-2}$. Errors are not displayed. Crosses indicate the data points with errors less than 25% both in $N_H{\sc i}$ and b (Sample A), while open circles indicate the lines with errors greater than 25% (Sample B consists of crosses and open circles together). At $<\!z\!>$ = 2.87, no Sample A can be defined due to the lack of errors in the line list. Thus, Sample B was used as a substitute for Sample A when compared to Sample A from QSOs at other z. The solid line, the dot-dashed line and the dashed line at each panel indicate the power law fit to Sample A, to Sample B and to the $<\!z\!>$ = 3.75 forest for comparisons, respectively. At $<\!z\!>$ = 2.87 the solid line is not present since no Sample A can be defined.

In practice, defining $b_{\rm c}(N_H{\sc i})$in an objective manner is not trivial due to the finite number of available absorption lines, sightline-to-sightline cosmic variances, limited S/N, and unidentified metal lines. Among several methods proposed to derive $b_{\rm c}(N_H{\sc i})$, we have adopted the following three: the iterative power-law fit, the power-law fit to the smoothed b distribution, and the b distribution. We refer the reader to other papers for more methods to derive $b_{\rm c}(N_H{\sc i})$ (Hu et al. 1995; McDonald et al. 2000; Theuns & Zaroubi 2000).

In our analysis, we divide the data points into 2 groups: Sample A and Sample B. Sample A consists of the lines in the range $N_H{\sc i} = 10^{12.5 -14.5} \ {\rm cm}^{-2}$ with errors less than 25% in both $N_H{\sc i}$ and b in order to avoid ill-fitted values from VPFIT. Sample B consists of all the lines with $N_H{\sc i} = 10^{12.5 -14.5} \ {\rm cm}^{-2}$ regardless of errors. The criteria for Sample A and Sample B are chosen to compare our results with the previous results by Schaye et al. (2000) and to investigate whether it is reasonable to include the Q0302-003 line list for which error estimates are not given. As no errors are available for Q0302-003, no Sample A can be defined at $<\!z\!> \ = 2.87$. Note that including the relatively few lines with $N_H{\sc i} = 10^{14.5 -16} \ {\rm cm}^{-2}$ does not change the results significantly. There is hardly any overlap in z, except for J2233-606 and HE 2217-2818. Since one of our aims is to probe the z-evolution of $b_{\rm c}(N_H{\sc i})$, we analyze the b distribution of each line of sight individually to derive $b_{\rm c}(N_H{\sc i})$.

4.3.2 The iterative power law fit

Since the equation of state is a power law, it is reasonable to fit $N_H{\sc i}$-$b_{\rm c}$ with one. We did so, using the bootstrap method described by Schaye et al. (1999), iterating until convergence was reached. After each iteration, those points were excluded that lay more than one mean absolute deviation above the fit. Finally, the lines more than one mean absolute deviation below the fit were also taken out and the final power law fit, $b_{\rm c} = c_{\rm0,p} \, N_H{\sc i}^{\Gamma_{\rm T}-1}$, was carried out. The procedure was repeated over 200 bootstrap realizations in order to determine the full probability distribution of the parameters of the cutoff. As noted by Schaye et al. (2000), the power law fit requires over 200 available lines to reach stable fit parameters.

Figure 7 shows the iterative power law fit in the $\log N_H{\sc i}$-b distributions. The noticeable difference between Sample A (cross symbols) and Sample B (cross symbols and open circles) occurs at $N_H{\sc i} \le 10^{13} \ {\rm cm}^{-2}$. These lines usually come from blends or from weak, asymmetric absorption lines. Table 4 lists the fitted parameters, such as $c_{\rm0,p}$and $(\Gamma _{\rm T}-1)$, including $b_{\rm c}$ values at the fixed column density $N_H{\sc i} = 10^{13.5} \ {\rm cm}^{-2}$, $b_{\rm c, 13.5}$, for Sample A and Sample B.

The power law fit between Sample A and Sample B does not give a significant difference except at $<\!z\!> \ = 2.13$ and at $<\!z\!> \ = 3.75$, for which several lines with $b \le 20$ km s^-1and $N_H{\sc i} \le 10^{13} \ {\rm cm}^{-2}$ contribute to a different power law fit for Sample B. This suggests that using the Q0302-003 line list at $<\!z\!> \ = 2.87$ without error bars does not severely distort our conclusions. Note that the power law fit at $<\!z\!> \ = 2.87$ might be less steep with a higher intercept, if the same general behavior of errors also occurs for the Q0302-003 forest (larger errors at $b \le 20$ km s^-1 or $b \ge 40$ km s^-1). Due to the small number of lines (47 lines for Sample A and 56 lines for Sample B) at $<\!z\!> \ = 1.61$, the power law fit should be taken as an upper limit on $b_{\rm c}(N_H{\sc i})$ and indeed it provides the highest $b_{\rm c, 13.5}$ among all the z bins. For both Sample A and Sample B, there is a weak trend of increasing $b_{\rm c, 13.5}$ as z decreases, except at $<\!z\!> \ = 2.87$which shows a higher $b_{\rm c, 13.5}$ value than at the adjacent z ranges (see Sect. 6.2 for further discussion). On the other hand, the power law slope $(\Gamma _{\rm T}-1)$ is rather ill-defined with z with a possible flatter slope at $<\!z\!> \ = 3.75$ than at z < 3.1.

Figure 8 shows the power law fit to Sample A at $z \sim 2.1$ (small filled circles; 242 lines from J2233-606 and HE 2217-28118) and at $<\!z\!> \ = 3.75$ (open squares; 209 lines) over $N_H{\sc i} = 10^{12.5 -14.5} \ {\rm cm}^{-2}$ (upper panel) and over $N_H{\sc i} = 10^{13-14.5} \ {\rm cm}^{-2}$ (lower panel). The fitted parameters are given in Table 4. For both $N_H{\sc i}$ ranges, the slopes of $b_{\rm c}(N_H{\sc i})$ are steeper at $z \sim 2.1$ than at $<\!z\!> \ = 3.75$. This result, however, is certainly biased by the lack of lines with $b \le 15$ km s^-1 and $N_H{\sc i} \le 10^{13.4} ~ {\rm cm}^{-2}$ at higher z, due to the severe line blending.

 

 

Table 4: The power law fit to the $N_H{\sc i}$-b distributions.

	Sample A					Sample B
$<\!z\!>$	# $^{{\rm a}}$	$\log (c_{\rm0,p})$	$(\Gamma _{\rm T}-1)$	$b_{\rm c, 13.5}$ (km s-1) $^{{\rm b}}$		# $^{{\rm a}}$	$\log (c_{\rm0,p})$	$(\Gamma _{\rm T}-1)$	$b_{\rm c, 13.5}$ (km s-1) $^{{\rm b}}$
1.61	47	$-0.92 \pm 0.09$	$0.17 \pm 0.01$	$24.5 \pm 2.4$		56	$-0.87 \pm 0.10$	$0.17 \pm 0.01$	$24.9 \pm 1.1$
1.98	103	$-0.49 \pm 0.11$	$0.14 \pm 0.01$	$21.4 \pm 0.5$		146	$-0.43 \pm 0.12$	$0.13 \pm 0.01$	$21.4 \pm 0.6$
2.13	139	$0.11 \pm 0.08$	$0.09 \pm 0.01$	$20.1 \pm 0.6$		181	$-0.70 \pm 0.11$	$0.15 \pm 0.01$	$20.6 \pm 0.5$
2.66	140	$-0.55 \pm 0.10$	$0.14 \pm 0.01$	$19.6 \pm 1.1$		204	$-0.13 \pm 0.10$	$0.10 \pm 0.01$	$18.7 \pm 0.5$
2.87	-	-	-	-		223	$-0.89 \pm 0.08$	$0.16 \pm 0.01$	$20.4 \pm 0.8$
3.75	209	$0.51 \pm 0.06$	$0.06 \pm 0.01$	$18.7 \pm 1.2$		271	$0.71 \pm 0.10$	$0.04 \pm 0.01$	$17.1 \pm 0.9$
2.1 $^{{\rm c}}$	242	$-0.22 \pm 0.08$	$0.11 \pm 0.01$	$19.6 \pm 0.6$		327	$-0.73 \pm 0.08$	$0.15 \pm 0.01$	$20.4 \pm 0.5$
2.1 $^{{\rm d}}$	156	$-0.64 \pm 0.13$	$0.15 \pm 0.01$	$21.2 \pm 0.7$		187	$-0.78 \pm 0.12$	$0.16 \pm 0.01$	$20.9 \pm 0.6$
3.75 $^{{\rm e}}$	188	$0.09 \pm 0.10$	$0.09 \pm 0.01$	$18.9 \pm 1.0$		233	$-0.06 \pm 0.09$	$0.10 \pm 0.01$	$18.1 \pm 0.9$

$^{{\rm a}}$

The number of absorption lines used for the fit.

$^{{\rm b}}$

The errors were estimated from the difference between the mean $b_{\rm c, 13.5}$ and the minimum/maximum $b_{\rm c, 13.5}$from the 200 bootstrap realizations for the given $N_H{\sc i}$-bpairs. The typical $1\sigma$ of the $b_{\rm c, 13.5}$ distribution from the 200 bootstrap realizations is $\sim$0.3 kms^-1, which is underestimated (cf. Schaye et al. 2000).

$^{{\rm c}}$

Combined line lists from J2233-606 and HE 2218-2817. The power law fit was carried out for $N_H{\sc i} = 10^{12.5 -14.5} \ {\rm cm}^{-2}$.

$^{{\rm d}}$

Combined line lists from J2233-606 and HE 2218-2817. The power law fit was carried out for $N_H{\sc i} = 10^{13-14.5} \ {\rm cm}^{-2}$.

$^{{\rm e}}$

For $N_H{\sc i} = 10^{13-14.5} \ {\rm cm}^{-2}$.

Figure 8: The $\log N_H{\sc i}$-b diagrams with the power law fits at $z \sim 2.1$ (only for J2233-606 and HE 2217-2818; small filled circles) and at $<\!z\!> \ = 3.75$ (open squares). At both redshifts, the data points with errors less than 25% in both $N_H{\sc i}$and b are displayed. The upper panel shows the fit to the column density range $N_H{\sc i} = 10^{12.5-14.5} ~ {\rm rm}^{-2}$, while the lower panel is to the range $N_H{\sc i} = 10^{13-14.5} ~ {\rm rm}^{-2}$. The vertical dotted lines in each panel represent the column density range over which the power law fit was carried out. The solid line shows the power law fit at $z \sim 2.1$, and the dashed line at z=3.75. A deficit of lines with $N_H{\sc i} \le 10^{13} ~ {\rm cm}^{-2}$ and $b \le 20$ kms-1observed at $<\!z\!> \ = 3.75$ is due in part to the fact that all the fitted lines with $N_H{\sc i} \le 10^{13} ~ {\rm cm}^{-2}$ and $b \le 20$ kms-1 have errors larger than 25% and in part to the severe line blending which limits the detection of weak lines.

4.3.3 The power law fit to the smoothed b distribution

Bryan & Machacek (2000) presented a method to measure $b_{\rm c}(N_H{\sc i})$ from a power law fit to a smoothed b distribution, sorting absorption lines by $N_H{\sc i}$ and then dividing them into groups containing similar numbers of lines. The b distribution in each group was then smoothed with a Gaussian filter with a smoothing constant $\sigma_{\rm b}$:

$$S_{{\rm b},j} (b) = \sum_{i} \exp(-(b_{i}-b)^{2}/2\sigma_{\rm b}^{2}),$$ (2)

where $S_{{\rm b},j} (b)$ is the smoothed density of lines in each group jand i indicates the lines in the group. Then, the location of the first peak in the derivative of $S_{{\rm b},j} (b)$ defines the lower cutoff at the average column density, $N_{H{\sc i},j}$, for the jth group.

Figure 9 shows the $\log N_H{\sc i}$-b diagram at each z with the $b_{\rm c}(N_H{\sc i})$ points for each group (filled circles) measured from the smoothed b distributions. We use the smoothing constant $\sigma_{\rm b} = 3$ kms^-1. However, $S_{{\rm b},j} (b)$ is largely insensitive to the smoothing constant. In general, 30 lines were included in each group except for the last group at higher $N_H{\sc i}$ for which typically smaller numbers of lines were available. For this same reason, at $<\!z\!> \ = 1.61$groups of $\sim$16 lines were used. The solid line represents the robust least-squares power law fit to filled circles: $b_{\rm c} (N_H{\sc i}) = c_{\rm0,s} N_H{\sc i}^{\ \Gamma_{\rm T}-1}$. Table 5 lists the parameters of the power law fit to the smoothed bdistributions.

We find that the power law fit to the smoothed b distribution produces in general a lower intercept and a steeper slope than the iterative power law fit. It also produces smaller $b_{\rm c, {\rm 13.5}}$ values. Direct comparison of Figs. 9 with 7 indicates that $b_{\rm c}(N_H{\sc i})$ measured from the smoothed b distribution can be considered as a lower limit on the real $b_{\rm c}(N_H{\sc i})$, while $b_{\rm c}(N_H{\sc i})$ from the iterative power law fit can be considered as an upper limit on the real $b_{\rm c}(N_H{\sc i})$.

As with the iterative power law fit, $b_{\rm c, {\rm 13.5}}$ measured from the smoothed b distribution increases continuously as z decreases, except at $<\!z\!> \ = 2.87$, where $b_{\rm c, {\rm 13.5}}$ is higher than at the adjacent redshifts (see Sect. 6.2 for further discussion). The slope $(\Gamma _{\rm T}-1)$ measured from the smoothed b distribution also does not show any well-defined trend with z.

 

 

Table 5: The power law fit to the smoothed b distributions.

$<\!z\!>$	$\log(c_{\rm0,s})$	$(\Gamma _{\rm T}-1)$	$b_{\rm c, 13.5}$
			(km s-1)
1.61	$-0.92 \pm 0.13$	$0.16 \pm 0.04$	$20.3 \pm 1.1$
1.98	$-1.42 \pm 0.02$	$0.20 \pm 0.01$	$19.1 \pm 1.0$
2.13	$-0.93 \pm 0.11$	$0.16 \pm 0.03$	$18.8 \pm 1.1$
2.66	$-0.73 \pm 0.22$	$0.14 \pm 0.06$	$16.6 \pm 1.2$
2.87 $^{{\rm a}}$	$-1.42 \pm 0.08$	$0.20 \pm 0.02$	$19.0 \pm 1.1$
3.75	$0.16 \pm 0.07$	$0.08 \pm 0.02$	$16.4 \pm 1.1$
2.1 $^{{\rm b}}$	$-0.71 \pm 0.14$	$0.15 \pm 0.04$	$19.2 \pm 1.0$
2.1 $^{{\rm c}}$	$-0.50 \pm 0.34$	$0.13 \pm 0.09$	$19.7 \pm 1.1$
3.75 $^{{\rm d}}$	$-0.30 \pm 0.13$	$0.11 \pm 0.03$	$15.8 \pm 1.0$

$\parbox{7cm}{ $^{{\rm a}}$\space For Sample B since Sample A cannot ... ....\\ $^{{\rm d}}$\space At $N_H{\sc i} = 10^{13-14.5} \ {\rm cm}^{-2}$ . }$

Figure 9: The $\log N_H{\sc i}$-b diagrams for Sample A at each z (Sample B at $<\!z\!> \ = 2.87$) with the robust least-squares power law fit to the smoothed b distribution with the Gaussian smoothing constant 3 kms-1. Small open squares represent the data points for Sample A (Sample B at $<\!z\!> \ = 2.87$), while filled circles represent the cutoff b values estimated from the smoothed b distributions with $\sim$30 lines in each group (see the text for the details). The solid line represents the robust least-squares power law fit to filled circles.

4.3.4 The $\mathsfsl{b}$ distributions

Assuming that absorption lines arise from local optical depth ($\tau$) peaks and that $\ln \tau$ is a Gaussian random variable, Hui & Rutledge (1999) derived a single-parameter b distribution:

$${\rm d}n/{\rm d}b = B_{\rm HR} \, \frac{{b_\sigma}^4}{b^5}\, \exp\left(-\frac{{b_\sigma}^4}{b^4}\right),$$ (3)

where n is the number of absorption lines, $B_{\rm HR}$ is a constant and $b_{\rm\sigma}$is a parameter determined by the average amplitude of the fluctuations and the effective smoothing scale.

Figure 10 shows the observed b distributions at each z. The noticeable difference between Sample A (solid lines) and Sample B (dot-dashed lines) occurs at $b \le 20$ km s^-1 or $b \ge 40$ km s^-1. These spurious lines are usually introduced by VPFIT to fit the noise so that the overall profile of H I forest complexes could be improved. The dashed line represents the best-fitting Hui-Rutledge b distribution, while the dotted line represents the b parameter for which the Hui-Rutledge b distribution function vanishes to 10^-4, $b_{\rm HR}$, i.e. the truncated b value for the Hui-Rutledge bdistribution function. The parameter $b_{\rm HR}$ cannot be considered equivalent to the cutoff $b_{\rm c}$since it is derived from the b distribution without assuming the $b_{\rm c}$ dependence on $N_H{\sc i}$. It is more sensitive to smaller b values in the b distribution, which are in general coupled with lower $N_H{\sc i}$. Table 6 lists the relevant parameters describing the Hui-Rutledge b distribution for Sample A, such as the constant $B_{\rm HR}$, $b_\sigma$, $b_{\rm HR}$ and the median b values at different column density ranges.

It is hard to specify subtle differences among the bdistributions: while the modal b value and the $b_{\rm HR}$ value have a minimum at $<\!z\!> \ = 3.75$, they have a maximum at $<\!z\!> \ = 2.87$. The $<\!z\!> \ = 2.87$ forest also has the broadest b distribution. However, this large $\sigma(b)$could be in part due to a different fitting program and in part due to a lack of information on the errors. Other parameters, such as $b_\sigma$, $b_{\rm HR}$, and $\sigma(b)$, appear to be fairly constant with z.

Figure 10: The b distribution of the Ly$\alpha$ forest at each z. While solid lines are for Sample A, dot-dashed lines are for Sample B (no Sample A at $<\!z\!> \ = 3.75$). The dashed line and the dotted line represent the best-fitting Hui-Rutledge function and $b_{\rm HR}$, respectively. The $b_{\rm med}$ value in the panels indicates the median b value at the corresponding z for Sample A. The number in parentheses indicates the $1\sigma (b)$ value from the Gaussiandistribution.

Figure 11: The b distribution as a function of z for Sample A (Sample B at $<\!z\!> \ = 3.75$). The horizontal dashed line indicates b = 20 kms-1 which is a $N_H{\sc i}$-independent $b_{\rm c}$at $z \sim 2.8$ (Hu et al. 1995). Circles, crosses, diamonds, stars, triangles and squares are from HE 0515-4414, J2233-606, HE 2217-2818, HS 1946+7658, Q0302-003 and Q0000-263, respectively. There is an indication of increasing $b_{\rm c}$ with decreasing z at $z \sim 3.7$. At z < 3.1, $b_{\rm c}$ is not clearly defined.

Figure 11 shows the b distribution with z. This diagram does not assume a $b_{\rm c}$ dependence on $N_H{\sc i}$, but is sensitive to a local $b_{\rm c}(N_H{\sc i})$ variance. At z < 3.1, there is no clear indication of the behavior of the lower cutoff b values as a function of z. However, there is a clear indication of a trend with z of the lower cutoff b values over 3.5 < z < 3.9. In Fig. 11, there are distinct regions at 1.8 < z < 2.4. The apparent cutoff values in b at 2.2 < z < 2.4 and at 1.8 < z < 1.9are clearly higher than at 1.9 < z < 2.2. The 2.2 < z < 2.4 region towards HE 2217-2818 corresponds to a $\sim$44$\, h^{-1}$ Mpc void (the region B in Fig. 14), which might suggest enhanced ionization due to a nearby QSO or processes of galaxy formation (Theuns et al. 2000a).

 

 

Table 6: The parameters of the b distributions.

$<\!z\!>$	$B_{\rm HR}$	$b_\sigma$	$b_{\rm HR}$	$b_{\rm med}^{{\rm a}}$	$b_{\rm med}^{{\rm b}}$
		(km s-1)	(km s-1)	(km s-1)	(km s-1)
1.61	7.37	23.01	12.61	28.14	34.56
1.98	7.98	23.83	13.04	26.04	29.10
2.13	7.46	23.61	12.95	25.34	29.57
2.66	6.82	24.09	13.25	28.30	30.10
2.87	7.09	27.75	15.30	28.74	34.05
3.75	6.72	22.41	12.31	28.90	30.70

$^{{\rm a}}$

For lines with $N_H{\sc i} = 10^{13.1-14} \ {\rm cm}^{-2}$ from Sample A (Sample B at $<\!z\!> \ = 2.87$).

$^{{\rm b}}$

For lines with $N_H{\sc i} = 10^{13.8-16} \ {\rm cm}^{-2}$ from Sample A (Sample B at $<\!z\!> \ = 2.87$).

4.4 The two-point velocity correlation function

The Ly$\alpha$ forest contains information on the large-scale matter distribution and the simplest way to study it is to compute the two-point velocity correlation function, $\xi(\Delta v)$. The correlation function compares the observed number of pairs ( $N_{\rm obs}$) with the expected number of pairs ( $N_{\rm exp}$) from a random distribution in a given velocity bin ($\Delta v$): $\xi(\Delta v) = N_{\rm obs}(\Delta v)/N_{\rm exp}(\Delta v) -1$, where $\Delta v = c\,(z_{2}-z_{1})/[1+ (z_{2}+z_{1})/2]$, z₁ and z₂ are redshifts of two lines and c is the speed of light (Cristiani et al. 1995; Cristiani et al. 1997; Kim et al. 1997).

Studies of the correlation function of the Ly$\alpha$ forest have generally led to conflicting results even at similar z. Some studies find a lack of clustering (Sargent et al. 1980 at 1.7 < z < 3.3; Rauch et al. 1992 at $z \sim 3$; Williger et al. 1994 at $z \sim 4$), while others find clustering at scales $\Delta v \le 350$ km s^-1 (Webb 1987 at 1.9 < z < 2.8; Hu et al. 1995 at $z \sim 2.8$; Kulkarni et al. 1996 at $z \sim 1.9$; Lu et al. 1996 at $z \sim 3.7$; Cristiani et al. 1997 at $z \sim 3.3$).

Figure 12: Evolution of the two-point correlation function with redshift for Ly$\alpha$ lines with column densities above $N_H{\sc i} = 10^{12.7} ~ {\rm cm}^{-2}$. Short-dashed and long-dashed lines represent the $1\sigma$ and $2 \sigma$ Poisson errors.

Figure 12 shows the velocity correlation strength at $\Delta v < 4000$ km s^-1. To obtain sufficient statistics, the analysis was carried out in three redshift bins: 1.5<z<2.4 (HE 0515-4414, J2233-606, and HE 2217-2818), 2.5<z<3.1 (HS 1946+7658 and Q0302-003) and 3.5<z<4.0 (Q0000-263).

In our approach $N_{\rm exp}$ was estimated averaging 1000 numerical simulations of the observed number of lines, trying to account for relevant cosmological and observational effects. In particular a set of lines was randomly generated in the same redshift interval as the data according to the cosmological distribution $\propto (1+z)^{\gamma}$, with $\gamma = 2.4$ (see Sect. 4.2). The results are not sensitive to the value of $\gamma$ adopted and even a flat distribution (i.e. $\gamma =0$) gives values of $\xi$ that differ typically by less than 0.02. Line blanketing of weak lines due to strong complexes was also accounted for. Lines with too small velocity splittings, compared with the finite resolution or the intrinsic blending due to the typical line widths-the so-called "line-blanketing'' effect (Giallongo et al. 1996), were excluded in the estimates of $N_{\rm exp}$.

Clustering is clearly detected at low redshift: at 1.5<z<2.4 in the 100 km s^-1 bin, we measure $\xi = 0.4\pm 0.1$ for lines with . There is a hint of increasing amplitude with increasing column density: in the same redshift range $\xi = 0.35\pm 0.08$ for lines with . The trend is not significant but agrees with the behavior observed at higher redshifts (Cristiani et al. 1997; Kim et al. 1997). Unfortunately the number of lines observed in the interval 1.5<z<2.4does not allow us to extend the analysis to higher column densities, although groups of strong lines are occasionally evident (e.g. the range 3230-3270 Å in HE 0515-4414).

The amplitude of the correlation at 100 km s^-1 decreases significantly with increasing redshift from $0.4 \pm 0.1$ at 1.5<z<2.4, to $0.14 \pm 0.06$ at 2.5<z<3.1 and $0.09\pm0.07$ at 3.5<z<4.0.

 

 

Table 7: Voids at $z \sim 2$.

QSO	Region	Wavelength	$<\!z\!>$	$\Delta z$	Comoving size $^{{\rm a}}$	$x_{\rm gap}$	$P_{>}(x_{\rm gap})$
		(Å)			(h-1 Mpc)
HE 0515-4414	A	3088-3161	1.570	0.060	61.1	5.7	0.045
HE 2217-2818	A	3504-3579	1.913	0.062	54.3	8.4	0.012
HE 2217-2818	B	3878-3946	2.218	0.056	43.5	8.0	0.018

$\parbox{13cm}{ $^{{\rm a}}$\space For $q_{\rm 0}=0.1$ . }$

4.5 Voids

Voids along the three low-redshift lines of sight were searched for. For comparison with previous results (Carswell & Rees 1987; Crotts 1987; Ostriker et al. 1988), we identify a void as a region without any absorption stronger than $N_H{\sc i} \sim 10^{13.5} \ {\rm cm}^{-2}$ over a comoving size of at least $30\,h^{-1}$ Mpc (assuming $q_{\rm0}=0.1$).

Figures 13-14 show the voids detected in the spectrum of HE 0515-4414 and HE 2217-2818, respectively. No significant void was found in the spectrum of J2233-606. The wavelength range used for searching for voids has been selected to be redward of the QSO's Ly$\beta$ emission line and 3000 km s^-1 blueward of the QSO's Ly$\alpha$ emission to avoid the proximity effect. The wavelength range searched for voids is larger than that used to study the Ly$\alpha$ forest in other sections.

Table 7 lists the dimensions of the voids, as well as the probability of finding a void larger than their comoving size. The probability was calculated assuming a Poisson distribution of the local forest. In this case, the probability of finding a void larger than a given size $x_{\rm gap}$ is $P_{>}(x_{\rm gap}) = 1 - (1-\exp^{-x_{\rm gap}})^{n}$, where $x_{\rm gap}$ is the line interval in the unit of the local mean line interval and n is the number of lines with $N_H{\sc i} \ge 10^{13.5} \ {\rm cm}^{-2}$(Ostriker et al. 1988). The joint probability of finding two voids with a size larger than $40 \, h^{-1}$ Mpc at $z \sim 2$, as observed in the spectrum of HE 2217-2818, is of the order of $2 \times 10^{-4}$. The results correspond very well to the probability estimates derived from the simulations described above.

There are different ways to produce a void in the forest: a large fluctuation in the gas density of absorbers, enhanced UV ionizing radiation from nearby faint QSOs or star-forming galaxies, feedback processes (including shock heating) from galaxy formation (Dobrzycki & Bechtold 1991; Heap et al. 2000; Theuns et al. 2000a). We recall here that the void B in the spectrum of HE 2217-2818 corresponds to a region of above-average Doppler parameter (Sect. 4.3.3). It will be interesting to carry out deep imaging around HE 2217-2818 to identify QSO candidates and investigate whether a local ionizing source is responsible for the $\sim$ $50\,h^{-1}$ Mpc voids.

Figure 13: The spectrum of HE 0515-4414 with the void at z = 1.570. See the text for the details.