All Figures

$\begin{figure} \par\includegraphics[angle=-90,width=8.4cm,clip]{6566fig1.ps} \end{figure}$ Figure 1: Contour plots for the parameters of the temperature-density relation derived from the simulated data. The crosses represent the position of the true values of $\gamma = 1.3$ and $\log T_0 = 4.3$ while the stars indicate the estimated minimum. The outer contours give the joint $1\sigma$ (solid line) and $2\sigma$ (dotted line) confidence levels, respectively. The inner contour represents the $1\sigma$ confidence level for one free parameter, i.e. the projection of the inner contour to the axes leads to the $1\sigma$ errors of $\log T_0$ and $\gamma$ , respectively. The signal-to-noise ratio of the artificial He II spectra is indicated in each panel. The confidence levels in case of S/N = 5 ( upper left panel) are outside the presented parameter range.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.4cm,clip]{6566fig2.ps} \end{figure}$ Figure 2: Results for the fit of the artificial data. The panels show the derived values for $\log T_0$ , $\gamma$ , and $\log\eta$ ( from the left to the right). The error bars correspond to the inner $1\sigma$ confidence contours of Fig. 1. The dotted lines indicate the true values. For comparison the $\eta$ values estimated under the assumption of pure turbulent line broadening are presented as triangles in the right panel.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.4cm,clip]{6566fig3.ps} \end{figure}$ Figure 3: Contour plots for the parameters of the temperature-density relation derived from the artificial data using three different H I line lists for the He II fitting (see text). The crosses represent the position of the true values of $\gamma = 1.3$ and $\log T_0 = 4.3$ while the stars indicate the estimated minimum. Contour levels are given as in Fig. 1.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig4.ps} \end{figure}$ Figure 4: Illustration of the importance of the H I fit. The upper panel shows a portion of the artificial H I data (histogram-like) and two different fits where the saturated feature at $4007~{\rm \AA}$ is modelled with one (dashed line) or two (solid line) Doppler components, respectively. The inset in the upper panel illustrates the only marginal difference between both fits. The lower panel shows the artificial He II data (histogram-like) and the best fits in case of the one component approach (dashed line; $\log T_0 \approx 4.25$ , $\gamma \approx 1.08$ ) or two component fit (solid line; $\log T_0 \approx 4.25$ , $\gamma \approx 1.24$ ), respectively. The dot-dashed line represents the best fit using the perfect line list ( $\log T_0 = 4.30$ , $\gamma = 1.30$ ).
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig5.ps} \end{figure}$ Figure 5: Results from HE 2347-4342 for $\log\eta$ (open circles; upper panel), $\log T_0$ ( middle panel), and $\gamma$ ( lower panel). The triangles in the upper panel present the resulting $\eta$ value assuming pure turbulent broadening. For clarity error bars are not given for the turbulent case, but are of the same size as those of the temperature-density relation fit. The $1\sigma$ uncertainties of $\log T_0$ and $\gamma$ would comprise the whole presented parameter range. Fit sections of $\sim$ $5~{\rm\AA}$ have been selected a priori. Sections marked with a cross represent redshift intervals without extremely saturated absorption features. The dotted lines in the lower panels indicated the expected value interpolated from the measurements of Schaye et al. (2000).
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig6.ps} \end{figure}$ Figure 6: As Fig. 5 but for HS 1700+6416.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig7.ps} \end{figure}$ Figure 7: Correlation between the difference in the quality of the fit, i.e. the $\chi ^2$ , and the difference in the $\eta$ value. The better the fit taking into account a temperature-density relation in comparison the the fit assuming pure turbulent broadening, the larger is the resulting $\log\eta$ for the best fit. The dotted line represents a linear fit to the Ly $\alpha$ forest data, i.e. HS 1700+6416 and HE 2347-4342 at redshifts z < 2.75.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig8.ps} \end{figure}$ Figure 8: Relation between H I column density $\log N_{H {\sc i}}$ and the absorber's temperature $\log T$ according to Eq. (6). The relations are shown for $\log T_0 = 4.3$ (solid lines) and $\log T_0 = 4.0$ (dotted lines) and $\gamma = 1.7$ , 1.5, 1.3, and 1.1.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig9.ps} \end{figure}$ Figure 9: Distribution of the parameters of the temperature-density relation estimated individually for the sight lines towards HS 1700+6416 (black) and HE 2347-4342 (light; z < 2.75). The value of $\log T_0 = 4.3$ has been fixed when estimating $\gamma$ ( left panel), while $\gamma = 1.3$ when $\log T_0$ has been optimized ( right panel).
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fg10.ps} \end{figure}$ Figure 10: Best fit towards HE 2347-4342 in the redshift range $2.58 \le z \le 2.73$ ( upper panel). Fitted parameters are the temperatures of the absorbers ( lower panel), the He II/H I column density ratio $\eta$ ( middle panel), and the scale of $\eta$ variation. The resulting temperatures are presented as open circles in the lower panel. However, most of them have large error bars and are actually upper limits. No temperature is given if turbulent line broadening is preferred (see also text). At $z \sim 2.69$ two saturated H I lines appear to be stronger than the corresponding He II features in the line center leading to a poor fit in this range.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fg11.ps} \end{figure}$ Figure 11: Best fit and resulting parameters towards HS 1700+6416 in the redshift range 2.58 < z < 2.75; lines and symbols as in Fig. 10.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fg12.ps} \end{figure}$ Figure 12: Distribution of the estimated parameters in the redshift range 2.58 < z < 2.74 towards HE 2347-4342 and HS 1700+6416. Presented are the distributions of the line temperature $\log T$ ( left panel), the He II/H I column density ratio $\log\eta$ per scale ( middle panel, thick line), and the scale size $\Delta z$ of $\eta$ variations ( right panel, thick line). The thin-lined histograms in the right panels indicate the distributions of $\log\eta$ and $\Delta z$ , respectively, in the case of turbulent line widths $b_{He {\sc ii}} = b_{H {\sc i}}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fg13.ps} \end{figure}$ Figure 13: Line temperature $\log T$ versus H I column density $\log N_{H {\sc i}}$ for the combined sample of HE 2347-4342 and HS 1700+6416 in the redshift range 2.58 < z < 2.74. Only lines that lead to realistic best-fit temperatures are indicated. The mean redshift of the considered line sample is $z = 2.663 \pm 0.082$ . The dotted line indicated a temperature-density relation with $\gamma = 1.17$ and $\log T_0 = 4.23$ as interpolated from the results of Schaye et al. (2000).
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.4cm,clip]{6566fig3.ps} \end{figure}$	Figure 3: Contour plots for the parameters of the temperature-density relation derived from the artificial data using three different H I line lists for the He II fitting (see text). The crosses represent the position of the true values of $\gamma = 1.3$ and $\log T_0 = 4.3$ while the stars indicate the estimated minimum. Contour levels are given as in Fig. 1.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig6.ps} \end{figure}$	Figure 6: As Fig. 5 but for HS 1700+6416.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig8.ps} \end{figure}$	Figure 8: Relation between H I column density $\log N_{H {\sc i}}$ and the absorber's temperature $\log T$ according to Eq. (6). The relations are shown for $\log T_0 = 4.3$ (solid lines) and $\log T_0 = 4.0$ (dotted lines) and $\gamma = 1.7$ , 1.5, 1.3, and 1.1.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fig9.ps} \end{figure}$	Figure 9: Distribution of the parameters of the temperature-density relation estimated individually for the sight lines towards HS 1700+6416 (black) and HE 2347-4342 (light; z < 2.75). The value of $\log T_0 = 4.3$ has been fixed when estimating $\gamma$ ( left panel), while $\gamma = 1.3$ when $\log T_0$ has been optimized ( right panel).
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{6566fg11.ps} \end{figure}$	Figure 11: Best fit and resulting parameters towards HS 1700+6416 in the redshift range 2.58 < z < 2.75; lines and symbols as in Fig. 10.
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