4 Discussion

Converting $(\Gamma-1)$ and $b_{\rm c}(13.6)$ to the corresponding $(\gamma-1)$ and T₀ depends on many uncertain parameters, such as the UV background and the reionization history. If the UV background is dominated by QSOs without any extra heating, T₀ decreases and $(\gamma-1)$ increases as z decreases, until they approach asymptotic values. The $b_{\rm c}(13.6)$ value, however, still increases with decreasing z, until it approaches an asymptotic value and finally decreases again at z < 2 (Schaye et al. 1999).

Instead of converting to T₀, we compare our $b_{\rm c,i}(13.6)$with Figs. 3 and 4 of Schaye et al. (2000) from their simulations assuming the QSO-dominated Haardt-Madau UV background, $J_{\rm HM}$ (Haardt & Madau 1996). Our $b_{\rm c}(13.6)$ is in agreement with their simulated $b_{\rm c}(13.6)$values without extra He II heating, although the difference increases as zdecreases (ours being a factor of 1.1 lower at $z \sim 2$). It is not clear what causes their numerical simulations without the He II reionization produce their $b_{\rm c}(13.6)$similar to ours which suggest the extra heating. One of the explanations could be a weaker effect on the forest from the He II reionization than their simulations suggest. When $T_{\rm0}$is converted from the same simulations (their Fig. 2), $T_{\rm0}$ decrease as z decreases.

For $(\gamma-1)$, we assume the conversion law between $N_{\mbox{H~{\sc i}}}$ and $\delta$ by Schaye (2001) assuming $J_{\rm HM}$ and $T\sim 59.2 \, b^{2}$ for thermally broadened lines. The conversion law is defined by

$$\begin{eqnarray*}N_{\mbox{H~{\sc i}}} \sim 2.7 \times 10^{13} \, (1+\delta)^{1.5... ....5}\, \left(\frac{f_{\rm g}}{0.16}\right)^{0.5} \ {\rm cm}^{-2}, \end{eqnarray*}$$ (2)

where $T_{\rm0} \equiv T_{\rm0,4} \times 10^{4}$ K, the H I photoionization rate $\Gamma_{{\mbox{H~{\sc i}}}} \equiv \Gamma_{{\mbox{H~{\sc i}}}, 12} \times 10^{12}$ s^-1, $\Omega_{\rm b}$is the baryon density, h is the Hubble constant divided by 100, and $f_{\rm g}$ is the fraction of the mass in gas (Schaye 2001). We read $T_{\rm0}$ from Fig. 3 of Schaye et al. (2000) and $\Gamma_{{\mbox{H~{\sc i}}}}$ from Fig. 8 of Haardt & Madau (1996), while we assume $\Omega_{\rm b}\,h^{2} = 0.02$and $f_{\rm g} = 0.16$. At $<z> \ = 2.1$, 3.3 and 3.8, $(\gamma-1) \sim 0.417$, 0.386 and 0.441. Assuming $J_{\rm HM}$, there is no clear z-evolution of $(\gamma-1)$ within large uncertainties. Our values and Schaye et al.'s ( $(\gamma-1) \sim$ 0.4, 0.35 and 0.25 at $<z> \ = 2.1$, 3.3 and 3.8 assuming $J_{\rm HM}$) agree at $<z> \ =$ 2.1, 3.3, while our value is a factor of 1.8 larger than theirs at <z> = 3.8. Our $(\gamma-1)$ values agree with those of McDonald et al. at the similar z ranges within uncertainties, although their simulations do not assume $J_{\rm HM}$.