6 Discussion

6.1 The line number density per unit redshift of the Ly$\alpha$ forest

For a given column density range, combined HST and ground-based observations have provided evidence for a change in the line number density evolution at $z \sim 1.7$: a rapid evolution at z > 1.7 and a slow evolution at z < 1.7 (Impey et al. 1996; Riediger et al. 1998; Weymann et al. 1998; Davé et al. 1999). These observations have led to a speculation of two distinct populations in the Ly$\alpha$ forest: a rapidly evolving population which dominates at higher z and a slowly evolving population which dominates at lower z.

Our results at $N_H{\sc i} = 10^{13.64-16}~ {\rm cm}^{-2}$ suggest that the transition from the stronger evolution to the weaker evolution in dn/dz occurs at $z \sim 1.2$ (UM18 fits in the picture and is not an outlier), rather than at $z \sim 1.7$as suggested by previous observations and numerical simulations. To be conservative, our results show that dn/dz at $N_H{\sc i} = 10^{13.64-16}~ {\rm cm}^{-2}$ continues to decrease at a similar rate from $z = 4 \rightarrow 1.5$, with a suggestion of slowing down in the evolution towards lower z.

The physics of the Ly$\alpha$ forest at z > 2 is determined mainly by the Hubble expansion and the ionizing background, $J_\nu$ (or the H I photoionization rate, $\Gamma_H{\sc i}$). If the forest is "fixed" in comoving coordinates for q₀ = 0.5 and $\Lambda = 0$, the observed number density of the Ly$\alpha$ forest is proportional to $[(1+z)^{5} \, \Gamma_H{\sc i}^{-1}(z)]^{\beta-1} \, (1+z)^{-3/2}$, where $\beta$ is from $f(N_H{\sc i}) \propto N_H{\sc i}^{-\beta}$. This implies that dn/dz for a given column density threshold decreases as z decreases (Miralda-Escudé et al. 1996; Davé et al. 1999). If we assume $\beta \sim 1.5$, dn/d $z \propto (1+z)\,\Gamma_H{\sc i}^{-0.5}(z)$. For a constant $\Gamma_H{\sc i}$, this is much lower than the observed index of dn/dz, $2.19 \pm 0.27$, suggesting structure evolution and/or $\Gamma_H{\sc i}$ evolution in dn/dz.

Recent numerical simulations suggest that a decrease of $\Gamma_H{\sc i}$ at z < 2plays a more important role to change the slope in dn/dz at $z \sim 1.7$ due to the decreasing QSO luminosity function at z < 2 (Davé et al. 1999; Riediger et al. 1998; Theuns et al. 1998; Zhang et al. 1998). The discrepancy between our observations and simulations could be due to limited box sizes at z < 2 in most simulations (losing large-scale power), to numerical resolutions (underestimating $\tau_H{\sc i}$or the number of lines at lower z) or to incorrect $\Gamma_H{\sc i}$. If we take the results from most simulations that $\Gamma_H{\sc i}$ is the main drive of the slope change in dn/dz, this discrepancy could simply indicate that $\Gamma^{-1}_H{\sc i}$ in most simulations, i.e. the QSO-dominated Haardt-Madau $\Gamma^{-1}_H{\sc i}$(Haardt & Madau 1996), is underestimated at z < 2 and that $\Gamma^{-1}_H{\sc i}$ at z < 2changes more slowly than a QSO-dominated $\Gamma^{-1}_H{\sc i}$, i.e. there is a non-negligible contribution from galaxies at z < 2.

6.2 The z-evolution of b $_{\mathsfsl c}$( ${\mathsfsl{N}}_{{\mathsf H}\!\!{\mbox{\fontsize{6}{8} \selectfont\sffamily\bfseries I}}}$)

Assuming a truncated Gaussian b distribution with a lower $N_H{\sc i}$-independent $b_{\rm c}$, Kim et al. (1997) concluded that $b_{\rm c}$over $N_H{\sc i} = 10^{12.8-16}~{\rm cm}^{-2}$increases as z decreases: 15 km s^-1 at $z \sim 3.7$, 17 km s^-1at $z \sim 3.3$, 20 km s^-1 at $z \sim 2.9$, and 22-24 km s^-1at $z \sim 2.3$. This result has been explained by an additional heating due to the on-going He II reionization, although the high $b_{\rm c}$ value at $z \sim 2.3$does not agree with any theoretical explanations (Kim et al. 1997; Haehnelt & Steinmetz 1998; Theuns et al. 2000b; Schaye et al. 2000).

Subsequent studies on the z-evolution of $b_{\rm c}$ have led to contradictory results. While $b_{\rm c}$ is clearly dependent on $N_H{\sc i}$ (Kirkman & Tytler 1997; Zhang et al. 1997), $b_{\rm c}(N_H{\sc i})$ and the mean b value at $<\!z\!> \ = 2.7$(Kirkman & Tytler 1997) and at $<\!z\!> \ = 1.7$ (Savaglio et al. 1999) does not show any noticeable difference compared with at z > 3. Combining the observations with the numerical simulations, Schaye et al. (2000) found that $b_{\rm c}$ at the fixed overdensity $\delta=0$, $b_{\delta =0}$, increases from $z \sim 4.5$ ( $b_{\delta=0} \sim 14.5$ kms^-1) to $z \sim 3$ ( $b_{\delta=0} \sim 19.5$ kms^-1) due to He II reionization at $z \sim 3$and then decreases from $z \sim 3$ ( $b_{\delta=0} \sim 19.5$ kms^-1) to $z \sim 1.8$ ( $b_{\delta=0} \sim 14$ kms^-1). Ricotti et al. (2000) also found a similar increase in $b_{\delta =0}$ at $z \sim 3$, although their $b_{\delta =0}$at z > 2.8 and at z < 2.8 can be considered to be constant at $b_{\delta=0} \sim 14.5$ kms^-1 and at $b_{\delta=0} \sim 21$ km s^-1, respectively. On the other hand, adopting a slightly different approach for identifying absorption lines instead of the Voigt profile fitting, McDonald et al. (2000) found that there is no $b_{\rm c}$ evolution over 2.1 < z < 4.4 at the slightly higher overdensity $\delta=0.4$.

Figure 21 shows the cutoff b values at the fixed column density $N_H{\sc i} = 10^{13.5} ~ {\rm cm}^{-2}$, $b_{\rm c, {\rm 13.5}}$, from the two power law fits in Sect. 4.3 as a function of z. Keep in mind that $b_{\rm c}(N_H{\sc i})$from the iterative power law fit is an upper limit, while $b_{\rm c}(N_H{\sc i})$ from the smoothed b distribution is a lower limit. The $b_{\rm c, {\rm 13.5}}$ value shows a slight increase with decreasing zfrom both power law fits, with a possible local $b_{\rm c, {\rm 13.5}}$maximum at $z \sim 2.9$ (with the caveat that the line list of Q0302-003 is generated by a different fitting program with respect to the other line lists. This could introduce an artificial result at $z \sim 2.9$ as shown in Sect. 5.4, although its redshift range suggests an influence of additional heating if He II reionization does occur at $z \sim 3$). When all the values from both fits are averaged, $b_{\rm c, {\rm 13.5}} = 17.6 \pm 1.6$ at $z \sim 3.75$ is smaller than $b_{\rm c, {\rm 13.5}} = 19.9 \pm 1.2$at $z \sim 2.1$, but the difference is significant only at the $1.44\sigma$ level.

In simulations, $b_{\rm c}$ is usually measured at a fixed overdensity $\delta$ rather than at a fixed column density. Translating an overdensity into the corresponding column density is not trivial and depends on many uncertain parameters, such as the ionizing background and the reionization history. If the simple law between $\delta$ and $N_H{\sc i}$ by Davé et al. (1999) is assumed, then $\delta$ becomes:

$$N_H{\sc i} \sim \left[0.05\, (1+\delta) \, 10^{0.4 \,z}\right]^{1.43} \times 10^{14} \ {\rm cm}^{-2}.$$ (7)

For $\delta=0$, the corresponding $N_H{\sc i}$ is $1.92 \times 10^{14}$, $6.04 \times 10^{13}$, $4.56 \times 10^{13}$, $2.28 \times 10^{13}$, $1.87 \times 10^{13}$ and $1.15 \times 10^{13} \ {\rm cm}^{-2}$ at $<\!z\!> \ =$ 3.75, 2.87, 2.66, 2.13, 1.98 and 1.61, respectively. Although this conversion does not include the effects of the He II reionization, we assume that it is correct at least on a relative scale at $z \sim 3.75$and at $z \sim 2.1$, where the He II reionization would not affect the temperature of the IGM as strongly as at $z \sim 3$. Therefore, we only discuss the relative behavior of $b_{\delta =0}$as a function of z, in particular, our $b_{\delta =0}$ at $z \sim 2.1$with other results mentioned above.

For the iterative power law fit (the power law fit to the smoothed b distribution), the b value at $\delta=0$, $b_{\delta =0}$, is 20.8 (17.2), 20.1 (17.2), 19.4 (17.8), 20.5 (17.5), 22.4 (21.6) and 20.4 (18.8) km s^-1 at $<\!z\!> \ =$ 3.75, 2.87, 2.66, 2.13, 1.98 and 1.61, respectively. In the second panel of Fig. 21, $b_{\delta =0}$ is fairly constant with z as $b_{\delta=0} \sim$ 17-20 km s^-1, with a possible local maximum $b_{\delta=0} \sim$ 22 km s^-1at $z \sim 2.9$. The observed behavior of $b_{\delta =0}$ is qualitatively in agreement with the results from McDonald et al. (2000). While the observations agree with the fairly constant $b_{\delta =0}$ at z < 3 derived by Ricotti et al. (2000), they do not show the abrupt increase of $b_{\delta =0}$ across $z \sim 3$ as large as $\sim$7 km s^-1 found by Ricotti et al. The observations at $z \sim 3.75$ and at $z \sim 2.1$agree with the results by Schaye et al. (2000) which show similar $b_{\delta =0}$ at $z \sim 3.75$and at $z \sim 2.1$. In addition, the observations do not show a strong decrease of $b_{\delta =0}$ from $z \sim 3$ to $z \sim 2$as large as $\sim$4 km s^-1 as found by Schaye et al. (2000). However, note that, considering the large error bars of Schaye et al. (2000), the significance of the decrease of $b_{\delta =0}$from $z \sim 3$ to $z \sim 2$ is not very strong and their result is not in disagreement with ours. It should also be recalled that we are using the scaling law between $\delta$ and $N_H{\sc i}$estimated from the QSO-dominated Haardt-Madau $\Gamma_H{\sc i}$. If this QSO-dominated UV background is underestimated at z < 2as suggested by the evolution of the absorption line number density, the actual $b_{\delta =0}$ at z < 2.4 can be higher than $b_{\delta =0}$ in Fig. 21.

The third panel of Fig. 21 shows the median b values as a function of z measured for two column density ranges: $N_H{\sc i} = 10^{13.1-14}~ {\rm cm}^{-2}$ and $N_H{\sc i} = 10^{13.8-16} ~{\rm cm}^{-2}$. It is rather difficult to interpret the z-dependence of the median bvalues. It could be constant at 1.5 < z < 4 with a small cosmic variance at $z \sim 2$. On the other hand, it could be decreasing with z at z < 3.1, if we discard the median b values at $<\!z\!> \ = 1.61$, based on a small number of absorption lines. Although simulations correctly predict the shape of the observed b distributions, the predicted median b values are typically 5-10 km s^-1smaller than the observed ones at all z(Bryan & Machacek 2000; Machacek et al. 2000; Theuns et al. 2000b).

The bottom panel of Fig. 21 shows the power law slope of the $N_H{\sc i}$-$b_{\rm c}$ distribution, $(\Gamma _{\rm T}-1)$, as a function of z (see Eq. (1)). No particular trend is apparent and $(\Gamma _{\rm T}-1)$ shows very little evolution at z < 3.1. Note that the lower $(\Gamma _{\rm T}-1)$ at $<\!z\!> \ = 3.75$ is in part due to the lack of lines with b < 15 kms^-1 and $N_H{\sc i} \le 10^{13.4} ~ {\rm cm}^{-2}$ (Fig. 8). When $(\Gamma _{\rm T}-1)$ is averaged over all the measured values, $(\Gamma_{\rm T}-1) = 0.16 \pm 0.03$ at z < 3.1 is larger than $(\Gamma_{\rm T}-1) = 0.07 \pm 0.02$ at $<\!z\!> \ = 3.75$. Due to the lower $(\Gamma _{\rm T}-1)$ at $<\!z\!> \ = 3.75$, it also seems clear from Fig. 8 that gas at lower overdensities is cooler at z < 3.1 than at higher z(keep in mind that a fixed column density corresponds to a larger gas overdensity as z decreases due to the Hubble expansion).

If we assume Eq. (7) again and $T\sim 59.2 \, b^{2}$ for thermally broadened lines, $(\gamma_{\rm T}-1) \sim 2.857 \, (\Gamma_{\rm T}-1)$. When this simple conversion law is assumed, $(\gamma_{\rm T}-1) = 0.46 \pm 0.10$ at z < 3.1 and $(\gamma_{\rm T}-1) = 0.18 \pm 0.05$ at $z \sim 3.75$. These $(\gamma_{\rm T}-1)$ could be considered to be consistent with the results by McDonald et al. (2000) within their error bars, although their error bars at z =3.9 ( $(\gamma_{\rm T}-1) = 0.42 \pm 0.45$) and at z=3 ( $(\gamma_{\rm T}-1) = 0.30 \pm 0.30$) are rather large. Their $(\gamma_{\rm T}-1)=0.5 \pm 0.15$ at $z \sim 2.4$ agrees with our $(\gamma_{\rm T}-1) = 0.46$. Our $(\gamma_{\rm T}-1)$ is marginally in agreement with the results from Ricotti et al. (2000) and from Schaye et al. (2000). Their $(\gamma_{\rm T}-1)$ values are lower at $z \sim 2$ and higher at $z \sim 3.7$, but within the error bars.

Figure 21: The z-evolution of $b_{\rm c, {\rm 13.5}}$, $b_{\delta =0}$, median b and $(\Gamma _{\rm T}-1)$from the two power law fits in Sect. 4.3. Stars and triangles represent the parameters measured from the iterative power law fit and from the smoothed b distribution, respectively. Vertical bars represent 1$\sigma$ errors. In the third panel, error bars in the x-axis represent the z ranges over which the median b values were estimated. Solid lines represent the median b values over $N_H{\sc i} = 10^{13.1-14}~ {\rm cm}^{-2}$, while dotted lines represent the median b values over $N_H{\sc i} = 10^{13.8-16} ~{\rm cm}^{-2}$.

6.3 The optical depth distribution function

While the one-point function of the flux is more closely related to observations, the one-point function of the optical depth is usually calculated in simulations (Zhang et al. 1998; Machacek et al. 2000). In the simulation by Machacek et al. (2000), the H I optical depth $\tau$ at which the maximum $\tau P(\tau )$ occurs, $\tau_{{\rm max}}$, is $\tau \sim 0.8$ at z=4, $\tau \sim 0.09$ at z=3, and $\tau \sim 0.013$ at z=2. Figure 22 shows $\tau P(\tau )$ as a function of $\tau$, which is calculated from the observed spectra at z < 2.4and the spectra generated with noise at z > 2.4. The $\tau_{{\rm max}}$ value is $\tau \sim 0.6$ at $<\!z\!> \ = 3.75$, $\tau \sim 0.13$ at $<\!z\!> \ = 2.87$, and $\tau \sim 0.08$ at $<\!z\!> \ = 2.66$, respectively. The observed $\tau_{{\rm max}}$ shows a behavior similar to the simulated results by Machacek et al. (2000). However, $\tau_{{\rm max}}$ converges to $\tau \sim 0.04$ at z < 2.4, not showing any z-dependence. Also there is no z-dependence of $\tau P(\tau )$ at $\tau \le 0.04$ and at z < 3.1.

The optical depth $\tau \sim 0.04$ corresponds to $F \sim 0.96$, while $\tau \sim 0.013$ corresponds to $F \sim 0.99$, almost to the continuum level. As z decreases, the number of pixels with F = 0.96-0.99 increases. These pixels are noise-dominated by the limited S/N and the continuum fitting uncertainty. Therefore, instead of showing the expected z-dependence of $\tau_{{\rm max}}$ and $\tau P(\tau )$, the observed $\tau_{{\rm max}}$ and $\tau P(\tau )$ approach to an asymptotic $\tau_{{\rm max}} \sim 0.04$ value at z < 2.4and an asymptotic $\tau P(\tau )$ at $\tau \le 0.04$ and at z < 3.1, respectively.

At $0.1 < \tau < 3$ (or 0.05 < F < 0.9), the observed $\tau P(\tau )$ is simply a different way of viewing the one-point function of the flux. As z decreases, $\tau P(\tau )$ decreases due to the expansion of the universe. At $\tau > 4$ (or F < 0.05), $\tau P(\tau )$starts to converge again since it typically samples saturated regions, again dominated by noise.

Figure 22: The z-evolution of $\tau P(\tau )$ as a function of $\tau$. The $\tau P(\tau )$ values are calculated from the observed spectra at z < 2.4 and from the spectra generated with noise at z > 2.4.

 

 

Table 9: The lower bounds on $\Omega _{\rm b}$.

z	$\overline {\tau }_H{\sc i}$	$\Omega_{\rm b}^{{\rm a}}$	$\Omega_{\rm b}^{{P(F),{\rm b}}}$
		( $\times h^{-1.75}$)	( $\times h^{-1.5}$)
1.61	0.086	$\ge$0.010	$\ge$0.020
1.98	0.161	$\ge$0.016	$\ge$0.021
2.13	0.131	$\ge$0.012	$\ge$0.021
2.66	0.234	$\ge$0.014	$\ge$0.020
2.87	0.275	$\ge$0.015	$\ge$0.020
3.75	0.733	$\ge$0.013	$\ge$0.013

$^{{\rm a}}$

For T = 6000 K, $\Gamma = 0.9 \times 10^{-12} ~ {\rm s}^{-1}$ (at $z \sim 1.6$), $1.4 \times 10^{-12} ~ {\rm s}^{-1}$ (at $z \sim 2$-3), $0.64 \times 10^{-12} ~ {\rm s}^{-1}$ (at $z \sim 3.75$), and $\Omega_{0}=1$(Haardt & Madau 1996; Hui & Gnedin 1997; Weinberg et al. 1997).

$^{{\rm b}}$

For T = 6000 K, $\Gamma = 0.9 \times 10^{-12} ~ {\rm s}^{-1}$ (at $z \sim 1.6$), $1.4 \times 10^{-12} ~ {\rm s}^{-1}$ (at $z \sim 2$-3), $0.64 \times 10^{-12} ~ {\rm s}^{-1}$ (at $z \sim 3.75$), $\Omega_{0}=1$, $\beta=1.58$, and $H(z)=H_{\rm0} \Omega_{\rm0}^{\frac{1}{2}} (1+z)^{\frac{3}{2}}$(Haardt & Madau 1996; Hui & Gnedin 1997; Weinberg et al. 1997).

6.4 The baryon density $\Omega_\mathsf{b}$

We derived the baryon density, $\Omega _{\rm b}$, from two properties of the Ly$\alpha$ forest, $\overline {\tau }_H{\sc i}$(including all the Ly$\alpha$ forest regardless of $N_H{\sc i}$) and P(F) (only for the forest with $N_H{\sc i} \le 10^{16} ~ {\rm cm}^{-2}$).

If $\overline{\tau}_H{\sc i} \ \le 1.24$, the lower limits on the baryon density become

$$\begin{eqnarray*}\Omega_{\rm b}^{\overline{\tau}_H{\sc i}} \ge 5.1 \times 10^{4}... ...rline{\tau}_H{\sc i})) (1+z)^{-2.5} (1 + \Omega_{\rm0}z)^{0.25} \end{eqnarray*}$$

$$\ \ \ \ \ \ \ \ \ \ \times h^{-1.75} T_{0}^{0.35} \, \Gamma_H{\sc i}^{0.5},$$ (8)

where all the symbols have their usual meanings in this study (Weinberg et al. 1997).

Table 9 lists the lower limits on $\Omega _{\rm b}$at different z, together with the parameter values used for calculating $\Omega_{\rm b}^{\overline{\tau}_H{\sc i}}$. It should be noted that the temperature used to calculate Eq. (8) is smaller than the values derived in Sect. 6.2, but in the present discussion we are only interested in the lower bounds on $\Omega _{\rm b}$. These lower limits on $\Omega_{\rm b}^{\overline{\tau}_H{\sc i}}$ are consistent with $\Omega_{\rm b} =0.0125 \, h^{-2}$ from the Big Bang nucleosynthesis analysis (Copi et al. 1995). These values also indicate that about 90% of all baryons reside in the Ly$\alpha$ forest at 1.5 < z < 4.

The lower bounds on the density parameter from the one-point function, $\Omega_{\rm b}^{P(F)}$, are given by Weinberg et al. (1997) as

$$\begin{eqnarray*}\Omega_{\rm b}^{P(F)} & \ge & 0.021 h^{-3/2}\!\! \left(\! \frac... ...{\gamma}} P(F)\, {\rm d}F/F]^{\beta_{\gamma}/2}}{0.70} \!\right) \end{eqnarray*}$$

$$\begin{eqnarray*}\ \ \ \ \ \ \ \ \ \ \times \left(\frac{4}{1+z}\right)^{3} \left... .../2} \left( \frac{T_{\rm0}}{10^{4} ~ {\mathrm K}} \right )^{0.35} \end{eqnarray*}$$

$$\ \ \ \ \ \ \ \ \ \ \times \left( \frac{\Gamma}{10^{-12} \ {\rm s}^{-1}} \right)^{1/2},$$ (9)

where $\beta_{\gamma} \equiv (2{-}0.7\gamma_{\rm T})^{-1} =1.6{-}1.8$, where $\gamma_{\rm T}$ is the power law index of the equation of state (Weinberg et al. 1997).

Table 9 lists the lower bounds on $\Omega_{\rm b}^{P(F)}$ along with the parameter values used to calculate $\Omega_{\rm b}^{P(F)}$. The $\Omega_{\rm b}^{P(F)}$ values are larger than $\Omega_{\rm b}^{\overline{\tau}_H{\sc i}}$ since $\overline {\tau }_H{\sc i}$ is not the true mean H I opacity, but the effective opacity which underestimates the true opacity when absorption lines become saturated. The lower $\Omega _{\rm b}$ limits from P(F) are about a factor of $1.6 \, h^{0.5}$ larger than the Big Bang nucleosynthesis analysis, $\Omega_{\rm b} =0.0125 \, h^{-2}$.

Our new lower bounds on $\Omega _{\rm b}$ are a factor of 1.5 smaller than some of the previous results, $\Omega_{\rm b} = 0.017$- $0.03 \, h^{-2}$ (Rauch et al. 1997; Zhang et al. 1998; Burles et al. 1999; Kirkman et al. 2000; McDonald et al. 2000), but still consistent with them within the error bars. However, our lower $\Omega _{\rm b}$ bounds are not consistent with the derived $\Omega_{\rm b} = 0.005 \, h^{-2}$- $0.01 \, h^{-2}$ from the high D/H measurements (Songaila et al. 1994; Rugers & Hogan 1996).