1 Introduction

The Ly$\alpha$ forest imprinted in the spectra of high-z QSOs arises from the fluctuating low-density intergalactic medium (IGM), highly photoionized by the metagalactic UV background. Since the universe expands adiabatically and the Ly$\alpha$ forest is in photoionization equilibrium with the UV background, the temperature of the Ly$\alpha$ forest as a function of z provides a unique and powerful tool to probe the physical state of the IGM and the reionization history of the universe (Hui & Gnedin 1997; Schaye et al. 1999; Ricotti, Gnedin & Shull 2000; McDonald et al. 2000).

For a low-density (the baryon overdensity $\delta < \$$\sim$10), photoionized gas, the temperature of the gas is shown to be tightly correlated with the overdensity of the gas. This relation, i.e. the equation of state, is defined by $T=T_{0}(1+\delta)^{\gamma-1}$, where T is the gas temperature in K, T₀ is the gas temperature in K at the mean gas density and $(\gamma-1)$ is a constant at a given redshift z. Both T₀ and $(\gamma-1)$ are a function of z, depending on the thermal history of the IGM (Hui & Gnedin 1997).

This equation of state, however, is not directly observable. Instead of T and $(1+\delta)$, observations only provide the neutral hydrogen column density $N_{\mbox{H~{\sc i}}}$ (in cm^-2) and the Doppler parameter b (in km s^-1) of the forest absorption lines. In practice, a lower cutoff envelope in the $N_{\mbox{H~{\sc i}}}$-b distribution is used to probe the upper limit on the temperature of the IGM since the forest lines could be broadened by processes other than the thermal broadening. Translating a $N_{\mbox{H~{\sc i}}}$-b envelope into a $(1+\delta)$-T relation depends on many physical assumptions, such as the ionizing UV background $J_{\nu}$(Miralda-Escudé et al. 1996; Schaye et al. 1999).

This minimum Doppler cutoff $b_{\rm c}(N_{\mbox{H~{\sc i}}})$ can be described by

$$\log(b_{\rm c}) = \log(b_{0}) + (\Gamma-1) \, \log(N_{\mbox{H~{\sc i}}}),$$ (1)

where $\log(b_{0})$ is the intercept of the cutoff in the $\log (N_{\mbox{H~{\sc i}}})$-$\log b$ diagram and $(\Gamma-1)$ is the slope of the cutoff (Schaye et al. 1999).

 

 

Table 1: Analyzed QSOs.

QSO	$z_{\rm em}^{{\rm a}}$	mag $^{{\rm a}}$	$\lambda\lambda$ (Å)	$z_{\rm Ly\alpha}$	# of lines $^{{\rm b}}$	Comments
Q1101-264	2.145	16.0	3500-3778	1.88-2.11	69	UVES SV, a damped system at z=1.8386
J2233-606	2.238	17.5	3500-3890	1.88-2.20	88	UVES Commissioning I
HE1122-1648	2.400	17.7	3500-4091	1.88-2.37	179	UVES SV, split into 2$^{\rm c}$
HE2217-2818	2.413	16.0	3510-4100	1.89-2.37	159	UVES Commissioning I, split into 2$^{\rm d}$
HE1347-2457	2.534	16.8	3760-4100	2.09-2.37	91	UVES SV, incomplete observations
Q0302-003	3.281	18.4	4808-5150	2.96-3.24	107	UVES Commissioning I, incomplete observations
Q0055-269	3.655	17.9	4850-5598	2.99-3.60	264	UVES, Sep. 20-22, 2000, split into 2$^{\rm e}$
Q0000-263	4.127	17.9	5450-6100	3.48-4.02	209	Lu et al. (1996), split into 2$^{\rm f}$

$^{{\rm a}}$  Taken from the SIMBAD database. The magnitude of HE1347-2457 is from NED.
$^{{\rm b}}$  For $N_{\mbox{H~{\sc i}}} = 10^{12.5-14.5} {\rm cm}^{-2}$. Only for lines with the errors less than 25% in both $N_{\mbox{H~{\sc i}}}$ and b.
$^{\rm c}$  For Sample B, the spectrum is split into 3500-3800 Å (87 lines) and 3800-4091 Å (92 lines).
$^{\rm d}$  For Sample B, the spectrum is split into 3510-3800 Å (76 lines) and 3800-4100 Å (83 lines).
$^{\rm e}$  For Sample B, the spectrum is split into 4850-5220 Å (110 lines) and 5220-5598 Å (154 lines).
$^{\rm f}$  For Sample B, the spectrum is split into 5450-5820 Å (106 lines) and 5820-6100 Å (103 lines).

From observations alone, both no z-evolution of $N_{\mbox{H~{\sc i}}}$-independent $b_{\rm c}$(Kirkman & Tytler 1997; Savaglio et al. 1999) and increasing $b_{\rm c}$ with decreasing z (Kim et al. 1997) have been claimed. Results from simulations combined with observations have also claimed both no-z evolution of T₀ and $(\gamma-1)$(McDonald et al. 2000) and a z-evolution (Ricotti et al. 2000; Schaye et al. 2000; Kim et al. 2001a). Deriving $b_{\rm c}(N_{\mbox{H~{\sc i}}})$ from observations depends on many factors such as the method of line deblending, the number of available absorption lines, the metal-line contamination, and the method of fitting the lower $N_{\mbox{H~{\sc i}}}$-b envelope (Hu et al. 1995; Kirkman & Tytler 1997; Bryan & Machacek 2000; McDonald et al. 2000; Ricotti et al. 2000; Shaye et al. 2000; Kim et al. 2001a). The different approaches and the limited numbers of lines have led, in part, to the contradicting results on the evolution of $b_{\rm c}(N_{\mbox{H~{\sc i}}})$ in the literature.

Here, using a new, increased dataset from 7 QSOs observed with the VLT/UVES combined with the published data on one QSO obtained with Keck/HIRES, we present the evolution of the Doppler cutoff $b_{\rm c}(N_{\mbox{H~{\sc i}}})$at three redshifts $<z> \,=$ 2.1, 3.3 and 3.8. In particular, five QSOs at $<z> \,=$ 2.1 enable us to study the cosmic variance of $b_{\rm c}(N_{\mbox{H~{\sc i}}})$ and to improve a determination of $b_{\rm c}(N_{\mbox{H~{\sc i}}})$ at lower z for the first time. In Sect. 2, we briefly describe the data used in this study. The analyses of the observations are presented in Sect. 3. The discussion is in Sect. 4 and the conclusions are summarized in Sect. 5. In this study, all the quoted uncertainties are $1\sigma$ errors.

 

 

Table 2: The power-law fits to $b_{\rm c}(N_{\mbox{H~{\sc i}}})$.

Sample A
<z>	$\log N_{\mbox{H~{\sc i}}}$	# of lines	$\log(b_{\rm0,i})$	$(\Gamma -1)_{\rm i}$	$b_{\rm c,i,}(13.6)$	$\log(b_{\rm0,s})$	$(\Gamma-1)_{\rm s}$	$b_{\rm c,s}(13.6)$
2.1	13.0-14.5	349	$-0.745 \pm 0.089$	$0.150 \pm 0.006$	$19.8 \pm 0.8$	$-0.495 \pm 0.074$	$0.131 \pm 0.002$	$19.1 \pm 1.1$
3.3	13.0-14.5	275	$-0.413 \pm 0.116$	$0.122 \pm 0.008$	$17.2 \pm 1.0$	$0.279 \pm 0.473$	$0.072 \pm 0.127$	$18.4 \pm 1.5$
3.8	13.3-14.5	152	$-0.948 \pm 0.125$	$0.159 \pm 0.008$	$16.8 \pm 1.6$	$-2.699 \pm 0.159$	$0.285 \pm 0.042$	$14.9 \pm 1.1$
2.0$^{{\rm a}}$	13.0-14.5	176	$-0.214 \pm 0.180$	$0.111 \pm 0.013$	$19.1 \pm 1.1$	$-1.078 \pm 0.173$	$0.172 \pm 0.047$	$18.0 \pm 1.1$
2.2$^{{\rm a}}$	13.0-14.5	173	$-0.954 \pm 0.132$	$0.167 \pm 0.010$	$19.7 \pm 1.0$	$0.400 \pm 0.110$	$0.066 \pm 0.030$	$19.9 \pm 1.1$
3.1$^{{\rm b}}$	13.0-14.5	157	$0.645 \pm 0.379$	$0.048 \pm 0.027$	$19.9 \pm 1.5$	$-0.366 \pm 0.103$	$0.122 \pm 0.028$	$19.6 \pm 1.1$
3.4$^{{\rm b}}$	13.0-14.5	118	$-0.330 \pm 0.053$	$0.117 \pm 0.009$	$17.5 \pm 0.8$	$-0.745 \pm 0.131$	$0.142 \pm 0.035$	$15.5 \pm 1.1$
3.6$^{\rm c}$	13.3-14.5	74	$0.572 \pm 0.331$	$0.054 \pm 0.023$	$20.4 \pm 0.8$	$1.092 \pm 0.007$	$0.014 \pm 0.002$	$19.0 \pm 1.0$
3.9$^{\rm c}$	13.3-14.5	78	$-0.697 \pm 0.133$	$0.143 \pm 0.010$	$17.5 \pm 0.7$	$-0.499 \pm 0.235$	$0.122 \pm 0.062$	$14.5 \pm 1.1$
Sample B (averaged for the individual members)
2.1	13.0-14.5	...	$-0.044 \pm 0.506$	$0.102 \pm 0.037$	$21.4 \pm 0.8$	$0.053 \pm 0.731$	$0.090 \pm 0.053$	$19.2 \pm 1.1$
3.3	13.0-14.5	...	$0.617 \pm 0.966$	$0.051 \pm 0.067$	$20.3 \pm 2.7$	$0.283 \pm 0.970$	$0.072 \pm 0.068$	$18.3 \pm 2.7$
3.8	13.3-14.5	...	$-0.063 \pm 0.898$	$0.098 \pm 0.062$	$19.0 \pm 2.1$	$0.182 \pm 1.350$	$0.078 \pm 0.094$	$17.4 \pm 3.1$
Results from Schaye et al. (2000) (their sample corresponds to our Sample B)
$\sim$3.1	12.5-14.5	...	...	$\le$0.150	$\sim$23-24	...	...	...
$\sim$3.1	12.5-14.5	...	...	$\sim$0	$\sim$20-22	...	...	...
$\sim$3.8	12.5-14.8	...	...	$\le$0.150	$\sim$18-19	...	...	...

$^{{\rm a}}$  In order to have a similar number of lines at $z \sim 2.0$ and $z \sim 2.2$, the $<z> \ = 2$ sample consists of the line parameters from Q0011-264, J2233-606, HE1122-1648 at 3500-3760 Å and HE2217-2818 at 3510-3760 Å. The the $<z> \ = 2.2$ sample consists of the line parameters from Q1347-2457, HE1122-1648 at 3760-4100 Å and HE2217-2818 at 3760-4100 Å.

$^{{\rm b}}$  The $<z> \ = 3.1$ sample and the $<z> \ = 3.4$ sample are taken from Q0302-003 and Q0055-269 at 4850-5220 Å, from Q0055-269 at 5220-5598 Å, respectively.

$^{\rm c}$  The $<z> \ = 3.6$ sample and the $<z> \ = 3.9$ sample are taken from 5450-5820 Å and from 5820-6100 Å, respectively.