3 The ionising background at 912 Å

In Fig. 3 we show the modelled UV background, $J(\nu,z)$, at the Lyman limit as a function of redshift (solid lines) for the flat universe with $\Omega _{\rm m}=1$. The total background is shown as the sum of the QSO contribution (the same in each model; dotted line) and the galaxy contribution (scaled with $f_{\rm esc}$; dashed lines). As predicted by other authors, for large values of $f_{\rm esc}$ the ionising background produced by galaxies dominates over the flux from QSOs (Giallongo et al. 1997; Devriendt et al. 1998; Shull et al. 1999).

At high redshift, the value of the UV background is constrained by the analysis of the proximity effect, i.e. the decrease in the number of intervening absorption lines that is observed in a QSO spectrum when approaching the QSO's redshift (Bajtlik et al. 1988). Using high resolution spectra, Giallongo et al. (1996) derived $J(912~{\rm\AA})=5.0_{-1}^{+2.5} \times 10^{-22}\;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$ for 1.7 < z < 4.1 (see also Giallongo et al. 1999). Larger values are obtained by Cooke et al. (1997), $J(912~{\rm\AA})=1.0_{-0.3}^{+0.5} \times 10^{-21}\;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$ for 2.0 < z < 4.5. A recent re-analysis of moderate resolution spectra by Scott et al. (2000) has lead to $J(912~{\rm\AA})=7.0_{-4.4}^{+3.4} \times 10^{-22}\;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$ for the same redshift range. We show these measurements in Fig. 3 with a shaded area: the spread of the measurements obtained with different methods and data gives an idea of the uncertainties associated with the study of the proximity effect.

Measurements of the ionising background at low redshift are not less uncertain. Kulkarni & Fall (1993) reported a first tentative detection of the proximity effect in a sample of 13 QSOs at z<1 observed with HST. They obtained $J(912~{\rm\AA})= 6_{-4}^{+30} \times 10^{-24}\;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$, the large uncertainties due to the small number of available absorbers. Vogel et al. (1995) derived a $2\sigma$ upper limit J(912 Å) $<8.0 \times 10^{-23}\;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$, by studying H$\alpha$ emission in a high latitude Galactic cloud. This upper limit is shown in Fig. 3.

We must remember here that our model does not take into account the Lyman-continuum emission from recombination in Ly$\alpha$ clouds. Haardt & Madau (1996) have shown that radiative recombination provides an important contribution to the ionising background. Using only QSOs as source of ionising radiation, they obtain J(912 Å)  $=5 \times 10^{-22} \;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$ at z=2.5, well within the shaded area in Fig. 3. Similar results are obtained by Fardal et al. (1998). Although the cloud contribution to the background depends on the adopted emissivity and intergalactic absorption, we have obtained a rough estimate of its importance by using UV background spectra kindly provided by F. Haardt. At z=3, the models of Haardt & Madau (1996) are a factor 1.7 higher than for the case of a purely absorbing medium; at z=0, the factor reduces to 1.3. A simple linear interpolation between these two points reproduces the actual data for 0<z<5 within 5%. The models shown in Fig. 3 are multiplied for this z-dependent factor.

Figure 4: Number density evolution of the Ly$\alpha$ forest with $N_H{\sc i}=10^{13.64-16} \;{\rm cm^{-2}}$, for the two cosmologies adopted in this paper. Dotted lines refer to the evolution compatible with an ionising UV background due only to QSOs. Solid lines show the evolution when both QSOs and galaxies contribute to the background, for the models with $f_{\rm esc}=0.05$ (upper line), 0.1 and 0.4 (lower line). Data points come from several observations in the literature for the column density range $N_H{\sc i}=10^{13.64-16}$ cm-2, as given by Kim et al. (2001). The modelled evolution has been normalized to the observed evolution in the redshift range 2<z<3.

Several models can produce a value of J(912 Å) compatible with one of the measurements from the proximity effect at $z\sim 3$ shown in Fig. 3, from a a simple QSO-dominated background to models with $f_{\rm esc}\sim 0.2$. However, a more stringent condition, , is required not to exceed the local upper limit of Vogel et al. (1995). Steidel et al. (2001) used their composite spectrum of Lyman-break galaxies and the UV emissivities of Steidel et al. (1999) to derive the ionising flux at z=3. They obtained J(912 Å) $=1.2\pm0.3\times 10^{-21}\;{\rm erg\; cm^{-2}\; s^{-1}\; Hz^{-1}\; sr^{-1}}$, a value consistent with our model for $f_{\rm esc}=0.4$. If Lyman-break galaxies with a spectrum similar to that observed by Steidel et al. (2001) dominate the UV background, they will produce an ionising flux higher than the local and high-redshift estimates.

Similar results are obtained in the $\Omega_{\rm m}=0.3$, $\Omega_\Lambda=0.7$ universe (not shown). Because of Eq. (2) and of the factor we have used in Sect. 2.3 to derive the emissivity for the $\Lambda$-cosmology, the contribution of galaxies to the emissivity is exactly the same in both the universe models adopted here. The QSOs contribution, instead, depends on an independent fit of the luminosity function (Sect. 2.2). In the new cosmology, the UV background produced by QSOs is slightly larger than the value for the Einstein-De Sitter model, by about 30% at the peak of the QSOs contribution.