4 The evolution of the Lyman forest

Further constraints can be obtained by studying the effect of the evolution of the ionising background on the evolution of absorbers. Davé et al. (1999) have studied the evolution of the low-redshift Ly$\alpha$ forest in a hydrodynamic cosmological simulation, adopting an UV ionising background with the same redshift evolution as that of Haardt & Madau (1996). They found a sharp transition at z=1.7 for the number density evolution, ${\rm d}N/{\rm d}z$. The change in evolution is primarily due to the drop in the UV ionising background, resulting from the decline in the QSO population. The formation of structure by gravitational growth plays only a minor role in the evolution. In the absence of structure evolution, it is possible to derive an analytical approximation for the evolution of ${\rm d}N/{\rm d}z$ with $J_\nu$ and the Hubble expansion. For clouds in photo-ionisation equilibrium with the background, it is easy to show that the evolution of lines above a given threshold in column density can be written as (Davé et al. 1999)

$$\left(\frac{{\rm d}N}{{\rm d}z}\right)_{N_H{\sc i}>N_H{\sc i}... ...}\; \left[\frac{(1+z)^5}{\Gamma_H{\sc i}(z)}\right]^{\beta-1},$$ (12)

where H(z) is the Hubble parameter and $\beta$ the coefficient of the power-law distribution of clouds with column density (Eq. (7)). $\Gamma_H{\sc i}(z)$ is the photo-ionisation rate

$$\Gamma_H{\sc i}(z)=\int_{\nu_0}^{\infty} \frac{4 \pi J(\nu,z)}{h\nu} \;\sigma_H{\sc i}(\nu) {\rm d}\nu,$$ (13)

with $\nu_0$ the frequency of the Lyman limit and $\sigma_H{\sc i}(\nu)=\sigma_H{\sc i}(\nu_0/\nu)^3$ the H I photo-ionisation cross-section.

In Fig. 4 we show ${\rm d}N/{\rm d}z$ for Lyman forest clouds in the column density range $N_H{\sc i}=10^{13.64-16} \;{\rm cm^{-2}}$. Data points come from several sources in the literature and from new high resolution VLT/UVES spectra of three QSO (Kim et al. 2001). For each of our models, we have computed $\Gamma_H{\sc i}(z)$ and we have derived the evolution of ${\rm d}N/{\rm d}z$ according to Eq. (12), for the two cosmologies adopted in this paper. The evolution has been normalized to the observed values for 2<z<3. In this redshift range, the UV background is nearly flat for any of the models and it is easy to show, from Eq. (12), that ${\rm d}N/{\rm d}z\propto (1+z)^{5\beta-6.5}$, independently of the cosmology. By fitting the observed data, Kim et al. (2001) have derived ${\rm d}N/{\rm d}z\propto (1+z)^{2.19}$ for z>1.5. A value $\beta\approx 1.7$ can well reproduce the evolution derived from the observations. This is consistent with fits of the density distribution, that give $\beta= 1.68\pm 0.15$over $N_H{\sc i}=10^{14-16} \;{\rm cm^{-2}}$ at $z\sim 1$ (Kim et al. 2001). On the other hand, weaker lines ( ) are known to have a flatter distribution in column density ( $\beta\sim 1.4{-}1.5$; Giallongo et al. 1996; Kim et al. 2001). This will produce a slower redshift evolution, as observed in this column density range for z>1.5( $\gamma\sim 1$; Kim et al. 2001). We remind here that in Sect. 2.1 we have used $\beta=1.46$, that provides a good description of the density distribution over a much larger column density range.

The analysis of Kim et al. (2001) shows that the change in evolution occurs at $z\approx 1$, rather than at $z\approx 1.7$, as previously suggested (Weymann et al. 1998). In Fig. 4 the break at $z\approx 1$ can be reproduced if the contribution of galaxies to the background is dominant. This is because of the rapid decrease of the star-formation rate (and of $\Gamma_H{\sc i}(z)$) below this redshift (Fig. 2). The photo-ionisation rate of a pure QSOs background, instead, peaks at $z\sim 2.5$ and has a slower evolution with z. It is interesting to note that the modelled evolution is closer to the observed data for the $\Lambda$-cosmology. For the Einstein-De Sitter universe, ${\rm d}N/{\rm d}z$ grows for z<1, which is not observed. However, the discrepancy may be mitigated when the effect of the formation of structures on ${\rm d}N/{\rm d}z$ is taken into account (Davé et al. 1999).

The modelled ${\rm d}N/{\rm d}z$ do not depend on our approximation of a purely absorbing intergalactic medium, since the contribution of cloud emission to $\Gamma_H{\sc i}(z)$ is nearly constant with redshift (see Fig. 6 in Haardt & Madau 1996).