2 The ultraviolet background

The mean specific intensity of the Ultraviolet background $J(\nu_{\rm obs},z_{\rm obs})$, as seen at a frequency $\nu_{\rm obs}$ by an observer at redshift $z_{\rm obs}$, can be derived from

 

$$J(\nu_{\rm obs},z_{\rm obs})= \displaystyle\frac{1}{4\pi} \int_{z_{\rm obs}}^{\infty} {(1+z_{\rm obs})^3 \over (1+z)^3} \epsilon(\nu,z)$$

$$\times \;\; {\rm e}^{-\tau_{\rm eff}(\nu_{\rm obs},z_{\rm obs},z)} \frac{{\rm d}l}{{\rm d}z} {\rm d}z,$$ (1)

where $\nu=\nu_{\rm obs}(1+z)/(1+z_{\rm obs})$, $\epsilon(\nu,z)$is the proper space-averaged volume emissivity, $\tau_{\rm eff}(\nu_{\rm obs},z_{\rm obs},z)$ is the effective optical depth at $\nu_{\rm obs}$ of the Intergalactic Medium (IGM) between redshifts $z_{\rm obs}$ and z, and ${\rm d}l/{\rm d}z$the proper line element (Madau 1991, 1992; Haardt & Madau 1996).

The emissivity $\epsilon(\nu,z)$ should include a contribution both from direct sources of UV radiation (e.g. QSOs and galaxies) and from the IGM clouds themselves, through continuum radiative recombination of the gas (Haardt & Madau 1996). For the sake of simplicity, we consider here the case of a purely absorbing IGM, thus omitting radiative recombination. The effect of this omission will be discussed later.

The line element can be written as

$$\frac{{\rm d}l}{{\rm d}z}=\frac{c}{H(z) (1+z)},$$ (2)

where c is the velocity of light and $H(z)=H_0 [\Omega_{\rm m} (1+z)^3 + \Omega_\Lambda]^{1/2}$ is the Hubble parameter for a flat universe ( $\Omega=\Omega_{\rm m}+\Omega_\Lambda=1$).

   
2.1 Opacity

The effective optical depth $\tau_{\rm eff}$ through the IGM is defined as ${\rm e}^{-\tau_{\rm eff}}=\langle {\rm e}^{-\tau} \rangle$, where the mean is taken over all the lines of sight from the redshift of interest. For a Poisson distribution of discrete absorbers (Paresce et al. 1980; Moller & Jakobsen 1990; Madau 1991, 1992),

 

$$\tau_{\rm eff}(\lambda_{\rm obs},z_{\rm obs},z)$$ =

$$\int_{z_{\rm obs}}^z {\rm d}z' \int_0^{\infty} {\rm d}N_H{\sc i} \; f(N_H{\sc i},z') \left(1{-}{\rm e}^{-\tau(\lambda')}\right) ,$$ (3)

where $f(N_H{\sc i},z')= \partial^2 N / \partial N_H{\sc i}\partial z'$ is the distribution of absorbers as a function of redshift and column density of the atomic hydrogen $N_H{\sc i}$, and $\tau(\lambda')$ is the optical depth of an individual cloud for ionising radiation at a wavelength $\lambda'=\lambda_{\rm obs}(1+z_{\rm obs})/(1+z)$. For 228 Å $<\lambda'\leq 912$ Å, the main contribution to absorption of UV photons is given by the ionisation of H I, therefore

$$\tau(\lambda')= \tau_H{\sc i} (\lambda')= N_H{\sc i}\, \sigma_H{\sc i} \, \left(\frac{\lambda'}{912~\mbox{\AA}}\right)^3,$$ (4)

where $\sigma_H{\sc i}=6.3 \times 10^{-18}$ cm^-2 is the photo-ionisation cross-section at the Lyman limit for H I (Osterbrock 1989).

The ionisation of He I at 504 Å is not considered: He I being almost completely ionised, its contribution to the total opacity is negligible (Haard & Madau 1996). For completeness, we have included the contribution of He II ionisation to the opacity, although it does not affect the results presented in this paper. He II is ionised for $\lambda'\leq228$ Å: thus,

$$\tau(\lambda')= \tau_H{\sc i}+ N_He{\sc ii}\, \sigma_He{\sc ii} \, \left(\frac{\lambda'}{228~\mbox{\AA}}\right)^3,$$ (5)

with $\sigma_He{\sc ii}=1.58 \times 10^{-18}$ cm^-2 (Osterbrock 1989). The column density $N_He{\sc ii}$ can be derived from $N_H{\sc i}$ by solving the radiative transfer within a cloud. When clouds are optically thin at 228 Å, a simple solution can be found (Haard & Madau 1996):

$$N_He{\sc ii}\,\approx\, 1.8\, N_H{\sc i}\, \frac{J(912~\mbox{\AA})}{J(228~\mbox{\AA})}\cdot$$ (6)

While Eq. (6) does not hold for optically thick clouds, its use in the optically thick case can still provide correct estimates for the cosmic opacity, when $\lambda_{\rm obs} > 228$ Å (Haard & Madau 1996). Since the He II contribution to $\tau_{\rm eff}$ depends on the UV background, we have solved Eq. (1) iteratively for every value of $z_{\rm obs}$.

For the redshift and column density distribution of absorption lines we have adopted the usual form

$$f(N_H{\sc i},z)= \left(\frac{A}{10^{17}}\right) \left(\frac{N... ...sc i}}{10^{17}{\rm\;\;cm^{-2}}}\right)^{-\beta} (1+z)^\gamma.$$ (7)

Fits of the absorption line distribution show that the index $\beta$varies for different ranges of $N_H{\sc i}$ (Fardal et al. 1998; Kim et al. 2001). However, it is possible to describe the cloud distribution with $\beta=1.46$ over several decades in $N_H{\sc i}$ (Petitjean et al. 1993). For simplicity, we have adopted this single value over the whole column density range considered here. Kim et al. (2001) combined high resolution VLT/UVES observations of 3 QSOs with literature data and derived a line number density per unit redshift ${\rm d}N/{\rm d}z=9.06 (1+z)^{2.19}$, for the column density range $N_H{\sc i}=10^{13.64-16}~{\rm cm^{-2}}$and z>1.5. At lower redshifts, HST observations from the QSO absorption line key project show a slower evolution with redshift, with ${\rm d}N/{\rm d}z=34.7\; (1+z)^{0.16}$ (Weymann et al. 1998). The change in evolution, according to the results of Weymann et al. (1998) and Kim et al. (2001), occurs at $z\sim 1$. In this paper, we use $\gamma=2.19$ to describe the redshift evolution of the Ly$\alpha$ forest ( $10^{13} \leq N_H{\sc i}/ {\rm cm^{-2}} \leq 1.58\times 10^{17}$) at z>1, and $\gamma=0.16$ at $z\leq 1$. By choosing A=0.13 and A=0.50, the integral of Eq. (7) over $N_H{\sc i}=10^{13.64-16}~{\rm cm^{-2}}$ reproduces the results of Kim et al. (2001) and Weymann et al. (1998), respectively. The distribution of Lyman Limit systems is derived in an analogous way from Storrie-Lombardi et al. (1994). For $1.58\times 10^{17} \leq N_H{\sc i}/{\rm cm^{-2}} \leq 10^{20}$, we use A=0.17 and $\gamma=1.55$. The parameters adopted for the distribution of absorbers are summarized in Table 1.

 

 

Table 1: Parameters adopted for the distribution of absorbing clouds (Eq. (7)).

A	$\beta$	$\gamma$	$N_H{\sc i}/{\rm cm^{-2}}$
0.50	1.46	0.16	$10^{13} {-} 1.58\times 10^{17}$	$z\leq 1$
0.13	1.46	2.19	$10^{13} {-} 1.58\times 10^{17}$	z> 1
0.17	1.46	1.55	$1.58\times 10^{17} {-} 10^{20}$

Figure 1: Distance in redshift $\Delta z$ between an observer at $z_{\rm obs}$ and a point at $z=z_{\rm obs}+\Delta z$ for which $\tau _{\rm eff}(\lambda _{\rm obs},z_{\rm obs},z)=1$, as a function of $z_{\rm obs}$. Plots are shown for radiation observed at the Lyman edge and for $\lambda _{\rm obs} = 600$ Å. The plot does not include the contribution to the opacity of the He II ionisation, that will affect the results at 600 Å. Assuming $J(912~\mbox{\AA})/J(228~\mbox{\AA})=50$ (the minimum value in our simulations, when only QSOs contribute to the background), $\Delta z$ at 600 Å will decrease to ${\approx } 2$ in the range 0<z<0.5.

As already pointed out by many authors (Madau 1991, 1992; Haardt & Madau 1996), the UV background becomes more dominated by local sources as the redshift increases. This can be seen in Fig. 1, where we show the distance in redshift $\Delta z=z-z_{\rm obs}$ corresponding to an effective optical depth $\tau _{\rm eff}(\lambda _{\rm obs},z_{\rm obs},z)=1$, as a function of $z_{\rm obs}$. For radiation at $\lambda_{\rm obs}=912$ Å $\Delta z$ decreases from 1.8 at $z_{\rm obs}=0$ to 0.08 at $z_{\rm obs}=5$. The same trend can be seen for $\lambda _{\rm obs} = 600$ Å but with larger values of $\Delta z$ (less absorption), because of the dependence of the H I ionisation cross-section on the wavelength. Since only radiation from local sources can easily reach $z_{\rm obs}$, it is not necessary to compute the integral in Eq. (1) up to $z=\infty$ (or to the maximum z for which UV emitting sources are available). We have used in our calculation $z_{\rm max}=5$.

The absorber distribution we have adopted produces opacities that are very similar to those of Haardt & Madau (1996). Fardal et al. (1998, see also Shull et al. 1999) have derived values for A, $\beta$ and $\gamma$ by fitting the distribution of absorption lines in several ranges of column density. The opacity provided by their model is smaller than the one presented here. For example, at $z_{\rm obs}=3$ we reach $\tau_{\rm eff}(912~{\rm\AA})=1$ for $\Delta z=0.18$, while it is $\Delta z=0.25$ for model A2 in Fardal et al. (1998). For the same emissivities, the opacity of Fardal et al. (1998) will result in a UV background higher than ours. The difference increases with $z_{\rm obs}$ and reaches 0.1 dex at $z_{\rm obs}=3$.

   
2.2 QSO emissivity

The QSO contribution to the UV emissivity has been derived from the QSO luminosity function, for which we have adopted the double power-law Pure Luminosity Evolution model (Boyle et al. 1988)

$$\phi(L,z)=\frac{\phi^\star}{ L^\star(z)\left[ \left(\frac{L}{... ...beta_1}+ \left(\frac{L}{L^\star(z)}\right)^{\beta_2} \right]},$$ (8)

where $\beta_1$ and $\beta_2$ are the faint- and bright-end of the luminosity function. A few functional forms have been adopted for the redshift evolution of the break luminosity $L^\star(z)$. Using a sample of over 6000 QSOs with 0.35 <z < 2.3, Boyle et al. (2000) find that the evolution is well fitted by a second-order polynomial of the form

$$L^\star(z)=L^\star(0)\; 10^{k_1 z + k_2 z^2}.$$ (9)

At redshifts $z\sim 2$, the luminosity evolution stops and the comoving number density of QSOs remains constant up to $z\sim 3$(Boyle et al. 1988). At z > 3 the QSO number density declines dramatically. A recent study by Fan et al. (2001) on a sample of 39 high-redshift QSOs from the Sloan Digital Sky Survey, suggests that the number density declines as ${\rm e}^{-1.15z}$ in the redshift range 3.6 < z < 5.

Boyle et al. (2000) provide the best-fitting parameters of the B-band luminosity function for the two cosmologies adopted in this paper. The B-band proper emissivity can then be derived through the integral

$$\begin{array}{lcl} \epsilon(\nu_B,z)&=& (1+z)^3 \; \left.\int... ...0^{k_1 z + k_2 z^2} \; \epsilon(\nu_B,0), \nonumber \end{array}$$ (10)

where the factor (1+z)³ is used to transform comoving into proper number densities. For the flat Einstein-De Sitter universe and the $\Lambda$-cosmology, we have derived $\epsilon(\nu_B,0)=1.2\times 10^{24}$and $7.1\times 10^{23} \;h_{70}\;{\rm erg\; s^{-1}\; Hz^{-1}\; Mpc^{-3}}$, respectively. $L_{\rm min}$ scales with z as $L^\star$ and is chosen to correspond to $M_{B}^{\rm max}=-22$ at z=3, for both cosmologies ( $M_{B}^{\rm max}\approx -18$ at z=0; Madau 1992). Since Eq. (9) well describes a flat luminosity evolution for $z\sim 2$, we have adopted the emissivity of Eq. (10) up to z=3. For z>3 we have adopted the exponential decline of Fan et al. (2001).

Boyle et al. (2000) derived the B-band luminosity function from observations in the QSOs UV restframe, applying the K-correction for the composite QSO spectrum of Cristiani & Vio (1990). For consistency, we have used the same spectrum to derive the UV emissivity for $\lambda >1050$ Å. For $\lambda< 1050$ Å we have used a power-law, $\epsilon(\nu)\propto \nu^{-1.8}$, as measured on a sample of radio-quiet QSOs observed with HST (Zheng et al. 1997).

We have also derived the QSO emissivity from the work of La Franca & Cristiani (1997). They fitted the luminosity function on a sample of 326 objects (the Homogeneous Bright QSO Survey) by using a different luminosity evolution. Results obtained with this emissivity are very similar to those for the emissivity discussed above and are not presented in this paper.

Figure 2:  Comoving UV emissivity in a flat $\Omega _{\rm m}=1$ universe, from literature (dots) and from the spectral synthesis model described in Sect. 2.3 (lines). The model emissivity has been corrected for internal dust extinction, using the Calzetti's (1997) attenuation law with E(B-V)=0.1. Because of Eq. (11) and of the redshift-independent dust correction, emissivity at $\lambda =1500$ Å is proportional to the SFR history adopted in the model (scale on right ordinate).

   
2.3 Galaxy emissivity

The galactic emissivity in the ionising UV has been derived following the method outlined by Madau et al. (1998). The comoving UV emissivity at $\lambda\ge 1500$ Å (rest frame) can be derived from galaxy surveys as a function of the redshift. Because UV light is mainly produced by short-lived OB stars, it is possible to convert the UV emissivity into a star-formation history of the universe. If we assume that the mean luminosity evolution of the galaxies in the universe can be described with a single galactic spectrum compatible with the derived star-formation history, a stellar population synthesis model can be used to derive the emissivity at any wavelength.

We have used the latest version of the Bruzual & Charlot (1993) stellar population synthesis models, updated with a new set of stellar evolutionary tracks and spectral libraries (Bruzual & Charlot 2001; see also Liu et al. 2000). A Salpeter (1955) IMF with $0.1 < M_\star/M_\odot < 100$ has been adopted. We have chosen the quasi-empirical library of stellar spectra for solar metallicity, derived from observations for $\lambda>1150$ Å and from model of stellar atmospheres for $\lambda<1150$ Å. The conversion factor between UV luminosity and SFR has been derived from the adopted stellar model, in the limit of continuous star-formation (Madau et al. 1998; Kennicutt 1998). For radiation at $\lambda =1500$ Å we obtain

$$\frac{L_{\rm UV}}{SFR}=7.9\times 10^{27}\;\; \frac{{\rm erg\; s^{-1 } \; Hz^{-1}}}{M_\odot\; {\rm yr^{-1}}}\cdot$$ (11)

Conversion factors for other choices of the IMF and stellar libraries among those provided by the Bruzual & Charlot model typically differ over a range of ${\approx} 0.3$ dex. The same span is also observed when comparing conversion factors obtained with different models (Kennicutt 1998).

The star-formation history at z<2 has been calibrated on the UV comoving emissivities tabulated by Madau et al. (1998) for a flat $\Omega _{\rm m}=1$ universe. Emissivities at $\lambda=2800$ Å have been derived from the Canada-France Redshift Survey (Lilly et al. 1996) in the range 0.2 < z < 1 and from a HDF-north sample of objects with optical photometric redshifts for 1<z<2 (Connolly et al. 1997). Madau et al. (1996) have derived the emissivity at $\lambda =1500$ Å from objects selected in two redshift ranges at $z\sim 3$ and $z\sim 4$, from object selected on the HDF-north with the UV dropout technique. The lower emissivities at z> 2 suggested that the star formation rate has reached a maximum at $z\sim 1 {-} 2$. However, a different picture emerged from the ground-based survey of Steidel et al. (1999). They selected Lyman break galaxies on a larger area than the HDF, refining the colour selection criteria with spectroscopic observations of a few object in the sample. The derived emissivity (at $\lambda=1700$ Å) for $z\sim 3$ is still consistent with the Madau et al. (1996) value, while the emissivity at $z\sim 4$ does not show a steep decline. A value for the emissivity at $\lambda=1700$ Å in the redshift bin 2.5 < z < 3.5 can be derived from the luminosity function fitted by Poli et al. (2001) on a combined ground-based and HST database. The datapoints are shown in Fig. 2 as a function of redshift. We have corrected the emissivities of Steidel et al. (1999) and Poli et al. (2001) to include all objects with luminosities from 0 to $\infty$. The large errors in the data derived from Steidel et al. (1999) and Poli et al. (2001) reflect the uncertainties in the faint-end slope $\alpha$ of the Schechter (1976) luminosity function. Steidel et al. (1999) derive $\alpha=-1.60 \pm 0.13$ and Poli et al. (2001) $\alpha=-1.37\pm0.20$. In the Madau et al. (1998) tabulation, $\alpha=-1.3$ was used.

A smooth star-formation history has been derived from the observed UV emissivities in Fig. 2, by using Eq. (11) and correcting for dust internal extinction according to the Calzetti's (1997) attenuation law. At high redshift, we have adopted the flat star-formation history suggested by the work of Steidel et al. (1999). Synthetic galactic spectra have then been produced with the Bruzual & Charlot (2001) code. In Fig. 2 we show the evolution of the modelled UV emissivity at 1500 Å (solid line) and at 2800 Å (dotted line), for a colour excess E(B-V)=0.1. The model is compatible with both data at 2800 Å and z<2 and data at 1500 Å (and 1700 Å) and z>2. For the chosen E(B-V), the model is also consistent with the evolution of the emissivity in the optical-NIR regime for z<2, as tabulated by Madau et al. (1998, not shown in Fig. 2).

Unfortunately, it has not been possible to repeat the same procedure to model the emissivity (and star-formation history) in $\Lambda$-cosmology. Only Poli et al. (2001) present a luminosity function derived assuming $\Omega_{\rm m}=0.3,\Omega_\Lambda=0.7$. Steidel et al. (1999) give the emissivity for the objects visible in their survey, but they do not provide a luminosity function for an extrapolation to fainter luminosities. Therefore, we have used the model of the emissivity for $\Omega _{\rm m}=1$ and scaled it with a redshift-dependent correction: for sufficiently small redshift bin, it can be shown that the ratio of the emissivities in the flat $\Lambda$- and Einstein-De Sitter cosmologies is $\sqrt{0.7+0.3(1+z)^3}/(1+z)^{1.5}$. The emissivity derived in this way is consistent with the data of Poli et al. (2001). Because of Eq. (11), the same ratios applies to the star-formation histories.

The synthetic spectrum has then been used to derive the emissivity for the ionising UV. Due to the absence of observations, synthetic spectra rely on models of stellar atmospheres for $\lambda\leq 912$ Å. Charlot & Longhetti (2001) compared stellar spectra from different models and concluded that discrepancies on the ionising flux are not higher than 0.1 dex. The uncertainty on our modelled emissivity also depends on the uncertainties in the determination of the SFR. To quantify the uncertainties in the adopted model, we have computed the effect on the emissivity of the variation of the basic ingredients of the Bruzual & Charlot (2001) model (IMF, metallicity, stellar libraries). We have used the same description for the UV emissivity at 1500 Å as in the main model (solid line in Fig. 2) and converted it into a star-formation history by using a conversion factor appropriate for the selected IMF and stellar spectra. The synthetic spectra obtained in this way typically differ by less than 0.2 dex at the ionisation limit and 0.3 dex at 600 Å.

Spectra at $\lambda\leq 912$ Å also need to be corrected for the internal absorption by the galaxy interstellar medium. We describe this correction with the parameter $f_{\rm esc}$, i.e. the fraction of Lyman-continuum photons that can escape into the IGM without being absorbed by the interstellar medium, either gas or dust. A wide range of values can be found in the literature for $f_{\rm esc}$, derived both from models of radiative transfer and observations of H I recombination lines ( $5\%<f_{\rm esc}<60\%$; for a review, see Barkana & Loeb 2001). UV observations of local starbursts suggest $f_{\rm esc}\approx 5\%$(Leitherer et al. 1995; Hurwitz et al. 1997; Heckman et al. 2001). Steidel et al. (2001) analysed a composite spectrum of 29 Lyman-break galaxies at $z\sim 3.4$. They derived a ratio between the flux densities at 1500 Å and 900 Å $f_{1500}/f_{900}=4.6\pm 1$, after correcting for the differential absorption due to the intervening IGM. The f₁₅₀₀/f₉₀₀ ratio for the unattenuated synthetic spectrum that we have used is very similar, $f_{1500}/f_{900}\approx 5.3$. If we assume that 40% of the radiation at 1500 Å is absorbed by dust (as obtained from the Calzetti's attenuation law with E(B-V)=0.1), the observed f₁₅₀₀/f₉₀₀ ratio is equivalent to $f_{\rm esc}\approx 0.4$ (if the internal absorption in the Lyman continuum do not change significantly with $\lambda$). Because of the increase of the disk density with the redshift, $f_{\rm esc}$ is expected to decrease with z; it is also found to depend heavily on the details of the distribution of the sources and the gas, i.e. whether the stars and/or gas are clumped or not (Wood & Loeb 2000). In this work, we will use a wavelength and redshift independent $f_{\rm esc}$, by which we multiply the synthetic spectrum at $\lambda\leq 912$ Å. We will show results for $f_{\rm esc}=0.05$ and 0.40, to cover the range of values suggested by local and $z\sim 3.4$ observations, and for an intermediate value, $f_{\rm esc} = 0.1$.

Finally, the galactic emissivity has been converted from comoving to proper, multiplying by (1+z)³. The total emissivity in Eq. (1) is the sum of the QSOs and the galaxy contribution.

Figure 3:  UV background at $\lambda =912$ Å for the models with $f_{\rm esc}$=0.05, 0.1 and 0.4 (solid lines), in a flat $\Omega _{\rm m}=1$ universe. Also shown are the separate contribution of the QSOs (dotted line) and of the galaxies (dashed lines, each corresponding to a value of $f_{\rm esc}$). The shaded area refer to the Lyman limit UV background estimated from the proximity effect (Giallongo et al. 1996; Cooke et al. 1997; Scott et al. 2000). The arrow shows an upper limit for the local ionising background (Vogel et al. 1995). The datapoint at z=3 is derived from a composite spectrum of Lyman-break galaxies (Steidel et al. 2001). The models and the datapoint of Steidel et al. (2001) have been multiplied by a z-dependent factor, to take into account the cloud emission (see text).