A&A 444, 461-479 (2005)
G. Vladilo1 - C. Péroux2
1 - Osservatorio Astronomico di Trieste - Istituto Nazionale di Astrofisica, Via Tiepolo 11, 34131 Trieste, Italy
2 - European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching-bei-München, Germany
Received 1 July 2004 / Accepted 4 October 2005
We present a new study of the effects of quasar obscuration on the statistics of Damped Ly (DLA) systems. We show that the extinction of any Galactic or extragalactic H I region, , increases linearly with the column density of zinc, with a turning point , above which background sources are suddenly obscured. We derive a relation between the extinction of a DLA system and its H I column density, , metallicity, Z, fraction of iron in dust, , and redshift, z. From this relation we estimate the fraction of DLA systems missed as a consequence of their own extinction in magnitude-limited surveys. We derive a method for recovering the true frequency distributions of and Z in DLAs, and fZ, using the biased distributions measured in the redshift range where the observations have sufficient statistics ( ). By applying our method we find that the well-known empirical thresholds of DLA column densities, atoms cm-2 and atoms cm-2, can be successfully explained in terms of the obscuration effect without tuning of the local dust parameters. The obscuration has a modest effect on the distribution of quasar apparent magnitudes, but plays an important role in shaping the statistical distributions of DLAs. The exact estimate of the bias is still limited by the paucity of the data (40 zinc measurements at ). We find that the fraction of DLAs missed as a consequence of obscuration is 30% to 50%, consistent with the results of surveys of radio-selected quasars. By modelling the metallicity distribution with a Schechter function we find that the mean metallicity can be 5 to 6 times higher than the value commonly reported for DLAs at .
Key words: ISM: dust, extinction - galaxies: ISM - galaxies: high-redshift - quasars: absorption lines
Quasar absorption line systems probe the diffuse gas in the Universe over cosmological scales and at various stages of evolution. Among these absorbers, the Damped Ly systems (hereafter DLAs) have the highest H I column densities ( atoms cm-2) and are believed to originate in H I regions of high-redshift galaxies (Wolfe et al. 1986). Also, they have higher metallicities than any other class of quasar absorbers ([M/H] dex; Pettini et al. 1994) and as such are expected to contain dust. If present, the dust in DLAs will absorb and scatter the radiation of the background quasar, dimming its apparent magnitude ("extinction'' effect) and changing the slope of its spectral distribution ("reddening''). In magnitude limited surveys, the extinction may obscure the quasar, leading to an observational selection effect known as obscuration bias (Ostriker & Heisler 1984; Fall & Pei 1989, 1993).
The bias can lead to misleading conclusions on the nature of the galaxies associated with the DLA systems (herefater DLA galaxies), because the H I regions most enriched by metals and dust would be systematically missed in the surveys. In addition, some properties of the high-redshift Universe that can only be derived from studies of DLAs, would also be affected by the bias. One example is the comoving mass density of neutral gas, , and its evolution with redshift, which is an indicator of gas consumption due to star formation (Wolfe et al. 1995). Another example is the mean cosmic metallicity inferred from abundance studies of DLAs (Pettini et al. 1997), which is related to the star formation rate in the early Universe (Pei & Fall 1995, hereafter PF95; Pei et al. 1999).
Whether the obscuration effect is important, depends on the actual amount of dust in DLAs. The first evidence of dust, in the form of reddening, was reported by Pei et al. (1991). The complex variability of the quasar continuum makes it very difficult to determine the reddening in individual cases. However, from a statistical comparison of quasars with and without foreground DLAs one can search for a systematic change of the continuum slope. In this way, Pei et al. found a systematic difference, suggestive of reddening, using a simplified power-law fit to the continuum distribution.
The detection or reddening motivated detailed studies of quasar obscuration. Fall & Pei (1993, hereafter FP93) presented an analytical method of computing the obscuration from the observed luminosity function of quasars, the typical dust-to-gas ratio of DLAs and the empirical distribution of H I column densities. By modelling the radial distribution of the neutral gas in DLA galaxies, they concluded that a large fraction of quasars may be obscured in optically selected samples. The large uncertainty of the reddening measurements prevented reaching firm conclusions on the dust-to-gas ratios and on the magnitude of the bias.
An alternative approach to measure the amount of the dust in DLAs consists in measuring the differential depletions between refractory and volatile elements (Pettini et al. 1994; Hou et al. 2001; Prochaska & Wolfe 2002; Ledoux et al. 2003; Vladilo 2002, 2004). The depletions can be converted into dust-to-gas ratios, yielding an approximate estimate of the quasar extinction (Vladilo et al. 2001b; Prochaska & Wolfe 2002). This method, as well as the reddening measurements, only give information on the detected DLAs, leaving open the possibility that more dusty H I regions remain undetected.
A more direct probe of the obscuration effect is a comparison of DLA statistics in optically selected and radio selected quasar samples, given that the latter are unaffected by dust extinction. Ellison et al. (2001) compiled a homogeneous sample of radio selected quasars and searched for DLAs towards every target, irrespective of its optical magnitude (CORALS survey). They concluded that dust-induced bias in previous magnitude limited surveys may have led to underestimating the H I mass density, , and the number density per unit redshift interval, n(z), of DLAs by at most a factor of two at . In addition, they found tentative evidence that n(z) is greater in fainter quasars, as expected by the obscuration effect. In a follow-up study of CORALS metallicities, Akerman et al. (2005) find . This value is higher than in a control sample, as expected by the obscuration, but only at level. In an extension of the CORALS survey at 0.6 < z < 1.7 Ellison et al. (2004) find n(z) = 0.16 +0.08-0.06using MgII absorbers as DLA candidates. This value is consistent with magnitude-limited estimates at the same z, but the error permits a factor of 2.5 difference in the sense predicted by the obscuration.
The results of the CORALS survey are not yet conclusive since the original sample of Ellison et al. had limited statistics, in particular at the high values of where the effect is expected to be more important. This weak evidence of obscuration, together with the non-detection of reddening in a large sample of quasars with and without candidate DLAs, recently reported by Murphy & Liske (2004), are conveying the impression that the obscuration effect is not important.
Yet, indirect evidence for the existence of the bias comes from other studies. The analysis of [Zn/H] versus by Boissé et al. (#Boiss&) showed that none of the DLAs with zinc detection known at the time have a metallicity above the threshold [Zn/H] , corresponding to atoms cm-2. They proposed that this threshold could be attributed to obscuration effect, assuming that metal column density tracks the dust column density.
While this assumption has not been proven by subsequent investigations, the threshold proposed by Boissé et al. has been invoked to reconcile several predictions of galactic evolution models (Prantzos & Boissier 2000; Hou et al. 2001) and cosmological simulations (Cen et al. 2003) with the observed properties of DLAs.
Another potential evidence of the bias is the lack of DLAs with H I column densities atoms cm-2. In this case, the obscuration has been invoked to explain the significant fraction of model galaxy disks predicted to have higher H I column densities (Churches et al. 2004). As an alternative explanation, it has been proposed that hydrogen may undergo a sudden transition from the atomic to the molecular form above a critical column density threshold (Schaye 2001), in which case the absorber would not be detected as a DLA system.
In the present work we address several open questions concerning the existence and importance of the obscuration bias. In Sect. 2, we perform a careful investigation of all the factors that determine the extinction of a Galactic or extragalactic H I region. In Sect. 3 we report some indirect evidence of obscuration effects. In Sect. 4 we present a mathematical formulation aimed at recovering the frequency distributions of H I column densities and metallicities of DLAs from the observed distributions affected by obscuration bias. The first implementation of this method is described in Sect. 5 and the results are discussed in Sect. 6.
We adopt the simplified notation and , ignoring contributions from ionization states other than the dominant one (see Vladilo et al. 2001a, and references therein).
We start by considering a refractory and a volatile
with interstellar abundance ratio by number
The ratio of the total column densities (gas plus dust)
of these two elements along an interstellar line of sight will be
and NE the column densities of an element E in the dust
and in the gas, respectively. The fraction of atoms of E in dust form will be
From these definitions we have
Combining these relations we have
Finally, we consider the relation between the interstellar extinction in magnitudes
at the wavelength ,
and the column density of dust grains.
For spherical grains with extinction efficiency factor
(see Spitzer 1978, Chap. 7) this relation is
The factor is approximately constant by definition of volatile and refractory elements. In a medium of constant composition, such as the local ISM, also the abundance is constant. Therefore, for a given type of dust, the term is constant and the expression (4) implies that the interstellar extinction increases linearly with the gas-phase column density of any volatile element measured in the same line of sight.
Differences in the shape, size, density, composition and extinction efficiency of the grains will induce a scatter in the relation. The dependence of on the grain size is linear, while in Eqs. (2) and (3) it was cubic and quadratic. Since the grain size distribution is a key factor in determining the wavelength dependence of the interstellar extinction (see e.g. Draine 2003), the mild dependence of on suggests that the relation between and may be similar even in interstellar regions with different types of extinction curves.
By adopting zinc as the volatile element and considering the extinction in
the V band (
m), we obtain from Eq. (4)
To estimate aV we gathered the measurements obtained from HST (Roth & Blades 1995) and IUE (Van Steenberg & Shull 1988) high-resolution spectra. The IUE data are more numerous, but have lower resolution and signal-to-noise ratio. To reduce the impact of these differences we selected the IUE measurements with total error <0.3 dex in . To minimize the contamination by stellar absorptions on top of the IUE Zn II interstellar lines (Van Steenberg & Shull 1988), we selected only IUE spectra of fast-rotating stars ( 200 km s-1). We then scaled the IUE column densities to match the Bergeson & Lawler (1993) Zn II oscillator strengths adopted in the HST data and in the current research. As far as the extinctions are concerned, we used the AV values given by Guarinos (1991). As a result, we obtained a sample of 11 lines of sight. The mean value of extinction per unit zinc column density that we derive is mag cm2 and mag cm2 for the HST and IUE sub-samples (4 and 7 lines of sight, respectively).
|Figure 1: Extinction AV=RV EB-V (mag) versus Zn II column density (atoms cm-2) in Galactic interstellar lines of sight. Filled diamonds: HST data from Roth & Blades (1995). Empty diamonds: IUE data from Van Steenberg & Shull (1988). Curve: extinction law (7) with mag cm2 derived empirically as explained in the text.|
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In order to enlarge the sample we also considered the color excess EB-V, which is more commonly measured than AV. The ratio of general-to-selective extinction, RV = AV / EB-V, is approximately constant, with a mean value in the diffuse ISM of the Milky Way (see e.g. Draine 2003). We used this property to derive a mean value from reddening and zinc column density measurements. The EB-V data were taken from the extinction catalog of Savage et al. (1985). The resulting sample includes 24 lines of sight, shown in Fig. 1. The mean value of the HST sub-sample (9 lines of sight) is mag cm2, and of the IUE sub-sample (15 lines of sight), mag cm2. We did not find significant correlations between and the parameters of UV extinction provided by Savage et al. (1985), such as the strength of the emission bump or the steepness of the UV rise. The lack of these correlations gives us a justification for averaging the results obtained from lines of sight with different types of dust.
The results obtained from the EB-V analysis are identical to those obtained from the more limited sample of direct measurements of AV. The larger dispersion of the IUE sub-sample is probably due to the lower quality of the data. We adopted the weighted average of the IUE and HST sub-samples, which yields mag cm2, a value almost identical to that of the HST sub-sample. The relation (7) obtained from this estimate of aVis displayed in Fig. 1 (solid curve). The exponential rise of the extinction is due to the fact that the Zn II column density is plotted on a logarithmic scale. In this scale the rise of the extinction is mild at low values of , but very fast at high values. The turning point marking the transition between these two regimes can be estimated from the condition . This criterion yields . Above this turning point the extinction undergoes a dramatic rise, acting as a barrier against detection of faint background sources. This barrier is remarkably similar to the detection threshold proposed by Boissé et al. (#Boiss&). This motivated us to investigate the relation between extinction and metal column density in DLA systems.
The relations (4), (5) and (6)
are valid in the ISM of any galaxy, including DLA systems,
which are H I regions of high-redshift galaxies.
Applying these relations to a DLA system and to the local ISM, and
adopting = Zn,
= Fe and
Relative abundances of metals do not show strong variations in DLAs, despite significant changes of the absolute metallicity Z. The Fe/Zn ratio is very close to solar in metal-poor stars with metallicities typical of DLA systems (Mishenina et al. 2002; Gratton et al. 2003; Nissen et al. 2004; see discussion in Vladilo 2004). Deviations from the solar ratio are measured in stars with extremely low metallicities. Such deviations do not affect the present work in any case, because the dust fraction and the extinction vanish at very low metallicity, as we now discuss. We adopt therefore .
The depletion of DLAs increases with metallicity (Ledoux et al. 2003).
In particular, the dust fraction of iron undergoes a fast rise between
between metallicities [Zn/H] dex and -1 dex, from
values close to zero up to values typical
of the Milky-Way warm interstellar gas
Here we model this trend with the analytical expression
The dependence on the changes of the grain properties enters through the factor G, which scales as (Eq. (6)). This term is mostly determined by the physical conditions of the medium, rather than by its chemical enrichment, with the possible exception of the abundance by mass of iron in the grains, . At the very early stages of chemical evolution we may expect variations of as a consequence of non-solar relative abundances of metals. However, these effects do not affect our estimate of the extinction since, at the very low metallicities typical of these early stages ( ), the factor vanishes in any case, yielding a null extinction.
The variations of G due to changes of the physical conditions can be assessed by sampling different regions in a medium of constant composition, such as the local ISM. The low scatter of aV derived in the previous section indicates that the scatter of G in the local ISM is low, even if the MW sample includes lines of sight with different physical conditions. To explain this qualitatively, we can imagine that variations of the grain size may compensate variations of the dust density . For instance, in a harsh interstellar environment, volatile elements tend to leave the grains and we may expect an increase of because carbon, the element potentially most abundant in the grains, is volatile and has low atomic mass. At the same time, we may expect that larger grains will be more easily destroyed in a harsh environment and the mean will decrease, countering the increase of in Eq. (6).
These general considerations apply to any type of interstellar environment, suggesting that G may not vary dramatically in DLAs with moderately
low abundances (
). To test
this hypothesis we computed G in the Small Magellanic Cloud (SMC)
from the expression
Equation (9) is valid in the rest frame of the H I region. To derive the extinction in the observer's frame we need to consider the wavelength dependence of the extinction. In the rest frame of the DLA system the extinction at the wavelength will be , where is, by definition, the normalized extinction curve. In the observer's frame the same extinction will appear at , where z is the absorption redshift.
|Figure 2: Extinction of a quasar at a fixed wavelength in the observer's frame, plotted versus the redshift of an intervening DLA system with cm-2. Solid and dashed curves represent MW-type and SMC-type extinction curves, respectively (Cardelli et al. 1988; Gordon et al. 2003). Curves left to right: photometric bands , , and , respectively.|
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In conclusion, the observer's frame extinction of a quasar
due to an intervening DLA system
at redshift z is
Since for , the net effect of the cosmological redshift is an amplification of the rest-frame extinction at . At this amplification compensates for the reduction of in systems of moderately low metallicity and for the mild reduction of G in SMC-type dust. Only at very low metallicity the extinction becomes negligible since tends to vanish. Overall, the extinction per unit column density of zinc atoms predicted for most DLAs is similar, or even larger, than that typical of the Milky-Way ISM.
Examples of quasar extinction estimated with Eq. (11) are plotted in Fig. 2 as a function of z, keeping fixed the other terms. This type of representation puts in evidence the amplification of the extinction with increasing redshift. Each curve was computed at a constant equal to the effective wavelength of the SLOAN Digital Sky Survey (SDSS) photometric bands , and (Fukugita et al. 1996). We adopted the extinction law by Cardelli et al. (1988; hereafter CCM law) with RV=3.1for the Milky-Way extinction curve and the mean SMC bar data by Gordon et al. (2003) for the SMC curve. The term was estimated at [Zn/H] dex. The total zinc column density was fixed at atoms cm-2, roughly two times the zinc column density threshold proposed by Boissé et al. (#Boiss&). For the redshift interval typical of most DLAs ( ) the predicted extinction is 1 mag, a value sufficient to obscure quasars in spectroscopic surveys.
|Figure 3: Diamonds: column densities versus in DLAs. Horizontal lines: DLA definition threshold ( bottom) and empirical upper boundary ( top) of . Vertical lines: detection limit ( left) and obscuration threshold ( right) of . Dashed lines: lines of constant metallicity. Circles: Milky Way data of Fig. 1. Star: GRB-DLA from Savaglio et al. (2003).|
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To search for some extinction effect related to the zinc column density we collected from the literature all DLAs with available measurements. The resulting sample, shown in Fig. 3, includes 41 measurements, most of which derived from Hires/Keck and UVES/VLT spectra. References to these data can be found in Vladilo (2004; Table 1), with the exception of 4 systems (without Fe II lines) published by Pettini et al. (1994, 1997) and Boissé et al. (#Boiss&).
The updated sample includes more than twice the measurements of the one discussed by Boissé et al. (#Boiss&) but, in spite of this increase, column densities above the original threshold (line AB in the figure) are not found. The lack of DLAs with atoms cm-2is surprising, since Milky Way lines of sight with Zn II measurements do show values above this limit in a significant fraction of cases (Fig. 3). Studies of gamma-ray bursts (GRBs) observed inmediately after the explosion demonstrate that zinc column densities above the threshold exist and can be detected in extragalactic absorbers (Savaglio et al. 2003; the only case with measured is indicated with a star in Fig. 3).
These results confirm that some selection effect is preventing the detection of values of above the threshold when the background source is faint. The remarkable similarity between the DLA cutoff at atoms cm-2 and the ISM turning point at suggests that the rapid rise of the extinction above the turning point can be responsible for the cutoff. This is consistent with the conclusion of the previous section that the extinction per unit zinc column density of many DLAs is similar to that of the local ISM.
|Figure 4: Normalized frequency distribution of quasar apparent magnitudes for the sub-sample of DLAs with (shaded histogram) and (light histogram).|
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Table 1: Results.
If the extinction increases with , we expect to find evidence of this effect also below the threshold, in the range where we do detect DLAs. To search for this evidence we analysed the frequency distribution of quasar apparent magnitudes of the zinc sample. We used the V magnitude, which is measured in most cases. In 4 quasars without V measurements we adopted the magnitude of nearby optical bands. In practice, we compared the behaviour of the two sub-samples with below and above the median value, . The differences between the two distributions, shown in Fig. 4, are consistent with a rise of the quasar extinction with increasing . The analysis of the three bins with more statistics indicates that the maximum of the sub-sample with high (shaded histogram) is shifted by 1 bin relative to the maximum of the other sub-sample (light histogram). A Kolmogorov-Smirnov test shows that there is only a 9.7% level of probability that the two distributions are drawn from the same parent population. The shift to fainter magnitudes suggests that even at dust extinction is already present, affecting more the high- sub-sample. The magnitude of this effect can only be explained by invoking a redshift amplification of the extinction, as predicted by relation (11), since the extinction expected at rest-frame is only 0.35 mag at (Fig. 1).
In Eq. (11) the variation of the depletion in DLAs is accounted for by the term , which is determined by the metallicity. The term G is approximately constant, at least in galaxies with metallicity similar to those of the SMC and Milky Way. Possible variations of Gin galaxies of very low metallicity do not affect the prediction of the extinction since, in any case, vanishes when . For these reasons, relation (11) allows us to estimate the extinction of a DLA system using, in practice, only its H I column density and metallicity. We use this property to derive a mathematical formulation of the obscuration bias based on the study of the distribution functions of and Z in DLA systems. In practice, we determine the fraction of DLAs that are missed as a consequence of their own extinction. We then assume that multiple DLA absorbers in a given line of sight play a negligible role in the estimate of the obscuration bias. In this way, as we show in the Appendix, we derive a mathematical formulation of the bias which only makes use of observable statistical distributions of DLAs and quasars. To quantify the obscuration effect we need a good statistics of the observed distributions and this requirement limits the redshift range where our method can be presently applied, since the bulk of DLAs data are currently concentrated in .
An advantage of our formulation is that it does not require a knowledge of the intrinsic luminosity function of quasars or the geometrical distribution of the neutral gas in DLA galaxies. We refer to FP93 for a mathematical treatment of the effect of quasar obscuration which describes the relation with the quasar luminosity function and the geometrical distribution of the gas in the intervening galaxies.
We consider the DLAs in the redshift interval
that are detectable in a survey with limiting magnitude .
number of such DLAs with column densities between
the number of those with
metallicity between Z and
and fZ are defined in absence
of obscuration bias and we call them the "true'' distributions.
When we consider the effect of the
extinction generated by these DLAs,
the number of detectable systems
the "biased'' distributions.
In the Appendix we derive the following relations between true and biased
(see Eqs. (A.12) and (A.13)),
The "bias function''
that transforms the true distributions
into the biased ones is given by the relation
(see Eq. (A.5))
In the next section we show how to estimate n(m; z), while in Sect. 4.3 we present a procedure for deriving the true distributions from Eqs. (12) and (13), starting from the observed distributions.
In deriving these equations we made the assumption that the distributions of and Zare statistically independent. We discuss this assumption in Sect. 4.4.
number of quasars with apparent magnitude
observable beyond redshift z all over the sky.
This number is defined
in absence of quasar obscuration and we call
the "true'' distribution.
Our goal is to infer the true distribution from the observed one,
which must equal
the distribution biased by the obscuration effect,
The relation between true and biased distribution is discussed
in the Appendix (Sect. A.2),
where we derive the equations Eq. (A.19), i.e.
We define the obscuration fraction using the mathematical relations
between the true and biased distributions.
We call total obscuration fraction
We call obscuration fraction,
the fraction by number of systems obscured at a particular value
of column density or metallicity.
For instance, the fraction of systems obscured at a specific
The true functions and fZ are the unknown quantities in Eqs. (12) and (13). The procedure that we follow to solve these equations consists in (i) adopting trial functions and fZ, (ii) estimating , (iii) computing and from the equations and (iv) comparing these biased distributions with the observed ones. These steps are repeated by changing the trial functions until the predicted biased functions match the observed ones. We briefly describe each step of this procedure.
In the first step, we adopt analytical functions in parametrical form as the trial functions and fZ. Once we choose the shape, the problem of determining the true functions and fZ is equivalent to that of determining the values of their parameters.
In the second step we use the trial functions and fZto derive the distribution of extinctions from Eq. (11). We use this extinction distribution in Eq. (16). We then iterate Eqs. (15) and (16), starting with a trial function n(m; z), until we find a n(m;z) that yields a perfect agreement between and the distribution measured from the surveys.
In the third step we use to derive from Eq. (14) and, finally, the biased distributions and from Eqs. (12) and (13).
In the last step we compute the deviation between the predicted distributions and and the corresponding observed distributions.
By iterating the steps we determine the parameters of and fZwhich allow and to best fit the empirical distributions. The search is done via minimization. Since we are not guaranteed a priori of the unicity of the solution, we carefully investigate the variation of in the parameter space to make sure the minimum is unique.
The mathematical formulation presented above relies on the assumption that and Z are statistically independent of each other in the true population of DLAs. The local ISM offers an example in support of this independence, since a broad range of H I column densities, including the high values typical of DLAs, are found at constant metallicity, . Similarly, a DLA system of given metallicity will display a distribution of H I column densities, if sampled along random lines of sight. This distribution will reflect the mass spectrum of interstellar clouds, which is likely to be determined by some physical mechanism, such as interstellar turbulence (see discussion in Khersonsky & Turnshek 1996 and refs. therein), rather than by the metallicity. In this sense, it is reasonable to assume that and fZ are independent distributions.
Independent support of these arguments comes from the hydrodynamic simulations of DLAs evolving in CDM cosmology (Cen et al. 2003), which indicate that metallicity and hydrogen column density are almost completely uncorrelated. Similar results are found from the cosmological SPH simulations performed by Nagamine et al. (2004).
Notwithstanding, a trend is expected because the metallicity correlates with the star formation rate, which in turn increases with the gas density in the host galaxy. Therefore, lines of sight with high may have a tendency to be associated with higher metallicities. A weak evidence for this trend was found by Cen et al. (2003), but the effect is particularly small at redshift . If the metallicity is correlated with , the effect of the obscuration is more important than in the ideal case considered above. In this case the application of our mathematical formulation gives a conservative estimate of the bias.
Competing with the obscuration effect, is the possible gravitational magnification of the background quasar by the absorber itself. Many studies over the years have attempted to estimate the magnitude of this phenomenon for both DLAs (Smette et al. 1997; Le Brun et al. 2000) and metal absorption lines (Vanden Berk et al. 1996; Ménard & Péroux #M&). The results show that for the DLAs selected at optical wavelengths, gravitational effects are small. In the present application of our method, based on DLAs selected at optical wavelength, we therefore neglect the magnification effect.
We describe the statistical distribution functions adopted here as a first example of implementation of the procedure. For the intrinsic distributions we only need to adopt a functional dependence (i.e. a shape), leaving to the procedure the task of determining the parameter values.
We consider the sub-sample of DLA systems with available measurements of Zn II column densities, for which it is possible to derive metallicities corrected for depletion effects (Vladilo 2004). The differences between the corrected metallicities and the [Zn/H] values taken at face value are small, with a mean value of +0.12 dex. Given the size of the zinc sub-sample (40 systems; see list of references in Table 1 of Vladilo 2004) we can estimate the frequency distribution of metallicities with a Poisson statistical error 10% in only a few bins. To make the best use of the available data we selected the systems in the redshift range , where most of the measurements have been obtained so far. The mean redshift of the sample (28 systems) is . To derive the empirical distribution of we used the surveys by Storrie-Lombardi & Wolfe (2000) and Péroux et al. (#P&), which have more statistics than the zinc sub-sample. Selecting the DLAs in the range we obtained a sample of 59 systems.
After binning the data in and [Zn/H] = , we transformed the derived distributions in linear space, and Z, for later comparison with the distributions and fZ. In computing the biased distributions with our procedure, we took into account the fact that the typical limiting magnitude of the surveys ( ) is somewhat larger than that of the metallicity surveys ( ), owing to the different requirements in spectral resolution. In fact, Eqs. (A.12) and (A.13) can be solved independently of each other, inserting in each case the appropriate value of .
To derive the empirical distribution of quasar magnitudes we used the data of the Sloan Digital Sky Survey (SDSS) (Schneider et al. 2003). The distribution was determined for two photometric bands well representative of the visual part of the spectrum, namely the and bands, with effective wavelengths m and m, respectively (Fukugita et al. 1996). In each band we binned the data at steps of 0.5 mag, counting all the quasars located beyond the typical absorption redshift of our sample (in practice, we adopted ). An example of empirical distribution computed for the band is shown in Fig. 5 (diamonds with error bars). After binning the data, we performed a polynomial fit to the observed distribution, up to the magnitude 21.5, for which the statistics and the completeness of the SDSS sample are still good. This fit was adopted as a smooth model of the observed distribution in the magnitude range . An example of fit is shown in Fig. 5 (solid line). The limit is sufficient for studying the effects of quasar obscuration on the statistics on DLAs since the typical limiting magnitude of DLA surveys is currently .
|Figure 5: Diamonds: distribution of the SDSS apparent magnitudes for quasars in the interval . Smooth lines: example of a pair of model distributions in which the "true'' model (dashed line) is determined from the biased model that fits the observations (solid line), once the distribution of DLA extinctions and are specified (see Sect. 4.1.1).|
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One way to approximate the true shape of is to assume a geometrical distribution of the gas in DLA galaxies. For planar disks with radial exponential profiles , the expected frequency distribution is for and for (FP93). For central column densities cm-2or much larger, as inferred by FP93, this gives for most of the column-density range typical of DLAs.
A realistic approximation of should also take into account the fact that, in a given galaxy, the interstellar clouds show a spectrum of masses, , and sizes, , which is not accounted for by a smooth radial profile. A large number of studies indicate that the mass spectrum of Milky-Way interstellar clouds follows a power law , with over a 6 orders of magnitude range of masses (Scalo & Lazarian 1996 and refs. therein). From this mass spectrum, together with a relation between the internal density of the cloud, n, and the cloud size we can infer a column-density distribution. Adopting with (Scalo & Lazarian 1996 and refs. therein) we find that the column-density distribution should follow a power law with . If this column-density spectrum applies to DLAs, we would expect a decline of column densities faster than that predicted by the exponential profile model.
Both the exponential profile model and the cloud mass-spectrum hypothesis suggest that the true distribution may be approximated with a power law. The hydrodynamic computations of DLAs by Cen et al. (2003) yield a simulated true distribution of consistent with this conclusion.
In light of these considerations, we adopt
the simple law
for the true distribution, with
to be determined by the method itself.
If this is a good approximation,
the resulting biased distribution
must reproduce the observed fast decline of the number of DLAs
with column density, which
is commonly fitted by a Schechter-type function
The arguments leading to the choice of fZ can be summarized as follows.
The observed distribution of [Zn/H] measurements
peaks at [Zn/H]
dex and does not show
systems below -2 dex
or above solar metallicity (Pettini et al. 1997).
Although this distribution is likely biased at
both ends, we do expect a genuine
decline in the number of DLAs below and
above some critical values of metallicity.
dex the zinc sample
shows a decrease of the number of DLAs
in a region of the
plane not affected by any bias
(the triangle DEF in Fig. 3).
At high metallicity it is reasonable to predict a decrease since we
expect a natural decline of the
number of systems with higher and higher star formation rates.
These arguments indicate that the true
distribution of [Zn/H] starts from a negligible value
at low metallicity, shows a rise around -1.5 dex
and declines after reaching a maximum.
We have been particularly careful in modelling
this trend with an analytical function
because the results, and in particular the mean metallicity,
depend on the adopted functional form.
To choose the function we started by making an educated
guess of the shape of the metallicity distribution based
on our knowledge of the statistics of galaxies and DLAs.
We then made sure that the adopted function satisfies two requirements:
(1) that it is a "conservative'' one, i.e. a function that does not overpredict
the effects of the obscuration;
(2) that the predicted extension of the function at high metallicities,
where the obscuration is important,
is not affected by the poor knowledge of the low end of the distribution.
After considering several possibilities,
we adopted the function
To make an educated guess for the shape of the distribution we started from the metallicity-luminosity relation in galaxies. Evidence is building up that this relation is valid not only in the local Universe (e.g. Lamareille et al. 2004), but also in DLA systems (Ledoux et al. 2005). The empirical relation between the logarithmic metallicity and the absolute magnitude implies that the linear metallicity Zincreases linearly with the luminosity. The luminosity in turn follows a Schechter distribution both in the local Universe (e.g. Cuesta-Bolao & Serna 2003), at redshift (e.g. Blanton et al. 2003) and, apparently, up to (Poli et al. 2003). Combining these different pieces of evidence, it is reasonable to assume that Z may follow a Schechter distribution in DLA systems.
|Figure 6: Frequency distributions of of H I column densities ( left panel) and metallicities ( right panel) in DLAs. Dashed lines: true distributions. Solid lines: distributions biased by quasar obscuration. Circles: empirical distributions obtained from DLA surveys. By imposing that the biased distributions best fit the observed ones, we infer the true distributions (see Sect. 4.4). The solution shown in the figure was derived for DLAs with a MW-type extinction and quasars observed in the band.|
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The requirement (1) of a "conservative'' function is equivalent to make sure that decline at high metallicity is fast. In fact, the faster the decline of the "true'' distribution, the lower the estimated number of missed DLAs. Luckily, the Schechter satisfies this requirement since it provides a very fast decline at . In fact, the decline is faster than in any polynomial in Zand in a lognormal distribution.
The requirement (2) implies that we should avoid functions for which a single parameter specifies the behaviour at both ends of the distribution. The Schechter function satisfies this requirement since the parameter , which in practice controls the extension of the distribution at low metallicities, has a negligible effect on the exponential decrease at . For a lognormal distribution, instead, the width parameter is affected by the fit of the distribution at the low-metallicity end, which is poorly constrained by the observations. The resulting error on the width parameter will affect the extension of the distribution at high metallicities and therefore the estimate of the obscuration effect. The same considerations apply to any distributions that, like the lognormal one, are specified by a centroid and a width. For these types of functions, the uncertain behaviour of the low end of the metallicity distribution affects the determination of the obscuration effect.
In Table 1 we summarize the results obtained from the application of our procedure. The quasar extinctions were estimated in the photometric bands and for an absorption redshift z=2.3. The differences between the results obtained for the two bands give an estimate of the uncertainty due to the lack of a homogeneous set of visual magnitudes for all the quasars of the adopted surveys. The obscuration fraction in the band is more conservative than that in the band, given the increase of the extinction with decreasing effective wavelength. The results for the band ( m) are more comparable to those derived in previous studies of the obscuration bias, which considered the B band ( m; Ellison et al. 2001).
Two types of dust extinction curves were considered, namely the MW curve by Cardelli et al. (1988) and the SMC curve by Gordon et al. (2003). The presence or absence of the 2175 Å extinction bump in DLAs is irrelevant for most of our results, because at z=2.3the extinction curves in the and bands do not sample the bump (Fig. 2). The only marginal exception is the combination of the band with the MW-type extinction. If the bump is completely absent in DLAs, the MW results become closer to the SMC results for the band.
For each combination of extinction type and photometric band we derived the best-fit values of the parameters via minimization in the intervals , , and . The best-fit parameters are listed in Table 1. We made sure the minimum is unique by careful inspection of the variation of in the specified parameter space. The typical fit errors of , and are , and dex, respectively.
In Fig. 6 we show the true and biased distributions (dashed and solid lines) of the best-fit solution for the case of MW-type extinction and band. The empirical distributions (circles with error bars) are well reproduced by the computed biased distributions. Similar results are found for all the cases considered in Table 1.
The effect of DLA extinction on the shape of distribution function of quasar apparent magnitudes can be seen in Fig. 5, where we show the adopted distribution (dashed line) and the predicted biased distribution (solid line) for the band and MW-type extinction. The agreement between and the empirical distribution (diamonds) demonstrates the capability of the procedure to account for the bias in a self-consistent way. The results are very stable for variations of of the parameter around the adopted value . Figure 5 shows that the obscuration effect does not change substantially the shape of the distribution of apparent magnitude of the quasars, at least within the current values of limiting magnitude of the surveys. This is due to the fact that a large fraction of DLAs has relatively low H I column density, as shown in Fig. 6 and, as a consequence, low extinction. The modest bias of the quasar statistics does not imply that the bias is unimportant for the DLAs statistics, as we discuss below.
Figure 6a is reminiscent of Fig. 8 by FP93 where these authors compare the model and empirical distributions of and k. In fact, the dust-to-gas ratios k is defined by FP93 as an extinction per unit H I column density and is therefore almost equivalent to a metallicity, as one can see in our Eq. (11). The fact that the metallicity distribution is much better constrained by the observations than the k distribution is an advantage of our approach. In FP93 the model distribution of is derived using an exponential H I radial profile; in our work the model is meant to be the simplest analytical approximation of the true distribution in the range probed by the observations.
From the analysis of the H I column density distribution we derive two main results. First, the typical value of that we find, , is remarkably similar to that measured in quasar absorbers of lower column densities, not affected by obscuration bias (Tytler 1987; Petitjean et al. 1993; Storrie-Lombardi & Wolfe 2000). Second, the biased distribution derived from a simple power law successfully reproduces the shape of the empirical distribution. Both results indicate that the fast decline of the number of absorbers observed in the DLA regime is due to the obscuration effect, the true distribution of column density being instead consistent with a simple power law, as for the majority of quasar absorbers. Quite interestingly, the typical values of that we derive are intermediate between the predictions of the exponential profile model of FP93, , and the values expected from the mass-spectrum of interstellar clouds, (see Sect. 5.2.2). This seems to indicate that both effects should be taken into account in the full theoretical treatment of the true distribution . The hydrodynamic simulations of DLAs by Cen et al. (2003) yield a slope slightly below 2, broadly consistent with our results.
The approximation of the true distribution with a simple power law should not be extrapolated in the range of column densities not sampled by the observations ( ). An intrinsic faster decline in that range must occur in order to avoid an infinite value of (see Sect. 6.5). A faster decline is expected in the exponential profile model of disk gas distribution when approaches the central value (FP93). Also the onset of physical mechanisms specific of very high density environments may induce a sharp drop of the distribution at very values of (Schaye 2001). The present results suggest that the genuine fast decline of the distribution may lie beyond the range probed by the observations. In principle, one could model this intrinsic decline with a Schechter function, but the position of the "knee'' would be unconstrained. In the following we introduce an upper cutoff in the column density distribution when we need to estimate quantities affected by the high end of the distribution.
The results on the metallicity distribution depend on the adopted model for the shape of the distribution. As discussed in Sect. 5.3, we believe that the results obtained from a Schechter function are more conservative and reliable than those derived from other functions (polynomials, lognormal or other functions parametrized in terms of "centroid'' and "width''). In addition to the considerations given in Sect. 5.3 we note here that the position of the "knee'' is sufficiently well constrained by the observational data (Fig. 6b).
|Figure 7: Distribution functions of the logarithmic metallicity normalized to unity. Curves: true distribution of DLAs at inferred with our method. Dashed (dotted) lines: MW-type (SMC-type) extinction; red (green): results for the () band. Histograms: Milky-Way stars belonging to the thick disk (shaded area) and thin disk (thick line) populations.|
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|Figure 8: a) Fraction of obscured DLAs and b) number density of DLAs predicted as a function of the limiting magnitude for the best-fit solutions of Table 1. Dashed and dotted lines: MW-type and SMC-type extinction, respectively. Red and green: results for the and bands. Stars with error bars: CORALS results from Ellison et al. (2001).|
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The main result of the analysis of the metallicity distribution is that a non-negligible fraction of DLAs exist with near solar metallicity. In previous work the metallicity distribution of DLAs has been compared with that of different stellar populations of the Milky Way to cast light on the nature of DLA galaxies. Pettini et al. (1997) found that the [Zn/H] distribution observed in DLAs does not resemble that of any Milky-Way population. The true metallicity distribution of DLAs inferred in the present work changes significantly these conclusions. In Fig. 7 we compare the DLA distributions obtained for the best-fit solutions listed in Table 1 with the stellar distributions published by Wyse & Gilmore (1995) for the thick and thin disk of the Milky Way (the same template used by Pettini et al. 1997). One can see that the true DLA distribution envelops those of both disk populations. This result is consistent with the paradigm that DLA galaxies are progenitors of present-day disk galaxies, even though there is plenty of room in the distribution for contributions from metal-poor galaxies. We cannot derive more information from these results because the exact shape of the true distribution is uncertain, particularly in the tails, owing to the poor statistics of the surveys. In any case, the present results bring fresh support to the most commonly accepted paradigm on the nature of DLA galaxies, suggesting that most disks may already be in place at , consistent with what happened in the Milky Way.
In Table 1 we list the total obscuration computed adopting atoms cm-2as upper limit of integration in Eq. (17). This is equivalent to truncate the H I distribution in the range of column densities not probed by the observations. The true obscuration factor might be higher than if DLA systems exist also beyond such limit. An estimate of this uncertainty can be appreciated in Table 1, where we also give , obtained integrating the distributions up to atoms cm-2.
The total obscuration fraction lies in the range for the magnitude limit typical of high-resolution surveys ( ). For spectroscopic surveys of moderate resolution, with , we obtain . These figures are underestimated by only if DLAs exist with the same power law, up to atoms cm-2.
The fraction of missing systems that we find is consistent with the range 0.23 to 0.38 estimated by PF95 at .
Our estimate of total obscuration fraction can be compared with that obtained by Ellison et al. (2001) from the analysis of an unbiased sample of radio-selected quasars. By comparing their number density of DLAs, n(z), with that obtained by Storrie-Lombardi & Wolfe (2000) from an optically-selected (biased) sample, Ellison et al. conclude that the unbiased n(z) is about larger than the biased one. Since the Storrie-Lombardi & Wolfe survey has a magnitude limit typical of moderate resolution surveys, this result should be compared with our obscuration fractions derived for , which indicate that the true number is between and larger than the biased one. The general agreement with the result found by Ellison et al. represents an important test of validity of our procedure.
One prediction of the obscuration bias is that the fraction of missed DLAs must decrease with increasing limiting magnitude of the survey. The variation predicted by our computations is shown in Fig. 8a for the different cases considered in Table 1. The obscured fraction decreases, but does not vanish at least up to , where . This variation of versus offers an important observational test based on the measurement of the number density of DLAs, n(z), at increasing values of in an unbiased sample. Ellison et al. did find an increase of n(z) with which supports the existence of the bias. The number density increases in the range , but flattens at , while we predict that it should keep increasing at least up to . However, taking into account the experimental error bars there is no real discrepancy. To demonstrate this, we plot in Fig. 8b for an assumed total density . One can see that these predictions agree with the measurements of n(z) performed by Ellison et al. If , the true number of systems would be larger than the biased n(z) measured by Storrie-Lombardi & Wolfe. This figure of obscuration is still consistent with our estimate. The only "disagreement'' with Ellison et al. would be on the interpretation of the n(z) versus plot, for which we claim that there is a steady increase rather than a plateau. More stringent measurements are crucial for clarifyng this issue. It is clear, in any case, that the combination of our treatment of the bias with studies of unbiased samples offers a powerful tool for a quantitative estimate of the obscuration effect.
By using our best-fit
we can compute the total contribution of the gas in DLAs
to the critical density of the Universe,
To estimate the missed fraction of H I mass, we compare these values of with , the biased value obtained integrating our H I sample with the commonly adopted Schechter function (19). Since is a conservative estimate, this comparison indicates that at the true comoving mass of DLAs is at least a factor of 2 higher than that obtained by magnitude-limited surveys.
The values of that we derive agree well with the value measured by Ellison et al. (2001) from the analysis of their "dust-free'' sample. In addition, they agree with the value derived at by Cen et al. (2003) from their hydrodynamic simulations.
The column-density weighted metallicity of DLA systems
is used to measure the degree of metal
enrichment of the population of DLAs as a whole (Pettini et al. 1999) and
to estimate the mean cosmic metallicity of the high-redshift
Universe (Pei & Fall 1995; Cen et al. 2003).
The expression commonly used in literature to measure the weighted metallicity
from a finite set of column densities,
can be put in the form
In Table 1 we list the mean metallicity, normalized in the usual form , obtained for a Schechter metallicity distribution. The mean metallicity lies between -0.44 and -0.27 dex below the solar level for the different dust models and photometric bands considered in Table 1. Considering the total error of -0.3 dex in our estimate (fit errors plus statistical errors of the empirical distributions) we conclude that the mean metallicity is -0.7 dex at 1 level. These values are higher than the direct measurements of weighted metallicity of DLAs, which give -1.1 dex at the typical redshift of our sample (Pettini et al. 1999; Prochaska et al. 2003). This large difference is probably be due to the fact that the weighted metallicity is extremely sensitive to the number of high column-density DLAs included in the average, which are the DLAs more affected by obscuration. Our results are marginally consistent with the mean weighted metallicity dex obtained from the CORALS metallicity survey (Akerman et al. 2005).
|Figure 9: Fraction of obscured DLAs as a function of a) H I column density, b) metallicity Z, and c) Zn II column density for the best-fit solutions of Table 1. Dashed and dotted lines: caption as in Fig. 8.|
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Quite interestingly, the high mean metallicity that we derive may help to solve the "missing metal'' problem pointed out by Wolfe et al. (2003) in their work on the Star Formation Rates (SFRs) in DLAs: the mass of metals produced by , inferred from their SFRs, is 30 times larger than detected in absorption in DLAs.
Our method allows us to estimate, for the first time, the fraction of obscured DLAs as a function of H I column density and metallicity Z, and also as a function of the total zinc column density. These obscuration fractions , and , shown in Fig. 9, should only be interpreted in a statistical sense. However, they provides us a powerful indication of how the statistical distributions of DLAs are distorted by the obscuration bias. In particular, they provide a means of quantifying the existence of observational cutoffs induced by the bias itself.
The analysis of , shown in panel c, gives a quantitative explanation for the lack of DLAs systems above the obscuration threshold in Zn II column density originally proposed by Boissé et al. (#Boiss&). We estimate that over of DLAs are missed at atoms cm-2, and this fraction rises rapidly to if the increases by 0.5 dex. The threshold is rather independent of the adopted input parameters. Only for peculiar extinction curves without UV rise (gray extinction) we may expect that the threshold can be crossed.
We can use our results to estimate how many DLAs should be observed in an unbiased survey in order to detect cases with high metal column density. From the typical distribution that we infer, we estimate that the number of DLAs in the range between the detection limit and the Boissé's threshold ( ) is about 24 times larger than the number of systems in a range of similar extension above the threshold ( ). This means that one is not guaranteed to detect one DLA above the threshold even with 30 unbiased spectroscopic observations. This is currently an observational challenge because the high extinction of the DLAs above the threshold makes very hard to perform high-resolution spectroscopy, even for the quasars which lie at the bright end of the true distribution of apparent magnitudes.
A natural consequence of the cutoff at high metal column densities, in conjunction with the artificial DLA threshold atoms cm-2, is the existence of an observational cutoff at high metallicities. From panel b one can see that about of systems are missed at solar metallicity and it would be practically impossible to detect systems with oversolar metallicity, if they exist. Also worth of mention is the fast rise of the obscuration with increasing metallicity, which distorts dramatically our perception of the metallicity distribution of DLA systems. The fact the different curves plotted in panel b lie close to each other indicates that these results are rather independent of the parameters adopted.
Also the distribution of H I column densities is severely distorted by the extinction bias, as shown in panel a. The curves of obtained for different parameters are not tightly close to each other, but the general behaviour is quite similar in all cases. The obscuration starts from a low value in the sub-DLA regime and rises steadily over the DLA regime, approaching at atoms cm-2. This is the approximate value of the observational threshold for detection of high column density DLAs. The rise of , in conjunction with the genuine decrease of the true distribution , provides a natural explanation for the non-detection of DLAs at atoms cm-2. This makes it unnecessary to invoke the sudden transition from atomic to molecular hydrogen proposed by Schaye (2001) in order to explain this empirical cutoff. In any case, if fully molecular clouds exist as predicted by Schaye, they would be characterized by a high column density of dust grains. The fact that quasar absorbers with have not yet been detected may indicate that also this type of absorber is missed due to obscuration.
Starting from the theoretical relation between the interstellar extinction, , and the column density of a volatile metal, , we investigated the detailed relation between and in DLAs. We derived in this way the Eq. (11), which gives the quasar extinction in the observer's frame as a function of the DLA hydrogen column density, , metallicity, , fraction in dust of iron, , and a factor G, which describes the dust grain properties. For the first time, the metallicity evolution of the dust-to-metal ratio in DLAs is explicitly accounted for in this type of relation, using an expression obtained from a previous study of depletions (Vladilo 2004). We argue that the factor G may not vary substantially in interstellar environments with moderately low metallicity ( ) and derive a value of 60% the Milky-Way value for SMC-type dust. Possible variations of G at the early stages of chemical evolution do not affect the extinction because vanishes when .
We searched for empirical evidence of the rise of the extinction with metal column density in DLAs. We did find an increase of the quasar magnitude with , consistent with this expectation (Fig. 4).
Starting from Eq. (11), we derived a mathematical formulation aimed at estimating how the frequency distributions of H I column densities and metallicities of a sample of DLAs are biased as a result of the obscuration generated by the DLAs of the same sample. We ignore the obscuration due to low-redshift DLAs along the same line of sight, showing that this contribution is relatively low. At variance with the formulation of PF95, where a constant dust-to-metal ratio was adopted for all DLAs, we adopted a metal-dependent dust-to-metal ratio . We cross-checked the validity of our formulation making use of the equations derived by FP93 for the case of a power law quasar luminosity function and constant dust-to-gas ratio (Sect. A.1.2).
We presented a practical procedure for recovering the unbiased distributions of column densities and metallicities, and fZ, using the empirical distributions obtained from magnitude-limited surveys. The unbiased distributions are modelled with simple analytical expressions in parametric form. The bias induced by the extinction on the quasar magnitude distribution is accounted for self-consistently by the procedure. A unique characteristic of the method is the possibility of recovering the true metallicity distribution starting from an educated guess of the functional form of the distribution. We have shown that using a Schechter function for the metallicity distribution appears to provide more conservative and reliable results than other functions (e.g. lognormal or polynomials) for the estimate of the bias (Sect. 5.3).
We applied our method to the sample of DLAs in the redshift interval , with mean redshift , where the bulk of spectra are available. The extinctions were computed in the and photometric bands of the SDSS, considering both a MW-type and an SMC-type average extinction curve. We found that the effect of the obscuration on the quasar magnitude distribution is modest (e.g. Fig. 5). On the other hand, the bias plays an important role in shaping the statistical distributions of DLAs.
The unbiased distribution function of H I column densities is successfully approximated by a power law , with . This simple law, in conjunction with the bias effect, is able to fit the observed decline of the number DLAs with up to atoms cm-2. The faster drop of the distribution expected at very high column densities (FP93; Schaye 2001) probably lies outside the range currently probed by the observations. The slope is remarkably similar to the value which characterizes most of quasars absorbers with lower column densities, for which the extinction bias is negligible. The value of the slope suggests that the mass spectrum of interstellar clouds may play an important role in shaping the distribution.
The observed distribution of metallicities is reproduced modelling the unbiased distribution with a Schechter function , with and dex. Once converted into logarithmic scale, this function peaks at [Zn/H] dex, indicating that a non-negligible fraction of DLAs exist with near solar metallicity. The true metallicity distribution of DLAs envelops the distributions of the thick-disk and thin-disk Milky-Way stellar populations. This result is in line with the paradigm that DLA galaxies are progenitors of present-day disk galaxies. However, in the inferred distribution there is also room for a relevant contribution from metal-poor galaxies.
The mean weighted metallicity that we derive , dex, is significantly higher than the column-density weighted metallicity dex measured at . This high value of mean metallicity may help to solve the discrepancy between the metallicity observed in DLAs and that predicted on the basis of their SFRs (Wolfe et al. 2003).
The fraction by number of obscured DLAs, , decreases with increasing limiting magnitude of the survey, . We obtain at , a limit representative of high resolution spectroscopic surveys, carried out with 4-m class telescopes. The corresponding figures for high-resolution surveys in 10-m class telescopes are ( ).
An important result of our work is the explanation (Fig. 9c) of the observational limit atoms cm-2 (Boissé et al. #Boiss&) in terms of very simple physics (Sect. 2), with no tuning of the local dust parameters. Alternative explanations would require an additional mechanism able to produce the exact same limit predicted by the extinction. The simplicity of the extinction model favours the existence of the obscuration. The existence of the threshold atoms cm-2 is fundamental for reconciling the predictions of galactic models (Prantzos & Boissier 2000; Hou et al. 2001; Churches et al. 2004) and cosmological simulations (Cen et al. 2003; Nagamine et al. 2004) with the observations of DLAs. Only for gray extinction curves, without UV rise, we may expect that the threshold can be crossed.
Our results on the magnitude of the obscuration effect are broadly consistent with those presented by FP93 and PF95, with some important differences. The modest bias of quasar statistiscs that we find is at variance with the predictions of FP93, which allowed up to 70% of quasars to be obscured. Contrary to FP93, we do not expect a significant contribution of DLA extinction at redshift in spite of the rise of with z. In fact, our model of dust fraction vanishes at very low metallicity so that the extinction (11) vanishes at , when the metallicity .
A crucial test of our results is the comparison with studies of radio-selected samples of quasars/DLAs. Our determinations are consistent with the estimates of obscuration fraction, number density and obtained from the CORALS survey (Ellison et al. 2001). Our prediction of mean metallicity is marginally consistent with the recent estimate of mean weighted metallicity of the CORALS sample (Akerman et al. 2005).
Accurate estimates of the obscuration effect will be possible as soon as the samples with metallicity measurements become sufficiently large for precise statistical analysis.
This work has benefitted from interactions with Irina Agafonova, Patrick Boissé, Miriam Centurión, Sergei Levshakov, Paolo Molaro, Pierluigi Monaco, Sara Ellison and Paolo Tozzi. We warmly thank Arthur Wolfe for his remarks on the first part of the manuscript.
We call the number of neutral hydrogen layers with column density and metallicity located in the spherical shell of redshift in the infinitesimal solid angle in a random direction (we assume that the gas layers are distributed isotropically).
We call the true number of quasars with apparent magnitude in the shell of emission redshift all over the sky (we assume an isotropic distribution of quasars). We define , where z is the absorption redshift of a foreground H I layer and the maximum redshift of the quasars in the survey. For simplicity of notation we omit hereafter the dependence on . Therefore is the total number of quasars with magnitude observable beyond redshift z all over the sky. This is the true number in absence of quasar obscuration.
We call the average solid angle subtended in the observer's optical band by quasars in the redshift interval .
With the above definitions, the number of DLAs
in front of one single background quasar with
We then consider the quasar extinction
due to the DLAs in
ignoring other sources of extinction.
From this we define the bias function
The relation between the true distribution
and the distribution
biased by its own DLAs is
To make practical use of Eq. (A.6)
we need to understand the relation between the biased distribution
and the observed distribution of DLAs in
that we call
To derive an expression for the observed distribution
we must consider all the sources of obscuration,
including the DLAs at
that lie in the
same lines of sight of the DLAs at redshift z.
If these additional sources of obscuration are negligible,
then the expression (A.6) is a good
approximation of the observed distribution (see Appendix A.1.1), i.e.
To compare the distributions with the observations
we integrate Eq. (A.6)
over the interval
close to the peak of the observed distribution.
If the statistical functions of interest show a smooth variation
inside the interval we can approximate them
with their value at
We now transform the distribution into a distribution of column densities, , and a distribution of metallicities, fZ, defined in such a way that is the true number of detectable DLAs with column densities between and in the interval and the true number of detectable DLAs with metallicity between Z and in the same redshift interval. We call and the corresponding biased distribution functions.
To proceed further we assume that
and Z are independent
(this assumption is discussed in Sect. 4.4).
This implies that
We substitute this expression in the right hand of Eq. (A.8).
We then integrate that equation in metallicity and obtain
We start by deriving the relation between the true number of detectable DLAs in absence of obscuration, , and the number observed when all sources of obscuration are considered, .
At redshift z the observed number equals the true number
minus the fraction of DLAs
missed as a consequence of their own extinction,
minus the fraction of DLAs at redshift z
missed due to the extinction of other DLAs at
that lie in the same line of sight,
From a recent summary (Ellison et al. 2004) of the unbiased number density of DLAs, , we calculate that the probability of intersecting 2 DLAs in a random line of sight up to is 0.15. We take this value as an estimate of .
By comparing the number densities of biased and unbiased surveys there is room for no more than 50% of missed DLAs at (Ellison et al. 2001, 2004). This represents an upper limit to because the extinction in the observer's frame tends to decrease with decreasing redshift and therefore the missed fraction tends to diminish at lower redshifts.
All together, we expect therefore at and lower values at lower redshifts. We conclude that the approximation (A.7) is good, even if it may slightly underestimate the obscuration fraction.
To test our formulation we cross-checked its predictions in a particular case considered by Fall & Pei (1993). Namely, we compared the predictions of our Eq. (A.6), calculated at constant metallicity Z, with those of Eq. (21) by FP93, which is calculated at a constant dust-to-gas ratio k. This test is equivalent to insert in Eq. (A.12) a Dirac delta distribution of metallicities. To make this comparison we transformed the power-law distribution of quasar luminosities used by FP93 [their Eq. (8)] into a magnitude distribution n(m;z)for our Eq. (A.5). We adopted and as common values in both equations. In Eq. (21; FP93) we adopted a value of k=0.04 consistent with our choice of G=1, and for this particular test. In Fig. A.1 we compare the resulting ratio f0(N,z)/ft(N,z) predicted from Eq. (21) by FP93 with the corresponding ratio predicted from our Eq. (A.6), which equals the term of Eq. (A.5). Our predictions (empty circles) perfectly match those obtained with the equations derived by FP93 (filled diamonds). Our results are independent of provided and . For these conditions are well satisfied.
|Figure A.1: A test of validity of the equations presented in Sect. 4.1. Circles: predictions of our Eq. (A.5). Diamonds: predicted from Eq. (21) by Fall & Pei (1993). The adopted parameters are given in Appendix A.1.2.|
We derive the relation
between the "true'' magnitude distribution of quasars, n(m; z),
and the observed one, which is biased by the obscuration effect.
We consider only the obscuration due to DLA systems.
We start by defining
the observed (biased)
number of quasars with magnitude
We then divide the quasars according to the number iof DLAs that lie in their direction.
the fraction of quasars at redshift that have i intervening DLAs.
Depending on the value of i, the distribution
of apparent magnitudes of the quasars will be different.
the biased number of quasars with i foreground DLAs,
We now assume that
the fraction of quasars with two or more
DLAs is negligible (see A.1.1), i.e. that
From the condition
We now integrate Eq. (A.18) in
over the interval
where zis the absorption redshift and
the maximum redshift of the quasars in the survey.
vary smoothly in the interval, so that it can be
approximated with its mean value
We now derive an independent relation
The apparent magnitude of the quasars with one DLA,
m, equals the
sum of the "true'' magnitude in absence of extinction, ,
plus the DLA extinction in the observer's frame, .
The frequency distribution of the sum
is given by the convolution of the frequency distributions
since these are independent variables.
The fraction is estimated from statistical surveys. From the statistics of quasars with one finds, in the redshift path observable from ground (, ), that 40% of the quasars have one single DLA (see Péroux et al. #P&). The true fraction will be slightly higher than 0.4owing to the limitations of the wavelength coverage of the spectra. We adopt here .
We assume that can be approximated with the distribution of extinctions in the redshift range where most DLAs are observed, . In this way the problem of estimating is brought back to that of estimating the distributions of column densities and metallicities in the same redshift range, and fZ. Assuming that and Z are statistically independent variables, on the basis of Eq. (11) we obtain the extinction distribution by convolving and fZ.
The normalization factor K is determined from the condition , which, from the integration in of Eq. (A.19), is equivalent to . These conditions guarantee that the total number of quasars is independent of the bias, since the only effect of the extinction is a redistribution of the apparent magnitudes.