6 DLA statistics-number density and $\Omega_\mathsf{DLA}$

One of the key areas of interest in the study of DLAs is how the population evolves with time. If DLAs represent the bulk of the galaxy population at a given epoch, then studying their properties as a function of redshift will be a powerful method for tracing galaxy evolution. For example, there is strong evidence for redshift evolution in the column density distribution function, f(N) (Storrie-Lombardi & Wolfe 2000). However, since the number density is well represented by a power law there appears to be no change in the product of space density and absorber cross section. As discussed in the Introduction, the evolution of $\Omega _{\rm DLA}$ is still unclear, particularly at $z_{\rm abs} < 1.5$ where statistics are poor.

Due to the relatively small size of the CORALS survey, it is not possible to investigate the evolution of DLA statistics for this sample. However, since our main objective is to ascertain whether or not a significant fraction of gas has gone undetected, it is sufficient for us to restrict our determination of $\Omega _{\rm DLA}$ to the range $1.8 < z_{\rm abs} < 3.5$ where there appears to be little evolution. This requires us to omit two DLAs from our sample (B1251-407a and b), and restrict our statistical analysis to the remaining 17 DLAs.

6.1 DLA number density

This is simply the total number of DLAs divided by the total intervening redshift interval covered, $\Delta z$, given by

$$\Delta z = \sum_{i = 1}^{n} (z_{i}^{\max} - z_{i}^{\min})$$ (1)

where the summation is over the n QSOs in the sample. For CORALS, $z_{\min} = 1.8$ and $z_{\max} = 3.5$ or z₃₀₀₀ (the redshift corresponding to a v= 3000 km s^-1), whichever is smaller:

$$z_{3000} = \left[ (z_{\rm em} + 1) \sqrt{ \frac{(~b - 1)}{(-b - 1)} } \right] - 1$$ (2)

and

$$b = \frac{v}{c} = 0.01.$$ (3)

In cases where a Lyman limit system is present in the QSO spectrum within the redshift range considered,

$$z_{\min} = \frac{(z_{\rm LLS} + 1) \times 912}{1216} - 1.$$ (4)

The total redshift interval for the CORALS sample is $\Delta z = 55.46$ within which we detect 17 DLAs. Thus, n(z) = 0.31^+0.09_-0.08 at a mean absorption redshift $\langle z_{\rm abs} \rangle= 2.37$ (the errors were calculated using the prescription by Gehrels 1986 for small number statistics).

Taken at face value, the estimate of n(z) from the CORALS survey is $\sim$50% larger (at the same $\langle z_{\rm abs} \rangle$) than that found in previous surveys. For example, Storrie-Lombardi & Wolfe (2000) deduced n(z)=0.055(1+z)^1.11 = 0.21 (no errors quoted). However, this difference is only marginally significant, since the two determinations of n(z)are within $\sim$1$\sigma$ of each other.

Figure 3: The mass density of neutral gas, $\Omega _{\rm DLA}$, in DLAs. Open circles and squares are measurements from the latest compilations by Péroux et al. (2001b) and Rao & Turnshek (2000) respectively. The solid circle is the value from the CORALS survey presented here for the redshift interval $1.8 < z_{\rm abs} < 3.5$.

Figure 4: Distribution of H I column densities in the large sample of Storrie-Lombardi & Wolfe (2000) and in the CORALS survey (shaded histogram). Due to the relatively small number of QSOs, and therefore DLAs, included in CORALS, we do not fully sample the N(H I) distribution.

6.2 The mass density of neutral gas in DLAs

The mass density of neutral gas in DLAs as a fraction of the closure density is expressed as

$$\Omega_{\rm DLA} = \frac{H_0 \mu m_{\rm H}}{c \rho_{\rm crit}} \int_{N_{\min}}^{N_{\max}} N f(N) {\rm d}N$$ (5)

where $\mu$ is the mean molecular weight (= 1.3), $m_{\rm H}$ is the mass of a hydrogen atom, and f(N) is the column density distribution. We avoid making a priori assumptions about the functional shape of f(N) by adopting the approximation by Storrie-Lombardi et al. (1996)

$$\int_{N_{\min}}^{N_{\max}} N f(N,z) {\rm d}N =\frac{ \sum_{i} N_i({\rm H~I})} { \Delta X}$$ (6)

so that

$$\Omega_{\rm DLA} = \frac{H_0 \mu m_{\rm H}}{c \rho_{\rm crit}} \frac{ \sum_{i} N_i({\rm H~I})}{ \Delta X}$$ (7)

from which the mass density of neutral gas traced by DLAs is obtained by direct summation of the individual values of neutral hydrogen column density. The redshift path, $\Delta X$, which takes into account co-moving distances, is given (in our adopted cosmology) by

$$\Delta X = \sum_{i} \frac{2}{3}[(1+z_{\max,i})^{\frac{3}{2}} - 1] - \frac{2}{3}[(1+z_{\min,i})^{\frac{3}{2}} - 1].$$ (8)

From Eqs. (7) and (8) with the values of N(H I) listed in Table 3 we deduce $\log \Omega_{\rm DLA} h = -2.59^{+0.17}_{-0.24}$ (with errors calculated as described by Storrie-Lombardi et al. 1996). As can be seen from Fig. 3, we again find that the CORALS value of $\Omega _{\rm DLA}$ is in good agreement with previous determinations. Thus, the most straight-forward conclusion from the CORALS survey is that existing magnitude limited samples of QSOs do not seriously underestimate the number of DLAs, nor their overall mass content; by and large, they seem to provide a fair census of neutral gas at high redshift. On the other hand, given the current statistical uncertainties, the data still admit a moderate degree of dust bias. The $1 \sigma$ limits on both n(z)and $\Omega _{\rm DLA}$ include the possibility that analyses based on optically selected samples may have underestimated both quantities by a factor of $\sim$2. This possibility is further highlighted when we consider the distribution of values of N(H I) in the CORALS survey (Fig. 4). Given the small size of our survey, we may well be missing DLAs at the high column density end of the distribution simply through small number statistics. We return to this point in Sect. 6.3 below. We note that although it is impossible to constrain the H I column density distribution for the CORALS sample, we do detect 2 relatively rare high N(H I) systems. Therefore, although we do not find evidence for a previously excluded population of very high column density absorbers, with improved statistics it would be of great interest to re-assess the N(H I) distribution function.

We also consider another effect. As explained in Sect. 2, many previous surveys have included in their statistical analyses candidate DLAs, identified on the basis of the equivalent width of the Lyman $\alpha$ line rather than by profile fitting to a damped profile. It is worthwhile examining the overestimate of $\Omega _{\rm DLA}$ which may result from this approximation. For the 17 DLAs for which we have obtained our own spectra, we measure the equivalent width and compare the implied column density to that determined by fitting the Lyman $\alpha$ line. In most cases we find the two techniques to be in very good agreement, certainly within the errors associated with each method. There are only three exceptions (the DLAs in B1055-301, B1251-407a and B2314-409) where the equivalent width determination leads to a much higher N(H I) than the line fit. Inspection of Fig. 2 shows that this is due to extended absorption around the DLA. Although fits of these DLAs were not straight-forward, this process was facilitated by higher spectral resolution and coverage of metal lines which provide additional guidance in the shape of the wings and $z_{\rm abs}$. Had we used the values of N(H I) deduced from the equivalent widths for the entire CORALS sample, we would have over-estimated $\Omega _{\rm DLA}$ by 20%. This discrepancy would have been further increased if extended blends of lines which do not include a DLA, such as those present in the spectrum of B1251-407, were mistakenly included. Nevertheless, the over-estimate is not large and, given the increasing body of accurate measurements of N(H I) in DLAs, we think it very unlikely that this effect could be masking a higher degree of dust bias than that indicated by inspection of Fig. 3.

Finally, for completeness, we calculate $\Omega _{\rm DLA}$ in the redshift range $3.5 < z_{\rm abs} < 4.0$ where we detect two DLAs despite the fact that with CORALS we only sample a total interval $\Delta z=1.26$. The error bars are naturally very large, but all the same it is intriguing that $\log \Omega_{\rm DLA} h$ seems to remain high at -2.37^+0.24_-0.59 in contrast with the slight down-turn suggested by the work of Storrie-Lombardi & Wolfe (2000) and Péroux et al. (2001b). It will be very interesting to see how better statistics will impact upon the CORALS value of $\Omega _{\rm DLA}$ at the highest redshifts since the present determination would suggest an increase in the importance of dust bias with increasing redshift.

  
6.3 DLA statistics as a function of B-band magnitude

In their preliminary analysis of this sample, Ellison et al. (2000) found tentative evidence that $\Omega _{\rm DLA}$ was higher towards fainter QSOs, consistent with the effect expected from a dust bias. We re-examine this point in Fig. 5, which shows cumulative statistics for the CORALS DLAs as a function of the B-band magnitude of the background QSOs. Since there are relatively few bright QSOs in our sample, we also show the statistics for the LBQS DLA survey (Wolfe et al. 1995) which has a limit , using the column densities reported by Wolfe et al. (1995) and Storrie-Lombardi & Wolfe (2000). We confirm the initial conclusion by Ellison et al. (2000) that $\Omega _{\rm DLA}$ increases as fainter QSOs are observed, but stabilises at $B \simeq 20.0$. Thus, we do not find a population of high N(H I) DLAs which is only revealed when faint ( ) QSOs are observed. However, closer inspection of the data emphasises the need to extend our survey in order to fully sample the column density distribution function, particularly at the high column density end. The increase in $\Omega _{\rm DLA}$ between the $B \leq 19.0$ and $B \leq 19.5$ bins is almost entirely due to a single DLA with N(H I) = $3.5 \times 10^{21}$ cm^-2. Similarly, the increase between the $B \leq 19.5$ and $B \leq 20.0$ bins is caused by a single DLA with N(H I) = $4.5 \times 10^{21}$ cm^-2. Between them, these two systems account for over half of the neutral gas in the entire DLA sample. A K-S test that compares the distribution of N(H I) among DLAs towards B < 20 QSOs with those from the large sample of Storrie-Lombardi & Wolfe (2000) provides inconclusive results, i.e. that the samples are indistinguishable at only the 1$\sigma$ level. Taken as a whole, the CORALS DLAs are inconsistent (again only at the 1$\sigma$ level) with the N(H I) distribution of Storrie-Lombardi & Wolfe (2000). As found by Ellison et al. (2000) for a sub-sample of the CORALS sample, we confirm that for the complete sample that there is no strong correlation between magnitude and redshift out to $z \sim 3.5$. Therefore, these trends are not likely to be associated with the evolution (apparent or real) of the properties of the QSOs themselves.

Figure 5: DLA statistics as a function of magnitude. The quantities plotted are for the cumulative values in the bins $\leq$ the B-band magnitude on the x-axis. Bottom panel: cumulative histogram of number of DLAs. The dashed area shows the number of LBQS DLAs used to determine improved statistics towards bright QSOs. Middle and top panels: stars show the values for CORALS DLAs, whilst the solid square shows the same quantity for the LBQS sample $(B\la 19)$; this point has been offset on the x-axis for clarity).

In addition to the cumulative values of n(z) shown in Fig. 5, we calculate the number density of DLAs towards QSOs with $B \geq 20$ and B < 20, and find n(z) = 0.38^+0.20_-0.14 (at $\langle z_{\rm abs} \rangle = 2.54$) and 0.27^+0.11_-0.08 (at $\langle z_{\rm abs} \rangle = 2.25$) respectively. For the B < 20 subset, this value is consistent with the number density found by Storrie-Lombardi & Wolfe (2000). Again, we see that there is an excess of DLAs in faint QSOs, but only at the $\sim$1$\sigma$ significance level.