PRECIOUS METALS IN SDSS QUASAR SPECTRA. I. TRACKING THE EVOLUTION OF STRONG, 1.5 < z < 4.5 C iv ABSORBERS WITH THOUSANDS OF SYSTEMS

We began our survey with the 105,783 sightlines in the SDSS DR7 QSO catalog Schneider et al. (2010; also see Richards et al. 2002 for the SDSS quasar selection). SDSS spectra have wavelength coverage 3820 Å ⩽ λ ⩽ 9200 Å and resolution varying from R = 1850 to 2200 (or 162 km s⁻¹ to 136 km s⁻¹), and the reduced spectra are binned to a log-linear scale of 69 km s⁻¹ pix⁻¹. We immediately excluded the 6214 broad-absorption line (BAL) QSOs detailed in Shen et al. (2011).

We limited the sightlines to those with z_QSO ⩾ 1.7 and median signal-to-noise ratio 〈S/N〉 ⩾ 4 pixel⁻¹ in the wavelength range sensitive to C iv absorbers (see Figure 1). This range depended on the quasar redshift and the SDSS wavelength coverage.

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Previous, smaller surveys for C iv doublets have searched from the Lyα λ1215 to C iv emission wavelengths. However, we removed a portion of this path length close to the Lyα emission to avoid possible confusion between the C iv doublet and any O i λ1302, Si ii λ1304 pair. These latter two have a wavelength separation similar enough to the C iv doublet that automated search algorithms naturally bring in a large fraction of false positives. Since we had so much available path length available, we simply excluded a comfortable region around the O i, Si ii "forest" in addition to the Lyα forest (i.e., λ_r ⩾ 1310 Å in the rest frame of the quasar). To exclude absorbers intrinsic to the QSO or affected by local ionization, clustering, and/or enrichment, we set the upper wavelength bound to 1548 Å (1 + z_QSO)(1 + δv_QSO/c),⁷ where we initially used δv_QSO = −3000 km s⁻¹, but for the main analyses, the limit is δv_QSO = −5000 km s⁻¹.

After the redshift, 〈S/N〉, and coverage cuts, we had 26,168 sightlines to search (see Table 1). Later, we excluded 138 sightlines as "visual BAL" QSOs for a final count of 26,030 spectra to analyze.

We generated individual quasar continua for all ≈26, 000 spectra in the sample using an algorithm that combined principle-component analysis (PCA) fits, low-frequency b-spline correction, and automated outlier pixel and absorption-line exclusion. Since we are interested in C iv lines only, we limited the fit to λ_r ⩾ 1230 Å, redward of the Lyα forest. We discuss details of the method below so interested users may replicate it; the codes are also publicly available in the xidl software library.⁸ Figure 2 shows example fits for eight spectra of varying 〈S/N〉 level.

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First, every spectrum was iteratively fit using a 50-vector basis set of PCA "eigenspectra" from the SDSS DR1 quasar sample (Yip et al. 2004). The fit coefficients and their errors were determined using the algorithms of Connolly & Szalay (1999, Equations (4)–(6)), with spectral pixels weighted by their inverse variance when quantifying the goodness of fit. The formal per-pixel error of the PCA continuum fit was negligible compared to the spectral shot noise.

A pixel exclusion mask was then generated upon each successive fit iteration. The initial mask consisted of the lowest 30% of pixels in successive 50 pixel windows, to filter strong absorption features and avoid biasing the continuum low on the critical initial fit. Then, for each subsequent iteration, a fresh mask was constructed that excluded pixels falling either less than −2σ or more than 3σ from the working continuum. This mask was further modified to include the neighboring pixels of each deviant pixel below the working continuum and exclude regions with three consecutive pixels above. The latter correspond to systematically errant regions of the fit that should actually be included, so they were unmasked automatically. This cycle was repeated with successive PCA fits until: (1) the mask converged so successive fits did not change; (2) the unmasked flux and continuum fit had median difference less than 0.001%; or (3) 10 iterations passed.

For high-S/N spectra, low-frequency residuals in the PCA-normalized spectra become noticeable and affect our ability to search for weak C iv lines in the best data. We therefore performed a secondary correction to spectra with 〈S/N〉 ⩾ 8 pixel⁻¹, re-fitting the PCA-normalized spectrum with an additional third-order b-spline, with breakpoint spacing of 25 pixels (or 1725 km s⁻¹). As before, we used inverse-variance weighting and clipped outliers using the same thresholds. We refer below to this PCA plus b-spline fit as the "hybrid continuum."

The b-spline systematic error was estimated as the median difference between the b-spline and the PCA-normalized (masked) spectrum in bins matching the breakpoint spacing. This error was typically of the order of 1% of the shot noise in the data; it exceeded the formal PCA fit error, and we added the two in quadrature to produce a total continuum error.

The sigma clipping methods described above are effective at identifying narrow features, instrumental artifacts, or defects in the spectra. However, they are not always optimal for finding true absorption lines, which are always below the continuum; they can also be kinematically complex and bias the fit. Since we have prior information about the characteristic widths of cosmological absorption, we develop methods in Section 2.3 to machine-identify candidate absorption lines with these characteristics for the actual survey. We ran each PCA-normalized spectrum through a single pass of the absorption-line finder, using the resulting catalogs to generate a master absorption mask for final hybrid-continuum fitting.

The full procedure for automated feature finding is described in Section 2.3; briefly, we convolve both the data and error arrays with the instrumental response profile, and search for absorption features with S/N ⩾3.5 per resolution element. For 〈S/N〉 ⩽ 16 pixel⁻¹ spectra, we masked out all such features; in high-S/N cases, we also enforced that the unconvolved spectral pixels at the line center must deviate from the continuum by >5σ. Pixels falling within ±600 km s⁻¹ of such features were added to the exclusion set. The mask consisting of automatically identified absorption lines was fed into the hybrid-continuum fit as a static mask.

This procedure produced excellent continuum fits for the vast majority of spectra. When it failed, the most common reasons were poorly measured z_QSO, strong intrinsic absorption, BALs, and foreground emission-line galaxy spectra superimposed on the quasar spectra. Instead of fixing these cases interactively, which would compromise our objective methodology and continuum error estimates, we chose to leave them in the sample and let the effects be accounted in our automated completeness and contamination tests (see Section 3).

We limited our candidate search to C iv doublets where two automatically detected lines were within ±150 km s⁻¹ of the characteristic velocity separation $\delta v_{\rm C\,\scriptsize{IV}}= 498\,{\rm km\,s}^{-1}$. The large uncertainty cut allowed heavily blended lines into our candidate list, but it also made the Mg ii doublet our largest contaminant since $\delta v_{\rm Mg\,\scriptsize{II}} = 767\,{\rm km\,s}^{-1}$.

Our automated feature-finding algorithm is based on Prochter et al. (2006). The hybrid-normalized flux and error arrays were convolved with a Gaussian kernel with FWHM equivalent to a resolution element of the SDSS spectrograph, i.e., σ_G = 1 pixel. The error array included the continuum fit error. The regions where the convolved signal-to-noise $({\rm S/N)_{{\rm conv}}}$ is greater than or equal to 3.5 per resolution element were identified as absorption features and saved to the sightline's line list. We masked out the few pixels (≈5 Å) around the strong skylines at 5579 Å and 6302 Å.

Candidate C iv doublets were compiled by pairing automatically detected lines with the characteristic velocity separation $\delta v_{\rm C\,\scriptsize{IV}}\pm 150\,{\rm km\,s}^{-1}$ and by identifying isolated, automatically detected lines that were broad enough to be a candidate doublet by themselves. As discussed in Section 2.1, the search region is set so that the C iv candidate is redward of the O i, Si ii "forest" (λ_r ⩾ 1310 Å) and blueward of the QSO by 3000 km s⁻¹. For any unpaired line, we re-ran the automated feature finder with a 2.5σ_conv cutoff and searched for a partner candidate 1550 line.

To identify isolated, broad lines that should be candidate C iv doublets, we used the automated procedure for measuring wavelength bounds (described in Section 2.4) to find lines with widths $\delta v_{\rm lin} \ge 1.5\,\delta v_{\rm C\,\scriptsize{IV}}$.

Ultimately, we identified 29,789 candidates, 6346 of which were from the broad-line search.

The redshift, equivalent width, and column density of a line depend on the definition of its wavelength bounds. The bounds were automatically defined by where the convolved S/N array stopped decreasing from the perspective of the automatically detected centroid, and the bounds were not allowed to exceed the midpoint between themselves and neighboring lines. Thus, for C iv lines, the inner wavelength bounds were not allowed to exceed the midpoint of the doublet.

The redshift, equivalent width, and column density errors included the estimated error due to the hybrid continuum. Redshifts were measured from the flux-weighted centroids. Equivalent widths were measured by simply summing the absorbed flux within the bounds.

We used the apparent optical depth method (AODM; Savage & Sembach 1991) to estimate column densities. Systems that are saturated (discussed below) formally have column densities that are lower limits (binary flag f_N = 2). Column density measurements that are less than 3σ_N are flagged as upper limits (f_N = 4).

With doublets, we have two measurements of the column density, and we use both to set $N_{\rm \mathrm{C\,\scriptsize{IV}}}$, as done in Cooksey et al. (2010). For doublets where both lines are measurements (not limits), $N_{\rm \mathrm{C\,\scriptsize{IV}}}$ is the inverse-variance weighted mean of the two values. If one line is a measurement, $N_{\rm \mathrm{C\,\scriptsize{IV}}}$ is set to its value. The column density limit is set to the inverse-variance weighted mean when both doublet limits are the same kind. When the doublet limits bracket a range, $N_{\rm \mathrm{C\,\scriptsize{IV}}}$ is set to the unweighted average with an error that reflects the range (f_N = 8). In the remaining cases, $N_{\rm \mathrm{C\,\scriptsize{IV}}}$ is set to the better measurement/limit and flagged f_N = 16.

The AODM systematically overestimates the true column densities in low-S/N spectra (Fox et al. 2005). On the other hand, most doublets that can be detected in the low-resolution SDSS spectra are saturated, as evidenced by the doublet ratios of near unity; therefore, the AOD column densities would formally be lower limits. We define the doublet to be unsaturated when the equivalent width ratio W_r,1548/W_r,1550 > 2 − σ_R, where σ_R is the ratio error due to the uncertainties in the measured equivalent widths. All other doublets are flagged as saturated with column densities that are lower limits; by this criterion, approximately 84% of the final catalog are saturated (see Table 3).

The candidate C iv selection relied solely on the characteristic wavelength separation of the doublet. However, true, unblended C iv doublets have a well-defined doublet ratio W_r,1548/W_r,1550 in the range of one to two in the saturated and unsaturated regimes, respectively. The resolved profiles of the doublet lines are very similar in the absence of blending. Most C iv absorbers are associated with Lyα absorption, though "naked" systems may exist (Schaye et al. 2007). There are frequently other metal lines associated with the C iv absorption, such as the Si iv and/or Mg ii doublets. In addition, outside of the Lyα (and O i, Si ii) forest, line confusion, and blending are less severe, and the C iv doublet is one of the more common metal lines.

We developed a graphical user interface to present all this information for each system to assist us in rating the 29,789 candidates. We leveraged our experience with C iv absorption systems and the varying amounts of information available, depending on redshift, to assign each candidate a rating:

• 0.  
  definitely false—either can definitively identify the lines as other metal lines or as spurious bad pixels masquerading as absorption;
• 1.  
  likely false—though cannot name alternate metal-line identification, better data would probably confirm this as not C iv absorption;
• 2.  
  likely true—though sparse supporting evidence (e.g., associated lines), better data would probably confirm this as C iv absorption; or
• 3.  
  definitely real—associated with other lines and/or shows clear correlations in line profiles.

A single author was trusted to accurately assign ratings of 0 or 3; these doublets were not viewed again. The more ambiguous cases (ratings 1 and 2) were reviewed by at least one additional author until consensus was achieved. We grouped the doublets into systems with δv_abs < 250 km s⁻¹.

Though we excluded the BAL QSOs from Shen et al. (2011), we found several sightlines with strong, self-blended, highly blueshifted C iv absorbers. Since we focused on the intergalactic absorption systems, we excluded these 138 sightlines from further analysis and labeled them "visual BAL" QSOs in Table 2 (binary flag f_BAL = 12). In addition, the extremely strong absorption lines in these spectra corrupted the automated continuum fitting algorithm.

Table 2. Sightline Summary

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
QSO ID	R.A.	Decl.	zQSO	〈S/N〉	fBAL	ΔXmax	$\mathcal {N}_{\rm cand}$	$\mathcal {N}_{{\rm C\,\scriptsize{IV}}}$	$\delta X_{\rm C\,\scriptsize{IV}}$
				(pixel−1)
52235-0750-082	00:00:09.38	+13:56:18.4	2.2342	5.94	0	1.33	0	0	...
52143-0650-199	00:00:09.42	−10:27:51.9	1.8449	4.68	0	0.83	1	1	0.028
52203-0685-198	00:00:14.82	−01:10:30.7	1.8877	5.16	0	1.24	0	0	...
52203-0685-439	00:00:15.47	+00:52:46.8	1.8516	8.64	0	2.16	0	0	...
54389-2822-315	00:00:24.83	+24:57:03.3	3.2137	9.21	0	1.48	0	0	...
51791-0387-167	00:00:39.00	−00:18:03.9	2.1249	6.61	0	1.75	0	0	...
52143-0650-178	00:00:50.60	−10:21:55.9	2.6404	8.68	0	1.56	4	2	0.083
52203-0685-154	00:00:53.17	−00:17:32.9	2.7571	7.55	0	1.15	0	0	...
52902-1091-546	00:00:57.58	+01:06:58.6	2.5551	11.83	0	1.78	3	2	0.084
51791-0387-093	00:00:58.22	−00:46:46.5	1.8973	13.27	0	0.84	1	1	0.037
52203-0685-134	00:01:25.14	+00:00:09.4	1.9739	5.32	0	1.52	0	0	...

Notes. Column 1 is the adopted QSO identifier from the spectroscopic modified Julian date, plate, and fiber number. Columns 2 through 4 are from the DR7 QSO catalog (Schneider et al. 2010). Column 5 is the median S/N measured in the region searched for C iv. The binary BAL flag f_BAL in Column 6 indicates which sightlines were considered BALs by at least one author (4) and which were confirmed by the authors as BALs to exclude (8). Column 7 is the maximum comoving path length available in the sightline. Columns 8 and 9 give the number of candidate and confirmed C iv doublets, respectively. Column 10 is the path length blocked by the $\mathcal {N}_{{\rm C\,\scriptsize{IV}}}$ doublets in the sightline.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The final catalog includes all doublets with rating  ⩾ 2. From the initial 29,789 candidates, 743 were in sightlines excluded as visual BAL QSOs. In the remaining 29,046 candidates, we found 16,459 real C iv doublets (see Table 3). A sample of doublets are shown in Figure 3. Ultimately, we analyzed the 14,772 with δv_QSO < −5000 km s⁻¹ (see Section 4.1).

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Table 3. C iv System Summary

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
QSO ID	zQSO	z1548	Wr,1548	Wr,1550	C(Wr,1548)	$\log N_{\rm \mathrm{C\,\scriptsize{IV}}}$	fN
			(Å)	(Å)		(log (cm−2))
52143-0650-199	1.8449	1.52755	0.567 ± 0.147	0.981 ± 0.162	0.562 ± 0.206	>14.30 ± 0.14	3
52143-0650-178	2.6404	2.23461	0.425 ± 0.059	0.303 ± 0.058	0.363 ± 0.092	>14.10 ± 0.06	3
		2.43947	1.050 ± 0.078	0.653 ± 0.072	0.860 ± 0.023	>14.53 ± 0.03	3
52902-1091-546	2.5551	2.34348	0.734 ± 0.202	0.898 ± 0.198	0.713 ± 0.192	>14.75 ± 0.14	3
		2.42902	1.053 ± 0.191	0.733 ± 0.196	0.860 ± 0.066	?14.63 ± 0.20	7
51791-0387-093	1.8973	1.77015	0.388 ± 0.097	0.479 ± 0.098	0.305 ± 0.132	>14.01 ± 0.13	3
52235-0750-550	2.6383	2.16695	0.581 ± 0.123	0.594 ± 0.127	0.576 ± 0.161	>14.23 ± 0.09	3
54389-2822-423	2.7668	2.36223	0.738 ± 0.087	0.332 ± 0.088	0.716 ± 0.073	14.34 ± 0.06	1
54327-2630-423	2.7251	2.23946	0.489 ± 0.100	0.390 ± 0.102	0.458 ± 0.151	>14.16 ± 0.11	3
54452-2824-554	2.5566	2.08349	0.837 ± 0.040	1.050 ± 0.042	0.781 ± 0.025	>14.55 ± 0.02	3
		2.08788	0.971 ± 0.039	0.398 ± 0.039	0.836 ± 0.014	14.46 ± 0.02	1
		2.14618	0.769 ± 0.041	0.462 ± 0.037	0.738 ± 0.030	>14.39 ± 0.03	3
		2.21361	0.381 ± 0.039	0.254 ± 0.041	0.294 ± 0.059	>14.00 ± 0.05	3
		2.38185	0.395 ± 0.039	0.284 ± 0.039	0.317 ± 0.062	>14.05 ± 0.05	3
52143-0650-561	1.7593	1.63718	0.393 ± 0.049	0.185 ± 0.053	0.314 ± 0.075	13.99 ± 0.07	1
52991-1489-581	1.8536	1.69764	1.013 ± 0.090	0.812 ± 0.088	0.849 ± 0.030	>14.55 ± 0.04	3
52203-0685-567	2.0921	1.70301	1.165 ± 0.186	0.669 ± 0.180	0.877 ± 0.038	14.57 ± 0.08	1
		1.75102	0.875 ± 0.121	0.827 ± 0.122	0.799 ± 0.070	>14.54 ± 0.07	3

Notes. For each sightline (identified in Columns 1 and 2), every confirmed doublet is listed by the redshift of its C iv 1548 line (Column 3). The rest equivalent widths of the C iv lines are given in Columns 4 and 5. In Column 6, we give the completeness fraction for the doublet from the whole survey average. The C iv column density (Column 7) is the combined value from the AODM measurements in both lines. The binary column density flag f_N is composed of 1 = good to analyze, 2 = lower limit, 4 = upper limit, 8 = unweighted average, and 16 = default to line value with greater significance (Column 8).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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We assessed the effects of blending and bad continua by inspecting 4500 doublets chosen at random. We flagged doublets where the 1548, 1550, or both lines were blended or otherwise problematic (e.g., the bounds were unrealistic) and where the continua ought to be adjusted locally (e.g., due to strong absorption or neighboring emission lines). The majority of our analysis depended on W_r,1548, so it mattered most what fraction of the 1548 lines were troublesome.

Over 90% of the random sample had "perfect" treatment of the 1548 line, and over 95% had acceptable continuum fits. The problematic doublets showed some redshift dependence due to the confused nature in the skyline region λ_obs ≳ 7500 Å or z₁₅₄₈ ≳ 3.8, where the fraction of problematic 1548 lines jumped from ≈10% to ≈20%. The stronger lines (W_r,1548 ≳ 2 Å) were more often blended. These systems tended to be self-blended and would generally be problematic to surveys using boxcar summation for equivalent widths, such as ours. To maintain the largely automated, and hence objective and repeatable, nature of the survey, we decided to accept the imperfections and assess the strength of our results in light of them.

We parameterized the simulated doublets to produce absorption lines that look like the diversity of profiles observed in the SDSS spectra. We created a Voigt profile library with a uniform distribution of equivalent widths, since we aimed to generate uniform errors on detection completeness as a function of W_r,1548. Each component had a column density that was sampled linearly from $11 < \log N_{\rm \mathrm{C\,\scriptsize{IV}}}< 14$ and a Doppler parameter randomly selected in the range 5 km s⁻¹ < b < 40 km s⁻¹. For each system, one primary component was generated, and a random draw from a Poisson distribution, with mean of μ^Poiss_comp, determined the number of additional components for the system. The velocity offset for each component was drawn from a Gaussian distribution with standard deviation σ^Gauss_δv, clipped to keep all components within the maximum offset δv_max relative to the system redshift.

The bulk of our profiles were generated with μ^Poiss_comp = 4 or 7, σ^Gauss_δv = 100 km s⁻¹, and δv_max = 300 km s⁻¹, which were roughly based on the component properties measured by Boksenberg et al. (2003). However, to reproduce the diversity of profiles found in the SDSS data, we had to modify the parameters (e.g., larger μ^Poiss_comp and δv_max), and we limited the alternate parameters to the larger equivalent width regime (e.g., W_r ⩾ 1.2 Å), where Boksenberg et al. (2003) did not have many systems.

We generated over 10⁵ Voigt profiles. Then we randomly selected or duplicated (in extreme equivalent width regimes) profiles to uniformly sample 0.05 Å < W_r,1548 < 3.25 Å in bins of 0.05 Å, resulting in a library of 32,000 profiles.

We have two Monte Carlo completeness tests, referred to as basic and user. The basic test included all of the automated procedures, from continuum fitting to candidate selection. It was applied to the largest fraction of sightlines (≈30%). We tested the user bias by visually inspecting the candidates in ≈6% of the sightlines. The subsamples were set by requiring at least one sightline in each bin of Δz_QSO = 0.25 for z_QSO ⩾ 1.7 and Δ〈S/N〉 = 0.25 pixel⁻¹ for 〈S/N〉 ⩾ 4 pixel⁻¹. For the basic test, with more sampled sightlines, we continued sampling from the sightlines in these Δz_QSO and Δ〈S/N〉 bins, so that the bins with more sightlines had more tested for completeness. Extra high-redshift sightlines were included in the user test in increase the statistics.

We chose to input the simulated doublets into the actual spectra in order to sample realistic data. We "cleaned" each spectrum by removing a random 30% of the automatically detected absorption features. The flux between the wavelength bounds (see Section 2.4) was replaced by continuum with a scatter drawn from neighboring pixels, and the new error reflected those same pixels. By leaving the remaining absorption features in, we were able to measure the effects of blending and of absorption lines on the automated continuum fit and line-finding algorithm.

We set a fiducial absorber redshift density $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dz=5$ to determine how many simulated doublets to input into the cleaned spectrum. This translated to typically injecting one to three doublets per loop, and the test iterated until at least 1000 simulated doublets were input per sightline. We did not change the error array when injecting profiles, which slightly but systematically lowered the S/N of the simulated doublets.

For every sightline in the basic test, a random sample of profiles were drawn from our library and assigned redshifts to fall within the limits of the current sightline. The spectrum was "cleaned" to start and every 50th loop thereafter. We injected one to three simulated doublets, fit the hybrid continuum (see Section 2.2), and ran the automated candidate identification algorithm (see Section 2.3). Any injected profile was flagged as recovered if the automatically identified candidate bounds spanned the observed wavelength of the simulated system. For any profile that was not recovered, we measured the flux-weighted redshift, equivalent width, and AOD column density at the expected location. The input and recovered information was stored for later processing (see Section 3.3).

The user completeness test served a dual purpose. It tested the effects of human bias (e.g., we were less likely to accept real doublets at high redshift due to poor sky subtraction at ≳ 7000 Å) and our accepted false-positive rate (i.e., how often we rated non-C iv lines as C iv absorption). The steps were largely the same for one sightline in the user test as in the basic, but we "cleaned" the profile every iteration and an author rated the automatically detected candidates. However, we modified the simulated doublets to have an unphysical characteristic wavelength separation of 4.8 Å or 924 km s⁻¹ so that any candidate was most likely either simulated or spurious. The fake λλ1547, 1552 profile library was truncated to W_r,1547 ⩽ 1 Å. We did not simulate other absorption lines because, fundamentally, we are conducting a blind C iv survey.

We combined the completeness tests in bins of redshift and onto grids of equivalent width so that we correct any detected absorber based on its completeness fraction. The estimated completeness fraction for a given grid point (z_g, W_g) is simply $\mathcal {N}_{\rm accept}/\mathcal {N}_{\rm input}$, the fraction of input C iv doublets that are recovered automatically and accepted by the user, in the given grid cell. The two completeness tests separately measured the recovering and accepting effects so that the final completeness fraction is estimated by

$$\begin{equation} C(z_{\rm g},W_{\rm g}) = \frac{ \mathcal {N}_{\rm rec}(z_{\rm g},W_{\rm g}) }{ \mathcal {N}_{\rm input}(z_{\rm g},W_{\rm g}) }\frac{ \mathcal {N}_{\rm accept}(z_{\rm g},W_{\rm g}) }{ \mathcal {N}_{\rm rec}(z_{\rm g},W_{\rm g}) }, \end{equation} \tag{ 1 }$$

where the right-hand product can be thought of as C_basicC_user. The completeness grid C(z_g, W_g) collapses to a curve C(W_g) in a fixed redshift bin. The basic completeness uncertainty $\sigma _{C_{\rm basic}}$ is estimated by the Wilson score interval for a binomial distribution (Wilson 1927). This confidence interval estimator is well behaved for small $\mathcal {N}_{\rm input}$ and/or for extreme completeness fractions.

The statistics on the user completeness were naturally smaller than for the basic, so we fit C_user(W_r) with the following model:

$$\begin{equation} C_{\rm user}(W_{\rm r}) = C_{0} (1-e^{\beta (W_{\rm r}- W_{0})}) \end{equation} \tag{ 2 }$$

in two redshift bins. The dividing z = 2.97 corresponded to the beginning of the skyline region at ≈7000 Å, with its resulting decrease in C_user due to confusion, and matched the start of the highest redshift bin. The fit uncertainties were estimated from Monte Carlo re-sampling of the equivalent width errors, and the final error on C(W_r) was propagated from $\sigma _{C_{\rm basic}}$ and $\sigma _{C_{\rm user}}$. We extrapolated the fits to larger equivalent widths when we calculated the full completeness fraction. The user completeness test is discussed in detail in Section 3.4 below.

We scaled C(z_g, W_g) by the fraction of the total path length not obscured by doublets with greater equivalent widths, in the redshift bin. All C iv lines blocked <2% of the total survey path length (see Table 2).

In actuality, $\mathcal {N}_{\rm input}$ and $\mathcal {N}_{\rm rec}$ contain the information only for the profiles and sightlines actually sampled in the basic completeness test. Any sightline that did not have a measurement for one or more quantities was accounted for with the average of the sightlines actually tested in the requested Δz_QSO and Δ〈S/N〉 bin.

The unblocked comoving path length⁹ for each grid cell is calculated by simply multiplying the completeness fraction by the total path length available in each redshift bin:

$$\begin{eqnarray} \Delta X(W_{\rm g}) & = & \Delta X(z_{\rm g}) C(z_{\rm g},W_{\rm g}) \\ \sigma _{\Delta X}^2{}_{(W_{\rm g})} & = & \Delta X(z_{\rm g})^2 \sigma _{C(z,W)}^2 {\rm.} \nonumber \end{eqnarray} \tag{ 3 }$$

In Figure 4, the black curves are the grid values ΔX(W_g) and errors.

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The unblocked comoving path length to which our survey is sensitive for any given detected equivalent width W_i is interpolated from the grid of ΔX(W_g):

$$\begin{eqnarray} \Delta X(W_{\rm i}) & = & {\rm interpol}(\Delta X(W_{\rm g}),\ W_{\rm g},\ W_{\rm i}) \nonumber\\ \sigma _{\Delta X}^2 & = & {\rm interpol}(\sigma _{\Delta X}^2{}_{(W_{\rm g})},\ W_{\rm g},\ W_{\rm i}) \nonumber \\ && + (\Delta X(W_{\rm i}) - \Delta X(W_{\rm i} \pm \sigma _{W_{\rm i}}))^2. \end{eqnarray} \tag{ 4 }$$

The error on ΔX(W_i) accounts for the error in our completeness correction (σ_ΔX) and for the uncertainty in our equivalent width measurement ($\sigma _{W_{\rm i}}$). We plot each doublet's ΔX(W_r,1548) in gray in Figure 4. Clearly, the uncertainty in equivalent width dominates the error in the completeness correction.

In all redshift bins, the typical completeness fraction reached 50% by W_r,1548 ≈ 0.6 Å, a value we frequently use as the minimum in subsequent analyses.

The completeness curves do not monotonically increase to 100% at large equivalent widths (see Figure 4). Instead, they roll off at W_r,1548 ≈ 2 Å. This feature results from the incompleteness in the broad (self-blended) C iv search and, to a much lesser extent (<5%), from broad profiles being overfit by the continuum algorithm, which led them to being missed in the automated search.

The user completeness test measured our ability to correctly rate real, automatically detected doublets ("true positives") as well as the rate at which we include spurious pairs of lines as true doublets ("accepted false negatives"). We injected $\mathcal {N}_{\rm input} = 5021$ fake doublets with W_r < 1 Å and δv_QSO < −5000 km s⁻¹, and $\mathcal {N}_{\rm rec} = 3070$ were automatically recovered. Of these, we correctly rated $\mathcal {N}_{\rm accept} = 2489$. The left panels of Figure 5 show the trends of the user completeness with redshift, spectrum 〈S/N〉, and equivalent width.

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Generally, C_user decreases with increasing redshift and decreasing 〈S/N〉. Skylines become numerous at λ ≳ 7000 Å, and poor sky subtraction leaves features in the spectrum that mimic absorption lines (see Figure 2). In the visual verification step, a larger fraction of real doublets are rejected due to the severe confusion. Lower S/N spectra induces the same effect.

There were an additional 966 spurious candidates brought forth by the automated search, of which we incorrectly accepted 121. The accepted false-positive fraction peaks at ≈20% at z ≈ 3 and 〈S/N〉 ≈ 10. However, our acceptance fraction grows sharply at W_r ≲ 0.6 Å and plateaus to 18% and 12% for z < 2.97 and ⩾2.97, respectively. The estimated comoving line densities of accepted false positives with W_r ⩾ 0.6 Å are $d\mathcal {N}_{\mathrm{afp}}/dX= 0.022$ and 0.020 for the low- and high-redshift bins, respectively.

We detail how we applied corrections for the accepted false-positive rate in the next section.

The equivalent width frequency distribution f(W_r) is the number of detections $\mathcal {N}_{\rm obs}(W_{\rm r})$ per rest equivalent width bin ΔW_r per the total comoving path length available, in the given equivalent width bin, ΔX(W_r):

$$\begin{eqnarray} f(W_{\rm r}) & = & \frac{\mathcal {N}_{\rm obs}(W_{\rm r})}{\Delta W_{\rm r}\,\Delta X(W_{\rm r})} \nonumber\\ \sigma _{f(W_{\rm r})}^2 & = & f(W_{\rm r})^2 \left (\left(\frac{\sigma _{\mathcal {N}_{\rm obs}}}{\mathcal {N}_{\rm obs}(W_{\rm r})} \right)^2 + \left (\frac{\sigma _{\Delta X}}{\Delta X(W_{\rm r})} \right)^2 \right). \end{eqnarray} \tag{ 5 }$$

The error on $\mathcal {N}_{\rm obs}$ is estimated from a Poisson distribution if $\mathcal {N}_{\rm obs} < 120$ and from a Gaussian approximation if $\mathcal {N}_{\rm obs} \ge 120$. For ΔX(W_r) and σ_ΔX, we used Equation (4) with the center of the equivalent width bin being W_i and $\sigma _{W_{\rm i}} = 0.5\Delta W_{\rm r}$.

We used the maximum likelihood analysis of Cooksey et al. (2010) to fit f(W_r) with an exponential:

$$\begin{equation} f(W_{\rm r}) = k e^{\alpha W_{\rm r}} \end{equation} \tag{ 6 }$$

(see Figure 6). The normalization k and scale α were simultaneously fit, and the errors were estimated by the maximum extent of the 1σ error ellipse on the likelihood surface (see Figure 7). All frequency distributions were fit over the range $0.6\,{\rm \mathring{\rm{A}}}\le W_{\rm r}{}_{,\mathrm{1548}} \le {\rm max}[W_{\rm r}{}_{,\mathrm{1548}}+\sigma _{W_{\rm r}}]$. The results were not strongly dependent on the choice of the upper limit. The exponential model is a very good description of the data, and the best-fit parameters are given in Table 4.

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Table 4. C iv Results Summary

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
〈z〉	zlim	$\mathcal {N}_{\rm obs}$	ΔXmax	$W_{50\%}$	$d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dz$	$d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$	$\Omega _{\mathrm{C\,\scriptsize{IV}}}$	$\mathcal {N}_{\rm fit}$	k	α	χ2red
				(Å)			(× 10−8)		(Å−1)	(Å−1)
1.96274	[1.46623, 4.54334]	14772	38624	0.52	0.92+0.02−0.01	0.275+0.004−0.004	1.71+0.20−0.20	8918	3.49+0.31−0.30	−2.62+0.04−0.04	0.577
1.55687	[1.46623, 1.60986]	1428	3962	0.63	0.95+0.03−0.03	0.332+0.011−0.010	2.18+0.39−0.39	1025	3.87+0.94−0.82	−2.52+0.11−0.12	1.560
1.65971	[1.61004, 1.69993]	1473	3420	0.57	1.00+0.04−0.03	0.336+0.012−0.011	2.10+0.32−0.32	956	3.77+0.95−0.82	−2.48+0.12−0.13	0.876
1.73999	[1.70035, 1.78000]	1449	2987	0.53	1.08+0.04−0.04	0.356+0.013−0.012	2.17+0.28−0.28	902	4.46+1.24−1.06	−2.61+0.14−0.14	1.672
1.82360	[1.78007, 1.86985]	1568	3067	0.51	1.09+0.04−0.04	0.351+0.012−0.011	2.11+0.23−0.24	932	4.49+1.20−1.04	−2.63+0.13−0.13	1.291
1.91460	[1.87000, 1.95998]	1424	2665	0.49	1.12+0.04−0.04	0.355+0.013−0.012	2.15+0.23−0.23	824	4.53+1.28−1.09	−2.63+0.14−0.14	1.832
2.01778	[1.96002, 2.07997]	1523	2983	0.46	1.04+0.04−0.04	0.322+0.012−0.011	1.86+0.17−0.17	852	5.18+1.76−1.51	−2.87+0.15−0.15	2.838
2.15320	[2.08009, 2.23997]	1490	2878	0.45	1.04+0.04−0.04	0.312+0.012−0.011	1.92+0.17−0.17	807	4.01+1.21−1.04	−2.63+0.14−0.14	3.267
2.35608	[2.24015, 2.50914]	1450	3341	0.50	0.96+0.04−0.03	0.276+0.011−0.010	1.61+0.18−0.18	775	4.26+1.62−1.37	−2.82+0.15−0.16	1.296
2.72298	[2.51028, 2.96976]	1503	5673	0.61	0.83+0.03−0.03	0.221+0.009−0.008	1.44+0.22−0.22	969	2.48+0.74−0.65	−2.49+0.12−0.12	0.837
3.25860	[2.97005, 4.54334]	1464	7632	0.59	0.59+0.02−0.02	0.145+0.006−0.005	0.87+0.13−0.13	876	1.82+0.75−0.65	−2.61+0.13−0.13	1.548

Notes. Summary of the most common redshift bins and data used for the various analyses. Columns 1 and 2 give the median, minimum, and maximum redshifts for the observed number of doublets (Column 3), and the maximum comoving path length in the redshift bin is given in Column 4. The 50% completeness limit from the Monte Carlo tests is in Column 5. The redshift and comoving absorber line densities for W_r ⩾ 0.6 Å are in Columns 6 and 7. In Column 8, the $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ from summing the mass in the W_r ⩾ 0.6 Å absorbers is a lower limit, since the majority of absorbers are saturated. The frequency distribution was fit with an exponential f(W_r) = kexp (αW_r) for $\mathcal {N}_{\rm fit}$ absorbers with W_r ⩾ 0.6 Å (Column 9), and the best-fit parameters are given in Columns 10 and 11. The reduced χ² from the best fit and f(W_r) (in bins with ≈100 doublets each) is given in Column 12.

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The best-fit parameters show smooth redshift evolution with respect to the constant $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ curves in Figure 7. The comoving line density is simply the integral of the frequency distribution from some limiting equivalent width W_lim to infinity, and substituting in our exponential model, we see

$$\begin{equation} \frac{\displaystyle d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}}{\displaystyle dX} |_{\rm fit} = \frac{\displaystyle -k}{\displaystyle \alpha } e^{\alpha W_{\rm lim}} {\rm.} \end{equation} \tag{ 7 }$$

By fixing $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$, we can solve for the required normalization k for any given α. The 1σ ellipses are elongated in the direction of the constant $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ curves, and redshift evolution can be seen by tracking systematic change perpendicular to these curves. The lowest five redshift bins (1.46 ⩽ z < 1.96) fall along roughly the same constant $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ curve. The next three highest redshift bins (1.96 ⩽ z < 2.51) have slightly smaller line densities. Then there is almost a factor of two drop over the highest two redshift bins (2.51 ⩽ z < 4.55).

Since the accepted false-positive rate is essentially constant at W_r ⩾ 0.6 Å (see Section 3.4), the frequency distribution of accepted false positives is a scaled-down version of the measured frequency distribution. Therefore, we scale the original f₀(W_r) as follows:

$$\begin{equation} f(W_{\rm r}) = \left (1 - \frac{\displaystyle d\mathcal {N}_{\mathrm{afp}}/dX}{\displaystyle d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX} \right) f_{0}(W_{\rm r}) \end{equation} \tag{ 8 }$$

and propagate the errors. This results in a decrease of ≈6%–12%, depending on the redshift. For the exponential fits, we scale the best-fit normalization k₀ in a similar fashion:

$$\begin{equation} k = \left(1 - \frac{\displaystyle d\mathcal {N}_{\mathrm{afp}}/dX}{\displaystyle (d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX)_{\rm fit}} \right) k_{0} {\rm ,} \end{equation} \tag{ 9 }$$

but the denominator is the integrated line density from the exponential model (Equation (7)). We report the propagated errors in Table 4 and in the text. However, since we cannot compute the change in the likelihood surface, we only shift the ellipses in Figures 7 and 8.

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4.1.1. Effect of Blending and "Intrinsic" Absorbers

We directly measured the absorber line density for doublets with W_r ⩾ W_lim as follows:

$$\begin{eqnarray} \frac{ d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}}{ dX} (W_{\rm r}\ge W_{\rm lim}) & = & \frac{\displaystyle \mathcal {N}_{\rm C}(W_{\rm r}) }{\displaystyle \Delta X(z)} \nonumber\\ \sigma _{d\mathcal {N}/dX}^{2} & = & \left(\frac{ \sigma _{\mathcal {N}_{\rm C}{} } }{ \Delta X(z)} \right)^2. \end{eqnarray} \tag{ 10 }$$

The completeness-corrected number of absorbers in any given bin is the completeness-weighted sum of the observed absorbers in the bin:

$$\begin{eqnarray} \mathcal {N}_{\rm C}(W_{\rm r}) & = & \sum _{W_{\rm i} \ge W_{\rm r}- 0.5\Delta W_{\rm r}}^{W_{\rm i} < W_{\rm r}+ 0.5\Delta W_{\rm r}} \frac{1}{C(W_{\rm i})} \nonumber\\ \ms \ms \sigma _{\mathcal {N}_{\rm C}}^2 & = & \sigma _{\mathcal {N}_{\rm obs}}^{2} + \sum _{W_{\rm i} \ge W_{\rm r}- 0.5\Delta W_{\rm r}}^{W_{\rm i} < W_{\rm r}+ 0.5\Delta W_{\rm r}} \left (\frac{ \sigma _{C(W_{\rm i})} }{ C(W_{\rm i})^2 } \right)^2. \end{eqnarray} \tag{ 11 }$$

Again, $\mathcal {N}_{\rm obs}$ is the contribution of the actual observed number of absorbers in the given W_r bin, and the completeness-corrected number $\mathcal {N}_{\rm C} \ge \mathcal {N}_{\rm obs}$. The error $\sigma _{\mathcal {N}_{\rm obs}}$ is estimated from a Poisson distribution if $\mathcal {N}_{\rm obs} < 120$ and from a Gaussian approximation if $\mathcal {N}_{\rm obs} \ge 120$.

We subtract $d\mathcal {N}_{\mathrm{afp}}/dX$ from all quoted $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ values, in the appropriate redshift bins (see Section 3.4) and add the errors in quadrature. For W_r ⩾ 0.6 Å, $d\mathcal {N}_{\mathrm{afp}}/dX= 0.020^{+0.003}_{-0.002}$ (1.4 < z < 4.6); 0.022^+0.005_−0.004 (z < 2.97); and 0.020^+0.004_−0.003 (z ⩾ 2.97).

We present $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ for different equivalent width limits in Figure 9. The line density shows little differential evolution based on $W_{\rm r}{}_{,\mathrm{\rm lim}}$, as expected from the consistent shape of the frequency distributions over time (Figure 6). For W_r ⩾ 0.6 Å, the inverse-variance weighted average $\bar{d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX} = 0.350\pm 0.005$ for 1.46 ⩽ z < 1.96 (or the lowest five redshift bins). This average is 2.37 ± 0.09 times larger than the highest redshift $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ at z = 3.76. Though the magnitude of the increase is modest, the detection is a >20σ result. Thus, the line density grows consistently and smoothly from z₁₅₄₈ = 4.5 → ≈ 1.74, then plateaus at $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX\approx 0.34$ until z = 1.46, as expected from the best-fit f(W_r) parameters (see Figure 7).

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There is a known bias in the SDSS quasar color selection that leads to an excess of Lyman-limit systems at 3 ≲ z_QSO ≲ 3.5 (Prochaska et al. 2010; Worseck & Prochaska 2011), which likely increases the incidence of strong metal-line absorption systems at these redshifts. Therefore, there is potential for the decrease in the highest SDSS bin to be even larger, but the effect of the color selection on the C iv sample is beyond the scope of this paper.

To fairly compare to other surveys, we applied various W_r cuts to our complete sample, chosen to match the corresponding cuts of prior surveys.

To compare f(W_r), we converted published column density frequency distributions fits from Songaila (2001) and D'Odorico et al. (2010) to equivalent width frequency distributions by assuming the linear curve of growth and mapping $f(N_{\rm \mathrm{C\,\scriptsize{IV}}})$ directly to f(W_r,1548): $f(W_{\rm r}) = f(N_{\rm \mathrm{C\,\scriptsize{IV}}}) dN_{\rm \mathrm{C\,\scriptsize{IV}}}/ dW_{\rm r}$. We adjusted for cosmology as necessary. A single-component cloud becomes saturated and nonlinear at $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\approx 14$, which translates to W_r,1548 ≈ 0.6 Å, where we typically are ≈50% complete.

While we have good statistics on the rare, strong absorbers, we suffer from incompleteness at W_r ≲ 0.6 Å. Songaila (2001) and D'Odorico et al. (2010) were smaller, higher-resolution, higher-S/N studies, and they were complete to very low W_r but suffered from sample variance at larger equivalent widths. However, Figure 10 shows that our results are consistent with each other in the overlap region. Our best-fit parameters to the exponential f(W_r) in the D'Odorico et al. (2010) and Songaila (2001) redshift bins were, respectively: α = −2.65^+0.04_−0.04 Å⁻¹ and k = 3.72^+0.37_−0.35 Å⁻¹ for 1.6 ⩽ z ⩽ 3.6 with 7884 systems; and α = −2.58^+0.13_−0.13 Å⁻¹ and k = 2.29^+0.80_−0.69 Å⁻¹ for 2.9 ⩽ z ⩽ 3.54 with 878 systems.

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We compared the SDSS redshift and comoving line densities with the literature (Steidel 1990; Barlow & Tytler 1998; Cooksey et al. 2010; D'Odorico et al. 2010; Simcoe et al. 2011). For D'Odorico et al. (2010), we used their best-fit $f(N_{\rm \mathrm{C\,\scriptsize{IV}}})$ values to calculate $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ and $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dz= (d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX)(dX/dz)$ for $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\ge 14$. For Barlow & Tytler (1998) and Steidel (1990), we estimated $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX\ = (d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dz)(dX/dz)^{-1}$. The Steidel (1990) results match ours well for the range where the author was fairly complete but with ≈20% uncertainty, compared to our ≈2% errors (see Figure 11). Extending the redshift coverage by including the low-redshift measurements of Barlow & Tytler (1998) and Cooksey et al. (2010) and the high-redshift values from Simcoe et al. (2011), we see that $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ has steadily increased by, roughly, a factor of 10 from z₁₅₄₈ = 6 → 0.

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We show the best measurements of $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ for W_r ⩾ 0.6 Å and 0 < z < 6.5 in the top panel of Figure 12.¹⁰ The comoving line density relates to the comoving volume density of absorbing clouds n_com and their physical cross-section σ_phys:

$$\begin{equation} \frac{\displaystyle d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}}{\displaystyle dX} = \frac{\displaystyle c}{\displaystyle H_{0}} n_{\rm com}\sigma _{\rm phys}{\rm.} \end{equation} \tag{ 12 }$$

Thus, the roughly order-of-magnitude increase in $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ from high-to-low redshift means the product n_comσ_phys has increased by a factor of ≈10. We know the metallicity of the universe has steadily increased over cosmic time, so the possible number of C iv-absorbing clouds (i.e., n_com) has likely increased. At least some C iv absorption traces galaxy halos at low (Chen et al. 2001) and high redshift (Adelberger et al. 2005; Martin et al. 2010; Steidel et al. 2010). Since galaxies (and their halos) have likely grown over cosmic time, increases in both σ_phys and n_com appear to contribute to the increase in $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$.

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Adelberger et al. (2005) reported that almost all $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\ge 14$ absorbers arise within ≈80 kpc of Lyman-break galaxies (LBGs) at 2 ≲ z ≲ 3. The evidence included individual strong absorber-LBG pairs; strong absorption in stacked spectra of close background galaxies, shifted to the rest frame of the foreground galaxies; and similar LBG-C iv cross-correlation and LBG autocorrelation functions, suggesting they have the same spatial distribution.

Steidel et al. (2010) increased the LBG–LBG pairs for stacking analysis, and they measured average W_r,1548 = 0.13 ± 0.05 Å and 1.18 ± 0.15 Å at distances b = 63 kpc and 103 kpc, respectively. Interpolating between these two measurements, an average W_r,1548 = 0.6 Å system would reside at b ≈ 85 kpc. The Steidel et al. (2010) LBG sample went as faint as ≈0.3 L*, assuming the luminosity function of Reddy & Steidel (2009), and since they used stacks of galaxy spectra, their analysis included the effects of partial covering fractions.

Since the W_r ⩾ 0.6 Å C iv absorbers in our sample have $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\gtrsim 14$, we used the comoving number density of UV-selected galaxies, to estimate the typical galaxy-C iv cross-section over time with Equation (12). We measured n_{com, UV} with the UV luminosity functions from Oesch et al. (2010), Reddy & Steidel (2009), and Bouwens et al. (2007), which covered several smaller redshift bins spanning 0.5 < z < 2, 1.9 < z < 3.4, and 3.8 ≲ z ≲ 5.9, respectively. For each luminosity function, we integrated the best-fit Schechter function down to 0.5 L* or ≈0.75 mag fainter than the published M*_UV, which ranged between ≈ − 19 and −21 mag. We estimated the n_{com, UV} errors with Monte Carlo simulations.

The UV-selected galaxy number density, n_{com, UV}, may increase by a factor of two to three from z ≈ 6 → 0, but the uncertainties are large (see Figure 12, middle panel). Applying n_{com, UV} and $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ to Equation (12), we estimated σ_phys, the physical galaxy-C iv cross-section (Figure 12, lower panel). For some redshift bins, multiple luminosity functions could be used in conjunction with our $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$, and we show all resulting values but highlight the preferred values with filled symbols.

The galaxy-C iv cross-section shows no evolution over the SDSS redshift range within the errors, which are dominated by the 20%–60% uncertainties in n_{com, UV}. Assuming the cross-section is due to a spherical halo that projects with 100% C iv covering fraction, the halo radius would be $R_{\rm phys} = \sqrt{\sigma _{\rm phys}/\pi } \approx 50\,\mathrm{kpc}$ for 1.5 ≲ z ≲ 4.5. This distance agrees with Adelberger et al. (2005) and allows for, e.g., a non-unity covering fraction or limiting to brighter UV-selected galaxies, both of which would increase R_phys.

There is an approximately 10-fold increase in σ_phys from z ≈ 6 → 0 when we include the $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ measurements from Simcoe et al. (2011) and Cooksey et al. (2010), respectively. This increase in the galaxy-C iv cross-section is comparable to that of $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ over the same redshift range. Thus, if the W_r ⩾ 0.6 Å C iv absorbers were only tracing galaxy halos, the redshift evolution of $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ could be solely due to the halos filling up with triply ionized carbon, through some combination of physical growth, increased metallicity, and/or evolution in the ionizing background.

However, the uncertainty in the z < 1 cross-section estimate is large and consistent with an increase of only two to three compared with the z ≈ 6 value. In this case, the roughly order-of-magnitude increase in $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ could equally be due to modest increases in n_{com, UV} and σ_phys.

We emphasize that the preceding discussion assumes all W_r ⩾ 0.6 C iv systems at 0 < z < 6 are only found in the circum-galactic media of UV-selected galaxies. Any intergalactic C iv contribution to $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ at any redshift would cause us to overestimate the galaxy-C iv cross-section.

Evolution in the ultraviolet background (UVB) possibly explains the seeming spike in σ_phys at z ≈ 5. Simcoe (2011) showed that C iv is a preferred, if not dominant, transition of carbon at z = 4.3 compared to lower redshift, assuming the UVB of Haardt & Madau (2001) or Faucher-Giguère et al. (2009). Though the redshift window of this effect is small (Δz ≈ 0.5), the increase in C iv-ionizing photons would increase the C iv cross-section of UV-selected galaxies and may explain the spike in $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ and σ_phys. In general, triply ionized carbon increasingly becomes a disfavored transition of carbon with decreasing redshift, given a standard model of the UVB (Oppenheimer & Davé 2008; Simcoe 2011). The UVB likely dominates the ionization flux at the expected tens of kiloparsecs from the host galaxies. Thus, the increasing number of C iv absorbers toward lower redshift indicates an increasing enrichment of the gas, to more than compensate for the decreasingly favored triply ionized state.

C iv absorption could also be a galactic wind signature (see Steidel et al. 2010). However, the $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ redshift evolution does not mimic the evolution of $d\mathcal {N}_{\mathrm{Mg\,\scriptsize{II}}}/dX$ for W_r,2796 ≳ 1 Å absorbers, and these strong Mg ii doublets are often analyzed as wind tracers (Weiner et al. 2009; Rubin et al. 2010; Bordoloi et al. 2011; but also see Gauthier et al. 2010 and Kacprzak et al. 2011). Strong Mg ii absorbers evolve strongly with redshift by increasing with increasing redshift (Nestor et al. 2005; Prochter et al. 2006), peaking at z ≈ 3, and then decreasing at higher redshifts (Matejek & Simcoe 2012). The latter study showed that the evolution in $d\mathcal {N}_{\mathrm{Mg\,\scriptsize{II}}}/dX$ tracks the cosmic star formation rate (Bouwens et al. 2010), using the scaling relation of Ménard et al. (2011), reinforcing the idea that strong Mg ii absorbers arise in galactic winds.

We see no decrease at lower redshifts for strong C iv absorbers (see Figure 9). If strong C iv and Mg ii absorbers were both tracing winds, then, to account for their different redshift evolution, either they must probe outflows in different ways or strong C iv absorption traces an additional medium, such as halo gas, that contributes significantly to its $d\mathcal {N}_{\mathrm{C\,\scriptsize{IV}}}/dX$ evolution.

The C iv doublet redshifts into optical passbands for 1.5 ≲ z ≲ 5.5 and has been observed extensively with large, ground-based telescopes (Songaila 2001; Boksenberg et al. 2003; Schaye et al. 2003; Scannapieco et al. 2006; D'Odorico et al. 2010). Surveys of ultraviolet quasar spectra taken with HST cover the z < 1 C iv systems (Danforth & Shull 2008; Cooksey et al. 2010). More recently, improved infrared spectrographs have pushed C iv surveys to z ≈ 6 (Ryan-Weber et al. 2009; Becker et al. 2009; Simcoe et al. 2011). Typically these studies have focused on the evolution of $\Omega _{\mathrm{C\,\scriptsize{IV}}}$, the C iv mass density relative to the closure density. The first moment of the column density frequency distribution $f(N_{\rm \mathrm{C\,\scriptsize{IV}}})$ is related to $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ as follows:

$$\begin{equation} \Omega _{\mathrm{C\,\scriptsize{IV}}}= \frac{H_{0}\,m_{\rm C}}{c\,\rho _{c,0}} \int _{N_{\rm \rm min}}^{N_{\rm \rm max}} f(N_{\rm \mathrm{C\,\scriptsize{IV}}}) N_{\rm \mathrm{C\,\scriptsize{IV}}}dN_{\rm \mathrm{C\,\scriptsize{IV}}}, \end{equation} \tag{ 13 }$$

where H₀ = 71.9 km s⁻¹ Mpc⁻¹ is the Hubble constant today; m_C = 2 × 10⁻²³ g is the mass of the carbon atom; c is the speed of light; and the critical density ρ_{c, 0} = 3H²₀(8πG)⁻¹ = 9.77 × 10⁻³⁰ g cm⁻³ for our assumed Hubble constant.

The earliest (optical) studies found that $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ was relatively constant for 2 ≲ z ≲ 4.5 (Songaila 2001, 2005; Boksenberg et al. 2003; Schaye et al. 2003; Scannapieco et al. 2006). Cooksey et al. (2010) showed that $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ increased by at least a factor of four from z ≈ 3 to z < 1. Using better optical spectra, D'Odorico et al. (2010) found that $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ increased smoothly from z = 3 to z = 1.5 and mapped well onto the z < 1 values.

However, all these studies measured a power-law shape for the $f(N_{\rm \mathrm{C\,\scriptsize{IV}}})$, which formally corresponds to an infinite C iv mass density. These surveys were limited to small numbers of available sightlines, typically less than 50. The rarest systems are the strongest, and these dominate the mass density measurement when the distribution is a power law. Hence the small-number statistics on the strongest C iv absorbers limited the quoted $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ to be for $12 \lesssim \log N_{\rm \mathrm{C\,\scriptsize{IV}}}\lesssim 15$, with one survey pushing to $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\approx 16$ (Scannapieco et al. 2006). Indeed, scaling the Scannapieco et al. (2006) $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ to $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\le 15$ reduces their value by 45% (Cooksey et al. 2010).

Our analysis, however, provides good statistics on the rare, strong systems that dominate the SDSS sample. The observed $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ can be approximated by the sum of the detected C iv absorbers (Lanzetta et al. 1991). The total column density in a given redshift bin is simply

$$\begin{eqnarray} &&N_{\rm \rm tot} = \sum _{\mathcal {N}_{\rm obs}} N_{\rm \mathrm{C\,\scriptsize{IV}}}\nonumber\\ &&\sigma _{N_{\rm \rm tot}}^{2} = \sigma _{\mathcal {N}_{\rm obs}}^{2} + \sum _{\mathcal {N}_{\rm obs}} \sigma _{N_{\rm \mathrm{C\,\scriptsize{IV}}}}^{2}, \end{eqnarray} \tag{ 14 }$$

where, again, we factor in the counting uncertainty for the number of detections $\sigma _{\mathcal {N}_{\rm obs}}$. Then, we estimated the mass density relative to the critical density as

$$\begin{eqnarray} \Omega _{\mathrm{C\,\scriptsize{IV}}}& = & \frac{H_{0}\,m_{\rm C}}{c\,\rho _{c,0}} \frac{\displaystyle N_{\rm \rm tot}}{\displaystyle \langle \Delta X(W_{\rm r}) \rangle } \nonumber\\ \sigma _{\Omega }^{2} & = & \Omega _{\mathrm{C\,\scriptsize{IV}}}^{2} \left(\left (\frac{\displaystyle \sigma _{N_{\rm \rm tot} }}{ \displaystyle N_{\rm \rm tot} } \right)^{2} + \left (\frac{\displaystyle \sigma _{\langle \Delta X\rangle } }{ \displaystyle \langle \Delta X(W_{\rm r}) \rangle } \right)^{2} \right). \end{eqnarray} \tag{ 15 }$$

Since the completeness corrections were compiled in equivalent width space, we used the median ΔX(W_r) available to each of the absorbers in the given bin, and the error was the standard deviation of the detections. The small scaling to account for the accepted false-positive distribution follows that of the frequency distribution (Equation (8)).

In Figure 13, we plot $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ over redshift from several studies for $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\ge 14$, adjusted for the new limits and cosmology (see Cooksey et al. 2010 for details). The direct SDSS measurements are lower limits since we have predominately saturated absorbers and use the AODM to estimate column densities. However, we can still exclude values below the limits, including the Songaila (2001) measurements at z < 3.¹¹ This reinforces the D'Odorico et al. (2010) result that $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ smoothly increases. In an upcoming paper, we will be combining the D'Odorico et al. (2010) and current data sets to fit $f(N_{\rm \mathrm{C\,\scriptsize{IV}}})$ and measure $\Omega _{\mathrm{C\,\scriptsize{IV}}}$ for $\log N_{\rm \mathrm{C\,\scriptsize{IV}}}\gtrsim 12$ via Equation (13).

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