A DEEP SEARCH FOR FAINT GALAXIES ASSOCIATED WITH VERY LOW-REDSHIFT C iv ABSORBERS. II. PROGRAM DESIGN, ABSORPTION-LINE MEASUREMENTS, AND ABSORBER STATISTICS

Our C iv sample is composed of systems detected in 89 sightlines targeted by three HST programs using the Cosmic Origins Spectrograph (COS; Green et al. 2012): COS-Halos (Werk et al. 2012; Tumlinson et al. 2013), COS-Dwarfs (Bordoloi et al. 2014), and the COS Absorption Survey of Baryon Harbors (CASBaH, Tripp et al. 2011; Meiring et al. 2013). Information about the QSOs included in this study is presented in Table 1, and we will generally refer to them with abbreviated versions of their names in this table. All of these data use the G130M and G160M gratings, covering the wavelength range 1100–1800 Å, and were reduced as described by Meiring et al. (2011). Due to the differing goals of each survey, the spectra possess various signal-to-noise ratios (S/N), and the S/N varies greatly across the wavelength range of each spectrum (see Section 3.3); the COS-Halos and COS-Dwarfs data have typical S/N values of ∼11 per ∼18 km s⁻¹ resolution element, while CASBaH spectra have S/N ∼30 per resolution element (also ∼18 km s⁻¹). The COS-Halos survey targeted 42 QSO sightlines that pass within 150 kpc of L ∼ L∗ galaxies with various stellar masses, star-formation rates, and impact parameters at redshifts z = 0.15–0.35 that bring the O vi λλ 1031.7, 1037.8 Å doublet and Lyα 1215.7 Å lines into the G130M/G160M bandpasses. The COS-Dwarfs survey similarly targeted sightlines that pass near known galaxies with selected star-formation properties and masses, but the 43 galaxies selected were specifically L < 0.1 L∗ galaxies. Also, the galaxies selected for COS-Dwarfs are at a lower redshift range, z_gal = 0.02–0.08, ideal for detecting the C iv doublet with COS. Lastly, the highest S/N spectra in our data set comes from the CASBaH survey (PI: Tripp), which targeted higher-redshift QSOs to measure transitions from high ions such as Ne viii, Mg x, and Si xii. These nine sightlines were not targeted based on preselected proximal galaxies, but deep follow-up galaxy environment data is currently being obtained around them (Meiring et al. 2011).

Table 1.  QSO Sample for the Blind C iv Survey

QSO Name	α (J2000)	δ (J2000)	${z}_{{\rm{qso}}}$
	(degrees)
SDSS J001224.01–102226.5	3.1001	−10.3740	0.228
SDSS J004222.29–103743.8	10.5929	−10.6288	0.424
SDSS J015530.02–085704.0	28.8751	−8.9511	0.165
SDSS J021218.32–073719.8	33.0764	−7.6222	0.174
SDSS J022614.46+001529.7	36.5603	0.2583	0.615
PHL 1337	38.7808	−4.0349	1.437
SDSS J024250.85–075914.2	40.7119	−7.9873	0.377
SDSS J025937.46+003736.3	44.9061	0.6268	0.534
SDSS J031027.82–004950.7	47.6159	−0.8308	0.080
SDSS J040148.98–054056.5	60.4541	−5.6824	0.570
FBQS 0751+2919	117.8013	29.3273	0.915
SDSS J080359.23+433258.4	120.9968	43.5496	0.449
SDSS J080908.13+461925.6	122.2839	46.3238	0.657
SDSS J082024.21+233450.4	125.1009	23.5807	0.470
SDSS J082633.51+074248.3	126.6396	7.7134	0.311
SDSS J084349.49+411741.6	130.9562	41.2949	0.990
SDSS J091029.75+101413.6	137.6240	10.2371	0.463
SDSS J091235.42+295725.4	138.1476	29.9571	0.305
SDSS J091440.38+282330.6	138.6683	28.3918	0.735
SDSS J092554.43+453544.4	141.4768	45.5957	0.329
SDSS J092554.70+400414.1	141.4779	40.0706	0.471
SDSS J092837.98+602521.0	142.1583	60.4225	0.296
SDSS J092909.79+464424.0	142.2908	46.7400	0.240
SDSS J093518.19+020415.5	143.8258	2.0710	0.649
SDSS J094331.61+053131.4	145.8817	5.5254	0.564
SDSS J094621.26+471131.3	146.5886	47.1920	0.230
SDSS J094733.21+100508.7	146.8884	10.0858	0.139
SDSS J094952.91+390203.9	147.4705	39.0344	0.365
SDSS J095000.73+483129.3	147.5031	48.5248	0.589
SDSS J095915.65+050355.1	149.8152	5.0653	0.162
SDSS J100102.55+594414.3	150.2606	59.7373	0.746
SDSS J100902.06+071343.8	152.2586	7.2289	0.456
SDSS J101622.60+470643.3	154.0942	47.1120	0.822
SDSS J102218.99+013218.8	155.5791	1.5386	0.789
PG1049–005	162.9643	−0.8549	0.359
SDSS J105945.23+144142.9	164.9385	14.6953	0.631
SDSS J105958.82+251708.8	164.9951	25.2858	0.662
SDSS J110312.93+414154.9	165.8039	41.6986	0.402
SDSS J110406.94+314111.4	166.0289	31.6865	0.434
SDSS J111239.11+353928.2	168.1630	35.6578	0.636
SDSS J111754.31+263416.6	169.4763	26.5713	0.421
SDSS J112114.22+032546.7	170.3092	3.4297	0.152
SDSS J112244.89+575543.0	170.6870	57.9286	0.906
SDSS J113327.78+032719.1	173.3658	3.4553	0.525
SDSS J113457.62+255527.9	173.7401	25.9244	0.710
PG1148+549	177.8353	54.6259	0.975
SDSS J115758.72–002220.8	179.4947	−0.3725	0.260
PG1202+281	181.1754	27.9033	0.165
SDSS J120720.99+262429.1	181.8375	26.4081	0.324
PG1206+459	182.2417	45.6765	1.163
SDSS J121037.56+315706.0	182.6565	31.9517	0.389
SDSS J121114.56+365739.5	182.8107	36.9610	0.171
SDSS J122035.10+385316.4	185.1463	38.8879	0.376
SDSS J123304.05–003134.1	188.2669	−0.5262	0.471
SDSS J123335.07+475800.4	188.3962	47.9668	0.382
SDSS J123604.02+264135.9	189.0168	26.6933	0.209
SDSS J124154.02+572107.3	190.4751	57.3520	0.583
SDSS J124511.25+335610.1	191.2969	33.9361	0.711
SDSS J132222.68+464535.2	200.5945	46.7598	0.375
SDSS J132704.13+443505.0	201.7672	44.5847	0.331
SDSS J133045.15+281321.4	202.6881	28.2226	0.417
SDSS J133053.27+311930.5	202.7220	31.3252	0.242
PG1338+416	205.2533	41.3872	1.214
SDSS J134206.56+050523.8	205.5274	5.0900	0.266
SDSS J134231.22+382903.4	205.6301	38.4843	0.172
SDSS J134246.89+184443.6	205.6954	18.7455	0.383
SDSS J134251.60–005345.3	205.7150	−0.8959	0.326
SDSS J135625.55+251523.7	209.1065	25.2566	0.164
SDSS J135712.61+170444.1	209.3026	17.0789	0.150
PG1407+265	212.3496	26.3059	0.940
SDSS J141910.20+420746.9	214.7925	42.1297	0.873
SDSS J143511.53+360437.2	218.7980	36.0770	0.429
SDSS J143726.14+504555.8	219.3589	50.7655	0.783
LBQS 1435–0134	219.4512	−1.7863	1.308
SDSS J144511.28+342825.4	221.2970	34.4737	0.697
SDSS J145108.76+270926.9	222.7865	27.1575	0.064
SDSS J151428.64+361957.9	228.6194	36.3328	0.695
SDSS J152139.66+033729.2	230.4153	3.6248	0.126
PG1522+101	231.1022	9.9748	1.328
SDSS J154553.48+093620.5	236.4729	9.6057	0.665
SDSS J155048.29+400144.9	237.7012	40.0291	0.497
SDSS J155304.92+354828.6	238.2705	35.8079	0.722
SDSS J155504.39+362848.0	238.7683	36.4800	0.714
SDSS J161649.42+415416.3	244.2059	41.9046	0.440
SDSS J161711.42+063833.4	244.2976	6.6426	0.229
SDSS J161916.54+334238.4	244.8189	33.7107	0.471
PG1630+377	248.0047	37.6306	1.476
SDSS J225738.20+134045.4	344.4092	13.6793	0.594
SDSS J234500.43–005936.0	356.2518	−0.9933	0.789

A machine-readable version of the table is available.

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We used a multi-step visual identification process to search for C iv systems within our 89 spectra observed with the G130M and G160M gratings, which enable coverage of the C iv doublet at z ≲ 0.16. To aid in the search for the C iv doublet, we created a user interface to scroll through a spectrum, select a candidate λ 1548 feature, and view the alignment of the 1548 and 1550 features in velocity space and in their apparent column density profiles (Savage & Sembach 1991; Sembach & Savage 1992). The apparent column density (as a function of velocity) is defined as follows:

$$\begin{eqnarray}&&N(v)=\displaystyle \frac{{m}_{e}c}{\pi {e}^{2}f\lambda }\tau (v),\end{eqnarray} \tag{ 1 }$$

where m_e is the electron mass, c is the speed of light, λ is the wavelength, and f is the line oscillator strength. τ(v) is the apparent optical depth at a given velocity and is defined as

$$\begin{eqnarray}&&\tau (v)={\rm{ln}}\displaystyle \frac{{I}_{c}(v)}{I(v)},\end{eqnarray} \tag{ 2 }$$

where I_c(v) is the continuum intensity and I(v) is the observed intensity.

If neither of the lines is blended with interloping absorption from another species and if the lines are not saturated, the apparent column density profiles should align. However, if the apparent column density corresponding to the 1550 line appears to be greater over a portion of the profile, the 1548 line may be saturated. If the 1548 line apparent column density appears greater than that of the 1550 line, the 1548 line may be blended with an interloper; otherwise, the candidate absorber may be rejected because ${f}_{1548}{\lambda }_{1548}/{f}_{1550}{\lambda }_{1550}\sim 2,$ and a greater apparent column density for the 1548 line is unphysical. In cases of possible blending, we accepted the candidate as a detection if the two apparent column density profiles were aligned in regions of the profile unaffected by a conspicuous interloper.

To further scrutinize the candidate systems, we searched for other common lines, such as Lyα , C ii, and Si iii, within ±400 km s⁻¹ of the "systemic redshift" determined by the C iv absorption. However, we emphasize that the presence of these other lines was not a necessary criterion for inclusion in our sample; the purpose was merely to offset some doubt initially held about the system, such as the apparent column density profiles being imprecisely aligned, which could occur merely due to known errors in the COS wavelength calibration (up to ±40 km s⁻¹ according to Wakker et al. 2015). Then, each system was checked for significance of detection, where we required 3σ detections for both the 1548 and 1550 lines. If a suspected C iv doublet was possibly blended with an interloping line, we used Voigt profile fitting to first fit the contaminant and then assess whether enough optical depth remained in the line profile to account for the presence of C iv. The candidates passing these tests were then checked for common misidentifications, notably interloping Lyman series lines and higher-z O i 1302, Si ii 1304 pairs, which have a similar wavelength separation to the C iv doublet.

Once we had established our independent sample of C iv candidates, we flagged those systems that were at the redshifts of the galaxies targeted in the original COS-Halos and COS-Dwarfs surveys, as those targeted systems would not compose a blind sample and were thus not included in the analyses presented here. Although none were detected, the survey design also called for omitting absorbers within 5000 km s⁻¹ of the QSO redshifts, based on the Tripp et al. (2008) finding of a proximate O vi absorber overabundance within 5000 km s⁻¹ of observed QSO redshifts. Lastly, we compared our identifications with those from the COS-Halos and COS-Dwarfs studies for verification. A previous detection by those studies was not required because their analysis was primarily focused on absorption associated with their targeted galaxies. Finally, we were left with a 42-absorber sample (see Figure 1).

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For our equivalent width (EW) and column density measurements, we fit local continua within ±500 km s⁻¹ of each line center using Legendre polynomials whose order was determined by an F-test (Sembach & Savage 1992), typically 3rd or 4th order. We then measured the EW of each line and calculated the integrated apparent column density, assuming the line was unsaturated and therefore on the linear part of the curve of growth.

We then normalized the spectrum flux by the fitted continuum and fit Voigt profiles to the absorption lines to measure column densities, Doppler parameters, and velocity centroids using our own software based on a Levenberg–Marquardt optimization algorithm. COS possesses a unique line spread function (Ghavamian et al. 2009) with large wings, and we accounted for this in the profile fitting process. In several cases, multiple absorption components were evident for a given species, and we attempted to fit the minimum number of components to account for the optical depth in each profile, including any apparent asymmetry, upon visual inspection. Doublet components were fit simultaneously as were interloping lines where blending from other species was evident. This blending could be observed, for example, by odd features in the (mostly) aligned apparent column density profiles of the doublet or by large asymmetries in the line profiles. In certain cases, line profiles were partially or completely obscured by the geocoronal O i λ1302 Å emission lines, preventing their measurement.

The resulting Voigt profile fitting measurements are presented in Table 2, and plots of the absorber sample spectra along with our profile fits are presented in the Appendix. Likely due in part to the resolution of our data, certain lines yield invalid Doppler b-value measurements, and these values are omitted from the table. However, measurements of b(C iv) < 10 km s⁻¹ may also be unreliable. For components of species with no unsaturated lines that would yield reliable Voigt profile fitting measurements, we report the integrated apparent column density profiles as lower limits.

Our statistics calculations require measuring the total path over which we may detect the C iv doublet. We express the total path in terms of two quantities: the redshift path length Δz and the comoving path length ΔX (Bahcall & Peebles 1969). The redshift path is simply the sum of the redshift ranges over which the doublet is detectable in each spectrum to a limiting EW or column density, but this calculation must account for varying S/N across the spectrum and absorption lines from systems at other redshifts, especially strong Lyα lines. Therefore, we must ignore those segments where the S/N is insufficient to reveal lines of a given strength.

To account for the decreasing sensitivity to lower column densities, authors using automatic line identification methods may also assume a constant path length across all column densities and employ Monte Carlo completeness corrections (e.g., Simcoe et al. 2011) in the derived statistics. Our survey comprises a visually identified absorber sample, and the method presented here alternatively measures a variable path length from the data S/N; the two methods should produce consistent results.

Because a varying redshift describes a varying comoving volume, we also employ the comoving path length X(z), which is defined as follows:

$$\begin{eqnarray}&&X(z)=\displaystyle \frac{2}{3{{\rm{\Omega }}}_{M}}\sqrt{{{\rm{\Omega }}}_{M}\times {(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}-\displaystyle \frac{2\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}{3{{\rm{\Omega }}}_{M}}.\end{eqnarray} \tag{ 3 }$$

For the Δz and ΔX calculation, we first convert the wavelength at each pixel to z and X, respectively, assuming the C iv λ1550 line were to fall at each location (i.e., z = 1550.77/λ − 1). We then sum over the regions in each spectrum (in terms of z or X) where the λ1550 would be detectable and finally sum the detectable regions in all spectra.

A line's detectability is a function of the line strength and of the S/N of the data at the line's location. Therefore, we calculate at each wavelength position the limiting rest EW (W_lim), a threshold above which lines may be detected. The W_lim is defined as follows:

$$\begin{eqnarray}&&{W}_{{\rm{lim}}}(\lambda )=\displaystyle \frac{3{\sigma }_{W(\lambda )}}{1+{z}_{\mathrm{abs}}},\end{eqnarray} \tag{ 4 }$$

where σ_W(λ) is the uncertainty of the observed EW summed in quadrature over a number of pixels. The number of integrated pixels is taken as a typical width (in pixels) of a line with EW W_lim, between 8 and 18 pixels for the absorbers in our sample and wider than the full width at half maximum of the spectra. The following expression defines σ_W(λ):

$$\begin{eqnarray}&&{\sigma }_{W(\lambda )}^{2}=\displaystyle \sum _{i}{\left({\rm{\Delta }}\lambda (i)\left[\displaystyle \frac{{\sigma }_{I({\lambda }_{i})}}{I({\lambda }_{i})}\right]\right)}^{2}\end{eqnarray} \tag{ 5 }$$

where Δ λ(i) is the pixel width (in Å), I(λ_i) is the continuum flux at pixel i, and ${\sigma }_{I({\lambda }_{i})}$ is the flux uncertainty at pixel i. We chose the number of integrated pixels based on typical widths of lines from our data with measured EWs similar to the threshold W_lim, e.g., the average profile of an ∼80 mÅ line was 12 pixels wide. An example W_lim calculation for the spectrum of J1342+1844 is shown in Figure 2.

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The W_lim was measured using these equations centered at each pixel of every spectrum in our data set in multiple passes using the number of pixels as described above for W_lim values commensurate with the column density bins used to compute the column density distribution function (see Section 4.1). The absorber column density range we are probing ($13\lesssim {\rm{log}}\;N({\rm{C}}\;{\rm{IV}})\lesssim 15\;{\mathrm{cm}}^{-2}$) falls in the region of the curve of growth where the lines of the doublet begin to saturate. Therefore, we mapped EW to column density by creating 1000 synthetic C iv doublets with S/N ∼ 12 (a typical S/N for our QSO spectra), measuring the resulting EW, and fitting the resulting W_r–log N(C iv) relation from the simulated data with a 4th-order polynomial. We then measured the path length at each $13\lesssim {\rm{log}}\;N({\rm{C}}\;{\rm{IV}})\lesssim 15\;{\mathrm{cm}}^{-2}$ at intervals of Δ(log N(C iv)) = 0.05 ${\mathrm{cm}}^{-2}.$ In accordance with our blind survey criteria, we did not include regions of the spectra within ±500 km s⁻¹ of the COS-Halos and COS-Dwarfs target galaxy redshifts (the CASBaH survey is intrinsically blind) or ±5000 km s⁻¹ of the QSO redshift in our path lengths. The path lengths measured for our data set and employed for the following statistics calculations are shown in Figure 3 and tabulated in Table 3; the limiting EW values corresponding to these column density bins are listed in Table 3. We also list in Table 3 the cumulative number of absorbers per unit redshift path length as a function of N(C iv), $d{\mathcal{N}}/{dz}$, defined as follows:

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathcal{N}}}{{dz}}(N({\rm{C}}\;{\rm{IV}}))=\displaystyle \sum _{i}\displaystyle \frac{{\mathcal{N}}({N}_{i}({\rm{C}}\;{\rm{IV}}))}{{\rm{\Delta }}z({N}_{i}({\rm{C}}\;{\rm{IV}}))},\end{eqnarray} \tag{ 6 }$$

where ${\mathcal{N}}({N}_{i}({\rm{C}}\;{\rm{IV}}))$ is the number of C iv absorbers in the i-th column density bin, ${\rm{\Delta }}z({N}_{i}({\rm{C}}\;{\rm{IV}}))$ is the redshift path length for detecting absorbers with ${N}_{i}({\rm{C}}\;{\rm{IV}}),$ and the summation is carried out for column density bins with ${N}_{i}({\rm{C}}\;{\rm{IV}})\geqslant N({\rm{C}}\;{\rm{IV}}).$

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Table 3.  Limiting Equivalent Widths and Path Lengths Calculated as Functions of Column Density

log N(C iv) (cm−2)	${W}_{{\rm{1550}}}$ a (mÅ)	Δz	ΔX	$d{\mathcal{N}}/{dz}$ b
13.0	19	0.1	0.1	⋯
13.1	24	0.5	0.5	⋯
13.2	29	0.8	0.8	⋯
13.3	35	1.1	1.3	${7.80}_{-1.4}^{+2.4}$
13.4	43	1.7	1.9	${6.79}_{-1.2}^{+2.1}$
13.5	54	1.8	2.0	${6.79}_{-1.2}^{+2.1}$
13.6	67	2.8	3.1	${5.51}_{-1.0}^{+1.6}$
13.7	83	4.0	4.4	${4.59}_{-0.8}^{+1.4}$
13.8	102	4.3	4.7	${3.22}_{-0.6}^{+1.1}$
13.9	124	5.5	6.1	${2.59}_{-0.5}^{+0.9}$
14.0	150	7.1	7.9	${2.43}_{-0.5}^{+0.8}$
14.1	178	8.7	9.6	${1.56}_{-0.4}^{+0.7}$
14.2	208	9.9	11.1	${0.91}_{-0.3}^{+0.6}$
14.3	241	10.9	12.1	${0.52}_{-0.2}^{+0.5}$
14.4	275	11.6	12.9	${0.17}_{-0.1}^{+0.4}$
14.5	310	12.0	13.4	${0.08}_{-0.1}^{+0.3}$
14.6	345	12.3	13.8	${0.08}_{-0.1}^{+0.3}$
14.7	379	12.5	14.0	⋯
14.8	412	12.7	14.2	⋯
14.9	442	12.8	14.3	⋯
15.0	468	12.9	14.4	⋯

Notes.

^aEquivalent width corresponding to N(C iv) in Column 1. ^bCumulative number of detected absorbers per redshift path length (see Equation (6)).

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We begin our analysis by deriving the column density distribution function, or f(N), of the C iv absorber sample. A fundamental observable measured for a sample of absorption systems, f(N) is useful for comparisons to theoretical predictions and describing the incidence and mass density of an absorber sample. One may evaluate f(N) in discrete column density bins ${\rm{\Delta }}N({\rm{C}}\;{\rm{IV}})$ as follows:

$$\begin{eqnarray}&&f(N)=\displaystyle \frac{{{\mathcal{N}}}_{{\rm{abs}}}(N({\rm{C}}\;{\rm{IV}}))}{{\rm{\Delta }}N({\rm{C}}\;{\rm{IV}}){\rm{\Delta }}X(N({\rm{C}}\;{\rm{IV}}))},\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Delta }}X(N({\rm{C}}\;{\rm{IV}}))$ is the comoving path length corresponding to the threshold set by the individual column density bin, and ${{\mathcal{N}}}_{{\rm{abs}}}(N({\rm{C}}\;{\rm{IV}}))$ is the number of absorbers within the specified column density bin.

Our binned evaluation of f(N(C iv)) is presented in Figure 4, assuming Poisson uncertainties in ${{\mathcal{N}}}_{{\rm{abs}}}(N({\rm{C}}\;{\rm{IV}}))$ for error estimation. Similar to previous work, we observe a steep decline in f(N) with increasing N(C iv). Earlier studies have frequently modeled f(N) as a single power-law,

$$\begin{eqnarray}&&f(N)={k}_{P}{\left(\displaystyle \frac{N({\rm{C}}\;{\rm{IV}})}{{10}^{14}\;{\mathrm{cm}}^{-2}}\right)}^{{\alpha }_{P}}\end{eqnarray} \tag{ 8 }$$

where α_P is the power law index, and k_P is the normalization constant. For small samples, this functional form has offered a good description for the limited data. Following Cooksey et al. (2010), we used a maximum likelihood analysis to find the best-fit parameters for this model: $\mathrm{log}{k}_{P}=-13.76\pm 0.07$ and α_P = −2.07 ± 0.15 (68% c.l.). For this likelihood analysis, we adopted a saturation limit ${N}_{{\rm{sat}}}={10}^{14.3}\;{\mathrm{cm}}^{-2}$ and have analyzed the data set from ${N}_{{\rm{min}}}={10}^{13.3}\;{\mathrm{cm}}^{-2}$ to ${N}_{{\rm{max}}}={10}^{15}\;{\mathrm{cm}}^{-2}.$ This model is overplotted on Figure 4 and offers a fair description of the data.

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However, Cooksey et al. (2013) found a steep (approximately exponential) turnover in the EW distribution of strong C iv systems. Furthermore, Danforth et al. (2014), whose sample contains more $13.0\leqslant {\rm{log}}\;N({\rm{C}}\;{\rm{IV}})\leqslant 13.5$ absorbers than that presented here, argue that a broken power law better fits their measured f(N(C iv)) than a single power law. We were thus inspired to consider an alternative model, specifically, fitting f(N(C iv)) with a Γ function

$$\begin{eqnarray}&&f(N)={k}_{{\rm{\Gamma }}}{\left(\displaystyle \frac{N({\rm{C}}\;{\rm{IV}})}{{N}_{*}}\right)}^{{\alpha }_{{\rm{\Gamma }}}}\;\mathrm{exp}\left[\displaystyle \frac{-N({\rm{C}}\;{\rm{IV}})}{{N}_{*}}\right],\end{eqnarray} \tag{ 9 }$$

parameterized by a normalization constant⁶ k_Γ, a power-law exponent ${\alpha }_{{\rm{\Gamma }}},$ and a "break" column density ${N}_{*}.$ Once again, we perform a maximum likelihood analysis with N_min, N_max, and N_sat as above on a three-dimenstional grid of k_Γ, α_Γ, and N_* values. The best-fit Γ function is overplotted on the binned evaluations in Figure 4 and provides an excellent description of the observations. The correlation in these parameters is illustrated in Figure 5; as expected, we find significant degeneracy between k_Γ and N_* although the latter is rather tightly constrained. The power-law exponent is shallow and constrained at the 99.7% c.l. to be greater than α_Γ = −1.5.

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The best-fit Γ function is overplotted on the binned evaluations in Figure 4 and provides an excellent description of the observed f(N(C iv)). To statistically assess the goodness-of-fit between these models, we conducted one-sample Anderson–Darling comparison tests between our absorber sample and each model. The traditional, and often-employed, implementation of the Kolmogorov–Smirnoff (K–S) and/or Anderson–Darling tests, wherein the so-called "D statistic" follows the K–S distribution (the integrals of which yield p-values for the null hypothesis) do not apply in this situation because the parameters of the models to be tested are derived from the sample distribution to be compared. Therefore, we produced distributions of the D statistics calculated between the normalized empirical cumulative distribution function of 10⁵ bootstrap resamples of our C iv absorber sample and the continuous cumulative distribution function of each model. Potentially due to the small numbers of absorbers we have at the lowest and highest column densities, the Anderson–Darling test does not yield a small enough p-value to reject the power-law model for f(N(C iv)) with strong statistical confidence. However, we call attention to Figure 6, which shows the cumulative distribution of our C iv absorber sample as a function of N(C iv) alongside the cumulative distributions following from the Γ function and power-law models. Due to its superior reproduction of our absorber sample, we adopt the Γ functional form for f(N(C iv)) in the following statistics calculations.

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From any f(N) model, it is trivial to integrate from any N_min value to estimate the incidence of absorption systems

$$\begin{eqnarray}&&d{\mathcal{N}}/{dX}={\displaystyle \int }_{{N}_{{\rm{min}}}}^{\infty }f(N,X){dN}\end{eqnarray} \tag{ 10 }$$

For N_min = 10^13.3 cm⁻², we calculate ${dN}/{dX}=7.5$ for our preferred model. Evaluating the likelihood function of this model to a 68% confidence limit, we find the rms in the resultant dN/dX values to be 1.1 (≈15%) which is consistent with the Poisson uncertainty of an N = 42 sample.

The mass density of triply ionized carbon is quantified by the Ω_{C iv} statistic, which is the ratio of the mass density of C iv to the critical density of the universe. Formally, Ω_{C iv} may be calculated from $f(N)$ as follows:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{C}}\;{\rm{IV}}}=\displaystyle \frac{{H}_{0}{m}_{C}}{c{\rho }_{c,0}}{\displaystyle \int }_{0}^{\infty }f(N({\rm{C}}\;{\rm{IV}}))N({\rm{C}}\;{\rm{IV}})\;{dN}({\rm{C}}\;{\rm{IV}})\ \ ,\end{eqnarray} \tag{ 11 }$$

where H₀ is the Hubble constant, ${\rho }_{c,0}=3{H}_{0}^{2}{(8\pi G)}^{-1}$ is the critical density, m_C is the mass of the carbon atom, and the other symbols have their typical meanings. In practice, Ω_{C iv} is typically estimated over a finite column density interval $N({\rm{C}}\;{\rm{IV}})=[{N}_{{\rm{min}}},{N}_{{\rm{max}}}].$ This is required for simple power-law models which will diverge at either high or low N(C iv). Furthermore, f(N) is generally only constrained over a finite N(C iv) interval.

In Figure 7, we present two evaluations of Ω_{C iv}: (1) the integration of our Γ-model for f(N) with ${N}_{{\rm{min}}}={10}^{13.3}\;{\mathrm{cm}}^{-2}$ and ${N}_{{\rm{max}}}={10}^{15}\;{\mathrm{cm}}^{-2};$ and (2) the summed evaluation of all C iv systems in our statistical analysis (which spans from ${10}^{13.3}\;{\mathrm{cm}}^{-2}$ to ${10}^{14.6}\;{\mathrm{cm}}^{-2}$):

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{C}}\;{\rm{IV}}}=\displaystyle \frac{{H}_{0}{m}_{C}}{c{\rho }_{c,0}}\displaystyle \sum _{i}\displaystyle \frac{N{({\rm{C}}\;{\rm{IV}})}_{i}}{{\rm{\Delta }}X(N{({\rm{C}}\;{\rm{IV}})}_{i})}.\end{eqnarray} \tag{ 12 }$$

These two evaluations are in excellent agreement and yield central values of 10.0×10⁻⁸ and 9.7×10⁻⁸, respectively. We further emphasize that the shallow power-law α_Γ derived from the Γ-model and its exponential decrease at high N(C iv) imply that our estimation is rather insensitive to the choice of N_min and N_max.

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We have assessed the uncertainty in Ω_{C iv} from sample variance through a bootstrap estimation of the summed evaluation (Equation (12)). Specifically, we have randomly sampled the observed distribution of N(C iv) values with 10,000 trials (allowing for duplications) and evaluated Equation (12) for each trial. Remarkably, the rms of the resultant distribution of Ω_{C iv} values is small: $\sigma ({{\rm{\Omega }}}_{{\rm{C}}\;{\rm{IV}}})=0.35\times {10}^{-8}.$ A set of 500 trials is overplotted in gray on Figure 7. We caution, however, that this bootstrap analysis may not sufficiently capture the sample variance in the highest N(C iv) systems that contribute ≈20% to Ω_{C iv}. Furthermore, this summed evaluation does not correct for line-saturation, unlike our maximum likelihood analysis of f(N). On the other hand, we find excellent agreement between the Ω_{C iv} evaluations and have confidence that the effects of line-saturation are small. We adopt a 15% uncertainty from systematic error and report a final estimate of ${{\rm{\Omega }}}_{{\rm{C}}\;{\rm{IV}}}=10.0\pm 1.5\times {10}^{-8}.$

Previous analysis of HST spectral data sets have presented estimates for Ω_{C iv} at z ≈ 0. Cooksey et al. (2010) reported ${{\rm{\Omega }}}_{{\rm{C}}\;{\rm{IV}}}$ = 7.0 ×  10⁻⁸ (≈30% error) from a sample of 19 absorbers at z < 0.6 discovered in STIS and GHRS observations. Tilton et al. (2012) analyzed a larger set of STIS spectra and reported an Ω_{C iv} value over twice as large from 29 absorbers at z < 0.12. That estimation was revised downward by Shull et al. (2014), who adopted a different approach to estimating Ω_{C iv} using binned evaluations of $f(N).$ From their analysis of the COS linelist posted to HST/MAST by Danforth et al. (2014), they report ${{\rm{\Omega }}}_{{\rm{C}}\;{\rm{IV}}}={10.1}_{-2.4}^{+5.2}\times {10}^{-8}$ (the methodology for their error estimation was not specified). Aside from the original Tilton et al. (2012) estimate, which was later revised, these various Ω_{C iv} evaluations are in good agreement with our new analysis.

To place our result in an evolutionary context with previous authors' findings, we convert their Ω_{C iv} values to our adopted integration limits and cosmology per Appendix C of Cooksey et al. (2010) where necessary. Figure 8 shows Ω_{C iv} values spanning z ∼ 6 to the present. The present (z = 0) value of Ω_{C iv} shows a clear increase over that of earlier cosmic time (z > 4), but the evolution since $z\sim 2$ is modest and not statistically significant.

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As we have alluded above, the measurements we report are key observables that can be predicted by cosmological hydrodynamic simulations, which must reproduce not only key properties of galaxies themselves, such as the stellar mass function and evolution of the star formation density (Madau et al. 1998) but also observed properties of the CGM and IGM. In fact, galaxies, the CGM, and the IGM are intimately related in the simulations, as gas infall and outflows are required to produce the observed global properties of the galaxy population; these processes in turn produce observational signatures in the CGM and IGM (Bordoloi et al. 2011; Fumagalli et al. 2011; Bouché et al. 2012) only probed by absorption line spectroscopy. As the scale and sophistication of cosmological simulations have progressed, the key factors driving their ability to reproduce actual observations are often encapsulated in "sub-grid" physics, prescriptions for processes such as supernova feedback and galactic winds that are not directly resolved. We attempt to use the preferred sub-grid variants of the simulations compared below, e.g., the vzw wind model of Oppenheimer et al. (2012), which is a momentum conserving wind prescription where the velocities of the outflowing particles scale as the internal velocity dispersion of the galaxy and which produces galaxies that more closely match observations than a constant-velocity wind model.

As the C iv doublet is among the most prominent metal-line spectral features, the Ω_{C iv} and f(N(C iv)) derived from simulation data are routinely used as metrics of the self-consistency of sub-grid processes invoked to reproduce observations of galaxies and the IGM. In Figure 9, we show our measured column density distribution function alongside predictions from three cosmological simulations. Cosmological hydrodynamical simulations largely use yields from stellar population synthesis models to establish global metallicities, and the uncertainties from these models can introduce uncertainties of ±0.3 dex in the column density distribution function. Given this uncertainty, both the Schaye et al. (2015) and Oppenheimer et al. (2012) models are largely consistent with our observations over the column density range probed, yet it is quite striking how well the Oppenheimer et al. (2012) predictions seem to follow our measured distribution function. Most importantly, both simulations predict downturns relative to a single power law at low and high column densities, corroborating both the results of Cooksey et al. (2013) that f(N(C iv)) deviates from a power law and our adoption of a Γ function form. The Cen & Chisari (2011) results do not extend to to the highest column density regime covered by our observations and the other simulations, but they predict a smooth downturn (relative to a power law) at the lowest column densities. The greatest dispersion among the models occurs at log N(C iv) $\lesssim \;13.0\;{\mathrm{cm}}^{-2},$ but much higher S/N spectroscopy is required to constrain the behavior of f(N(C iv)) in this regime.

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Figure 10 shows the predicted evolution of Ω_{C iv} by Oppenheimer et al. (2012) and Cen & Chisari (2011) from z = 2.5 to the present alongside observational measurements. The Oppenheimer et al. (2012) simulations predict minimal evolution in Ω_{C iv}, but the uncertainties dwarf any differences between the observational constraints. However, Cen & Chisari (2011) predict a steady upward evolution in Ω_{C iv}. The behaviors of these predictions differ most significantly between z ∼ 1.5 and z ∼ 0.5, where the observational constraints are insufficient to favor one model over the other. Regardless, the predicted Ω_{C iv} values at z ∼ 0 are consistent with our reported measurement.

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As an indicator of the physical conditions in the gas, the Doppler b value provides another observable in the absorption line data by way of Voigt profile fitting. As Cen & Chisari (2011) remark, directly comparing the b values from any simulation versus those measured in observed absorbers is complicated by differences in measurement procedures (we derive b(C iv) from Voigt profile fitting); the problem is exacerbated in systems with many components spread over a wide velocity range. However, our sample is primarily composed of systems with simple velocity structures (one or two components spread over <100 km s⁻¹), so our data often provide reliable b-values that can be compared with simulation predictions. We compare our sample with the predicted b(C iv) distributions from Oppenheimer et al. (2012), who produced mock spectra for direct comparison with COS observations, in Figure 11. The distributions shown here from Oppenheimer et al. (2012) employ their preferred feedback prescription including and excluding a model for turbulence. While their turbulence model improves the reproduction of O vi absorber observations, the model overpredicts the line widths of observed C iv absorbers. Our data suggest that the media probed by C iv absorption at z ∼ 0 may not be highly turbulent or that the turbulence model does not properly treat the media where C iv is observed. We return to the discussion of these predictions in Section 5.2.

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Also shown in Figure 11 are the predicted b(C iv) distributions from Cen & Chisari (2011) for two column density bins: log N(C iv)$\ =\ 13\mbox{--}15\;{\mathrm{cm}}^{-2}$ and log N(C iv)$\ =\ 14\mbox{--}15\;{\mathrm{cm}}^{-2},$ ranges over which our survey is sensitive. Their data are not intended for direct comparison with observations, and we only present them here for discussion. They report an increasing mean b(C iv) with increasing N(C iv), and as seen in Figure 11, the b-value distribution for their log N(C iv)$\ =\ 14\mbox{--}15\;{\mathrm{cm}}^{-2}$ bin (cyan circles in Figure 11) shows a higher mean b(C iv) than that of the wider N(C iv) range (blue circles). Cen & Chisari (2011) physically explain this trend by lower column C iv absorbers residing in "quiescent" environments. If this interpretation is valid, the dearth of low-N(C iv), high-b(C iv) absorbers in our sample (see Section 6) may in fact be of physical origin rather than observational bias.

The spectral resolution of the G130M and G160M gratings coupled with Voigt profile fitting enable the velocity centroids of unsaturated individual lines to be localized within $\lt 10\;\mathrm{km}\;{{\rm{s}}}^{-1}$, although the COS wavelength calibration introduces additional uncertainties. Even where lines are blended, fitting the individual profiles can attempt to "deblend" the constituents, albeit with increased uncertainty. All of the analyses in this section (this velocity comparison and the column density relationships to follow) utilize the individual components resulting from Voigt profile fitting. Thus, we begin by examining the velocity differences between fitted components of various species identified within the same system.

We examined the component structure of all species in each system and grouped together components of different species with the smallest velocity separation without yet imposing a maximum velocity offset. For example, in the absorber shown in Figure 12 (the z_abs = 0.07174 system in the spectrum of J1342-0053), we identified three components in the absorption profiles of Si ii, C ii, N ii, and Si iii, two components in the profiles of C iv and H i, and only one component in O i. For this absorber, we grouped together the Si ii, C ii, N ii, and Si iii components at δ v = 8, −10, 5, and 0 km s⁻¹, respectively. The C ii, C iv, H i, Si ii, N ii, O i, and Si iii components at δ v = −114, −114, −139, −123, −110, −110, and −113 km s⁻¹, respectively, were placed in a second group. The C ii, N ii, Si ii, and Si iii components at −39, −49, −43, and −35 km s⁻¹, respectively, were placed in a third group. We proceeded in this manner until all apparently related lines in the 42 absorbers in our sample had been assigned to a group. We note that in certain instances, the data are inadequate for this exercise; for example, we see that the H i Lyα line in Figure 12 is strongly saturated, thus precluding reliable measurements of the quantities required for this analysis. The C iv component at −17 km s⁻¹ in Figure 12 illustrates another limitation of the COS data for this purpose: we can see from Figure 12 that the C iv absorption at −17 km s⁻¹ has a clear correspondence with the lower ionization stages at similar velocities, but the group 1 and group 3 components appear to be blended in the C iv profile, and the C iv data are not good enough to enable a two-component decomposition of the feature at −17 km s⁻¹. The C iv components in this feature could be broader than the corresponding components of lower-ionization and/or more massive species, or noise in the C iv profile could have smeared together the components. With higher S/N and/or higher resolution data, it should be possible to better constrain the component structure of features such as the C iv component at −17 km s⁻¹ in Figure 12.

Figure 13 shows distributions of velocity offsets between C iv and a variety of ions as grouped by the above procedure; the species detected in our data cover a range of ionization stages, from C ii and O i (low ions) to Si iii and Si iv (intermediate ions) to O vi and N v (high ions). In some instances, Figure 13 confirms expected outcomes, i.e., species that should be aligned (e.g., C iv and Si iv) are aligned. However, Figure 13 also reveals some surprising alignments. For example, the only three detections of O i in our sample show close alignments with components of C iv, even though the ionization potentials greatly differ: E(O i $\;\to \;$ ii) = 13.6 eV and E(C iii $\;\to$ C iv) = 47.9 eV (Kramida et al. 2014). Likewise, the majority of the C ii absorption lines are well aligned with the C iv lines. For C ii, we have a larger sample, and we find that the distribution of velocity differences is centered on 0 km s⁻¹, and 79% of the C ii lines are aligned with C iv to within $\pm 20\;\mathrm{km}\;{{\rm{s}}}^{-1}$. Furthermore, our limited sample of O vi detections shows decent alignment with C iv components; in contrast, Lehner et al. (2014) find that O vi and C iv components often show quite distinct kinematics from one another in their z ∼ 3 sample of Lyman limit and damped Lyα systems. Larger low-redshift samples at higher resolution would enable a further investigation of possibly evolving kinematic relationships between these species over the age of the universe.

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It is clear from Figure 13 that not all component pairs classified in the above manner are well aligned. However, it is important to recognize that the spectral resolution, line-spread function, and well-known wavelength calibration problems of COS (Wakker et al. 2015) limit our ability to accurately measure velocity centroids. Therefore, we impose the following requirement for individual component pairs in order to include them in our subsequent analyses of aligned absorption lines:

$$\begin{eqnarray}&&\delta {v}_{\mathrm{XY}}\leqslant \sqrt{{\sigma }_{v}^{2}({\rm{X}})+{\sigma }_{v}^{2}({\rm{Y}})+{\sigma }_{v}^{2}({\rm{COS}})},\end{eqnarray} \tag{ 13 }$$

where δv_XY is the velocity offset between components of species X and Y, and σ_v(X) and σ_v(Y) are the uncertainties in the component velocity centroids of species X and Y, respectively, from Voigt profile fitting. The final term, σ_v(COS), accounts for known (but poorly understood) errors in the COS wavelength solution that well exceed the 15 km s⁻¹ resolution of the instrument (Wakker et al. 2015). Efforts by several teams are underway to solve for corrections to these errors, but the effects appear to be highly wavelength dependent and nonlinear. We have adopted σ_v(COS) = 25 km s⁻¹, commensurate with offsets we have measured in our spectra between multiple transitions of the same ion that should be perfectly aligned but are not (T. M. Tripp et al. 2015, in preparation).

Using the components grouped in accord with Equation (13), we next examine the correlations in column density of one species with another. The aforementioned limitations in our H i measurements (due to strong saturation of the only available H i lines) inhibit our ability to analyze the absorbers with the usual photoionization or collisional ionization models, but we can nevertheless gain insights on the nature of these systems by comparing unsaturated species. For example, if two given species generally are located in the same (cospatial) gas phase, then one might expect their column densities to be strongly correlated if the various clouds have the same relative abundance patterns. Conversely, if two species have a physical relationship but are not necessarily cospatial (e.g., if the C iv arises in an interface layer on the surface of a low-ionization phase), then the column densities of those species might be poorly correlated despite being kinematically well aligned. We note that it has already been shown that the population of O vi absorbers that are well aligned with H i exhibit very weak correlation between N(O vi) and N(H i) (Tripp et al. 2008), and we might anticipate a similar situation with C iv.

Figures 14–16 show the column density measurements of individual species components associated in velocity as described above. The points marked by yellow squares indicate unsaturated detections of both species, and magenta squares indicate that one of the two species is saturated. Here, we have flagged a line as saturated if ≥10% of the pixels in the line profile have flux values that are less than their noise values. Upper limits are indicated where one or both species was not detected at a coincident velocity to components identified in one or more other species. Certain points in the ion–ion plot reflect only upper limits for both species, and we have excluded these cases from the statistical analysis below .

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To quantitatively assess whether the species shown in Figures 14–16 are correlated, Table 4 shows the ion–ion column density correlation statistics calculated using the Kendall Tau rank correlation method, which incorporates censored data (upper/lower limits) to calculate the probability of the null hypothesis that no correlation exists between the two variables (smaller values of P_τ in Table 2 correspond to greater confidence of rejecting the null hypothesis that the quantities are not correlated). We have imposed a criterion for saturation upon the column density measurements as described above. However, these column densities may be better constrained than merely assigning lower limits, as Voigt profile fitting (used to measure column densities) is able to measure mildly saturated lines with adequate accuracy. Therefore, we calculate the Kendall tau correlation statistics flagging the measurements of the "saturated" lines as both lower limits and detections, which are reflected in P_{τ ss} and P_{τ sd}, respectively.

Table 4.  Correlation Statistics of Ion Column Densities

Ion 1	Ion 2	τssa	Pτ ssb	τsdc	Pτ sdd	ΦI(1) (eV)e	ΦI(2) (eV)f	$| {\rm{\Delta }}{\rm{\Phi }}|$ (eV)
C iv	C ii	0.4286	0.0559	0.3744	0.1314	47.8	11.3	36.5
C iv	O i	0.0585	0.7698	0.0	1.0	47.8	0.0	47.8
C iv	Si iii	0.4729	0.0242	0.3744	0.1353	47.8	16.3	31.5
C iv	Si iv	0.5867	0.0091	0.4933	0.0483	47.8	33.5	14.3
C iv	O vi	1.1667	0.0154	1.2778	0.0159	47.8	113.9	66.1
C iv	N v	0.3922	0.0244	0.4052	0.0224	47.8	77.5	29.7
C ii	Si iii	1.0292	0.0014	1.076	0.0009	11.3	16.3	5.0
C ii	Si iv	0.8182	0.0172	0.8182	0.0172	11.3	33.5	22.2
Si ii	Si iv	0.2821	0.4017	0.2821	0.4017	8.1	33.5	25.4
Si iii	Si iv	0.8382	0.0057	0.8971	0.0034	16.3	33.5	17.2
C ii	Si ii	1.1048	0.0041	1.1048	0.0041	11.3	8.1	3.2
Si ii	Si iii	0.8235	0.014	0.8235	0.0145	8.1	16.3	8.2
C iv	Si ii	0.2333	0.2427	0.1267	0.5993	47.8	8.1	39.7

Notes.

^aThe Kendall tau correlation coefficient between the column densities of Ion 1 and Ion 2, assuming that lines flagged as saturated yield lower limits for their column densities. ^bProbablility that a correlation does not exist between the column densities of Ion 1 and Ion 2, assuming that lines flagged as saturated yield lower limits for their column densities. ^cThe Kendall tau correlation coefficient between the column densities of Ion 1 and Ion 2, assuming that lines flagged as saturated yield reliable column densities. ^dProbablility that a correlation does not exist between the column densities of Ion 1 and Ion 2, assuming that lines flagged as saturated yield reliable column densities. ^eEnergy to attain ionization state of Ion 1. ^fEnergy to attain ionization state of Ion 2.

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With the exception of cases in which our sample is very small (e.g., O i, N v), most of the Kendall-Tau tests in Table 4 suggest that the column densities of most of these species are correlated, albeit in some cases weakly. However, we find that the likelihood of a correlation is generally very strong (i.e., small P_τ) between species with small ionization energy differences ($| {\rm{\Delta }}{\rm{\Phi }}| \lt 20$ eV). The most notable exception occurs for the C iv–O vi pair, which shows a ∼98% probability for the existence of a correlation even though these species have the greatest separation in ionization potential ($| {\rm{\Delta }}{\rm{\Phi }}| =66.1$ eV). This result should be interpreted cautiously as we only have a small sample of O vi lines, but this could occur if the C iv and O vi arise in a collisionally ionized hot phase. We note that our survey is C iv selected and therefore includes, by design, gaseous systems with ionization conditions conducive to maintaining sufficient quantities of C iv ions for detection. Given the small sample sizes of certain ion pairs, we caution against overinterpreting these correlation results; some peculiarities clearly arise in Table 4, such as the apparent correlations for C ii–Si ii and C ii–Si iv but the lack thereof for Si ii–Si iv.

The decreasing likelihood of correlation with increasing differences in ionization potential may indicate that C iv has some type of physical relationship with lower ionization stages but nevertheless arises in a phase that is distinct from the low ions. This would help to explain the poor agreement between the observed C iv b-values and the predicted b-values from Oppenheimer et al. (2012) when turbulence is included (purple points in Figure 11): Oppenheimer et al. (2012) were motivated to add turbulence to their model based on the nonthermal broadening required by aligned O vi and H i absorbers under the assumption that the b-values of O vi and H i can be jointly used to solve for the temperature and nonthermal broadening. If the O vi and H i lines arise in physically distinct phases, then the nonthermal broadening found this way may not be valid. Indeed, the addition of this "turbulence" is not supported by the measured C iv b-values in our sample (see Figure 11). More sophisticated modeling of multiphase absorbers may be required for these CGM absorbers. It would also be helpful to expand the sample of C iv absorbers to improve the statistical significance of these analyses.