Mg ii Absorption at 2 < Z < 7 with Magellan/Fire. III. Full Statistics of Absorption toward 100 High-redshift QSOs^*

We fit an automatically generated continuum to each flux-calibrated spectrum via custom IDL routines. These routines first generate an initial mask of absorption features identified by pixel fluxes near zero. The masked spectrum is then split into segments of width 1250 km s⁻¹, which are median filtered to remove narrow absorption features. Each segment is allocated two knots for a cubic spline interpolation fit to the continuum across the full spectrum. The knots' locations are determined from the statistics of the median filtering process. The spline fit was iterated between two and five times with rejection of outlying pixels, to achieve convergence of the fit.

We then searched the continuum-normalized data for cosmological absorbers using a matched filter, composed of two Gaussians separated by the intrinsic Mg ii doublet spacing. An initial candidate catalog was constructed with redshifts located at S/N $\gt 5$ peaks in the data/kernel cross-correlation. This was repeated for a set of Gaussian kernels with FWHM between 37.5 and 150 km s⁻¹ to match systems of different intrinsic width. Since the matched filter returns many false positive detections (principally caused by OH sky line residuals), we subjected each candidate to a set of consistency checks to eliminate obviously spurious systems. These are described in detail in Paper I; they are accomplished by explicitly fitting individual Gaussians to each component of the doublet, and then verifying from the fit parameters that (1) ${W}_{2796}\gt {W}_{2803}$ (within measurement errors); (2) the FWHM exceeds FIRE's resolution element, but is not larger than 25 pixels (313 km s⁻¹, chosen empirically to minimize BAL contamination and continuum errors); (3) the amplitude of the Gaussian fit must be net positive and exceed the local noise rms; and (4) single systems cannot have broad kinematic components separated by more than three times the total FWHM, and are instead split into separate absorbers in the sample. While criterion (3) might at first seem superfluous given the $5\sigma$ threshold for creating the parent catalog, it proves helpful in screening some narrow positive peaks in the filtered spectrum located immediately adjacent to narrow negative peaks (from sky lime residuals in crowded bandheads).

Each Mg ii candidate that survived this screening was visually inspected, and the accepted systems were incorporated into the final sample presented in Table 2 and plotted in Figure 2. We measured rest-frame equivalent widths and associated errors by direct summation of the unweighted spectral pixels (and quadrature summation of the error vector) rather than parameterized fits. Table 2 also reports a velocity width ${\rm{\Delta }}v$ for each system corresponding to the interval on either side of the the line centroid where the normalized absorption profile remains below unity.

Table 2.  Summary of Absorption Properties for the FIRE Mg ii Sample

Index #	Sightline	z	${W}_{r}(2796)$	$\sigma (2796)$	${\rm{\Delta }}v$ d
			(Å)	(Å)	(km s−1)
1	Q0000–26	2.1839	0.140	0.032	167.8
2	Q0000–26	3.3900	1.340	0.029	232.3
3	BR0004–6224	3.7765	0.988	0.054	193.8
4	BR0004–6224	3.2037	0.601	0.043	114.6
5	BR0004–6224	3.6946	0.250	0.047	154.7
6	BR0004–6224	2.9598	0.470	0.086	167.8
7	BR0016–3544	2.7825	0.566	0.034	180.8
8	BR0016–3544	3.7571	1.321	0.038	384.5
9	BR0016–3544	2.9485	0.146	0.031	86.7
10	BR0016–3544	2.8184	4.116	0.059	686.1
11	SDSS J0040–0915	4.4268	0.204	0.023	128.2
12	SDSS J0040–0915	4.7396	0.831	0.027	154.7
13	SDSS J0040–0915	2.6715	0.582	0.057	346.6
14	SDSS J0042–1020	3.6297	1.129	0.031	308.7
15	SDSS J0042–1020	2.7550	2.141	0.023	334.0
16	SDSS J0054–0109	4.9975	0.476	0.063	283.3
17	SDSS J0054–0109	2.4471	0.159	0.065	55.9
18	SDSS J0100+28	4.5192	0.834	0.029	308.7
19	SDSS J0100+28	5.3389	0.147	0.005	128.2
20	SDSS J0100+28	3.3376	0.286	0.021	114.6
21	SDSS J0100+28	5.1084	1.300	0.014	193.8
22a	SDSS J0100+28	3.0515	0.151	0.012	167.8
23b	SDSS J0100+28	4.2230	1.971	0.035	598.3
24	SDSS J0100+28	2.3255	1.364	0.009	180.8
25	SDSS J0100+28	4.3479	0.060	0.008	167.8
26	SDSS J0100+28	6.1437	0.415	0.006	86.7
27	SDSS J0100+28	2.9001	0.114	0.005	128.2
28	SDSS J0100+28	2.5620	0.147	0.009	114.6
29	SDSS J0100+28	2.5819	0.098	0.011	232.3
30	SDSS J0100+28	2.7501	0.264	0.009	167.8
31	SDSS J0100+28	4.6435	0.149	0.010	114.6
32	SDSS J0100+28	6.1118	0.300	0.005	114.6
33	SDSS J0106+0048	3.7290	0.854	0.016	154.7
34	VIK J0109–3047	2.9695	0.454	0.074	100.8
35	VIK J0109–3047	5.0011	0.335	0.042	128.2
36	SDSS J0113–0935	3.6167	0.581	0.046	180.8
37	SDSS J0113–0935	3.5446	0.231	0.039	128.2
38a,c	SDSS J0113–0935	3.1140	0.257	0.031	141.5
39	SDSS J0113–0935	2.8252	0.188	0.029	114.6
40c	SDSS J0127–0045	1.9785	0.168	0.021	296.0
41	SDSS J0127–0045	3.7282	0.863	0.014	167.8
42	SDSS J0127–0045	3.1688	0.270	0.019	346.6
43	SDSS J0127–0045	2.5881	1.568	0.027	535.5
44	SDSS J0127–0045	2.9458	2.272	0.040	397.1
45	SDSS J0140–0839	2.2408	0.415	0.027	141.5
46	SDSS J0140–0839	3.2122	0.093	0.014	141.5
47a	SDSS J0140–0839	3.0815	0.565	0.021	114.6
48	SDSS J0157–0106	3.3860	1.332	0.083	409.7
49	SDSS J0157–0106	2.6311	0.734	0.079	206.7
50	SDSS J0157–0106	2.7980	0.510	0.052	359.3
51	PSO J029–29	4.8762	0.289	0.028	114.6
52	PSO J029–29	3.6086	1.219	0.054	180.8
53	PSO J029–29	4.9864	2.966	0.102	472.7
54	ULAS J0203+0012	3.7110	0.267	0.045	154.7
55a	ULAS J0203+0012	4.3129	0.830	0.095	154.7
56	ULAS J0203+0012	4.9770	0.916	0.105	193.8
57	ULAS J0203+0012	4.4818	0.548	0.195	128.2
58	SDSS J0216-0921	2.4363	0.433	0.056	232.3
59	SDSS J0231-0728	5.3391	0.699	0.056	296.0
60a	SDSS J0231–0728	3.1113	0.518	0.052	86.7
61	SDSS J0231–0728	4.8840	1.322	0.133	409.7
62	SDSS J0231–0728	3.4298	0.431	0.037	167.8
63	ATLAS J025–33	5.3153	1.007	0.050	154.7
64	ATLAS J025–33	2.6666	0.470	0.024	154.7
65	ATLAS J025–33	2.7340	0.591	0.017	128.2
66	ATLAS J025–33	2.4460	2.183	0.031	409.7
67	BR0305–4957	3.3545	0.576	0.016	154.7
68	BR0305–4957	2.5023	0.322	0.027	206.7
69	BR0305–4957	2.6295	1.127	0.021	245.1
70	BR0305–4957	4.4669	1.789	0.016	283.3
71b,c	BR0305–4957	4.2120	2.047	0.018	761.4
72	BR0305–4957	3.5916	1.503	0.020	232.3
73	VIK J0305–3150	2.4962	2.707	0.122	397.1
74	VIK J0305–3150	4.6202	0.401	0.040	128.2
75	VIK J0305–3150	2.5652	2.638	0.124	573.2
76	VIK J0305–3150	3.4650	0.256	0.035	100.8
77	BR0322–2928	2.2291	0.617	0.023	128.2
78	BR0331–1622	2.5933	0.230	0.024	100.8
79	BR0331–1622	2.2952	1.804	0.076	460.1
80	BR0331–1622	2.9277	1.311	0.055	359.3
81	BR0331–1622	3.5566	0.714	0.039	154.7
82a	SDSS J0332–0654	3.0618	0.686	0.113	245.1
83	SDSS J0338+0021	2.2947	1.103	0.091	128.2
84	BR0353–3820	1.9871	3.142	0.036	548.1
85	BR0353–3820	2.7537	4.519	0.020	824.0
86	BR0353–3820	2.6965	0.357	0.018	180.8
87	PSO J036+03	4.6947	0.295	0.027	167.8
88	PSO J036+03	3.2745	0.710	0.028	180.8
89a	BR0418–5723	2.9780	1.896	0.080	334.0
90	BR0418–5723	2.0305	1.533	0.074	245.1
91	DES0454–4448	2.5264	1.566	0.050	257.9
92	DES0454–4448	2.3174	2.350	0.047	384.5
93	DES0454–4448	3.7234	0.407	0.048	180.8
94	DES0454–4448	3.3932	0.842	0.082	206.7
95	DES0454–4448	3.5017	0.176	0.043	100.8
96	DES0454–4448	3.4500	0.582	0.020	128.2
97	DES0454–4448	2.7565	0.370	0.023	114.6
98	PSO J065–26	3.5381	1.923	0.133	346.6
99	PSO J065–26	3.4480	1.902	0.029	257.9
100a	PSO J065–26	2.9829	1.315	0.070	257.9
101	PSO J071–02	2.7732	0.747	0.042	167.8
102	PSO J071–02	4.9944	1.059	0.073	257.9
103	PSO J071–02	5.1735	2.738	0.111	371.9
104a	SDSS J0817+1351	2.9946	1.185	0.102	232.3
105	SDSS J0817+1351	3.4648	0.293	0.055	193.8
106c	SDSS J0818+0719	2.2049	0.353	0.047	193.8
107c	SDSS J0818+0719	2.0832	0.217	0.028	167.8
108	SDSS J0818+1722	3.5629	0.607	0.078	128.2
109	SDSS J0818+1722	5.0649	0.834	0.063	128.2
110	SDSS J0818+1722	4.4309	0.478	0.053	180.8
111	SDSS J0824+1302	2.7919	0.327	0.055	154.7
112	SDSS J0824+1302	4.8110	0.224	0.035	100.8
113	SDSS J0824+1302	3.5872	0.234	0.071	86.7
114	SDSS J0824+1302	4.4716	0.866	0.027	167.8
115	SDSS J0824+1302	4.8308	0.659	0.047	114.6
116	SDSS J0836+0054	2.2990	0.565	0.022	232.3
117	SDSS J0836+0054	3.7443	2.509	0.016	510.4
118	SDSS J0842+1218	5.0481	1.813	0.146	245.1
119	SDSS J0842+1218	2.3921	1.437	0.251	193.8
120	SDSS J0842+1218	2.5397	2.157	0.098	384.5
121	SDSS J0949+0335	3.3105	2.026	0.044	296.0
122	SDSS J0949+0335	2.2888	2.834	0.065	472.7
123	SDSS J1015+0020	2.0588	3.161	0.133	510.4
124b	SDSS J1015+0020	3.1040	3.862	0.072	773.9
125	SDSS J1015+0020	2.7103	1.417	0.073	296.0
126	SDSS J1015+0020	3.7299	0.489	0.029	141.5
127	SDSS J1020+0922	3.4786	0.117	0.016	128.2
128	SDSS J1020+0922	2.7485	0.635	0.024	141.5
129	SDSS J1020+0922	2.5933	0.482	0.027	128.2
130	SDSS J1020+0922	2.0461	0.381	0.046	114.6
131	SDSS J1030+0524	2.1881	0.315	0.021	371.9
132	SDSS J1030+0524	4.5836	1.839	0.033	321.3
133	SDSS J1030+0524	4.9481	0.455	0.023	141.5
134	SDSS J1030+0524	5.1307	0.146	0.013	55.9
135a	SDSS J1037+0704	3.1373	0.349	0.062	193.8
136	J1048–0109	6.2215	1.647	0.163	232.3
137	J1048–0109	3.7465	0.952	0.061	167.8
138	J1048–0109	4.8206	0.890	0.037	154.7
139	J1048–0109	3.4968	2.221	0.076	434.9
140	J1048–0109	3.4133	0.547	0.031	167.8
141	SDSS J1100+1122	3.7566	1.342	0.055	232.3
142	SDSS J1100+1122	2.7825	0.691	0.054	193.8
143	SDSS J1100+1122	2.8225	0.570	0.063	206.7
144	SDSS J1100+1122	4.3959	1.866	0.101	257.9
145a	SDSS J1101+0531	4.3431	3.118	0.264	460.1
146	SDSS J1101+0531	4.8902	0.346	0.074	154.7
147	SDSS J1101+0531	3.7191	0.820	0.063	257.9
148	SDSS J1110+0244	2.1188	2.957	0.043	460.1
149	SDSS J1110+0244	2.2232	0.193	0.024	141.5
150	SDSS J1115+0829	3.4045	0.731	0.034	154.7
151	SDSS J1115+0829	3.5427	1.557	0.172	219.5
152	SDSS J1115+0829	2.3209	0.359	0.037	55.9
153	ULAS J1120+0641	4.4725	0.298	0.015	128.2
154	ULAS J1120+0641	2.8004	0.178	0.041	71.9
155	SDSS J1132+1209	2.7334	0.180	0.031	206.7
156	SDSS J1132+1209	2.9568	1.210	0.072	206.7
157	SDSS J1132+1209	4.3801	0.968	0.098	193.8
158	SDSS J1132+1209	2.4541	0.333	0.049	180.8
159	SDSS J1132+1209	5.0162	0.249	0.027	114.6
160	ULAS J1148+0702	4.3673	4.784	0.112	371.9
161	ULAS J1148+0702	2.3858	2.600	0.287	359.3
162	ULAS J1148+0702	3.4936	4.822	0.194	899.2
163	PSO J183–12	4.8709	0.503	0.019	114.6
164	PSO J183–12	2.1068	0.710	0.024	245.1
165	PSO J183–12	2.2972	0.341	0.021	100.8
166	PSO J183–12	2.4058	0.225	0.030	128.2
167	PSO J183–12	2.4308	1.574	0.023	346.6
168	PSO J183–12	3.3956	1.069	0.032	283.3
169	SDSS J1253+1046	4.7930	0.394	0.052	100.8
170a	SDSS J1253+1046	3.0282	1.010	0.037	193.8
171	SDSS J1253+1046	2.8565	0.169	0.030	100.8
172	SDSS J1253+1046	4.6004	0.882	0.108	154.7
173	SDSS J1257–0111	2.4894	0.223	0.019	154.7
174	SDSS J1257–0111	2.9181	0.955	0.020	180.8
175	SDSS J1305+0521	2.7527	0.375	0.040	128.2
176	SDSS J1305+0521	2.3023	1.976	0.122	346.6
177	SDSS J1305+0521	3.2354	0.337	0.026	128.2
178	SDSS J1305+0521	3.6799	1.749	0.069	270.6
179	SDSS J1306+0356	3.4898	0.607	0.033	167.8
180	SDSS J1306+0356	2.5328	2.813	0.115	535.5
181	SDSS J1306+0356	4.8651	2.804	0.068	180.8
182	SDSS J1306+0356	4.6147	0.547	0.089	128.2
183	ULAS J1319+0950	4.5681	0.420	0.062	128.2
184	SDSS J1402+0146	3.2772	1.085	0.021	180.8
185	SDSS J1408+0205	2.4622	1.349	0.047	219.5
186	SDSS J1408+0205	1.9816	2.174	0.063	334.0
187	SDSS J1408+0205	1.9910	0.830	0.038	219.5
188	SDSS J1411+1217	5.0552	0.193	0.016	86.7
189	SDSS J1411+1217	2.2367	0.647	0.040	193.8
190	SDSS J1411+1217	5.2501	0.295	0.015	128.2
191	SDSS J1411+1217	5.3315	0.182	0.016	100.8
192	SDSS J1411+1217	3.4773	0.343	0.020	86.7
193	SDSS J1411+1217	4.9285	0.659	0.024	128.2
194	PSO J213–02	4.9125	0.623	0.030	128.2
195	PSO J213–02	4.7777	0.295	0.028	114.6
196	Q1422+2309	1.9720	0.163	0.020	128.2
197	SDSS J1433+0227	2.7717	0.726	0.018	128.2
198b	SDSS J1436+2132	2.9070	4.309	0.030	610.9
199	SDSS J1436+2132	4.5211	0.964	0.166	193.8
200b	SDSS J1444–0101	4.4690	2.002	0.173	472.7
201	SDSS J1444–0101	2.8103	0.599	0.059	141.5
202	SDSS J1444–0101	2.7967	0.264	0.044	114.6
203	CFQS1509–1749	3.2662	0.940	0.018	180.8
204a	CFQS1509–1749	3.1272	0.878	0.076	245.1
205	CFQS1509–1749	3.3925	5.679	0.056	811.5
206	SDSS J1511+0408	2.0394	2.978	0.090	359.3
207	SDSS J1511+0408	2.2771	2.756	0.081	485.3
208	SDSS J1511+0408	3.3588	1.464	0.067	397.1
209	SDSS J1511+0408	2.2310	1.825	0.051	321.3
210	SDSS J1511+0408	2.0230	1.129	0.053	232.3
211	SDSS J1532+2237	2.6116	1.725	0.032	245.1
212	SDSS J1532+2237	2.7414	0.862	0.033	283.3
213	SDSS J1538+0855	3.4979	0.165	0.012	346.6
214	SDSS J1538+0855	2.6383	0.282	0.027	154.7
215	PSO J159–02	6.2376	0.458	0.045	257.9
216	PSO J159–02	2.2465	0.163	0.027	71.9
217	PSO J159–02	3.6695	2.269	0.115	460.1
218	PSO J159–02	3.7422	0.681	0.047	257.9
219	PSO J159–02	6.0549	0.436	0.065	167.8
220a	PSO J159–02	4.3426	0.222	0.046	141.5
221	SDSS J1601+0435	3.5007	1.467	0.129	308.7
222	SDSS J1606+0850	2.7636	3.433	0.128	548.1
223	SDSS J1606+0850	4.4426	0.464	0.041	100.8
224	SDSS J1611+0844	3.7767	0.801	0.053	206.7
225	SDSS J1611+0844	2.0144	0.506	0.059	100.8
226a	SDSS J1611+0844	3.1454	2.662	0.213	422.3
227	SDSS J1611+0844	3.3861	0.464	0.038	141.5
228c	SDSS J1616+0501	1.9809	2.115	0.050	270.6
229	SDSS J1616+0501	3.2747	0.853	0.021	180.8
230	SDSS J1616+0501	2.7409	1.188	0.026	193.8
231	SDSS J1616+0501	3.3955	0.916	0.055	141.5
232	SDSS J1616+0501	3.4507	0.584	0.017	128.2
233	SDSS J1616+0501	3.7327	1.866	0.057	321.3
234	SDSS J1620+0020	2.9106	1.159	0.055	270.6
235	SDSS J1620+0020	3.7515	1.601	0.070	232.3
236	SDSS J1620+0020	3.6200	1.366	0.066	397.1
237	SDSS J1620+0020	3.2726	0.988	0.047	167.8
238	SDSS J1621–0042	2.6780	0.189	0.019	100.8
239a	SDSS J1621–0042	3.1057	1.013	0.013	232.3
240	SDSS J1626+2751	2.8288	1.260	0.041	206.7
241	SDSS J1626+2751	4.4619	0.829	0.014	219.5
242	SDSS J1626+2751	4.4968	1.673	0.019	283.3
243	SDSS J1626+2751	4.5682	0.561	0.025	128.2
244a	SDSS J1626+2751	4.3108	3.188	0.050	434.9
245	SDSS J1626+2751	2.4822	0.300	0.032	141.5
246	SDSS J1626+2751	3.6826	0.833	0.011	167.8
247	SDSS J1626+2751	2.1320	3.679	0.091	321.3
248	PSO J167–13	3.3889	0.581	0.036	141.5
249	PSO J183+05	6.0643	0.653	0.096	141.5
250	PSO J183+05	3.2071	0.803	0.042	180.8
251	PSO J183+05	3.4184	0.533	0.077	219.5
252	PSO J209–26	5.2021	0.643	0.025	154.7
253	PSO J209–26	2.9505	0.631	0.048	206.7
254	PSO J209–26	5.2758	0.299	0.020	100.8
255	SDSS J2147–0838	2.2863	1.058	0.049	206.7
256	PSO J217–16	4.6420	1.261	0.044	219.5
257a	PSO J217–16	5.3571	2.489	0.029	359.3
258	PSO J217–16	2.4166	0.501	0.050	128.2
259	VIK J2211–3206	3.6302	1.416	0.092	257.9
260	VIK J2211–3206	3.7144	3.505	0.068	623.4
261	SDSS J2228–0757	3.1754	0.287	0.038	71.9
262	PSO J231–20	2.4191	1.115	0.090	257.9
263	SDSS J2310+1855	3.2998	0.856	0.058	257.9
264	SDSS J2310+1855	2.3510	0.789	0.052	193.8
265	SDSS J2310+1855	2.2430	1.523	0.068	334.0
266	VIK J2318–3113	2.9030	0.887	0.075	219.5
267	BR2346–3729	3.6922	0.371	0.019	128.2
268	BR2346–3729	2.8300	1.665	0.054	270.6
269	BR2346–3729	2.9226	0.535	0.041	167.8
270	BR2346–3729	3.6188	0.422	0.036	141.5
271a	VIK J2348–3054	4.2996	2.567	0.118	384.5
272	VIK J2348–3054	6.2682	0.564	0.062	167.8
273	PSO J239–07	5.3238	0.287	0.024	141.5
274	PSO J239–07	5.1209	0.193	0.022	114.6
275	PSO J239–07	4.4276	0.193	0.018	141.5
276	PSO J242–12	2.6351	0.543	0.075	114.6
277	PSO J242–12	2.6880	0.620	0.087	180.8
278	PSO J242–12	4.3658	0.646	0.086	154.7
279	PSO J242–12	4.4351	0.671	0.041	141.5
280	PSO J308–27	2.8797	0.229	0.032	71.9
⋯	BR0004–6224	2.663	0.260	0.045	58.0
⋯	BR0004–6224	2.908	0.596	0.047	83.3
⋯	SDSS J1030+0525	2.780	2.617	0.069	583.9
⋯	SDSS J1306+0356	4.882	1.941	0.079	248.8
⋯	SDSS J1402+0146	3.454	0.341	0.016	173.3
⋯	Q1422+2309	3.540	0.169	0.011	130.0
⋯	SDSS J2310+1855	2.243	1.441	0.050	292.1

Notes.

^aPoor telluric region. ^bMissed by automated search algorithm. ^cNot identified in Paper I. ^d ${\rm{\Delta }}v$ is defined as the total velocity interval about each line centroid within which the absorption profile remains below the fitted continuum.

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For consistency, we have redone the line finding for the sightlines presented in Paper I. A complete list of these doublets and their continuum-normalized profiles are included in Table 2 and Figure 2, respectively. Differences in user acceptances/rejections are noted in the table: in general, as the visual inspection step was carried out by a different user than in the original survey (SC and MM, respectively), we tended to be more optimistic in accepting borderline candidates for Mg ii doublets. These tendencies are reflected in the user-rating calibration, discussed later. In addition, we serendipitously identified five systems excluded by the automated search algorithm. These are reported and flagged in the table of absorbers, but they are omitted from calculations of the Mg ii population statistics, because the statistical calculations account for such missed systems via incompleteness simulations. In the process of the visual identification, we also identified five Mg ii absorbers that were not included in our sample due to their proximity to the background quasar; these are listed with their associated properties in Table 3. The proximate absorbers in the two PS1 quasars are of particular interest and will be discussed in detail in forthcoming work (E. Banados 2017, in preparation).

Table 3.  Proximate Mg ii Systems

Index #	Sightline	z	${W}_{r}(2796)$	$\sigma (2796)$	${\rm{\Delta }}v$
			(Å)	(Å)	(km s−1)
1	PSO J065-26	6.122	2.346	0.038	553.4
2	SDSS J0140-0839	3.703	0.584	0.015	216.0
3	SDSS J1436+2132	4.522	0.973	0.189	332.8
4	SDSS J1626+2751	5.178	1.416	0.022	518.6
5	PSO J183+05	6.404	0.774	0.053	356.6

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We ran a large Monte Carlo simulation to quantify the the completeness of the automated line-finding algorithm. For each QSO, 10,000 simulated Mg ii doublets with equivalent widths uniformly distributed between 0.05 and 0.95 Å and random redshifts were injected into the spectrum (from which the real doublets were previously removed and replaced with noise) and then subjected to the automated line-finding algorithm. The rates at which these simulated doublets were recovered were then binned into an automated completeness grid by redshift and equivalent width (with dz = 0.02 and dW = 0.01 Å) for each QSO, which we will call ${L}_{q}(z,W)$. These computationally intensive simulations were run on the antares computing cluster at the MIT Kavli Institute.

A subset of the automatically simulated doublets were inspected visually to evaluate the efficacy of the human inspection step in our doublet-finding procedure. In particular, the user may either reject a real Mg ii system or accept a false positive, thus requiring a correction to our statistical calculations. We inspected 1000 such simulated doublets, with the important difference that the user-test systems had a slightly larger velocity spacing than legitimate Mg ii doublets. This ensures that any "doublets" identified by the machine are either artificially injected (and should therefore be accepted) or correlated noise (and should be rejected).

While inspecting these false-spacing doublets, we identified three very large absorbers, likely not due to Mg ii. These were manually excised and masked from our Monte Carlo data so that only injected doublets and correlated noise factored into the user ratings calculation. The user then either accepted or rejected the remaining candidate doublets, and the success rates at which the user identified real systems and rejected false positives were used to calculate a total completeness for each QSO.

As discussed in Paper I, the time-consuming nature of visual inspection precludes the use of finely grained bins in W_r and z, but we found that the acceptance rate for real systems and false positives depended primarily on the S/N of the candidate doublets. They can be parameterized with S/N as follows:

$$\begin{eqnarray}&&{P}^{\mathrm{Mg}{\rm{II}}}(s)={P}_{\infty }(1-{e}^{-s/{s}_{c}})\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{P}^{\mathrm{FP}}(s)=\left\{\begin{array}{l}{P}_{\max }^{\mathrm{FP}}\left(\tfrac{s}{{s}_{p}}\right),\quad s\leqslant {s}_{p}\\ {P}_{\max }^{\mathrm{FP}}\left(\tfrac{s-{s}_{f}}{{s}_{p}-{s}_{f}}\right),\quad s\gt {s}_{p}\end{array}\right.,\end{eqnarray} \tag{ 2 }$$

where ${P}^{\mathrm{Mg}{\rm{II}}}$ and ${P}^{\mathrm{FP}}$ are the acceptance rates for real systems and false positives, respectively, and P_∞, s_c, ${P}_{\max }^{\mathrm{FP}}$, ${s}_{p},$ and s_f are free parameters fit by maximum-likelihood estimation (MLE). Plots of the user acceptance rates are given in Figure 3. Comparing the ratings of SC and MM, it is apparent that SC correctly identified a higher fraction of Mg ii doublets at low S/N, but this comes at the expense of a higher false positive rate. After proper calibration these tendencies should cancel, and indeed we will find very similar statistical results as Paper I in areas where both may be compared.

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For each individual QSO, the user acceptance rates were then estimated as functions of the equivalent width W_r to give ${A}^{\mathrm{Mg}{\rm{II}}}(W,z)$ and ${A}^{\mathrm{FP}}(W,z)$, the acceptance rates for real systems and false positives, respectively. The total completeness fraction for each QSO q, ${C}_{q}(z,W)$, was then calculated as the product of the automated completeness fraction and the user acceptance rate, namely

$$\begin{eqnarray}&&{C}_{q}(z,W)={A}^{\mathrm{Mg}{\rm{II}}}(z,W){L}_{q}(z,W).\end{eqnarray} \tag{ 3 }$$

The total pathlength-weighted completeness for our survey, thus calculated, is shown in Figure 4. The acceptance rate for false positives are not included in this step, but rather are accounted for directly in our calculations of the population statistics. Unlike in Paper I, the grid now extends to a maximum redshift of z = 7, picking up path at $z\gt 6.1$ where Mg ii re-emerges into the K band atmospheric window.

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Given the completeness grid calculated in Figure 4, we can calculate the redshift path density $g(z,W)$ of our survey (i.e., the total number of sightlines at redshift z for which Mg ii absorbers with equivalent width greater than W_r can be observed) as

$$\begin{eqnarray}&&g(z,W)=\displaystyle \sum _{q}{R}_{q}(z){C}_{q}(z,W),\end{eqnarray} \tag{ 4 }$$

where R_q(z) is equal to one within the redshift search limits (but outside the redshifts excluded due to poor telluric corrections) and zero everywhere else. This function is shown in the top panel of Figure 5; the bottom panel indicates the survey path g(W), defined as

$$\begin{eqnarray}&&g(W)=\int g(z,W){dz}.\end{eqnarray} \tag{ 5 }$$

Here the increase in completeness toward higher W_r is reflected in the rising path probed at larger equivalent width. The converged value at $g(W)\sim 150$ toward large equivalent width indicates a high completeness, and an average redshift coverage of ${\rm{\Delta }}z\sim 1.5$ per sightline for our 100 objects. The total survey path of Paper I was approximately 80, so we have roughly doubled the path by doubling the number of QSOs observed.

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We employed the same formalism described in Paper I to account for incompleteness and false positives in our statistical results. Briefly, in a given redshift and equivalent width bin k, the corrected (true) number of systems can be calculated from the number ${\breve{N}}_{k}$ of detected systems in that bin as

$$\begin{eqnarray}&&{N}_{k}=\displaystyle \frac{{\breve{N}}_{k}(1-{\bar{A}}_{k}^{F})-{\bar{A}}_{k}^{F}{\breve{F}}_{k}}{{\bar{C}}_{k}-{\bar{L}}_{k}{\bar{A}}_{k}^{F}},\end{eqnarray} \tag{ 6 }$$

where ${\breve{F}}^{k}$ is the number of rejected candidates, ${\bar{C}}_{k}$ is the average completeness, ${\bar{L}}_{k}$ is the automated line identification finding probability, and ${\bar{A}}_{k}^{F}$ is the user acceptance rate for false positives, each calculated for the kth bin. These fractions are calculated from the previously discussed automated completeness tests and user-rating calibrations. An important caveat is that the average completeness of a redshift and equivalent width bin is weighted according to the number distribution ${d}^{2}N/{dzdW}$, which must in principle be determined from the true number of systems N_k. Here we follow the discussion in Paper I and apply the simplifying assumption that ${d}^{2}N/{dzdW}$ is constant across each bin to resolve the apparent circularity.

Table 4 lists completeness-corrected values for the rest equivalent width frequency distribution ${d}^{2}N/{dzdW}$, an absorption line analog of the galaxy luminosity function. These values are plotted in Figures 7 and 8, where the equivalent width distribution is binned across the full survey redshift range and then split into four redshift intervals, respectively. The error bars for each point account both for Poisson fluctuations in the number count in each bin (which dominate the error budget), and for uncertainty in the completeness-adjusted pathlength. The latter term reflects errors in our completeness estimates, which are much smaller by design because of the large number of simulated doublets in the completeness and user rejection tests.

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Table 4.  Mg ii Equivalent Width Distribution, Full Sample, and Redshift Cuts

$\langle {W}_{r}\rangle$	${\rm{\Delta }}{W}_{r}$	$\bar{C}$	Number	${d}^{2}N/{dzdW}$
(Å)	(Å)	$( \% )$
$z=1.90\mbox{--}6.30$
0.42	0.05–0.64	46.0	130	1.539 ± 0.215
0.94	0.64–1.23	77.7	66	0.591 ± 0.082
1.52	1.23–1.82	79.7	34	0.298 ± 0.055
2.11	1.82–2.41	79.7	21	0.185 ± 0.042
2.70	2.41–3.00	79.7	15	0.134 ± 0.035
4.39	3.00–5.78	79.7	14	0.026 ± 0.007
$z=1.95\mbox{--}2.98$
0.42	0.05–0.64	42.0	56	1.371 ± 0.336
0.94	0.64–1.23	74.4	19	0.410 ± 0.108
1.52	1.23–1.82	76.7	14	0.308 ± 0.089
2.11	1.82–2.41	76.7	9	0.205 ± 0.071
2.70	2.41–3.00	76.7	8	0.187 ± 0.067
4.39	3.00–5.78	76.7	7	0.033 ± 0.013
$z=3.15\mbox{--}3.81$
0.41	0.05–0.64	54.8	31	1.183 ± 0.284
0.94	0.64–1.23	85.5	22	0.691 ± 0.152
1.53	1.23–1.82	87.1	12	0.370 ± 0.109
2.12	1.82–2.41	87.1	6	0.182 ± 0.076
2.71	2.41–3.00	87.1	1	0.031 ± 0.031
4.39	3.00–5.78	87.1	3	0.020 ± 0.011
$z=4.34\mbox{--}5.35$
0.41	0.05–0.64	52.1	32	1.840 ± 0.397
0.94	0.64–1.23	84.8	18	0.719 ± 0.176
1.53	1.23–1.82	86.6	6	0.236 ± 0.097
2.12	1.82–2.41	86.6	3	0.118 ± 0.069
2.71	2.41–3.00	86.6	3	0.118 ± 0.069
4.39	3.00–5.78	86.6	1	0.008 ± 0.008
$z=6.00\mbox{--}7.08$
0.93	0.05–1.53	52.7	6	0.979 ± 0.446
2.26	1.53–3.00	68.2	1	0.131 ± 0.132
4.39	3.00–5.78	68.2	0	<0.070

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With only seven systems in the highest redshift bin, the fractional errors on each point and the associated fit parameters are large. However, one can read off from these figures that the density of lines at ${W}_{r}\lt 1$ Å, even at $z\gt 6$, is quite comparable to lower redshift. At higher equivalent width (${W}_{r}\gtrsim 2$ Å), there is weak indication of a deficit compared with lower redshift, but the statistical errors on these points are significant and the fit parameterizations should therefore be interpreted with caution.

We fit the equivalent width distribution using maximum-likelihood estimation to the exponential form

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}N}{{dzdW}}=\displaystyle \frac{{N}_{* }}{{W}_{* }}{e}^{-W/{W}_{* }}\end{eqnarray} \tag{ 7 }$$

by first fitting W_* and then setting the overall normalization N_* such that the calculated number of systems in our survey is recovered. These fits are plotted as dashed lines in the figure of the frequency distribution. A list of the fit parameters is given in Table 5.

Table 5.  Maximum-likelihood Fit Parameters for Exponential Parameterization of the W_r Distribution

$\langle z\rangle$	${\rm{\Delta }}z$	W*	N*
		(Å)
0.68a	0.366–0.871	0.585 ± 0.024	1.216 ± 0.124
1.10a	0.871–1.311	0.741 ± 0.032	1.171 ± 0.083
1.60a	1.311–2.269	0.804 ± 0.034	1.267 ± 0.092
2.52	1.947–2.975	0.840 ± 0.092	2.226 ± 0.081
3.46	3.150–3.805	0.806 ± 0.105	1.864 ± 0.069
4.80	4.345–5.350	0.618 ± 0.097	2.227 ± 0.131
6.29	5.995–7.080	0.500 ± 0.148	2.625 ± 0.452
3.47	1.947–6.207	0.798 ± 0.055	2.051 ± 0.046

Note.

^aParameter fits from Nestor et al. (2005).

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Figure 9 displays evolution in the characteristic equivalent width W_* with redshift. For comparison, we have added the equivalent parameters provided in Nestor et al. (2005) and Seyffert et al. (2013) at lower redshifts, though it is important to note that Seyffert et al. only include systems with ${W}_{r}\gt 1\,\mathring{\rm A}$  in their fits. Throughout the following analysis we use these two samples as our low-redshift references, even though many other Mg ii surveys have been performed on the SDSS QSO sample (Prochter et al. 2006; Lundgren et al. 2009; Quider et al. 2011; Zhu & Ménard 2013; Chen et al. 2015; Raghunathan et al. 2016). The main motivation for our choice is that Nestor et al. (2005) probes the smallest equivalent widths (comparable to our measurements) in the SDSS data, while Seyffert et al. (2013) uses identification and analysis techniques most similar to our methods. However, these results are broadly consistent with other works in the literature where they may be compared.

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Our results confirm the trend noted in Paper I: at higher redshifts W_* does not continue its growth with redshift at earlier times. Rather, it peaks at around z = 2–3, after which it begins to decline. In direct terms, this corresponds to a similar peak in the incidence of strong Mg ii absorbers around z = 2–3, with a dropoff toward early epochs in the strong systems relative to their weaker counterparts.

The zeroth moment of the frequency distribution gives the line density of Mg ii absorption lines dN/dz, as plotted in Figure 10. We include low-redshift points from Mg ii surveys of the SDSS (Nestor et al. 2005; Seyffert et al. 2013) for comparison. For completeness we have also performed MLE fits of the form

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dz}}={N}_{* }{(1+z)}^{\beta }\end{eqnarray} \tag{ 8 }$$

on our high-redshift points, where the normalization N_* is fixed such that dN/dz integrated by redshift with the survey path density $g(z,W)$ recovers the number counts of our survey; these fits are shown as dashed lines in Figure 10. These results and parameter fits are listed in Tables 6 and 7, respectively.

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Table 6.  Mg ii Absorption Line Density dN/dz

$\langle z\rangle$	${\rm{\Delta }}z$	$\bar{C}$	Number	dN/dz	dN/dX
		$( \% )$
$0.3{\rm{\mathring{\rm A} }}\lt {W}_{r}\lt 0.6$ Å
2.236	1.947–2.461	46.2	11	0.308 ± 0.168	0.100 ± 0.054
2.727	2.461–2.975	62.2	18	0.499 ± 0.137	0.149 ± 0.041
3.460	3.150–3.805	69.4	15	0.336 ± 0.091	0.090 ± 0.024
4.806	4.345–5.350	66.2	13	0.391 ± 0.113	0.091 ± 0.026
6.291	5.995–7.085	43.1	4	1.173 ± 0.667	0.241 ± 0.137
$0.6{\rm{\mathring{\rm A} }}\lt {W}_{r}\lt 1.0{\rm{\mathring{\rm A} }}$
2.236	1.947–2.461	63.7	5	0.133 ± 0.081	0.043 ± 0.026
2.723	2.461–2.975	78.9	10	0.232 ± 0.079	0.069 ± 0.024
3.463	3.150–3.805	83.7	21	0.398 ± 0.090	0.107 ± 0.024
4.802	4.345–5.350	82.6	17	0.412 ± 0.103	0.096 ± 0.024
6.289	5.995–7.085	62.5	1	0.211 ± 0.213	0.043 ± 0.044
${W}_{r}\gt 1.0{\rm{\mathring{\rm A} }}$
2.236	1.947–2.461	68.7	24	0.751 ± 0.172	0.244 ± 0.056
2.722	2.461–2.975	83.4	22	0.505 ± 0.113	0.150 ± 0.034
3.463	3.150–3.805	87.1	26	0.471 ± 0.096	0.127 ± 0.026
4.801	4.345–5.350	86.6	15	0.346 ± 0.092	0.081 ± 0.022
6.287	5.995–7.085	68.2	1	0.193 ± 0.195	0.040 ± 0.040

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Table 7.  Maximum-likelihood Estimates of the Line Density Evolution ${dN}/{dz}={N}^{* }\,{(1+z)}^{\beta }$

$\langle {W}_{r}\rangle$	${\rm{\Delta }}{W}_{r}$	${\rm{\Delta }}z$	β	N*
(Å)	(Å)
1.17a	1.00–1.40	0.35–2.3	${0.99}_{-0.22}^{+0.29}$	${0.51}_{-0.10}^{+0.09}$
1.58a	1.40–1.80	0.35–2.3	${1.56}_{-0.31}^{+0.33}$	${0.020}_{-0.05}^{+0.05}$
1.63a	1.00+	0.35–2.3	${1.40}_{-0.16}^{+0.16}$	${0.08}_{-0.05}^{+0.15}$
2.08a	1.40+	0.35–2.3	${1.74}_{-0.22}^{+0.22}$	${0.036}_{-0.06}^{+0.06}$
2.52a	1.80+	0.35–2.3	${1.92}_{-0.32}^{+0.30}$	${0.016}_{-0.03}^{+0.06}$
0.45	0.30–0.60	1.9–6.3	−0.345 ± 0.616	0.722 ± 0.653
0.79	0.60–1.00	1.9–6.3	0.821 ± 0.505	0.090 ± 0.069
1.80	1.00+	1.9–6.3	−1.020 ± 0.475	2.298 ± 1.561

Note.

^aParameter fits from Prochter et al. (2006), with corresponding upper and lower 95% confidence intervals. This survey's results include $1\sigma$ errors.

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The line density dN/dz can further be converted to the more physical comoving line density dN/dX. Here we divide the distribution into two equivalent width bins separated at W_r = 1 Å, and five redshift bins to illustrate differences in evolutionary trends.

As with the equivalent width distribution, error bars include a Poisson contribution from the number of systems in each bin, and an additional (much smaller) contribution from uncertainty in the completeness values used to adjust the survey pathlength (dz or dX). The overall accuracy of the FIRE survey points is likely limited by statistical errors—even with 100 sightlines we average just 10–25 absorbers per bin, corresponding to a 20%–30% uncertainty. In contrast, the low redshift studies from SDSS have thousands of absorbers per redshift bin and therefore have errors dominated by systematic effects not explicitly quantified in these studies (and therefore not captured in the figure). As argued by Seyffert et al. (2013), these likely arise from differences in continuum fitting procedures and algorithms for measuring W_r, and the use of a sharp W_r cutoff when defining samples used to derive dN/dX. Comparison of different Mg ii surveys from SDSS QSOs suggests a systematic scatter of ∼10%–15%, far larger than the $\sim 1 \%$ random errors (Seyffert et al. 2013) These different errors must be considered when comparing in regions of overlap, such as the bottom panel of Figure 11.

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The larger survey confirms and strengthens two key findings of Paper I by both reducing Poisson errors on points at $z\lt 5$ and adding new redshift coverage at $z\gt 6$. First, the comoving absorption density (i.e., the frequency) of typical Mg ii systems with $0.3\lt {W}_{r}\lt 1$ Å remains remarkably constant from z = 0.5 to z = 7 (i.e., all redshifts that have been searched).

This can only be true if the product of the comoving volume density of absorbers n(z), multiplied by the physical cross section of each absorber σ, also remains a constant. If Mg ii absorbers at high redshift are associated with luminous galaxies like their low-redshift counterparts, then circumgalactic gas must therefore have a substantial cross section for heavy element absorption, even very early in these galaxies' evolutionary history. Our previous work suggested this result to $z=5.5;$ the new sightlines presented here exhibit the exact number of Mg ii one would expect from simple extrapolation of this trend to z = 6.5, when the universe was 850 Myr old.

The second key finding from Paper I confirmed here is a firm evolution in the frequency of strong Mg ii absorbers at ${W}_{r}\gt 1$ Å. This trend is in marked contrast to the weaker systems, and is consistent with the evolution in W^* of the frequency distribution ${d}^{2}N/{dXdW}$. We find just one strong system at $z\gt 6$, again consistent with expectations extrapolated from lower z. The decline of nearly an order of magnitude from the peak at $z\sim 2.5$ suggests that further searches for strong systems at $z\sim 7$ and beyond are likely to require many sightlines toward faint QSOs; however, the weaker systems may well remain plentiful.

In the time since initial submission of this paper, two other relevant manuscripts have been posted describing Mg ii searches in the near-IR. We comment briefly here on comparisons of these studies with our work.

Codoreanu et al. (2017) searched a sample of four high-S/N spectra obtained with VLT/XShooter for Mg ii; because of the exceptional data quality, this search is more sensitive to weak absorption lines, but its shorter survey pathlength leads to larger Poisson uncertainties in bins of higher equivalent width. In the regions where our samples are best compared ($0.3\lt {W}_{r}\lt 1.0$ Å), the agreement in number density is very good. Our larger sample size reveals evidence for evolution at ${W}_{R}\gt 1$ Å not visible in their data; however, their higher sensitivity reveals numerous weak systems (${W}_{r}\lt 0.3$ Å). While we report some such systems in Table 2, our overall completeness was not sufficient to claim robust statistics on these absorbers. Their analysis reveals an excess of weak Mg ii systems relative to an extrapolation of the exponential frequency distribution, as found at lower redshift. The trend of number density with redshift for these weak systems is broadly consistent with no evolution, though increased sample size could reveal underlying trends.

Separately, Bosman et al. (2017) performed an ultra-deep survey for Mg ii along the line of sight to ULAS1120+0641, also covered in our sample. They recover the two systems in our sample, and further recover three systems with ${W}_{r}\lt 0.3$ Å at $z\gt 6$ not detected by our search (because of our lower S/N, particularly in regions of strong and/or blended telluric absorption and emission). The number of weak systems uncovered in this sightline tentatively suggests that the frequency distribution may transition to a power-law slope at low column densities where our survey would have correspondingly low completeness.

In Paper I, we discussed the hypothesis that strong Mg ii absorption is linked closely with star forming galaxies, using the scaling relation presented in Ménard et al. (2011) to convert Mg ii equivalent widths into an effective contribution to the global star formation rate. This integral is dominated by the strongest absorbers in the sample, which peak strongly in number density near z ∼ 2–3, similar to the SFR rate density.

The conversion method relies on a correlation observed in SDSS-detected Mg ii systems between W_r and [O ii] luminosity surface density measured in the same fiber as the background QSO:

$$\begin{eqnarray}&&\langle {{\rm{\Sigma }}}_{{L}_{{\rm{O}}{\rm{II}}}}\rangle \propto {\left(\displaystyle \frac{{W}_{r}}{1\mathring{\rm A} }\right)}^{1.75}.\end{eqnarray} \tag{ 9 }$$

By integrating the Mg ii equivalent width distribution ${d}^{2}N/{dWdX}$, weighted by the function in Equation (9), one obtains a volumetric luminosity density of [O ii], which can then be converted into a star formation rate density using the [O ii] —SFR scaling relations of Zhu et al. (2009). As discussed in Paper I, one should keep in mind the possibility raised by López & Chen (2011) that the correlation in Equation (9) could arise from the decline in W_r with impact parameter, coupled with differential loss of [O ii] flux from the SDSS fiber, rather than a physical link between the SFR and Mg ii absorption strength. However, in light of the observed evolution in dN/dX for strong systems, the large velocity spreads seen in the strongest systems, and the link between star formation and Mg ii seen in individual galaxies (Bouché et al. 2007; Noterdaeme et al. 2010), we explore this possibility while acknowledging its possible limitations.

Figure 12 presents an updated version of this calculation, with smaller errors from our new and larger sample, and an additional point at $z\gt 6$ from our new high-redshift sightlines. Despite the caveats presented in Paper I about the methodology of the Mg ii -SFR conversion (López & Chen 2011), and the application of low-redshift scalings at these early epochs, the agreement between the Mg ii-inferred SFR and the values measured directly from deep fields remains remarkable. This suggests that at least the strongest Mg ii systems in our surveys derive their large equivalent widths (i.e., their velocity structure) from processes connected to star formation.

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The persistence of Mg ii at ${dN}/{dX}\approx 0.2$ absorbers per comoving pathlength at z ∼ 5.5–6.5 merits further examination, because it implies that some CGM gas was enriched very early in cosmic history—indeed, well before galactic stellar populations were fully relaxed. In Paper I we explored whether known high-redshift galaxy populations could plausibly account for the observed number of Mg ii systems, supposing that radial scaling relations of W_r and covering fraction measured at $z\lt 0.5$ apply at early times. These calculations essentially integrate down a mass function or a luminosity function to obtain a number density of halos, and then seed these with Mg ii gas using a radial prescription. Mg ii absorption statistics calculated in this way are sensitive to the lower limit of integration, as well as the value assumed for the low-mass (or faint-end) slope.

In that work, we first examined the predictions of a halo-occupation distribution model from Tinker & Chen (2010a). These authors integrate the halo mass function down to a fixed, redshift-independent cutoff below which it is assumed that galaxies do not harbor Mg ii in their CGM. The cutoff is chosen to match the evolution in number statistics below $z\lt 2$, but substantially underpredicts the Mg ii incidence rate at higher redshift. This likely results from the evolving mass function; since halos have lower masses at early times, a higher percentage of galaxies miss the (redshift-independent) mass cut.

In the red lines of Figure 13 we approximate this calculation, using the dark matter mass function extracted from the Illustris cosmological simulation (Torrey et al. 2015). We consider two models for halo cross section. The first, simplest model specifies that all halos have a constant absorption radius of 90 proper kpc, and geometric covering factor $\kappa =0.5$ within this volume (dashed red line). The second model (solid red line) assumes that a halo's absorption radius scales with mass, as in Churchill et al. (2013). We integrate the cross-section weighted mass function for each model down to a redshift-independent mass cut that matches the low redshift line densities of Nestor et al. (2005). These two scalings require mass cuts at 10¹¹ and 10¹⁰ solar masses, respectively. This model is only slightly simpler than that of Tinker & Chen (2010b), who also included a radial scaling of W_r and a varying absorption efficiency with halo mass. However, we verified that both methods reproduce the same basic result: strict allocation of Mg ii absorption by halo mass underpredicts dN/dX at high redshift.

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Alternatively, one can specify a parameterized halo geometry and then explore how far down one must set the minimum mass limit of integration to reproduce the flat trend of dN/dX for that model. The evolution of this minimum integration mass is shown in Figure 14, again for a fixed halo radius of R = 90 proper kpc (red points) and using Churchill's mass–radius scaling (black points). The minimum required halo mass declines by two orders of magnitude between redshifts $z=1\mbox{--}6$ when cross sections scale as in Churchill et al. (2013), and by an order of magnitude even when cross sections do not scale with mass. If these radial scalings apply at early times, then the observed incidence rate of Mg ii requires absorption from smaller mass halos in the early universe.

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While straight halo mass cuts are conceptually simple, we are not limited to this criterion. Numerous authors have investigated the density of halos as a function of both stellar mass and star formation rate (SFR). In fact, for Paper I we found that dN/dX was reproduced better at high redshift using weighted integrals of the luminosity function rather than the mass function. This was a purely empirical calculation, which used observed luminosity functions that required corrections for observations different redshifts, filters, and systematic survey completeness.

In Illustris, we have additional direct access to the star formation history of each simulated galaxy. Since the average SFR at fixed halo mass is larger at earlier times (Behroozi et al. 2013), we may integrate instead down to a fixed, redshift-independent SFR, which corresponds to lower dark matter halo mass at higher redshift. This achieves the desired effect of seeding smaller halos with Mg ii at early times.

The blue lines in Figure 13 show the result of this calculation for Illustris, using a constant-radius 90 pkpc halo for objects above SFR $\gt \,0.5\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (blue dashed line) and Churchill's mass-dependent radial scaling for objects with SFR $\gt \,0.02\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (solid blue line). As before, the minimum SFRs are selected to fit low redshift (Nestor et al. 2005) measurements. This methodology increases dN/dX by an order of magnitude or more at high redshifts, partially mitigating the discrepancy with a redshift-independent, fixed-mass bound on the integration. However, there is no single value for the SFR cutoff that fits all redshifts; the value chosen here is a compromise but predicts too many Mg ii absorbers at low redshift and slightly too few at early times.

5.2.1. Can Low-mass Galaxies Yield Enough Magnesium to Enrich Gaseous Halos?

At face value the small halo masses at high redshift in Figure 14—corresponding to even smaller stellar masses—require us to consider whether the these objects' stellar populations could plausibly produce enough magnesium to fill their intra-halo media at the radii required for observation.

If we define the galaxy yield η as the ratio of magnesium mass in the circumgalactic halo to the galaxy's stellar mass, then for a mass-independent halo radius

$$\begin{eqnarray}\eta & \equiv & \displaystyle \frac{{M}_{{Mg}}}{{M}_{* }}\\ & = & \displaystyle \frac{1}{{M}_{* }}\kappa \pi {R}^{2}{N}_{\mathrm{Mg}{\rm{II}}}{f}^{-1}{m}_{\mathrm{Mg}{\rm{II}}}\\ & = & 2.5\times {10}^{-5}{\left(\displaystyle \frac{R}{90\mathrm{kpc}}\right)}^{2}{\left(\displaystyle \frac{{M}_{* }}{{10}^{10}{M}_{\odot }}\right)}^{-1}\left(\displaystyle \frac{{N}_{\mathrm{Mg}{\rm{II}}}}{{10}^{13}\ {\mathrm{cm}}^{-2}}\right)\\ & & \times \left(\displaystyle \frac{\kappa }{0.5}\right){\left(\displaystyle \frac{f}{0.1}\right)}^{-1},\end{eqnarray} \tag{ 10 }$$

where κ represents the Mg ii absorption covering factor, R is the gaseous halo radius, f is the Mg ii ionization fraction, ${m}_{\mathrm{Mg}{\rm{II}}}=24.3u$ is the mass of a magnesium ion, and ${N}_{\mathrm{Mg}{\rm{II}}}={10}^{13}$ cm⁻² represents the typical column density of a modestly saturated absorption component. For our lowest-mass halos at $z\sim 6\mbox{--}7$, Figure 13 provides a lower integration limit of ${M}_{h}\sim {10}^{10.2}\,{M}_{\odot }$, corresponding to a stellar mass of ${10}^{8}\,{M}_{\odot }$ (Behroozi et al. 2013). For these inputs the required galactic yield of $\eta =0.0025$ is larger than the the IMF-weighted stellar magnesium yield of ∼0.001 (Saitoh 2017). Although numerical simulations do require that a significant fraction of the stellar yield is returned to the halo (Peeples et al. 2014), it is still the case that a ${10}^{8}\,{M}_{\odot }$ stellar population produces too little Mg ii to fill a halo to 90 kpc with observable Mg ii , by a factor of a few.

This discrepancy is reduced if we invoke a larger number of lower-mass halos, with radii scaled as $R\propto {M}^{\gamma }$ as calculated in Figure 13. Then, the required yield becomes

$$\begin{eqnarray}&&\eta =2.5\times {10}^{-5}{\left(\displaystyle \frac{{M}_{* }}{{10}^{10}{M}_{\odot }}\right)}^{2\gamma -1}\left(\displaystyle \frac{{N}_{\mathrm{Mg}{\rm{II}}}}{{10}^{13}\,{\mathrm{cm}}^{-2}}\right)\left(\displaystyle \frac{\kappa }{0.5}\right){\left(\displaystyle \frac{f}{0.1}\right)}^{-1}.\end{eqnarray} \tag{ 11 }$$

For $\gamma \sim 0.39$ (Churchill et al. 2013) and a ${M}_{* }\sim {10}^{7}{M}_{\odot }$ stellar mass (associated with the ${M}_{h}\sim {10}^{8}{M}_{\odot }$), $\eta =0.00011$, roughly an order of magnitude smaller than the IMF-weighted magnesium yield, leaving a comfortable margin to account for the difference between the strict stellar yield and the galactic yield of Mg mass ejected into the halo.

As a final, crude consistency test, we explore whether individual small halos have the correct combination of size and density to produce observable Mg ii absorption lines. For an average chord length through the halo of $\bar{l}=\tfrac{4}{3}R$, and the mass-scaled halo radius relation provided previously, one obtains an order-of-magnitude estimate of the column density:

$$\begin{eqnarray}&&{N}_{\mathrm{Mg}{\rm{II}}}=4\times {10}^{13}\ {\mathrm{cm}}^{-2}\left(\displaystyle \frac{\eta }{0.001}\right){\left(\displaystyle \frac{f}{0.1}\right)}^{-1}{\left(\displaystyle \frac{{M}_{* }}{{10}^{7}{M}_{\odot }}\right)}^{1-2\gamma }.\end{eqnarray} \tag{ 12 }$$

This column density is sufficient to produce saturated absorption, though in any realistic model of the halo one expects cool gas to be more highly structured (Crighton et al. 2015), leading to a lower covering fraction but slightly more variant total column densities in individual sightline samples.

Taken together, these results point to a modest tension for models where Mg ii is hosted at high redshift by massive galaxies with R ∼ 100 kpc gas envelopes; in contrast, models populating Mg ii in galaxies with ∼10–100× smaller stellar mass but 3–10× smaller gas envelopes comfortably accommodate the observations for reasonable heavy element yields.

Such objects would be qualitatively distinct from the L^* galaxies hosting Mg ii in the low-redshift universe, although they could evolve over time into such massive systems. If they are not yet dynamically relaxed, it may be the case that the observed Mg ii is not solely a by-product of winds from the halo's internal stellar population, but rather combines winds with material stripped through interactions during the initial assembly of the halo. In this case, some fraction of the heavy elements producing observed absorption may never have been in the halo center, reducing the requirements on wind transport during epochs where the Hubble time was $\lt 1\,\mathrm{Gyr}$.

The statistics presented in the previous section made reference to cosmological simulations of galaxy formation (specifically the Illustris simulation) but employed a simple analytic model to predict the likelihood of absorption by a given galaxy's CGM. This model utilizes covering fractions derived from low redshift observations to derive a binomial hit/miss rate, and has no power to predict equivalent widths or absorber kinematics (which are closely correlated).

These same simulations incorporate sophisticated hydrodynamic solvers and therefore can be used—at least in principle—to calculate line densities and frequency distributions directly without resort to assumptions about covering fraction. Indeed, these CGM statistics can serve as an independent check on the simulations' feedback prescriptions, beyond the present day galaxy mass function and star formation main sequence (which the simulations reproduce by design). In practice, however, computational limitations of the simulations make direct predictions of the cosmological evolution of cool gas quite difficult. In this section, we use a simple analysis of the Illustris simulation (Genel et al. 2014; Vogelsberger et al. 2014b; Sijacki et al. 2015) to demonstrate some of the challenges.

Figure 15 depicts the Mg ii absorber frequency distribution in our z = 3.15–3.81 redshift bin, along with the predicted frequency distributions found in three different runs at these redshifts used for resolution convergence testing in Illustris. The simulation boxes are 75 h⁻¹ comoving Mpc on a side, with 18.1, 2.3, and $0.3\times {10}^{9}$ hydro cells and a minimum cell size of 48, 98, and 273 pc, respectively (Vogelsberger et al. 2014b). The simulation tracks metallicity, temperature, density, and velocity for each cell in the simulation. We calculate absorption profiles in post-processing using the methodology and code described in Bird et al. (2015). In short, the ionization balance for each cell is calculated using the UV background spectrum of Faucher-Giguère et al. (2009) at the appropriate redshift, applied to a grid of ionization fractions calculated using CLOUDY (Ferland et al. 1998). Because Mg ii has an ionization potential of 1.1 Ryd, neutral hydrogen can shield it efficiently from ionizing radiation. Since its ionization potential is very close to that of hydrogen, we make a simple correction for self-shielding of absorbing structures using the formalism of Rahmati et al. (2013). Absorbers were identified via instances where the simulated spectra dip 5% below continuum values. Absorption troughs within 500 km s⁻¹ of each other were grouped together and identified as single absorbers. The equivalent widths of such absorbers were then calculated by integrating 500 km s⁻¹ past the most extremal components of each absorber.

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Figure 15 shows that the simulations produce too few Mg ii absorbers except for the weakest values of W_r. There is a marginal increase in the normalization of the predicted frequency distribution as the simulation resolution is increased. However, the slope of the simulated distribution remains steeper than the observed slope at all resolutions. Even at ${W}_{r}\sim 1$ Å, the universe has 3–10× more absorbers than the simulations; at larger equivalent widths the discrepancy spans many orders of magnitude, since the box contains few or no systems at ${W}_{r}\gtrsim 2$ Å.

The simulations' relatively coarse mesh resolution, required to simulate a large cosmological volume, likely contributes to this deficit of strong absorbers. The spatial resolution of a single particle can be approximated as

$$\begin{eqnarray}&&l=7.1\,\mathrm{kpc}{\left(\displaystyle \frac{{m}_{\mathrm{res}}}{{10}^{6}{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{\rho }{{10}^{-3}{\mathrm{cm}}^{-3}}\right)}^{-1/3},\end{eqnarray} \tag{ 13 }$$

where ${m}_{\mathrm{res}}$ is the gas mass resolution of the simulation and ρ is the gas density. Even with the aggressively high particle count in the Illustris simulation, the baryon mass resolution is ${m}_{\mathrm{res}}=1.26\times {10}^{6}{M}_{\odot }$, which gives an associated particle spatial resolution of ∼7 kpc. While the dense interstellar medium will have higher spatial resolution, the comparatively diffuse CGM can only be expected to have a spatial resolution of the order of tens of kiloparsecs. The coarse resolution can affect predictions of absorption statistics in two important ways.

First, limited simulation spatial resolution may suppress important small scale fluctuations in density, temperature, and velocity. In the case of the CGM in Illustris-1, any density/temperature fluctuations or bulk/turbulent velocity fluctuations with spatial scales less than ∼10 kpc at typical CGM gas densities will not be captured. For the lower resolution Illustris-2 and Illustris-3 simulations, the CGM spatial resolution is degraded further. Indeed the higher resolution runs exhibit an increase in the overall normalization of the Mg ii frequency distribution, indicating that numerical convergence has not yet been attained in regions producing Mg ii absorption. Moving to yet higher resolution may lead to a continued increase in the overall normalization of the Mg ii frequency distribution as smaller-scale density fluctuations continue to be resolved.⁸

Evidence for unresolved simulated CGM velocity substructure comes from the deficit of strong absorbers. In these systems, individual absorption components are saturated, so the Mg ii equivalent width is a proxy for velocity dispersion. Simple tests show that the Illustris Mg ii absorbers easily meet the column density criterion for saturation. They simply did not have enough velocity substructure to generate large equivalent widths. The existence of unresolved simulated CGM velocity substructure could easily follow from the limited spatial resolution employed in Illustris. As with density fluctuations, bulk/turbulent velocity distributions on spatial scales less than a few cell sizes (i.e., tens of kpc) are unlikely to be captured.

The second and more subtle issue concerns the position of Mg ii absorbing gas in the density–temperature plane. As a ubiquitous constituent of the CGM, Mg ii traces material in the transition range between the IGM and the ISM. To simplify and speed calculations in these regions with short cooling timescales, most numerical codes also transition somewhere in this density regime to a subgrid physics model that will necessarily compromise some of their predictive power for studying gas physics. Figure 16 shows the phase diagram in the temperature–density plane for Illustris, with a color scale indicating the cross-section weighted Mg ii distribution function in different parts of the plane (i.e., the fraction of all Mg ii cross section in the box contained within a bounded logarithmic interval of n and T). More than 80% of the aggregate Mg ii cross section in Illustris resides within the contour defined by an ionization fraction ${f}_{\mathrm{Mg}{\rm{II}}}=0.1$. Here gas is either generally under-resolved (Equation (13)) or has transitioned onto an effective equation-of-state implemented in Illustris (represented by a thin line at lower right for $n\gt 0.13$ cm⁻³).

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Particles on the effective equation-of-state are modeled using a two phase medium of cool clouds embedded in a hot tenuous phase, and these particles are assigned a star formation rate and associated IMF. In practice, these cells are treated as current or future ISM constituents, and simplified in exactly the region where the phase diagram indicates that Mg ii should be strongest in the CGM.

These shortcomings could possibly be mitigated for the study of individual absorbers by using a galaxy scale simulation with similar resolution and/or refinement (Shen et al. 2012; Churchill et al. 2015; Muratov et al. 2015). By transitioning to subgrid physics at smaller scales these smaller boxes track CGM gas physics with higher fidelity but are less useful for studying cosmological statistics and number counts.

As an intermediate step, we tested whether the observed equivalent width distribution could be reproduced by artificially inflating the line-of-sight velocity dispersions of Illustris Mg ii in post-processing. We find that this can be achieved by uniformly broadening the Voigt profiles of simulation particles contributing to our simulated spectra, effectively adding an additional velocity dispersion associated with unresolved bulk flows of $b\sim 40$ km s⁻¹ (Figure 17). This procedure has no effect on the equivalent width of unsaturated profiles, but for saturated systems, W_r is increased by spreading the absorption over a wider velocity range.

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The driving source for this unresolved turbulence is left unspecified. However, the strongest (${W}_{r}\gt 1$ Å) absorbers that most clearly reveal unresolved velocity substructure peak in incidence at z = 2–3, coincident with the global star formation rate (Figure 12).

Although this prescription works for any single redshift, Illustris also predicts redshift evolution in the overall normalization of the frequency distribution, in contrast to the observations (Figure 18). Yet despite these detailed discrepancies, the simulations do produce approximately the correct total number of Mg ii absorbers—they simply allocate too many to low values of W_r. A natural interpretation is that Mg and even Mg ii is broadly being produced and distributed in a physically plausible spatial pattern within the simulation box, but is insufficiently stirred. The large-volume simulations would then provide accurate overall number counts, but smaller volumes (Joung et al. 2012; Armillotta et al. 2016; Brüggen & Scannapieco 2016) will be the most promising avenue to resolve outstanding questions about the micro-physics of enrichment, turbulence, and the interaction of intra-halo gas with material flowing into and out of the central disk.

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