ALMA CENSUS OF FAINT 1.2 mm SOURCES DOWN TO ∼ 0.02 mJy: EXTRAGALACTIC BACKGROUND LIGHT AND DUST-POOR, HIGH-z GALAXIES

The four maps of the field data were obtained by our ALMA single-pointing observations. Two out of the four maps were taken in the ALMA cycle-0 Band 6 observations for the spectroscopically confirmed Lyα emitter (LAE) at z = 6.595, Himiko (Ouchi et al. 2013) and an LAE at z = 6.511, NB921-N-79144 (Ouchi et al. 2010, PI R. Momose). See Ono et al. (2014) for a summary of these observations. We also utilize two maps of newly obtained (ALMA cycle 2) field data of Bands 6 and 7 taken for spectroscopically confirmed LAEs at z = 5.7 (NB816-S-61269; Ouchi et al. 2008) and at z = 7.3 (NB101-S-2904; Konno et al. 2014, M. Ouchi et al. 2016, in preparation). The NB816-S-61269 observations were carried out in the ALMA program of #2012.1.00602.S (PI: R. Momose) on 2014 May 20, Jun 19, and July 7 with 43 12 m antennae array in the range of 18–650 m baseline. The FWHM of the primary beam was 206. The available 7.5 GHz bandwidth with four spectral windows was centered at an observed frequency of 290.8 GHz (i.e., ∼1.03 mm). J0217+0144 and J2258−279 were observed as a flux calibrator, while J0006−0623 was used for bandpass calibrator. Phase calibration was generally performed by using observations of J0217+0144 and J0215−0222. The total on-source observed time was ∼2.3 hr. The NB101-S-2904 data were taken in the ALMA program of #2012.1.00088.S (PI: M. Ouchi) on 2014 July 22 and August 6, 7, 14, and 18, with 55 12 m antennae array in the extended configuration of 18–1300 m baseline. The FWHM of the primary beam was 253. The center of the observed frequency is 237.9 GHz (i.e., ∼1.26 mm). J0238+166, J2258−279, and J0334−401 were used for primary flux calibrators. Bandpass and phase calibrations were performed with J0241−0815/J0238+1636 and J0215−0222/J0217+0144, respectively. The total on-source integration time was ∼5.2 hr.

To increase the number of ALMA sources, we make full use of ALMA archival data of cycles 0 and 1 that became public by 2015 June. We select all of the available ALMA Band 6/7 data sets fulfilling the following three criteria: (1) the 1σ noise level is ≲100 μJy beam⁻¹ for the continuum maps; (2) no bright (≥50σ) sources like active galactic nucleus (AGN), QSO, and SMG are included in the maps; and (3) the galactic latitude is high enough to avoid Galactic objects. The second criterion is important to reduce the chance for selecting residual side lobes of the bright sources that might remain even after the CLEAN algorithm on the Common Astronomy Software Applications (casa; McMullin et al. 2007) package. Moreover, the second criterion significantly reduces the systematic uncertainties from galaxy clustering because it is reported that there is a clear excess of submm number counts around bright sources such as AGNs and SMGs (Silva et al. 2015; Simpson et al. 2015a).

2.2.1. Field Data

2.2.2. Cluster Data

Basically, the data are reduced with CASA version 4.3.0 in a standard manner with the scripts used for the data reduction provided by the ALMA observatory. In this process, we also use previous CASA versions from 3.4 to 4.2.2, if we find problems in the final image that the noise level is significantly higher than the calibrated products provided by the ALMA observatory, or if there remain striped patterns. Exceptionally for the cycle 0 data with the high noise level or the patterned noise, we use the calibrated data produced by the ALMA observatory. Similarly for the data of ALMA #2011.0.00648.S (PI: Ohta), we use a re-calibrated data provided by Seko et al. because they find that the coordinates of a phase calibrator of #2011.0.00648.S is wrong (∼03), which causes positional offsets (Hatsukade et al. 2015; A. Seko et al. 2016, in preparation).

Our CASA reduction has three major steps: bad data flagging, bandpass calibration, and gain calibration with flux scaling. In the first step, we remove the data of the shadowed antennas and the edge channels of spectral windows. We also do not use the unreliable data such with jumps or a low phase/amplitude gain. We apply the flaggings that are shown in the scripts, but no additional flaggings. This is because we find that the flaggings in the scripts are good enough for our scientific goals. The second step is the bandpass calibration. After we calibrate the phase time variation on the bandpass calibrator scan, we obtain the bandpass calibration in the phase and amplitude. The final step is the gain calibration. We estimate the time variations of the phase and amplitude on the phase calibrators, and then transfer the calibration to the target source by applying the linear interpolation with the results of the phase calibrators. The water vapor radiometer is used for the correction of the short-time phase variations that cannot be traced by the phase calibrators. We estimate flux scaling factors, using solar system objects and bright QSOs with the flux models of "Bulter-JPL-Horizons 2012" and ALMA Calibrator Source Catalog, respectively. The difference of the opacity between the flux calibrators and the target sources is corrected with the system noise temperature (T_sys) measurements. The systematic flux uncertainty is typically 10% in bands 6 and 7 (ALMA proposer's guide⁴ ).

We perform Fourier transformation for the original UV-data to create "dirty" maps. The continuum maps are made using all the line-free channels of the four spectral windows. We process the maps with the auto CLEAN algorithm down to the depth of 3σ noise levels of the dirty maps, using natural weighting. The final cleaned maps achieve angular resolutions of 045 × 033 −188 × 090. The sensitivities of these ALMA maps range from 8.5 to 95.2 μJy beam⁻¹ before the primary beam corrections.

Because the observations of the cluster data were taken place in four different epochs, we recalculate the data weights with the statwt task in CASA based on its visibility scatters which include the effects of integration time, channel width, and systematic temperature. After applying the data weights to the data sets, we combine these two data sets with the concat task in CASA. Since the mosaic mapping observations were conducted for the cluster, the noise levels vary by positions. To estimate the noise levels of the cluster data, we divide the cluster's mosaic map in nine regions as shown in Figure 1. The data of the nine regions achieve the 1σ noise levels in the range of 38.4–40.6 μJy.

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We conduct source extraction for our ALMA maps before primary beam corrections with SExtractor version 2.5.0 (Bertin & Arnouts 1996). The source extraction is carried out in high-sensitivity regions. For the single-pointing maps presented in Sections 2.1 and 2.2.1, we use the regions with the primary beam sensitivity greater than 50%. For the mosaic map shown in Section 2.2.2, we perform source extraction where the relative sensitivity to the deepest part of the mosaic map is greater than 50%.

We identify sources with a positive peak count above the 3.0σ level, assuming the sources are not resolved out in our ALMA maps. The catalog of these sources is referred to as the 3.0σ detection catalog. From our 3.0σ detection catalog, we remove the objects located at the map centers that are the main science targets of the archived-data ALMA observations.

The 3.0σ detection catalog should include many spurious sources, because the detection is simply determined with a peak pixel count. To evaluate the number of the spurious sources, we conduct negative peak analysis (Hatsukade et al. 2013; Ono et al. 2014; Carniani et al. 2015) for the deep (A), medium deep (B), and cluster (C) data sets. We multiply −1 to each pixel value of our ALMA maps, and perform the source extraction for negative peaks in the same manner as those in Section 3.1. The numbers of negative and positive peaks in our ALMA maps are shown in Figure 2. The excess of the positive to negative peak numbers is regarded as the real source numbers. We model the negative and positive peak distributions of Figure 2 with the functions of signal-to-noise ratio S/N,

$$\begin{eqnarray}&&{N}_{{\rm{np}}}^{{\rm{c}}}={a}_{{\rm{n}}}\times {10}^{-{b}_{{\rm{n}}}\times {\rm{S/N}}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{N}_{{\rm{pp}}}^{{\rm{c}}}={a}_{{\rm{n}}}\times {10}^{-{b}_{{\rm{n}}}\times {\rm{S/N}}}+{a}_{{\rm{p}}}\times {10}^{-{b}_{{\rm{p}}}\times {\rm{S/N}}},\end{eqnarray} \tag{ 2 }$$

where ${N}_{{\rm{np}}}^{{\rm{c}}}$ and ${N}_{{\rm{pp}}}^{{\rm{c}}}$ are the cumulative numbers of negative and positive peaks, respectively, and a_n, b_n, a_p, and b_p are free parameters.

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We estimate spurious source rates f_sp defined by the ratio of negative to positive peak numbers,

$$\begin{eqnarray}&&{f}_{{\rm{sp}}}({\rm{S/N}})=\displaystyle \frac{{N}_{{\rm{np}}}^{{\rm{c}}}({\rm{S/N}})}{{N}_{{\rm{pp}}}^{{\rm{c}}}({\rm{S/N}})}.\end{eqnarray} \tag{ 3 }$$

We evaluate f_sp with the best-fit functions of Equations (1) and (2), and present f_sp in Figure 3.

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Figures 3 indicates that the field data of deep (A) and medium deep (B) have an almost identical distribution of the spurious source rates, while the cluster data (C) shows spurious source rates higher than (A) and (B) at a given S/N. This result would suggest that spurious source rates do not depend on the data depth, but the mapping modes. The complex distribution of the depths in the mapping data of (C) would have the relation of S/N and a spurious source rate that is different from the smooth distribution of the depth in the single-pointing data of (A) and (B). We use sources down to an S/N whose spurious source rate is ≲70%, and adopt 3.4, 3.4, and 3.9σ levels for our selection limits of the data (A), (B), and (C), respectively. Applying these selection limits to our 3.0σ detection catalog, we obtain a source catalog consisting of 133 sources: 122 from the field data (A and B) and 11 from the cluster data (C).

The source catalog of our 133 faint ALMA sources is presented in Table 3.

Table 3.  Source Catalog

ID	Map ID	R.A.(J2000)	decl.(J2000)	${S}_{{\lambda }_{{\rm{obs}}}}^{{\rm{obs}}}$	${S}_{1.2\;{\rm{mm}}}^{{\rm{corr}}}$	S/N	fsp
				(mJy)	(mJy)
(1)	(2)			(3)	(4)	(5)	(6)
1	1	337.049805	−35.169643	0.06 ± 0.02	0.07 ± 0.02	3.6	0.57
2	1	337.050232	−35.164864	0.41 ± 0.01	0.68 ± 0.07	39.3	0.00
3	1	337.050293	−35.170044	0.07 ± 0.02	0.09 ± 0.02	4.0	0.34
4	1	337.053314	−35.165119	0.06 ± 0.01	0.12 ± 0.02	5.7	0.00
5	1	337.054413	−35.167576	0.05 ± 0.01	0.07 ± 0.02	4.1	0.27
6	1	337.054718	−35.166332	0.05 ± 0.01	0.06 ± 0.02	3.5	0.66
7	2	148.887314	18.150942	0.05 ± 0.01	0.04 ± 0.02	3.6	0.63
8	2	148.891403	18.151211	0.06 ± 0.02	0.06 ± 0.02	3.4	0.71
9	3	34.446007	−5.008159	0.05 ± 0.01	0.05 ± 0.02	3.5	0.70
10	3	34.446648	−5.007715	0.05 ± 0.01	0.04 ± 0.02	3.4	0.70
11a	4	34.437332	−5.492185	0.06 ± 0.02	0.04 ± 0.01	3.7	0.53
12a	4	34.437649	−5.494778	0.06 ± 0.02	0.03 ± 0.01	3.5	0.67
13a	4	34.437805	−5.494339	0.06 ± 0.01	0.03 ± 0.01	3.8	0.48
14a	4	34.439056	−5.494986	0.06 ± 0.02	0.03 ± 0.01	3.6	0.60
15a	4	34.440075	−5.494945	0.07 ± 0.02	0.04 ± 0.01	3.6	0.63
16	5	121.593948	−41.376675	0.10 ± 0.03	0.08 ± 0.04	3.6	0.60
17	5	121.596870	−41.375671	0.06 ± 0.02	0.08 ± 0.02	3.6	0.63
18	5	121.599480	−41.373417	0.08 ± 0.02	0.10 ± 0.03	3.7	0.55
19	5	121.599564	−41.375568	0.07 ± 0.02	0.08 ± 0.03	3.6	0.61
20	5	121.600471	−41.375629	0.08 ± 0.02	0.09 ± 0.03	3.6	0.60
21	7	336.940674	−35.119549	0.11 ± 0.03	0.13 ± 0.04	3.8	0.48
22	8	201.001526	27.413574	0.10 ± 0.03	0.11 ± 0.04	3.5	0.70
23	8	201.001526	27.413574	0.10 ± 0.03	0.11 ± 0.04	3.5	0.70
24	8	201.001526	27.413574	0.10 ± 0.03	0.11 ± 0.04	3.5	0.70
25	9	325.760986	6.904217	0.13 ± 0.04	0.17 ± 0.05	3.5	0.68
26	9	325.762177	6.902022	0.14 ± 0.04	0.17 ± 0.05	3.6	0.62
27	9	325.763336	6.905892	0.08 ± 0.02	0.10 ± 0.03	3.4	0.72
28	9	325.767029	6.902850	0.15 ± 0.04	0.16 ± 0.05	3.6	0.60
29	10	352.281616	−3.033988	0.13 ± 0.04	0.12 ± 0.04	3.4	0.71
30	10	352.284851	−3.031080	0.16 ± 0.03	0.16 ± 0.03	5.9	0.00
31	10	352.285248	−3.030138	0.17 ± 0.04	0.17 ± 0.04	4.6	0.08
32	10	352.285309	−3.035150	0.11 ± 0.03	0.10 ± 0.03	3.8	0.47
33	11	200.929321	27.340736	0.15 ± 0.04	0.13 ± 0.04	3.9	0.40
34	12	150.547440	1.910628	0.13 ± 0.04	0.13 ± 0.05	3.7	0.55
35	13	351.831116	6.267834	0.13 ± 0.03	0.12 ± 0.03	4.2	0.22
36	13	351.833496	6.267574	0.16 ± 0.04	0.15 ± 0.04	3.7	0.53
37	14	13.760888	1.771644	0.10 ± 0.03	0.08 ± 0.02	4.0	0.35
38	14	13.761566	1.768838	0.16 ± 0.04	0.12 ± 0.04	3.7	0.52
39	14	13.763390	1.770106	0.11 ± 0.03	0.08 ± 0.03	3.7	0.52
40	14	13.764014	1.770272	0.13 ± 0.03	0.09 ± 0.03	3.7	0.50
41	15	149.775772	1.736944	0.16 ± 0.03	0.18 ± 0.04	5.1	0.02
42	15	149.775955	1.738510	0.15 ± 0.04	0.17 ± 0.05	3.9	0.41
43	15	149.776794	1.736661	0.09 ± 0.03	0.07 ± 0.03	3.6	0.63
44	18	337.254547	14.950516	0.17 ± 0.05	0.14 ± 0.04	3.8	0.48
45	18	337.256073	14.954236	0.19 ± 0.03	0.21 ± 0.03	6.1	0.00
46	19	149.722565	2.606706	0.11 ± 0.03	0.10 ± 0.04	3.4	0.70
47	19	149.724045	2.602218	0.21 ± 0.05	0.28 ± 0.07	4.1	0.26
48	20	149.792572	2.103762	0.12 ± 0.03	0.10 ± 0.04	4.0	0.30
49	21	32.553406	−4.939113	0.12 ± 0.03	0.11 ± 0.03	3.9	0.37
50	21	32.553787	−4.939454	0.11 ± 0.03	0.15 ± 0.03	3.7	0.56
51	21	32.555397	−4.937203	0.13 ± 0.03	0.13 ± 0.04	3.7	0.54
52a	22	150.127991	2.288839	0.13 ± 0.04	0.08 ± 0.03	3.4	0.72
53	23	359.017120	19.607481	0.10 ± 0.03	0.08 ± 0.03	3.4	0.71
54	24	150.406784	2.034714	0.14 ± 0.04	0.12 ± 0.04	3.5	0.64
55	25	150.393936	1.998264	0.15 ± 0.04	0.16 ± 0.05	3.7	0.56
56	26	34.100346	−5.154171	0.15 ± 0.04	0.14 ± 0.05	3.5	0.64
57	26	34.100658	−5.156616	0.16 ± 0.05	0.17 ± 0.06	3.5	0.66
58	26	34.102978	−5.155593	0.15 ± 0.04	0.15 ± 0.05	3.4	0.70
59	27	34.034069	−5.104988	0.17 ± 0.04	0.18 ± 0.06	3.8	0.45
60	28	34.472076	−4.711596	0.26 ± 0.07	0.27 ± 0.09	3.5	0.65
61	28	34.473747	−4.713997	0.16 ± 0.04	0.20 ± 0.06	3.6	0.62
62	29	34.392979	−5.043759	0.17 ± 0.04	0.19 ± 0.06	3.7	0.51
63	29	34.396526	−5.042582	0.21 ± 0.06	0.23 ± 0.07	3.6	0.59
64	29	34.396526	−5.046781	0.25 ± 0.07	0.26 ± 0.09	3.4	0.72
65	31	34.612003	−4.588018	0.25 ± 0.07	0.28 ± 0.08	3.5	0.69
66	31	34.612686	−4.583063	0.36 ± 0.07	0.43 ± 0.09	4.9	0.04
67	31	34.613739	−4.588505	0.31 ± 0.09	0.31 ± 0.10	3.5	0.68
68	31	34.613930	−4.585264	0.19 ± 0.06	0.18 ± 0.06	3.5	0.70
69	33	34.272991	−4.859830	0.29 ± 0.08	0.34 ± 0.11	3.5	0.68
70	33	34.274231	−4.860201	0.32 ± 0.09	0.43 ± 0.12	3.6	0.59
71	33	34.274761	−4.859237	0.26 ± 0.07	0.32 ± 0.09	3.8	0.46
72	34	34.305080	−5.067344	0.29 ± 0.07	0.39 ± 0.10	3.9	0.41
73	34	34.305538	−5.069685	0.23 ± 0.06	0.33 ± 0.09	3.6	0.61
74	34	34.305553	−5.067293	0.29 ± 0.07	0.22 ± 0.09	4.2	0.19
75	34	34.307556	−5.071889	0.33 ± 0.09	0.48 ± 0.12	3.6	0.61
76	34	34.309822	−5.069085	0.29 ± 0.08	0.23 ± 0.11	3.5	0.65
77a	35	230.738800	−0.125928	0.35 ± 0.10	0.23 ± 0.07	3.5	0.69
78a	35	230.739777	−0.125576	0.28 ± 0.08	0.20 ± 0.06	3.6	0.59
79a	35	230.743118	−0.127994	0.28 ± 0.08	0.18 ± 0.06	3.5	0.66
80	36	34.327038	−5.131583	0.21 ± 0.06	0.27 ± 0.09	3.8	0.48
81	37	34.401154	−5.152167	0.33 ± 0.10	0.37 ± 0.15	3.4	0.72
82	37	34.401531	−5.154098	0.30 ± 0.08	0.34 ± 0.13	3.7	0.55
83	37	34.403275	−5.149917	0.36 ± 0.11	0.61 ± 0.18	3.4	0.71
84a	38	231.036255	−0.176025	0.39 ± 0.11	0.22 ± 0.08	3.5	0.67
85a	38	231.037949	−0.178142	0.22 ± 0.06	0.15 ± 0.04	3.7	0.54
86	40	34.414261	−5.201880	0.34 ± 0.10	0.56 ± 0.16	3.5	0.65
87	40	34.419544	−5.199385	0.31 ± 0.08	0.52 ± 0.14	3.8	0.46
88	41	149.590530	2.807018	0.28 ± 0.07	0.23 ± 0.05	4.2	0.21
89	41	149.591949	2.805352	0.26 ± 0.07	0.21 ± 0.05	4.0	0.31
90	42	150.688171	2.681118	0.37 ± 0.07	0.26 ± 0.06	5.4	0.01
91	45	34.194702	−5.056259	0.43 ± 0.12	0.39 ± 0.15	3.5	0.71
92	45	34.197121	−5.059778	0.31 ± 0.08	0.37 ± 0.10	3.7	0.55
93	46	150.409714	2.358208	0.30 ± 0.09	0.32 ± 0.09	3.5	0.71
94	47	149.770050	1.804996	0.32 ± 0.09	0.31 ± 0.10	3.4	0.72
95	48	34.411190	−4.745051	0.30 ± 0.08	0.14 ± 0.09	3.8	0.44
96	48	34.411701	−4.745893	0.31 ± 0.08	0.34 ± 0.09	4.0	0.31
97	49	150.600494	2.444365	0.47 ± 0.13	0.53 ± 0.14	3.6	0.59
98	51	34.440269	−4.912953	0.39 ± 0.11	0.40 ± 0.12	3.6	0.60
99	51	34.441357	−4.912045	0.32 ± 0.08	0.29 ± 0.09	3.9	0.43
100	51	34.441635	−4.913539	0.46 ± 0.13	0.45 ± 0.14	3.6	0.62
101	51	34.442795	−4.911066	0.53 ± 0.08	0.56 ± 0.11	6.2	0.00
102	51	34.443920	−4.909602	0.41 ± 0.12	0.35 ± 0.13	3.5	0.65
103	52	150.151459	1.751314	0.49 ± 0.14	0.50 ± 0.13	3.6	0.64
104	54	34.759140	−4.832398	0.32 ± 0.09	0.37 ± 0.11	3.5	0.71
105	55	150.022736	2.027032	0.53 ± 0.08	0.46 ± 0.08	6.8	0.00
106	57	34.141144	−5.232085	0.47 ± 0.13	0.42 ± 0.15	3.6	0.60
107	58	34.303791	−5.161064	0.47 ± 0.14	0.58 ± 0.16	3.4	0.73
108	59	150.201645	2.021740	0.57 ± 0.15	0.72 ± 0.22	3.7	0.53
109	60	34.250214	−4.804325	0.34 ± 0.09	0.34 ± 0.10	3.9	0.39
110	60	34.250927	−4.802006	0.37 ± 0.11	0.39 ± 0.13	3.5	0.68
111	61	34.744495	−4.855921	0.35 ± 0.10	0.27 ± 0.12	3.5	0.70
112	61	34.747295	−4.858103	0.37 ± 0.10	0.26 ± 0.11	3.8	0.45
113	62	150.757401	1.700786	0.50 ± 0.14	0.57 ± 0.21	3.5	0.70
114	62	150.758331	1.703298	0.42 ± 0.12	0.52 ± 0.17	3.6	0.60
115	62	150.760971	1.703856	1.01 ± 0.10	1.17 ± 0.18	9.7	0.00
116	62	150.762024	1.698676	0.57 ± 0.16	0.85 ± 0.24	3.5	0.69
117	62	150.762161	1.701653	0.34 ± 0.10	0.37 ± 0.14	3.5	0.68
118	62	150.762741	1.700690	0.43 ± 0.12	0.61 ± 0.17	3.6	0.62
119	62	150.763138	1.702965	0.47 ± 0.13	0.70 ± 0.19	3.6	0.62
120	64	34.587593	−5.318766	0.32 ± 0.09	0.21 ± 0.10	3.6	0.59
121	64	34.589180	−5.315989	0.61 ± 0.17	0.69 ± 0.21	3.5	0.65
122	65	34.387825	−5.220457	0.41 ± 0.11	0.42 ± 0.13	3.7	0.51
123	67	197.860764	−1.337225	0.25 ± 0.06	0.03b ± 0.01	4.3	0.53
124	67	197.862061	−1.342226	0.24 ± 0.06	0.00b ± 0.00	4.3	0.54
125	67	197.864731	−1.343368	0.27 ± 0.06	0.03b ± 0.01	4.7	0.32
126	67	197.869086	−1.337143	0.31 ± 0.08	0.08b ± 0.03	4.1	0.60
127	67	197.871590	−1.345783	0.30 ± 0.06	0.02b ± 0.01	5.1	0.20
128	67	197.874725	−1.326138	0.22 ± 0.05	0.02b ± 0.01	4.3	0.51
129	67	197.881638	−1.323127	0.26 ± 0.05	0.05b ± 0.02	4.8	0.29
130	67	197.882812	−1.356609	0.28 ± 0.07	0.05b ± 0.02	4.1	0.64
131	67	197.883881	−1.355241	0.30 ± 0.08	0.06b ± 0.03	4.0	0.65
132	67	197.884125	−1.334885	0.30 ± 0.07	0.13b ± 0.05	4.2	0.58
133	67	197.886185	−1.328756	0.24 ± 0.05	0.04b ± 0.01	5.1	0.19

Notes. (1) Source ID. (2) Map ID that corresponds to the one in Table 2. (3) Peak flux density of the SExtractor measurement (Section 3.1) with the primary beam correction at the observed wavelength. (4) The best estimate source flux density at 1.2 mm. The source flux is estimated with the 2D Gaussian-fitting routine of imfit in CASA (Section 3.4) with the primary beam correction and the flux scaling to 1.2 mm (Section 3). For the sources found in the cluster data, the lensing magnification corrections are also applied (Section 3.5). The lensing magnification factors are estimated at the ALMA flux peak positions. The error bar includes the random noise, 10% of ALMA system's flux measurement uncertainty, and the lensing magnification uncertainty (Section 4.1). (5) Signal-to-noise ratio of the peak flux density. (6) spurious source rate that is defined by Equation (3).

^aSource identified in the Band 7 map. Unless otherwise specified, the source is found in the Band 6 map. ^bThe lensing magnification is corrected. The ID 124 source has 0.0020 ± 0.0010 mJy for the best estimate source flux density at 1.2 mm with the primary beam and lensing magnification corrections.

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Note that A1689-zD1, an LBG at z = 7.5 behind A1689 (Watson et al. 2015), is not included in our source catalog because A1689-zD1 is located outside of the primary beam with a >50% sensitivity that is one of our selection criteria. Nevertheless, we have looked at the position of A1689-zD1 in our data, and found that there is a source with an S/N of 4 whose flux is the same as the one derived by Watson et al. (2015) within the ∼1σ uncertainty. We confirm the ALMA source detection at the position of A1689-zD1.

We estimate the detection completeness by Monte Carlo simulations. We put a flux-scaled synthesized beam into a map as an artificial source on a random position. These artificial sources have S/Ns ranging from 3.0 to 7.0 with an S/N step of 0.2 dex. If an artificial source is extracted within a distance of the synthesized beam size from the input position, we consider that the source is recovered. These completeness estimations are conducted in the data before primary beam corrections, because we perform the source extractions in the real data (Section 3.1) uncorrected for primary beam attenuations. For all of our ALMA maps, we iterate this process 100 times at an S/N bin. Figure 4 displays our completeness estimates for three sub-datasets. We fit the function of $1-\mathrm{exp}(a{\rm{S/N}}-b)$ to our completeness estimates, where a and b are free parameters. The best-fit functions are also shown in Figure 4.

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We examine the flux boosting that is caused by the confusions of the undetected faint sources (Austermann et al. 2009, 2010). We use the detected objects of the Monte Carlo simulations conducted for the completeness estimates, and measure the output flux densities at the input positions. We find that the ratios of the input to output flux densities are almost unity with a variance of only 5% over the wide S/N range of our ALMA maps. This small flux boosting is probably found because the angular resolution of ALMA is high. We thus conclude that the effect of flux boosting is negligibly small.

We estimate the source fluxes of our 133 objects from the integrated flux densities measured with the two-dimensional (2D) Gaussian-fitting routine of imfit in CASA. Because the integrated flux density values for low-S/N sources would include large systematic uncertainties originating from the spatially correlated positive noise at around the source position, we use the integrated flux densities of imfit only for the sources which are identified as "resolved" with an S/N of ≥5. For the rest of the sources, we use the peak fluxes of the imfit best-fit Gaussian measurements. We confirm that the peak flux values of imfit, S_imfit, agree with those of SExtractor, S_SEx, within the 1σ error, S_imfit/S_SEx = 0.93 ± 0.15.

Because the sources in the cluster data of A1689 would be distorted by the gravitational lensing effects, we make a low-resolution map of A1689 produced with a 1'' circular Gaussian UV-taper that is the same as the one produced by Watson et al. (2015). The taped beam size is 131 × 112. In this map, we perform the imfit flux measurements (as described above) at the source positions determined in the original map. Here we define flux measurements in the tapered and non-tapered maps as S_taper and S_{no taper}, respectively. We then obtain S_taper/S_{no taper} = 1.01 ± 0.19 on average, if we omit two sources showing signatures of source confusions by neighboring objects. However, the strongly lensed source of A1 shows a systematically high fraction of S_taper/S_{no taper} = 1.23. Thus, we decide to use S_taper for the cluster data of A1689, except for those two confusion sources. We also make low-resolution maps for some of the field data, and perform the same test of tapering. Because we find that S_taper and S_{no taper} are almost identical in these data, we use S_{no taper} for the field data.

We test the reliability of our flux measurements, performing flux recovery simulations. First, we select the typical data sets of ID22 and ID55 that have beam sizes of ∼08 and ∼12, corresponding to the two peaks of bimodal size distribution for our data. We then create 200 input sources of elliptical Gaussian model profiles with the uniform distribution of S/Ns in 3–7 and major axis sizes of 02 and 05. Here, the input source sizes are chosen from recent ALMA high-resolution studies. Ikarashi et al. (2014) and Simpson et al. (2015b) claim that the median sizes of individual submm sources are ∼02–03 in FWHM. Simpson et al. (2015b) also find that the stacked image of submm sources with a low S/N of 4–5 has a size of FWHM $0\buildrel{\prime\prime}\over{.} {35}_{-0.10}^{+0.17}$. Thus, we investigate the model sources of 02 and 05 that are taken from the median individual source size and the 1σ upper limit stacked source size. We add these model sources to the ID22 and ID55 maps at random positions. Finally, we obtain flux densities of the model sources in the same manner as our flux measurements for the real sources, and compare these flux measurements with the input source fluxes. In the case of the 02 input source size, we find that the flux recovery ratio (defined by the ratio of the output to input fluxes) is 1.0 ± 0.1 (1.0 ± 0.2) for the ID22 (ID55) data. Similarly, in the case of the 05 input source size, we obtain 0.8 ± 0.1 (0.9 ± 0.2) for the ID22 (ID55) data. These results suggest that the flux recovery ratios are nearly unity within the 1–2σ uncertainties, and that our flux measurements are reliable.

Note that missing fluxes of interferometric observations are negligible. ALMA proposer's guide⁵ shows that there exist missing flux effects for sources with ≃20–30 at Band 6/7 for the shortest baseline configuration that we use. The source sizes of ≃20–30 are significantly larger than our ALMA sources whose angular sizes are ≲05, and the missing fluxes should be negligible.

We construct a mass model for the cluster data (C) of A1689 at z = 0.183. We make the mass model with the parametric gravitational lensing package GLAFIC (Oguri 2010) in the same manner as Ishigaki et al. (2015). Our mass model consists of three types of mass distributions: cluster scale halos, cluster member galaxy halos, and external perturbation. For evaluating the mass distributions, we make a galaxy catalog of A1689, conducting source extractions in the optical images of the g₄₇₅, r₆₂₅, and i₇₇₅ bands taken with the Hubble Space Telescope (HST). The cluster scale halos are estimated with the three brightest member galaxies in the core of the cluster. The cluster member galaxies are selected by the color criteria of

$$\begin{eqnarray}{r}_{625} & \lt & 24,\\ {g}_{475}-{r}_{625} & \lt & -\displaystyle \frac{1}{18}({r}_{625}-24)+1.3,\\ {g}_{475}-{r}_{625} & \gt & -\displaystyle \frac{1}{18}({r}_{625}-24)+0.7,\end{eqnarray} \tag{ 4 }$$

where the source extraction and photometry are carried out with SExtractor. The external perturbation is calculated with the theoretical model under the assumption that the perturbation is weak (e.g., Kochanek 1991).

Using the positions of the multiple images presented in the literature (Coe et al. 2010; Diego et al. 2015), we optimize free parameters of the mass profiles based on the standard χ² minimization to determine the best-fit mass model. Figure 1 presents the best-fit mass model. In the best-fit mass model, all of the offsets between the model and observed positions of multiple images are within 10 in the image plane. We then calculate magnification factors μ at the positions of our sources. For the μ estimates, we assume our source redshift of z = 2.5, which is the same as the one used in the flux scaling shown in Section 3. This assumption does not change our statistical results, because most of mm sources in the target fields at the high-galactic latitude should reside at z > 1, where the μ values do not depend much on a redshift. Here we calculate the magnification factors for sources at z = 7.5 that correspond to the redshift of A1689-zD1, and evaluate the differences between the magnification factors for sources at z = 2.5 (μ_2.5) and z = 7.5 (μ_7.5). We find that the average difference, ${\rm{\Delta }}\mu \equiv ({\mu }_{7.5}-{\mu }_{2.5})/{\mu }_{2.5}$, is 0.36 ± 0.25 for our 11 sources in A1689. Although the difference is not large, we include the lensing magnification difference of 0.36 into the errors of the intrinsic flux estimates for our 11 sources in A1689. Accordingly, we propagate these uncertainties to our major results such as number counts shown in Section 4.

We estimate survey areas of our data of A, B, and C, and present these estimates in Figure 5. The survey areas are defined by the high-sensitivity regions that are detailed in Section 3.1. Because the sensitivities of our ALMA maps are not spatially uniform, the survey areas depend on the flux densities.

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In the field data of A and B, the sensitivity of the primary beam decreases with increasing radius from a map center. In other words, the detectable intrinsic flux densities (corrected for the primary beam attenuation) increase from the center to the edge of the data. First, we find the radius where sources with the intrinsic flux densities can be detected at our selection limit S/N. Then, we calculate areas with the radius given in each map, and sum up these areas to obtain the total survey areas.

In the cluster data of C, the spatial distribution of the sensitivity is not expressed by a simple function of radius, because the cluster data are taken by mosaic mapping. Moreover, there are cluster-lensing magnification effects that allow us to detect intrinsically faint sources. We make an effective magnification map, multiplying at each position the sensitivity and the magnification factor estimated with the mass model of Section 3.5. We then calculate the survey area of the C data where a source with an intrinsic flux density can be observed above the selection limit.

We derive differential number counts at 1.2 mm.

For the number counts, we use the 122 sources identified in the ALMA Band 6 maps covering the 1.2 mm band, because there is a possibility that the 11 sources in Band 7 would include unknown systematics in the flux scaling to 1.2 mm fluxes (Section 3). To derive the number counts, we basically follow the methods used in the previous ALMA studies (Hatsukade et al. 2013; Ono et al. 2014; Carniani et al. 2015). A contribution from an identified source with an intrinsic flux density of S to the number counts, ξ, is determined by

$$\begin{eqnarray}&&\xi (S)=\displaystyle \frac{1-{f}_{{\rm{sp}}}(S)}{C(S){A}_{{\rm{eff}}}(S)},\end{eqnarray} \tag{ 5 }$$

where C is the completeness, and A_eff is the survey area. For C, A_eff, and f_sp (Equation (3)), we use the values derived in Section 3. Then, we calculate a sum of the contributions for each flux bin,

$$\begin{eqnarray}&&n(S)=\displaystyle \frac{{\rm{\Sigma }}\xi (S)}{{\rm{\Delta }}\mathrm{log}\;S},\end{eqnarray} \tag{ 6 }$$

where Δ log S is the scaling factor of 0.25 for the 1 dex width logarithmic differential number counts. We estimate the errors that include both Poisson statistical errors of the source numbers and flux uncertainties of the identified sources. Because the numbers of our sources are small, we use the Poisson uncertainty values presented in Gehrels (1986) which are applicable for the small number statistics. The flux uncertainties are composed of the random noise and the ALMA system's flux measurement uncertainty whose typical value is ∼10% (Section 2.3). Moreover, for our 11 sources in the A1689 data, the intrinsic flux estimates include the lensing magnification uncertainties (see Section 3.5). To evaluate these flux uncertainties, we perform Monte Carlo simulations. We make a mock catalog of the faint ALMA sources whose flux densities follow the Gaussian probability distributions whose standard deviations are given by the combination of the random noise and the system (plus the lensing magnification) uncertainties. We obtain the number counts for each mock catalog in the same manner as those for our real sources. We repeat this processes 1000 times, and calculate the standard deviation of the number counts per flux bin caused by the flux uncertainties. Combining the standard deviation of the flux uncertainties with the Poisson errors, we finally obtain the 1σ uncertainties of the number counts.

The differential number counts and the associated 1σ uncertainties are shown in Figure 6 and Table 4. With the technique same as those deriving the differential number counts, we estimate the cumulative number counts that are summarized in Table 4.

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Table 4.  Differential and Cumulative Number Counts at 1.2 mm

S	log(n)	Ndiff
(mJy)	([Δ log S = 1]−1 deg−2)
(0.002)	${7.1}_{-7.1}^{+0.5}$	1
0.020	${5.5}_{-0.5}^{+0.3}$	2
0.036	${5.6}_{-0.4}^{+0.3}$	6
0.063	${5.3}_{-0.4}^{+0.2}$	15
0.112	${5.1}_{-0.2}^{+0.2}$	25
0.200	${4.9}_{-0.2}^{+0.1}$	27
0.356	${4.7}_{-0.2}^{+0.1}$	29
0.633	${4.2}_{-0.2}^{+0.2}$	15
1.124	${3.4}_{-1.3}^{+0.5}$	2
S	log(n(>S))	Ncum
(mJy)	(deg−2)
(0.002)	${6.6}_{-1.1}^{+0.5}$	122
0.015	${5.5}_{-0.4}^{+0.3}$	121
0.027	${5.3}_{-0.3}^{+0.2}$	119
0.047	${5.0}_{-0.2}^{+0.2}$	113
0.084	${4.8}_{-0.2}^{+0.1}$	98
0.150	${4.6}_{-0.1}^{+0.1}$	73
0.267	${4.2}_{-0.2}^{+0.2}$	46
0.474	${3.7}_{-0.3}^{+0.2}$	17
0.843	${2.8}_{-0.5}^{+0.6}$	2

Note. The 1σ uncertainties are estimated from the combination of the number count Poisson statistical errors and the flux uncertainties (see the text). The column N_diff presents the numbers of our 1.2 mm sources in each flux bin, while the N_cum column denotes the cumulative source numbers down to the flux densities. For our analysis, we do not use the faintest bin data indicated with the parentheses (see the text).

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In Figure 6, we find that the faintest data point at ∼0.002 mJy (red open circle) is composed of a source with a very high magnification factor of μ > 100. The highly magnified sources would potentially include large systematic uncertainties originated from the mass modeling. Moreover, the faintest data point has the large error bar that does not provide important constraints on the number count measurements. Thus, we do not use the faintest data point in the following discussion. Except for this very high magnification factor object, the faintest and brightest sources in our sample have intrinsic fluxes of 0.018 and 1.2 mJy, respectively. Thus, our study covers the flux density range of ∼0.02–1 mJy.

We compare our number counts to previous measurements at the ∼1.2 mm band. In the flux range close to our study, there are two types of previous studies, blank field observations with ALMA and cluster observations for lensed sources with single-dish telescopes. For the measurements of 1.1 and 1.3 mm bands that are different from 1.2 mm band, we scale the flux densities with the methods described in Section 3.

4.2.1. Blank Field Observations

4.2.2. Cluster Observations for Lensed Sources

In Section 4.1, we obtain the 1.2 mm number counts with our faint ALMA source. Because our sample lacks sources brighter than 1.2 mJy, we use previous studies of bright sources in the literature to investigate the source contributions to the IR EBL. There are a number of studies that investigate the bright sources (Figure 6), but we use the number counts at 2–10 mJy obtained from a survey for six blank fields with a single-dish telescope instrument of AzTEC (Scott et al. 2012), which shows reliable number counts at 1.1 mm that is close to the 1.2 mm band. Although there is another systematic survey for bright sources from the single-dish observations with LABOCA (Weiß et al. 2009), we do not use their data for our analysis, due to their observation wavelength of 870 μm far from our 1.2 mm band. It should be noted that the number counts of bright sources are still under debate (e.g., Chen et al. 2013; Karim et al. 2013; Simpson et al. 2015a), because the recent high-resolution ALMA observations reveal multiple sources in a beam of single-dish observations (Hodge et al. 2013). Nevertheless, our results below do not depend on the bright source number counts, because the contribution from the bright sources is not large (Section 1).

The source contributions to the IR EBL can be calculated by integrating the number counts down to a flux density limit. To characterize the shape of the number counts, we use the Schechter function form (Schechter 1976) in the same manner as the previous studies,

$$\begin{eqnarray}&&\phi (S){dS}={\phi }_{*}{(\displaystyle \frac{S}{{S}_{*}})}^{\alpha }\mathrm{exp}(-\displaystyle \frac{S}{{S}_{*}})d(\displaystyle \frac{S}{{S}_{*}}),\end{eqnarray} \tag{ 7 }$$

where ϕ_*, S_*, and α are the normalization, characteristic flux density, and faint end slope power-law index, respectively. The logarithmic form of n(S)d(log S) = ϕ(S)dS is given by

$$\begin{eqnarray}&&n(S)=(\mathrm{ln}\ 10){\phi }_{*}{(\displaystyle \frac{S}{{S}_{*}})}^{(\alpha +1)}\mathrm{exp}(-\displaystyle \frac{S}{{S}_{*}}).\end{eqnarray} \tag{ 8 }$$

Using Equation (8), we conduct χ² fitting to the differential number counts derived from our and the previous observations shown with the filled symbols in Figure 6 which are our measurements and the AzTEC (Scott et al. 2012) and ALMA 1.1 mm (Carniani et al. 2015) results. Here we do not use the measurements of Hatsukade et al. (2013), Ono et al. (2014), Oteo et al. (2015), or the 1.3 mm results of Carniani et al. (2015) that are presented with the open symbols of Figure 6, because these studies use ALMA data covered by our study or the sample does not meet our selection criteria (Section 2.2). We confirm that the entire ALMA 1.1 mm data of Carniani et al. (2015) are different from ours. Including the bright number counts at 2–10 mJy (see above), we perform Schechter function fitting with our number counts and those from the 1.1 mm observations with AzTEC (Scott et al. 2012) and ALMA (Carniani et al. 2015). We vary three Schechter parameters, and search for the best-fit parameter set of (ϕ_*, S_*, α) that minimizes χ². The best-fit function and parameter set are presented in Figure 6 and Table 5, respectively. Figure 6 and the χ²/dof of 16.9/19 (Table 5) indicate that the number counts are well represented by the Schechter function down to ∼0.02 mJy.

Table 5.  Best-fit Parameters

Schechter
S*	ϕ*	α		χ2/dof
(mJy)	(103 deg−2)
(1)	(2)	(3)		(4)
${2.35}_{-0.16}^{+0.16}$	${1.54}_{-0.26}^{+0.29}$	$-{2.12}_{-0.06}^{+0.07}$		16.6/19
DPL
${S}_{*}^{{\rm{DPL}}}$	${\phi }_{*}^{{\rm{DPL}}}$	αDPL	βDPL	χ2/dof
(mJy)	(103 deg−2)
(5)	(6)	(7)	(8)
${4.43}_{-0.49}^{+0.34}$	${0.26}_{-0.07}^{+0.05}$	${2.39}_{-0.06}^{+0.05}$	${6.20}_{-1.11}^{+0.48}$	12.8/18

Note.  (1)–(3) Best-fit parameter set for the Schechter function. (4) χ² over the degree of freedom. (5)–(8) Best-fit parameter set for the DPL function.

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We also carry out fitting of the double-power-law (DPL) function in the same manner as the Schechter function fitting. The DPL function has four free parameters of ${\phi }_{*}^{{\rm{DPL}}}$, ${S}_{*}^{{\rm{DPL}}}$, α^DPL, and β^DPL (see, e.g., Ono et al. 2014). The best-fit parameters are presented in Table 5. The χ²/dof value for the DPL fitting is nearly unity (12.8/18), and the DPL also well represents the number counts. Because the Schechter and DPL functions show similarly good fitting results, we use the Schechter function fitting results for simplicity in the following discussion.

With the best-fit Schechter parameter set, we calculate the integrated flux densities, ${\int }_{{S}_{{\rm{limit}}}}^{\infty }\;S\phi (S){dS}$, down to the flux limit of S_limit. Because the faintest bin of our number count data covers down to 0.015 mJy (Figure 6), we choose S_limit = 0.015 mJy. We calculate the integrated flux density to be ${22.9}_{-5.6}^{+6.7}$ Jy deg⁻² at 1.2 mm. Figure 7 shows the integrated flux density. We evaluate the fraction contributed to the total IR EBL measurement at 1.2 mm (${22}_{-8}^{+14}$ Jy deg⁻²; Fixsen et al. 1998) which is obtained by the observations of the COBE Far Infrared Absolute Spectrophotometer, and find that the resolved fraction in our study is ${104}_{-25}^{+31}\%$. This result suggests that our study fully resolves the 1.2 mm EBL into the individual sources within the uncertainties.

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Our result also indicates that the EBL contribution of sources with ≲0.02 mJy are negligibly small. It would suggest that there is (1) a flattening of number counts slope α, or (2) a truncation of the number counts at a flux S_trunc below ≃0.02 mJy. Using our resolved fraction of ${f}_{{\rm{res}}}={104}_{-25}^{+31}\%$, we constrain α and S_trunc in the scenarios (1) and (2). For scenario (1), we extrapolate the Schechter function from S_limit to 0 mJy with a power law of $n(S)\propto {S}^{({\alpha }_{0-{S}_{{\rm{limit}}}}+1)}$, where ${\alpha }_{0-{S}_{{\rm{limit}}}}$ is a power-law slope at 0 mJy to S_limit, and integrate the Schechter function with the power-law extrapolation. We constrain the ${\alpha }_{0-{S}_{{\rm{limit}}}}$ value for the condition that the integrated flux density does not exceed the 1σ upper limit of the EBL measurement. We thus determine that the lower limit of ${\alpha }_{0-{S}_{{\rm{limit}}}}\gtrsim -1.0$. For scenario (2), we extrapolate our best-fit Schechter function below S_limit. We find S_trunc ≃ 0.004 mJy (corresponding to SFR_IR ∼ 1.6 M_⊙ yr⁻¹; Kennicutt 1998) where the 1σ lower error of f_res reaches 100% of the EBL contribution fraction. Because scenarios (1) and (2) are two extreme cases, our results would suggest that the faint end slope at ≲0.02 mJy is flatter than the best-fit Schechter function slope of $\alpha =-{2.12}_{-0.06}^{+0.07}$ (Table 5), and that there is a number counts truncation, if any, at ≲0.02 mJy.

We estimate a galaxy bias of our faint ALMA sources by the counts-in-cells technique (e.g., Adelberger et al. 1998; Robertson 2010)

$$\begin{eqnarray}&&{b}_{{\rm{g}}}^{2}\approx \displaystyle \frac{{\sigma }_{N}^{2}-\bar{N}}{{\bar{N}}^{2}{\sigma }_{V}^{2}(z)},\end{eqnarray} \tag{ 9 }$$

where ${\sigma }_{N}^{2}$ is the standard deviation of the detected source counts per field, ${\sigma }_{V}^{2}$ is the matter variance averaged over the survey volumes V, and $\bar{N}$ is the average source counts per field.

In this counts-in-cells analysis, we use 66 maps of the field data (A) and (B). Because the cluster data (C) includes lensing effects of magnification and survey volume distortion as well as the different spurious source rate (Section 3.2), we do not include the cluster data (C) in the sample of our counts-in-cells analysis. The survey volume is calculated by using the average effective area per field (Section 3.6). For the redshift distribution to calculate the volume, we again assume a median redshift of z = 2.5 (Section 3) and the top hat function covering z = 1–4. For ${\sigma }_{N}^{2}$ and $\bar{N}$, we do not use the simple detection numbers, but the corrected detection numbers obtained with Equation (5), which allows us to remove the effects of spurious sources. We calculate ${\sigma }_{V}^{2}$ with the analytic structure formation model (e.g., Mo & White 2002). To estimate the 1σ uncertainty of b_g, we carry out Monte Carlo simulations. We first make 1000 mock catalogs of faint ALMA sources consisting of source counts that follow the Poission probability distribution function whose average values agree with our observed source counts. Then, we calculate galaxy bias values with each mock catalog, and repeat this process for the 1000 mock catalogs. We define a 1σ error of our observational b_g estimate by the 68 percentile of the b_g distribution obtained from the 1000 mock catalog results. Because the 1σ error is large, we cannot obtain the measurement of b_g but place a 1σ upper limit of b_g < 3.5 for our faint ALMA sources. This 1σ upper limit agrees with the previous b_g estimate for faint ALMA sources, b_g < 4, given by Ono et al. (2014).

We estimate the dark halo masses of our faint ALMA sources with our result of b_g < 3.5 with the analytics structure formation model of Sheth & Tormen (1999), assuming the one-to-one correspondence between galaxies and dark halos. We obtain the 1σ upper limit dark halo mass of ≲5 × 10¹²M_⊙.

Figure 8 shows the galaxy biases of our faint ALMA sources and a variety of high-z galaxy populations; bright SMGs, K-selected galaxies including passively evolving BzK galaxies (pBzKs) and sBzKs, distant red galaxies (DRGs), and UV-selected galaxies composed of BX/BM galaxies, LBGs, and LAEs. Some of these high-z galaxy studies show no galaxy bias estimates, but only the galaxy correlation lengths r₀ and the power-law indices of the two point spatial correlation function γ. For these results, we calculate the galaxy biases in the same manner as Ono et al. (2014) from b_g = σ_8,gal/σ₈(z), where σ₈(z) is a matter fluctuation in spheres of comoving radius of 8 h⁻¹ Mpc, and σ_8,gal is a galaxy fluctuation. The value of σ_8,gal is derived with (see Equation (7.72) in Peebles 1993)

$$\begin{eqnarray}&&{\sigma }_{8,\mathrm{gal}}^{2}=\displaystyle \frac{72}{(3-\gamma )(4-\gamma )(6-\gamma ){2}^{\gamma }}{(\displaystyle \frac{{r}_{0}}{8\ {h}^{-1}\;{\rm{Mpc}}})}^{\gamma }.\end{eqnarray} \tag{ 10 }$$

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In Figure 8, the galaxy bias estimates for the most of SMGs, DRGs, and pBzKs are higher than the upper limit of our faint ALMA sources. On the other hand, K-selected star-forming galaxies of sBzKs have halo masses comparable to those of our faint ALMA sources. The UV-selected LBGs and LAEs also have the galaxy bias values similar to our faint ALMA sources. Thus, we find that our faint ALMA sources are not similar to SMGs, DRGs, and pBzK with masses of ≳10¹³M_⊙, but sBzKs, LBGs, and LAEs with masses ≲10¹³M_⊙ (Figure 8). These clustering results suggest a possibility in a statistical sense that a large fraction of our faint ALMA sources could be mm counterparts of sBzKs, LBGs, and LAEs.

We investigate optical-NIR counterparts of our faint ALMA sources, and characterize the sources with the multi-wavelength properties, especially the optical-NIR colors. We search for the optical-NIR counterparts in the regions of SXDS and A1689 that have rich multi-wavelength data. In these two regions, there are a total of 65 faint ALMA sources; 54 and 11 sources in SXDS and A1689 regions, respectively. For the optical-NIR counterpart studies  conducted so far, this is the largest sample of faint mm sources down to ∼0.02 mJy, while there exist studies, such as ALESS and COSMOS, that use a comparably large but relatively bright sample (Hodge et al. 2013; Scoville et al. 2015).

Table 6 summarizes the multi-wavelength data of SXDS and A1689 used in this study. The multi-wavelength data cover from X-ray to far-IR radiation. In the SXDS region, we use published photometry catalogs in 0.5–10 keV (Ueda et al. 2008), u* (S. Foucaud et al. 2016, in preparation), B, V, R, i', z' (Furusawa et al. 2008), J, H, K (UKIDSS DR10; O. Almaini et al. 2016, in preparation), 3.6–24 μm (Spitzer UKIDSS Ultra Deep Survey; SpUDS), 250–500 μm (Herschel Multi-tiered Extragalactic Survey; HerMES, Oliver et al. 2012), and 1.4 GHz (Simpson et al. 2006) as well as the photometric redshift (z_phot) catalog (Williams et al. 2009). In the A1689 region, we investigate published photometry catalogs of g₄₇₅, r₆₂₅, i₇₇₅, z₈₅₀, J₁₁₀, H₁₆₀ (Hubble Source Catalog), 3.6–24 μm (Spitzer Enhanced Imaging Products; SEIP), 250–500 μm (HerMES), and the photometric redshift z_photo and spectroscopic redshift z_spec catalogs (Limousin et al. 2007; Coe et al. 2010; Diego et al. 2015).

We use the u* to K (g' to H) band images in the SXDS (A1689) region for the optical-NIR (0.4–2 μm) counterpart identification, because these data have a good spatial resolution. The optical-NIR counterparts are defined by the criterion that the distance between optical-NIR and ALMA source centers is within a 10. If an ALMA source meets this criterion in any of optical-NIR bands, we regard that the ALMA source has an optical-NIR counterpart. The 10 radius is chosen, because the absolute positional accuracy (APA) of ≃10 is guaranteed in our ALMA data (see, e.g., optical-NIR data for ≲03; see, e.g., Furusawa et al. 2008). Regarding the ALMA's APA, the knowledge base page of ALMA Science portal shows that the APA is smaller than the synthesized beam width.⁶ Because the synthesized beam widths of our data are ≲10, the 10 radius is applied.

Using the SXDS+A1689 source catalog of the 65 faint ALMA sources, we identify a total of 17 optical-NIR counterparts. Fifteen and two sources are identified in SXDS and A1689 and are referred to as S1–15 and A1–2, respectively. Figure 9 shows the postage stamps of these identified sources, and Table 7 summarizes their photometric properties. The photometry values in Table 7 are total magnitudes. The u* magnitudes are MAG_AUTO values given by our SExtractor photometry. Similarly, BV R_ci'z' magnitudes are MAG_AUTO values listed in the Furusawa et al. (2008) catalogs. The Spitzer photometry values are estimated from the SEIP aperture flux densities with the aperture correction⁷ , while the HST photometry values (for A1 and A2) are the total magnitudes calculated from MAG_APER values of our SExtractor measurements and the aperture correction.⁸ Note that there is an offset of 12 (106) between the flux peaks of the optical-NIR counterpart and S9 (S15) ALMA source. Because S9 and S15 have an extended optical-NIR profile whose major component falls in the ALMA-emitting region, the optical-NIR source is selected as a counterpart of the ALMA source.

Table 7.  Photometry of the Optical-NIR Counterparts

SXDS
ID	u*	B	V	Rc	i'	z'	J	H	K	3.6 μm	4.5 μm	5.8 μm	8.0 μm	24 μm	250 μm	350 μm	500 μm
S1	⋯	24.46	24.24	23.85	23.52	22.88	⋯	⋯	⋯	20.02	19.94	20.11	20.12	>18.56	>14.35	>14.55	>14.15
S2	⋯	25.91	26.18	26.59	26.71	25.01	⋯	⋯	⋯	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S3	27.24	26.69	26.35	25.63	25.29	25.25	24.80	24.43	24.41	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S4	25.76	25.61	25.30	25.00	24.47	24.05	23.21	22.92	22.66	21.68	21.84	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S5	26.31	25.11	24.70	24.76	24.58	24.75	24.53	24.05	24.22	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S6	25.69	24.84	23.76	22.79	21.96	21.55	20.81	20.42	20.08	19.61	20.06	19.95	20.24	>18.56	>14.35	>14.55	>14.15
S7	>28.23	27.14	27.42	27.30	27.32	26.62	>26.18	>25.31	>25.94	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S8a	24.57	>28.93	>28.56	>28.33	>28.24	>27.22	>26.18	>25.31	>25.94	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S9b	>28.23	25.30	23.86	22.96	21.63	20.93	20.24	20.22	20.45	20.40	20.78	20.46	>22.81	>18.56	>14.35	>14.55	>14.15
S10	>28.23	26.68	26.25	26.25	25.65	25.07	25.90	24.60	25.67	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S11	26.00	25.14	24.66	24.53	24.02	23.62	22.36	21.76	21.26	19.91	19.69	19.31	19.51	15.95	>14.35	>14.55	>14.15
S12	23.45	23.22	22.96	22.94	22.65	22.45	21.55	21.18	20.88	19.89	19.66	19.35	19.62	16.66	>14.35	>14.55	>14.15
S13	>28.23	>28.93	28.50	>28.33	>28.24	>27.22	>26.18	>25.31	25.22	22.55	22.73	22.33	>22.81	>18.56	>14.35	>14.55	>14.15
S14	>28.23	25.96	26.09	25.68	>28.24	>27.22	>26.18	>25.31	>25.94	>25.27	>25.16	>22.85	>22.81	>18.56	>14.35	>14.55	>14.15
S15	23.87	23.49	23.40	23.08	22.84	22.42	21.72	21.25	21.16	20.27	20.12	20.32	19.89	>18.56	>14.35	>14.55	>14.15
Abell 1689
ID	⋯	⋯	g475	r625	i775	z850	J110	H160	⋯	3.6 μm	4.5 μm	5.8 μm	8.0 μm	24 μm	250 μm	350 μm	500 μm
A1	⋯	⋯	26.61	25.78	24.79	25.36	24.29	23.55	⋯	18.25	18.45	18.72	19.10	16.94	>14.55	>14.75	>14.35
A2	⋯	⋯	27.35	26.28	25.84	25.59	>26.75	24.36	⋯	>25.35	>24.85	>23.95	>23.45	>20.05	>14.55	>14.75	>14.35

Notes. The u* magnitudes are MAG_AUTO values given by our SExtractor photometry. Similarly, ${{BVR}}_{c}{i}^{\prime }{z}^{\prime }$ magnitudes are MAG_AUTO values listed in the Furusawa et al. (2008) catalogs. The Spitzer photometry values are total magnitudes estimated from the SEIP aperture flux densities with the aperture correction, while the HST photometry values (for A1 and A2) are the total magnitudes calculated from MAG_APER values of our SExtractor measurements. The aperture corrections for the Spitzer and HST photometry are provided by the Spitzer and HST websites, respectively (see the text). The lower limits correspond to the 3σ levels. The S1 and S2 objects are not observed in the u* band and  the JHK bands.

^aThe secure photometry cannot be carried out due to the blending with a bright neighboring objects for the B-500 μm bands. See Figure 9. ^bThe positions of the optical-NIR counterpart are far from the ALMA source center by ∼12.

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Because spurious sources are included in the 65 faint ALMA sources, we calculate the real ALMA source number in the SXDS (A1689) region. In the SXDS (A1689) region, we subtract the expected spurious source number of 31 (5) from the SXDS (A1689) source number of 54 (11), and estimate the real ALMA source number to be 23 (6). The total real ALMA source number in these two regions is 29 (=23 + 6). Thus, the counterpart-identification fraction that is the ratio of the total optical-NIR counterpart ALMA sources to the real ALMA sources is estimated to be 59% (=17/29). We estimate the probability of the chance projection that a foreground or background object is located by chance in the 10 radius circle of an ALMA source, and obtain ∼10% in the P-value (Downes et al. 1986). Multiplying this probability with the number of optical-NIR counterparts, 17, we expect that ∼2 objects would be the foreground/background contamination. Even if we include this contamination effect, a half of our faint ALMA sources, 52% (=[17–2]/29), have an optical-NIR counterpart.

Interestingly, there is a difference of the counterpart-identification fractions between SXDS and A1689 regions, although the depths of the optical-NIR data are comparable in these two regions (27–28 mag in optical and 25–26 mag in NIR; see Table 6). The counterpart-identification fraction in the SXDS region is 65% (=15/23), while the one in the A1689 region is only 33% (=2/6). Including the 1σ statistical uncertainties, these fractions are 65 ± 22% and 33 ± 30% for the SXDS and A1689 regions, respectively. We carefully examine whether this difference is just an artifact originated from, for example, a systematically high spurious rate and cluster member galaxy obscurations in the cluster region. However, there are no hints of notable artificial effects in the data. Chen et al. (2014) also report a similarly low identification rate of MIR/radio counterparts in A1689 based on their observations with Submillimeter Array (SMA). It should be noted that the intrinsic mm flux densities of our sources in the A1689 region are ≃0.01–0.1 mJy (magnification corrected) that is about an order of magnitude fainter than those of our sources in the SXDS region, ≃0.1–1 mJy.

There are two possible explanations for the difference between SXDS and A1689. The first explanation is made by the negative k-correction. Due to the faint flux densities, the mm sources of the A1689 would reside at a redshift higher than the SXDS sources on average, since the abundance of the dusty population decreases toward the epoch of the first generation galaxies. Because the effect of the negative k-correction is significant, the ratio of mm to optical-NIR fluxes is higher for high-z sources than low-z sources. In this case, the counterpart-identification fraction would be small in the A1689 region. The second explanation is the A1689's lensing distortion effects in conjunction with the intrinsic offsets between mm and optical-NIR emitting regions within a galaxy. Chen et al. (2015) report that there is an offset between submm and optical-NIR emitting regions typically by ∼04. If there is an intrinsic ∼04 offset in the source plane of A1689, the image distortion effects of lensing would magnify the offset to >10. We thus estimate the probability that the offsets between mm and optical-NIR sources in the source plane are magnified to >10 in the image plane by Monte Carlo simulations. We place 100 artificial sources in the source plane of A1689 at around an ALMA source in A1689. The positions of these artificial sources follow the Gaussian distributions whose standard deviation is 04. Using our mass model, we calculate the offsets between the ALMA source and the artificial sources in the image plane. We repeat this process for 11 ALMA sources in A1689. We find that ∼50% of the artificial sources show >10 offsets from the ALMA source in the image plane. Because the number of optical counterparts in A1689 is 2, our simulation results suggest that a total of 4 (=2/[50%]) optical counterparts should exist in the case of no lensing distortion effects (plus the intrinsic offsets). Thus, the true counterpart-identification fraction of A1689 is 67%(=4/6) that is very similar to the counterpart-identification fraction of SXDS (65 ± 22%). If the lensing distortion effects (plus the intrinsic offsets) would provide 2 (=4–2) sources whose optical counterparts have an offset from the ALMA sources by >10 in the image plane, we would find such ALMA sources whose optical counterparts are located at the position just beyond 10 in our data. We refer the distribution of the artificial optical counterparts from our simulations, and search for potential optical-NIR counterparts whose distances from the ALMA sources are >10. We find such potential optical counterparts in four sources of A1689, some of which may be real optical counterparts, and confirm the hypothesis that the combination of the lensing distortion effects and the intrinsic offsets can be the major reason for the difference of the counterpart-identification fractions between the cluster of A1689 and the blank field of SXDS.

Table 7 lists the photometry of the Spitzer (3.6–24 μm) and Herschel (250–500 μm) data. Because the spatial resolutions of the Spitzer and Herschel data are poor, we determine the Spitzer and Herschel counterparts by visual inspection that allows sources residing at a position slightly beyond the 1'' search radius. Although no sources are identified in the Herschel bands, about half of our faint ALMA sources are detected in the Spitzer data. These results would indicate that our faint ALMA sources are very high-redshift sources (z > 4). However, it should be noted that the Herschel data can detect dusty starbursts whose SFRs are very high, > a few 100M_⊙ yr⁻¹ even at z ∼ 2–3.

Here we examine whether we simply miss red spectral energy distribution (SED) objects that are not detected in optical-NIR (0.4–2 μm) bands but only in the Spitzer and Herschel bands. We search for Spitzer and Herschel counterparts of our faint ALMA sources with no optical-NIR counterparts, adopting the visual inspection technique described above. We find no Spitzer and Herschel counterparts that have undetectable optical-NIR fluxes. Again, these results do not suggest that our faint ALMA sources are dusty starbursts.

We cross-match the catalogs of our faint ALMA sources and the radio 1.4 GHz and X-ray 0.5–10 keV sources. We find no counterparts within a 10–30 radius around our faint ALMA sources. Thus, our faint ALMA sources have no signatures of AGN.

To summarize these counterpart-identification results with the Spitzer and Herschel as well as radio and X-ray data, there are signs of neither dusty starbursts nor AGNs in our faint ALMA sources. Note that this cross-match analysis is completed only in SXDS because there are no radio and X-ray data available in A1689.

We identify two optical counterparts of A1 and A2 that are lensed by the A1689 cluster. The optical image at the position of A1 is presented in Figure 10. There is a small positional offset of ∼08 between the optical and mm emission, but this small difference can be explained by the positional uncertainty of the ALMA image (Section 6.1).⁹

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This optical counterpart of A1 is one of three known multiple images that is called 5.2, while the rest of two multiple images are called 5.1 and 5.3 (Limousin et al. 2007). We investigate our ALMA image at the optical positions of 5.1 and 5.3 sources. In the ALMA image, the 5.1 source is marginally detected at the 3σ level, while the 5.3 source has no signals above the 3σ level. Table 8 summarizes the physical properties of these three multiple images. The difference of mm emissivities between these multiple images is discussed in Section 6.1.

Table 8.  Multiple Images at z = 2.60 with the ALMA Counterpart

ID	ID	R.A.	decl.	${S}_{\text{1.3 mm}}^{{\rm{obs}}}$	${S}_{\text{1.3 mm}}^{{\rm{corr}}}$	${S}_{\text{1.2 mm}}^{{\rm{corr}}}$	g475	r625	i775	z850	μ
(L07)	(This Work)	(J2000)	(J2000)	(mJy)	(mJy)	(mJy)	(magAB)
(1)	(2)	(3)		(4)	(5)	(6)	(7)				(8)
5.1	⋯	13:11:29.064	−1:20:48.64	0.18 ± 0.06a	0.049 ± 0.021	0.067 ± 0.029	26.29	25.53	25.37	25.17	5.7
5.2	A1	13:11:29.224	−1:20:44.24	0.30 ± 0.06	0.015 ± 0.006	0.020 ± 0.008	26.61	25.78	25.58	25.36	22.8
5.3	⋯	13:11:34.120	−1:20:20.96	<0.28	<0.093	<0.13	26.61	25.85	25.85	25.60	3.0

Note. (1) and (2) ID names given in Limousin et al. (2007; L07) and this work, respectively. (3) R.A. and decl. in the optical data. (4) Peak flux density SExtractor measurement (Section 3.1) with the primary beam correction. If no significant 1.32 mm fluxes are detected at the optical source positions, 3σ upper limits are presented. (5) Observed flux density at 1.32 mm on the UV-taper image that are measured by 2D Gaussian-fitting routine of imfit in CASA with the primary beam correction. The magnification uncertainty (Section 3.5) is included in the error. (6) Source flux density at 1.2 mm with the flux scaling from 1.32 to 1.2 mm (Section 3). (7) Total magnitudes in the HST/ACS bands. (8) Magnification factor at the peak of the optical flux.

^aMarginal detection of the 3σ level that is below our source selection limit for the cluster data (>3.9σ; see Section 3.2).

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Although the 5.2 source, i.e., A1, does not have a spectroscopic redshift, the redshift of the 5.1 and 5.3 sources is z = 2.60 determined by spectroscopy (J. Richard et al. 2015, in preparation; see Limousin et al. 2007). We thus regard that the redshift of A1 is z = 2.60. We calculate the intrinsic 1.2 mm flux of A1 from our 1.3 mm data with the flux scaling factor from 1.3 mm to 1.2 mm (Section 3), and obtain 0.020 ± 0.008 mJy which is corrected for the gravitational lensing magnification of μ = 22.8 and the primary beam sensitivity. Chen et al. (2014) report that the 5.1 and 5.3 sources are not detected above the 3σ level by their SMA 870 μm observations, but only the 5.2 source. Chen et al. (2014) measure the flux of the 5.2 source, and find the intrinsic flux of 5.2 source is 0.085 ± 0.035 mJy at 870 μm, which corresponds to 0.033 ± 0.014 mJy at 1.2 mm with the k-correction from 870 μm to 1.2 mm (Section 3). This 1.2 mm flux of 0.033 ± 0.014 mJy is consistent with our measurement of 0.020 ± 0.008 mJy within the 1σ uncertainties.

The optical-NIR counterpart of A2 is identified in the optical bands with a lensing magnification factor of μ = 4.9. The intrinsic 1.2 mm flux of A2 is estimated to be 0.077 ± 0.035 mJy which are corrected for the gravitational lensing magnification and the primary beam attenuation. Neither z_spec nor z_photo of A2 are available in the literature.

We compare these A1689 sources with optical-NIR counterparts (Table 8) and the previously known lensed sources, Cosmic Eylash (e.g., Swinbank et al. 2010) and SMMJ16358 (e.g., Kneib et al. 2004), identified by single-dish observations. Cosmic Eyelash and SMMJ16358 have intrinsic SFRs of ∼210 and 500 M_⊙ yr⁻¹, respectively. These SFRs correspond to 0.5 and 1.3 mJy at 1.2 mm under the assumption of the $\mathrm{SFR}-{L}_{{\rm{IR}}}$ relation (Kennicutt 1998). On the other hand, our optical-NIR counterparts have intrinsic fluxes of ∼0.02–0.08 mJy at 1.2 mm. Thus, our ALMA study has identified sources with intrinsic fluxes ≳10 times fainter than these lensed sources.

Although there are no optical-NIR counterparts, we find a very strongly lensed source candidate, A3, that shows ALMA flux double peaks dubbed A3a and A3b. The upper panel of Figure 11 shows the ALMA image of A3. The A3 source is very close to the critical line for a z = 3 source (Figure 11), and the critical line is located between the double peaks of A3. More quantitatively, we investigate whether these double peaks can be explained by multiple images of a single lensed source behind A1689. Using our mass model, we calculate the positions of multiple images made by a lensed source at z = 2–6 (cyan lines in the upper panel of Figure 11). We find that multiple images at z ∼ 3 reproduce the positions of the double peaks very well. The best-fit redshift is z = 3.07, and the magnification factor is very high, 159. The bottom panel of Figure 11 is the HST optical images. No optical counterparts of A3 are found. We summarize the properties of A3 in Table 9.

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Table 9.  Very Strongly Lensed Source Candidate

ID	R.A	decl.	${S}_{\text{1.3 mm}}^{{\rm{obs}}}$	${S}_{\text{1.3 mm}}^{\mathrm{corr}}$	${S}_{\text{1.2 mm}}^{\mathrm{corr}}$	g475	r625	i775	z850	μ
	(J2000)	(J2000)	(mJy)	(mJy)	(mJy)	(magAB)
(1)	(2)		(3)	(4)	(5)	(6)				(7)
A3	13:11:26.897	−1:20:32.02	0.24 ± 0.06	0.0015 ± 0.0007	0.0020 ± 0.0010	>28.56	>28.33	>28.24	>27.22	159.5

Note. Columns (1)–(7) correspond to columns (2)–(8) in Table 8. These values are for the A3a component (Figure 11). Because A3a and A3b are confused in the UV-taper image, these fluxes are obtained on the image without UV-taper.

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We investigate color (i.e., SED) and luminosity properties of the optical-NIR counterparts, S1–15 and A1–2, to understand the population of the faint ALMA sources with the detectable optical-NIR continuum emission. First of all, we refer the catalog of photo-z estimated from the optical-NIR SEDs (Williams et al. 2009), and find that S11, S12, and S15 have photo-z values of ${1.54}_{-0.02}^{+0.08}$, ${1.57}_{-0.02}^{+0.07}$, and ${1.45}_{-0.02}^{+0.03}$, respectively. However, the photo-z values of the S4 and S6 sources are ${0.77}_{-0.03}^{+0.23}$ and ${0.67}_{-0.02}^{+0.01}$, respectively. We thus regard that the S4 and S6 sources reside at low redshift (z < 1), and find that ∼10% (=2/17) optical-NIR counterparts are low-z objects. Because the majority of mm sources are thought to be high-z galaxies and AGNs (Casey et al. 2014), the rest of  the 15 (=17–2) optical-NIR counterparts are candidates of high-z galaxies. We thus compare colors and luminosities of these high-z galaxy candidates with those of high-z sources. The major high-z populations are SMGs, LBGs (including BX/BM), DRGs, BzKs, LAEs, and AGNs. Since the SMGs, DRGs, and LAEs generally have weak optical continua that cannot be reliably compared with our optical-NIR counterparts, we focus on the comparisons with LBGs (BX/BM) and BzKs.

6.3.1. Comparison with the LBG BX/BM Populations

The BX and BM galaxies, generally included in the LBG population, reside in the redshift ranges of 2.0 ≲ z ≲ 2.5 and 1.5 ≲ z ≲ 2.0, respectively (Steidel et al. 2004). Similarly, there are LBGs at z ∼ 3. These galaxies are defined by the selection criteria of ${U}_{n}-G$ and G − R colors. We apply these color selection criteria to our optical-NIR counterparts, but the photometry bands of our data are different from the U_n, G, and R bands. Following the procedures of Ly et al. (2011), we alternatively use a color set of U − BV and ${BV}-{R}_{c}{i}^{\prime }$. The BV and ${R}_{c}{i}^{\prime }$ photometry are defined by

$$\begin{eqnarray}{BV} & = & -2.5\mathrm{log}[\displaystyle \frac{{x}_{1}{f}_{B}+(1-{x}_{1}{f}_{V})}{3630\;\mu \mathrm{Jy}}],\\ {R}_{c}{i}^{\prime } & = & -2.5\mathrm{log}[\displaystyle \frac{{x}_{2}{f}_{R}+(1-{x}_{2}{f}_{{i}^{\prime }})}{3630\;\mu \mathrm{Jy}}],\end{eqnarray} \tag{ 11 }$$

where f_X is the flux density per unit frequency (erg s⁻¹ cm⁻² Hz⁻¹) in a band "X" corresponding to B, V, R_c, and i'. The coefficients of x₁ and x₂ are 0.314 and 0.207, respectively. With these photometric relations and the selection colors (Steidel et al. 2004), the color criteria of BX objects are re-written as

$$\begin{eqnarray}{BV}-{R}_{c}{i}^{\prime } & \geqslant & -0.2,\\ U-{BV}\ & \geqslant & {BV}-{R}_{c}{i}^{\prime }+0.23,\\ {BV}-{R}_{c}{i}^{\prime }\ & \leqslant & 0.2(u-{BV})+0.4,\\ U-{BV}\ & \leqslant & {BV}-{R}_{c}{i}^{\prime }+1.0.\end{eqnarray} \tag{ 12 }$$

Similarly, the criteria of BM objects are

$$\begin{eqnarray}{BV}-{R}_{c}{i}^{\prime } & \geqslant & -0.2,\\ U-{BV} & \geqslant & -0.1,\\ {BV}-{R}_{c}{i}^{\prime } & \leqslant & -0.382(U-{BV})+0.853,\\ {BV}-{R}_{c}{i}^{\prime } & \leqslant & 0.70(U-{BV})+0.280,\\ U-{BV} & \geqslant & {BV}-{R}_{c}{i}^{\prime }+0.23.\end{eqnarray} \tag{ 13 }$$

We compare these color criteria with our optical-NIR counterparts. Among the 17 optical-NIR counterparts, 10 are observed with the photometric system of ${u}^{*}{{BVR}}_{c}{i}^{\prime }$ necessary for comparison. Removing the two obvious low-z objects, S4 and S6, we use eight optical-NIR counterparts. Figure 12 displays the two color diagram of these eight sources with the color selection criteria. Six sources fall in or near the selection windows of z ∼ 2 BX/BM or z ∼ 3 LBGs, where we include the S11 source at the border of the selection window. The S9 source is placed at the upper right corner of Figure 12. These colors of the S9 source are similar to z ≳ 4 LBGs (see, e.g., Steidel et al. 2003). It is likely that S9 source is a LBG at z ≳ 4. On the other hand, S3 clearly escapes from the selection windows. We thus find that seven out of eight optical-NIR counterparts are have colors consistent with either z ∼ 2 BX/BM or z ≳ 3 LBGs. Including the two low-z objects of S4 and S6 sources, 70% (=7/[8 + 2]) of the optical-NIR counterparts meet the LBG (BX/BM) color criteria. We then compare magnitudes of the 10 optical-NIR counterparts with the known z ∼ 2 BX/BM and z ≳ 3 LBGs, omitting u* magnitudes that would include a Lyman break whose central wavelength is very sensitive to a source redshift. We find that all of the 10 optical-NIR sources have R_c magnitudes in the range of 23–27 mag (Table 7), which is the typical brightness of z ∼ 2 BX/BM and z ≳ 3 LBGs (Steidel et al. 2003, 2004). We thus conclude that a majority (∼70%) of the 10 optical-NIR counterparts belong to LBG (BX/BM) population.

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6.3.2. Comparison with the BzK Populations

In contrast with the BX/BM galaxies and LBGs, the BzK galaxy population includes not only blue star-forming galaxies, but also moderately reddened, dust-poor, star-forming galaxies (also known as sBzK) and passive galaxies (also known as pBzK) at 1.4 ≲ z ≲ 2.5 (Daddi et al. 2004). We compare the color criteria of the BzK population with our optical-NIR counterparts. There are 10 out of 17 optical-NIR counterparts that are located in the SXDS region with the photometric measurements of the B, z', and K bands (Table 7).¹⁰ Removing the two obvious low-z objects, S4 and S6, we use eight optical-NIR counterparts for comparison.

Because our photometry system (SXDS) is different from that of the BzK studies (Daddi et al. 2004), we again apply corrections to the color selection criteria for BzK galaxies, which are suggested by Yuma et al. (2012),

$$\begin{eqnarray}{(B-z)}_{{\rm{Daddi+04}}} & = & {(B-z)}_{{\rm{SXDS}}}+0.3,\\ {(z-K)}_{{\rm{Daddi+04}}} & = & {(z-K)}_{{\rm{SXDS}}}+0.1.\end{eqnarray} \tag{ 14 }$$

Figure 13 presents the B − z and z − K colors of the eight optical-NIR counterparts. We find that five out of the eight optical-NIR counterparts meet the color criteria of sBzK, while the rest of the three sources, S3, S9 and S10, have colors different from sBzK or pBzK galaxies whose redshift range is z ≃ 1.4–2.5. Because S3 also shows the colors that are different from those of BX/BM galaxies and LBGs (Section 6.3.1), S3 would reside at z ≲ 1.4, which is out of the BzK redshift window. Although the colors of S9 and S10 are similar to those of stars, in Section 6.3.1, the BX/BM galaxy and LBG selections suggest these two sources have colors consistent with z ≳ 3 galaxies. Thus, S9 and S10 would be galaxies at z ≳ 3 which escape from the BzK redshift window. In this way, we confirm that the results of the BzK selection are consistent with those of BX/BM galaxies and LBGs. In summary, including the two low-z objects, S4 and S6 sources, we find that 50%(=5/[8 + 2]) of the optical-NIR counterparts are sBzK galaxies at z ≃ 1.4–2.5. We then compare magnitudes of these optical-NIR counterparts (Table 7) and find that the optical and NIR magnitudes are comparable with the sBzK galaxies (Daddi et al. 2004).

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6.3.3. Comparison with the AGN Population

The six counterparts of S1, S6, S11, S12, S15, and A1 are detected in the mid-IR of 8 and 24 μm bands. Because mid-IR bright objects are prominent candidates of AGNs that have strong hot dust emission, we compare the colors of the six mid-IR bright counterparts and a typical AGN (Mrk 231), as well as a starburst (Arp 220). Figure 14 shows the colors of the five mid-IR bright counterparts, Mrk 231, and Arp 220 in the diagnostic color diagram of S_{8.0 μm}/S_{4.5 μm} versus S_{24 μm}/S_{8.0 μm} that separates AGN and starburst (SB) populations (Ivison et al. 2004). In Figure 14, the dashed and dotted lines indicate the color tracks of Mrk 231 and Arp 220 in the redshift range of 0.5–3.0, which is taken from Figure 4 of Ivison et al. (2004). All of our five mid-IR bright counterparts are placed near the color track of Arp 220, and far from the one of Mrk 231, which suggests that these six counterparts are not AGN but starbursts.

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6.3.4. IRX-β Relation

To evaluate the SED properties quantitatively, we calculate the IR-to-UV luminosity ratio IRX(≡L_IR/L₁₆₀₀) and the rest-frame UV continuum slope β for six faint ALMA sources, S4, S6, S11, S12, S15, and A1, which have spectroscopic or photometric redshifts. Here L_IR is the integrated IR flux density defined in a range of rest-frame 8–1000 μm. The β values are estimated with two broadband magnitudes, m₁ and m₂:

$$\begin{eqnarray}&&\beta =-\displaystyle \frac{{m}_{1}-{m}_{2}}{2.5\mathrm{log}({\lambda }_{{\rm{c}}}^{1}/{\lambda }_{{\rm{c}}}^{2})}-2,\end{eqnarray} \tag{ 15 }$$

where ${\lambda }_{{\rm{c}}}^{1}$ and ${\lambda }_{{\rm{c}}}^{2}$ are the central wavelengths of the two broadband filters. We use (m₁, m₂) = (u*, B), (u*, B), (V, i'), (V, i'), (V, i'), and (r₆₂₅, z₈₅₀) for S4, S6, S11, S12, S15, and A1, respectively. These IRX and β values are shown in Figure 15, together with those of nearby (U)LIRGs and optically selected star-forming galaxies at z ∼ 2. In Figure 15, we also plot the 5.1 source of the multiple image (Table 8) which is marginally detected (3σ). For the lensed sources of A1 and 5.1, we assume that the positional offsets between the submm and optical sources are real, and use the lensing magnification values estimated at the submm (optical) source positions to derive the intrinsic submm (optical) luminosities.

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In Figure 15, all of the six faint ALMA sources fall in the region nearly enclosed by the Calzetti and SMC law curves. The six faint ALMA sources are placed clearly below the (U)LIRGs that have large IRX values at a given β. The clear difference between the faint ALMA sources and (U)LIRGs suggests that the faint ALMA sources are not miniature (U)LIRGs with IR luminosities fainter than (U)LIRGs by 1–2 order(s) of magnitude. Note that these results are based on the 59% of the faint ALMA sources with the optical-NIR counterparts. The rest of the 41% of the faint ALMA sources that have no optical-NIR counterparts could have IRX values as high as those of (U)LIRGs, because these sources have low L₁₆₀₀ values that are suggested by the fact of no optical-NIR counterparts.

These six faint ALMA sources are composed of four high-z sources (S11, S12, S15, and A1) and two low-z sources (S4 and S6), where we define high-z and low-z by z ≥ 1 and z < 1, respectively. Four high-z sources have the IRX-β relation similar to those of optically selected star-forming galaxies at z = 2 (Reddy et al. 2012). One low-z source of S4 is also placed at the region of z = 2 optically selected galaxies in Figure 15, while the other low-z source of S6 has large IRX and β values. It implies that majority of high-z sources are similar to z = 2 optically selected galaxies, but that low-z sources would be widely distributed in the IRX-β plot.

Figure 15 shows that the multiple images of 5.2 (i.e., A1) and 5.1 have similar IRX-β values. This fact confirms that these sources are multiple images of a single high-z source.

6.3.5. What are the Optical-NIR Counterparts of the Faint ALMA Sources?

We summarize the results of Sections 6.3.1–6.3.3 in Table 10. We find, in Sections 6.3.1 and 6.3.2, that 7 out of the 10 optical-NIR counterparts (including the low-z galaxies) meet the color criteria of BX/BM LBG and sBzK galaxy populations (see Table 10), and that the magnitudes of the counterparts are consistent with these populations. The rest of the three counterparts are low-z objects (two objects) and probably the foreground/background object of the chance projection (one object; Section 6.1). These results suggest that the majority, but not all, of the optical-NIR counterparts of our faint ALMA sources belong to the sBzKs and BX/BM LBGs populations. Moreover, the optical-NIR counterparts show neither AGN nor (U) LIRG signatures (Sections 6.3.3 and 6.3.4).

Table 10.  Properties of our Optical-NIR Counterparts

ID	R.A.	decl.	λobs	${S}_{{\lambda }_{{\rm{obs}}}}^{{\rm{obs}}}$	${S}_{\text{1.2 mm}}^{{\rm{corr}}}$	zphoto	zspec	sBzK	pBzK	BX/BM+LBG	AGN
	(J2000)	(J2000)	(mm)	(mJy)	(mJy)
			(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
S1	34.612686	−4.583063	1.22	0.36 ± 0.07	0.43 ± 0.09	⋯	⋯	⋯	⋯	⋯	N
S2	34.613739	−4.588505	1.22	0.31 ± 0.09	0.31 ± 0.10	⋯	⋯	⋯	⋯	⋯	⋯
S3	34.437805	−5.494339	1.03	0.056 ± 0.015	0.032 ± 0.010	⋯	⋯	N	N	N	⋯
S4	34.446648	−5.007715	1.26	0.047 ± 0.014	0.044 ± 0.016	${0.77}_{-0.03}^{+0.23}$	⋯	⋯	⋯	⋯	⋯
S5	34.446007	−5.008159	1.26	0.049 ± 0.014	0.050 ± 0.017	⋯	⋯	Y	N	Y	⋯
S6	34.387825	−5.220457	1.25	0.41 ± 0.11	0.42 ± 0.13	${0.67}_{-0.02}^{+0.01}$	⋯	⋯	⋯	⋯	N
S7	34.411190	−4.745051	1.23	0.30 ± 0.08	0.14 ± 0.09a	⋯	⋯	Y'	N	Y'	⋯
S8b	34.305553	−5.067293	1.30	0.29 ± 0.07	0.22 ± 0.09	⋯	⋯	⋯	⋯	⋯	⋯
S9	34.194702	−5.056259	1.25	0.43 ± 0.12	0.39 ± 0.15	⋯	⋯	N	N	Y'	⋯
S10	34.197121	−5.059778	1.25	0.31 ± 0.08	0.37 ± 0.10	⋯	⋯	N	N	Y'	⋯
S11c	34.442795	−4.911066	1.23	0.53 ± 0.10	0.56 ± 0.11	${1.54}_{-0.02}^{+0.08}$	⋯	Y	N	Y'	N
S12	34.441357	−4.912045	1.23	0.32 ± 0.09	0.29 ± 0.09	${1.57}_{-0.02}^{+0.07}$	⋯	Y	N	Y	N
S13	34.747295	−4.858103	1.25	0.37 ± 0.10	0.26 ± 0.11	⋯	⋯	⋯	⋯	⋯	⋯
S14	34.274231	−4.860201	1.30	0.32 ± 0.09	0.43 ± 0.12	⋯	⋯	⋯	⋯	⋯	⋯
S15	34.587593	−5.318766	1.25	0.32 ± 0.09	0.21 ± 0.10	${1.45}_{-0.02}^{+0.03}$	⋯	Y	N	Y	N
A1d	197.871590	−1.345783	1.32	0.30 ± 0.06	0.020 ± 0.008e	⋯	2.60	⋯	⋯	⋯	N
A2	197.869086	−1.337143	1.32	0.31 ± 0.08	0.077 ± 0.033e	⋯	⋯	⋯	⋯	⋯	⋯

Notes. (1) Wavelength in the observed frame. (2) Peak flux density of the SExtractor measurement (Section 3.1) with the primary beam correction at the observed wavelength. (3) The best estimate source flux density at 1.2 mm. The source flux is estimated with the 2D Gaussian-fitting routine of imfit in CASA (Section 3.4) with the primary beam correction and the flux scaling to 1.2 mm (Section 3). For A1 and A2 sources, the lensing magnification corrections are also applied. The lensing magnification factors are estimated at the ALMA flux peak positions of A1 and A2. (4) Photometric redshift estimated by Williams et al. (2009). (5) Spectroscopic redshift obtained by Limousin et al. (2007). (6)–(9) "Y" indicates the objects that meet the color selection criteria of sBzK, pBzK, BX/BM, and AGN populations, respectively. "N" represents the sources escaping from the color space of the selection criteria. "Y'" is presented for the sources having colors consistent with the populations.

^aThe photometry of imfit may be biased due to the systematic noise. The SExtractor peak flux measurement suggests 0.32 ± 0.08 mJy. ^bSame as the AS2 source in Hatsukade et al. (2015). ^cSame as the AS4 source in Hatsukade et al. (2015). ^dSame as the Chen-5 source in Chen et al. (2014). ^eThe magnification uncertainty (Section 3.5) is also included in the error.

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We thus conclude that the majority of our faint ALMA sources with a detectable optical continuum (∼25 mag) are UV-bright star-forming galaxies of BX/BM LBGs and sBzKs with no AGN. This conclusion is supported by the clustering analysis results in Section 5 that presents a weak clustering signal of b_g < 3.5 comparable to those of LBGs and sBzKs.

Note that we find the optical-NIR counterparts in 59% of our faint ALMA sources (Section 6.1). In other words, we understand that roughly about a half of the faint ALMA sources are LBGs and sBzKs. For the remaining half of the faint ALMA sources, the physical origin is unclear. Our study identifies a new issue of faint ALMA sources that should be addressed in the future studies.