An ALMA Survey of the SCUBA-2 Cosmology Legacy Survey UKIDSS/UDS Field: Number Counts of Submillimeter Galaxies

Our survey (the ALMA-SCUBA-2 Ultra Deep Survey field survey; hereafter AS2UDS) is based on a complete sample of 850 μm sources selected from the S2CLS map of the UDS field (Geach et al. 2017). The S2CLS UDS map covers an area of 0.96 deg², with noise levels below 1.3 mJy and a median depth of σ₈₅₀ = 0.88 mJy beam⁻¹ with 80% of sources having σ₈₅₀ = 0.86–1.02 mJy beam⁻¹. Between Cycles 1, 3, and 4, we observed all 716 > $4\sigma$ sources from the SCUBA-2 map, giving an observed flux density limit of S₈₅₀ ≥ 3.4 mJy, or a deboosted flux density of ${S}_{850}^{\mathrm{deb}}\geqslant 2.5$ mJy (Geach et al. 2017).

As a pilot project in ALMA Cycle 1 (Project ID: 2012.1.00090.S), 30 of the brightest sources from an early version of the SCUBA-2 map (data taken before 2013 February) were observed in Band 7 (Simpson et al. 2015a, 2015b, 2017). This early version of the map had a depth of σ₈₅₀ ∼ 2.0 mJy ⁻¹ and subsequent integration time scattered three of these sources below our final sample selection criteria, leaving 27 of these original single-dish detected sources in our final sample. In Cycles 3 and 4 (Project ID: 2015.1.01528.S and 2016.1.00434.S, respectively), we observed the remaining 689 single-dish sources in the final S2CLS catalog. To cross-calibrate the data, a fraction of these sources were observed twice in Cycles 3 and 4 or twice in Cycle 4.

Full details of the data reduction and source detection will be presented in S.M. Stach et al. (2018, in preparation), but here we provide a brief overview. Our ALMA targets were observed in Band 7 (344 GHz ∼ 870 μm), where the frequency closely matches the central frequency of the SCUBA-2 filter transmission and the FWHM of the ALMA primary beam at this frequency (173) comfortably covers the whole of the SCUBA-2 beam (147 FWHM). Cycle 1 observations were carried out on 2013 November 1, Cycle 3 between 2016 July 23 and August 11, and Cycle 4 between 2016 November 9 and 17 and 2017 May 6.

The phase center for each pointing was set to the SCUBA-2 positions from the S2CLS DR1 submillimeter source catalog (Geach et al. 2017), with observations taken with 7.5 GHz bandwidth centered at 344 GHz using a single continuum correlator setup with four basebands. Observations of 40 s were employed with the aim to yield 03 resolution maps with a depth of σ₈₇₀ =  0.25 mJy beam⁻¹. However, the Cycle 3 observations were taken in a more extended ALMA configuration, yielding a median synthesized beam of 019 FWHM.

Calibration and imaging were carried out with the Common Astronomy Software Application (casa v4.6.0; McMullin et al. 2007). For source detection, we created "detection" maps by applying a 05 FWHM Gaussian taper in the uv-plane, to ensure sensitivity to extended flux from our SMGs that might fall below our detection threshold, as well as improving efficiency for selecting extended sources. This downweighting of the long baseline information results in final "detection" maps with a mean synthesized beam size of 073 × 059 for Cycle 1, 056 × 050 for Cycle 3 and 058 × 055 for Cycle 4.

The clean algorithm was used to create the continuum maps using multi-frequency synthesis mode with a natural weighting to maximize sensitivity. We initially created a dirty image from the combined spectral windows (SPWs) for each field and calculated the rms noise values. The fields were then initially cleaned to 3σ and then masking ellipses are placed on sources above 4σ and the sources are then cleaned to 15, σ. The final cleaned, uv-tapered detection maps have mean depths of σ₈₇₀ = 0.25 mJy beam⁻¹ for Cycle 1, σ₈₇₀ =  0.34 mJy beam⁻¹ for Cycle 3 and σ₈₇₀ = 0.23 mJy beam⁻¹ in Cycle 4, the differences here largely being due to the varying resolutions of the observations in each ALMA cycle.

For source detection, sextractor was initially used to find >2σ peaks within the "detection" maps. Noise estimates were then calculated from the standard deviation in the integrated fluxes in 100 randomly placed 05 diameter apertures in each map. These were then used, along with the 05 diameter flux measured for each detection, to determine the signal-to-noise ratio (S/N) of the sources. As we used an aperture smaller than the beam size, the mean 05 aperture depths in the detection maps are approximately a factor of two deeper than the noise per beams quoted above (with the caveat of a corresponding aperture correction).

The choice of the size of the detection aperture and the S/N cut for the sample selection was made based on a trade-off between purity and depth of the catalog. The final catalog consists of the 695¹⁸ sources that have a 05 aperture S/N ≥ 4.3 and fall within the primary beam of the ALMA maps. This threshold and aperture size was chosen to give us a 98% purity rate, ${P}_{{\rm{r}}}$ (2% contamination), calculated as follows:

$$\begin{eqnarray}&&{P}_{{\rm{r}}}=\displaystyle \frac{{N}_{p}-{N}_{n}}{{N}_{p}},\end{eqnarray} \tag{ 1 }$$

where N_p is the number of positive sources detected above the chosen S/N limit (i.e., 695) and N_n is the number of sources detected above the same limit in the inverted detection maps (made by multiplying the detection maps by −1, Figure 1).

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We confirm the behavior of the noise in our maps by comparing our number of "negative" sources from the inverted maps at our selected S/N threshold against that expected from a simple Gaussian distribution of independent synthesized beams (Dunlop et al. 2016). In AS2UDS, for our average restored beam size, there are roughly ∼ 450,000 independent beams across the 716 ALMA pointings. For Gaussian statistics we would then expect ∼8 "negative" sources at 4.3σ. However, as noted by Dunlop et al. (2016), based on Condon (1997), Condon et al. (1998), there are effectively twice as many statistically independent noise samples as one would expect from a naive Gaussian approach due to the non-independence of pixel values in synthesized imaging. This would result in an expected ∼ 16 "negative" sources, or 2.3% ± 0.5%, which is consistent with the number we detect.

For each of the detected sources, we then derived a 10 diameter aperture flux density from the primary beam corrected maps; these flux densities are aperture corrected and flux deboosted using the same methodology as Simpson et al. (2015a), as briefly described below.

To calculate the completeness and flux deboosting factors for our ALMA catalog, we inserted model sources into simulated ALMA maps and determined the properties of those that were recovered. We start with simulated noise maps, to make these as realistic as possible we used 10 residual maps output from casa (i.e., an observed ALMA map where the source flux from any detected sources has been removed). The maps were selected to match the distribution in observed σ₈₇₀ for all 716 AS2UDS pointings. Model sources with flux densities drawn from a steeply declining power-law distribution with an index of −2, consistent with Karim et al. (2013) and Simpson et al. (2015a), and intrinsic FWHM sizes drawn uniformly from a range 0''–09, were convolved with ALMA synthesized beams and inserted into 60,000 simulated noise maps. Then we applied our source detection algorithm and measured recovered fluxes as detailed above, with a successful recovery claimed for detections within the size of a synthesized beam, i.e., 06, from the injected model source position.

The result of these simulations is that we estimate our catalog is 98% ± 1% complete for all our simulated sources at S₈₇₀ ≥ 4 mJy, with the incompleteness exclusively arising from the most extended simulated sources (intrinsic FWHM >06). As found in Franco et al. (2018), our simulated maps show the intrinsic sizes of the submillimeter galaxies strongly effects the completeness fractions at low S/N. But, at our 4 mJy threshold, we are only missing a small number of the most extended galaxies. We note that our simulated sources had sizes which were uniformly distributed up to 09, whereas previous studies suggest median submillimeter sizes of ∼03 (Tacconi et al. 2006; Simpson et al. 2015b); therefore, the 98% ± 1% completeness is probably conservative.

We estimate the flux boosting, the effect of noise fluctuations in the overestimation of a source's flux density, by calculating the ratio of the flux density for each recovered simulated source to the original input flux density. The fact that noise in the maps is approximately Gaussian, combined with the steep counts of faint sources, means that we find that fluxes are typically overestimated in the lower flux bins. However, again brighter than S₈₇₀ ≥ 4 mJy, the flux deboosting becomes a minor correction with a median correction factor of 0.98 ± 0.04 for the SMGs considered in this paper.

The complete catalog of SMGs from AS2UDS, with full descriptions of the source extraction, flux density measurements and flux deboosting will be presented in Stach et al. (2018, in preparation).

We start by determining the fraction of the original SCUBA-2 sources fluxes that are recovered in the sources we detect in the corresponding maps from ALMA. In the flux regime of interest in this paper, S₈₇₀ ≥ 4 mJy, we find that we recover a median fraction of ${97}_{-2}^{+1}$% of the original SCUBA-2 flux from SMGs detected within the ALMA primary beam pointing of the corresponding SCUBA-2 parent source.

With respect to the "blank" maps, both the noise properties of the SCUBA-2 sources which resulted in "blank" maps and the noise properties of the ALMA observations of these maps are indistinguishable from those where ALMA detected an SMG. This suggests that these "blank" maps are not simply due to variations in the quality of the input catalog or follow-up observations. Similarly, it could be that many of the "blank" map sources are due to spurious false positives in the S2CLS parent sample. We test this by stacking Herschel/SPIRE maps at the locations of the 108 "blank" map sources, ranked in five bins of their SCUBA-2 flux. We recover emission in all the SPIRE bands (250, 350 and 500 μm) with flux densities between 7–20 mJy for all five flux bins. Even for the faintest 10% of SCUBA-2 sources with corresponding "blank" ALMA maps, we still recover SPIRE detections at 250 and 350 μm. Hence, we are confident that the majority of the "blank" maps are a result of genuine non-detections in ALMA and not false positive sources in the S2CLS map. However, these "blank" maps do typically correspond to fainter single-dish sources: the median flux of the "blank" maps is S₈₅₀ = 4.0 ± 0.1 mJy, compared with S₈₅₀ = 4.5 ± 0.1 mJy for the whole sample. Thus, it is possible that a strong increase in flux boosting in the original S2CLS catalog at S/N of ≲4–4.5σ (S₈₇₀ ∼ 3.6–4.0 mJy) may play a part in explaining why ALMA detects no SMGs in these maps. To remove this concern, in our analysis, we only consider the number counts brighter than S₈₇₀ ≥  4 mJy.

We conclude that with the sensitivities of our ALMA maps we can detect S₈₇₀ = 4 mJy SMGs in even the shallowest AS2UDS maps across the entirety of the primary beam. In addition, based on our simulated ALMA maps described above, we have shown we have with reliable measured flux densities for the complete sample of 299 S₈₇₀ ≥  4 mJy SMGs in the AS2UDS catalog presented here.

In Figure 2, we show the cumulative and differential number counts of the 299 870 μm selected SMGs from AS2UDS to a flux limit of S₈₇₀ = 4 mJy. Both the cumulative and differential number counts are normalized by the area of the S2CLS UDS map from which the original targets were selected: 0.96 degree². While the ALMA completeness factors are minimal for AS2UDS, the number counts do have to be adjusted for the incompleteness of the parent S2CLS survey. We correct our counts by factoring in the estimated incompleteness of the catalog of the S2CLS UDS map from Geach et al. (2017), who reported that the parent sample is effectively complete at ≥5 mJy, dropping to ∼88% at ≥4.5 mJy and ∼ 83% at ≥4 mJy.

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As in Karim et al. (2013), the errors are calculated from both the Poissonian error and the individual flux uncertainties added in quadrature, where the flux uncertainty error is the standard deviation of the mean of the counts for each bin based on 1,000 resamples of the catalog, assigning random flux densities to each source within their individual error margins, Table 1. We also compare these counts with those from the published parent single-dish counts of the S2CLS UDS field (Geach et al. 2017), and the earlier ALESS survey (Karim et al. 2013). To convert the S2CLS 850 μm counts to a common S₈₇₀, we use a factor of S₈₇₀/S₈₅₀ = 0.95, derived from a redshifted (z = 2.5), composite spectral energy distribution (SED) for SMGs from the ALESS survey (Swinbank et al. 2013), although we note that this correction is smaller than the estimated absolute calibration precision from S2CLS of 15% (Geach et al. 2017).

Table 1.  AS2UDS Number Counts

S870	$N(\gt {S}_{870}^{{\prime} })$ a	dN/dSb
(mJy)	(deg−2)	(mJy−1 deg−2)
4.5	${385.3}_{-7.7}^{+21.1}$	${168.5}_{-7.9}^{+14.8}$
5.5	${216.7}_{-6.6}^{+17.3}$	${110.5}_{-4.1}^{+12.1}$
6.5	${106.2}_{-3.5}^{+11.4}$	${52.6}_{-2.6}^{+8.3}$
7.5	${53.6}_{-2.5}^{+8.4}$	${24.1}_{-1.9}^{+6.0}$
8.5	${29.6}_{-1.9}^{+6.5}$	${9.5}_{-1.1}^{+4.2}$
9.5	${20.0}_{-1.8}^{+5.7}$	${9.4}_{-1.1}^{+4.2}$
10.5	${10.5}_{-1.2}^{+4.4}$	${5.2}_{-0.9}^{+3.5}$
11.5	${5.3}_{-0.9}^{+3.5}$	${3.1}_{-0.7}^{+3.0}$
12.5	${2.1}_{-0.6}^{+2.8}$	⋯
13.5	${2.1}_{-0.6}^{+2.8}$	${1.0}_{-0.5}^{+2.4}$
14.5	${1.0}_{-0.5}^{+2.4}$	⋯

Notes.

^a ${S}_{870}^{{\prime} }={S}_{870}-0.5{\rm{\Delta }}S$ where ΔS is 1 mJy. ^b"–" denotes fluxes where there is no change in the cumulative counts between the lower flux bin and the current bin.

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Compared to a single power-law fit, the number counts of SMGs show a steepening decline at brighter fluxes. As a result, the best fit to the differential number counts is with a double power-law function with the form

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dS}}=\displaystyle \frac{{N}_{0}}{{S}_{0}}{\left[{\left(\displaystyle \frac{S}{{S}_{0}}\right)}^{\alpha }+{\left(\displaystyle \frac{S}{{S}_{0}}\right)}^{\beta }\right]}^{-1},\end{eqnarray} \tag{ 2 }$$

where N₀ describes the normalization, S₀ the break flux density, and α and β the two power-law slopes. For our AS2UDS data, the best-fit parameters found are ${N}_{0}={1200}_{-300}^{+200}$ deg⁻², S₀ = 5.1 ± 0.7 mJy, $\alpha ={5.9}_{-0.9}^{+1.3}$ and β = 0.4 ± 0.1.

At S₈₇₀ ≥  4 mJy, we derive a surface density of ${390}_{-80}^{+70}$ deg⁻², corresponding to one SMG per ∼ arcmin² or one source per ∼130 ALMA primary beams at this frequency. Figure 2 shows a systematic reduction in the surface density of SMGs compared with the single-dish estimate at all fluxes. This reduction from the SCUBA-2 counts to AS2UDS is statistically significant for sources fainter than S₈₇₀ = 8 mJy, with a reduction of a factor of 28% ± 2% at S₈₇₀ ≥  4 mJy and 41% ± 8% at S₈₇₀ ≥  7 mJy. At the very bright end (S₈₇₀ ≥ 12 mJy), the number of SMGs is so low (just two in our ∼ 1 deg² field) that the reduction in the relative number counts is poorly constrained, 30% ± 20%. Our bright-end reduction does agree with that seen in Hill et al. (2018), where they found a 24% ± 6% reduction between 11–15 mJy in their SMA follow-up counts compared with the original SCUBA-2 parent sample. This agreement is unsurprising as a large number of their sources are drawn from our ALMA survey of the UDS field. We also note that, as with our earlier pilot study of UDS in Simpson et al. (2015a), that we do not see an extreme drop-off of the counts above S₈₇₀ ∼ 9 mJy as was suggested from the smaller-area ALESS survey (Karim et al. 2013).

As we discuss below, the main factor that appears to be driving the the systematically lower counts of SMGs from interferometric studies, compared with the single-dish surveys, is that a fraction of the brighter single-dish sources break up into multiple fainter sources (with flux densities of S₈₇₀ ≲ 1–4 mJy) in the interferometer maps and thus fall below the single-dish limit adopted for our counts. This effect has been termed "multiplicity" (Karim et al. 2013; Simpson et al. 2015a). An additional factor is the 12 ALMA "blank" maps of S2CLS sources brighter than ${S}_{870}^{\mathrm{deb}}\geqslant 4$ mJy, which also contribute to lowering the normalization of the number counts. These S2CLS sources, have a mean S/N of 5.8 ± 0.8, and are therefore unlikely to be spurious SCUBA-2 detections and our Herschel/SPIRE stacking confirms this; instead, the most likely explanation for their ALMA non-detection is "extreme" multiplicity, where the single-dish source breaks up into several faint SMGs below the detection limit of our ALMA maps. For these brighter SCUBA-2 sources with "blank" ALMA maps this would require that the single-dish source breaks up into ≥4 sources to result in a non-detection.

There are differing claims in the literature regarding the influence of multiplicity of SMGs on single-dish submillimeter surveys. This is a result of both the differing depths of the interferometric studies used to investigate this issue and the different definitions of "multiplicity" adopted in these works. Our survey has a relatively uniform sensitivity of σ₈₇₀ ∼ 0.25 mJy beam⁻¹, therefore we adopt a fixed S₈₇₀ limit to identify multiple SMGs. We follow Simpson et al. (2015a) and define a multiple map as any field with more than one S₈₇₀ ≥  1 mJy SMG within our ALMA Band 7 primary beam (i.e., within ∼ 9'' of the original SCUBA-2 detection locations). At the redshift of SMGs, this corresponds to borderline U/LIRG systems, L_IR ≥ 10¹² L_⊙ which have SFRs of the order of 10² M_⊙ yr⁻¹ (Swinbank et al. 2013). We also believe this is a more physical choice than, for example, using the relative submillimeter brightness of the two sources to decide if they constitute a "multiple," as the relative fluxes may have little relevance to their other physical properties (e.g., mass or redshift) that are essential to understand their significance.

In our full sample, we have maps with more than one S₈₇₀ ≥ 1 mJy SMG in 79 of the 716 observations (11% ± 1%). We note that at 1 mJy our ALMA observations are not complete, therefore this sets the multiplicity as a lower limit. The surface density of S₈₇₀ ∼ 1 mJy SMGs is ∼1 arcmin⁻², as estimated from unbiased ALMA surveys (Aravena et al. 2016; Dunlop et al. 2016). Hence, we expect to find one S₈₇₀ ∼ 1 mJy SMG per ∼19 ALMA primary beams or in ∼5% of the maps, compared to the observed rate of ∼11% (one per nine ALMA maps). We note, however, that the presence of a secondary source in these maps may act to increase the likelihood of the inclusion of that map into our sample by boosting the apparent SCUBA-2 flux into the S2CLS catalog. To address this potential bias, we estimate the multiplicity rate for the 179 brighter single-dish sources with deboosted SCUBA-2 flux densities of ${S}_{850}^{\mathrm{deb}}\geqslant 5$ mJy. The rate of multiples in these brighter SCUBA-2 sources is much higher 51/179 (28% ± 2%), suggesting that the presence of a detected secondary SMG in faint single-dish sources does not strongly influence the inclusion of that single-dish source into our parent catalog. Instead, the influence of multiplicity in faint single-dish sources is more likely to be seen through the presence of "blank" maps. Hence, we also place an upper limit on the multiplicity in our full survey by assuming that all of the blank fields are a result of the blending of multiple faint SMGs, giving 187/716 (26% ± 2%) multiples.

As implied above, the multiplicity appears to depend on the single-dish flux; as expected, as the inclusion of emission from other SMGs within the beam can only act to increase the apparent flux of the (blended) single-dish source. As described in Section 1, early observations suggested that roughly a third of S₈₅₀ > 5 mJy single-dish sources could be blends of multiple SMGs, with this rate increasing to 90% for S₈₇₀ > 9 mJy (e.g., Karim et al. 2013). As shown in Figure 3, for AS2UDS, we find a frequency of multiplicity (ignoring "blank" maps) of 28% ± 2% for ${S}_{850}^{\mathrm{deb}}\geqslant 5$ mJy rising to ${44}_{-14}^{+16}$% at ${S}_{850}^{\mathrm{deb}}\geqslant 9$ mJy.

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In Figure 3, we also plot the fractional contribution of each secondary and tertiary ALMA SMG (ranked by flux density) to the total recovered ALMA flux density of all the SMGs for each field with multiple SMGs. The mean fraction of the total flux contributed by the secondary component is 34% ± 2% with no significant variation of this fraction as a function of the original deboosted SCUBA-2 source flux. The 64% ± 2% contribution from the primary components in maps with multiple SMGs is broadly consistent with the semi-analytic model of Cowley et al. (2015), which suggested that ∼ 70% of the flux density in blended sources would arise from the brightest component.

3.3.1. Physical Association of the Multiple SMGs

Based on our Cycle 1 pilot study, Simpson et al. (2015a) showed that the number density of secondary SMGs in the maps of their 30 bright SCUBA-2 sources was 80 ± 30 times that expected from blank-field number counts, suggesting that at least a fraction of these SMGs must be physically associated. Using our large sample, we now seek to test this further. The most reliable route to test for physical association between SMGs in the same ALMA map would be to use spectroscopic redshifts for the SMGs. However, as the current spectroscopic coverage of SMGs in AS2UDS is sparse, we instead exploit photometric redshifts to undertake this test. We use the photometric redshift catalog constructed from the UKIDSS DR11 release (W. Hartley et al. 2018, in preparation), where a full description of the DR11 observations will be given in O. Almaini et al. (2018, in preparation). These photometric redshifts are derived from twelve photometric bands (U, B, V, R, I, z, Y, J, H, K, [3.6], [4.5]) and applied to 296,007 K-band-detected sources using eazy (Brammer et al. 2008); details of the methodology can be found in Simpson et al. (2013). The accuracy of these photometric redshifts is investigated in W. Hartley et al. (2018, in preparation) from comparison with the ∼6,500 sources in the UKIDSS DR11 catalog that have spectroscopic redshifts, finding $| {z}_{\mathrm{spec}}-{z}_{\mathrm{phot}}| /(1+{z}_{\mathrm{spec}})\,=0.019\pm 0.001$ with a median precision of ∼9%. Around 85% of the ALMA maps fall in regions of the UDS with high-quality photometric redshifts and these are considered in the following analysis.

In Figure 4, we plot the distribution of the differences in photometric redshifts (Δz_phot) for pairs of SMGs in those single-dish maps with multiple ALMA-detected SMGs. We limit our analysis to SMGs that fall within the region with high-quality photometric redshifts and which have K-band detections within 06 radius from the ALMA positions (497 of the 695 SMGs) for both sources in the map. This yields 46 pairs of SMGs (92 SMGs in total) from the 164 SMGs in the 79 maps with multiple SMGs. We find that 52% of these pairs (24/46) have Δz_phot <  0.25. We note that 2'' diameter apertures were employed for the photometry in the DR11 catalog, therefore the Δz_phot was additionally calculated for only pairs that are separated by greater than 2'', thus removing the possibility of neighbors contaminating photometry and thus photometric redshifts. This still results in 53% of pairs having Δz_phot <  0.25 (23/43).

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To assess the significance of this result, we next quantify whether the 24 pairs of blended SMGs with Δz_phot < 0.25 is statistically in excess of expectations for 46 random SMG pairs. To do this, we determine the expected distribution of Δz_phot for pairs of SMGs randomly selected from the 497 SMGs with high-quality photometric redshifts across the full field, and plot this in Figure 4. To perform this test, we sample the random distribution of our unassociated SMGs 10,000 times, each time drawing 46 pairs, and test how frequently >52% of these are found to have Δz_phot <  0.25. This analysis shows that the median fraction of random pairs with Δz_phot <  0.25 is 20% ± 2% compared with the 52% for the actual pairs of SMGs. This strongly suggests that a significant fraction of the single-dish sources that resolve into multiple optically bright (e.g., those with photometric redshifts) SMGs are in fact physically associated galaxies on projected angular scales of ∼10–100 kpc scales. If we assume that all pairs without photometric redshifts for both SMGs are physically unassociated, a conservative estimate, then comparing with the total number of ALMA fields with multiple SMGs, we can place a lower limit of at least 30% (24 pairs out of 79) on the fraction of all multiple SMG fields arising from closely associated galaxies. This is consistent with previous spectroscopic studies of SMG multiples e.g., ∼40% of SMG pairs physically associated combining the estimates from Wardlow et al. (2018) and Hayward et al. (2018). Of course, to truly test this requires a spectroscopic redshift survey of a much large sample of these multiple SMG systems.