A DEEP 1.2 mm MAP OF THE LOCKMAN HOLE NORTH FIELD

The raw bolometer time stream data were reduced using Robert Zylka's MOPSIC¹² pipeline, which is distributed in parallel with IRAM's GILDAS package. MOPSIC is the standard package for reducing deep MAMBO on-the-fly maps (see, e.g., Greve et al. 2004, 2008; Voss et al. 2006; Bertoldi et al. 2007). We now briefly outline the steps of the MOPSIC reduction pipeline; for further details see Greve et al. (2004). The pipeline removes spikes in the time stream data stronger than 5 × the instantaneous bolometer rms noise. It also subtracts a third-order polynomial baseline in time and performs correlated signal filtering on the bolometer time streams to identify and remove foreground atmospheric emission that affects many bolometers simultaneously. Each bolometer is correlated with an annulus of neighboring bolometers within a 150'' radius, and the average signal of the twelve most highly correlated bolometers is subtracted away. The filtered time streams are then binned into 35 × 35 pixels, and a signal map is reconstructed using the SAA algorithm. The individual signal maps are combined into a mosaic image by averaging the map flux density at each pixel weighted by the local inverse variance. Our "optimally filtered" signal map (Figure 2) was created by applying a PSF-matched filter to the final mosaic image (Section 5.1).

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In addition to the signal map, the MOPSIC pipeline also produces a weight map that is locally proportional to the inverse variance in the signal map. By enforcing that the Gaussian distribution of the signal-to-noise (S/N) map pixel distribution has a standard deviation of unity, we normalize the weight map so that it can be used to find the local rms noise, $\sigma =1/\sqrt{W}$, in the image (see Figure 1). Using the weight map as a guide to find the local rms noise for a detection is more robust than using the nearby pixels themselves, because locally the pixels are affected by the negative residual sidelobes of SAA reconstruction as well as those of other bright nearby sources.

Because of the telescope's effective triple-beam PSF, each source in the field injects negative as well as positive flux into the map. To generate realizations of source-free maps, hereafter referred to as "noise maps," we removed the negative and positive flux from undetected as well as bright sources using two different techniques. We go on to use the different results for different purposes.

We constructed the first type of noise map with a technique common in MAMBO data analysis (see, e.g., Greve et al. 2004, 2008; Bertoldi et al. 2007), using the data reduction pipeline to scramble the known locations of the bolometers within the image plane. During reconstruction of the time stream data, this has the effect of smearing the flux from any one source into an area on the sky of approximately 200 arcmin², reducing the intensity of the source's peak flux contributions by a factor of ∼10³ and making the peak flux contribution from our strongest sources ∼200 times fainter than the rms noise. Because the telescope's chopping ensures that the mean of the map is zero, there is no residual baseline increase as the negative flux contributions are identically smoothed. These "shuffled noise maps" are simple to construct, but it is cumbersome to produce large numbers of them since each requires a full reduction of the data using the MOPSIC pipeline. Therefore, we use the shuffled noise maps only to estimate the noise of our "full" data during source extraction (Section 5.1) as well as in the Monte Carlo simulation of completeness (Section 5.5).

We needed to develop a different technique for creating noise maps in order to quickly generate thousands of independent noise realizations of chopped data for our P(D) analysis. For this we subtracted subsets of the data that are "jackknifed" in the sense that we remove fractions of the original data first. One full image of our field is created using the data from only one bolometer in the array at a time. All bolometers other than the one of interest are masked away after the correlated signal filter is applied, so the data still receive the benefit of correlated sky noise subtraction. Two half-sets of these images are then selected at random and subtracted from each other to produce one realization of noise. This technique is similar to the jackknifing by scan commonly used for AzTEC data (see, e.g., Scott et al. 2008, 2010; Perera et al. 2008; Austermann et al. 2010), in that each jackknifed subset uses the scanning information of every available map. Use of this information is especially important for our chopped data if we are to remove negative flux artifacts from the triple-beam PSF as well as positive flux. These "jackknifed noise maps" are more amenable to mass production and are guaranteed to remove all contributions from a source however faint, so they are used in our pixel flux distribution (PFD) analysis (Section 4) and in our Monte Carlo simulations to estimate numbers of spurious detections (Section 5.4).

The PFDs for S/N maps created with both jackknifed and shuffled noise exhibit random Gaussian noise to high precision (Figure 3), with reduced chi-square for standard normal distribution fits of 1.0 ± 0.2 and 1.2 ± 0.2, respectively.

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Our simulated sky maps are constructed by populating noise maps with simulated sources. Careful construction of these maps is important for the fluctuation analysis described below (Section 4), for which our method relies entirely on our ability to authentically reproduce the signal from the MAMBO array so as to faithfully reproduce the PFD. Thus, when adding sources into a noise map, we need to take into account the position-dependent negative sidelobes as well as the position-independent positive flux profile for each injected source.

To handle the varying PSF properly, we take an approach similar to that of Greve et al. (2008) and model the changes in the PSF explicitly as a function of position. We use the MOPSIC pipeline script map_negres.mopsic, which will calculate the expected negative residual pattern on the sky in equatorial coordinates for a given set of observations and an ideal, gridded, input source model. As an input we used an array of ideal Gaussian point source profiles, each with 11'' FWHM, spanning the entire field and spaced as closely as possible without having the sidelobes overlap. This minimum spacing is set by our larger chop throw (42'' for all of our "best" data and the overwhelming majority of our "full" data set). The result is an array showing the full PSF near any location in the map (see Figure 4). Because the ∼84'' spacing is less than the 300'' extent of each individual map and the 120'' separation between map pointing centers, the PSF morphologies change slowly from one to the next. We thus generate an authentic point source response in a simulated map by using the closest available PSF relative to the position of a given injected source.

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We extracted sources from our "best" and "full" maps in a three-step process. First, we minimized the chi-square statistic for a two-dimensional Gaussian profile with an 11'' FWHM fit at each pixel center. This minimization was achieved quickly by using a matched-filter convolution (see, e.g., Serjeant et al. 2003). Given a signal image S_ij, an image of the local rms noise σ_ij, and a smaller array describing the telescope's PSF P_xy, the chi-square statistic for a source with flux F located at pixel (i, j) can be written as

$$\begin{equation} \chi ^2(F|i,j) = \sum _{xy} \left(\frac{S_{i-x,j-y} - F P_{xy}}{\sigma _{i-x,j-y}}\right)^2. \end{equation} \tag{ 6 }$$

We ignored the position-dependent negative sidelobes when we applied the matched filter and used only the central Gaussian profile, since the negative residual flux reaches only ⩽5% of the peak positive intensity (see Figure 4) in the map interior. Additionally, our significant detections are on average farther away from each other than the largest chop throw used during the observations (48''), so their effect on our flux measurements will be less than 5% of our strongest sources' flux densities (i.e., ≲ 0.25 mJy).

By finding the minimum of χ²(F|i, j) as a function of F and determining the associated uncertainty ΔF (Serjeant et al. 2003), we obtained

$$\begin{equation} \frac{d\chi ^2}{dF}=0 \longrightarrow \frac{F}{\Delta F} = \frac{ \sum _{xy} S_{i-x,j-y} W_{i-x,j-y} P_{xy} }{\sqrt{{\sum _{xy} W_{i-x,j-y} P_{xy}^2}}} \end{equation} \tag{ 7 }$$

as the S/N of each pixel, in terms of the weight map W defined in Section 3.1. Next, we located the centroid of each source to sub-pixel precision by fitting the PSF to the region in the original signal map at the location of each significant peak in the S/N map, allowing the position of the Gaussian to vary. Figure 7 shows the uncertainty in this best-fit centroid position, derived via Monte Carlo simulations. For the typical flux densities of our significant detections, the average offset between injected and recovered centroids is 1''–3''. Finally, we computed the best-fit flux density by taking the matched-filter-weighted average of the flux map, this time with the PSF kernel shifted by interpolation to the more precise location of the source centroid. After the flux density of each source was recorded, the source was removed from the map by subtracting the flux-scaled PSF from the source location before we searched for the next most significant detection.

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By propagating the uncertainty in the signal image through the χ²-minimization process, Serjeant et al. (2003) have shown that the uncertainty in the resulting best-fit flux density at position (i, j) is

$$\begin{equation} \Delta F(i,j) =\frac{1}{\sqrt{\sum _{xy}\,W_{i-x,j-y}\,P_{x\,y}^{\,2}}}. \end{equation} \tag{ 8 }$$

We find that this expression consistently underestimates the uncertainty in our map. The reason is simply that the derivation by Serjeant et al. (2003) implicitly assumes that the Gaussian noise in the flux image is spatially uncorrelated. In our images, the noise is correlated on a length exactly matching the FWHM of the telescope PSF, and pure noise fluctuations can be amplified along with the real point sources. In order to correct for this underestimate we (1) produce an optimally filtered image and weight map, (2) use this filtered image and weight map to make an S/N ratio map, and (3) rescale the filtered weight map such that the standard deviation of the S/N map equals unity. This empirical calibration corrects for the effects of correlated noise and produces a map with accurate post-filtered flux density uncertainties. The accuracy of the method is evidenced by its matching the predicted number of positive excursions in a correlated Gaussian field as a function of S/N level (see Section 5.4).

The quoted 1σ errors (see columns S^Best_ν and S^Full_ν in Table 2) for each detection correspond to the rescaled version of ΔF evaluated using the shuffled noise map at the location of the source (see Section 3.2). This noise map describes the Gaussian noise of the observations more faithfully than the original signal map, which overestimates the noise by ∼5% due to the positive and negative sidelobes from bright sources. Because our noise is well above the estimated confusion limit (Section 5.3), Gaussian random fluctuations are the dominant source of uncertainty in our measurements.

As discussed in Section 2, control software problems during our first two semesters of observations undermined our confidence in the reliability of the resulting maps. To assess whether the "full" map could be trusted for bright source extraction, we performed two comparisons between our "full" data and observations with pristine calibration. For the first comparison, we carried out the source extraction steps described in Section 5.1 for both the "best" and the "full" data sets and compared the properties of the sources recovered from each. Specifically, we began by choosing the eight sources with S/N ⩾ 4.5σ detections in our "best" map: above this threshold, we expect to see fewer than one spurious detection (Section 5.4). All eight of these sources are recovered with ⩾5.0σ significance in the "full" map. Figure 8 shows the locations of these sources in the field, and the locations of the additional ⩾5.0σ detections in the "full" map. Each of the eight sources increases in significance between the "best" and the "full" maps. Additionally, all but one of the "full" map's 17 ⩾5.0σ sources are identified in the "best" data at lower significance. One source kept the same significance because it lies in the northeast corner of the field, where observations in the first two semesters contribute little additional sensitivity. Further, for these 17 sources, the ratio of the flux densities in the "best" and "full" maps is consistent with unity (Figure 9).

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Next, we compared our "full" map to MAMBO on–off photometry by Fiolet et al. (2009) of Spitzer-selected high-redshift starburst candidates in the LHN. We tabulated the map flux densities at the positions of the 13 galaxies in their sample that lie within our map's footprint (two of these turn up as significant detections in our "full" map; we use these sources' non-deboosted flux densities here) and compared them to the flux densities reported by Fiolet et al. (2009). We found that the flux densities from the two significant detections as well as those from 10 of the 11 non-detections are consistent to within 1σ (see Figure 10; one source is only consistent to within ∼1.5σ).

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These two successful consistency checks lead us to conclude that the errors in our first two semesters' data are not at a level that compromises point source detection, at least for high-significance sources. We have therefore proceeded to define our source catalog on the basis of the "full" map. Since the fluctuation analysis described in Section 4 relies on the authentic reproduction of the field's noise properties and low-S/N fluctuations, we have restricted this analysis to the "best" data only.

Random Gaussian noise is the uncertainty in the total flux density inside any single beam on the sky due to random fluctuations, while confusion noise is an additional uncertainty in the flux density of a single source due to the contributions of faint sources within that beam. The "confusion limit" is defined as the flux density threshold at which confusion noise significantly affects the measured flux density of a source, and is commonly taken to be the flux density above which the integrated number counts of all brighter sources reach ≃ 0.033 per beam (Condon 1974). In our map, this rule gives ≃ 0.9 mJy using θ_B = 156 (in the smoothed ↔ filtered version of our "full" map) and assuming our best-fit number counts (Section 4).

We have also made a direct estimate of the confusion noise by comparing the noise in the central regions of the filtered "best" map to the same region in a series of filtered jackknifed noise maps. Since the jackknifed noise maps remove confused as well as bright sources, the increase in average rms noise in this relatively uniform region indicates our map contains confusion noise at the level of σ_C ≃ 0.24 mJy beam⁻¹. As a consistency check, we have also estimated the confusion noise by generating simulated maps with source populations following our best-fit number counts from 0.05 mJy up to the confusion limit of 0.9 mJy. Due to the central limit theorem, these faint and confused maps with zero mean have roughly Gaussian PFDs and provide approximations of the confusion noise, assuming our model of the number counts. The standard deviation in these maps is 0.21 mJy beam⁻¹, in agreement with the measured confusion noise within the uncertainties of the number count model. We therefore adopt the measured value of σ_C ≃ 0.24 mJy as our estimate of the confusion noise. The average uncertainty in the flux density of a source in our catalog is 0.62 mJy, indicating that confusion does not dominate our noise budget.

We estimated the number of spurious detections as a function of S/N by running our source extraction algorithm on various noise maps (see Section 3.2). We tested jackknifed noise maps, shuffled noise maps, and simple Gaussian random numbers. The Gaussian random numbers had a spatially varying standard deviation matched to the weight map of the observations. Figure 11 shows the mean total numbers of spurious detections found in 10³ jackknifed and Gaussian number noise maps, and in 10² shuffled noise maps, as a function of S/N. All three styles of noise map are consistent with each other in their ability to produce spurious detections with S/N ⩾ 3.0. This result confirms that for the purposes of extracting high-significance detections, the shuffled noise maps are just as "source-free" as the jackknifed noise maps. Additionally, both are consistent with a Gaussian distribution down to 3.0σ (and likely consistent with Gaussian noise at all S/N, as implied by the PFD histograms in Section 3). The overplotted curve in Figure 11 shows the expected number of excursions above a given S/N level in any isotropic and homogenous Gaussian random field, derived (and thus only formally valid) for high excursions (see, e.g., Chapter 6 of Adler 1981). The agreement at high S/N indicates that our noise maps and source extraction algorithm are well behaved. For both the "best" and "full" maps, we expect 0.8 (5.4) spurious sources with S/N ⩾ 4.5 (4.0).

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A source at high risk of being a spurious detection can be identified by calculating the total probability that the deboosted flux density is ⩽0 mJy (see, e.g., Austermann et al. 2010; Scott et al. 2010), hereafter referred to as P(S < 0). Using the threshold of P(S < 0) ⩾ 0.10 used by Austermann et al. (2010), we identify only one (ID 33) high-risk spurious detection in our S/N > 4.0 sample (see Table 2).

We estimate the completeness in our data using the Monte Carlo method of searching for injected sources of varying flux density. The inhomogeneous noise in our map means that sources of identical flux density have different probabilities of being detected in different locations. We account for this effect statistically by performing the completeness simulation assuming sources have equal probability of being located anywhere in the map during the injection process. Although the high-redshift star-forming galaxy population that our observations trace is likely to exhibit clustering, with the brightest galaxy mergers occurring in the most massive dark matter halos (see, e.g., Weiß  et al. 2009), the large Δz interval to which millimeter selection is sensitive tends to weaken angular clustering signatures (see however Section 5.8). We inject model sources with known flux density into our original signal map one at a time at random positions and search for them using the same source extraction algorithm used to create our source list. If an artificial source is recovered with S/N ⩾ 4.0 within 11'' of the injected location, it is considered detected. The injection process was repeated 10³ times for each flux density in a logarithmic grid from 1.0 mJy to 10.0 mJy; the average recovery percentage is shown in Figure 12. Our map is 80% complete at 3.7 mJy and 50% complete at 2.6 mJy. We also tabulated the angular separations between the injected and recovered source positions to characterize the uncertainties in the positions of our actual significant detections (ignoring telescope pointing errors). For S_1.2 mm ≳ 2 mJy, the average position error 〈Δθ〉 ⩽ 3'' (see Figure 7).

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To correct the measured flux densities of our detections for the effect of "flux boosting," we use the Bayesian technique described in Coppin et al. (2005), which because of its versatility in handling both chopped and unchopped data has been adapted for use at many wavelengths (e.g., Coppin et al. 2006; Greve et al. 2008; Scott et al. 2008, 2010; Perera et al. 2008; Austermann et al. 2010). Using Bayes's theorem and the prior information of the number count functional form found from our P(D) analysis (Section 4), the probability that a source has true flux density S₀ given a measurement S with uncertainty σ is equal to

$$\begin{equation} P(S_0|S,\sigma) = \frac{P(S,\sigma | S_0) P(S_0)}{P(S,\sigma)}, \end{equation} \tag{ 9 }$$

where P(S, σ|S₀) is the posterior probability, P(S₀) is the prior flux density distribution, P(S, σ|S₀) is the likelihood, and P(S, σ) is the prior measurement distribution. P(S, σ) is independent of S₀, so only acts to normalize the expression such that ∫P(S₀|S, σ)dS₀ = 1; hereafter, it will be ignored. We have shown that the uncertainty in our map is dominated by Gaussian random noise, so P(S, σ|S₀) takes the form

$$\begin{equation} P(S,\sigma | S_0) \propto {\rm exp}\left[-\frac{\left(S-S_0\right)^2}{2\sigma ^2}\right]. \end{equation} \tag{ 10 }$$

To estimate the prior flux distribution in the map (P(S₀)), we assembled a PFD containing the pixels from 10⁴ noise-free random sky realizations that used the best-fit number count parameters from our P(D) analysis. The peak value and 68% double-sided confidence intervals of the resulting posterior probability function (P(S₀|S, σ)) were found numerically for each measured flux density and uncertainty. Figure 13 shows four examples of the deboosting process, each in a different regime of source S/N. If the S/N is too low, and the integration of the confidence intervals does not converge, we instead use an analytic formula to estimate the deboosted flux density. For this, we generalize the formalism of Hogg & Turner (1998) to a Schechter function and locate the maximum of the posterior flux distribution:

$$\begin{equation} P(S_0|S,\sigma) \propto S_{0}^{-\delta ^{\prime }}\,\exp \left[ -\frac{S_{0}}{S_{\rm exp}^{\prime }}- \frac{\left(S-S_0\right)^2}{2\sigma ^2}\right], \end{equation} \tag{ 11 }$$

where δ' and S'_exp are the power-law slope and exponential scale factor of the Schechter function, respectively. By solving for S₀ when the derivative of the above expression vanishes, we find the highest posterior probability to be achieved for

$$\begin{equation} S_{\rm true} = \frac{S \, S_{\rm exp}^{\prime }-\sigma ^2 + \sqrt{ (\sigma ^2-S \, S_{\rm exp}^{\prime })^2 -4 \,\delta ^{\prime }\,S_{\rm exp}^{\prime 2}\sigma ^2 }}{2 S_{\rm exp}^{\prime }}. \end{equation} \tag{ 12 }$$

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To ensure that our adaption of the Bayesian method of flux deboosting returns a properly corrected estimate of the true number counts, we performed a Monte Carlo simulation to directly calculate the observed number counts of random sky realizations populated with source distributions following our best-fit number counts (see, e.g., Coppin et al. 2006). Figure 14 shows the results of this consistency check. This simulation demonstrates that the Bayesian method of flux deboosting performs well in recovering the original injected number counts. The residual scatter of the average recovered number counts around the average input model in Figure 14 demonstrates the level of systematic error in the algorithm, which is significantly smaller than the statistical error of our differential number count estimate (see Figure 15).

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While our catalog of detections includes all sources with S/N > 4.0, we use only detections with S/N > 4.5 for our direct calculation of the number counts because, above this threshold, we expect to detect less than one spurious source (see Figure 11). Table 3 presents integral and differential number counts after correction for completeness and flux boosting. Figure 15 shows our directly calculated number counts, along with the 95% confidence regions for the best-fit power-law and Schechter function models of the differential number counts found from the P(D) analysis. These two independent methods of estimating the number counts are in agreement with each other. This consistency is encouraging because the P(D) analysis and the direct estimate of number counts depend on the faint and bright pixel values in different ways.

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The group of sources in the southeast corner of our field, as well as the large void in the center, prompted us to perform a clustering analysis to determine whether or not the distribution of sources in our map is statistically clustered or not. To perform the analysis, we used the Landy & Szalay (1993) correlation function estimator:

$$\begin{equation} w(\theta)=\frac{{\it DD}-2{\it DR}+{\it RR}}{{\it RR}} \end{equation} \tag{ 13 }$$

with variance

$$\begin{equation} \langle \Delta w(\theta) \rangle ^2 \simeq \frac{(1+w(\theta))^2}{{\it RR}} \end{equation} \tag{ 14 }$$

(Gawiser et al. 2006). In the equations above, DD, RR, and DR represent the normalized numbers of unique galaxy–galaxy, random–random, and galaxy–random pairs with angular separations θ ± dθ/2. This estimator is used frequently in extragalactic deep field analyses (e.g., Borys et al. 2003; Scott et al. 2006; Weiß  et al. 2009) and has been shown to have nearly Poisson variance and zero bias (Landy & Szalay 1993). We take into account the geometric boundary of the map and the variation in detectability with position by generating the random locations with the same Monte Carlo algorithm used for the P(D) analysis. We inject ensembles of sources following our best-fitting number counts into a noise map at random locations and use the positions of sources detected with S/N > 4.0 as our random coordinates. This Monte Carlo technique is important for ensuring that we do not misinterpret depth variation in the map as a clustering signal.

To confirm that this technique is unbiased, we also performed the full clustering analysis on only random positions to check that we recovered a flat w(θ) = 0 response (see Figure 16). For small separations (≲ 2'), however, it turns out that w(θ) does not return zero in our data: depending on the position within the map, the negative sidelobes can suppress the flux densities of nearby sources enough to lower their S/N ratios below the detection threshold. This effect begins to have an effect at ≃ 2 × the chop throw (of which the maximum used in any semester was 48'') and has a strong effect at separations ⩽1 × chop throw. Because this effect suppresses the detection of RR and DD pairs but not DR pairs, the zero-clustering baseline for chopped data like ours is less than zero at these small angles. In order to assess the clustering in the map while taking this bias into account, we measure the effective clustering relative to the zero-clustering baseline for these separations (≲ 2').

The result of our Monte Carlo clustering analysis is shown in Figure 16. We find a small clustering signal when using all detections with S/N > 4.0 that agrees reasonably well with the angular correlation function measured by Scott et al. (2006), who combined many different SCUBA 850 μm blank field maps, and that shows stronger clustering (albeit at lower S/N) than the correlation function measured by Weiß  et al. (2009) at 870 μm in the ECDF-S. Williams et al. (2011) have analyzed the clustering of 3.0–3.5σ 1.1 mm detections in a 0.72 deg² map of the COSMOS field with ASTE/AzTEC, concluding that it is difficult to recover reliable clustering parameters for SMGs from maps whose angular resolution and total area are limited. This result argues for caution in interpreting our clustering analysis, although we do benefit to an extent from MAMBO's relatively high angular resolution. An interesting feature in our correlation function is the spike near θ ≃ 4'. This signal is due to the rich group of sources in the southeastern corner of the map, all at typical relative spacings of a few arcminutes from each other (see also Section 7.4). It may be noteworthy that Weiß  et al. (2009) find a ∼2.4 σ spike above their best-fitting model of ω(θ) at a scale of ∼5', near where Williams et al. (2011) also detect a slight positive excess in ω(θ). When performing the analysis on only our (27) most significant sources with S/N > 4.5, we find no significant clustering signal.

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Following the analysis of, e.g., Borys et al. (2003) for the SCUBA "Supermap," we also use the method of Scott & Tout (1989) to analyze the cumulative distribution of nearest neighbors to test whether our galaxy positions are consistent with being drawn from a random distribution (see Figure 17). Because the nearest neighbor analysis is sensitive to the total number of positions used, we use the 41 most significant detections in each Monte Carlo realization, instead of all of those detections with S/N > 4.0 as in the correlation function analysis. Because the number counts rise quickly, the S/N of the least significant discovered source varies, but is always close to 4.0. A Kolmogorov–Smirnov test rules out the null hypothesis that our significant detections are drawn from a random position distribution at the 95% confidence level, implying that the source locations in the map (e.g., defining the southeastern clump and the central void) are not arranged randomly.

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We used the NRAO VLA Sky Survey and the deep SWIRE number counts of Condon et al. (1998) and Owen & Morrison (2008), respectively, in the calculation of P to assess the significance of 20 cm counterparts. The 20 cm VLA pointing of the LHN has a central rms sensitivity of 2.7 μJy, rising to ∼4–5 μJy near the edges of our MAMBO map. When we compare our 41 S/N > 4.0 detections to the 5σ 20 cm catalog of Owen & Morrison (2008), 44% (18) have robust counterparts and 41% (17) have likely counterparts. We have also reexamined the 20 cm map in the vicinity of the remaining MAMBO sources and have identified one additional robust counterpart (ID 9), two likely counterparts (ID 28 and 36), and one unlikely counterpart (ID 20) at the 4σ–5σ level. We also deblended one likely counterpart into one robust and one unlikely counterpart (ID 17). After including these additional sources, 49% (20) of our MAMBO detections have robust counterparts, 44% (18) have likely counterparts, and 7% (3) have unlikely or no detected counterparts. We performed a Monte Carlo simulation to test the reliability of our P values and found that 4.9% ± 0.2% of randomly chosen positions within our MAMBO field have a likely radio counterpart (P < 0.05) within 8'', confirming the validity of the high number of robust associations. We expect ∼5 spurious detections above S/N > 4.0; thus, the handful of sources with unlikely or no radio counterparts may be spurious detections if they do not lie at a very high redshift. One MAMBO source (ID 32) with an unlikely (P_20 cm = 0.056) radio counterpart also has a likely (P_250 μm = 0.016) Herschel counterpart (Magdis et al. 2010), arguing against its being a spurious detection.

We have extracted 50 cm flux densities from the GMRT map (F. N. Owen et al. 2011a, in preparation) with the same technique used at 20 cm and 50 cm (see Owen & Morrison 2008; F. N. Owen et al. 2011b, in preparation). The uncertainties listed in Table 2 reflect the local rms noise in the image and do not include a ∼3% calibration error or a spatially varying GMRT pointing error. Two of the 50 cm detections are heavily blended with bright neighbors, so for these counterparts we report only tentative fluxes. Of the 40 tabulated 20 cm counterparts (including the two with P > 0.05), all 40 have 50 cm counterparts. The one 20 cm non-detection (within 8'') is also a 50 cm non-detection.

To search for 90 cm counterparts, we used the 90 cm radio catalog of F. N. Owen et al. (2011b, in preparation), which has an rms sensitivity of 10 μJy. Of our 41 MAMBO sources, 24% (10) have 90 cm counterparts. Each of the ten 90 cm counterparts is also detected at 50 and 20 cm with P_20 cm < 0.05.

In addition to SWIRE 24 μm observations of the LHN (3σ depth of 209 μJy), there exist deeper Spitzer/MIPS data with a 3σ depth of 18 μJy (F. N. Owen et al. 2011b, in preparation). We searched for 24 μm counterparts to our MAMBO detections in this deeper MIPS image. To calculate P statistics for 24 μm counterparts, we used the counts of Béthermin et al. (2010). A Monte Carlo simulation of the 24 μm P-statistic finds that 4.7% ± 0.3% of random positions yield a counterpart with P < 0.05. Within our sample of 41 MAMBO sources, 20% (8) have robust 24 μm counterparts, and 29% (12) have likely counterparts.

Only one source (MM J104522.8+585558 = ID 26) has a likely X-ray counterpart (CXOSW J104523.6+585601; Wilkes et al. 2009). The X-ray source is at a distance of 72 and has a broadband (0.3–8.0 keV) flux of (2.5 ± 1.1) × 10⁻¹⁵ erg cm⁻² s⁻¹ (Polletta et al. 2006). By using the 2.5–7 keV flux of 1.58 × 10⁻¹⁵ erg cm⁻² s⁻¹ together with the Chandra/SWIRE counts from 2–8 keV (Wilkes et al. 2009), we can set an upper limit on the probability of chance association of P ≲ 0.02.

Previous deep surveys at 1.2 mm using MAMBO (e.g., Greve et al. 2004; Bertoldi et al. 2007) have returned directly calculated 1.2 mm number counts in the Lockman Hole East (LHE), ELAIS-N2, and COSMOS fields. The parameters of these surveys are listed in Table 4; we compare their results to our directly calculated counts, as well as to our best-fit P(D) models, in Figure 15. We find that our power-law slope is consistent with their results, but our results have a lower overall normalization. This difference in normalization might be due to the different methods used in the number count calculations. We have used the Bayesian method of flux deboosting presented in Coppin et al. (2005), and only include our most significant detections in the calculation. The analyses of Greve et al. (2004) and Bertoldi et al. (2007) use the method of injecting sources into noise maps to determine their flux deboosting correction, and include sources with lower S/N in their number count calculation. In principle, any S/N cutoff would be acceptable for the latter calculation as long as the completeness correction uses the same threshold; however, lower S/N thresholds will lead to more spurious detections. Both of these effects could be contributing to their higher normalization. However, considering the relatively large error bars on all the measurements and the internal variation among the ELAIS-N2, LHE, and COSMOS data sets themselves, the results are still nearly consistent.

Because a single power-law parameterization of the number counts is commonly used to compare the results of deep surveys, we begin by noting that our best-fit power-law index (δ = 3.14^+0.14_{− 0.18}) is consistent with the results of surveys at other wavelengths that fit their number counts using a similar (single power-law) model. Coppin et al. (2006) find 850 μm power-law indices of δ = 2.9 ± 0.2 and δ = 3.0 ± 0.3 in the LHE and Subaru/XMM-Newton deep fields, respectively. Using Bolocam data at 1.1 mm, Laurent et al. (2005) estimate a power-law index of δ = 3.16 from directly calculated counts in the Lockman Hole (LH). However, Maloney et al. (2005) performed a P(D) analysis on the same 1.1 mm Bolocam data and find δ = 2.7^+0.18_{− 0.15}. Although it is well within the 1 σ uncertainties of the Laurent et al. (2005) result, the latter slope differs from ours by >2σ. In this case, differences in the methods of our P(D) analyses might be the differentiating factor. Maloney et al. (2005) also used chopped observations in their analysis, for example, but ignored the effects of chopping on the PFD. It is possible that by not including the negative residual flux in their P(D) analysis, they required many fewer faint sources to match the pixel distribution of the real data (and therefore derived a shallower power-law slope). However, a value of δ ≃ 2.7 is also the best-fit single power-law slope found by Scott et al. (2010) for their P(D) analysis of unchopped 1.1 mm AzTEC data in the GOODS-S field.

In Figure 15 (see also Table 4), we also compare our results to those for deep field observations at 1.1 mm by AzTEC of the COSMOS (Scott et al. 2008), GOODS-N (Perera et al. 2008), GOODS-S (Scott et al. 2010), SHADES (Austermann et al. 2010), and AKARI, SSA-N2, and SXDF (Hatsukade et al. 2011) deep fields. (For clarity, the observations of Hatsukade et al. (2011) and Perera et al. (2008) are not shown in the plot because their data points lie within the scatter of the other AzTEC observations.) We also compare our results to the extremely deep SMG counts measured in lensing fields at 850 μm by Knudsen et al. (2008) as well as the recent wide map by Weiß  et al. (2009) using LABOCA at 870 μm in the ECDF-S. In order to compare our number counts directly to the results of these surveys at other wavelengths, we rescale their flux densities. Our choice of rescaling factor is based on the direct comparisons between S_850 μm, S_1.1 mm, and S_1.2 mm for galaxies in the GOODS-N field. The average flux density ratio for sources with robust SCUBA and AzTEC detections in the GOODS-N field is $S_{850\,\rm \mu m}/S_{1.1\,\rm mm}\simeq 1.8\hbox{--}2.0$ (Perera et al. 2008; Chapin et al. 2009). When comparing SCUBA and MAMBO detections, Greve et al. (2008) find S_850μm/S_1.2 mm ≃ 2.5. By co-adding the MAMBO and AzTEC observations in the GOODS-N field into a map at an effective wavelength of λ = 1.16 mm, Penner et al. (2011) find an average value of S_1.16 mm/S_1.1 mm ∼ 0.88 and S_1.16 mm/S_1.2 mm ∼ 1.14. All of these results are consistent with a single modified blackbody spectrum, for β = 1.5 and T_d = 30 K, observed at z ≃ 2.5. Therefore, we adopt this fiducial galaxy model when comparing fluxes at different wavelengths and use $S_{850\,\rm \mu m}/S_{1.2\,\rm mm}=2.3$, $S_{870\,\rm \mu m}/S_{1.2\,\rm mm}=2.2$, and $S_{1.1\,\rm mm}/S_{1.2\,\rm mm}=1.2$. Our directly calculated counts are in excellent agreement with the rescaled results of the AzTEC surveys. Additionally, our prediction for the shape of the number counts below our sensitivity threshold, afforded by our P(D) analysis, agrees well with the deepest AzTEC number counts and is even in rough agreement with the deepest SMG counts by Knudsen et al. (2008).

Figure 15 also compares our results to various number count predictions derived from backward evolution models that incorporate multi-waveband observations of number counts and redshift distributions as well as limits imposed by the CIB light. We have restricted this comparison to models that offer predictions at wavelengths of 1.2 mm (Béthermin et al. 2011) or at 1.1 mm (Valiante et al. 2009; Rowan-Robinson 2009; Marsden et al. 2011), to which we can apply the rescaling described above. Although flux scaling will generally not provide a precise representation of a model's predictions at 1.2 mm, the extrapolation from 1.1 mm to 1.2 mm is fairly modest. At flux densities equal to or less than those of our significant detections, we find that our observations are generally consistent with all model predictions, although the Valiante et al. (2009) model slightly overpredicts our P(D) curve near 1 mJy. At the high flux density limit, all models uniformly overpredict the counts from our best-fitting Schechter function model, while remaining consistent with the predictions from our power-law result (which is only marginally compatible with the CIB; see Figure 5). However, we cannot draw any conclusions from this apparent discrepancy, as our P(D) analysis cannot constrain the differential counts at flux densities greater than those of our brightest detections.

Here, we investigate the question of whether our radio counterpart identification rate (R_1.4 GHz) of ≃  93⁺⁴_{− 7}% (38/41) in the LHN is intrinsically greater than is seen in other surveys, or if it is simply a function of the increased 20 cm sensitivity in this field. We compare our identification rate to those found for previous deep surveys at 850 μm, 870 μm, 1.1 mm, and 1.2 mm. Table 5 lists recent millimeter and submillimeter deep field surveys from the GOODS-N, LHE, SXDF, COSMOS, and ECDF-S fields (Borys et al. 2003, 2004; Ivison et al. 2007; Bertoldi et al. 2007; Schinnerer et al. 2007; Perera et al. 2008; Chapin et al. 2009; Weiß  et al. 2009; Biggs et al. 2011), along with their 20 cm radio counterpart identification rates and 20 cm map sensitivities. Because the surveys have different definitions of "significant" (sub)millimeter detections, different data reduction techniques, and different standards for radio counterpart associations, we marginalize over all of these variables by looking at the average radio counterpart identification rate, and the average 20 cm map sensitivity. Using the surveys listed in Table 5, we find $\langle \sigma _{1.4\,\rm GHz} \rangle \simeq \,7.2\, \rm \mu Jy$ and 〈R_1.4 GHz〉 ≃ 57%.

If we imagine that our field had a sensitivity $\langle \sigma _{1.4\,\rm GHz} \rangle \simeq \,7.2\,\mu \rm Jy$, six of our likely radio counterparts would fall below the 4.0 σ limit of $S_{1.4\,\rm GHz} < 29\,\mu \rm Jy$ and would not be detected. Four additional likely 20 cm counterparts would appear at the 4σ–5σ level and would be at high risk of not being detected due to the usual completeness effects. Therefore, our radio counterpart identification rate would be 68⁺⁸_{− 9}% to 78⁺⁷_{− 8}%. This range is only marginally greater than the average value of 57%, and well within the scatter of the previous surveys. Therefore, we attribute our high radio identification rate to the extremely sensitive VLA map of this field, rather than to unusual properties of 1.2 mm selected sources at this depth.

Because we expect ∼5 spurious detections among our 41 sources with S/N > 4.0 and we find only 2–3 detections with unlikely or no radio counterparts, there is little room left to accommodate a substantial, extremely high-redshift (z > 5) population of radio-undetected SMGs (see also Ivison et al. 2005). This work suggests that with a deep enough radio image, perhaps all SMGs might have their radio counterparts identified, auguring well for upcoming deep surveys that exploit the dramatically expanded correlator bandwidth of the EVLA.

We find that 7.3^+6.7_{− 4.0}% (3/41) of our detections have two likely radio counterparts (MAMBO ID 3, 15, and 39). If we consider the fact that 5% of all randomly chosen positions within our MAMBO map will have counterparts within 8'' with P ⩽ 0.05, then we would expect to find a double radio counterpart rate of ∼4.6% from chance associations. Previous studies have found that ∼10% of SMGs host multiple likely radio counterparts (see, e.g., Ivison et al. 2002, 2007; Pope et al. 2006), probably due to the effects of confusion within the submillimeter/millimeter image, physical interactions, or the extended jets of radio-loud active galactic nuclei (AGNs). Although our SMG sample in the LHN is too small to be able to constrain the fraction of multiple radio counterparts to better than ±5%, we note that the pair separations of the radio counterparts are 21, 77, and 74, and that two of the three MAMBO sources have deboosted flux densities in the top 25% of our sample. These results may be in agreement with the trend identified in Ivison et al. (2007) that multiple radio counterparts are preferentially associated with the brightest SMGs and have pair angular separations Δθ ≃ 2''–6''.

As listed in Table 2 and detailed in the Appendix, of our 41 significant individual detections, two have optical spectroscopic redshifts (Polletta et al. 2006; Owen & Morrison 2009a), two have mid-IR spectroscopic redshifts (Fiolet et al. 2010), and two have high-quality photometric redshifts based on Herschel far-IR photometry that we will denote in what follows as "z'_phot" (Magdis et al. 2010). For those of the remaining 35 sources with robust or likely radio counterparts, we generally adopt the optical photometric redshifts (denoted z_phot in what follows) determined by Strazzullo et al. (2010) for the radio catalog of Owen & Morrison (2008). The exception to this rule comes for {z_phot} to which Strazzullo et al. (2010) assign a goodness-of-fit quality flag of "C"; these redshifts are less reliable, and in particular are more likely to manifest catastrophic errors. For such sources, as well as for the one 1.2 mm detection that lacks a radio counterpart altogether, we instead derive our own redshift estimates (z_α) using the radio–submillimeter spectral index redshift indicator of Carilli & Yun (1999):

$$\begin{equation} \alpha ^{350}_{1.4} = -0.24 -[ 0.42 \times (\alpha _{\rm radio} - \alpha _{\rm submm}) \times \log _{10}(1+z_\alpha) ]\quad \end{equation} \tag{ 16 }$$

(see also Carilli & Yun 2000; Yun & Carilli 2002). For α_radio we use, in order of priority and availability, α^90 cm_20 cm, α^50 cm_20 cm, or −0.68. We use $\alpha ^{50\,\rm cm}_{20\,\rm cm}$ only for sources with clean, unblended 50 cm detections. For these unblended 50 cm counterparts, we find an average value of $\langle \alpha ^{50\,\rm cm}_{20\,\rm cm} \rangle = -0.68\pm 0.06$ (see Figure 18), in agreement with the average spectral index of SMGs in the LHE field of $\langle \alpha ^{50\,\rm cm}_{20\,\rm cm} \rangle = -0.75\pm 0.06$ (Ibar et al. 2009, 2010). We adopt this mean value of $\alpha ^{50\,\rm cm}_{20\,\rm cm}$ (−0.68) for redshift determination of sources with only 20 cm radio counterparts, or whose 50 cm counterparts are confused. For α_submm, we use the spectral index at 1.2 mm of the fiducial high-redshift dusty galaxy spectral energy distribution (SED; α_submm = 3.2), as motivated in Section 7.1. For our detection with no likely radio counterpart, we estimate a redshift lower bound by using the local S_20 cm 4σ upper limit in Equation (16).

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Figure 19 illustrates why we exclude C-quality photometric redshifts from our catalog. Plotted is the α³⁵⁰_1.4 spectral index of the detections as a function of z_phot. The points are coded according to photometric redshift quality flag (Strazzullo et al. 2010). The points with the best photometric redshift fit quality (AA) are shown as black circles, followed by blue squares (A), green diamonds (B), and red triangles (C). The shaded region shows the Carilli & Yun (1999) relation for −α_radio = 0.52–0.80, where −0.52 represents the median $\alpha ^{90\rm \,cm}_{20\rm \,cm}$ spectral index in the LHN field (F. N. Owen et al. 2011b, in preparation) and −0.80 is the fiducial synchrotron value (Condon 1992). The overplotted lines show the empirical relations recovered by redshifting the SEDs of nearby star-forming galaxies M82 and Arp 220 (Klein et al. 1988; Scoville et al. 1991). The highest quality photometric redshifts agree with their galaxies' spectral indices in that they either follow the Carilli & Yun relation, or are consistent with an M82 or Arp 220 SED. In contrast, the C-quality photometric redshifts are scattered almost uniformly in z for a given α³⁵⁰_1.4, demonstrating their lack of robustness.

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Figure 20 shows the redshift distribution of our catalog, including all spectroscopic, photometric, and α³⁵⁰_1.4 estimated redshifts. It is apparent in Figure 20 that the 17 spectroscopic and high-quality (Herschel and AA/A/B-grade optical) photometric redshifts are biased toward lower redshifts. The median redshift for this 41% of our sample is z_median = 2.26, with an inter-quartile range of 1.72–2.90. For all galaxies, z_median = 2.90, with an inter-quartile range of 2.33–3.70. This systematic bias has two causes. First, the highest-redshift galaxies have the faintest counterparts, and will necessarily be detected in fewer optical bands, which results in a poorer fit. This trend is in contrast to the full radio catalog, for which the median z_phot is ∼1 and the fraction of photometric redshifts with AA/A/B quality (∼85%) is much higher than for our MAMBO sources. Second, the SEDs in the Strazzullo et al. (2010) galaxy template library are most representative of nearby galaxies, potentially resulting in a poor fit if they are applied to high-z galaxies whose SEDs are not included in that library.

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The median redshift for our sample (z_median = 2.90) is larger than the median redshift determined by Pope et al. (2006) for a complete sample of 850 μm selected SMGs with spectroscopic redshifts (z_median = 2.0). Although our redshift distribution has greater uncertainties because it relies heavily on photometric redshifts and spectral index redshift estimates, it is in agreement with the results of Chapin et al. (2009), who have shown that, with high statistical significance, galaxies in a sample selected at 1.1 mm are detected at higher redshift (z_median = 2.7) than those selected at 850 μm. Our median redshift is also greater than that of the sample of 68 galaxies selected at 870 μm from the LABOCA survey of the ECDF-S. Using 17-band optical through mid-IR photometry, Wardlow et al. (2011) find z_median = 2.2.

The results from the w(θ) and nearest neighbor analyses (Section 5.8) suggest that our sources are clustered to some degree. The spike in w(θ) on ∼4' scales is an intriguing result that is consistent with the visual impression of Figure 2 (i.e., the southeastern overdensity and the central void) and hints at the existence of LSS in this field. To investigate whether the spatial distribution of our detections traces LSS that can also be seen at other wavelengths, we compare our source positions to the distribution of radio sources within the LHN. In order to compare our MAMBO sources to radio sources at comparable redshifts, we include only radio sources with 1.5 < z_phot < 4.5 (excluding all that only have a C-quality z_phot). Additionally, we only consider radio sources with sizes greater than 10. These larger sources will be preferentially gas-rich mergers with extended star formation or radio-loud AGNs, which we would naively expect to trace environmental overdensities on the basis of studies at lower redshift (e.g., Hill & Lilly 1991; Best 2004; Best et al. 2005; Wake et al. 2008). From a practical standpoint, they can also be detected over the full area of the MAMBO map, allowing for a fair comparison; sources with 20 cm sizes ⩽1'', in contrast, tend to be fainter, and therefore have systematically lower surface densities farther from the center of the VLA map.

In Figure 21, we plot our 1.2 mm source positions over a 3' resolution smoothed surface density map of the 307 radio sources in the Owen & Morrison (2008) VLA catalog that satisfy our selection cuts. We find that the distributions of the two populations agree quite well: (1) the density map shows a deficiency of radio sources at the location of our central MAMBO void, (2) every radio source density peak (〈Σ_20 cm〉 ⩾ 0.5 arcmin⁻²) is associated with at least one MAMBO detection, and (3) ∼66% of our MAMBO detections are located in regions with higher than average ($\langle \Sigma _{20\,\rm cm} \rangle \ge 0.34\,\rm arcmin^{-2}$) radio source density, whose area comprises only 45% of the total. This striking agreement seems to argue for a real physical correlation between our MAMBO detections and 20 cm radio galaxies at similar redshifts. Following Austermann et al. (2009), we have also compared our SMG catalog to an identical number of homogeneously distributed, randomly chosen positions. The experiment confirms the spatial correlation at a confidence level of 90%. However, when using random positions derived from our simulated maps that take into account the spatially varying sensitivity, we find that our MAMBO detections, although significantly correlated with each other, are not significantly spatially correlated with this sample of high-z 20 cm galaxies.

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At 1.2 mm, the CIB intensity is I_ν ≃ 15–24 Jy deg⁻² (Puget et al. 1996; Fixsen et al. 1998). By adding up the deboosted flux densities of our detections with S/N > 4.0, we recover ≃ 0.58 Jy deg⁻² of the CIB, or about ∼3%. Figure 5 shows that our best-fitting Schechter function estimate of the differential number counts is entirely consistent with the intensity of the CIB, while the power-law model is only marginally compatible with it. The analysis performed by Scott et al. (2010) on ASTE/AzTEC data in the GOODS-S field finds that the best-fit power-law model from their P(D) analysis can account for the CIB, although only if they integrate the counts past the cutoff used for that analysis. They also find that their Schechter function model is incompatible with the CIB, and can only recover ∼30% of the 1.1 mm background when integrated down to $S_{1.1\,\rm mm}^{\prime }=0\,\rm mJy$. These results may be due to their a priori choice of a faint-end power-law index δ' = 1.0. We fit for this parameter directly and find that a steeper value of δ' ≃ 1.86 is optimal for our data and produces enough faint sources to account fully for the CIB light at 1.2 mm.

Polletta et al. (2006) report an optical z_spec = 2.562 for this source. It also has a 70 μm counterpart with $S_{70\,\mu \rm m}=10.4\pm 1.7\,\rm mJy$ at a distance of 32, and a 160 μm counterpart with $S_{160\,\mu \rm m}=24.1\pm 1.9\,\rm mJy$ at a distance of 17 (F. N. Owen et al. 2011b, in preparation).

In addition to the robust 20 cm counterpart listed in Table 2, this source has an additional likely counterpart with S_20 cm = 30 μJy, separation 26, and P = 0.034, with which it is nearly blended. Neither radio counterpart has an estimated photometric redshift. Figure 22 shows 1.2 mm contours overlaid on a 20 cm cutout image that includes both counterparts.

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We identify this source with LHN8 in the Herschel catalog of Magdis et al. (2010), from which it is separated by 20 (P = 0.0084), and with SWIRE4_J104638.68+585612.5 = ID L14 in the Spitzer sample of Fiolet et al. (2010), who report a mid-IR z_spec = 2.03. Fiolet et al. (2009) report an on–off flux density measurement of $S_{\rm 1.2\,{\rm mm}} = 2.13 \pm 0.71\,{\rm mJy}$, which is consistent with our (non-deboosted) S_1.2 mm = 2.7 ± 0.5 mJy within the uncertainties.

The radio counterpart to this source (P = 0.0043) is not in the catalog of Owen & Morrison (2008) because it has an S/N ratio of 4.8.

We identify this source with LHN1 in the Herschel catalog of Magdis et al. (2010), from which it is separated by 34 (P = 0.0057), and with SWIRE4_J104556.90+585318.8 = ID L11 in the Spitzer sample of Fiolet et al. (2010), who report a mid-IR z_spec = 1.95 in good agreement with the optical z_phot = 1.80 reported by Strazzullo et al. (2010). Fiolet et al. (2009) report an on–off flux density measurement of $S_{\rm 1.2\,{\rm mm}} = 3.08 \pm 0.58\,{\rm mJy}$, which is consistent with our (non-deboosted) S_1.2 mm = 3.4 ± 0.6 mJy within the uncertainties. This source also has a 160 μm counterpart (F. N. Owen et al. 2011b, in preparation) with $S_{160\,\mu \rm m}= 11.8\pm 1.5\,\rm mJy$ at a distance of 43.

This source has two radio counterparts; the primary counterpart listed in Table 2 has z_phot = 2.76, while the second (with P = 0.032) has z_phot = 1.06. In the 20 cm map, we see a quadruple radio source (see Figure 23). At a low level of significance, the 1.2 mm emission appears to be elongated in the same direction as the radio source(s). This system also has a 160 μm counterpart (F. N. Owen et al. 2011b, in preparation) with $S_{160\,\mu \rm m}=12.7\pm 1.5\,\rm mJy$ at a distance of 68.

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This source is separated by only ∼15'' from MM J104611.9+590231 = ID 39. The catalog of Owen & Morrison (2008) includes a single radio source whose nominal position is midway between two very faint sources (one resolved, one unresolved) that are visible in the original 20 cm map (see Figure 24). These two sources, with peak flux densities of 15.6 μJy and 13.5 μJy, are only identifiable because they lie very close to the center of the VLA map, where the local rms noise is only 2.9 μJy beam⁻¹. After attributing the flux of the catalogued source to its two constituents, we find that one is a robust radio counterpart for the MAMBO source (10 separation with P = 0.0099), while the other is probably a chance association (33 separation with P = 0.077).

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This source has an unlikely 20 cm radio counterpart (P = 0.12) with S/N = 4.2 (S_20 cm = 15.0 ± 3.6 μJy), and is therefore not in the catalog of Owen & Morrison (2008), which includes only sources with S/N > 5.0.

This source has no radio counterpart within 8'' and no likely 24 μm counterpart, although it does have an X-ray counterpart (see Section 6.5).

This source has a likely 20 cm radio counterpart (P = 0.029) with S/N = 4.3 (S_20 cm = 24.8 ± 5.8 μJy), and is therefore not in the catalog of Owen & Morrison (2008), which includes only sources with S/N > 5.0.

We identify this source with SDSS J104555.49+590915.9, an optically bright galaxy for which Owen & Morrison (2009a) report an optical z_spec = 0.044. Figure 25 shows a red optical image overlaid with 1.2 mm contours, which at a low level of significance are elongated in the same direction as the galaxy's stars. This ≃ 20'' source is heavily resolved at 20 cm and 50 cm. It is also detected at 160 μm (F. N. Owen et al. 2011b, in preparation) with $S_{160\,\mu \rm m}= 15.8\pm 1.4\,\rm mJy$ (at a separation of 66), and at 250 μm (Herschel/SPIRE; A. Smith et al. 2011, in preparation) with $S_{250\rm \,\mu m}=133\pm 8\,\rm mJy$ (at a separation of 36).

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We identify this source with LHN3 in the Herschel catalog of Magdis et al. (2010), from which it is separated by 52 (P = 0.016). Magdis et al. (2010) estimate z'_phot = 2.40 on the basis of their PACS and SPIRE photometry, which we list in Table 2 rather than the Strazzullo et al. (2010) optical z_phot = 1.32, due to the close connection between far-IR and millimeter emission.

This source has a 20 cm radio counterpart (P = 0.041) with S/N = 4.9 (S_20 cm = 16.1 ± 3.3 μJy), and is therefore not in the catalog of Owen & Morrison (2008), which includes only sources with S/N > 5.0.

We identify this source with LHN4 in the Herschel catalog of Magdis et al. (2010), from which it is separated by 24 (P = 0.013). Magdis et al. (2010) estimate z'_phot = 1.72 on the basis of their PACS and SPIRE photometry, which we list in Table 2; this is in good agreement with the optical z_phot = 1.66 reported by Strazzullo et al. (2010).

This source is separated by only ∼15'' from MM J104610.4+590242 = ID 17. It also has a pair of likely radio counterparts within an 8'' search radius. Figure 24 shows 1.2 mm contours overlaid on a 20 cm cutout image. This source is not identified in the catalog of Owen & Morrison (2008).