GRAVITATIONAL LENS MODELS BASED ON SUBMILLIMETER ARRAY^* IMAGING OF HERSCHEL^†-SELECTED STRONGLY LENSED SUB-MILLIMETER GALAXIES AT z > 1.5

The first suggestion that wide-field surveys (i.e., covering ≳ 100 deg²) at submm or mm wavelengths would efficiently identify strongly lensed galaxies was made nearly two decades ago (Blain 1996), but it is only in the past few years, with the advent of Herschel and the SPT, that such surveys have reached the requisite survey area and sensitivity to discover them in large numbers. We select candidate strongly lensed galaxies from the two widest Herschel extra-galactic surveys: H-ATLAS and HerMES. The total area considered for the candidate selection is ≈300 deg² in H-ATLAS (comprising the full equatorial fields and ≈75% of the northern galactic pole field) and 94.8 deg² in nine independent fields in HerMES (for details of the fields, see Oliver et al. 2012). The total area surveyed by Herschel as part of H-ATLAS and HerMES is ∼1000 deg² (i.e., roughly a three-fold increase over the area considered for this paper).

An important aspect of candidate lens selection is source extraction and photometry for the Herschel Spectral and Photometric Imaging REceiver (SPIRE; Griffin et al. 2010) data. We summarize the relevant aspects of the methodology here and provide references where appropriate.

In the HerMES fields, source detection is achieved by applying the StarFinder code (Diolaiti et al. 2000) to the 250 μみゅーm images. Photometry is then computed using the HerMES XID pipeline (Roseboom et al. 2010), which allocates flux density based on the 250 μみゅーm position priors obtained with StarFinder.

In the H-ATLAS fields, sources are identified and flux densities are measured using the Multi-band Algorithm for source eXtraction (MADX; S. Maddox et al., in preparation). MADX first subtracts a smooth background, and then filters with the point-spread function (PSF) appropriate for each band. Next, >2.5σしぐま peaks are identified in the 250 μみゅーm map, and "first-pass" flux density estimates are obtained from the pixel values at these positions in each band. Sub-pixel positions are estimated by fitting to the 250 μみゅーm peaks, and more accurate flux-densities are estimated using bi-cubic interpolation to the accurate 250 μみゅーm position. In each band, the sources are sorted in order of decreasing flux density using the first-pass pixel values, and a scaled PSF is subtracted from the map before estimation of flux densities for any fainter sources. This step prevents faint source flux densities from being overestimated when they lie near brighter sources. Finally candidate sources are retained in the catalog if their flux densities are more than 5σしぐま in any of the three bands. The 5σしぐま flux density limits in H-ATLAS, including confusion noise (typically ≈6 mJy in all three SPIRE bands Nguyen et al. 2010), are 32 mJy at 250 μみゅーm, 36 mJy at 350 μみゅーm, and 45 mJy at 500 μみゅーm (Pascale et al. 2011; Rigby et al. 2011).

Although the HerMES and H-ATLAS teams use different methods to extract photometry, the sources that are the subject of this paper are all high signal-to-noise ratio (S/N), point sources as seen by Herschel. In this regime, we expect that the different methods should provide consistent flux density measurements.

The essence of the candidate lens selection technique is to identify objects that are bright at 500 μみゅーm. A complete description of the selection technique for the HerMES lens candidates is given by Wardlow et al. (2013). This paper includes the objects tabulated in Wardlow et al. (2013) as well as objects identified in H-ATLAS. We select objects that satisfy S₅₀₀ > 100 mJy, a regime that has been shown from previous studies of smaller areas to have relatively little contamination from unlensed SMGs (Negrello et al. 2010; Wardlow et al. 2013). The primary contaminant is local universe galaxies (z < 0.1). These galaxies are spatially resolved in SDSS imaging and therefore trivial to remove. There is also a small contamination from blazars, which are non-thermal emitters and are easily removed using data from the NVSS or the Very Large Array Faint Images of the Radio Sky at Twenty-Centimeters survey (FIRST; Becker et al. 1995).

A total of 13 objects in HerMES satisfy S₅₀₀ > 100 mJy and are not local galaxies or blazars (Wardlow et al. 2013). In the H-ATLAS fields, there are 91 objects that satisfy these criteria. Considering the combination of the two surveys, this results in a surface density on the sky of ≈0.26 deg⁻². This value lies between the values of 0.32 deg⁻² from Negrello et al. (2010) and 0.14 ± 0.04 deg⁻² from Wardlow et al. (2013), as expected since it represents a combination and extension of these previous efforts. Cosmic variance likely explains the difference in the surface densities of lenses between HerMES and H-ATLAS. A detailed calculation of this effect requires taking into account the cosmic variance of both the lensed SMGs and the lenses themselves and is beyond the scope of this paper.

Efforts are on-going to obtain a complete database of follow-up observations for this sample of 104 candidate lensed SMGs. The present paper focuses on a subset of 30 candidates with superb existing follow-up observations (hereafter, we refer to this as the "SMA subsample"). These targets were initially selected on the basis of strong 1.2 mm detections from the Max Planck Millimeter Bolometer (MAMBO) array (Kreysa et al. 1998) at the Institut de Radioastronomie Millimétrique (IRAM) 30 m telescope (H. Dannerbauer et al. in preparation). Subsequent follow-up efforts have now provided high-spatial resolution 880 μみゅーm imaging with the SMA, spectroscopic redshifts of the lensed SMGs obtained with GBT, CSO, CARMA, PdBI, and Herschel (Cox et al. 2011; Harris et al. 2012; Lupu et al. 2012, D. A. Riechers et al., in preparation; M. Krips et al., in preparation, George et al. 2013), and spectroscopic redshifts to the lenses obtained with the MMT, Gemini-S, or WHT. In addition, Keck-II Near InfraRed Camera 2 (NIRC2) laser guide star adaptive optics (LGSAO) imaging has been obtained for nearly half of the candidate lensed SMG sample (Wardlow et al. 2013; J. Calanog et al., in preparation). These datasets provide the information needed to confirm the lensing hypothesis and begin analysis of the source and lens properties. Table 1 provides basic positional data for the SMA subsample, including the International Astronomical Union names, short names to aid comparison with previous publications, positions measured from the SMA 880 μみゅーm image (see Section 2.2), and redshift measurements for the lens(es) (see Sections 2.3, 2.4, 2.5, 2.6, and references in the table) and background sources (references given in the table), where available.

Figure 1 shows the S₃₅₀/S₅₀₀ SPIRE colors as a function of S₅₀₀ flux density for all galaxies in the H-ATLAS phase I catalog with S/N >3 in all SPIRE bands (grayscale, logarithmic scaling), the full sample of 104 candidate Herschel lensed SMGs in HerMES and H-ATLAS (cyan squares), the SMA subsample with superb follow-up data that is the focus of this paper, and a sample of objects selected from the SPT survey with published lens models (Hezaveh et al. 2013). The SMA subsample is biased to higher S₅₀₀ values than the full sample. We therefore expect the lensing rate to be higher than in the full sample. The SPIRE colors (S₃₅₀/S₅₀₀ and S₂₅₀/S₃₅₀) are comparable between the full sample and the SMA subsample. The SMA subsample contains nearly all galaxies in the parent sample with S₅₀₀ > 170 mJy (the lone exception is H-ATLAS J1429−002, which has the highest S₃₅₀/S₅₀₀ ratio in the full lens candidate sample and is the subject of a study based on data from the Atacama Large Millimeter/submillimeter Array (ALMA) and other facilities; H. Messias et al., in preparation).

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SMA data were obtained over a period of multiple semesters from 2010 March through 2013 May with a total of 162 hr of on-source integration time. Each target was typically observed in multiple array configurations with t_int = 1–2 hr on-source per configuration. We used track-sharing of multiple targets per track to ensure the best possible uv coverage.

Observations took place in a range of conditions from superb (atmospheric opacities of τたう_225 GHz = 0.04, phase errors of Δでるた(ϕ_rms) = 10°) to good (atmospheric opacities of τたう_225 GHz = 0.1, phase errors of Δでるた(ϕ_rms) = 40°). Phase errors are estimated from a fixed monitoring system on a variety of baselines. For some of the observations, 1 or more antennas were unavailable, so in those cases the total number of antennas (N_ant) used was less than 8. Most notably, in early 2011, the lower sideband of the SMA 345 GHz receiver in antenna 1 was flagged due to significant instrumental noise. In general, we optimized the SMA single-polarization 345 GHz receivers for continuum detection by tuning to a frequency of νにゅー_LO ≈ 340 GHz. In some cases νにゅー_LO varied from this value by up to 10 GHz to avoid retuning in the middle of the night when the SMA switched from another program. Table 2 presents details regarding the SMA observations, including the date, array configuration, local oscillator frequency (νにゅー_LO), τたう_225 GHz, ϕ_rms, N_ant, t_int, and original reference (some of the data used in this paper were originally presented elsewhere).

The SMA receivers make use of an intermediate frequency coverage of 4 GHz, providing a total of 8 GHz bandwidth (considering both sidebands), with a center-to-center sideband separation of 12 GHz. The primary goal of the observations is to detect the continuum emission at the highest possible significance. When a spectroscopic redshift for a Herschel source was available, we tuned the receivers to the closest CO rotational transition, as long as the S/N of the continuum data would not be compromised by doing so. This was possible for 8 of our targets. Since the background galaxies lie at 1.5 < z < 4.5, our observations probe emission from the J = 10 (or higher) levels in lines that are typically faint in SMGs and therefore difficult to detect. We defer a discussion of the molecular line measurements based on the SMA data to a future publication.

Bandpass calibrators were chosen primarily based on their 880 μみゅーm flux densities (where possible, we used calibrators with S₈₈₀ > 5 Jy) and their observability at the beginning or end of each night. For absolute flux density calibration, Titan was used whenever possible, followed by Callisto, Neptune (in subcompact array only), and MWC349A (when no planets or moons were available). Amplitude and phase gain calibration was achieved by monitoring nearby (angular separation from science target of <15°), bright (S₈₈₀ > 0.5 Jy) quasars. Whenever possible, we used multiple quasars for gain calibration, including a fainter quasar (S₈₈₀ > 0.1 Jy) much closer to the science target (angular separation <5°) to provide an independent check of the reliability of the calibration, particularly phase transfer. We used the Interactive Data Language (IDL) MIR package to calibrate the uv visibilities.

We used the Multichannel Image Reconstruction, Image Analysis, and Display (MIRIAD) software package (Sault et al. 1995) to reconstruct and deconvolve the image from the visibilities. We used natural weighting to achieve maximum sensitivity for all targets. We combine visibility data from all available configurations for each target. The beam size and shape in the resulting images vary greatly from target to target. This is primarily because not every target was observed in all array configurations, but there is also some dependence on the declination of the target, because the SMA uv coverage is less complete at declinations near 0°. In general, we achieve spatial resolutions of ≈06 FWHM.

Figure 2 shows the SMA image of each object in the SMA subsample (red contours, beginning at ±3σしぐま and increasing by factors of $\sqrt{2}$) in comparison with the best available optical or near-IR image (grayscale; see Section 2.7). A detailed source-by-source description is deferred to Section 3.2. The position of the 880 μみゅーm emission centroid (estimated by-eye) for each source is presented in Table 1. There is no absolute significance to these centroid values, but they are necessary to undertake lens modeling.

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We use the SMA images in conjunction with knowledge of the redshifts of the lenses and sources (see Sections 2.3, 2.4, 2.5, and 2.6 for details) to characterize the nature of the lensing that is occurring in the SMA subsample. Those galaxies showing multiple images with a morphology typical of strong lensing and that have known distinct lens and source redshifts are given an A grade. Galaxies with obvious strong lensing morphology but with only a single known redshift measurement (either of the lens or the source) receive a B grade. We expect that all B grade systems are strong lenses, but without distinct redshift measurements we cannot be certain. Galaxies showing only a single image of the background source, but with known distinct lens and source redshifts, are given a C grade. Finally, an X grade is given to those objects where the SMA imaging and the available spectroscopic redshifts provide inconclusive evidence of lensing. Additional data are needed to determine whether lensing is occurring in these objects. Our grades are listed in Table 1.

We compute total flux densities at 880 μみゅーm within rectangular apertures customized to match the spatial extent of each object in the SMA images. Uncertainties on these measurements are derived by placing apertures of the same size and shape at random, non-overlapping locations within the SMA primary beam field of view (excluding regions containing flux density from the source) and computing the 1σしぐま root-mean-square variation (which is generally well-described by a Gaussian) in the distribution of aperture flux densities. The number of apertures varied from target to target, but was typically ≈100.

Long-slit spectroscopic observations using the MMT Red Channel Spectrograph (Schmidt et al. 1989) were conducted in the 2012A semester (PI: R. S. Bussmann) and provided data on H-ATLAS J125632.7+233625, H-ATLAS J132630.1+334410, H-ATLAS J133008.4+245900, and H-ATLAS J134429.4+303036. The total on-source integration times for these targets were 70–120 minutes apiece. The observations were obtained during dark time in near-photometric conditions and seeing was typically 10–15. Details of the observations for each target are given in Table 3.

We obtained lamp flats for all targets at the beginning of each night for flat fields. We obtained a sequence of up to six consecutive 10 minute exposures on each target. For wavelength calibration, before and after each of these sequences we obtained comparison lamp observations using a He/Ar/Ne comparison lamp. We checked the focus periodically throughout the night.

We used either the 270 or the 300 lines per mm grating with a central wavelength (λらむだ_central) of 6996 Å or 5504 Å. The slit width was chosen to be as small as atmospheric seeing allowed and was either 15 or 20. The spectral resolution at λらむだ_central (Δでるたλらむだ) was 10–30 Å.

The long-slit data were reduced using standard IRAF one-dimensional spectroscopy routines. The primary aim of these observations for the purpose of this paper is to obtain a spectroscopic redshift for the putative lensing system. We accomplished this task with the xcsao routine in IRAF, using as a template a 5 Gyr old simple stellar population from Bruzual & Charlot (2003) with solar metallicity and a Chabrier initial mass function (IMF). This template does not perfectly match the lensing galaxy spectra, but it is sufficient to determine a robust redshift. The lens redshifts are presented in Table 1. A more detailed exploration of the optical spectra is deferred to a subsequent publication, so we do not show the spectra in this paper.

Long-slit spectroscopic observations using the Gemini Multi-Object Spectrograph-South (GMOS-S; Hook et al. 2004) were conducted in queue mode during the 2012A semester as part of program GS-2012A-Q-52 (PI: R. S. Bussmann) and provided data on H-ATLAS J090740.0−004200 and H-ATLAS J141351.9−000026. The total on-source integration times for each of these targets were 1–4 hr. The observations were obtained during dark time in near-photometric conditions. Observational details are given in Table 3.

Some aspects of the observing strategy were common to both targets and followed the guidelines given by the Gemini observatory.³⁴ Flat field observations were interspersed between the science exposures at each wavelength setting. CuAr arc lamp exposures were taken for the purpose of wavelength calibration, using the same instrumental setup as for the science exposures. A 5 Å spectral dither between exposures was used to cover the gap in the GMOS-S chip, and we binned the CCD pixels by a factor of 4 in both the spatial and spectral directions, providing a spatial scale of 0288 pixel⁻¹ and a spectral pixel scale of 2.69 Å. We used the GG455 blocking filter. Aspects of the observing strategy that varied from target to target are given in Table 3.

The long-slit data were reduced using the IRAF Gemini GMOS reduction routines, following the standard GMOS-S reduction steps in the example taken from the Gemini observatory webpage.³⁵ We used the same procedure as outlined in Section 2.3 to measure a spectroscopic redshift using IRAF's xcsao task. The lens redshifts are presented in Table 1. We plan a detailed exploration of the optical spectra in future work, so we do not show the optical spectra in this paper.

Long-slit spectroscopic observations using the auxiliary port camera (ACAM; Benn et al. 2008) at the WHT were conducted in the 2011A semester (PIs: I. Pérez-Fournon and A. Verma) and provided three 1200 s exposures of H-ATLAS J083051.0+013224 on 2011 April 24. ACAM provides fixed-format spectroscopy, covering a spectral range of 3500–9400 Å (spectral resolution of ≈3.3 Å pix⁻¹) using a 400 lines mm⁻¹ transmission Volume Phase Holographic grating. The observing conditions were photometric and the seeing was 075. We have included the observational details for this target in Table 3 for completeness.

We obtained Tungsten lamp flats at the beginning of each night for flat fields. For wavelength calibration, we obtained comparison lamp observations using a CuNe lamp. We checked the focus periodically throughout the night using the ACAM imaging mode.

A position angle of 114 deg east of north was chosen to include several objects visible on the optical imaging close to the candidate lensing galaxy. The slit width was 10 and the corresponding spectral resolution provided by the grating was R ≈ 450, or 15 Å FWHM at 6750 Å.

The long-slit spectroscopic data were reduced using standard IRAF two-dimensional spectroscopy programs to correct for the distortions in the data and apply the wavelength and flux density calibration.

The ACAM spectroscopy, based on G-band and Mg absorption features, shows that the primary lens (i.e., the galaxy that is closest to the SMA emission centroid) is located at z_lens = 0.626 ± 0.001. A nearby disk-like galaxy was detected in the acquisition images and spatially resolved from the primary lens (the existence of this additional galaxy is confirmed by ancillary high-spatial resolution imaging from Keck-II adaptive optics imaging; see Section 2.7). In the ACAM long-slit spectra an emission line is detected at 7462 Å at the spatial location of the disk-like galaxy. We cannot associate this line with emission lines at the same redshift as the primary lens. The most plausible identification is [O ii]3727 at z = 1.002 ± 0.001, in which case the two galaxies are at different redshifts.

The optical and near-IR spectra of the southeastern lens in J114637−001132 (i.e., the galaxy denoted as "G2" by Fu et al. 2012) were obtained with the X-shooter spectrograph (Vernet et al. 2011) at the VLT. X-shooter provides simultaneous spectral coverage from 300 nm to 2.5 μみゅーm. The 1'' × 11'' slit was used in the "UVB" arm (300–560 nm), while the 09 × 11'' slits were used in the "VIS" arm (550–1020 nm) and "NIR" arm (1020–2480 nm), yielding a resolving power of λらむだ/Δでるたλらむだ of 4350, 7450 and 5300 in the three bands, respectively. The observations were obtained on 2012 September 18 and 19. The total integration time was 320 minutes and comprised individual exposures of 20 minutes each. The object was nodded along the slit by a few arcsec between one exposure and the next.

Data reduction was performed by following the standard steps of the public X-shooter pipeline (Goldoni et al. 2006). Sky emission lines were subtracted by exploiting temporally contiguous exposures in which the objects was nodded in a different position of the slit. After flat-fielding, the pipeline extracts the different orders of the echelle spectrum, which are then rectified, wavelength calibrated and merged. Then the spectra were calibrated in flux by using the observation of a spectrophotometric standard. The final mono-dimensional spectrum was extracted from an aperture of 1''.

The lensing galaxy G2 is clearly detected with several emission lines ([O iii],Hb,Ha,[N ii],[S ii], [O ii]) that imply a redshift of z_lens = 1.2247.

In all cases where the SMA has clearly resolved multiple images of the background source, there is no evidence for submm emission from the lens. This means that detection of the foreground lens requires observations at optical or near-IR wavelengths. This paper makes use of the best available optical or near-IR imaging to pinpoint the location of the lens and determine whether it comprises multiple galaxies. This imaging is shown in grayscale in Figure 2, and the telescope and filter used are given in the lower left corner of each panel. Fifteen objects use HST Snapshot imaging (marked as "HST F110W" in Figure 2), five use Keck-II/NIRC2-LGSAO imaging (marked as "Keck-II_NIRC2 Ks" in Figure 2), five use full-orbit HST imaging (marked as "HST F160W" in Figure 2), four use SDSS i-band imaging (marked as "SDSS i" in Figure 2), and two use WHT K_s-band imaging (marked as "WHT Ks" in Figure 2).

The focus of this paper is lens modeling of the SMA data. A detailed analysis of the optical and near-IR imaging will appear in a set of papers specific to the HST Snapshot imaging (S. Amber et al., in preparation; J. Calanog et al. in preparation), the full-orbit HST imaging (M. Negrello et al., in preparation, S. Dye et al., in preparation), and the Keck-II/NIRC2-LGSAO imaging (J. Calanog et al., in preparation) and the WHT K_s imaging (P. Martínez-Navajas et al., in preparation).

To facilitate comparison with existing surveys for lenses based on SDSS spectroscopy (e.g., Bolton et al. 2008; Brownstein et al. 2012), we compute i-band photometry using SDSS Data Release 9 (DR9) for all of the objects in the SMA subsample. The imaging aspect of DR9 provides five optical bands: u, g, r, i, and z. The 95% completeness levels for point sources are u = 22.0, g = 22.2, r = 22.2, i = 21.3, and z = 20.5 (AB mag), corresponding to flux densities of 5.7 μみゅーJy, 4.8 μみゅーJy, 4.8 μみゅーJy, 11.0 μみゅーJy, and 23.0 μみゅーJy, respectively. The median seeing in the images at r-band is typically 13.

We searched for counterparts in the DR9 catalog within a 2'' radius of each expected lens position based on the best available optical or near-IR imaging. If a counterpart was found, then it was assigned photometry directly from the DR9 catalog. If no counterpart was found, we used our own custom aperture photometry code to measure the 2σしぐま limiting flux density at the position of the target (note that at these wavelengths, the lens is typically much brighter than the source). We used a 4'' diameter circular aperture and computed the sky background in an annulus with an inner radius of 2'' and an outer radius of 5''. We measured the uncertainties by placing N random apertures (where N ≈ 300) of the same size and shape within 3' of the lens candidate (taking care to avoid any objects found in the DR9 catalog) and computing the 68% confidence interval of the dispersion in the measured flux densities. The i-band AB magnitudes are reported in Table 4 (limits indicate 2σしぐま values), along with the Herschel/SPIRE and SMA 880 μみゅーm measurements (the values reported in the table do not include absolute flux density calibration uncertainty of 7%).

The SMA is an interferometer, so the surface brightness map of each lensed SMG is obtained with incomplete sampling of the uv plane. This means that surface brightness is not necessarily conserved and that the pixel-to-pixel errors in the surface brightness map are correlated. For these reasons, it is important to compare model and data visibilities rather than surface brightness maps. We follow the methodology used in Bussmann et al. (2012), who presented the first lens model derived from a visibility-plane analysis of interferometric imaging of a strongly lensed SMG discovered in wide-field submm surveys. We summarize important details here and refer the interested reader to Bussmann et al. (2012) for further information.

We use the publicly available Gravlens software (Keeton 2001) to map emission from the source plane to the image plane for a given lensing mass distribution. To represent the lens mass profile, we use N_lens singular isothermal ellipsoid (SIE) profiles, where N_lens is the number of lensing galaxies found from the best available optical or near-IR imaging (a multitude of evidence supports the SIE as a reasonable choice; for a recent review, see Treu 2010).

The source(s) are assumed to have Sérsic profile morphologies (Sersic 1968). We always use a single Sérsic profile in our fits, with the exception of objects that are clearly only moderately lensed (i.e., singly imaged with μみゅー < 2) and that show evidence of multiple source-plane components in the SMA imaging. This is true for J022016.5−060143 and J084933.4+021443.

Each SIE is fully described by the following five free parameters: the position of the lens relative to the SMA emission centroid (Δでるたαあるふぁ_lens = αあるふぁ_lens − αあるふぁ₈₈₀ and Δでるたδでるた_lens = δでるた_lens − δでるた₈₈₀; these can be compared with the position of the optical or near-IR counterpart relative to the SMA emission centroid: Δでるたαあるふぁ_NIR = αあるふぁ_NIR − αあるふぁ₈₈₀ and Δでるたδでるた_NIR = δでるた_NIR − δでるた₈₈₀), the mass of the lens (parameterized in terms of the angular Einstein radius, θしーた_E), the ellipticity of the lens (_lens; defined as 1 − b/a), and the position angle of the lens (ϕ_lens; degrees east of north). When there is evidence for additional lenses from optical or near-IR imaging (see Figure 2), we estimate by-eye centroids for each lens (carrying an uncertainty of order 1 pixel, or 004 and 012 in the Keck-II/NIRC2-LGSAO and HST images, respectively) and fix the positions of the additional lenses with respect to the primary lens. Therefore, each additional lens has only 3 free parameters: θしーた_E, _lens, and ϕ_lens. We assume secondary, tertiary, etc., lenses are located at the same redshift as the primary lens, unless there is evidence against that assumption (e.g., J083051.0+013224).

Each Sérsic profile is fully described by the following seven free parameters: the position of the source relative to the primary lens (Δでるたαあるふぁ_s = αあるふぁ_s − αあるふぁ_lens and Δでるたδでるた_s = δでるた_s − δでるた_lens), the intrinsic flux density (S_in), the Sérsic index (n_s), the half-light semi-major axis length (a_s), the ellipticity (_s, defined as 1 − b/a), and the position angle (ϕ_s, degrees east of north).

The total number of free parameters for any given system is N_free = 5 + 3 × (N_lens − 1) + 7*N_source, where N_source is the number of Sérsic profiles used.

We adopt loose, uniform priors for all model parameters. The 1-σしぐま absolute astrometric solution between the SMA and optical/near-IR images is generally 02, so in our modeling efforts the prior on the position of the lens covers ±06 in both R.A. and Decl. (i.e., 3σしぐま in each direction). In Section 3.2, we discuss the level of agreement between the astrometry from the images and the astrometry from the lens modeling. For θしーた_E, the prior covers 01–6''. The ellipticities of the lens and source are always restricted to be <0.7. No prior is placed on the position angle of the lens or source. The intrinsic flux density is allowed to vary from 0.1 mJy to the total flux density observed by the SMA. The source position is allowed to vary by ±1'' relative to the position of the primary lens. The Sérsic index varies from 0.1 to 4.0. The half-light radius varies from 005 to 15.

For a given set of model parameters, Gravlens generates a surface brightness map of the lensed emission (note that no model of the emission from the lens is needed because the lenses are undetected in the SMA imaging). This surface brightness map can then be used as input to MIRIAD's uvmodel task, which produces a "simulated visibility" dataset (V_model) by computing the Fourier transform of the model lensed image and sampling the resulting visibilities to match the sampling of the actual observed SMA visibility dataset (V_SMA). The quality of fit for a given set of model parameters is determined from the chi-squared value (χかい²) according to the following equations:

$$\begin{equation} \chi ^2 = \chi ^2_{\rm real} + \chi ^2_{\rm imag}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \chi ^2_{\rm real} = \sum _{u, v} \frac{[Re(V_{\rm SMA}(u, v)) - Re(V_{\rm model}(u, v))]^2}{\sigma _{\rm real}^2(u, v) + \sigma _{\rm imag}^2(u, v)}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \chi ^2_{\rm imag} = \sum _{u, v} \frac{[Im(V_{\rm SMA}(u, v)) - Im(V_{\rm model}(u, v))]^2}{\sigma _{\rm real}^2(u, v) + \sigma _{\rm imag}^2(u, v)}, \end{equation} \tag{ 3 }$$

where σしぐま_real(u, v) = σしぐま_imag(u, v) is the 1σしぐま uncertainty level for each visibility and is determined from the system temperatures (this corresponds to a natural weighting scheme).

To sample the posterior probability density function (PDF) of our model parameters, we use emcee (Foreman-Mackey et al. 2013), a Markov chain Monte Carlo (MCMC) code that uses an affine-invariant ensemble sampler to obtain significant performance advantages over standard MCMC sampling methods (Goodman & Weare 2010).

We employ a "burn-in" phase with 250 walkers and 1000 iterations (i.e., 250,000 samplings of the posterior PDF) to identify the best-fit model parameters. This position is then used to initialize the "final" phase with 250 walkers and 20 iterations (i.e., 5,000 samplings of the posterior PDF) to determine uncertainties on the best-fit model parameters. The autocorrelation time for each parameter in a given ensemble of walkers and is of order unity for each parameter, implying that we have 5,000 independent samplings of the posterior PDF, more than enough to obtain a robust measurement of the mean and uncertainty on each parameter of the model.

During each MCMC iteration, we also measure the magnification factor at 880 μみゅーm, μみゅー₈₈₀ (we follow the nomenclature in the SMG literature here and use μみゅー to refer to the total magnification obtained by summing over all individual lensed components). Here, we describe how we measure μみゅー₈₈₀.

First, we take the unlensed, intrinsic source model and measure the total flux density (S_in) within an elliptical aperture (A_in) centered on the source with ellipticity and position angle equal to that of the source model and with a semi-major axis length of 2a_s. Second, we take the lensed image of the best-fit model and measure the total flux density (S_out) within the aperture A_out, where A_out is determined by using Gravlens to map A_in in the source plane to A_out in the image plane (using the lens parameters which correspond to the best-fit model). The magnification is then computed simply as μみゅー₈₈₀ = S_out/S_in. The best-fit value and 1σしぐま uncertainty are drawn from the posterior PDF, as with the other parameters of the model.

The choice of A_in has important implications for magnification measurements. For multiply imaged systems, apertures that are too large include in the source plane too much flux density that is far away from the caustic and relatively unmagnified. This is a particularly important issue for the models used here because the Sérsic index of the background source is a free parameter. The Sérsic index is partially degenerate with the half-light radius of the source, in the sense that good fits to the data can be obtained with a combination of small source size and small Sérsic index or large source size and large Sérsic index. In accordance with this, our model fits sometimes include relatively large sources where significant fractions of the unlensed flux density (≈10%–20%) arise from regions in the source plane far away from the caustic and hence contribute nothing to the observed lensed emission. This situation biases the estimate of μみゅー₈₈₀ below the true value. Conversely, apertures that are too small will omit flux density in the source plane that is detected at high significance in the SMA imaging, thus biasing the estimate of μみゅー₈₈₀ above the true value. Our choice—double the semi-major axis length of the source—represents a compromise between these two extremes.

In this section, we describe the basic characteristics of each object in the SMA subsample, including the position of the lens relative to the SMA 880 μみゅーm emission centroid, the lensing configuration (where applicable), and any unique notes for each object. Figure 3 shows the best-fit model in comparison with the SMA data for every lensed SMG with a robust lens model. Tables 5 and 6 present, for lenses and sources respectively, the model parameter mean values and 1σしぐま uncertainties as drawn from the posterior PDF for each parameter. Note that in some cases the posterior PDFs are non-Gaussian and therefore the best-fit model shown in Figure 3 does not always correspond perfectly to the model parameter mean values presented in Tables 5 and 6.

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Table 5. Gravitational Lens Model Results: Lens Properties

Object	ΔでるたαあるふぁNIR	ΔでるたδでるたNIR	Δでるたαあるふぁlens	Δでるたδでるたlens	θしーたE	lens	ϕlens	χかい2	NDOF
	('')	('')	('')	('')	('')		(deg)
J021830.5−053124	−0.17	−0.02	0.09 ± 0.04	−0.13 ± 0.05	0.44 ± 0.02	0.35 ± 0.10	156 ± 18	106027.4	114284
J022016.5−060143	2.1	0.6	2.07 ± 0.01	0.63 ± 0.01	0.27 ± 0.13	0.29 ± 0.13	114 ± 52	131464.2	133819
...	−4.76	0.3	−4.76	0.3	0.34 ± 0.14	0.30 ± 0.18	53 ± 54	...	...
J083051.0+013224	0.18	0.10	0.20 ± 0.01	0.00 ± 0.02	0.39 ± 0.02	0.43 ± 0.05	123 ± 3	97761.3	100561
...	0.54	0.595	0.54	0.595	0.43 ± 0.02	0.25 ± 0.07	47 ± 9	...	...
J084933.4+021443	−7.90	−0.60	−7.90	−0.60	1.41 ± 0.04	0.11 ± 0.06	62 ± 22	188965.5	168376
J085358.9+015537	0.04	0.02	0.09 ± 0.03	−0.03 ± 0.01	0.553 ± 0.004	0.06 ± 0.02	70 ± 12	160445.4	161200
J090302.9−014127	0.20	0.16	0.092 ± 0.006	0.03 ± 0.02	0.33 ± 0.02	0.39 ± 0.07	83 ± 5	124529.4	124378
J090311.6+003906	0.07	0.10	0.13 ± 0.02	0.03 ± 0.05	1.52 ± 0.03	0.34 ± 0.05	179 ± 4	198305.2	188164
J090740.0−004200	−0.03	0.17	0.01 ± 0.04	0.03 ± 0.04	0.59 ± 0.04	0.50 ± 0.08	44 ± 5	98244.2	91314
J091043.1−000321	0.22	−0.14	0.05 ± 0.02	−0.18 ± 0.01	0.95 ± 0.02	0.08 ± 0.03	152 ± 24	171966.9	159664
J091305.0−005343	−0.32	0.35	−0.45 ± 0.05	0.38 ± 0.08	0.43 ± 0.07	0.51 ± 0.13	43 ± 14	211928.5	200412
J103826.6+581542	−1.80	0.25	−1.1 ± 0.2	0.27 ± 0.09	2.0 ± 0.2	0.54 ± 0.11	18 ± 3	78239.2	76704
J105750.9+573026	−0.65	−1.19	−0.11 ± 0.05	−0.80 ± 0.08	3.86 ± 0.01	0.52 ± 0.01	14 ± 1	142740.4	120764
...	1.7	3.4	1.7	3.4	0.12 ± 0.01	0.54 ± 0.11	170 ± 9	...	...
J114637.9−001132	−0.40	0.02	−0.59 ± 0.02	0.02 ± 0.02	0.65 ± 0.02	0.26 ± 0.04	114 ± 19	90621.1	94863
...	1.154	1.270	1.154	1.270	0.67 ± 0.01	0.50 ± 0.04	68 ± 2	...	...
...	−3.076	−1.688	−3.076	−1.688	0.61 ± 0.03	0.33 ± 0.07	95 ± 12	...	...
...	−4.340	0.632	−4.340	0.632	0.51 ± 0.04	0.50 ± 0.10	77 ± 6	...	...
J125135.4+261457	0.04	0.13	0.05 ± 0.04	−0.25 ± 0.09	1.02 ± 0.03	0.46 ± 0.06	122 ± 1	72453.5	62912
J125632.7+233625	0.11	0.25	0.00 ± 0.01	0.25 ± 0.01	0.68 ± 0.01	0.31 ± 0.03	123 ± 1	62465.3	45880
J132427.0+284452	3.09	5.22	3.09	5.22	1.7 ± 0.4	0.34 ± 0.14	81 ± 16	228617.0	213957
...	−7.51	−9.66	−7.51	−9.66	2.2 ± 0.3	0.14 ± 0.09	88 ± 15	...	...
J132630.1+334410	0.68	−0.78	0.53 ± 0.05	−0.54 ± 0.05	1.80 ± 0.02	0.26 ± 0.04	66 ± 4	43605.2	38964
J133008.4+245900	−0.24	−0.02	−0.14 ± 0.03	−0.13 ± 0.02	0.88 ± 0.02	0.52 ± 0.03	81 ± 1	71948.3	52744
J133649.9+291801	...	...	0.00 ± 0.11	0.03 ± 0.16	0.40 ± 0.03	0.38 ± 0.14	120 ± 13	100742.9	95844
J134429.4+303036	−0.03	−0.10	−0.03 ± 0.02	−0.01 ± 0.02	0.92 ± 0.02	0.39 ± 0.06	172 ± 14	182936.2	171864
J141351.9−000026	−0.32	−2.65	−0.32 ± 0.03	−2.50 ± 0.03	1.13 ± 0.09	0.31 ± 0.12	118 ± 33	132668.6	123420
J142413.9+022303	−0.11	0.79	−0.27 ± 0.03	0.63 ± 0.03	0.57 ± 0.01	0.29 ± 0.01	62 ± 1	108899.5	144191
...	0.025	−0.327	0.025	−0.327	0.40 ± 0.01	0.06 ± 0.02	133 ± 14	...	...
J142823.9+352619	−0.14	−0.02	−0.01 ± 0.10	0.00 ± 0.11	0.10 ± 0.03	0.36 ± 0.18	87 ± 40	17284.0	17036
J142825.5+345547	0.25	−0.05	−0.20 ± 0.03	−0.04 ± 0.03	0.77 ± 0.03	0.46 ± 0.06	56 ± 5	93163.7	75448
J143330.8+345439	0.00	0.05	−0.07 ± 0.03	0.07 ± 0.02	0.28 ± 0.02	0.59 ± 0.08	104 ± 7	91554.9	117732

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Table 6. Gravitational Lens Model Results: Source Properties

IAU Name	Δでるたαあるふぁs	Δでるたδでるたs	ns	as	s	ϕs	μみゅー880 μみゅーm
	('')	('')		('')		(deg)
J021830.5−053124	−0.08 ± 0.03	0.22 ± 0.04	2.1 ± 0.9	0.33 ± 0.12	0.29 ± 0.13	82 ± 22	4.4 ± 1.0
J022016.5−060143	−1.54 ± 0.02	0.65 ± 0.02	2.1 ± 0.8	0.16 ± 0.05	0.25 ± 0.10	68 ± 34	1.5 ± 0.3
...	−2.56 ± 0.02	−0.82 ± 0.07	1.7 ± 0.8	0.30 ± 0.07	0.40 ± 0.13	45 ± 53	1.2 ± 0.1
...	−2.61 ± 0.02	−2.05 ± 0.02	1.4 ± 0.7	0.28 ± 0.06	0.19 ± 0.09	72 ± 44	1.2 ± 0.0
J083051.0+013224	−0.25 ± 0.01	0.08 ± 0.02	1.8 ± 0.3	0.14 ± 0.01	0.32 ± 0.06	20 ± 7	6.9 ± 0.6
J084933.4+021443	0.67 ± 0.03	−0.73 ± 0.03	2.1 ± 0.7	0.15 ± 0.03	0.20 ± 0.12	103 ± 35	2.8 ± 0.2
...	11.3	3.0	0.4 ± 0.3	0.25 ± 0.02	0.33 ± 0.10	24 ± 18	1.0
...	14.3	4.0	2.0 ± 0.9	0.17 ± 0.07	0.43 ± 0.17	93 ± 34	1.0
J085358.9+015537	−0.09 ± 0.03	0.001 ± 0.002	2.0 ± 0.7	0.06 ± 0.01	0.33 ± 0.14	83 ± 17	15.3 ± 3.5
J090302.9−014127	−0.06 ± 0.01	−0.01 ± 0.01	2.5 ± 0.6	0.42 ± 0.12	0.22 ± 0.12	116 ± 23	4.9 ± 0.7
J090311.6+003906	−0.26 ± 0.03	0.04 ± 0.05	2.2 ± 0.4	0.52 ± 0.10	0.35 ± 0.06	94 ± 11	11.1 ± 1.1
J090740.0−004200	−0.15 ± 0.03	0.14 ± 0.03	2.0 ± 1.1	0.16 ± 0.10	0.31 ± 0.14	73 ± 57	8.8 ± 2.2
J091043.1−000321	−0.017 ± 0.008	0.23 ± 0.02	1.5 ± 0.5	0.13 ± 0.02	0.35 ± 0.12	5 ± 20	10.9 ± 1.3
J091305.0−005343	0.42 ± 0.05	−0.37 ± 0.06	3.0 ± 0.6	0.76 ± 0.12	0.56 ± 0.13	129 ± 10	2.1 ± 0.3
J103826.6+581542	1.1 ± 0.2	−0.33 ± 0.06	3.0 ± 0.7	0.45 ± 0.18	0.44 ± 0.19	129 ± 32	7.1 ± 1.5
J105750.9+573026	−0.07 ± 0.04	0.76 ± 0.06	2.4 ± 0.8	0.57 ± 0.08	0.58 ± 0.11	122 ± 6	9.2 ± 0.4
J114637.9−001132	−0.50 ± 0.06	−0.28 ± 0.04	0.5a	0.38 ± 0.03	0.77 ± 0.03	107 ± 4	9.5 ± 0.6
J125135.4+261457	0.02 ± 0.04	0.22 ± 0.09	1.9 ± 0.6	0.15 ± 0.03	0.22 ± 0.10	109 ± 32	11.0 ± 1.0
J125632.7+233625	0.014 ± 0.006	−0.12 ± 0.01	1.1 ± 0.6	0.07 ± 0.01	0.37 ± 0.14	140 ± 21	11.3 ± 1.7
J132427.0+284452	−3.8 ± 0.4	−5.1 ± 0.6	2.4 ± 0.4	0.72 ± 0.09	0.69 ± 0.01	169 ± 17	2.8 ± 0.4
J132630.1+334410	−0.60 ± 0.03	0.60 ± 0.03	1.1 ± 0.3	0.22 ± 0.02	0.18 ± 0.09	150 ± 15	4.1 ± 0.3
J133008.4+245900	0.05 ± 0.01	0.23 ± 0.02	1.6 ± 0.7	0.09 ± 0.03	0.37 ± 0.10	129 ± 29	13.0 ± 1.5
J133649.9+291801	−0.04 ± 0.09	−0.05 ± 0.15	1.2 ± 0.5	0.19 ± 0.03	0.43 ± 0.12	125 ± 13	4.4 ± 0.8
J134429.4+303036	0.22 ± 0.02	0.04 ± 0.02	1.9 ± 0.5	0.24 ± 0.06	0.39 ± 0.07	100 ± 15	11.7 ± 0.9
J141351.9−000026	0.13 ± 0.13	1.40 ± 0.09	1.5 ± 0.5	0.30 ± 0.04	0.48 ± 0.12	62 ± 7	1.8 ± 0.3
J142413.9+022303	−0.24 ± 0.03	−0.53 ± 0.03	2.9 ± 0.3	0.64 ± 0.07	0.27 ± 0.09	77 ± 12	4.6 ± 0.5
J142823.9+352619	−0.00 ± 0.07	0.03 ± 0.08	1.9 ± 1.2	0.10 ± 0.07	0.33 ± 0.18	64 ± 62	3.0 ± 1.5
J142825.5+345547	0.05 ± 0.01	0.15 ± 0.03	0.7 ± 0.4	0.16 ± 0.04	0.50 ± 0.09	49 ± 10	10.3 ± 1.7
J143330.8+345439	−0.08 ± 0.02	−0.07 ± 0.02	1.9 ± 0.7	0.31 ± 0.06	0.55 ± 0.14	116 ± 13	4.5 ± 0.4

Note. ^an_s = 0.5 was assumed for this source.

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J021830.5−053124. SMA data from the compact and extended array configurations were originally presented in Ikarashi et al. (2011) and Wardlow et al. (2013). This paper presents new data obtained the in the very extended array, permitting the first resolved measurement of the 880 μみゅーm emission from this object. The best-fit lens model finds a position for the lens that is offset relative to that indicated in the HST image by 026 in R.A. and −011 in Decl. These values are near the expected level of 02–03 absolute astrometric uncertainty between the SMA and HST reference frames. The lensed emission is barely resolved by the SMA due to the small Einstein radius of the lens. The bulk of the source-plane emission originates outside the tangential caustic, favoring a two-image rather than four-image configuration (excluding the central demagnified image, which is never detected at the sensitivity levels probed in our data). This is consistent with the best-fit magnification factor of μみゅー₈₈₀ = 4.4 ± 1.0.

J022016.5−060143. This object is the subject of a detailed study by Fu et al. (2013). We present it here mostly for completeness, but also to test the validity of the model in Fu et al. (2013) using the visibility-plane lens modeling technique described in this paper. We use a two-component lens model and a three-component source model to reproduce the observed 880 μみゅーm emission. We enforce a minimum Einstein radius of θしーた_E > 02 for both lenses (corresponding to a minimum mass of M_lens > 1.6 × 10¹⁰ M_☉). This is intended to reproduce the prior on the lens masses used by Fu et al. (2013) based on the lens stellar masses and an assumed relation between the stellar mass and dark matter halo mass. The best-fit parameters of this model are statistically consistent with those found by Fu et al. (2013), despite a very different approach in modeling the data. In particular, we find a modest magnification factor is appropriate for each of the three components, with an average total magnification factor of 〈μみゅー_{880 μみゅーm}〉 = 1.3 ± 0.1.

J083051.0+013224. The lensed emission has an unusual configuration that reflects the complexity of the foreground mass distribution due to the presence of a secondary lensing galaxy <1'' northwest of the primary lensing galaxy. WHT/ACAM spectroscopy shows that the primary lens is located at z_lens = 0.626 based on G-band and Mg absorption features, while the secondary lens has very faint continuum and an emission line at observed frame 7462 Å. If this feature is [O ii]3727, then its redshift is z_lens2 = 1.002. The best-fit lens model for this system assigns each of the foreground galaxies approximately equal Einstein radii (≈04). Given the stated lens and source redshifts, this implies lens masses of ≈0.7 × 10¹¹ M_☉ and 1.6 × 10¹¹ M_☉, respectively.

J084933.4+021443. This object is the subject of a detailed study by Ivison et al. (2013). It is presented here for completeness and to ensure that the use of a visibility-plane lens model provides the same results as given in Ivison et al. (2013). Note that the center and extent of the image cutout shown in Figure 3 has been adjusted from the SMA emission centroid used in Figure 2 to include only the lens and the lensed source (dubbed "T" in Ivison et al. 2013). This is done simply to facilitate the comparison of model and data. An additional two sources used in the fitting process are not shown in this diagram. These two unlensed sources are modeled using the same visibility method but assuming no magnification by foreground objects. We find evidence for a larger magnification factor for "T" than Ivison et al. (2013): μみゅー₈₈₀ = 2.8 ± 0.2 instead of μみゅー₈₈₀ = 1.5 ± 0.2. A 2σしぐま emission peak located just northwest of the lens can be seen in Figure 3. If real, this emission peak supports the notion of a higher magnification factor for this object. Overall, however, our results are in broad agreement with those of Ivison et al. (2013).

J085358.9+015537. Although the image separations are small for this object (θしーた_E = 0553 ± 0004), the S/N is high. The source appears to lie very close to the caustic (which is itself small due to the low ellipticity of the lens), implying a high magnification factor μみゅー₈₈₀ = 15.3 ± 3.5.

J090302.9−014127. The image separations are close to the smallest values found in the SMA subsample (θしーた_E = 035 ± 002). There has been tentative evidence (3σしぐま) of CO(J = 5–4) emission from the lens at z_lens = 0.942 ± 0.004 (Lupu et al. 2012), but this has not been confirmed with subsequent, more sensitive observations that rule out lens redshifts from 0.922 to 0.944 (Omont et al. 2011). In the lens model presented here, the lens is assumed to be an insignificant submm emitter.

J090311.6+003906. Because of the high S/N and well-separated images of the background source (θしーた_E = 152 ± 003), the lens model is well-constrained for this object. However, the image of the residual visibilities shows emission at the ±3σしぐま level, possibly an indication of complexity in the source structure that is not captured by a single Sérsic profile.

J090740.0−004200. The counter image to the northwest of the lens is detected at the 4σしぐま level and provides good constraints on the lens model for this object.

J091043.1−000321. This is one of a handful of objects with emission at the >±2σしぐま level that can be seen in the surface brightness map made from the residual visibilities. An edge-on galaxy located 44 to the northwest with a position angle of 135° east of north could be responsible for an external shear that has not been included in the lens model. Alternatively, the residual flux density may reflect a more complicated source structure than can be represented by our choice of a single Sérsic profile.

J091305.0−005343. The best-fit lens models for this object are obtained when the source is relatively large and most of it is located outside the region in the source plane that produces multiple images. For this reason, the best-fit magnification factor is relatively low (μみゅー₈₈₀ = 2.1 ± 0.3).

J091840.8+023047. This object is unique in having no counterpart within 1'' of the 880 μみゅーm centroid in the HST/F110W Snapshot imaging. There is no obvious morphological signature of lensing based on the SMA data, so this may be a rare unlensed SMG. It is not included in Figure 3 since no lens model is available. Further investigation is needed to determine the nature of this object.

J103826.6+581542. This object was originally presented by Wardlow et al. (2013). We present it here for completeness and to test the validity of the model in Wardlow et al. (2013) using the visibility-plane lens modeling technique described in this paper. Our results for the size (045 ± 018 versus <05) and magnification factor of the background source (7.1 ± 1.5 versus $5.32^{+1.28}_{-1.06}$) are consistent with those reported in Wardlow et al. (2013). Since a redshift measurement for the background source remains unavailable, no further analysis is possible for this object.

J105712.2+565457. This object was originally presented by Wardlow et al. (2013). We present a slightly modified reduction of this object here. Instead of MIRIAD's mossdi task, which was used in Wardlow et al. (2013) to image this object, here we have shifted all of the visibility datasets to have the same phase center and then used MIRIAD's clean task. This resulted in slightly improved spatial resolution (068 × 057 versus 118 × 097), that is still not sufficient to distinguish cleanly separated images from a single lensed source. It is not included in Figure 3 since no lens model is available. Further investigation is needed to determine the nature of this object.

J105750.9+573026. This object was originally analyzed in a series of papers reporting its discovery, interstellar medium (ISM) properties, gas dynamics, and (on the basis of K_p-band imaging) lensing geometry (Conley et al. 2011; Scott et al. 2011; Riechers et al. 2011b; Gavazzi et al. 2011). We present new very extended array data here and compare the SMA and Keck lens models. The very extended array data are not as sensitive as the compact array data, so natural weighting provides a beam size of 099 × 094. This is insufficient to resolve the individual images of the lensed source seen in the image from the compact array only data and is further indication that the lens model is primarily constrained by the compact array data. Because the S/N and spatial resolution in the Keck image are superior to those of the SMA image, we fix the parameters of the lens to match those of the model found by Gavazzi et al. (2011). We find a similar magnification factor at 880 μみゅーm compared to K_p (9.2 ± 0.4 versus 10.9 ± 0.7). The model has difficulty reproducing the locations of the images seen in the SMA data. In addition, we find an offset in the position of the lens of 054 in R.A. and 04 in Decl. This represents a ≈2–3σしぐま discrepancy in R.A. and may be an indication that some of the assumptions in our model are over-simplifications.

J113526.3−014605. There is no counterpart detected in a 15-minute K_s integration with the WHT. The SMA 880 μみゅーm image is clearly resolved, but does not show individual, well-separated images of a lensed SMG. It is not included in Figure 3 since no lens model is available. Further investigation is required to determine whether strong lensing is occurring in this object. The main avenues for progress are higher-spatial resolution submm imaging (e.g., SMA very extended array) and deeper observations in the optical or near-IR to determine whether or not there is a lensing galaxy.

J114637.9−001132. This object was originally presented by Fu et al. (2012). Here, we present new SMA extended array data that resolve the lensed emission into five striking images of the SMG. The position and morphology of the lensed SMG in the source plane reported here are statistically consistent with those found by Fu et al. (2012). The presence of flux density at the 3σしぐま level in the map of the residual visibilities (see Figure 3) indicates our simple assumptions about the lens mass model (singular isothermal ellipsoids at the locations of galaxies identified in the Keck-II/NIRC2-LGSAO and HST/F110W imaging) and/or the background source Sérsic profile may be breaking down.

J125135.4+261457. The offset between the position of the lens from the lens model and from the WHT astrometry is 001 in R.A. and 038 in Decl. compared to the best-fit parameters from the lens model. The offset in Decl. is larger than expected given the astrometric uncertainty in aligning the SMA and WHT reference frames. On the other hand, the lens model correctly predicts the ellipticity and position angle of the lens potential (no priors were assumed for the shape of the lens potential), a strong indication that the lens model is robust.

J125632.7+233625. The S/N is very high in this object and the images of the lensed SMG are well-separated, making the parameters of the lens model very robust.

J132427.0+284452. This object is the subject of detailed studies by (George et al. 2013) and H. Fu et al. (in preparation) to explore its dust, gas, and stellar properties. In this paper, we use the Keck-II/NIRC2-LGSAO image to constrain the positions of the lenses and apply our visibility-plane lens modeling technique to the SMA data. We find that a two-lens mass model is needed to reproduce the observed data. These correspond directly to two galaxies detected at high significance in the Keck imaging. However, it should be noted that there are two nearby clusters detected in the Red-Sequence Cluster (RSC) survey (RCS J132427+2845.2 at z = 0.997 ± 0.017 and RCS J132419+2844.7 at z = 0.802 ± 0.018 Gladders & Yee 2005). The centers of these clusters are uncertain due to a lack of X-ray data and no clear brightest-cluster galaxy, but the RCS surface density map suggests that RCS J132427+2845.2 lies only 10'' away from lensed SMG. Our lens modeling here does not account for the presence of this cluster. Furthermore, the background source is not multiply imaged in the SMA data, so the constraints on the lens parameters are weak. The possibility of counter images falling outside of the SMA primary beam (FWHM of 36'' at 880 μみゅーm) is low due to the lack of such a counterpart in the SPIRE maps.

J132630.1+334410. Two well-separated images of the lensed SMG are obvious in the SMA data, indicating the source is strongly lensed but is not among the highest magnification sources. The lens modeling result is consistent with this (μみゅー₈₈₀ = 4.1 ± 0.3), suggesting a robust model fit.

J132859.3+292317. The nature of this object is unclear based on existing data. It is well-detected and clearly spatially resolved by the SMA at 880 μみゅーm. The problem is the lack of optical or near-IR imaging at a depth beyond what is achieved in SDSS, where the object is undetected in all bands. One plausible interpretation is that the object is lensed by a relatively low-mass foreground galaxy at intermediate redshift so that it is undetected in the SDSS images. Alternatively, it is conceivable that the object is not lensed and has an intrinsic submm flux density of S₈₈₀ = 51.8 ± 2.0 mJy. We consider this latter option unlikely since there are no known SMGs with intrinsic submm flux densities that high. However, without additional data to confirm this intuition, we do not consider this object further in our analysis. It is not included in Figure 3 since no lens model is available.

J133008.4+245900. Multiple, well-separated images are detected in the SMA data, consistent with the relatively large inferred magnification factor from the lens model (μみゅー₈₈₀ = 13.0 ± 1.5), suggesting a robust lens model has been obtained.

J133649.9+291801. This object is similar to J132859.3+292317 in that there is no significant detection in any of the SDSS optical bands, nor is deeper optical or near-IR imaging available. However, the SMA morphology provides evidence typical of strong lensing. Here, we assume that the object is strongly lensed.

J134429.4+303036. The lensed images are well-separated and well-detected in the SMA data. In fact, the S/N is so high that the map of the residual visibilities reveals emission at the ±3σしぐま level, likely indicating that some of our model assumptions are over-simplifications. Nevertheless, the model captures the vast majority of the SMA emission and therefore provides a fair representation of intrinsic source size and luminosity. This object is similar to J125135.4+261457 in that the lens model successfully predicts the ellipticity and position angle of the lens potential without any non-standard priors placed on these parameters.

J141351.9−000026. No counter image of this target is detected in the SMA data, an immediate indication that this object is not strongly lensed. The possibility of counter images falling outside of the SMA primary beam (FWHM of 36'' at 880 μみゅーm) is low due to the lack of such a counterpart in the SPIRE maps. The lens modeling confirms this, with μみゅー₈₈₀ = 1.8 ± 0.3. The lens is located 77 northeast of a brightest cluster galaxy (BCG). The lens redshift used here is from the BCG, and in order to derive the mass of the lens from its θしーた_E value, we have assumed that it is located at the same redshift as the BCG.

J142413.9+022303. This object is the subject of a detailed study by Bussmann et al. (2012). We do not reproduce the results of the previous work here because it used the same visibility-plane lens modeling technique outlined in this paper.

J142823.9+352619. This object was originally discovered in Spitzer mid-IR imaging of the Boötes Field (Borys et al. 2006) and has since been the subject of a great many follow-up observations, a thorough summary of which may be found in Wardlow et al. (2013). We present new SMA extended array data which do not resolve the source, further corroborating the idea that this object is very small and is not strongly lensed (i.e., μみゅー < 2). Since we know that there is an intervening galaxy along the line of sight to the Herschel source, we impose a minimum Einstein radius for the lens of θしーた_E > 01 (corresponding to a minimum lens mass of M_lens > 10¹⁰ M_☉). The constraints on the lens model are weak. One of the few robust claims we can make regarding this source is that it is very small (a_s < 02).

J142825.5+345547. This object is the subject of a detailed study by J. L. Wardlow et al. (in preparation). The lens model suggests the lens is located 020 west of the SMA emission centroid, but the peak of the edge-on spiral seen in the astrometrically-aligned HST image occurs 025 east of the SMA emission centroid. This is a difference of 045 in R.A. (the difference in Decl. is insignificant) and is larger than the expected 1σしぐま astrometric uncertainty of 02. However, estimating the peak position of the lensing galaxy is difficult because it is nearly exactly edge-on with a prominent dust lane and very little bulge. Moreover, the lensed images are well-resolved and well-detected in the SMA data, and the lens model correctly predicts the position angle and approximate ellipticity of the lens without any priors on these parameters. For these reasons, we consider the lens model for this object to be robust.

J143330.8+345439. This object was originally presented in Wardlow et al. (2013). We present a slightly modified reduction of this object here in which we shift all of the visibility datasets to have the same phase center and then use MIRIAD's clean task. This provides slightly improved spatial resolution that is still insufficient to identify clearly separated images of a single lensed source. Nevertheless, we use the knowledge that spectroscopic redshifts are available for both the lens and background source to infer that strong lensing is occurring. We then model this object using the same set of model parameters as applied to the other objects in the SMA subsample. Note that absolute astrometric calibration based on alignment to existing ground-based imaging is not available in the HST image for this object, but the lens model still finds a position for the lens that is in reasonable agreement (within 04) with that in the HST image.

J144556.1−004853. No lens or source redshift is available for this object. The SMA data resolve the 880 μみゅーm emission, but detect the source at relatively low S/N (S/N <6). Keck-II/NIRC2-LGSAO K_s-band imaging detects a counterpart that is located between the resolved components identified in the SMA data. This could be a detection of the lens, or it could simply be an intrinsic component of an unlensed SMG. Unlike J132859.3+292317, the total 880 μみゅーm flux density (S₈₈₀ = 9.0 ± 2.1 mJy) is not unprecedented for unlensed SMGs. It is not included in Figure 3 since no lens model is available. Further investigation is required to determine if strong lensing is occurring in this object.

Strong gravitational lensing permits the study of SMGs at higher spatial resolutions than would otherwise be possible (e.g., a highly magnified SMG at z = 2 has been studied at 100 pc resolution with the SMA; Swinbank et al. 2010). While this is a highly attractive feature of strong lensing, it does necessitate certain unique considerations when transferring conclusions regarding lensed SMGs to the unlensed population. Chief amongst these considerations is the size bias inherent in any flux-limited sample of lensed galaxies. This bias has been investigated in a quantitative manner by a number of authors (Serjeant 2012; Hezaveh et al. 2012; Wardlow et al. 2013), who find that those objects with the brightest apparent flux densities should also have higher magnification factors and smaller sizes, on average.

Objects with high magnification factors are preferentially selected to have small sizes by flux-limited surveys like those of Herschel and SPT. This is because the degree of magnification depends primarily on the fraction of the source that is close to the caustic. A source that is extended relative to the size of the caustic will inevitably have a significant fraction of its flux density emitted in a region that is not highly magnified, so the total magnification factor summed over the entire source is not critically dependent on the exact location of the source relative to the caustic. Conversely, a population of lensed sources which are intrinsically compact will have a bimodal distribution of magnification factors that depends primarily on how far from the caustic the source is located. If a source is not highly magnified, it will likely not be bright enough to appear in our sample.

Figure 5 demonstrates this degeneracy between size (shown as half-light radius or r_half, where $r_{\rm half} = a_{\rm s} \sqrt{(1-e_{\rm s})}$) and magnification factor (μみゅー₈₈₀) for the SMA subsample and a handful of objects from SPT (Hezaveh et al. 2013). Our determinations of these values for the SMA subsample are presented in Table 6. Nearly every lensed SMG with r_half < 01 is associated with μみゅー₈₈₀ > 10. However, there are a surprising number of sources with 01 < r_half < 02 and relatively modest magnification factors of 3 < μみゅー₈₈₀ < 10. We will return to the implications of the large number of low-magnification objects in Section 5.2.

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For a source at z = 2, these sizes correspond to a physical scale of ≈1 kpc, which is at the low end of sizes measured for unlensed SMGs (Tacconi et al. 2006). It must be noted, however, that there is a subtle bias when comparing the sizes of lensed and unlensed SMGs: if intrinsically low surface brightness regions are preferentially located farther from the caustic than high surface brightness regions (as expected for the reasons outlined above), then we expect the lensed SMGs to show smaller sizes than unlensed SMGs, because differential lensing makes detecting the low surface brightness regions more difficult than in the unlensed scenario.

Definitive measurements of the relative bias in the size measurements of lensed and unlensed SMGs will require spatially resolved observations of unlensed SMGs. The dashed vertical lines in Figure 5 illustrate both the advantages offered by lensing and the difficulties that must be overcome to assemble a statistically significant sample of unlensed SMGs with spatially resolved imaging. From right to left, the three lines indicate the maximum spatial resolutions available with the SMA, Cycle 1 ALMA, and full ALMA. It is only with the full ALMA and baselines >10 km that spatial resolution better than 01 can be achieved at these wavelengths—i.e., matching the best the SMA can do today for lensed SMGs discovered by Herschel.

The number counts of unlensed SMGs fall off dramatically at the bright end of the luminosity function (e.g., Barger et al. 1999; Coppin et al. 2006; Oliver et al. 2010; Clements et al. 2010). This is the central reason why wide-field surveys at (sub-)mm wavelengths are useful tools for discovering strongly lensed galaxies (e.g., Blain 1996). There are several key elements of astrophysical interest in models which predict the magnification factor as a function of (sub-)mm flux density for strongly lensed galaxies found in wide-field (sub-)mm surveys. These are discussed in detail elsewhere (Perrotta et al. 2002; Negrello et al. 2007; Paciga et al. 2009; Hezaveh & Holder 2011; Wardlow et al. 2013), so we provide only the briefest of summaries here. In short, they are the lens mass profile (typically assumed to match the analytical form of Navarro–Frenk–White (NFW; Navarro et al. 1997) or a singular isothermal sphere) and the number densities of lenses and (unlensed) sources as functions of mass and redshift.

Figure 6 shows the magnification factor as a function of the 500 μみゅーm flux density for each strongly lensed SMG in the SMA subsample. Recall that we are complete for S₅₀₀ > 170 mJy (see Section 2.1), except for one object which is the subject of a paper by H. Messias et al., (in preparation). The blue line is taken from Wardlow et al. (2013) and represents the predicted mean μみゅー as a function of S₅₀₀ for a complete sample of strongly lensed SMGs (μみゅー > 2). There are far more low-μみゅー₈₈₀ objects than expected based on the model. In fact, only two objects (J085358.9+015537 and J142825.5+345547) have μみゅー₈₈₀ values that are consistent at the 1σしぐま level with the model predictions.

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It is not presently clear why the model over-predicts the magnification factors at a given S₅₀₀ value. One possibility is that our assumption of a single, smooth Sérsic profile for the background source leads to underestimates in some cases of the magnification factor. For example, one might imagine that a multi-component, clumpy model for the source morphology could reproduce the observed data while yielding larger magnification factors on average. Testing such models is beyond the scope of this paper, but we acknowledge that this possibility exists.

If the discrepancy between model and data is not simply a product of limited data quality, there still exist several possible explanations. Investigation into the dependence of the predicted mean magnification on the various model parameters shows that this prediction is sensitive to a number of factors. The first of these is the shape of the intrinsic SMG number counts. The intrinsic counts are not well constrained at the bright end, primarily because of the contribution from lensed SMGs (Wardlow et al. 2013). Modest changes in the parameters of the Schechter function used to characterize the counts have a significant effect on the predicted mean magnification factor. This can be seen by a comparison of the dotted blue line in Figure 6, which traces the predicted μみゅー₈₈₀(S₅₀₀) curve from Wardlow et al. (2013) based on the observed number counts of SMGs, and the dashed purple line, which shows μみゅー₈₈₀(S₅₀₀) found from a satisfactory joint fit to the SMG number counts and the magnification measurements reported in Table 6. The Schechter function from the joint fit has a flatter slope (by 25%), a brighter characteristic flux density (by 15%), and a lower normalization (by 30%). It is also possible that a Schechter function provides an inadequate description of the bright end shape of the number counts (e.g., if blending is not properly taken into account; see Fu et al. 2013; Ivison et al. 2013; Hodge et al. 2013; Karim et al. 2013).

Another important parameter for predicting magnification factors is a fixed maximum magnification (μみゅー_max), that is intended to reflect both the intrinsic sizes of the background sources as well as the typical angular separation between the centers of the sources and the centers of the lenses. The green lines shown in Figure 6 are taken from Lapi et al. (2012) and show the effect of this parameter on μみゅー₈₈₀(S₅₀₀). These curves help to highlight the apparent bimodality in μみゅー₈₈₀ values in the SMA subsample. One speculative explanation for this is a bimodal distribution of sizes in SMGs (this is supported by the sizes from the lens modeling; see Table 6 and Figure 5) and hence a bimodal distribution in μみゅー_max. Larger samples and higher-spatial resolution are needed to confirm this intriguing possibility, which could plausibly result from SMGs comprising a population of rotating disks as well as major mergers. Such a study is beyond the scope of this paper and is therefore deferred to future publications.

All but two sources in this sample have short baseline SMA data (i.e., D < 50 m) that provide a robust total flux density at 880 μみゅーm. The exceptions are J132630.1+334410, which is resolved into two distinct images of the background source, and J142823.9+352619, which appears unresolved in extended array only data. We therefore expect that the SMA total flux density measurements are reliable for the entire sample. In conjunction with the Herschel/SPIRE photometry, these data can be used to measure the apparent (i.e., lensed) bolometric luminosity as well as the shape of the far-IR SED. In this section, we describe the methodology we use to undertake this task.

For galaxies at redshifts of 1.5 < z < 4.5, Herschel/SPIRE and SMA 880 μみゅーm photometry probe rest-frame 45 μみゅーm to 350 μみゅーm. At these wavelengths, the dominant contribution is thermal emission from dust heated by star-formation or an active galactic nucleus (AGN). We fit single-temperature, optically-thin, modified blackbody curves to the data, assuming an emissivity parameter of βべーた = 1.5 (Hildebrand 1983). Studies based on Infrared Astronomical Satellite data have shown that this simple model gives a useful measure of the heating conditions of the ISM in galaxies (Desert et al. 1990). For the highest redshift sources (z_source > 3.5), the 250 μみゅーm channel of SPIRE probes the Wien side of the modified blackbody curve. The observed 250 μみゅーm flux density is under-predicted by this model if there are alternative powering sources that drive mid-IR luminosity (e.g., hot dust from AGN or intense SF). If such powering sources exist, the best-fit dust temperatures will be artificially inflated to compensate for the stronger 250 μみゅーm emission. However, for the four objects in the SMA subsample at z > 3.5), we do not find evidence for significantly higher dust temperatures or poor fits to the data (see Table 7, indicating that a simple modified blackbody is a reasonable choice even for the high-redshift objects.

Table 7. Intrinsic (i.e., Unlensed) Properties of SMA Sample (Assuming βべーた = 1.5, Optically Thin Modified Blackbody, and L_FIR Integrated over 40–120 μみゅーm)

IAU Name	$\chi ^2_{\rm min}$	Tdust	log(Mdust)	log(LFIR)	rhalf	log(ΣしぐまFIR)
		(K)	(M☉)	(L☉)	(kpc)	(L☉ kpc−2)
J021830.5−053124	7.11	36 ± 1	9.49 ± 0.11	12.79 ± 0.09	2.03 ± 0.71	11.45 ± 0.41
J022016.5−060143	4.46	37 ± 1	9.10 ± 0.10	12.79 ± 0.05	1.14 ± 0.37	11.93 ± 0.34
...	...	...	9.07 ± 0.10	12.76 ± 0.05	2.09 ± 0.35	11.34 ± 0.15
...	...	...	9.00 ± 0.10	12.67 ± 0.05	2.09 ± 0.39	11.24 ± 0.17
J083051.0+013224	0.86	44 ± 1	9.16 ± 0.06	13.09 ± 0.05	0.85 ± 0.07	12.44 ± 0.07
J084933.4+021443	9.62	36 ± 1	8.92 ± 0.05	12.72 ± 0.04	1.10 ± 0.22	11.86 ± 0.18
...	...	...	9.31 ± 0.05	13.11 ± 0.04	1.69 ± 0.19	11.87 ± 0.10
...	...	...	8.76 ± 0.05	12.56 ± 0.04	1.04 ± 0.42	11.85 ± 0.51
J085358.9+015537	0.98	36 ± 1	8.91 ± 0.09	12.37 ± 0.09	0.41 ± 0.08	12.37 ± 0.17
J090302.9−014127	1.13	38 ± 1	9.29 ± 0.07	12.92 ± 0.05	3.05 ± 0.92	11.20 ± 0.30
J090311.6+003906	0.87	34 ± 1	9.18 ± 0.06	12.45 ± 0.04	3.30 ± 0.65	10.63 ± 0.18
J090740.0−004200	10.30	43 ± 2	8.73 ± 0.10	12.58 ± 0.11	1.09 ± 0.62	11.92 ± 0.74
J091043.1−000321	25.66	41 ± 1	8.69 ± 0.06	12.50 ± 0.06	0.89 ± 0.16	11.82 ± 0.16
J091305.0−005343	0.04	35 ± 1	9.56 ± 0.08	12.94 ± 0.07	4.14 ± 0.72	10.92 ± 0.15
J105750.9+573026	2.64	47 ± 1	8.83 ± 0.05	12.94 ± 0.03	1.83 ± 0.26	11.62 ± 0.13
J114637.9−001132	1.86	41 ± 1	9.12 ± 0.06	12.90 ± 0.04	1.59 ± 0.13	11.70 ± 0.07
J125135.4+261457	2.24	39 ± 1	9.02 ± 0.06	12.68 ± 0.05	0.93 ± 0.21	11.97 ± 0.20
J125632.7+233625	0.41	40 ± 1	9.10 ± 0.07	12.80 ± 0.06	0.40 ± 0.07	12.80 ± 0.16
J132427.0+284452	47.59	37 ± 1	9.39 ± 0.07	12.92 ± 0.07	3.44 ± 0.44	11.05 ± 0.11
J132630.1+334410	13.33	36 ± 1	9.48 ± 0.04	13.00 ± 0.04	1.57 ± 0.17	11.81 ± 0.09
J133008.4+245900	1.14	43 ± 1	8.78 ± 0.06	12.66 ± 0.05	0.55 ± 0.24	12.52 ± 0.56
J133649.9+291801	7.62	39 ± 1	9.15 ± 0.09	12.87 ± 0.08	0.87 ± 0.34	12.27 ± 0.41
J134429.4+303036	6.90	38 ± 1	9.06 ± 0.04	12.73 ± 0.05	1.50 ± 0.40	11.62 ± 0.26
J141351.9−000026	23.28	38 ± 1	9.50 ± 0.09	13.18 ± 0.08	1.73 ± 0.31	11.92 ± 0.16
J142413.9+022303	0.49	41 ± 1	9.36 ± 0.05	13.17 ± 0.03	3.79 ± 0.38	11.22 ± 0.09
J142823.9+352619	6.14	39 ± 2	9.19 ± 0.20	12.77 ± 0.20	0.71 ± 0.43	12.52 ± 0.84
J142825.5+345547	0.88	38 ± 1	8.85 ± 0.10	12.45 ± 0.09	0.89 ± 0.18	11.77 ± 0.19
J143330.8+345439	0.84	39 ± 1	9.34 ± 0.06	12.96 ± 0.05	1.59 ± 0.34	11.78 ± 0.19

Note. Error bars are pertinent only to our chosen model and therefore underestimate the true uncertainties.

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Another consideration that is important at high redshift is the influence of the cosmic microwave background (CMB) radiation. The CMB acts as an additional source of heating of the dust that shifts the SED to warmer temperatures and boosts the observed flux densities. However, for the dust temperatures and redshifts of the SMA subsample, this effect is insignificant (da Cunha et al. 2013) and we therefore do not incorporate it into our model fitting process.

The modified blackbody curve used here has the following form for the flux density, S_νにゅー, given a dust temperature, T_dust:

$$\begin{equation} S_\nu \propto \frac{\nu ^{3 + \beta }}{ {\rm exp}(h \nu / k T_{\rm dust}) - 1 }. \end{equation} \tag{ 4 }$$

An analysis of more complicated models that incorporate additional temperature components (e.g., Dunne & Eales 2001; Planck Collaboration et al. 2011) or allow the frequency at which the thermal emission becomes optically thick to vary (e.g., Hayward et al. 2012) is beyond the scope of this paper. This is because we are chiefly concerned with the apparent far-IR luminosities (μみゅーL_FIR) of the sources, which are relatively insensitive to the particular details of the chosen modified blackbody model. The dust temperature is also of interest here, but mainly for the purpose of comparison to existing samples of SMGs. Our choices here are well-matched to those that have been made previously, thereby facilitating direct comparisons to the literature (e.g., Magnelli et al. 2012).

For a given μみゅーL_FIR and T_dust, we compute model flux densities at the Herschel/SPIRE and SMA bands by multiplying the modified blackbody curve with the appropriate filter function and integrating (for the SMA, we assume a top-hat filter shape with 8 GHz of bandwidth centered on νにゅー_LO). Calibration uncertainties are accounted for by adding 7% uncertainty in quadrature to each photometric measurement. Confusion noise is also included, though it is largely insignificant at the flux densities of the lensed SMGs. We use the emcee software package to iterate over plausible ranges in μみゅーL_FIR (10¹⁰–10¹⁵ L_☉) and T_dust (20–100 K) values for each lensed SMG (see Section 3.1 for a description of the behavior of emcee). We use 100 walkers and 200 iterations in the "burn-in" stage to converge on the best-fit model, keeping in mind the additional 7% absolute flux density calibration uncertainty in the Herschel/SPIRE and SMA photometry. In the final stage, we use 100 walkers and 10 iterations for a total of 1000 sets of model parameters. These provide the shape of the posterior probability density functions for μみゅーL_FIR and T_dust, which are then used to compute the best-fit values and the 1σしぐま confidence intervals. We use our measurements of μみゅー₈₈₀ from Section 5 to obtain the intrinsic, unlensed FIR luminosity, L_FIR. Finally, we compute dust masses using the standard equation (Hildebrand 1983),

$$\begin{equation} M_{\rm dust} = \frac{1}{1+z_{\rm s}} \frac{S_{880} d_{\rm L}^2}{\kappa _d^{\rm rest} B(\nu ^{\rm rest}, T_{\rm dust})}, \end{equation} \tag{ 5 }$$

where $\kappa _{\rm d}^{\rm rest}$ is the mass absorption coefficient and B(νにゅー^rest, T_dust) is the value of the blackbody function for the given T_dust and computed at the rest-frame frequency νにゅー^rest. We obtain $\kappa _{\rm d}^{\rm rest}$ by interpolation of the values in Draine (2003) over the appropriate range in rest-frame wavelength. The uncertainty in $\kappa _{\rm d}^{\rm rest}$ is a factor of a few and dominates the total error budget for our dust mass estimates.

We follow the procedure outlined here for all SMGs with Herschel/SPIRE and submm photometry. Besides the objects in this paper, this also includes SMGs from Hezaveh et al. (2013) (with photometry and redshifts coming from Weiß et al. 2013; Bothwell et al. 2013) and Magnelli et al. (2012).

The results of our SED fitting procedure are given for the SMA subsample in Table 7, including the best-fit reduced chi-squared value ($\chi ^2_{\rm min}$), the dust temperature (T_dust), the dust mass (M_dust; error bars do not include the factor of a few uncertainty in the mass opacity coefficient), the FIR luminosity (L_FIR), the half-light radius (r_half), and the FIR luminosity surface density (Σしぐま_FIR). The error bars are pertinent only to our chosen model of a single optically thin modified blackbody. In this table, the magnification factor and uncertainty inferred from the lens model have been used to compute intrinsic values and their uncertainties. For 13 objects, $\chi ^2_{\rm min} > 2$, suggesting that the single-component modified blackbody model is an over-simplification in nearly half of the SMA subsample.

The T_dust–L_FIR diagram is useful for characterizing the shape and amplitude of the rest-frame far-IR SED of dusty galaxies. Figure 7 shows these parameters for the objects in the SMA subsample, a handful of SPT lensed SMGs (Hezaveh et al. 2013), and a sample of primarily unlensed SMGs (Magnelli et al. 2012). Some of the objects in this diagram are known to have multiple components in the source plane (e.g., J022016.5−060143). In these cases, we show the L_FIR value integrated over all components in the source plane. This is primarily done because our measurements of T_dust are based on Herschel photometry in large part, which does not resolve the individual components in the source plane.

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Objects in the SMA subsample populate the high-L_FIR, high-T_dust regime (median L_FIR = 7.9 × 10¹² L_☉, median T_dust = 39 K) compared to unlensed SMGs (median L_FIR = 2.8 × 10¹² L_☉, median T_dust = 32 K). The selection of the brightest objects found in wide-field surveys is the dominant reason for the high L_FIR values reported here. However, thanks to lensing, we probe a relatively wide range in L_FIR: 2.7–17.0 × 10¹² L_☉, despite selecting the brightest objects at 500 μみゅーm. The top axis in Figure 7 shows the star formation rates (SFRs) that are inferred based on the Kennicutt (1998) prescription for converting L_IR (computed from L_FIR assuming a bolometric correction factor of 1.91; Dale et al. 2001) to SFR assuming a Salpeter IMF. Finally, the high dust temperatures in the SMA subsample reflect the fact that a greater portion of the IR luminosity is emitted at short wavelengths in the highest luminosity sources. This result is consistent with what has been found previously for lensed SMGs (Harris et al. 2012) as well as for unlensed SMGs (Magnelli et al. 2012) and is one of the first indications that Herschel-selected lensed SMGs are not greatly dissimilar in their physical properties from unlensed SMGs (the same conclusion is reached by Harris et al. 2012).

One of the central themes in understanding the evolution of the most luminous galaxies during the epoch of peak SFR density in the universe (i.e., 1 < z < 4; e.g., Burgarella et al. 2013) is the role played by major mergers. It has been clear for decades that the most luminous galaxies at z ∼ 0—commonly known as ultra-luminous infrared galaxies (ULIRGs) and defined to have L_IR > 10¹² L_☉— are powered by major mergers (e.g., Armus et al. 1987; Clements et al. 1996; Murphy et al. 1996). However, it is also clear that such systems contribute only trivially to the SFR density in the universe today because they are so rare (Soifer et al. 1986). Since ULIRGs contribute an increasing fraction of the total SFR density as a function of redshift (e.g., Le Floc'h et al. 2005; Magnelli et al. 2011), some theoretical efforts have suggested that an increase in the merger rate in conjunction with strong inflows of gas from the intergalactic medium (IGM) at high redshift could allow the merger paradigm to explain a large fraction of the luminous galaxies at these epochs (e.g., Hopkins et al. 2010). Providing support for this theoretical paradigm are sub-arcsecond observations of CO emission in a handful of SMGs which show that a significant fraction of SMGs that are spatially resolved have multiple, similar mass components—i.e., they are major mergers (Tacconi et al. 2008; Engel et al. 2010; Ivison et al. 2011; Riechers et al. 2011a).

On the other hand, some recent theoretical attempts to simulate the formation of galaxies on cosmological scales (i.e., cubes that are ≈200 Mpc on a side) have found evidence that favors a model of galaxy formation in which smooth flows of gas from the IGM feed large, extended reservoirs of gas in disk galaxies (e.g., Kereš et al. 2009; Davé et al. 2010). There is even observational evidence based on dynamical models that disk-like systems exist at high redshift (Hodge et al. 2012). Ultimately, detailed dynamical models of statistically significant samples can resolve the dispute between the merger and disk paradigms. However, assembling the requisite datasets is extremely expensive in terms of telescope time. A far more feasible goal is spatially resolved observations of the dust emission in SMGs at high redshift.

The dust emission in SMGs is critically important because it represents reprocessed emission from young massive stars which provide a reliable measure of the instantaneous SFR (within the past ≈10 Myr). This assumes no significant contribution from a cold, diffuse ISM component (supported by our measurements of T_dust) and no significant contribution from an AGN (our use of the FIR luminosity integrated over 40–120 μみゅーm rest-frame is intended to mitigate this possibility). Measuring the size-scale and the luminosity of this dust can therefore in principle be used to contrast extended galaxy morphologies expected from accretion-fuelled disks (e.g., Dekel et al. 2009) and compact morphologies expected from dissipational mergers of gas-rich disks (e.g., Mihos & Hernquist 1996).

The left panel of Figure 8 shows the physical sizes (circularized radii, r_half) as a function of L_FIR for the SMA subsample and the SPT sample (Hezaveh et al. 2013). In this figure, objects with multiple components in the source plane are plotted individually, unlike in Figure 7. In doing this, we have assumed that the FIR luminosity in each component is proportional to its intrinsic flux density at 880 μみゅーm from the lens model (i.e., μみゅー_FIR ≈ μみゅー₈₈₀. This assumption is consistent with our modeling efforts because the only time we use multiple components in the source plane is when it is clear that only moderate lensing is occurring, in which case μみゅー_FIR ≈ μみゅー₈₈₀ is a valid approximation. Two comparison samples of unlensed LIRGs and ULIRGs from Rujopakarn et al. (2011) are shown: one at 0.5 < z < 2.5 and one at z ∼ 0. The sample of intermediate and high redshift LIRGs and ULIRGs relies on size measurements based primarily on radio observations (Chapman et al. 2004; Muxlow et al. 2005; Biggs & Ivison 2008; Casey et al. 2009), but also includes a handful of with mm size measurements (Tacconi et al. 2006, 2010; Daddi et al. 2010) and the assumption that the radio and far-IR sizes are correlated. Finally, Figure 8 also shows the median and 1σしぐま range in these values for a handful of unlensed SMGs with high-spatial resolution imaging by Tacconi et al. (blue rectangle; 2006).

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We find a wide range in sizes for the SMA subsample: the minimum, median, and maximum r_half values are 0.41 kpc, 1.53 kpc, and 4.16 kpc. In comparison, Rujopakarn et al. (2011) find values of 0.9 kpc, 2.6 kpc, and 8.0 kpc for the intermediate and high redshift sample. Part of this difference is a result of lensing, which lets us access spatial resolutions that are otherwise inaccessible (the radio observations used to measure sizes in the comparison sample have a typical angular resolution of 015, corresponding to a physical scale of ≈1 kpc at the redshifts of interest). Local LIRGs and ULIRGs have smaller sizes than their higher redshift counterparts (Rujopakarn et al. 2011), but the sizes of objects in the SMA subsample begin to overlap with those of the local LIRGs and ULIRGs. Very strong lensing (e.g., μみゅー ≳ 30) can reach ≈100 pc scale spatial resolution (Swinbank et al. 2010), but these occurrences must be rare because no such object is found in the SMA subsample.

The right panel of Figure 8 shows the FIR luminosity surface density (Σしぐま_FIR) as a function of L_FIR for the same set of objects as in the left panel. A wide range in Σしぐま_FIR is evident for the SMA subsample: the minimum, median, and maximum Σしぐま_FIR values are 0.05, 0.6, and 6.0 × 10¹² L_☉ kpc⁻², respectively. A horizontal line drawn at 1000 M_☉ yr⁻¹ kpc⁻² represents the theoretical limit for SFR surface density in a sustainable radiation-pressure supported disk (Thompson et al. 2005). A handful of objects in the SMA subsample and one object in the SPT sample lie near to or in excess of this limit, possibly indicating that they are in a very short-lived phase of evolution (e.g., the coalescence stage of a major merger). There are also, however, a number of objects with Σしぐま_FIR values more than an order of magnitude below the limit, suggesting that multiple modes of star-formation are viable in high redshift SMGs.

A spherically symmetric dust source radiating as a blackbody obeys the Stefan–Boltzmann law relating emitted flux density and the temperature of the dust. We use this fact to infer an alternative measure of the size of the lensed SMG, which we denote as r_SB:

$$\begin{equation} r_{\rm SB} = \sqrt{\frac{L_{\rm IR}}{4 \pi \sigma _{\rm SB} T_{\rm dust}^4}} \end{equation} \tag{ 6 }$$

where σしぐま_SB is the Stefan–Boltzmann constant. This quantity is similar in scope to the "effective radius" described in Greve et al. (2012) for lensed SMGs discovered by the SPT. Figure 9 shows the results of this analysis for the SMA and SPT samples. In this diagram, the error bars reflect the formal values obtained for our given set of model assumptions—e.g., the uncertainty introduced by the assumption of a single-temperature modified blackbody is not included. A dashed line traces the 1:1 relation between these two measures of the size (i.e., r_SB = r_half).

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A total of 15 out of 23 sources satisfy the unphysical criterion of r_SB > r_half. The best explanation for the large number of sources that violate the blackbody limit is that our T_dust values are underestimated by the assumption of optically thin emission. In fact, r_SB ≈ r_half suggests optically thick FIR emission. Adopting an optically thick model for the SED fitting would lead to larger dust temperatures by 25%–50%, an increase that is nearly sufficient for all of the sources in our sample to satisfy r_SB < r_half. The exact geometry of the source is unknown and is therefore an additional complicating factor. Nevertheless, this crude line of analysis is one indication that the FIR emission is optically thick in lensed SMGs discovered by Herschel, similar to local ULIRGs, which have r_half ≈ 2–3 × r_SB (Solomon et al. 1997). Only a handful of sources have large r_half values and low Σしぐま_FIR values typical of optically thin disks far from the Eddington limit (e.g., J091305.0−005343).