THE SCUBA-2 COSMOLOGY LEGACY SURVEY: ALMA RESOLVES THE REST-FRAME FAR-INFRARED EMISSION OF SUB-MILLIMETER GALAXIES

The data reduction and source extraction from our ALMA data is described in detail in J. M. Simpson et al. (in preparation). Here we give a brief description. The ALMA data were taken on 2013 November 1, as part of the Cycle-1 project 2012.1.00090.S. We observed all 30 sub-millimeter sources with ALMA, using 7.5 GHz of bandwidth centered at 344 GHz (870 μm; Band 7); the same frequency as the original SCUBA-2 observations. We used a "single continuum" correlator setup with four basebands of 128 dual-polarization channels each.

The phase centers for the ALMA pointings are the centroid position of the sub-millimeter source from the early SCUBA-2 map. The ALMA primary beam at this frequency is 173 FWHM, larger than the 148 FWHM of the SCUBA-2 beam and so is sufficient to recover all of the SMGs contributing to the single-dish sub-millimeter emission. The array configuration for our observations was such that the 26 ALMA 12 m antennae employed had a maximum baseline of 1250 m, and a median baseline of 200 m. This is in fact more extended than our requested compact C32–1 configuration and as a result yields a synthesized beam of 035 × 025 using Briggs weighting (robust parameter = 0.5). The maximum angular scale that our observations are sensitive to is 5'', which as we show in Section  3.1 is an order of magnitude larger than the FWHM of the sources in our sample.

The observations of our 30 targets were split into two 15 target blocks. Each block comprises seven or eight sub-blocks of 30 s observations of 10 targets, with the targets each observed five times. The targets were randomly assigned to different sub-blocks; i.e., one source might be observed in sub-blocks 1,3,5,6,7, and another in 1,2,3,4,6. Each block has a full set of calibration observations and each 5-min sub-block is separated by a 90 s phase calibration observation and a 30 s atmospheric calibration (taken on the phase calibrator). Hence, for each map we obtained a 150 s integration. Absolute flux calibration was derived from J 0238 + 166 and we used observations of the secondary phase calibrator J 0217 + 014, for phase referencing. The total integration time for the project was 2.6 hr.

The data were imaged using the Common Astronomy Software Application (casa version 4.2.1). As detailed in J. M. Simpson et al. (in preparation) the uv data were Fourier transformed to create a "dirty" image. We then applied the same procedure adopted by Hodge et al. (2013b): a tight clean box was placed around all >5σ emission in each map and these were cleaned down to a depth of 1.5σ. The final cleaned maps have a median angular resolution of 035 × 025 (P.A. ∼ 55 deg), using Briggs weighting (robust parameter = 0.5), and a median rms of σ₈₇₀ = 0.21 mJy beam⁻¹ (with a range from 0.19–0.24 mJy beam⁻¹). We inspect our maps and find that we do not detect any sources at sufficiently high signal-to-noise ratio (S/N) to reliably self-calibrate.

To construct the master catalog, and ensure that the extended flux from the SMGs is recovered, we repeated the imaging procedure described above, but using natural weighting, and applying a Gaussian taper to the data in the uv plane. Applying a Gaussian taper means that a lower weighting is given to visibilities at large distances in the uv plane, producing a map with a larger synthesized beam, hence with less emission resolved out, at the expense of higher noise. The resulting maps have a median angular resolution of 08 × 065, and median rms of σ₈₇₀ = 0.26 mJy beam⁻¹. Hence two sets of maps were created; "detection" maps with a synthesized beam of ∼08 FWHM and "high-resolution" maps at ∼03 FWHM.

To construct our catalog, we identify sources within the ALMA primary beam FWHM that are detected at >4σ in the 08 FWHM "detection" maps, and extract both the peak flux density and the total flux density in a 08 radius aperture for each SMG. We search for sources outside the ALMA primary beam FWHM, but do not find any statistically significant detections (see J. M. Simpson et al. in preparation). In total we identify 52 SMGs above 4σ in the 30 ALMA maps, at 08 FWHM resolution. These SMGs have a range of 870 μm flux density of 1–14 mJy and we recover the single–dish SCUBA-2 with a median ratio of S^SCUBA2/S^ALMA = 1.04 ± 0.05. As the analysis presented in this paper is focused on the size distribution of SMGs, we cut our SMG sample at a higher significance limit to provide sufficiently high S/Ns for the SMGs to reliably measure their sizes. As we show below, this corresponds to S/N ⩾ 10 (S₈₇₀ ≳ 4 mJy) at 08 resolution, which reduces our sample to 27 SMGs. In order to check the measured fluxes of these sources we also used the imfit routine in casa to model each SMG and find good agreement between the model and aperture derived flux densities (S^Model/S^Aperture = 1.02 ± 0.01). Where appropriate in the following analysis, we will also use the average properties of the remaining 25 SMGs with S₈₇₀ ≲ 4 mJy to test for trends with flux density across the whole sample.

Given the relatively bright ≳8 mJy flux limit used to select our target sample, we need to be aware of the potential influence of gravitational lensing (e.g., Blain 1996; Chapman et al. 2002).¹⁶ Indeed, we identify four examples of potential gravitationally lensed sources in our bright SMG sample: UDS 109.0, UDS 160.0, UDS 269.0, and UDS 286.0. All of these sources appear to be close to, but spatially offset from, galaxies at z ≲ 1 (Figure 1). If these sources are lensed then their apparent sizes need to be corrected for lens amplification. However, they all appear singly imaged (with no sign of multiple images at our sensitive limits) and as we have no precise estimates of their redshifts or the masses of the foreground lenses, it is impossible to reliably determine this correction. We therefore highlight these four SMGs in Figure 1, and note that the derived sizes for these sources are likely to be over-estimated. When discussing the properties of our sample we do not include the potentially lensed sources in the analysis, and so our final sample consists of 23 SMGs detected at S/N > 10σ.

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As we show in Section  3.1, most of the SMGs in our sample are resolved in the 03 resolution ALMA maps. However, we now perform two tests to ensure that an error in the calibration of the raw data does not drive our conclusion that the sources are resolved.

The observing strategy for our targets was such that each SMG was observed five separate times through the observing block, with phase calibration observations between each repeat observation. First, we test for variations in the source size as a function of time, as might be expected if the phase solution applied to the data does not correctly model fluctuations. To do so, we separately image each repeat observation of UDS 204.0, the brightest SMG in our sample (the integration time is sufficient to detect the source at 8σ–10σ in an individual scan). The SMG appears resolved in all five images, with an intrinsic source size in the range 052–063, and all sizes consistent within the associated 1σ uncertainties (∼005).

Second, we reclassify an observation of the phase calibrator as a science target observation in the observing sequence and repeat the calibration of the data set. If an error in the calibration of the raw data, due to phase variations, are causing the SMGs in our sample to appear resolved, then the phase calibrator will also appear resolved in this scan. We image this phase calibrator scan in the same manner as the SMG observations and model the emission using the imfit routine. In the individual scan the phase calibrator has an intrinsic size of FWHM = 006 ± 001, which is marginally higher that the intrinsic size measured in an image combining all scans (FWHM = 003 ± 001). However, by removing a phase calibrator scan we have introduced a 10 minute gap in our calibration observations, which is double that used throughout the observations. Hence any difference in the size of the phase calibrator should be classed as an upper limit on the uncertainty due to errors in the phase calibration of the data. Taken together these tests indicate that the SMGs in our sample do not appear resolved due to errors in the calibration of the raw data.

In our analysis in Section 3.1, we use the imfit routine in casa to fit an elliptical Gaussian model (convolved with the synthesized beam) for the 870 μm emission from each SMG. However, before applying this approach to the bright SMGs in our sample, we now test the reliability of the fits as a function of S/N using simple simulated galaxies. To do so, we create 20,000 elliptical Gaussian model sources with a uniform distribution of peak S/N from 4 to 30 (covering the range of SMGs in our full sample) and a major-axis size distribution that is uniform from 0'' to 1''. The model sources are distributed uniformly in S/N at 08 resolution, reflecting the selection function of our full SMG catalog. In order to use realistic noise maps we add these models to a randomly chosen position in one of the residual (source-subtracted) ALMA maps.

We use casa to fit an elliptical Gaussian to each injected model source and derive the best-fit size for each. We note that the imfit routine can return a point source solution if this is the best model. For each model parameter we calculate the fractional offset between the input model value and the recovered value. As expected we find that the precision of the recovered parameters is a function of S/N. To ensure that the sizes we derive for the SMGs are reliable we define a selection limit that the 1σ spread in ( FWHM^Model −FWHM^True)/FWHM^True < 0.3, i.e., 68% of the model sources are recovered with a fractional error less than 30%. We find that this requirement is met for sources that are detected at a S/N ⩾ 10, which is the justification for our use of this limit to define the bright SMG sample analyzed in this paper.

These simulations also indicate a small bias in the recovered size for sources with S/N between 10σ and 30σ of ( FWHM^Model −FWHM^True)/FWHM^True = 0.018 ± 0.002 (1σ spread of ±0.14), which we have not corrected for in the following analysis. The simulations also allow us to investigate the resolution limit of our maps, by determining the true size of a model source for which the imfit routine returns a point source best-fit solution. For SMGs with S/N = 10–30 we find that 90% of the model sources that are best fit by point source models usingimfit actually have a "true" size ⩽ 018 (i.e., smaller than half the size of the beam major axis). Hence, we can be confident that any SMGs that have a point source best-fit model have a size ⩽ 018, and we adopt this as an upper limit for the size of the unresolved SMGs.

During our analysis we use archival multi-wavelength information on this well-studied field. An extensive analysis of the multi-wavelength properties of the SMG sample will be presented in C.-J. Ma et al. (in preparation), but for the purposes of this paper we use two pieces of information from that work: constraints on the likely redshifts of the SMGs using photometric redshifts; and estimates of their far-infrared luminosities.

First, we use the photometric redshift catalog of this field produced by the UKIDSS UDS team, which is based on a K-band selected sample of sources. To briefly summarize, photometric redshifts are determined for each field source by fitting template spectral energy distributions (SEDs) to the observed UBVRi'z'JHK and IRAC 3.6 and 4.5 μm photometry, using the public SED fitting code eazy (Brammer et al. 2008). Excellent agreement is found between the photometric and spectroscopic redshifts for 2146 sources in the field, with a median dispersion of (z_phot − z_spec)/(1 + z_spec) = 0.03 (see Hartley et al. 2013; Mortlock et al. 2013).

Matching the SMG catalog to the photometric redshift sample finds 19 matches within 1'' of the 23 bright SMGs, which leaves four of the SMGs without redshift information. We also searched for IRAC counterparts for the four SMGs without photometric redshifts and find that three have detections in the IRAC imaging; however, the limited number of photometric bands available for all of these sources mean it is impossible to derive precise photometric redshifts. Simpson et al. (2014) argue that these NIR-faint SMGs typically lie at higher redshifts than the optical/NIR-brighter SMGs, and so we adopt their approach and assign these sources a redshift of z = 4 ± 1.

To derive the FIR luminosities of the SMGs we exploit the Herschel SPIRE imaging (Pilbratt et al. 2010; Griffin et al. 2010) of the field at 250, 350, and 500 μm, along with the precise position of the ALMA source to deblend and extract flux densities for the SMGs. Following the approach of Swinbank et al. (2014), they use a prior catalog of sources detected at 24 μm or 1.4 GHz, along with the ALMA source positions, to model the sources contributing to the map flux in the vicinity of each SMG. Having extracted flux densities or limits in the three SPIRE bands, along with the ALMA 870 μm flux density, they fit a library of model SED templates (see Swinbank et al. 2014), to the FIR-photometry and determine the best-fit template. We integrate the best-fit SED from 8 to 1000 μm to derive the total far-infrared luminosity of each SMG. If an SMG is undetected in the Herschel 250, 350, and 500 μm imaging, then the far-infrared luminosity is treated as an upper limit. We measure a median far-infrared luminosity for the 23 SMGs in our sample of L_FIR = (5.7 ± 0.7) × 10¹² L_☉, and a median dust temperature of T_d = 32 ± 3 K. We find that 5/11 and 10/14 SMGs detected at S/N = 5σ–10σ and S/N = 4σ–5σ, respectively, are undetected in the Herschel 250, 350, and 500 μm imaging. We stack the Herschel SPIRE maps for these subsets and derive an average L_FIR = $3.6^{+0.8}_{-0.9}$ × 10¹² L_☉ and L_FIR = $2.7^{+1.6}_{-0.6}$ × 10¹² L_☉ for the S/N = 5σ–10σ and S/N = 4σ–5σ subsets, respectively.

We show the ALMA maps with a 03 FWHM resolution synthesized beam for the 27 S/N ⩾ 10 SMGs in Figure 1. These are presented as contours overlaid on K-band grayscale images, to allow the reader to make a qualitative comparison of the size and shape of the contour corresponding to the half-peak-flux level and the corresponding size for the synthesized beam. We see evidence in a majority of the SMGs that the sub-millimeter emission appears to be more extended than would be expected for an unresolved source with the same peak flux, with several examples also showing structure on smaller scales (e.g., UDS 74.0, UDS 216.0, UDS 218.0, UDS 361.0, UDS 408.0).

To quantitatively test if the SMGs are resolved in our data we initially perform two non-parametric tests. First we compare the peak flux of each source in the 08 FWHM observations to the higher resolution 03 FWHM imaging (note that the maps are calibrated in Jy beam⁻¹). The peak flux of the SMGs is lower in the 03 resolution maps, with a median ratio of $S^{0.3}_{\rm pk}$/$S^{0.8}_{\rm pk} =$ 0.65 ± 0.02, and a 1σ range of ± 0.04. A drop in the peak flux density between the sources imaged at different resolutions indicates that flux from each SMG is indeed more resolved in the higher-resolution imaging. We note that a 40% reduction in the peak flux between the 03 and 08 resolution imaging corresponds to an intrinsic source size of ∼03.

As a second test we compare the total flux density in an aperture for each source to the peak flux density at both 03 and 08 FWHM resolution. We convert each map into units of Jy pixel⁻¹ and measure the "total" flux density in a 08 radius aperture. The ratio of the peak-to-total flux density is 0.50 ± 0.03 at 03 FWHM resolution, compared to 0.83 ± 0.03 at 08 FWHM. This supports the conclusion that the bulk of the bright SMGs are resolved by ALMA at 870 μm with a 03 FWHM synthesized beam, and suggests that the average angular size of the population is 03–04. We note that the aperture fluxes of the SMGs measured in the 03 and 08 imaging are in good agreement, with a median ratio of $S^{0.3}_{\rm Aper}$/$S^{0.8}_{\rm Aper} =$ 1.02 ± 0.03.

We next investigate whether the SMGs appear resolved in the uv plane, rather than in the image plane, compared to the calibrator sources used for our observations (which are expected to be unresolved). For each source we align the phase center of the map to the position of the SMG or calibrator source, and extract the amplitudes for each source as a function of uv distance. The amplitudes represent the observed flux of the source on different angular scales in the image plane, with the longest baseline providing information on the smallest angular scales that our observations are sensitive to. For an ideal point source the amplitudes should be constant with uv distance. As Figure 2 demonstrates we do indeed recover an effectively flat distribution for both the phase and bandpass calibrators.

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In Figure 2, the amplitudes as a function of uv distance for the eight SMGs, which are the sole sources detected in their respective ALMA maps (restricting the sample in this way removes any complications due to modeling and subtracting other sources in the primary beam). The amplitudes of seven of the SMGs clearly decrease with increasing uv distance. A single SMG, UDS 392.0, appears to be only marginally resolved on the longest baselines (our analysis in the image plane identifies this SMG as unresolved, see Figures 2 and 3). We fit the amplitudes for each SMG with a single Gaussian component and determine the equivalent size in the uv plane. The model fits in the uv and image planes are consistent, and derive a median size ratio between the image- and uv-derived sizes of FWHM^uv/FWHM^image = 0.9 ± 0.2 and a median flux density ratio S^image/S^uv = 1.1 ± 0.1. In 2–3 of the SMGs the flux density does not fall to zero on baselines ≳600 kλ, indicating that there is unresolved flux in these sources on an angular scale of ≲ 03, but this unresolved component comprises ≲10% of the total flux density.

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Given that our non-parametric tests indicate that the SMGs are resolved in our 03 imaging, we now chose to fit a more complex model to the sources. Using the imfit routine we determine the best-fit elliptical Gaussian model for the 23 SMGs in our bright sample (S/N > 10σ). The free parameters of the model are position; peak flux density; major and minor axis; and position angle. The median intrinsic size of the 23 SMGs is 030 ± 004 (deconvolved FWHM of the major axis), and one SMG is modeled as a point source; as the intrinsic sizes of the sources are comparable to the beam we choose to focus our analysis on the FWHM of the major axis. The median χ² for the elliptical Gaussian fit is 38.3 ± 1.4, and the number of degrees of freedom of the fit is 33 (we define beam to be the size of the resolution element). If we fit each SMG with a point source model then the median difference in χ² between the best-fit extended and point-source model is Δχ² = 20 ± 2 (see Figure 3).¹⁷

The top panel of Figure 4 shows the sizes of the SMGs as a function of their 870 μm flux density. The dispersion of the 870 μm sizes is small and there is no clear trend in the size of the dust emission region with 870 μm flux density. We note however that the two SMG with S₈₇₀ ∼ 12 mJy in the sample are around two times larger than the median of the sample. While this may be interpreted as a weak trend in the size of SMGs with 870 μm flux density we caution against a strong conclusion, given the limited sample size at these fluxes. We also show the sizes of two bright SMGs (S₈₇₀ = 13 mJy and 18 mJy) presented by Younger et al. (2010). These SMGs were observed with the SMA and both appear marginally resolved in the uv plane, with sizes of 06 ± 02 (deconvolved FWHM). While the size measurements for these sources have large associated uncertainties, they are consistent with the sizes presented here.

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To search for trends in our sample over a larger dynamical range we have also used the fainter SMGs detected in our ALMA maps. While the individual sizes measured for these galaxies have significant uncertainties, we can attempt to derive a typical size for samples of faint SMGs via stacking (this is reasonable as the synthesized beam does not vary significantly between observations; the beam major and minor axes vary by at most 003 and 002, respectively, across the full sample). We split the fainter 25 SMGs into roughly equal sub-samples detected at S/N = 5σ–10σ (median S₈₇₀ = 3.1 ± 0.5 mJy) and S/N = 4σ–5σ (median S₈₇₀ = 2.6 ± 0.3 mJy), containing 11 and 14 SMGs, respectively. We stack the 03 FWHM maps of these sub-samples and use imfit to measure the size of the stacked profile. We derive sizes of 030 ± 007 and 0.35$_{-0.10}^{+0.17}$'' for the two sub-samples, respectively. The errors on the measured size of each subset are derived by bootstrapping the maps included in the stacks and repeating the analysis. These results indicate that at least on average the 1–4 mJy SMGs in our full sample have the same size distribution as those with observed 870 μm flux densities of 4–15 mJy sources (see Figure 4).

The 23 SMGs detected at >10σ in our sample have a median angular FWHM of 030 ± 004. For each SMG we use the photometric redshift to convert the measured angular diameter into an intrinsic physical scale, and derive that the median physical size of the SMGs is 2.4 ± 0.2 kpc (FWHM of the major axis). As shown in Figure 4, the distribution has a narrow interquartile range of 1.8–3.2 kpc, indicating that the dispersion in the sizes of our sample of SMGs is small. We use the results of our stacking analysis to derive an average physical size of 2.4 ± 0.6 kpc and $2.5_{-0.7}^{+1.2}$ kpc, at a median redshift of z = 2.6 ± 0.4 and z = 4 ± 1, for the S/N = 5σ–10σ and S/N = 4σ–5σ subsets, respectively, indicating no trend in the size of SMGs with redshift. An important caveat of this result is that only one SMG in our sample has a photometric redshift of z ≳ 3.5. Four SMGs do not have a photometric redshift and it is likely that these sources lie at z ∼ 4 (see Simpson et al. 2014); however, the uncertainties on these redshifts are large. At z = 4 these SMGs have a distribution of physical FWHM from <1.3–2.1 kpc, and are consistent with the lower redshift SMGs in the sample. However, if they instead lie at z = 6, then the range of FWHM is <1.1–1.7 kpc, and these sources would be smaller than all but one z < 4 SMG.

Recently Weiß et al. (2013) discussed that evolution in the size of SMGs with redshift is a possible explanation for the discrepancy between the redshift distribution of lensed and unlensed SMGs. As 60%  of the SMGs presented by Weiß et al. (2013) lie at z > 3.5 we cannot investigate this claim further. However, we note that for the unlensed SMGs presented here, we do not find any evolution in the sizes of the 870 μm emitting region at z < 3.5.

4.1.1. Comparison of the Multi-wavelength Sizes of SMGs

We now compare the 870 μm sizes presented here to the sizes of SMGs measured from observations at different wavelengths. First, we compare our results to a sample of 12 SMGs (median S₈₅₀ = 6.8 ± 0.6 mJy) with deconvolved sizes measured from 05 resolution 1.4 GHz/VLA imaging (Biggs & Ivison 2008). In their analysis Biggs & Ivison (2008) fit an elliptical Gaussian model to the 1.4 GHz emission from each SMG, deriving a median angular size of FWHM = 064 ± 014. At the redshifts of these SMGs this angular size corresponds to a median physical size of FWHM = 5.3 ± 1.1 kpc. Four SMGs in their sample do not have a redshift and we set these to the median redshift of the sample (z = 2; consistent with the radio-detected SMGs in Simpson et al. 2014). Hence the 1.4 GHz sizes for the SMGs presented by Biggs & Ivison (2008) are about two times larger than the 870 μm sizes presented here.

It is important to initially note that the radio counterparts to the sub-mm emission presented in Biggs & Ivison (2008) were identified via a probabilistic approach using the far-infrared radio correlation. As shown by Hodge et al. (2013b) these identifications are 80%  reliable, and so we might expect that two or three of the radio sources presented in Biggs & Ivison (2008) are not the true counterpart to the sub-mm emission. However, we do not know if such mis-identifications are likely to be biased towards galaxies with larger, or smaller, sizes at 1.4 GHz.

Another possible explanation for the discrepancy between the radio and far-infrared sizes of SMGs is that the diffusion length of cosmic rays (∼1–2 kpc) is expected to be larger than the diffusion length of far-infrared photons (∼100 pc), due to magnetic fields efficiently transporting cosmic rays through the host galaxy. Murphy et al. (2008) present a study of 18 local star-forming galaxies with resolved imaging at 70 μm and 1.4 GHz. They show that the radio emission from these galaxies can be modeled as the convolution of the far-infrared emission with an exponentially declining kernel, with an e-folding length of the kernel of 1–2 kpc (see also Bicay & Helou 1990; Murphy et al. 2006). We convolve the median 870 μm size of the SMGs with this exponential kernel and find that the FWHM of the convolved profile is 3.8–5.2 kpc, showing that the radio sizes of Biggs & Ivison (2008) are likely consistent with our 870 μm sizes.

The galaxies presented by Murphy et al. (2008) are "normal" star-forming galaxies, with SFRs three orders of magnitude lower than the SMGs presented here. The e-folding length of the kernel has been shown to depend on both SFR density and morphology, with high SFR densities and irregular galaxies having lower e-folding lengths (Murphy et al. 2008). In both of these situations ordered magnetic fields are disrupted, allowing cosmic ray electrons to stream more freely out of the galaxy, resulting in a shorter scale length. As such, we caution that a scale length of 1–2 kpc might be an over-estimate of the appropriate scale length for the radio emission from SMGs.

Next we compare the 870 μm sizes of SMGs to the sizes of SMGs measured from ¹²CO emission lines. Engel et al. (2010) present a compilation (including new observations) of the sizes of SMGs measured from resolved ¹² CO (J = 3–2/4–3/6–5/7–6) emission lines (see also Tacconi et al. 2006, 2008). Deriving a median size for the sample is challenging, as the upper limits on the size of unresolved SMGs are typically similar to the sizes of the resolved sources. Treating the unresolved sources as point sources, the median size of the sample is FWHM = 2.3 ± 1.1 kpc, but this rises to 3.7 ± 0.6 kpc if upper limits are taken as the "true" source size. In our analysis we have not included the sizes taken from Bothwell et al. (2010), as it is not clear that these sizes have been deconvolved for the beam.

In the study by Engel et al. (2010) the ¹²CO-emission lines are often either unresolved, or marginally resolved, with sizes constrained at only 1σ–2σ. The sizes are also measured from ¹²CO emission lines across a range of rotational transitions, from J = 2–1 to 7–6, and we note that there is ongoing debate over whether emission from different transitions traces the same molecular gas (Weiß et al. 2007; Panuzzo et al. 2010; Bothwell et al. 2013; Narayanan & Krumholz 2014). Due to these issues we simply note that the sizes derived from moderate-J transitions of ¹²CO are broadly consistent with the median 870 μm size of the SMGs in our sample.

Finally, we compare our results to the sizes of SMGs derived from high-resolution HST NIR imaging. Chen et al. (2014) recently presented H-band/HST imaging (rest-frame optical imaging) of 48 SMGs. These SMGs were drawn from the ALESS sample (Hodge et al. 2013b), and have precise ≲ 03 identifications from ALMA 870 μm observations. Chen et al. (2014) fit a Sérsic profile to the H-band emission, deriving a median half-light radius of R_e = 4.4$^{+1.1}_{-0.5}$ kpc (see also Targett et al. 2013; Wiklind et al. 2014). As recommended by Chen et al. (2014), we have restricted their sample to the 17 SMGs with an apparent H-band magnitude <24 mag and a photometric redshift between z_phot = 1–3, which ensures that the size distribution is complete (median S₈₅₀ = 4.0 ± 0.6 mJy). The Sérsic profiles fitted by Chen et al. (2014) are more complicated than the Gaussian profiles used in our analysis, but we note that a Gaussian profile corresponds to a Sérsic profile with Sérsic index n = 0.5. If we convert our 870 μm sizes to a half-light radius we find that the SMGs have a median R_e = 1.2 ± 0.1 kpc in the sub-millimeter. As we show in Section  4.3.1 the half-light radius of the best-fit Sérsic profile to the stacked 870 μm emission from the SMGs is consistent with the median size from the elliptical Gaussian fit. The optical sizes of the SMGs are about four times larger than the 870 μm dust emitting region, and as can be seen in Figure 5 the distribution of NIR sizes has considerably more dispersion than the 870 μm sizes. The compact nature of the star formation means that it is plausible we are observing bulge growth in the form of obscured star formation. However, to test this hypothesis requires high-resolution dust and optical imaging of the same sample of SMGs to pinpoint the location of the star formation within the host galaxy.

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4.1.2. Comparison to Local U/LIRGs

The high SFRs of SMGs means that they could host regions with an extreme star formation density, which could result in very different star formation conditions to other galaxy populations. In an isolated star-forming region the radiation pressure from massive stars may provide sufficient feedback to regulate the further collapse of the giant molecular cloud. Andrews & Thompson (2011) show via the balance of radiation pressure from star formation with self-gravitation, that the maximum SFR surface density, assuming optically thick dust emission (τ > 1),¹⁸ and in the absence of nuclear heating via an active galactic nucleus, is

$$\begin{equation} {\rm SFR_{\rm max}} \gtrsim 11 f_{\rm gas}^{-0.5} f_{\rm dg}^{-1}\,\mathrel {M_{\odot }}{\rm yr}^{-1}\ {\rm kpc}^{-2}\,, \end{equation} \tag{ 2 }$$

where f_gas is the gas fraction in the star-forming region, and f_dg is the dust-to-gas ratio (see also Murray et al. 2005; Thompson et al. 2005; Murray et al. 2010). We adopt a dust-to-gas ratio of f_dg = 1/90 (Magnelli et al. 2012; Swinbank et al. 2014), but note that f_dg = 1/50 has also been suggested for star-forming galaxies (Kovács et al. 2006). Estimates of the dynamical masses (Swinbank et al. 2006; Alaghband-Zadeh et al. 2012), stellar masses (Hainline et al. 2011; Simpson et al. 2014) and gas masses (Bothwell et al. 2013) indicate that SMGs have typical gas fractions of ∼40%; however, this is a global property of the galaxy. As the size of the rest-frame optical emitting region in SMGs is around four times larger than the size of the star–forming region (Figure 5), it is likely that the gas fraction is considerably higher in the star-forming region. In our analysis we adopt a gas fraction of unity, i.e., gas is the dominant component, as this sets a lower limit on the maximum SFR surface density. If we adopt a gas fraction of 40%  then the maximum SFR surface density would rise by 60%. Adopting these values the maximum SFR surface density predicted by Andrews & Thompson (2011) is ∼1000 M_☉ yr⁻¹ kpc⁻².

The 23 SMGs in our sample have a median far-infrared luminosity of (5.7 ± 0.7) × 10¹² L_☉, which corresponds to a median SFR = 990 ± 120 M_☉ yr⁻¹ (for a Salpeter initial mass function (IMF)). To derive the SFR surface density for our sample we initially divide the SFR of each SMG by a factor of two (the measured size of each SMG corresponds to the half-light radius of the profile). We show our sample in Figure 6, and assuming a uniform disk profile for the gas and cool dust distribution the median SFR surface density of the SMGs is 90 ± 30 M_☉ yr⁻¹ kpc⁻². Only two SMGs have a SFR surface density above 500 M_☉ yr⁻¹ kpc⁻², and no SMGs exceed the maximum value predicted by Andrews & Thompson (2011).

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The SMGs in our sample are forming stars at surface rate densities an order of magnitude lower than the estimated Eddington limit. However, we stress that these surface rate densities are integrated across the star-forming region (see Meurer et al. 1997). In the well-studied lensed SMG SMM J2135-0102 (Swinbank et al. 2010b) it has been shown that the star formation is in four distinct "clumps" (FWHM = 100–300 pc), located within a gas disk with a half light radius ∼1 kpc (Swinbank et al. 2011; Danielson et al. 2011, 2013). If the star formation in the SMGs in our sample is "clumpy," then individual regions in the gas disk may be Eddington limited, but the integrated SFR surface density will appear sub-Eddington.

Meurer et al. (1997) show that a sample of ultraviolet-selected sources (at z = 0–3) have a 90th percentile global luminosity surface density of 2 × 10¹¹ L_☉ kpc⁻² (with an associated uncertainty of a factor of three), which they interpret as an upper limit on the global luminosity surface density of a starburst. The SMGs presented here have a median global luminosity surface density of ( 5.2 ± 1.7) × 10¹¹ L_☉ kpc⁻², a factor of three times higher than the value found by Meurer et al. (1997: see Figure 6). A possible explanation for this discrepancy is that Meurer et al. (1997) measure half-light radii for the galaxies in their sample from ultraviolet imaging, and assume that dust attenuation is uniform across the galaxy. As discussed in Section 4.1.1 we find that the rest-frame optical half-light radii of SMG are a factor of four larger than the 870 μm dust half-light radius. The discrepancy in the optical sizes and dust sizes of SMGs indicates that their is significant differential reddening in these sources. If the ultraviolet-selected sources in (Meurer et al. 1997) suffer from similar differential reddening, then the half-light radii measured in the ultraviolet are likely to overestimate the true extent of the star formation.

It has been suggested that an evolutionary sequence exists whereby SMGs transition into local elliptical galaxies via a quiescent galaxy phase at redshift z ∼ 2 (e.g., Lilly et al. 1999; Genzel et al. 2003; Blain et al. 2004; Swinbank et al. 2006; Tacconi et al. 2008; Swinbank et al. 2010a; Hickox et al. 2012; Toft et al. 2014; Simpson et al. 2014; Chen et al. 2014). This evolutionary scenario has been investigated by comparing properties, such as the stellar masses, spatial clustering, and space densities of SMGs, to the properties of the proposed descendants. However, each of these methods has significant uncertainties: the stellar masses of SMGs have been shown to be highly dependent on the assumed star formation history; the number of SMGs with measured dynamical masses is small and the samples are inhomogeneous; and the spatial clustering strength of single-dish identified sub-mm sources is biased by source blending.

Here we use sizes to test this evolutionary link, since a distinctive feature of the population of quiescent galaxies at redshift z = 1–3 is that they are extremely compact, with half-light sizes of ∼1 kpc ¹⁹ (e.g., Daddi et al. 2005; Zirm et al. 2007; Toft et al. 2007; Buitrago et al. 2008; van Dokkum et al. 2008; Newman et al. 2012; Patel et al. 2013; Krogager et al. 2013). A morphological analysis of SMGs thus provides an alternative route to investigating the proposed evolutionary sequence.

We construct a sample of spectroscopically confirmed quiescent galaxies at z ∼ 2, to compare to SMGs, by combining the samples presented by van Dokkum (2008) and Krogager et al. (2013). The galaxies in these samples have spectroscopically confirmed 4000 Å/Balmer breaks, indicative of an old (≳0.5–1 Gyr) stellar population, assuming no age–dependent dust reddening. Both studies use the code Galfit to fit a Sérsic profile to the H-band emission from each galaxy. The combined sample of 24 quiescent galaxies has a median half-light radius of R_e = 1.5 ± 0.2 kpc and a median Sérsic index of n = 3.3 ± 0.7. The stellar masses of these galaxies are estimated to be ≳ 10¹¹ M_☉ and the median redshift of the sample is z = 2.2 ± 0.1. The age of the stellar population in these quiescent galaxies (0.5–1 Gyr) means that their progenitors likely lie at z = 2.5–3 consistent with the redshift distribution of SMGs (Toft et al. 2014; Simpson et al. 2014). We note that the half-light radii of the spectroscopically confirmed quiescent galaxies in our comparison sample are consistent with the sizes measured for a large sample of color-selected quiescent galaxies at a similar redshift (z = 1.5–2.5; Patel et al. 2013).

As discussed in Section 4.1.1, Chen et al. (2014) show that the unobscured rest-frame optical light in SMGs has a median half-light size of R_e = 4.4$^{+1.1}_{-0.5}$ kpc and Sérsic index of n = 1.2 ± 0.3. In contrast, the quiescent galaxies in our comparison sample have a median half-light radius of R_e = 1.5 ± 0.2 kpc, significantly smaller than the rest-frame optical half-light radius of SMGs. The median Sérsic index of the SMGs is also significantly lower than the quiescent galaxies, indicating that a significant transformation of the light profile (and potentially stellar mass distribution) of SMGs is required if they are indeed the progenitors of the quiescent galaxies in our comparison sample. A number of studies have suggested that the star formation in SMGs is triggered by merger activity: in H-band/HST imaging (82 ± 9)% of SMGs have signatures of interactions and/or irregular morphologies (Chen et al. 2014), and the low Sérsic index n measured for the population is consistent with merging systems (Kartaltepe et al. 2012); dynamical studies of resolved Hα emission from SMGs find strong evidence of kinematically distinct multiple components (Swinbank et al. 2006; Alaghband-Zadeh et al. 2012); and a study of molecular line emission from SMGs finds that 75% of sources are ongoing mergers, with the remaining 25% classed as either disk or late stage mergers (Engel et al. 2010).

It is well established that mergers can induce torques that are effective at removing angular momentum from the gas in the system, allowing the gas to fall into the inner regions and potentially resulting in a kiloparsec-scale nuclear starburst (e.g., Barnes & Hernquist 1991; Mihos & Hernquist 1994). Recently, Hopkins et al. (2013) presented numerical simulations of two gas rich disk galaxies undergoing a major merger at high redshift. Hopkins et al. (2013) show that the major merger triggers an intense central starburst, with 50%  of the star formation activity concentrated within a diameter of ∼2 kpc. The simulations also demonstrate that the starburst increases the stellar surface density within the central 2 kpc diameter by two orders of magnitude. The size of the star-forming region in the SMGs presented in this work are consistent with simulations of major mergers. If the star formation in SMGs is indeed a centrally concentrated starburst then we expect a significant rise in the concentration of the stellar component, increasing the fraction of light contained in a de Vaucouleurs-like (n = 4) profile.

4.3.1. Sérsic Index

We now investigate whether there is evidence that the star formation in SMGs is more highly concentrated than the stellar component. First, we test whether our data have sufficiently high S/N to recover a more complex Sérsic profile.²⁰ We create 4000 model sources with a half-light size fixed at the median size of the SMGs, a range of Sérsic index from n = 0.5–4 and an S/N distribution that is uniform from 10σ–30σ. To simulate realistic noise we add these models to a randomly chosen position in one of the residual (source-subtracted) ALMA maps.

We fit a Sérsic profile to each model source using the code galfit (Peng et al. 2010), and find that the half-light size of the model sources is recovered accurately at all values of S/N and Sérsic index (${R_{\rm e}^{\rm rec}}$/${R_{\rm e}^{\rm input} =}$ 0.99 ± 0.01; 1σ dispersion ±0.2). In contrast, although the Sérsic index is recovered correctly on average (n^rec/n^input = 0.97 ± 0.02) the distribution has a 1σ dispersion of 0.4–2.2, indicating that measurements for individual sources are highly uncertain.²¹ Crucially, we find that the dispersion in the recovered Sérsic index is dependent on the S/N of the injected source; for S/N >20σ the 1σ dispersion is 0.6–1.8. As such we do not attempt to fit a Sérsic profile on a source-by-source basis, but instead stack the 870 μm emission from the 23 SMGs detected at >10σ and use galfit to fit a Sérsic model to the stacked profile. The best-fit model to the stacked profile has a Sérsic index of n = 2.5 ± 0.4 and a half-light radius of R_e = 017 ± 003, where the errors are derived by bootstrapping the maps used in the stacking.

The median Sérsic index of the 870 μm emission from the SMGs is higher than the Sérsic index of the unobscured rest-frame optical light in SMGs (Chen et al. 2014, but see also Targett et al. 2013), indicating that the ongoing star formation is more centrally concentrated. As we show in Figure 7, the half-light radius of the star formation in SMGs, median R_e = 1.2 ± 0.1 kpc, is comparable to the half-light radius of the quiescent galaxies in our comparison sample, R_e = 1.5 ± 0.2 kpc. If we combine the average Sérsic profile of the star formation and the pre-existing stellar population, under the simple assumption that the ongoing star-formation in SMGs doubles the luminosity of the stellar component, then the half-light radius of the resulting galaxy is ∼2 kpc. This estimated half-light radius of the post–starburst SMGs is still marginally larger than median half-light radius of z ∼ 2 quiescent galaxies. However, as shown by Chen et al. (2014) (82 ± 9)% of SMGs have signs of either merger activity or disturbed morphologies, and it is likely that the size measured for the ongoing merger is larger than the size of the galaxy at post-coalescence. In addition, the contribution of the stars formed in the SMG phase to the luminosity of the post-starburst galaxy is highly dependent on the mass of stars formed and the relative ages of the stars formed in the starburst and the pre-existing stellar population. The half-light radius of the ongoing, extreme, star formation in SMGs is comparable to the half-light radius of high-redshift compact quiescent galaxies, indicating that the ongoing star formation may explain the transformation required in the proposed evolutionary scenario from an intense starburst phase (SMG) to a local elliptical galaxies, via a compact quiescent galaxy at z ∼ 2.

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