An ALMA Spectroscopic Survey of the Brightest Submillimeter Galaxies in the SCUBA-2-COSMOS field (AS2COSPEC): Survey Description and First Results

We introduce an ALMA band 3 spectroscopic survey, targeting the brightest submillimeter galaxies (SMGs) in the COSMOS field. Here we present the first results based on the 18 primary SMGs that have 870 $\mu$m flux densities of $S_{870}=12.4-19.3$ mJy and are drawn from a parent sample of 260 ALMA-detected SMGs from the AS2COSMOS survey. We detect emission lines in 17 and determine their redshifts to be in the range of $z=2-5$ with a median of ${3.3\pm0.3}$. We confirm that SMGs with brighter $S_{870}$ are located at higher redshifts. The data additionally cover five fainter companion SMGs, and we obtain line detection in one. Together with previous studies, our results indicate that for SMGs that satisfy our selection, their brightest companion SMGs are physically associated with their corresponding primary SMGs in $\ge40$% of the time, suggesting that mergers play a role in the triggering of star formation. By modeling the foreground gravitational fields, $<10$% of the primary SMGs can be strongly lensed with a magnification $\mu>2$. We determine that about 90\% of the primary SMGs have lines that are better described by double Gaussian profiles, and the median separation of the two Gaussian peaks is 430$\pm$40 km s$^{-1}$. This allows estimates of an average baryon mass, which together with the line dispersion measurements puts our primary SMGs on the similar mass-$\sigma$ correlation found on local early-type galaxies. Finally, the number density of our $z>4$ primary SMGs is found to be $1^{+0.9}_{-0.6}\times10^6$ cMpc$^{-3}$, suggesting that they can be the progenitors of $z\sim3-4$ massive quiescent galaxies.


INTRODUCTION
Galaxies that are bright in far-infrared and submillimeter host sites of both vigorous star formation and active black hole growth (Sanders et al. 1988).Predominantly at z = 1 − 4 (e.g., Chapman et al. 2005;Ivison et al. 2007;Wardlow et al. 2011;Casey et al. 2014;Hodge & da Cunha 2020), recent studies focusing on submillimeter galaxies that are bright at 850 µm (S 850 1 mJy), or SMGs, found that they are some of the most massive systems during these epochs, sitting at the massive end of the stellar mass functions (Dye et al. 2008;Hainline et al. 2011;Micha lowski et al. 2012;Koprowski et al. 2016;Dudzevičiūtė et al. 2020), molecular gas mass functions (Bothwell et al. 2013;Birkin et al. 2021), dust mass functions (da Cunha et al. 2015(da Cunha et al. , 2021)), and also the halo mass functions (Hickox et al. 2012;Chen et al. 2016;Wilkinson et al. 2017;An et al. 2019;Stach et al. 2021).The large cold molecular gas reservoir (∼ 10 11 M ) estimated from their molecular line measurements is believed to provide the gas supply over ∼100 Myr timescales for their extensive starformation rates (SFRs) of ∼100-1000 M yr −1 , similar to the levels of star formation seen in nearby (hyper-)ultra-luminous infrared galaxies (HyLIRGs/ULIRGs; Sanders & Mirabel 1996).Supported by observational evidence including large-scale clustering, the current hypothesis is that SMGs are intimately linked to luminous quasars at similar redshifts, and likely followed by phases of compact quiescent galaxies and the local massive ellipticals (Lilly et al. 1999;Swinbank et al. 2006;Alexander & Hickox 2012;Toft et al. 2014;Dudzevičiūtė et al. 2020).The fact that the massive ellipticals dominate the stellar mass budget among the local massive galaxies ( M * ; Ilbert et al. 2013) suggests that the phases of SMGs could be responsible for the formation of most of their stellar masses (Thomas et al. 2010).
Their essential role in formation and evolution of massive galaxies highlights how incredibly little we understand SMGs even after over two decades since their first discovery (Smail et al. 1997;Barger et al. 1998;Hughes et al. 1998).Technical difficulties compounded with their extreme infrared luminosity have kept many issues unsettled regarding the basic physical properties of SMGs, both in theory and observations.The one issue, and arguably the only one, that has recently been considered more or less clear, is the SMG number densities, or the number counts.Observational results have now converged to within statistical uncertainties over a wide flux range after taking into account field-to-field variations (Karim et al. 2013;Simpson et al. 2015a;Oteo et al. 2016;Geach et al. 2017;Hill et al. 2018;Stach et al. 2018;Simpson et al. 2020).However this comes only after years of efforts in examining the impact of multiplicity 1 on counts deduced from single-dish measurements, which typically served as the basis of constructing the number counts (Wang et al. 2011;Barger et al. 2012;Chen et al. 2013;Hodge et al. 2013;Cowie et al. 2018).Models have also made significant progress, where measured number counts can now be reproduced, in many cases without having to invoke top-heavy stellar initial mass functions (Lacey et al. 2016;Safarzadeh et al. 2017;Béthermin et al. 2017;Casey et al. 2018;Lagos et al. 2020;Lovell et al. 2021).Although note that top-heavy stellar initial mass functions have been suggested in recent observational studies (Zhang et al. 2018;Dye et al. 2022).However, a variety of different techniques used to construct models has led to vastly different predictions on some of the most fundamental properties.
One outstanding example is redshift; While most models can reproduce the average redshift distribution (median of z ∼ 2.5; Chapman et al. 2005) of typical SMGs with 850 µm fluxes of S 850 ∼ 2 − 9 mJy, they predict opposite trends and correlations toward the brighter end; A school of models predict a median redshift of z < 2 at S 850 9 mJy (Lacey et al. 2016;Lagos et al. 2020) and another a median z > 3 (Figure 1; Béthermin et al. 2017;Casey et al. 2018;Lovell et al. 2021).In addition, some models predict more or less a linear correlation between median redshift and S 850 but others have no such correlation.The differences in these predictions are likely due to different treatments of physical processes such as the triggering of star formation and dust formation, or the adoption of phenomenological recipes driven by the observations themselves.Precise redshift measurements of bright SMG samples that are sufficiently large and complete would be powerful in testing these predictions, and therefore represent the next frontier in understanding the formation of massive galaxies across cosmic time.
In the pre-ALMA era, redshift measurements are focused on SMGs that in many cases have radio or mid-  (Cai et al. 2013;Negrello et al. 2017), Casey-Zavala (Casey et al. 2018;Zavala et al. 2021), SIDES (Béthermin et al. 2017), the semi-analytical model SHARK (Lagos et al. 2020), and the hydro-dynamical simulation SIMBA (Davé et al. 2019) coupled with dust radiative transfer modeling POWDERDAY (Narayanan et al. 2021;Lovell et al. 2021).In the bottom of the figure we show the histograms normalized of three SMG samples that are constructed via a similar approach, meaning ALMA band 7 continuum follow-up observations on flux limited samples of single-dish detected submillimeter sources.The histograms are arbitrarily normalized to show contrasts of different samples.AS2COSMOS (Simpson et al. 2020) has significantly more brighter SMGs than AS2UDS (Stach et al. 2018) and ALESS (Hodge et al. 2013).The flux range presented in this work is indicated, which is ideal for redshift measurements thanks to its brightness and test model predictions.
infrared detection, which was the technique used to identify their optical and near-infrared (OIR) counterparts after being discovered from single-dish submillimeter surveys (Ivison et al. 1998;Barger et al. 1999;Smail et al. 2000;Chapman et al. 2003;Chapman et al. 2005;Ivison et al. 2007;Wardlow et al. 2011;Umehata et al. 2014).Their redshifts were mostly estimated via simple flux ratios and in some cases OIR spectroscopic observations.They typically found a wide range of median redshifts between z = 2 − 3, with uncertainties on the order of 10-20%.In the ALMA era, where precise locations can be efficiently measured and counterpart identification is less biased, studies using well-defined samples have found more stable results of an overall median of z ∼ 2.5 − 2.6 ± 0.1, with a moderate increase in median as a function of flux density (Simpson et al. 2014;Cowie et al. 2017;Stach et al. 2019).However majority of these redshift are estimated via fittings of spectral energy distributions (SEDs), thus any possible systematic offsets due to model assumptions need to be understood with spectroscopic redshift measurements.Spectroscopic redshift measurements of SMGs are known to be difficult; With about two hours exposures from 8-10 m class OIR telescopes, the line detection rates were found to be about ∼30% (Casey et al. 2017;Cowie et al. 2017;Danielson et al. 2017).(Sub-)Millimeter spectral-scan observations targeting bright CO and [CI] lines appear more feasible for SMGs that are dusty but ISM rich.Indeed, with moderate time investment, this strategy has been successfully demonstrated on samples of very bright and typically strongly lensed sources discovered by Herschel or the South Pole Telescope (Vieira et al. 2013;Strandet et al. 2016;Neri et al. 2020;Bakx et al. 2020;Reuter et al. 2020;Urquhart et al. 2022).However, a generally lack of these very bright (S 850 25 mJy) SMGs in models motivates similar observations to be conducted on fainter samples which are less influenced by gravitational lensing (Birkin et al. 2021;Jones et al. 2021).
Recently, Simpson et al. (2020) published a AS2COSMOS catalog of 260 SMGs that is based on subarcsecond ALMA band 7 continuum follow-up observations of the brightest 182 single-dish detected submillimeter sources (S 850 > 6.2 mJy) drawn from the parent 1.6 degree 2 of the SCUBA-2 850 µm survey in the COSMOS field (S2COSMOS; Simpson et al. 2019 The spectral coverage of the five sets of tuning adopted for our ALMA observations, where each set is plotted the same color.Bottom: The spectral sensitivity curves of our ALMA data expressed in pixel-to-pixel r.m.s., which are made based on data cubes smoothed to a common uniform beam size and shape.The curves of all 18 sources are plotted with each being assigned to a random brightness of green, and this is to show the small scatter (a few percent) of data quality among each of our sample source thanks to the observing strategy.The overall r.m.s.median is 0.33 mJy beam −1 per 15.6 MHz (∼ 40 − 55 km s −1 ), with notable dips due to overlaps of spectral coverage between different tuning, except for the marked position due to the WVR leak, which was flagged during the QA2 process.
cially, considering the depths of both the initial SCUBA-2 survey and the ALMA follow-up observations, the catalog provides ∼20 and ∼60 SMGs that are 100% and >90% complete, at flux cuts of S 850 ∼ 12 mJy and S 850 ∼ 9 mJy, respectively (Figure 1; Simpson et al. 2020).Under similar flux cuts the bright sub-samples are a factor of > 3 and > 50 larger than those of AS2UDS (Stach et al. 2019) and ALESS (Hodge et al. 2013), the two previous largest uniform ALMA followup SMG surveys, representing a major step forward in constructing a sizable and highly complete bright SMG sample.Together with the fact that the model predictions differ the most in this regime, the brightest SMGs (effectively high-redshift HyLIRGs) from AS2COSMOS provides an ideal testbed for constraining models (Figure 1).
Here we present the first results of an ALMA spectroscopic survey of the brightest AS2COSMOS SMGs, or AS2COSPEC.We present sample selection and the ALMA data in Section 2. In Section 3 we present spectral extraction, as well as analyses on detailed modeling of the line profiles and redshift determinations.The implications and comparisons of our results are discussed in Section 4 and a summary is given in Section 5. Note throughout this paper S 850 and S 870 are used to represent the flux measurements at the representative wavelengths of SCUBA-2 and ALMA band 7 observations, respectively.In reality these measurements can be used interchangeably given the current precision of flux measurements.We assume the Planck cosmology: H 0 = 67.7 km s −1 Mpc −1 , Ω M = 0.31, and Ω Λ = 0.69 (Planck Collaboration et al. 2020).

Sample
Our sample is drawn from AS2COSMOS, a parent sample of 260 SMGs identified by ALMA band 7 continuum follow-up observations of a highly complete (99.5%), flux-limited (S 850 > 6.2 mJy) sample of the brightest submillimeter sources uncovered by the singledish SCUBA-2 survey at 850 µm in the COSMOS field over an area of 1.6 deg 2 (Simpson et al. 2019;2020).The ALMA band 3 observations presented in this paper targeted the brightest 18 AS2COSMOS SMGs that are at the flux range of S 870 = 12.4 − 19.3 mJy based on the ALMA measurements (Simpson et al. 2020).The main goals are to measure the redshift distributions, gravitational lensing magnifications, as well as to study the ISM properties via molecular lines such as CO and [CI].To compare to the previous large samples of ALMAidentified SMGs from follow-up observations of singledish detected submillimeter sources, in Figure 1 we show that there are only two SMGs that are similarly bright as our sample in the AS2UDS sample (Stach et al. 2018) and there is none in the ALESS sample (Hodge et al. 2013).Note that about one third of the sample was observed in millimeter by NOEMA and other ALMA datasets, and their measured redshifts were reported by Jiménez-Andrade et al. (2020), Simpson et al. (2020), andBirkin et al. (2021), which are consistent with our results.

ALMA data
The data were taken in November and December of 2019, under the project number 2019.1.01600.S.The observations were split into eleven execution blocks, each of which amounts about one and half hours worth of data for both science and calibrations.The default spectral scan setup was adopted in the ALMA Observing Tool (OT), meaning that each execution block contains data on all 18 SMGs with five spectral tunings covering continuously from 84.1 to 113.3 GHz (Figure 2), where each tuning with four spectral windows was carried out for exactly 26s.That is, across the whole frequency range observed, each spectral channel received totally about 4.8 minutes (26s times 11 executions) of science data on each SMG, except in the range of 97.7-101.3GHz, as well as overlapping edge channels, where the on-target exposure time roughly doubles.Each spectral window is tuned to a standard Time Division Mode (TDM) with 128 channels and a frequency width of 15.625 MHz, corresponding to a velocity width of about 40-55 km s −1 depending on the observed frequency.At least 43, up to 49, antennas were used for each execution block, with a baseline length ranging from 15 to 313 meters hence a nominal C43-2 configuration.Since the target SMGs are all located in the COSMOS field, the phase calibrator was always J1008+0029, and J1037-2934 was always used for the amplitude and bandpass calibrations.The weather conditions during these observations were standard band 3 weather, with a precipitable water vapor (PWV) ranging from 1.7 to 5.8 mm.
We adopt the calibrations performed in the second level of the quality assurance (QA2).Flagging and calibrations were done using casa pipeline version 5.6.1-8,which is also the version used for imaging.Additional manual flags were put in by the QA2 team to remove antennas or scans with poor performance, in particular a line at 91.66 GHz leaking from the local oscillator (LO) water vapor radiometer, which is marked in Figure 2. We reviewed the weblog record and confirmed the quality of the calibration results.
To image the visibilities, the calibrated visibilities were first continuum subtracted.To do that, the channels that contain line emissions need to be excluded for a proper first-order continuum fitting.The line channels to be masked were determined from the spectra extracted from the delivered pipeline-produced imaging cubes through typical single Gaussian fitting, spanning across +/-3 sigma centered at the Gaussian peaks based on the fits.The spectra were extracted using circular apertures centered at the 870 micron continuum positions (the phase centers), and their radius were determined by a curve-of-growth method.We then used Tclean to inverse Fourier transform and clean the continuum-subtracted visibilities.The visibilities were transformed to image cubes of 270×270 pixels in x (RA) and y (DEC) axis with a pixel size of 0. 42 (∼6-8 pixels per synthesized beam), and 1870 channels in the z (frequency) axis.We adopt natural baseline weighting across all spectral windows in order to maximize the signal-to-noise ratio (SNR) for line detection.To increase the efficiency of clean, we adopted the auto-masking approach (Kepley et al. 2020) in Tclean, which allows casa to generate masks for each channel in an iterative process based on a few parameters.To do that we set the usemask parameter to 'auto-multithresh'.For each channel, the first round of mask creation only includes pixels with SNR of four and above (noisethreshold = 4), and then the mask is allowed to expand in order to include lower signal-to-noise regions (lownoisethreshold = 2.5).The masked regions determined by auto-masking were then cleaned down to 2 sigma level (nsigma = 2 in Tclean).Examples of the cleaned channels and their associated masks are shown in Figure 3. Finally, given the same baseline weighting across the whole frequency range, the spatial resolution of each frequency channel differs slightly, up to In panel (d) we show that by normalizing all curves to a beam factor of one (grey), the median curve (solid) with the bootstrapped uncertainties appears stable and converging, which is consistent with the shape of the synthesized beam (dashed) and showing that the non-converging behavior of individual targets could be due to larger-scale noise structures.We therefore in principle adopt the synthesized beam as the aperture to extract spectra and use this median curve to correct the measured flux densities to total.Note that this median curve is stable to within a few percent against the choice of aperture size to which all the curves are normalized.
∼35% end-to-end.To allow more straightforward analyses and understanding of the data we applied smoothing on the reduced cubes by using the CASA routine imsmooth, setting the kernel to the common resolution, which is normally the largest beam size of the pre-smoothed cube.After this step, the spatial resolution of all cubes is uniform, with a synthesized beam FWHM of 4. 3×3. 5 and a small P.A. range of 69-72 degrees.The final spectral sensitivity achieved is uniform across all sources to within a few percents thanks to the observing strategy, however it differs among channels mainly due to the tuning coverage.We show the pixelto-pixel r.m.s. in Figure 2, and the overall median is 0.33 mJy beam −1 per 15.6 MHz (∼40-55 km s −1 ).The continuum of each source was also imaged with natural weighting, resulting into images with a synthesized beam FWHM of 3. 6×3.0 and a small P.A. range of 77-80 degrees.The sensitivity achieved for continuum is also uniform across different sources, 12-13 µJy beam −1 at the representative frequency 98.7 GHz.

Line detection and spectra extraction
To systematically search for line emission and quantify their significance, we ran the publicly available code LineSeeker, which was first written and used to search for line emission for the ALMA frontier field survey (González-López et al. 2017), and was later applied to the data taken for the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS; Walter et al. 2016;González-López et al. 2019).LineSeeker utilizes a matched filtering technique that combines spectral channels based on Gaussian kernels with a range of widths.The widths are chosen to match lines detected in real observations.The line candidates are determined to be significantly detected if their signal-to-noise ratios are larger than the most significantly detected negative signals in the cube, and we find a significance threshold of SNR>10 satisfies the aforementioned requirement in all cases.In this paper we focus on the spectra of the known SMGs in the AS2COSMOS catalog (Simpson et al. 2020), and the results from the search of serendipitous detection will be presented in a future work.For the 18 primary SMGs, 17 have yielded significant line detection based on LineSeeker, and AS2COS0037.1 is the only source that has none.
The spectra used for later analyses are extracted based on the coordinates presented in Simpson et al. (2020), which are based on ALMA 870 µm continuum observations.All our targets have their 3 mm continuum detected and we confirm that via 2D elliptical Gaussian fitting using imfit the coordinates of the 3 mm continuum best-fit peaks are consistent with those of 870 µm continuum.We therefore adopt the 870 µm coordinates given the significantly lower positional uncertainties.The results of imfit also suggest that the 3 mm continuum is not spatial resolved, which is expected since none strongly lensed SMGs like our targets are typically found to have sub-arcsecond sizes in rest-frame submillimeter (e.g., Simpson et al. 2015b;Ikarashi et al. 2015;Hodge et al. 2016Hodge et al. , 2019;;Chen et al. 2017;Fujimoto et al. 2018;Gullberg et al. 2019).
To determine the optimal aperture size for extracting the spectra and to derive the total line flux density, we employ the curve-of-growth analyses at the frequency of the emission lines and using the continuum subtracted 300 400 500 600  5. Top: An overview of the continuum-subtracted spectra of all our 18 primary SMGs in their rest-frame frequencies, along with the marked locations of strong molecular and atomic lines.Spectral lines are detected in 17 out of 18 targeted SMGs.SMGs with single line detection have their redshifts determined jointly with photometric redshifts, which can be shown to have ∼70-80% success rate based on those that have multiple line detection (Section 3.3; Birkin et al. 2021).AS2COS0037.1 is the only one SMG that is bright in optical and near-infrared but no lines detected in the scan, suggesting that based on its photometric redshift it is likely sitting within the known redshift gap for the band 3 scan.In this plot we therefore set the redshift of AS2COS0037.1 to 2.0 for demonstration purpose.Middle: Error-weighted stacked spectrum in signal-to-noise ratio, binned to 500 km s −1 per channel.The range in y-axis is from -4 to 4. AS2COS0037.1 is not included in the stack.No additional lines are detected in the stacked spectrum.Bottom: The number of spectra involved in the stack for each channel.
cubes.For each source, spectra are extracted based on a set of apertures that are centered at the 870 µm continuum position, shaped as a synthesized beam, and scaled by a range of factors in beam size from 0.5 to 4.5 with steps of 0.25.To estimate line flux density we conduct χ 2 model fitting for each spectrum extracted, and two models, single Gaussian and double Gaussian, are considered.Given the channel-to-channel variations in sensitivity, the fittings are weighted by the noise of each spectral channel, which is estimated by taking the standard deviation of a collection of 1000 flux density measurements at random positions at same frequency but with the same aperture as that used to extract the target spectra.The line flux density of each extracted spectrum is then computed by integrating the best-fit model curves.
Because of the two adopted Gaussian models the above approach results in two curve-of-growth curves for each target SMG.The selected examples are shown in Figure 4, in which we can first observe that the integrated line flux densities are not sensitive to the different underlying models.Secondly, while some curves converge to a certain flux density level, as expected from a curve-of-growth analysis, some appear to be not converging.We suspect that this is due to larger-scale noise structures, which have also been shown to be the case in other ALMA data sets (e.g., Novak et al. 2020).Indeed, by taking the median of all the curves (one from Figure 6.Spectra of all the detected line emissions from our primary SMGs, with their IDs indicated.All spectra are extracted using the aperture equivalent to the synthesized beam, and centered on the line centers of the best-fit models.The range of frequency plotted corresponds to a velocity range of ±2000 km s −1 .The best-fit single and double Gaussian models are plotted in dashed and solid curves, where the preferred models according to Table A.1 are plotted in solid.The red and blue components of the double Gaussian models are also shown.The gray bands indicate the r.m.s.levels.The transition of the line is also indicated if the redshift of the corresponding SMG is confirmed with more than one emission line.The inset of each spectrum shows the velocity-integrated emissions over the line FWHM, and it is shown in signal-to-noise ratio linearly scaled from -4 to 6 with a box size of 12.6 on a side.The contours are plotted at levels of [3,4,5,7,9,15] σ; They are in black if single Gaussian model is preferred, or red and blue if double Gaussian model is selected.The small ticks below the lines that are better described by double Gaussian models mark the frequency ranges for making the velocity-integrated signals for the red and blue components.each target SMG) normalized at a given aperture size, one beam in this case, we find that this median curve nicely converges beyond around three times the beam size.In addition, this converging median curve is stable to within a few percent against the choice of aperture size to which all the curves are normalized, suggesting that this median curve can be used as a common profile for correcting the line integrated flux densities to total.Note that there is a hint of extended emissions by comparing the median curve to the curve-of-grown of the synthesized beam, although the difference is not significant (Figure 4).The detailed size analyses will be presented in the forthcoming paper.However, to account for this possible extended emission we therefore adopt the median curve, instead of the curve of the synthesized beam for the correction to total (the difference is ∼20±10%).We adopt the aperture equivalent to one synthesized beam partly because all the curves appear to be stable till this point and beyond which they start to diverge (Figure 4).It also yields higher signal-tonoise ratios spectra compared to those extracted with larger aperture sizes, which help to determine a more appropriate model for the spectral profile (Section 3.2).
Given the above justifications, the spectra that we use for analyses from this point on are therefore extracted using the aperture equivalent to the synthesized beam, and their estimated line flux densities are corrected to total based on the median of the curveof-growth curves.For AS2COS0001.1, AS2COS0008.1, and AS2COS0028.1,where the nearby SMGs are also detected in a few cases and close enough to potentially affect the spectra, we re-image the cubes using the Briggs baseline weighting with a robust parameter of -1.52 , resulting into a ∼35% reduction of the beam size, allowing clear separations to the companions with at least one synthesized beam distance (∼3 ).The spectra of these three SMGs are then extracted based on the smaller beam and again corrected to total using the corresponding median curve3 .Finally for AS2COS0037.1, the only target SMG that we do not obtain significant line detection, its spectrum is extracted using the synthesized beam but no correction to total is applied.An overview of all the 18 spectra are shown in Figure 5, where we also show the error-weighted stacked spectrum in signal-tonoise ratio.No additional emission lines are detected in the stacked spectrum.We have also experimented lu-minosity weighted stacking and luminosity normalized stacking, however the results remain unchanged.

Model fitting
The first analysis that we performed on the extracted spectra was to determine which model, single or double Gaussian, better describes the data.To make a quantitative assessment we employ both the Bayesian information criterion (BIC) and a version of Akaike information criterion that is modified for the limited sample size (AICc).The analyses are documented in Appendix A, and the best-fit models are shown in Figure 6.In general we find that the results based on BIC and AICs are consistent to each other in 80% of the cases, and they all point toward the fact that about 90% of the primary SMGs have lines that are better described by double Gaussian profiles.The implications of this result are discussed later in the discussion section.
Moment calculations are adopted to compute line properties using best-fit single and double Gaussian models, meaning and the first moment (v), i.e., intensity-weighted central frequencies, are used to determine redshifts.The zeroth and the second moment will be used in the forthcoming paper to discuss line properties in detail.

Redshift determination
To determine redshifts, the target SMGs are grouped into three categories; SMGs with multiple line detection, those with only one line detection, and then finally those that do not have emission line detection.Other work in the literature are referenced to aid the redshift determinations, in particular those from Jiménez-Andrade et al. ( 2020), Mitsuhashi et al. (2021), and Daddi et al. (2022).The category to which each SMG belongs is given in Table 1.In total, ten (56%) SMGs have their redshifts determined by multiple emission lines, seven (39%) by single emission line, and one (6%) by its photometric redshift.
For the SMGs that only have one line detection, we find the solutions that are best matched to the photometric redshifts reported in Ikarashi et al. (2022), where as in Dudzevičiūtė et al. (2020) the SED code magphys is employed.We assume the line to be one of the CO transitions, which is justified by the fact that the frequency coverage of our data would have covered a CO line if the line is instead [CI].Since in SMGs [CI] is typically weaker than the nearby CO lines (J up = 4, 5; Spilker et al. 2014;Birkin et al. 2021), if the line is [CI] then the nearby CO lines would have been detected.We note that compared to other methods, magphys has the advantage of taking into account far-infrared and submillimeter photometry, allowing redshifts estimates for all AS2COSMOS SMGs, which are often times faint or undetected in optical and near-infrared wavebands where the known COSMOS catalogs such as COSMOS2015 (Laigle et al. 2016) and COSMOS2020 (Weaver et al. 2021) are based on.However given the rich information provided in the COSMOS catalogs and a good number of matches can be found on our target SMGs, we nevertheless exploit them later for detail assessments.
Lastly, for the only SMG that has no line detection by far, AS2COS0037.1, the photometric redshift reported by Ikarashi et al. as well as the latest COSMOS2020 catalog (1.93 +0.04 −0.07 and 1.98 +0.03 −0.15 based on the LePhare and EAZY codes, respectively; Weaver et al. 2021) suggest that this SMG could be located within the known redshift gap of ALMA band 3 spectral scan, which is z = 1.74 − 2.05.Indeed, given the depth reached by our data (∼0.3 mJy beam −1 ), assuming a single Gaussian model of 3 σ peak and a typical line width of 500 km s −1 , the line intensity limit is 0.5 Jy km s −1 .As- The outliers are defined as those that have their ∆z > 1, meaning that if only single line detected it is assigned to an incorrect CO transition based on photometric redshifts.The ID numbers of the two outliers are indicated.The SMGs that have multiple lines detected are marked as filled circles and those with only one line detected marked as empty ones.The one that is boxed is AS2COS0001.1, due to the reason that its photometry could be contaminated by a nearby foreground lens galaxy.Bottom: Histogram of differences between photometric and spectroscopic redshifts (∆z), based on SMGs that have multiple lines detected.Excluding the two outliers, we find that the median redshift difference is ∆z/(1 + zspec) = 0.04 ± 0.04 with an bootstrapped error.
suming the missing line is CO(3-2), the closest transition based on the photometric redshift, the line intensity limit corresponds to a limit on the CO(1-0) luminosity of L CO(1−0) = 2 − 4 × 10 10 K km s −1 pc 2 , assuming a latest r 31 ≡ L CO(3−2) /L CO(1−0) = 0.63 (Birkin et al. 2021).Given the infrared luminosity of ∼ 10 13 L (Liao et al. in prep.), this luminosity limit is well below the measured L IR − L CO(1−0) correlation and would have allowed detection of the line in 90% of the literature reported detection on SMGs (Birkin et al. 2021).We therefore adopt 1.90±0.05for AS2COS0037.1, which is simply the mean of the redshift gap with an uncertainty that makes the probability of it being outside the redshift gap equivalent to 3 σ.
In Table 1 we report the redshift solutions of our sample on individual lines as well as the final adopted values for each SMGs based on the methods stated above.We adopt the redshift results based on the selected model for each SMG listed in Table A.1, although we note that these results are not sensitive to the chosen single or double Gaussian models for the line fitting.
In Figure 7 we compare the photometric redshifts and the adopted spectroscopic redshifts for all target SMGs except for AS2COS0037.1.We find that based on the ten SMGs that have multiple line detection so unambiguous redshifts, three sources have their spectroscopic redshifts differ from photometric redshifts by more than one, effectively meaning that if only single line detected it would have been assigned to an incorrect CO transition based on photometric redshifts.This suggests a 70% success rate for our strategy of using photometric redshfits to aid determinations of spectroscopic redshifts for SMGs with single line detection.Although note that this success rate is estimated based on the higher redshift subset, which tends to have less reliable photometric redshifts.Thus the success rate for our sample SMGs that have one single line detection could be higher.
Our success rate is similar as that of a recent blind survey of emission lines on fainter SMGs (Birkin et al. 2021), in which they found about 80% success rate.Excluding the three outliers, we find that the median redshift difference is ∆z/(1 + z spec ) = 0.04 ± 0.04 with an bootstrapped error, and a scatter of σ ∆z/(1+zspec) = 0.06, suggesting that the photometric redshifts estimated via SED fitting using magphys can be accurate on average with good precision.However, we also find that the scatter is much larger than what can be inferred from the quoted uncertainties of the photometric redshifts, indicating that the uncertainties of photometric redshifts are underestimated.
Based on the assessments above, we expect that for the seven SMGs that have only one line detection, about 1-2 sources may have their real redshifts different from, and likely higher than, the adopted values (details given in the Appendix B).We take these into accounts when comparing our results to model predictions.

Physically associated pairs
Interferometric follow-up observations in the past decade have convincingly shown that submillimeter sources detected by single-dish observations tend to break up into multiple sources, with a probability strongly correlated with the flux density of single-dish measurements, from ∼10% for ∼3 mJy SMGs to >50% for SMGs at 10 mJy (Wang et al. 2011;Barger et al. 2012;Hodge et al. 2013;Simpson et al. 2015a;Cowie et al. 2018;Hill et al. 2018;Stach et al. 2019;Simpson et al. 2020).While the number of constituent SMGs can range from 2 to 4, it has also been shown that the flux densities measured from single-dish observations are dominated by the brightest constituent SMG, with a 70% contribution on average.
The relationship among these constituent SMGs is however unclear, in particular whether they are physically associated and undergoing dynamical interactions, or unrelated SMGs that are simply a result of line-ofsight chance projections.The determination of this relationship has profound implications in our understandings on the triggering mechanisms of star formation for SMGs.Observationally, statistical approaches using photometric redshifts and number density analyses have suggested that ∼30% of these pairs are physically associated (e.g., Stach et al. 2018;Simpson et al. 2020), which is in agreement with spectroscopic measurements of small samples of paired SMGs (Hayward et al. 2018;Wardlow et al. 2018).Theoretically, models have instead predicted that the majority of these pairs are physically unassociated, especially for the bright submillimeter sources with single-dish measured flux density of S 850 10 mJy (Hayward et al. 2013a;Cowley et al. 2015;Muñoz Arancibia et al. 2015).To definitely settle this issue spectroscopic follow-up observations are needed, and our sensitive spectral measurements on a complete sample of bright SMGs are suitable for providing further constraints.
Our sample include five pairs of SMGs and their 870 µm continuum images and 3 mm spectra are shown in Figure 8.
As stated in Section 3.1 and seen in Figure 8, the spatial resolutions of the naturally weighted cubes are too coarse to separate the SMG pairs of AS2COS0001.1/1.2,AS2COS0008.1/8.2, and AS2COS0028.1/28.2.We therefore re-image those cubes using Briggs weighting with a robust parameter of -1.5, which stands as a good balance between sensitivity and resolution after experimenting different baseline weightings.To increase the detectability of line emissions we also binned the channels to ∼150 km s −1 per channel, appropriate for the typical linewidth of SMGs which is about 500 km s −1 in FWHM (e.g., Bothwell et al. 2013;Birkin et al. 2021).The spectra are extracted using apertures centered on their reported 870 µm locations (Simpson et al. 2020) with sizes and shapes of their corresponding synthesized beam, shown at the corners of Figure 8.
We obtain a significant detection from AS2COS0008.2 and a marginal detection from AS2COS0001.2.The best-fit single Gaussian model suggests that the redshift of AS2COS0008.2 is 3.5739±0.0009assuming that it is the same line as AS2COS0008.1, and the significance of the detection is at >4 σ based on the velocityintegrated intensity over the FWHM of the best-fit model (Figure 8).Although marginal (∼ 2 σ over the FWHM) in our data, the same line from AS2COS0001.The velocity differences between these two pairs are within 500 km s −1 .At the projected distances of 20-30 kpc, their velocity differences would suggest that these two pairs are gravitationally bound systems, assuming that SMGs as suggested by clustering analyses reside within halos with masses of ∼10 13 M (Hickox et al. 2012;Chen et al. 2016;Wilkinson et al. 2017;An et al. 2019;Lim et al. 2020;Stach et al. 2021).The redshift differences of the two pairs are both ∆z < 0.02, a definition adopted by some models for physically associated pairs (Hayward et al. 2013a;Muñoz Arancibia et al. 2015).This suggests that 40% of the paired SMGs in our sample are physically associated.In addition, given that our observations are designed to detect lines from bright SMGs, deeper observations may reveal associated line emissions from the secondary SMGs of other pairs, and the true fraction of physically associated pairs could be higher.Indeed, as seen in Figure 8, among these four secondary SMGs, those not detected in line by our data are the faintest in 870 µm flux densities (<2 mJy), and based on recent measurements (Dudzevičiūtė et al. 2020;Birkin et al. 2021) we do not expect to detect lines on these faint SMGs given the sensitivity of our data.Our results are also consistent with Simpson et al. (2020), where based on the number counts analyses they estimated 62±7% of the > 3 mJy secondaries are physically associated with the primary SMGs.
While our results suggest a relatively high fraction (≥40%) of physically associated SMG pairs, the current  1 for the primary SMGs.The images are uniformly and linearly scaled between -1 and 2 mJy and the color bar plotted on top.The SMGs are boxes in white and their IDs are indicated.The synthesized beams of the 3 mm spectra adopted for each pair are shown in the bottom left corners and the black contours are signal-to-noise ratios in 3, 4, 5, and 7 σ for the line emissions averaged over the FWHM based on the best-fit single Gaussian model profiles shown in the bottom spectra.Middle and Bottom: 3 mm spectra binned to ∼150 km s −1 per channel for the primary (middle panels) and secondary (bottom panels) SMGs.These spectra are extracted using apertures centered on their reported 870 µm locations (Simpson et al. 2020) with sizes and shapes of their corresponding synthesized beam.The velocities are referenced to the redshifts of the primary SMGs (Table 1), and the 1 σ r.m.s.levels are plotted in gray.For the secondary SMGs, we obtain significant detection from AS2COS0008.2, marginal detection from AS2COS0001.2, and no detection from AS2COS0028.2, AS2COS0065.2, and AS2COS0090.2.There is no line detection from AS2COS0028.2 in both frequency ranges covering CO(3-2) and CO(4-3), and for demonstration purposes we only show the range of CO(3-2).The spectra of AS2COS0001.sample size is admittedly too small to have much constraining power.Furthermore, it is also not straightforward to compare our results against model predictions.First of all, the adopted criteria in defining pairs are different in each model.For example, Cowley et al. (2015).made predictions based on the brightest two constituent SMGs.On the other hand, Hayward et al. (2013a) considered all SMGs with 850 µm flux densities of >1 mJy, which is the flux level that is highly incomplete in the parent AS2COSMOS catalog.Secondly, our spectral measurements are incomplete in terms of including all pairs or multiples of submillimeter sources above a cer-tain flux density limit measured by SCUBA-24 , which is what model predictions are typically based on.To make fair comparisons, experiments that are tailored for these model predictions are needed.
In spite of these caveats, however, our results do provide constraints on this issue from another angle.Since our target SMGs are complete to their flux cuts, it can be stated that our results suggest that for SMGs brighter than 12.4 mJy at 870 micron, their brightest companion SMGs are physically associated with the primary SMGs in ≥40% of the time.

Redshift distribution
Taking the adopted redshifts in Table 1, the overall median of the 18 primary target SMGs is 3.3±0.3with a bootstrapping uncertainty, which is slightly larger than that of the photometric redshifts (3.1±0.2),simply due to the three outliers whose real redshifts are all higher than their photometric redshifts.Three target SMGs, AS2COS0001.1, AS2COS0002.1, and AS2COS0006.1, have been shown to be part of a z = 4.6 coherent Mpc scale structure (Mitsuhashi et al. 2021).
The uncertainty for the median caused by the measurement errors is on the order of 3 × 10 −4 , which is the standard deviation of a set of 1000 medians, each of which is a result of a set of randomly perturbed redshifts generated according to the measurements.The uncertainty for the median caused by the wrongly assigned line transitions for the SMGs with single line detection is estimated to be 0.1, which is again the standard deviation of a set of 1000 medians, each of which is a result of a set of redshifts that are generated based on the adopted redshifts but having randomly chosen 2 SMGs with single line that are randomly assigned to one transition up or down from the adopted one, providing that the new transition would not allow the coverage of another CO transition within the bandwidth.In short summary, the dominant contributor to the uncertainty of the median is the bootstrapping uncertainty, essentially reflecting the sample size and the underlying distribution.

Comparisons with previous measurements
Previously, redshift measurements of SMGs have been mostly focused on those with S 850/870 ∼ 2−10 mJy (e.g., Chapman et al. 2005;Ivison et al. 2007;Wardlow et al. 2011;Simpson et al. 2014;Cowie et al. 2017;Stach et al. 2019;Birkin et al. 2021).This is simply a result of the .Redshift distribution as a function of S870 based on uniformly selected SMG samples with ALMA follow-up observations.Spectroscopic redshifts of this work are plotted in black circles, and those for ALESS and AS2UDS subsamples are plotted in blue squares, which are reported by (Birkin et al. 2021).We also plot the spectroscopic redshifts of the SPT SMGs versus their intrinsic flux densities (Reuter et al. 2020).Typical uncertainties for the spectroscopic redshift samples are shown at the bottom right corner with matched colors.The background density contours show the distribution of photometric redshifts based on the three samples, ALESS (Simpson et al. 2014), AS2UDS (Stach et al. 2019), and AS2COSMOS (Ikarashi et al. 2022).The results of linear fittings on photometric redshifts and spectroscopic redshifts of the combined ALESS, AS2UDS, and our results are plotted as gray and blue bands, respectively, where the widths represent 68% confidence levels.The slope of the best fit model on the spectroscopic redshifts is 0.09±0.02,consistent with that of photometric redshifts and previous studies (Cowie et al. 2017;Stach et al. 2019;Simpson et al. 2020;Birkin et al. 2021).
typical survey depth and area, coupled with the shape of the number counts, that maximizes the number of detection in this flux range.The median redshifts deduced differ slightly in different studies but they generally lie in the range of 2.4-2.6, with an uncertainty of about 0.1-0.2.Together with our results this hints that brighter SMGs are on average located at higher redshifts.Since these previous studies used various methods to estimate or measure redshifts and they often covered different flux ranges.To better assess the correlation between flux density and redshift in Figure 9 we plot the most recent results from self-similar and well-defined samples, namely ALESS (da Cunha et al. 2015), AS2UDS (Stach et al. 2019), and AS2COSMOS (Simpson et al. 2020;Ikarashi et al. 2022).We investi- gate photometric and spectroscopic redshifts separately.
In Figure 9 the photometric redshifts of all three samples are combined and shown as density contours.In both cases the correlation between flux density and redshift can be observed.Indeed the Spearman correlation coefficients are 0.3 and 0.4 for photometric and spectroscopic redshifts, respectively, and the p-values are 0.001.The maximum likelihood linear fitting yields z median = (2.2 ± 0.3) + (0.09 ± 0.02) × S 870 for the sample with spectroscopic redshift, consistent with the best-fit based on photometric redshifts.Our results are also consistent with previous assessments regarding either the general trends or the slope (Ivison et al. 2002;Cowie et al. 2017;Stach et al. 2019;Simpson et al. 2020;Birkin et al. 2021).
Finally, we compare our results to the SPT SMGs presented by Reuter et al. (2020), where complete measurements of spectroscopic redshifts are reported on a welldefined millimeter selected SMG sample.The reported SPT SMGs have apparent flux densites of S 870 > 25 mJy and they are mostly strongly lensed.In Figure 9 we show their redshift distributions with respect to the intrinsic flux densities where gravitational magnifications are accounted for.Interestingly, the distribution of the SPT SMGs appears to follow our unlensed SMGs at S 870 10 mJy, below which the known selection effect of lensing biasing sources toward higher redshifts can be clearly seen.

Comparisons with models
We now compare our measurements to model predictions.Currently there are various schools of models that can reproduce basic properties of SMGs such as number counts and have made predictions for other properties such as redshifts.They include empirical models (e.g., Béthermin et al. 2012Béthermin et al. , 2017;;Casey et al. 2018;Zavala et al. 2021), semi-empirical models (e.g., Hayward et al. 2013b;Popping et al. 2020), semi-analytical models (e.g., Lacey et al. 2016;Lagos et al. 2020), hybrid models including both analytical and empirical recipes (e.g., Negrello et al. 2007;Cai et al. 2013;Negrello et al. 2017), and finally hydrodynamical simulations with various treatments of dust and its emissions (e.g., McAlpine et al. 2019;Hayward et al. 2021;Lovell et al. 2021).
In general, while good progress has been made over the years, it is still difficult to find in hydrodynamical simulations as many bright SMGs as those detected in observations.For example, imposing the flux cut of our sample, there are only two such sources in the model presented by Lovell et al. (2021), and similar or fewer numbers are seen in other hydrodynamical models (McAlpine et al. 2019;Hayward et al. 2021).The lack of SMGs brighter than our flux cut could be due to the fact that current simulations do not have sufficiently volumes, which are typically about 100 cMpc on a side.As a result, for a meaningful comparison we do not consider predictions based on hydrodynamical simulations in this section.
We list the model catalogs used for comparisons and their details in Table 2, including the area from which the catalogs are drawn and the methods used to generate these catalogs.Briefly, all these models employ certain methods to predict SFRs thus total infrared luminosity (L IR ), and adopt various templates or relations to model the infrared and submillimeter SEDs.For example, analytical solutions of SFRs are calculated in physically motivated models like SHARK and Cai-Negrello, where the predicted submillimeter flux densities are modeled using selected SED templates.For models like SIDES and Popping, SFRs are estimated by fitting their dark matter base empirical models to stellar mass functions with considerations of the star formation main sequence.SIDES then adopt SED template to model submillimeter fluxes, where Popping apply a relation between the 850 micron flux density and SFRs and dust masses based on dust radiative transfer codes.Casey-Zavala obtains L IR directly by modeling the infrared luminosity functions, and a modified black body plus mid-infrared power-law SED model was assumed together with a relation between dust temperature and L IR .The full descriptions on the methodologies used to generate these model catalogs can be found in the latest reference for each model listed in Table 2.Although note that for Cai-Negrello the model has been slightly updated with a new version of SED template, resulting in a better agreement with the recently reported number counts in far-infrared and submilllimeter (Negrello et al. in preparation).We assess their redshift distributions as the following.For SIDES since it is drawn from an area that has a size similar to our initial SCUBA-2 survey, we simply impose the flux cut of our sample and plot the cumulative distribution in Figure 10.For model catalogs that are drawn from areas that are much larger than 2 square degrees, we randomly draw the model catalogs 100 times based on the size of our surveyed area.For each draw we assess their cumulative distribution and in Figure 10 we plot the median of all the 100 draws as well as their 10-90 percentile range to roughly indicate the fluctuations of these distributions mainly due to their field-to-field variations and shot noise.
To compare our observational results to models.We generate a set of 100 randomly perturbed redshift distributions based on our measurements and their associated uncertainties, as well as the probability of wrongly assigned redshifts for SMGs with only single line detection, following the procedure described in the previous paragraph.The results are plotted in Figure 10 again with 10-90 percentile range.We then perform two-sample Kolmogorov-Smirnov (KS) test between each of the 100 perturbed observational distributions and each of those 100 drawn from each of the model catalogs, so a total of 10000 comparisons except for SIDES which only has 100.For each model catalog, we record the p-values of each comparison and compute the percentage of them having p < 0.05, a standard value that suggests the difference between the two samples is highly significant.Changing this p-value cut does not affect the conclusion.We show the results of these comparisons in Figure 10.Strikingly all models predict lower redshifts compared to our measurements, although from the statistical point of view the significance of their differences differ.Our measurements agree best with SIDES, however SIDES only provides one sight line.With a much larger volume perhaps the comparison against SIDES would yield a result that is similar to that from the comparison against Cai-Negrello.The most significant difference comes from the comparison against SHARK, where all sight lines that we draw do not agree with our measurements.
We can also compare them in median redshifts, which are 1.7±0.3 for SHARK, 2.9±0.7 for SIDES, 2.6±0.3 for Casey-Zavala, 2.1±0.4 for Popping, and 2.9±0.4 for Cai-Negrello.The error budget again includes both statistical and systematic uncertainties caused by field-to-field variations and the shot noise of simulations.Our measured median redshift of 3.3±0.3 is larger compared to all models, with the least agreement to SHARK, which is in 3.8 σ tension.These are the same conclusions we make based on more detail two-sample KS tests.Measurements based on a much larger sample size (i.e., a factor of few more) would allow meaningful tests on the surviving models.

Implications
The physical reasons for brighter SMGs lying at higher redshifts remain unclear.It is worth noting that among models that are used to compare with our results, SHARK is the only one that is created with a complete suite of physically motivated treatments including merger trees of cold dark matter and baryon physics.It is also intriguing to note that another self-consistent and physically motivated semi-analytical model, GAL-FORM (Lacey et al. 2016), made with different treatment of physics, has also predicted a low redshift (z < 2) for the flux range of our sample and also a negative correlation between S 850/870 and redshift (Cowley et al. 2015).
Other physically motivated models based on hydrodynamical simulations like EAGLE (McAlpine et al. 2019), Illustris/IllustrisTNG (Hayward et al. 2021), and SIMBA (Lovell et al. 2021) have recently made good progress in matching number counts at the typical flux regime ( 10 mJy).While they are yet to be capable of making significant amount of 10 mJy SMGs, in part due to the limited volume, many are successful in reproducing median redshifts of fainter SMGs, although not all can reproduce the trend between flux density and redshift (McAlpine et al. 2019).These efforts have pointed to the need to better understand the properties of dust, in particular the relationship between dust, metal, and gas masses such as the dust-to-metals ratios, in order to address the issue that models tend to under predict dust masses for SMGs.For example recent measurements of dust mass of SMGs have arrived to an average of about 6 − 7 × 10 8 M (Dudzevičiūtė et al. 2020;da Cunha et al. 2021), somewhat higher than what is seen in most models.In our forthcoming paper (Liao et al. in preparation) we will show that our sample SMGs have even higher dust masses.On the other hand, sub-grid physics regarding the feedback processes has been suggested to be the key in making enough amount of dust in models (Hayward et al. 2021).
Moving forward, observational constraints on various physical properties would be required to solve the puzzle of the formation of SMGs.For example, measuring dust-to-metal ratios for dusty galaxies through observations would be key in testing the predicted values from recent models.ALMA has been proven invaluable on the measurements of dust and gas masses, as well as dust properties such as dust emissivity index and temperature.Measurements of gas-phase metalicity, on the other hand, has been a challenging task for dusty galaxies due to them often being faint in rest-frame optical.One can always start with those that are significantly detected on the optical strong emission lines, although one needs to also bear in mind the possible biases toward the galaxies that may have the largest spatial offsets between dust and optical emissions, or the subset that is the least obscured.Far-infrared fine structure lines from [N ii] and [O iii] offer a promising way to measure gas-phase metalicity of dusty galaxies (e.g., Rigopoulou et al. 2018), but a future facility such as a sensitive farinfrared probe satellite would be required to conduct such measurements on a large sample at cosmic noon.Finally, upcoming facilities such as JWST could provide sub-kpc rest-frame near-infrared imaging, which would help constrain stellar mass and morphology, addressing the issue of the triggering of star formation which has also been a topic of debate.

Gravitational lensing
Strong gravitational lensing (lensing magnifications µ > 2) is found to be responsible for the exceptional apparent brightness (S 500 µm 100 mJy or S 1.4 mm 10 mJy or S 850 µm 30 mJy) of the majority of the submillimeter and milllimeter sources that are uncovered by shallow but wide surveys such as those conducted by the SPT and Herschel (Negrello et al. 2010(Negrello et al. , 2017;;Bussmann et al. 2013;Hezaveh et al. 2013;Wardlow et al. 2013;Spilker et al. 2016).Most models predict that the fraction of sources experiencing strong lensing drops significantly, from about 70% to almost zero, within a narrow range of flux density, which is about 80-100 mJy at 500 µm and 10-30 mJy at 850 µm (Wardlow et al. 2013;Negrello et al. 2017).Our statistical sample SMGs have their flux densities within this range and therefore represent an opportunity to test model predictions.
To estimate lensing magnifications of our sample SMGs we employ lenstool (Kneib et al. 1996;Jullo & Kneib 2009).For each SMG, we utilize the COS-MOS15 catalog (Laigle et al. 2016) and construct a list of neighboring galaxies that are located within 30 of the SMG and have had reported estimates of photometric redshifts and stellar masses.This angular distance is chosen such that beyond which the results do not change significantly.A gallery of the selected massive subsets of these neighboring galaxies are plotted for each SMG in Figure 11.To construct a model of foreground gravitational potential for each SMG, we assume a regular AS2COS0065.1 1.5/10.7 1.8/10.7 1.4/10.5 1.5/10.80.9/10.80.9/10.9 AS2COS0066.1  et al. 2005).We describe the procedures in obtaining these parameters in the following.We first run SExtractor (Bertin & Arnouts 1996) to obtain their structural parameters ellipticity and positional angle, and the primary image considered is the K-band image from UltraVista DR4, except for AS2COS0031.1 where it is outside the UltraVista footprint so we adopt the 3.6 µm image from Spitzer.The scale radius and characteristic velocity are related to the assumed NFW profile, which is expressed as where r s is the scale radius and ρ s is the characteristic density so that the characteristic velocity is σ s = 4/3Gr 2 s ρ s .ρ s can be related to critical density ρ c as ρ s = δ c ρ c , where the density contrast δ c = 200c 3 /(3ln(1+c)−3(c/1+c)).c is the concentration factor defined as the ratio between the halo radius r 200 (the mean density within which is 200ρ c ) and r s .The halo mass can therefore be deduced as M 200 = 800/3πr 3 200 ρ c .
That is, r s and σ s can be derived for a given combination of concentration factor and halo mass.
For each neighboring galaxy, we estimate its halo mass by adopting the stellar mass and redshift provided by COSMOS15 catalog and applying the latest stellar-tohalo mass relationships given by Legrand et al. (2019).For concentration we refer to the classical work of Wechsler et al. (2002) based on dark matter N-body simulations of ΛCDM cosmology, where the relationships between concentration, halo mass, and redshift are shown.
Each of the adopted input parameters has its own reported uncertainty or scatter, and they should be taken into account in order to properly assess the probability distributions of lensing magnifications.To do so, for each SMG we create 100 potential models where each constituent potential is created by randomly perturbing each of their parameters based on their quoted uncertainties.We run lenstool on these 100 potential models for each SMG and calculate the median and 1 σ percentile.The results are given in Table 3.
As a check, the lensing magnification of AS2COS0001.1 was also estimated by Jiménez-Andrade et al. ( 2020) with a different assumption of the density profile, and they found a factor of 1.5, consistent with our results.Notably, we find that if we only look at the mass distributions of a much closer surroundings, the Note: Details can be found in Section 4.3.Uncertainties less than 0.05 are omitted.magnifications of a few SMGs, such as AS2COS0002.1 and AS2COS0014.1, would be significantly underestimated.This is because in these cases there are a few very massive (up to 10 11.5 M ) foreground galaxies, and in the case of AS2COS0002.1 a number of galaxies including the very massive ones are located at z ∼ 1, suggesting a group environment.
Based on Table 3, we can see that by looking at their median only one out of the 18 SMGs in our sample can be classified as being strongly lensed with µ > 2, suggesting a relatively low strongly lensed fraction of < 10% compared with the brighter sources.This is in line with what has been predicted by some observations (Chapman et al. 2002) and models.We note that our methodology in constructing foreground gravitational potentials does not account for contributions from halos larger than galactic halos (i.e., group scale halos) thus it is possible that magnificantions of some SMGs may have been underestimated by a moderate amount.On the other hand, our methodology also does not account for underdense regions so the overall magnification is slightly biased high, and as a result many unlensed SMGs have µ = 1.1.Nevertheless, these two factors are expected to not affect our results significantly.
We compare our results with predictions of magnification for the flux range of our sample from SIDES and Cai-Negrello in Figure 12 and find that our results are in-consistent with SIDES but consistent with Cai-Negrello.This can be understood by the fact that while the Cai-Negrello model takes into account both the redshift distribution of model SMGs and the mass distributions in the foreground, the predictions from SIDES are simply drawn from some probability distributions according to the redshifts of the model SMGs, as stated in Bethermin et al. (2017).
We now expand the comparisons by including published measurements for a wide range of 870 µm flux density, including the SMA and ALMA follow-up studies of Herschel-selected sources (Bussmann et al. 2013(Bussmann et al. , 2015) ) as well as those based on the SPT survey (Spilker et al. 2016).For each of the SPT sources separate magnification factors were reported for each of their constituent components, so we compute their flux-weighted magnification as the nominal magnification for each SPT source (Figure 3 in Spilker et al. (2016)).It is also worth noting that while the Herschel-selected sample observed by the SMA is a flux limited sample, the one observed by ALMA was selected to have a nearby foreground source to maximize the probability of detecting strongly lensed objects.As a result the Herschel-selected sample observed by ALMA is likely biased high in lensing magnification for the same flux range.
The results are plotted in Figure 13, where we find that despite the possibility of having a bias, the Herschel-ALMA measurements are consistent with our results at the overlapping flux density range (10-20 mJy).We can also compare to the model predictions from Cai-Negrello.Note that in the Cai-Negrello model sources that are not strongly lensed (µ < 2) are assigned 1 for their magnification.We therefore adopt the same method for the measurements and compute the binned median with bootstrapped errors.We compare our results with the median of the model prediction with the same flux bins and we find they have good agreement overall.
Last but not least, having lensing magnification measurements of a flux limited sample means that we can compute the number counts of the strongly lensed SMGs.The cumulative number counts of the parent AS2COSMOS sample were presented in Simpson et al. (2020), and they are plotted in Figure 14.Since our sample is limited to 12.4 mJy at 870 µm, the cumulative counts for the strongly lensed SMGs can be simply scaled from the parent counts of the corresponding flux bin (12.1 mJy in this case) by dividing its value by our sample size.There is no source between 12.1 mJy and 12.4 mJy so no correction needs to be applied.To estimate the uncertainty, the Poisson error of 1 is adopted and propagated based on the uncertainty of the parent  (Bussmann et al. 2015), orange squares for a SMA study of a brighter Herschel-selected sample (Bussmann et al. 2013), and green diamonds show the results from the SPT survey (Spilker et al. 2016).The red circles are binned medians where magnifications of µ < 2 are assigned to 1, following the method that is used to generate the model catalog by Cai-Negrello (Cai et al. 2013;Negrello et al. 2017), which is shown as the red curve.Overall we find the measurements are consistent with model predictions across this two orders of magnitude flux range.
counts.The results are plotted in Figure 14 as the red point.As a check, the value is very close to one divided by the size of the footprint of the primary S2COSMOS catalog so one divided by 1.6 degree square.Overall we find again good agreement with the model prediction of Cai-Negrello.

Connecting to the descendants
As shown in Section 3.2, we find that the emission lines of 15 out of the 17 (∼90±30%) primary SMGs with line detection appear to be better modeled by double Gaussian profiles.This high fraction of SMGs exhibiting double Gaussian line profiles is in contrast with previous studies of other SMGs samples, which have typically re-ported fractions under 50% (Greve et al. 2005;Bothwell et al. 2013;Birkin et al. 2021).This could be in part due to our smaller sample size, or it could also be because of our sample SMGs being brighter so they may appear dynamically distinct from their fainter counterparts.Unless dispersion dominated, it is expected that the lines would appear double Gaussian in systems that exhibit velocity gradients, either due to orderly rotating disks or mergers.Spatially resolved dynamical studies are needed to make a more definite conclusion on their physical origins.and blue squares (Stockmann et al. 2020;Valentino et al. 2020), respectively.We also include results of fainter SMGs that also have spectroscopic redshift measurements (Birkin et al. 2021).Our results are consistent with these plotted samples, corroborating the hypothesis that these populations are linked in evolution.
On the other hand, we find that in our sample, for lines that are better described by double Gaussian, the median separation in velocity between the red and blue peaks is 430±40 km s −1 , which is consistent with previous studies using their double-peaked SMGs (Bothwell et al. 2013;Birkin et al. 2021), as well as recent spatially resolved observations of CO/[CII] lines on a few SMGs (Chen et al. 2017;Calistro Rivera et al. 2018;Litke et al. 2019).This measurement of red and blue velocity separations allow us to crudely estimate dynamical masses provided some reasonable assumptions.Dynamical mass estimate can be expressed as M dyn = C(v c /sin(i)) 2 × R/G, where G is the gravitational constant, v c is the circular velocity, R is the representative radius, C is the correction factor depending on the mass distributions and the adopted representative radius, and i is the inclination angle.This assumes a rotation dominated system, which is supported by recent high resolution and high sensitivity dynamical measurements of SMGs (e.g., De Breuck et al. 2014;Lelli et al. 2021;Rizzo et al. 2021).Assuming an exponential disk profile with a scale length h and a half-light radius r e , it has been shown that R = 3h = 1.79r e 5 is a good approximation for spatially unresolved CO observations (de Blok & Walter 2014).Under this assumption of an exponential disk C is about 2 (Binney & Tremaine 2008).We adopt r e = 3 kpc based on recent spatially resolved CO measurements of SMGs (Chen et al. 2017;Calistro Rivera et al. 2018) and v c half of the median velocity separation that we measure.Since v c in our case is an average over a sample of sources with random inclination angles, we adopt an average sin 2 (i) = 2/3 (Tacconi et al. 2008).Since the dynamical mass estimated this way only includes mass within the assumed r e , we multiply a factor of two to obtain the total dynamical mass.As a result, we estimate a total dynamical mass of M dyn = 3.5 ± 0.6 × 10 11 M .Accounting for the typical dark matter fraction at high redshifts, which was recently reported as 12% by Genzel et al. (2020), we deduce a total baryon mass of M baryon = 3.1 ± 0.6 × 10 11 M .Note that the observational constraints on dark matter fraction are obtained from galaxies at z ∼ 2, slightly lower than our sample SMGs.However, recent Illustris-TNG hydrodynamical simulations have shown that the dark matter fractions are expected to be lower at higher redshifts (Lovell et al. 2018).On the other hand, the simulations have predicted about a factor of two higher fractions than those obtained from observations.It is outside the scope of this paper to argue which is correct, although by adopting either a factor of two higher dark matter fraction or no dark matter our conclusions would not be significantly altered.Finally, in the following paper where we plan to present the properties of the ISM via CO/[CI] and dust SED measurement and modeling (Liao et al., in preparation), we will show that by including stellar, gas, and dust mass the averaged total baryon mass is consistent with the dynamical estimates presented in this work.
Following Birkin et al. (2021), The estimates of total baryon mass and the measurements of velocity dispersion via molecular lines allow us to assess the connection between our sample SMGs and their proposed descendants at lower redshifts, such as compact quiescent galaxies at similar redshifts or local massive early-type galaxies (ETGs; Lilly et al. 1999;Swinbank et al. 2006;Alexander & Hickox 2012;Toft et al. 2014;Dudzevičiūtė et al. 2020).This is under the assumption that the total mass and dynamical energy are mostly preserved along the evolution.
In Figure 15 we plot our sample as a combined median where σ is the median of all the line moment measurements presented in Section 3.2, which is 260±21 km s −1 .As a comparison measurements of local ETGs that are based on the MASSIVE (Davis et al. 2019) and ATLAS 3D (Cappellari et al. 2013) surveys are shown, as well as the quiescent galaxies (D n 4000> 1.55; Kauffmann et al. 2003;Wu et al. 2018) at z = 0.6 − 1.0 based on the LEGA-C survey (van der Wel et al. 2021;Wu et al. 2021) and some compiled z ∼ 2 − 4 spectroscopically confirmed quiescent galaxies (Stockmann et al. 2020;Valentino et al. 2020).The results from the similar analyses of mostly fainter SMGs presented by Birkin et al. (2021) are also shown.For ETGs and high-redshift quiescent galaxies we adopt their stellar masses for the total baryon masses given stellar mass dominates baryon mass in these populations, whereas for fainter SMGs in Birkin et al. (2021) stellar, gas, and dust masses are summed to deduce their baryon masses.Our sample of bright SMGs lies in the regions occupied by local ETGs and high-redshift quiescent galaxies, corroborating the proposed evolutionary link among these populations.
In addition, given that our sample is well-defined in both flux density and area, we can compute number densities of our sample SMGs and compare the results with those based on recently confirmed quiescent galaxies at similar redshifts, especially those at z > 4. The total number of our primary SMGs in the redshift bins of 2.1 < z < 3, 3 < z < 4, and 4 < z < 5.1 are 6, 6, and 5, respectively.The lower bound of the lowest redshift bin is determined by the upper bound of the redshift gap (Section 3.3), and the upper bound of the highest redshift bin is justified by the fact that given our wavelength coverage emission lines should have been detected from sources at z = 5.1 − 7.2.The sky area covered by our initial SCUBA-2 survey is 1.6 deg 2 (Simpson et al. 2019) so the volumes covered in the three redshift bins are all approximately 2.0 × 10 7 cMpc 3 .Thus the observed number density of SMGs with S 870 > 12.4 mJy is about 3 × 10 −7 cMpc −3 for all three redshift bins.However, assuming a typical lifetime of 100 Myr for our sam-ple SMGs (Liao et al. in preparation) the duty cycle corrected number densities are 3.3 +2.8  −1.8 × 10 −6 cMpc −3 , 1.9 +1.6 −1.1 × 10 −6 cMpc −3 , and 1.0 +0.9 −0.6 × 10 −6 cMpc −3 for the three redshift bins, respectively.The uncertainties include poisson errors and a 40% field-to-field variation (Simpson et al. 2020).Our number density estimate at z > 4 is similar to that presented by searches based on Herschel (Ivison et al. 2016) and ALMA 2 mm survey (Manning et al. 2021), although sample selections are different thus direct comparisons are not straightforward.
There has been an active discussion regarding the progenitors of recently confirmed z ∼ 3 − 4 quiescent galaxies, and extreme dusty star-forming galaxies such as SMGs at z > 4 have been proposed to be ideal candidate given their nature (e.g., Straatman et al. 2014;Valentino et al. 2020).From the number density, our results at z > 4 appears to be in agreement with those of z ∼ 3−4 quiescent galaxies estimated by Davidzon et al. (2017) and Girelli et al. (2019) (∼ 10 −6 cMpc −3 ) but lower by a factor of a few to ten compared to other estimates (∼ 10 −5 cMpc −3 ; Straatman et al. 2014;Schreiber et al. 2018;Merlin et al. 2019).The different number densities estimated for quiescent galaxies could be due to selection effects such as different mass limits, as well as the methods to classify quiescent galaxies.However, given that we are selecting the brightest SMGs we expect our number density estimates to be lower limits, and fainter SMGs at z > 4 are confirmed to exist (Section 4.2.1).We can conclude that our sample of SMGs have properties expected for the progenitors of some z ∼ 3 − 4 quiescent galaxies, especially the most massive ones with stellar mass of > 10 11 M .

SUMMARY
We report the first results of the AS2COSPEC survey, a millimeter spectroscopic survey using ALMA to target the brightest SMGs drawn from AS2COSMOS (Simpson et al. 2020), a sample constructed through ALMA follow-up continuum observations of a flux-limited (S 850 > 6.2 mJy) sample of submillimeter sources uncovered by the single-dish SCUBA-2 S2COSMOS survey (Simpson et al. 2019).We targeted the brightest 18 primary SMGs with their flux densities between 12.4-19.3mJy at 870 µm (L IR ∼ 10 13 L ; Liao et al. in preparation), along with five fainter companion SMGs that are located within the primary beam of our observations.The 18 primary SMGs represent a 100% complete selection considering both the original SCUBA-2 and the follow-up ALMA observations.Our findings are the following.
1.With about one hour of observations per target, we detect emission lines in CO or [CI] on 17 of the 18 primary SMGs.This high efficacy (> 90%) represents a major step forward in measuring spectroscopic redshifts of high-redshift luminous (L IR > 10 12 L ) dusty galaxies that are otherwise difficult to obtain from traditional optical and near-infrared spectrographs of 8-10 m class telescopes.
2. Among the five companion SMGs we obtain significant line detection from one, AS2COS0008.2.Together with the previously reported line detection toward AS2COS0001.2, we find that the velocity differences of these two pairs are both within 500 km s −1 , at the projected distances of 20-30 kpc, suggesting that they are gravitationally bound systems therefore confirmed physically associated pairs.Our results suggest that for SMGs with S 870 > 12.4 mJy, their brightest companion SMGs are physically associated with their corresponding primary SMGs in ≥ 40% of the time, suggesting that mergers play a role in the triggering of star formation in these exceptional sources.
3. The overall median redshift of the 18 primary SMGs is 3.3 ± 0.3, confirming that brighter SMGs are located at higher redshifts.By carefully comparing with various model predictions we find that although the significance differs our measured redshift distribution is higher than the predictions of those models, where the only model made with a complete suite of physically motivated treatments has the most significant disagreement, at 3.8 σ.We suspect that this may be because that the models lack halos with sufficiently massive gas reservoirs, which may be caused by either too efficient star formation or too much feedback at even earlier times.
4. By exploiting the rich ancillary data sets in the COSMOS field, we carefully model the foreground gravitational fields and find that only one of the 18 primary SMGs can be strongly lensed with a magnification µ > 2, suggesting a strongly lensed fraction of < 10%.Our results are consistent with previous studies focusing on Herschel-selected dusty galaxies, as well as the empirical model that interprets SMGs as proto-spheroidal galaxies.
5. With these high signal-to-noise spectra, we determine that about 90% of the primary SMGs have their emission lines better described with double Gaussian profiles.The median separation of the two line peaks of the best-fit double Gaussian models is 430±40 km s −1 , consistent with previous studies where redshifted and blueshifted components are spatially resolved.The average dynamical mass (3.5 ± 0.6 × 10 11 M ) estimated based on the velocity separation, together with the median line dispersion measured, puts our primary SMGs on similar mass-σ correlations found on local ETGs and high-redshift quiescent galaxies, corroborating the proposed evolutionary link among these populations.
6. Lastly, given the well-defined nature of our sample we estimate that the number density of our primary SMGs at z = 4 − 5.1 is 1 +0.9 −0.6 × 10 −6 cMpc −3 , after accounting for the duty cycle for which we assume a typical lifetime of 100 Myr.This suggests that our primary SMGs can be the progenitors of some z ∼ 3 − 4 quiescent galaxies, especially the most massive ones (> 10 11 M ).
We demonstrate the power of millimeter spectroscopic redshift surveys on a complete sample of bright SMGs in constraining theoretical models.In the forthcoming paper (Liao et al. in preparation) we will present the detailed analyses of the ISM properties of this effectively a well-defined sample of z ∼ 3 hyper luminous infrared galaxies (HyLIRGs; L IR 10 13 L ).These efforts are in line with future developments of ALMA, ngVLA, and AtLAST (Geach et al. 2019;Klaassen et al. 2020), where submillimeter and millimeter spectroscopic surveys are expected to bring fundamental breakthroughs to our understandings of the high-redshift, dust-enshrouded galaxies.

Chen et al.
We thank the reviewer for a constructive report that improves the manuscript.We would also like to thank Mathiew Bethermin, Caitlin Casey, Chris Hayward, Claudia Lagos, Chris Lovell, Mattia Negrello, Gergo Popping, Martin Sparre, and Jorge Zavala, for sharing their thoughts and catalogs about their models.We also thank Hsi-Wei Yen for the suggestions on the auto-masking approach, and Po-Feng Wu  The first and second columns after the ID column provide the reduced χ 2 values of each fit, and the last two columns show the preferred selection based on AICc and BIC, respectively.
Given their acceptable χ 2 we therefore adopt single Gaussian models for AS2COS0028.1 for their consistency and simplicity.
As a check, we perform simultaneous fitting on SMGs that have two lines detected in our data.In this exercise, each source has its redshift fixed for the two lines, and for the double Gaussian models the velocity separations of the two constituent Gaussian are also fixed.We find the results are consistent with our decisions.The final selected models are given in the last column of Table A.1, and they are further discussed in Section 4.4.

B. FURTHER COMPARISONS BETWEEN PHOTOMETRIC REDSHIFTS AND SPECTROSCOPIC REDSHIFTS
To gain more insights into the redshift assignment for the single line detected SMGs, we compare the adopted spectroscopic redshifts in Table 1 and the photometric redshifts provided by the two well tested OIR-base catalogs in the COSMOS field, COSMOS2015 (Laigle et al. 2016) and COSMOS2020 (Weaver et al. 2021).Focusing on the primary SMGs, matches to the two catalogs can be found on 14, where the corresponding matches are not exactly the same and most are found on SMGs with single line detection.The corresponding redshifts for the matched SMGs are plotted in Figure B.1 except for AS2COS0037.1 where no emission line is detected.Two versions of photometric redshifts are provided in the COSMOS2020 catalog and they are both shown.A few key features worth noting.First, the overall median in redshift difference is consistent with zero for all three sets of comparisons, suggesting that the redshift distribution discussed in Section 4.2 is not affected by the photometric catalogs adopted for determining redshifts for single line SMGs.Second, compared to Figure 7, OIR-base photometric redshifts tend to miss or underestimate redshifts for the highest redshift (z > 4) SMGs.This is expected as they tend to be faint or undetected in OIR images.Finally, AS2COS0009.1 has significantly higher photometric redshifts (∆z 1) in all three matches, suggesting that this source could be one of those that are expected to have incorrect redshift assignments in Table 1.Note AS2COS0090.1

Figure 1 .
Figure1.Predictions of median redshifts for SMGs with 850 µm flux density (S850) greater than certain flux cuts, based on empirical models of Cai-Negrello(Cai et al. 2013;Negrello et al. 2017),Casey-Zavala (Casey et al. 2018;Zavala et al. 2021), SIDES(Béthermin et al. 2017), the semi-analytical model SHARK(Lagos et al. 2020), and the hydro-dynamical simulation SIMBA(Davé et al. 2019) coupled with dust radiative transfer modeling POWDERDAY(Narayanan et al. 2021;Lovell et al. 2021).In the bottom of the figure we show the histograms normalized of three SMG samples that are constructed via a similar approach, meaning ALMA band 7 continuum follow-up observations on flux limited samples of single-dish detected submillimeter sources.The histograms are arbitrarily normalized to show contrasts of different samples.AS2COSMOS(Simpson et al. 2020) has significantly more brighter SMGs than AS2UDS(Stach et al. 2018) and ALESS(Hodge et al. 2013).The flux range presented in this work is indicated, which is ideal for redshift measurements thanks to its brightness and test model predictions.

Figure 3 .
Figure 3.An example of channel maps of AS2COS0023.1, each 20 on a side, where the velocities are referenced to the systemic redshift.The masked regions for tclean are enclosed within the white curves.This demonstrates the flexibility of auto-masking in tclean.

Figure 4 .
Figure 4. Example results of the curve-of-grown analyses, showing the curves converging (a), monotonically increasing (b) , and down turning (c), as a function of the size of the synthesized beam.Results from the single and double Gaussian models are both shown, as indicated in the legend.In panel (d) we show that by normalizing all curves to a beam factor of one (grey), the median curve (solid) with the bootstrapped uncertainties appears stable and converging, which is consistent with the shape of the synthesized beam (dashed) and showing that the non-converging behavior of individual targets could be due to larger-scale noise structures.We therefore in principle adopt the synthesized beam as the aperture to extract spectra and use this median curve to correct the measured flux densities to total.Note that this median curve is stable to within a few percent against the choice of aperture size to which all the curves are normalized.
Note: a Number of lines detected.ML stands for multiple lines, SL for single line, and NL for no line.b Redshift solutions for each line.c Redshift solutions for each source.d Mitsuhashi et al. (2021) e Frias Castillo et al. in prep.fDaddi et al. (2022)

Figure 7 .
Figure7.Top: Comparisons between photometric redshifts and spectroscopic redshifts on the 17 primary SMGs that are detected in lines, where the dashed line shows the one-to-one relation and the two dotted lines show ∆z ≡ |z photo − zspec| = 1, outside of which is considered outlier.The outliers are defined as those that have their ∆z > 1, meaning that if only single line detected it is assigned to an incorrect CO transition based on photometric redshifts.The ID numbers of the two outliers are indicated.The SMGs that have multiple lines detected are marked as filled circles and those with only one line detected marked as empty ones.The one that is boxed is AS2COS0001.1, due to the reason that its photometry could be contaminated by a nearby foreground lens galaxy.Bottom: Histogram of differences between photometric and spectroscopic redshifts (∆z), based on SMGs that have multiple lines detected.Excluding the two outliers, we find that the median redshift difference is ∆z/(1 + zspec) = 0.04 ± 0.04 with an bootstrapped error.

Figure 8 .
Figure8.Continuum images at 870 µm and 3 mm spectra of the five sets of paired SMGs.Top: Sub-arcsecond continuum images at 870 µm from AS2COSMOS(Simpson et al. 2020), with scale bars of 20 kpc marked based on the adopted redshifts in Table1for the primary SMGs.The images are uniformly and linearly scaled between -1 and 2 mJy and the color bar plotted on top.The SMGs are boxes in white and their IDs are indicated.The synthesized beams of the 3 mm spectra adopted for each pair are shown in the bottom left corners and the black contours are signal-to-noise ratios in 3, 4, 5, and 7 σ for the line emissions averaged over the FWHM based on the best-fit single Gaussian model profiles shown in the bottom spectra.Middle and Bottom: 3 mm spectra binned to ∼150 km s −1 per channel for the primary (middle panels) and secondary (bottom panels) SMGs.These spectra are extracted using apertures centered on their reported 870 µm locations(Simpson et al. 2020) with sizes and shapes of their corresponding synthesized beam.The velocities are referenced to the redshifts of the primary SMGs (Table1), and the 1 σ r.m.s.levels are plotted in gray.For the secondary SMGs, we obtain significant detection from AS2COS0008.2, marginal detection from AS2COS0001.2, and no detection from AS2COS0028.2, AS2COS0065.2, and AS2COS0090.2.There is no line detection from AS2COS0028.2 in both frequency ranges covering CO(3-2) and CO(4-3), and for demonstration purposes we only show the range of CO(3-2).The spectra of AS2COS0001.2 and AS2COS0008.2 are fitted with single Gaussian model profiles and the best fits are plotted as black curves.The same line for AS2COS0001.2 was more significantly detected by Jiménez-Andrade et al. (2020) using deeper NOEMA observations and the line peak deduced based on their results is shown in dashed blue line and marked as JA20.
Figure8.Continuum images at 870 µm and 3 mm spectra of the five sets of paired SMGs.Top: Sub-arcsecond continuum images at 870 µm from AS2COSMOS(Simpson et al. 2020), with scale bars of 20 kpc marked based on the adopted redshifts in Table1for the primary SMGs.The images are uniformly and linearly scaled between -1 and 2 mJy and the color bar plotted on top.The SMGs are boxes in white and their IDs are indicated.The synthesized beams of the 3 mm spectra adopted for each pair are shown in the bottom left corners and the black contours are signal-to-noise ratios in 3, 4, 5, and 7 σ for the line emissions averaged over the FWHM based on the best-fit single Gaussian model profiles shown in the bottom spectra.Middle and Bottom: 3 mm spectra binned to ∼150 km s −1 per channel for the primary (middle panels) and secondary (bottom panels) SMGs.These spectra are extracted using apertures centered on their reported 870 µm locations(Simpson et al. 2020) with sizes and shapes of their corresponding synthesized beam.The velocities are referenced to the redshifts of the primary SMGs (Table1), and the 1 σ r.m.s.levels are plotted in gray.For the secondary SMGs, we obtain significant detection from AS2COS0008.2, marginal detection from AS2COS0001.2, and no detection from AS2COS0028.2, AS2COS0065.2, and AS2COS0090.2.There is no line detection from AS2COS0028.2 in both frequency ranges covering CO(3-2) and CO(4-3), and for demonstration purposes we only show the range of CO(3-2).The spectra of AS2COS0001.2 and AS2COS0008.2 are fitted with single Gaussian model profiles and the best fits are plotted as black curves.The same line for AS2COS0001.2 was more significantly detected by Jiménez-Andrade et al. (2020) using deeper NOEMA observations and the line peak deduced based on their results is shown in dashed blue line and marked as JA20.
Figure9.Redshift distribution as a function of S870 based on uniformly selected SMG samples with ALMA follow-up observations.Spectroscopic redshifts of this work are plotted in black circles, and those for ALESS and AS2UDS subsamples are plotted in blue squares, which are reported by(Birkin et al. 2021).We also plot the spectroscopic redshifts of the SPT SMGs versus their intrinsic flux densities(Reuter et al. 2020).Typical uncertainties for the spectroscopic redshift samples are shown at the bottom right corner with matched colors.The background density contours show the distribution of photometric redshifts based on the three samples, ALESS(Simpson et al. 2014), AS2UDS(Stach et al. 2019), and AS2COSMOS(Ikarashi et al. 2022).The results of linear fittings on photometric redshifts and spectroscopic redshifts of the combined ALESS, AS2UDS, and our results are plotted as gray and blue bands, respectively, where the widths represent 68% confidence levels.The slope of the best fit model on the spectroscopic redshifts is 0.09±0.02,consistent with that of photometric redshifts and previous studies(Cowie et al. 2017;Stach et al. 2019;Simpson et al. 2020;Birkin et al. 2021).

Figure 10 .
Figure 10.Cumulative distributions of our measured redshifts in black and the distributions of each model catalog, summarized in Table 2, are plotted in each panel in blue with their associated model names indicated on top.The 10-90 percentile distributions for each curves are based on simulations described in Section 4.2.In each panel, the probability of finding p < 0.05 from a set of two-sample Kolmogorov-Smirnov tests based on simulations of the two comparison distributions are shown.

Figure 11 .
Figure 11.Thumbnail images in K-band (3.6 µm for AS2COS0031.1) of our 18 sample SMGs.The large dashed white circles with a radius of 30 and centered at our sample SMGs (magenta points) show the search area for constructing the foreground potential.Small unfilled boxes mark the neighboring foreground galaxies that have stellar masses > 10 10.5 M .The corresponding small values for each box are redshift in the left and stellar mass in logarithm base 10 in the right.

Figure 12 .Figure 13 .
Figure 12.Cumulative distributions of estimated lensing magnifications in black and those based on model predictions of SIDES and Cai-Negrello are plotted in blue with their corresponding model names indicated on top.The 10-90 percentile distributions for each curves are based on simulations done in a similar fashion as that described in Section 4.2.In each panel, the probability of finding p < 0.05 from a set of two-sample Kolmogorov-Smirnov tests based on simulations of the two comparison distributions are shown.

Figure 14 .
Figure14.Cumulative number counts of the parent AS2COSMOS sample in black circles, and the counts of strongly lensed (µ > 2) SMGs are plotted in red circles, which are scaled from the counts of the parent sample with a factor of 1/18, the strongly lensed fraction based on Table 3.The counts predictions based on Cai-Negrello model are plotted in black for total counts, blue for none strongly lensed counts, and red for strongly lensed counts.

Figure 15 .
Figure 15.Correlations between baryon mass (M baryon ) and velocity dispersion (σ).An overall median is shown in black circle for both parameters for our sample SMGs.A compiled sample of local ETGs including MASSIVE (Davis et al. 2019) and ATLAS 3D (Cappellari et al. 2013) is plotted in gray, and recent measurements of z = 0.6 − 1.0 and z ∼ 2 − 4 quiescent galaxies (QGs) are shown in green hexagons as 2D histogram (van der Wel et al. 2021; Wu et al. 2021) and blue squares(Stockmann et al. 2020;Valentino et al. 2020), respectively.We also include results of fainter SMGs that also have spectroscopic redshift measurements(Birkin et al. 2021).Our results are consistent with these plotted samples, corroborating the hypothesis that these populations are linked in evolution.
for sharing the stellar mass estimates of the LEGA-C sample.C.C.C. acknowledges support from the Ministry of Science and Technology of Taiwan (MOST 109-2112-M-001-016-MY3). IRS and AMS acknowledge support from STFC (ST/T000244/1).H. U. is supported by JSPS KAKENHI Grant No. 20H01953.AJB acknowledges funding from the "FirstGalaxies" Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 789056).The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under Grant No. 140.This paper makes use of the following ALMA data: ADS/JAO.ALMA#2019.1.01600.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ Software: astropy (Astropy Collaboration et al. 2013), casa (McMullin et al. 2007)

Table 1 .
Redshifts of the primary target SMGsIDz phot N a

Table 2 .
Model catalogs used for comparisons

Table A .
1. Best-fit model selection based on AICc and BIC Note: SG and DG are abbreviations for single Gaussian and double Gaussian models.