A dusty star-forming galaxy at z=6 revealed by strong gravitational lensing
Jorge A. Zavala Alfredo Montaña David H. Hughes Min S. Yun R. J. Ivison Elisabetta Valiante David Wilner Justin Spilker Itziar Aretxaga Stephen Eales Vladimir Avila-Reese Miguel Chávez Asantha Cooray Helmut Dannerbauer James S. Dunlop Loretta Dunne Arturo I. Gómez-Ruiz Michal J. Michalowski Gopal Narayanan Hooshang Nayyeri Ivan Oteo Daniel Rosa González David Sánchez-Argüelles Stephen Serjeant Matthew W. L. Smith Elena Terlevich Olga Vega Alan Villalba Paul van der Werf Grant W. Wilson Milagros Zeballos
astro-ph.GA

Dynamic band structure tuning of graphene moiré superlattices with pressure
    Cory R. Dean^1
    December 30, 2023
===========================================================================



  * Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Luis Enrique Erro 1, 72840, Puebla, Mexico
 
  * Department of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA 
 
  * CONACyT-Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, 72840, Puebla, Mexico
 
  * Department of Astronomy, University of Massachusetts, MA 01003, USA
 
  * European Southern Observatory, Karl Schwarzschild Strasse 2, Garching, Germany
 
  * Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
 
  * School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK
 
  * Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
 
  * Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
 
  * Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 70-264, 04510, CDMX, Mexico
 
  * Dept. of Physics & Astronomy, University of California, Irvine, CA 92697, USA
 
  * Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  
 
  * Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain 
 
  * Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, ul. Słoneczna 36, 60-286 Poznań, Poland
 
  * Department of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK
 
  * Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, The Netherlands 
 
 


 2

Since their discovery, submillimeter-selected galaxies<cit.> have revolutionized the 
field of galaxy formation and evolution. 
From the hundreds of square degrees mapped at
submillimeter wavelengths<cit.>,  
only a handful of sources have been confirmed to lie 
at z>5 (ref.<cit.>) and only two at z≥6 (ref.<cit.>).
All of these SMGs are rare examples of extreme starburst galaxies with star formation rates of
≳ 1000 M_⊙ yr^-1 and therefore are not representative of the general population
of dusty star-forming galaxies. Consequently, our understanding of the nature of these sources, 
at the earliest epochs, is still incomplete. Here we report the spectroscopic identification of a
gravitationally amplified (μ=9.3±1.0) dusty star-forming galaxy at z=6.027.
After correcting for gravitational lensing we derive an intrinsic less extreme star formation rate 
of 380±50 M_⊙ yr^-1 for this source, and find that its gas and dust properties are similar
to those measured for local Ultra Luminous Infrared Galaxies, extending the local trends to a 
poorly explored territory in the early Universe. The star-formation efficiency of this galaxy is
similar to those measured in its local analogues<cit.>, despite a ∼12 Gyr difference in cosmic time.


HATLAS J090045.4+004125 (α=09^h00^m45.8, δ=+00^∘41'23”; hereafter G09 83808, since it was detected in the GAMA 09hrs field) 
is part of a sub-sample of the Herschel ATLAS `500 μm-riser' galaxies<cit.> with  ultra-red
far-infrared (FIR) colours  of  S_500_μ m/S_250_μ m > 2 and S_500_μ m/S_350_μ m > 1, with a flux density threshold 
of S_500_μ m<80  mJy. 
The FIR colours of this source are consistent with thermal dust emission redshifted to z>4 and  represent a relatively simple selection criterion to find 
high-redshift galaxies. A similar selection  allowed  the identification of HFLS3<cit.>, an extreme starburst galaxy (even after corrected for 
gravitational amplification<cit.>) at z = 6.3, in the HerMES blank field survey<cit.>.

G09 83808 was observed, among other ultrared-Herschel dusty star-forming galaxies, as part of a follow-up program with the 
Large Millimeter Telescope Alfonso Serrano (LMT)
using the AzTEC camera, in order to obtain 
higher angular resolution (∼8.5 arcsec) continuum observations at 1.1 mm. A sub-sample of those galaxies detected as a single source in the AzTEC images 
(i.e. with no evidence of multiple components)
and with photometric redshifts of z>4, was selected for spectroscopic observations in the 3 mm band using the Redshift Search Receiver (RSR) on the LMT.
In the LMT/RSR spectrum of G09 83808 we identify three emission lines corresponding to ^12CO(6-5), ^12CO(5-4), and H_2O(2_11-2_02) (see Fig. 1).
Based on these lines we unambiguously determine the galaxy redshift to be z=6.0269±0.0006 (i.e. when the Universe was just 900 million years old).
Follow-up observations with The Submillimeter Array (SMA) telescope confirm this solution through the  detection of the redshifted [CII] ionized carbon line at 270.35 GHz (see Fig. 1).



High-angular resolution observations (0.24 arcsec ×0.13 arcsec, corresponding to a physical scale of ∼1 kpc at this redshift)
taken with the Atacama Large Millimeter/submillimeter Array (ALMA; see Methods section)
at ∼890 μ m reveal a double arc structure (in a partial Einstein ring configuration of radius ∼ 1.4 arcsec)
around a foreground galaxy at z=0.776 (see Fig. 2), implying 
strong gravitational amplification of the high-redshift background galaxy. 
Using these ALMA continuum observations to constrain the effects of gravitational lensing, modelling directly the visibilities in the uv plane
(see Methods section for additional details), we derive a gravitational amplification factor of μ=9.3±1.0. 
This amplification factor is used to derive the intrinsic physical properties of G09 83808.



Using the Herschel 250, 350, and 500 μ m photometry<cit.>, combined with the SCUBA-2 850 μ m<cit.> imaging and our AzTEC 1.1 mm
observations (see Table 1), we model the 
continuum spectral energy distribution (SED; see Figure 3). We estimate an infrared (IR, 8-1000 μ m) luminosity, L_ IR, of 3.8±0.5×10^12 L_⊙ 
(corrected for gravitational magnification) which implies a dust-obscured star formation rate (SFR) of 380±50 M_⊙ yr^-1 
(see Methods section for more information). This means that G09 83808 is a member of the Ultra Luminous Infrared Galaxy (ULIRGs<cit.>) population.  
This is one of the first SMG with an unambiguous spectroscopic redshift in this luminosity range at z≳5, 
lying between the extreme obscured starbursts<cit.> 
(≳ 1000 M_⊙ yr^-1) discovered at submm wavelengths and the UV/optical selected star-forming galaxies with follow-up detections at submm
wavelengths<cit.> (≲ 100 M_⊙ yr^-1).  Within this luminosity range only a handful of galaxies at z>6 are known, which 
were recently discovered around quasars<cit.> thanks to the serendepitously detection of a single emission line associated to [CII].



Although these galaxies are unreachable with the current generation of submm wide-area surveys<cit.> without the 
benefit of gravitational amplification, they can be found in the deepest
surveys recently achieved  with ground-based telescopes, such as the James Clerk Maxwell Telescope (JCMT) SCUBA-2 Cosmology Legacy Survey (S2CLS). However,
none of them has yet been spectroscopically confirmed. With the caveat of using the position of the dust SED peak as an estimation of redshift,
a study based on S2CLS observations<cit.> has derived a comoving space density of 3.2×10^-6  Mpc^-3 for sources with 300< SFR < 1000 M_⊙ yr^-1
at 5<z≲6 (i.e. in the range probed by our galaxy). With a duty-cycle correction of ≈ 40  Myr, 
as the gas depletion time scale measured for G09 83808 (see below) and other galaxies<cit.>, we estimate the corrected comoving space density 
of this population of galaxies to be  ≈2×10^-5  Mpc^-3, which perfectly matches that of massive quiescent galaxies at z≈3-4 (refs.<cit.>).
This suggests, that these ULIRG-type galaxies at 
5≲ z≲6 are the progenitors of these quiescent galaxies, which cannot be explained only by the rare extreme starburst galaxies (like HFLS3), 
since they are an order of magnitude less abundant<cit.>.



Based on the CO lines detected in the LMT/RSR spectrum we derive a molecular gas mass of M(H_2)=1.6±0.6×10^10 M_⊙ (see Methods section for 
details). This implies a gas depletion 
timescale of M(H_2)/SFR≈40 Myr, consistent with the value found for other SMGs at lower redshifts with ULIRG-luminosity<cit.>. 
G09 83808 shows a remarkable large gas mass fraction of f_ gas=M_ H_2/M_ dyn∼ 60% (see Methods secction), among the largest measured for
star forming galaxies at z≈2-3 (ref.<cit.>). The CO(6-5)/CO(5-4) line luminosity ratio of 0.4±0.1 
is in agreement with local ULIRGs (although lower than the average<cit.>), and implies  a CO ladder peaking at 
J≤5 (i.e. less excited than AGN-dominated galaxies<cit.>). These two CO transitions, as well as the H_2O line, 
lie (within the error bars) on their respective FIR/IR-line luminosity relations (L_ FIR∝ L_ CO(6-5)^0.93, L_ FIR∝ L_ CO(5-4)^0.97, and 
L_ H_2O∝ L_ IR^1.16) found for local ULIRGs and lower redshifts SMGs<cit.>. 
The star-formation efficiency (SFE) of our galaxy, estimated through the L'_CO-L_ IR relation (which describes the relationship
between the luminosity due to star formation and the gas content), is similar to local (U)LIRGs (see Fig. 4). Then, the same SFE
can be found across several decades of molecular gas masses from z=6 to z∼0 (i.e., during the last 12.8 Gyr of the Universe).
In addition, the estimated dust mass of M_d=1.9±0.4×10^8 M_⊙ results in a gas-to-dust ratio, δ_ GDR, of 80±30.
This is in agreement with the value estimated for HFLS3<cit.> and also with local (U)LIRGs<cit.> (δ_ GDR=120±28). 



The luminosity of the [CII] ionized carbon line detected with the SMA is 1.3±0.4×10^9 L_⊙ which corresponds to a [CII]/FIR ratio of 
3.4±1.1×10^-4, a value that is among the lowest measured for local (U)LIRGs and SMGs. As shown in Figure 4, our source follows the same [CII]
deficiency trend measured for local LIRGS<cit.> extending it to L_FIR≳ 10^12 L_⊙ and up to z=6. 
The [CII]/FIR ratio of G09 83808 is also consistent with the lowest values measured for lower-redshift SMGs and lies on a region where SMGs and AGN-host galaxies converge (Fig. 4).
It may be the case that other SMGs suffer from gravitiational amplification, which could help to reduce the large scatter 
since many of these galaxies should fall along the LIRG relation when corrected for magnification. However, the intrinsic scatter in the relation is high<cit.>, even for the 
local sample, and therefore, larger samples of SMGs are required to derive conclusions about the origin of the [CII] deficiency.



We confirm the existence of ULIRG-like galaxies within the first billion years of Universe's history. These sources may be more
representative of the dusty star-forming galaxy population at these epochs than the extreme starbursts previously discovered. 
Four emission-line-selected galaxies with similar luminosities and redshifts have been recently found around quasars<cit.>
(with the caveat of using just one line for redshift determination), however, the properties of these sources  may be affected by 
the companion quasar and therefore not representative of the whole population. Although G09 83808 shows similar properties to those 
measured in lower-redshift SMGs, its higher dust temperature (T_d=49±3 K) and compact morphology (R_1/2=0.6±0.1 kpc) resemble that of local ULIRGs.
For comparison, typical UV/optically-selected star-forming galaxies at z∼6 have SFRs ∼10 times lower and radii ∼1.7 times larger than G09 83808<cit.>.
This study is hence crucial for understanding the evolutionary path of SMGs and their link with local galaxies. 
Although a larger sample is needed to statistically estimate the properties of these sources and their contribution to the cosmic star formation history,
this galaxy suggests that star formation in dusty star-forming galaxies has been driven by similar physical processes during the last ∼12.8 Gyr .




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  * We thank Ian Smail for insightful comments that improved the quality of the paper. 
 JAZ acknowledges support from a mexican CONACyT studentship.
 RJI, LD and IO acknowledge support from ERC in the form of the Advanced Investigator Programme, 321302, COSMICISM.
 LD additionally acknowledges support from the ERC Consolidator Grant CosmicDust.
 HD acknowledges financial support from the Spanish Ministry of 
 Economy and Competitiveness (MINECO) under the 2014 Ramón y Cajal 
 program MINECO RYC-2014-15686.
 MJM acknowledges the support of the National Science Centre, Poland
through the POLONEZ grant 2015/19/P/ST9/04010 and  the European Union's Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No.
665778.
This work would not have been possible without the long-term
financial support from the Mexican CONACyT
during the construction and early operational phase of the
Large Millimeter Telescope Alfonso Serrano, as well as support
from the  US National Science Foundation via the University
Radio Observatory program, the Instituto Nacional de Astrofísica,
Óptica y Electrónica (INAOE), and the University of Massachusetts
(UMass). The Submillimeter Array is a joint project between the Smithsonian 
Astrophysical Observatory and the Academia Sinica Institute of Astronomy and 
Astrophysics and is funded by the Smithsonian Institution and the Academia 
Sinica. 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.


  Author Contributions JAZ led the scientific analysis and the writing of the paper, 
as well as the SMA follow-up proposal. RJI, EV, SE, AC, HD, JSD, LD, MJM, SS, IS, MWLS, and PW 
have contributed to the original Herschel proposals and source selection of the red sources, 
where this source was originally identify. AM, DHH, EV, IA, VAR, MC, DRG, ET, and OV  performed the selection of the 
sample for the LMT observations and lead the LMT proposals. MSY, GN, FPS, DS, GW, DSA, AV, and MZ carried out 
LMT data reduction and  interpretation. DW, MY, and AIGR assisted with the SMA observations and data reduction. 
JS, IO, HN have contributed to the data analysis and to fitting and modeling the results.
All the authors have discussed and contributed to this manuscript.

 
  Correspondence Correspondence and requests for materials
should be addressed to Jorge Zavala (email: zavala@inaoep.mx).


  Competing Interests The authors declare that they have no
competing financial interests.





















2


§ OBSERVATIONS AND DATA REDUCTION





 §.§ LMT observations




Continuum and spectroscopic observations were obtained using the Large Millimeter Telescope (LMT<cit.>, PI: D. Hughes), 
located on the summit of Volcán Sierra Negra 
(Tliltépetl), Mexico, at ∼4600 m.a.s.l. Observations were carried out during the Early Science Phase of the telescope using the 
1.1 mm continuum camera, AzTEC<cit.>, and the 3 mm spectrograph, Redshift Search Receiver (RSR<cit.>). During these observations only the inner
32-m diameter region of the telescope active surface was illuminated, which provided an effective beam size of ≈8.5 arcsec
at 1.1 mm and between 20-28 arcsec in the RSR 3 mm window (75 GHz - 110 GHz). 



AzTEC observations were performed on 2014 November 10 with an opacity of τ_225=0.07 and total on-source integration time of 11 min. Data reduction
were done following the AzTEC Standard Pipeline<cit.>. G09 83808 was detected with a 
S/N ≈20 with a flux density of S_1.1 mm=20.0±1.0 mJy. RSR observations were  subsequently taken at the AzTEC position  in 
two different periods: February 2016 and February 2017, along five different nights with an opacity range of τ_225=0.05-0.15 and a total integration 
time of 8 hrs. Pointing observations on bright millimetre sources were done every hour. Data reduction was performed using the Data Reduction
and Analysis Methods in Python (DREAMPY). The final spectra were obtained by averaging all scans using 1/σ^2 weights after flagging bad data. Finally,
to convert from antenna temperature units to flux, a factor of 7 Jy K^-1 was used<cit.>. 
The final spectrum shows three lines detected at 
S/N ≳5 associated to CO(6-5), CO(5-4) and H_2O(2_11-2_02) at z=6.0269. A cross-correlation template analysis<cit.>
also identifies this redshift as the best solution with a S/N =9.1. 
Figure 1 shows the final spectrum after a Savitzky-Golay filter<cit.> has been applied for better visualization (the filter does not modified any of the properties 
of the detected lines).



At the redshift of our source the [CII] 158 μm line (see below) falls within the AzTEC band pass and then contributes to the total flux density
measured at 1.1 mm. However, the contamination from the line is measured to be less than 2 per cent. Even if the [CII] line luminosity was as high as 1 per cent of 
the total IR luminosity, the contamination to the AzTEC measurement would be only ∼6 per cent, which is similar to the absolute flux calibration uncertainty. 
Therefore, and at least for this source, the contamination of the emission line to the  1.1 mm continuum flux density is less important than anticipated<cit.>.





 §.§ SMA observations




G09 83808 was observed with the Submillimeter Array (SMA, PI: J. Zavala) on Mauna Kea,
Hawaii, on 2017 April 03. The weather conditions were good, with an average 
atmospheric opacity of τ_225=0.07 and stable phase. The seven available
array antennas were in a compact configuration that provided baseline lengths
from 8 to 77 meters. The `345' receiver set was
tuned to provide spectral coverage ±(4-12) GHz from a LO frequency of
277.5 GHz, specifically to span a broad range around the estimated (redshifted)
[CII] line frequency of ∼270.5 GHz in the lower sideband. The SWARM
correlator provided uniform channel spacing of 140 kHz (∼0.16 km s^-1)
over the full bandwidth. The usable field of view is set by the FWHM primary
beam size of ∼47 arcsec at this
frequency.



The basic observing sequence consisted of a loop of 2 minutes each on the
gain calibrators J0909+013 (1.57 Jy) and J0831+044 (0.47 Jy) and
17.5 minutes on G09 83808. The track spanned an hour angle range of
-0.8 to 4.8 for the target source. Passband calibration was obtained
with observations of the strong quasar 3C279. The absolute flux scale was
set using observations of Callisto, with an estimated accuracy of 20%.
All of the basic calibration steps were performed using standard procedures
in the MIR software package. The calibrated visibilities were exported to
the MIRIAD software package for imaging and deconvolution. Within MIRIAD,
the task uvaver was used to combine the 4 correlator windows of the
lower sideband and to resample the visibilities to 50 km s^-1 spectral
resolution. The task uvlin was used for continuum subtraction, using
a linear fit to line-free channel ranges in the band. The task invert
provided Fourier inversion for both continuum and spectral line imaging,
followed by clean for deconvolution. The synthesized beam size obtained
with natural weighting was 2.5^”×2.3^”, p.a. 82^∘ for
the spectral line data cube, with rms noise 7.1 mJy per 50 km s^-1 bin.
The final spectrum (see Fig. 1) was then extracted from a rectangular region that
comprise all the continuum emission. We measured the continuum flux density of the 
source to be 21.5±3 mJy, in very good agreement with the AzTEC photometry.






 §.§ ALMA observations




The ALMA high-resolution 870  μ m observations used in this work were 
taken on 31 August 2015 (project 2013.1.00001.S, PI Rob Ivison; ref.<cit.> ), when the array was in a 
relatively extended configuration with baselines up to 1.6   km. The default
continuum spectral configuration was used, covering [335.49 - 339.49]  GHz and 
[347.49 - 351.49]  GHz. The data were calibrated using the ALMA pipeline, with no 
further manual flagging required. The calibrated visibilities were imaged by using Briggs 
weighting with robust = 0.5, which is a good compromise between sensitivity and angular 
resolution. The beam size is then ∼ 0.12^” and the continuum sensitivity is 
r.m.s.∼ 0.1   mJy   beam^-1. The visibility weighting in ALMA data is
generally only correct in a relative sense, while our subsequent lens modeling procedure 
(see `Lensing Model' below) requires an absolute estimate of the noise in the data. The data weights
are determined by differencing successive visibilities on the same baseline, polarization,
and frequency baseband. The ALMA data also serendipitously cover the frequency of the 
redshifted 122  μ m [NII] line; this line is not detected at >3σ significance.





§ LENSING MODEL




The lens model was created using the publicly-available  code<cit.>; details of the code 
are given in that work. Briefly, the lens mass profile is parameterized as a Singular Isothermal Ellipsoid, and the 
background source is modeled with a single elliptical Sérsic profile. The parameter space is explored using a 
Markov Chain Monte Carlo sampling method, generating a model lensed image at each proposed combination of lens and 
source parameters. The redshift of both sources is fixed at z=0.776 (based on X-Shooter/VLT observations<cit.>) and z=6.027, respectively. 
Because pixel values in interferometric images are correlated and subject to difficult-to-model
residual calibration errors, the proposed model image is inverted to the visibility domain and sampled at the uv 
coordinates of the ALMA data. We also allow for 
residual antenna-based phase calibration errors in the model which 
could be due to, for example, uncompensated atmospheric delays. The phase shifts of all antennas are <10 deg,
indicating that no significant antenna-based calibration problems remain. 



The lensed emission is reasonably well-fit by a single background Sérsic component, leaving peak residuals of
∼4σ (the source is detected at peak significance ∼20σ). These residuals may indicate that either the lens,
source, or both are more complex than the simple parametric forms we have assumed. We have verified that an additional 
background source component is not statistically motivated. The best-fit magnification of the source is 
μ_890μ m = 9.3 ± 1.0, with an intrinsic flux density S_890 μ m = 4.3 ± 0.5 mJy and half-light
radius 0.10 ± 0.01” (=0.6±0.1 kpc). This compact morphology resembles the sizes found for local ULIRGs<cit.> (∼0.5 kpc), 
which are smaller than the typical values in SMGs (∼1.8 kpc, ref:<cit.>).





§ SED FITTING AND DUST PROPERTIES




We fit different galaxy SED templates to the photometry of G09 83808 through a χ^2 minimization method.
We include the SED template of Arp220<cit.>, Cosmic Eyelash<cit.> (SMM J2135-0102), two average SMGs templates<cit.>,
and finally a composite SED of 24 μm-selected star-forming galaxy<cit.>. 
All the SED templates were fixed at z=6.027.
The Arp220 SED template gives us the best fit with χ_ red^2=0.7. Using this template we derive an IR 
(8-1000μ m) luminosity of 3.8±0.5×10^12 L_⊙ and a FIR (42.5-122.5μ m) luminosity of 2.3±0.3×10^12 L_⊙ 
(both corrected for gravitational amplification) . For comparison, if we adopt instead a SMGs average 
template (χ_ red^2=1.2) we obtain L_IR=3.0±0.4×10^12 L_⊙, which is in good agreement with the value derived using 
the Arp220 template. Using Kennicutt standard relation<cit.> for a Chabrier initial mass function (IMF)<cit.>, this IR luminosity corresponds
to a star formation rate (SFR) of 380±50 M_⊙ yr^-1, or to 570±70 M_⊙ yr^-1 if the most recent relation<cit.> is used. 
If we adopt instead the Kennicutt calibration<cit.> for a 
Salpeter IMF<cit.>, the SFR increases to 640±90 M_⊙ yr^-1, still below the range probed by other SMGs at z≳5.



We also use a modified blackbody function to fit our photometric measurements  described by

    S_ν∝{1- exp[-(ν/ν_0)^β]}B(ν,T_ d),

where S_ν is the flux density at frequency ν, ν_0 is the rest-frame frequency at which the emission becomes optically thick,
T_ d is the dust temperature,  β is the emissivity index, and B(ν,T_ d) is the Planck function at 
temperature T_d. To minimize the number of free parameters, the emissivity index is fixed (previous observational works 
suggest β=1.5-2; refs.<cit.>), as well as ν_0=c/100 μ m
(refs.<cit.>), where c is the speed of light. From the best fit (χ^2≈1.1) we derive T_d=49±3 K for 
β=1.8 and  T_d=52±3 K for β=1.5. For these dust temperatures and at the redshift of our source the CMB effects<cit.>
are not significant (Δ T≲1 K).



Assuming the dust  is isothermal, the dust mass, M_ d, is estimated from 

    M_ d=S_ν/(1+z) D_L^2/(1+z)κ_ν B(ν,T_ d),

where S_ν is the  flux density at frequency ν, κ_ν is the dust
mass absorption coefficient at ν, T_ d is the dust temperature,
and B(ν_,T_d) is the Planck function at temperature T_ d. The dust mass 
absorption follows the same power law as the optical depth, κ∝ν^β. 
Assuming normalization of κ_ d(850μ m)=0.07 m^2 kg^-1 (ref.<cit.>) and a dust temperature of 49±3 K,
we estimate a dust mass of M_d=1.9±0.4×10^8 M_⊙ after correcting for the CMB effects<cit.> (although this
correction is less than 5 per cent). These calculations do not include the uncertainties of the dust mass absorption coefficient, 
which could be at least a factor of 3 (ref. <cit.>). If we use instead a lower dust temperature of 35 K, the dust mass increases 
by a factor of ∼2.



We also fit the observerd photometry with the MAGPHYS<cit.> SED modelling code finding consistent results, within the error bars, 
with median values of SFR=360^+80_-70 M_⊙ yr^-1, L_IR=4.5±0.7×10^12 L_⊙, T_d=40^+4_-2 K, and M_d=4.2±0.7×10^8 M_⊙.





§ SPECTRAL LINE PROPERTIES




We calculate the line luminosity for each detected line following the standard relation<cit.> described by:

    L'_ CO=3.25× 10^7 S_COΔ V ν_obs^-2 D^2_L (1+z)^-3,

where L'_ CO is the line luminosity in K km s^-1 pc^2,  S_COΔ V is the 
velocity-integrated line flux in Jy km s^-1, ν_obs is the observed central frequency 
of the line in GHz and D_L is the luminosity distance in Mpc. The integrated flux, S_COΔ V,
is calculated as the integral of the best-fit Gaussian distribution, and its associated uncertainty 
through Monte Carlo simulations taking into account the errors in the Gaussian parameters (i.e. peak 
flux density and line width). To estimate the line luminosity in L_⊙, we use 
L=3×10^-11ν_r^3L', where ν_r is the rest frequency of the line<cit.>. All properties are summarized in
Table 1.





§ CO(1-0) LINE LUMINOSITY AND MOLECULAR GAS MASS




The molecular gas mass, M(H_2), can be derived using the CO luminosity to molecular gas mass conversion factor, α, 
following the relation

    M(H_2)=α L'_CO(1-0).




For the  L'_CO(1-0) line luminosity we adopt the average value of L'_CO(1-0)=2.0±0.8×10^10 K km s^-1 pc^-2
extrapolated from our CO(6-5) and CO(5-4) transitions and correcting for gravitational amplification. The extrapolation was done 
using average brightness ratios found for lower-redshift SMGs<cit.> (L'_CO(5-4)/L'_CO(1-0)=0.32±0.05, L'_CO(6-5)/L'_CO(1-0)=0.21±0.04),
this sample includes galaxies with similar luminosities to G09 83808 and are in agreement with those found for local ULIRGs<cit.> (within the large scatter).
On the other hand, if we use the relationship between the Rayleigh-Jeans specific luminosity and CO(1-0) luminosity<cit.>, 
L'_CO(1-0) [K km s^-1pc^2]=3.02×10^-21 L_ν [ erg s^-1 Hz^-1], we obtain a consistent line luminosity of 1±0.1×10^10 K km s^-1 pc^-2 (assuming 
a mass-weighted dust temperature of 35 K, which is different from the luminosity-weighted dust temperature determined from the SED fitting<cit.>).
Using the former value  and α=0.8 M_⊙ (K km s^-1 pc^-2)^-1,
which is appropriate for starburst galaxies<cit.> (although some studies suggest larger values<cit.>),
we derive a molecular mass of M(H_2)=1.6±0.6×10^10 M_⊙. 





§ DYNAMICAL MASS AND GAS MASS FRACTION




Dynamical mass has been derived using the `isotropic virial estimator', which has been shown to be appropriate for lower-redshift SMGs<cit.>:

    M_ dyn[M_⊙]=2.8×10^5 Δν_ FWHM^2[ km s^-1] R_1/2[ kpc],

where Δν_ FWHM is the integrated line FWHM, which has been assumed to be 400 km/s (as the average between the CO and [CII] lines), and R_1/2 
is the half-light radius of ∼0.6 kpc (derived from the lesing model of the continuum emission). This results in a dynamical mass of 
M_ dyn=2.6×10^10 M_⊙. 
Using this estimation we calculate a gas mass fraction of f_ gas=M_ H_2/M_ dyn≈60%.
This constrain the CO luminosity to molecular gas mass conversion factor to α≲1.4 M_⊙ (K km s^-1 pc^-2)^-1, otherwise the molecuar gas
mass would exceed the dynamical mass.


 
  Data Availability The datasets generated and analysed during this study are available from the corresponding 
 author on reasonable request. The ALMA observations presented here are part of the project 2013.1.00001.S  and the SMA
data correspond to the project 2016B-S078.


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