Dynamical properties of z \sim 4.5 dusty star-forming galaxies and their connection with local early type galaxies
Francesca Rizzo, Simona Vegetti, Filippo Fraternali, Hannah Stacey, Devon Powell
MMNRAS , 1–17 (2020) Preprint 12 February 2021 Compiled using MNRAS L A TEX style file v3.0
Dynamical properties of z ∼ . dusty star-forming galaxies andtheir connection with local early type galaxies Francesca Rizzo , , , ★ Simona Vegetti , Filippo Fraternali , Hannah Stacey , Devon Powell Max-Planck Institute for Astrophysics, Karl-Schwarzschild Str. 1, D-85748, Garching, Germany Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200, Copenhagen N, Denmark Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK-2100 Copenhagen Ø, Denmark University of Groningen, Kapteyn Astronomical Institute, Postbus 800, 9700 AV Groningen, The Netherlands
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
There is a large consensus that gas in high- 𝑧 galaxies is highly turbulent, because of a com-bination of stellar feedback processes and gravitational instabilities driven by mergers andgas accretion. In this paper, we present the analysis of a sample of five Dusty Star Form-ing Galaxies (DSFGs) at 4 (cid:46) 𝑧 (cid:46)
5. Taking advantage of the magnifying power of stronggravitational lensing, we quantified their kinematic and dynamical properties from ALMAobservations of their [CII] emission line. We combined the dynamical measurements obtainedfor these galaxies with those obtained from previous studies to build the largest sample of 𝑧 ∼ . 𝑉 / 𝜎 ,between 7 to 15. The relation between their velocity dispersions and their star-formation ratesindicates that stellar feedback is sufficient to sustain the turbulence within these galaxies andno further mechanisms are needed. In addition, we performed a rotation curve decompositionto infer the relative contribution of the baryonic (gas, stars) and dark matter components to thetotal gravitational potentials. This analysis allowed us to compare the structural properties ofthe studied DSFGs with those of their descendants, the local early type galaxies. In particular,we found that five out of six galaxies of the sample show the dynamical signature of a bulge,indicating that the spheroidal component is already in place at 𝑧 ∼ . Key words: galaxies: evolution – galaxies: high-redshift – galaxies: ISM – galaxies: kinematicsand dynamics – submillimetre: galaxies – gravitational lensing: strong
Within the framework of current galaxy formation and evolutionarymodels, galaxies grow by acquiring material through gas accretionand mergers (e.g., Dekel et al. 2009; Rodriguez-Gomez et al. 2016;Naab & Ostriker 2017). Feedback processes driven by active galac-tic nuclei (AGN) or star formation are able to temper the growthof galaxies through the heating or expulsion of gas (Hopkins et al.2012; Silk 2013; Nelson et al. 2019a). On the other hand, many ofthese feedback processes operate on physical scales that are wellbelow the resolution of current cosmological simulations, and are,therefore, usually parametrized using simple sub-grid prescriptionswhich are mostly calibrated to match observations in the local Uni-verse (e.g., Crain et al. 2015; Kim & Ostriker 2018; Vogelsbergeret al. 2020).Observational evidence able to give a consistent and quantitativepicture of how galaxies grow and evolve across cosmic time is stilllacking. For example, the importance of mergers in driving the stel- ★ E-mail: [email protected] lar mass growth, as well as in determining the resulting kinematicand chemical properties of galaxies, is still a matter of debate (e.g.Oesch et al. 2010; Satyapal et al. 2014; Eliche-Moral et al. 2018).The influence of feedback processes (e.g., outflows, turbulence),mergers and gas accretion in regulating the growth of stellar mass isobservationally challenging to evaluate, even in the local Universe,since complex physical mechanisms operating at different scalesneed to be identified and constrained (e.g., Cicone et al. 2016; Con-cas et al. 2017; McQuinn et al. 2019). Moreover, at high redshift,these challanges are exacerbated due to the limited angular reso-lution and signal-to-noise ratio (SNR) of the current observations(e.g., Chisholm et al. 2017; Stanley et al. 2019; Ginolfi et al. 2020).One of the ways to partly overcome these observational limitationsis to target strongly gravitationally lensed galaxies. The magnifi-cation provided by gravitational lensing increases the solid angleof background sources and hence their observed total flux (Schnei-der 2006). As a result, distant objects can be observed with eitherincreased spatial resolution (for spatially-resolved observations) orincreased SNR (for unresolved observations).Recently, Rizzo et al. (2020) showed the power of high resolu- © a r X i v : . [ a s t r o - ph . GA ] F e b F. Rizzo et al. tion observations in unveiling the dynamical properties of a lenseddusty star-forming galaxy (DSFG), SPT0418, at 𝑧 = .
2. In particu-lar, they found that SPT0418-47 has dynamical properties similar tothose of local spiral galaxies: it is rotationally supported and has alow level of turbulence, that is, it is dynamically cold. These featuresindicate that, unexpectedly, the high star-formation rate (SFR) andthe gas fraction measured for this DSFG do not drive high turbulentmotions nor affect the stability of the galaxy. Fraternali et al. (2020)obtained a similar result for two non-lensed DSFGs at 𝑧 ≈ .
5. Theglobal SFR and interstellar medium (ISM) properties of these threegalaxies are typical of the population of DSFGs (Hodge et al. 2015;Aravena et al. 2016; Gómez-Guijarro et al. 2018), suggesting thatsuch systems are common among star-forming galaxies at high red-shifts. However, reproducing the existence of galaxies with such alarge rotational velocity, SFR and content of cold gas remains chal-lenging for most numerical simulations and semi-analytic models(e.g. Dekel & Burkert 2014; Zolotov et al. 2015; Pillepich et al.2019; Dekel et al. 2020). Robust observational constraints on thespatially-resolved kinematic properties of star-forming galaxies at 𝑧 (cid:38) 𝑧 =
4, GN20, using CO observations fromthe Very Large Array. However, the long integration times (e.g., ≈
120 hours for GN20) needed to spatially resolve the faint COtransitions made very challenging the extension of this analysis toa large sample of 𝑧 (cid:38) 𝑧 ∼ 𝑃 / → 𝑃 / transition at 1900.5469 GHz (157.74 𝜇 m,[CII]) of the ionized carbon C+ (e.g., De Breuck et al. 2014; Joneset al. 2017; Smit et al. 2018; Neeleman et al. 2020; Rizzo et al.2020; Fraternali et al. 2020). The 158 𝜇 m [CII] emission line is,in fact, a powerful tool to investigate the gas physical conditions inthe distant Universe: it is typically the brightest fine-structure lineemitted in star-forming galaxies, representing ∼ . (cid:46) 𝑧 (cid:46)
5, we also gain insights into the formation of massivequiescent galaxies at lower redshift. Stellar population studies oflocal early type galaxies (ETGs) have shown, indeed, that morethan half of the stars in the most massive galaxies were formed at 𝑧 (cid:38) (cid:46) 𝑧 (cid:46) 𝑧 (cid:38) 𝑧 (cid:38) (cid:46) 𝑧 (cid:46) Λ CDM cosmology, with Hubble constant 𝐻 = 67.8 km/s/Mpc,matter density Ω m = 0.308, and vacuum energy density Ω Λ = 0.691from Planck Collaboration et al. (2016). 𝜇 m [C II] emission line: a tracer of the gaskinematics at high- 𝑧 The 158- 𝜇 m [CII] emission line can trace multiple phases of theISM, including the warm ionized, the warm and cold neutral atomic,and the dense molecular medium (Stacey et al. 2010b), due to thelower ionization potential of 11.3 eV of the atomic carbon withrespect to HI. However, several studies (Rigopoulou et al. 2014;Cormier et al. 2015; De Breuck et al. 2019) found that more than60 percent of the [CII] emission originates in the photodissociationregions, the external layers of molecular clouds heated by the far-ultraviolet photons emitted from OB stars. In these regions, bothatomic and molecular hydrogen, as well as electrons, can collision-ally excite the ground state of C+ ions producing the [CII] emissionline. Furthermore, the molecular gas can be an efficient emitterof [CII] rather than CO (Pineda et al. 2013; Nordon & Sternberg2016; Glover & Smith 2016). Specifically, in low-metallicity envi-ronments or the outer regions of molecular clouds, H self-shieldsand survives while CO can be easily photodissociated into C andC+ (e.g., Papadopoulos et al. 2002; Wolfire et al. 2010; Maddenet al. 2020). This wide range of physical conditions makes [CII] anexcellent tracer of the kinematics of high- 𝑧 star-forming galaxiesover large areas of their discs. The targets in this paper are five gravitationally lensed DSFGs (Ta-ble 1). We collect the sample by selecting from the ALMA publicarchive star-forming galaxies at 4 (cid:46) 𝑧 (cid:46) (cid:46) . (cid:46)
40 km/s) resolutions high enough to resolve theemission of the 158- 𝜇 m [CII] emission line (Table 1). In addition,we select observations with a median (over the spectral channels)SNR of the order of 10. The modelling code that we will apply tothese data requires, indeed, fairly high SNRs (Rizzo et al. 2018),since it fits the data in all spectral channels, without requiring anyprojections to first and second moment maps. This approach is fun-damental to obtain robust kinematic measurements, not affected by MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies the so-called beam smearing effect, which may cause strong bias tolow values of the rotation velocities and high values of the veloc-ity dispersions (Di Teodoro & Fraternali 2015). Furthermore, thesehigh SNRs guarantee that no spatial binning is required, so that wecan fully take advantage of the angular resolution of ALMA andprobe the kinematics of our sample on sub-kpc scales.The five lensing system of our selected sample were identified in theSouth Pole Telescope (SPT) survey (Carlstrom et al. 2011; Vieiraet al. 2010, 2013; Reuter et al. 2020) and have existing ancillaryspectroscopic and imaging data in the sub-mm and far-infraredwavelength range, from which the redshifts of the background galax-ies (Table 1) and the infrared luminosities (see Section 4.3) werederived (e.g., Weiß et al. 2013; Strandet et al. 2016; Reuter et al.2020; Aravena et al. 2016). The observations for each target have two spectral windows cover-ing the redshifted rest frequency of the [CII] line and two spectralwindows for the continuum. Each spectral window has 240 chan-nels and a 1.875 GHz bandwidth. In this paper, we make use ofthe calibrated measurement sets provided by the European ALMARegional Centre (Hatziminaoglou et al. 2015), that calibrated theraw visibility data using the ALMA pipeline in the CASA package(McMullin et al. 2007). These data were then inspected to confirmthe quality of the pipeline calibration and that no further flaggingwas required. We then performed one or two rounds of phase-onlyself-calibration on the continuum data, using solution intervals ofthe scan length or half the scan length and applied the complex gaincorrections from the continuum to the spectral windows containingthe line. The continuum is subtracted from the line spectral windowusing UVCONTSUB. The data are averaged into groups of betweenfour (e.g., SPT0345-47, SPT2132-58) and six (e.g., SPT0441-46,SPT2146-56) channels in order to increase the overall SNR. Thisprocedure results in channels with a typical velocity width of ≈ − (see Table 1).The targets are imaged with natural weighting of the visibilities anddeconvolved using CLEAN (Högbom 1974). In the panels a to c ofFig. 1 and Figs. A1-A4 in Appendix A, we show the spectral linemoment maps of the lensed galaxies. We note that these images areintended only for visualisation, as the analysis is performed on thevisibility data directly (see Section 3). In this section, we provide an overview of the lens-kinematicmodelling technique with which we derive the [CII] surfacebrightness distribution in each spectral channel, as well as thekinematic and dynamical properties of the lensed galaxy. Inaddition, we also recover the surface brightness distribution of thesources from the continuum dataset.To infer the lens mass models, the sources and their kinematicproperties, we make use of the Bayesian pixellated techniquedeveloped by Rizzo et al. (2018) and further extended to thevisibility domain by Powell et al. (2020). We refer the readers tothe above papers for a detailed description of the methodology.Here, we provide a summary and emphasise that our modellingtechnique enables us to simultaneously reconstruct the lensingmass distribution (see Section 3.1) and the kinematics of the source(see Section 3.2) from the same three-dimensional (3D, two spatialand one spectral dimension) data, by fitting directly in their native visibility plane. The dynamical analysis of the background galaxiesis described in Section 3.3.
We assume that the lens mass distribution is described by an el-liptical power-law profile with an external shear component. Theparameters defining the power-law profile are the surface mass den-sity normalization 𝜅 , the position angle 𝜃 , the axis ratio 𝑞 and theslope 𝛾 . The lens mass model parameters define the projected massdensity distribution, normalised to the critical density, as follows 𝜅 ( 𝑥, 𝑦 ) = 𝜅 (cid:16) − 𝛾 (cid:17) 𝑞 𝛾 − (cid:104) 𝑞 (cid:16) 𝑥 + 𝑟 (cid:17) + 𝑦 (cid:105) 𝛾 − . (1)The parameters defining the shear component are its strength Γ sh and position angle 𝜃 sh . In our methodology, the source is pixellated, that is, its surfacebrightness distribution is reconstructed on a grid that is adaptiveto the varying resolution given by the local lensing magnification(Vegetti & Koopmans 2009). For the sample studied in this paper,the minimum spatial resolutions range from ≈
20 to 130 pc andthe median spatial resolutions range from ≈
170 to 300 pc (seeAppendix B for further details).The kinematic-lens modelling technique adopted in this paper(Rizzo et al. 2018) makes use of a 3D kinematic model, that de-scribes a rotating disc, as a regularising prior to the pixellated sourcereconstruction. The kinematic model is defined by the geometricalparameters (inclination 𝑖 and position angle 𝑃𝐴 ) and the parametersdescribing the rotation and velocity dispersion curves. In particular,for all but one of the sources in the sample, we adopted a multi-parameter function for their rotation curve 𝑉 rot ( 𝑅 ) = 𝑉 t (cid:16) + 𝑅 t 𝑅 (cid:17) 𝛽 (cid:20) + (cid:16) 𝑅 t 𝑅 (cid:17) 𝜉 (cid:21) / 𝜉 . (2)In contrast, we found that for SPT2132-58 a simpler arctangentfunction, 𝑉 rot ( 𝑅 ) = 𝜋 𝑉 t arctan (cid:18) 𝑅𝑅 t (cid:19) , (3)was sufficient to fit the data to the noise level (see Section 4.1). Webelieve this to be related to the small Einstein ring ≈ . 𝜎 ( 𝑅 ) = 𝜎 𝑒 − 𝑅𝑅𝜎 . (4) To quantify how the different matter components contribute to thetotal galactic gravitational potential of the source galaxies, we per-form a rotation curve decomposition, similarly to what has beendone in Rizzo et al. (2020). The rotation velocity 𝑉 rot ( 𝑅 ) of the gas MNRAS000
170 to 300 pc (seeAppendix B for further details).The kinematic-lens modelling technique adopted in this paper(Rizzo et al. 2018) makes use of a 3D kinematic model, that de-scribes a rotating disc, as a regularising prior to the pixellated sourcereconstruction. The kinematic model is defined by the geometricalparameters (inclination 𝑖 and position angle 𝑃𝐴 ) and the parametersdescribing the rotation and velocity dispersion curves. In particular,for all but one of the sources in the sample, we adopted a multi-parameter function for their rotation curve 𝑉 rot ( 𝑅 ) = 𝑉 t (cid:16) + 𝑅 t 𝑅 (cid:17) 𝛽 (cid:20) + (cid:16) 𝑅 t 𝑅 (cid:17) 𝜉 (cid:21) / 𝜉 . (2)In contrast, we found that for SPT2132-58 a simpler arctangentfunction, 𝑉 rot ( 𝑅 ) = 𝜋 𝑉 t arctan (cid:18) 𝑅𝑅 t (cid:19) , (3)was sufficient to fit the data to the noise level (see Section 4.1). Webelieve this to be related to the small Einstein ring ≈ . 𝜎 ( 𝑅 ) = 𝜎 𝑒 − 𝑅𝑅𝜎 . (4) To quantify how the different matter components contribute to thetotal galactic gravitational potential of the source galaxies, we per-form a rotation curve decomposition, similarly to what has beendone in Rizzo et al. (2020). The rotation velocity 𝑉 rot ( 𝑅 ) of the gas MNRAS000 , 1–17 (2020)
F. Rizzo et al.
Table 1.
Summary of the observed targets. Columns one and two: IAU and short name. Column three: lens redshifts from Spilker et al. (2016), when available.Column four: source redshifts from Reuter et al. (2020). Column five: total on-source integration times. Column six: ALMA project codes (PI: K. Litke).Column seven: beam size of the [CII] observations. Column eight: channel width of the [CII] data cube.IAU Name Short Name 𝑧 lens 𝑧 source 𝑡 s Project code Beam size Channel widthmin arcsec × arcsec km s − SPT-S J011308-4617.7 SPT0113-46 – 4.23 13.9 2016.1.01499.S 0.35 × × × × × is, indeed, related to the total gravitation potential of the galaxy Φ by 𝑅 (cid:18) 𝜕 Φ 𝜕𝑅 (cid:19) 𝑧 = = 𝑉 = 𝑉 + 𝑉 , (5)where 𝑉 c is the circular velocity, and 𝑉 A is the asymmetric-driftcorrection that accounts for the pressure support due to the randommotions (see equation 6 in Rizzo et al. 2020). To derive 𝑉 A not onlythe velocity dispersion profile 𝜎 ( 𝑅 ) is needed, but also the spatialdistribution of the gas component, traced by the [CII] emissionline. To this end, we assume that the gas has a spatial distributiondescribed by an exponential profile, Σ gas = Σ exp (cid:18) − 𝑅𝑅 gas (cid:19) . (6)To measure the scale radius 𝑅 gas , we divide the zeroth-momentmap of the reconstructed sources (Fig. 1 and Figs. A1 to A4, paneld) in rings with centers, 𝑃𝐴 and 𝑖 defined by the values of thekinematic models. We then azimuthally average the values insideeach ring to obtain the surface density profile that is then fitted withthe exponential profile.We model the circular velocity as 𝑉 c = √︃ 𝑉 + 𝑉 + 𝑉 , (7)where 𝑉 star , 𝑉 gas , 𝑉 DM are the contributions of the stellar, gas anddark matter components to the circular speed for which we makethe following assumptions: • the stellar component is described by a Sérsic profile (LimaNeto et al. 1999; Terzić & Graham 2005). 𝑉 star is, therefore, 𝑉 star = √︄ 𝐺 𝑀 star
𝑅 𝛾 ( 𝑛 ( − 𝑝 ) , 𝑏 ( 𝑅 / 𝑅 e ) / 𝑛 ) Γ ( 𝑛 ( − 𝑝 )) (8)where 𝑀 star is the total stellar mass, 𝑅 e is the effective radius and 𝑛 is the Sérsic index. In equation (8), 𝛾 and Γ are the incompleteand complete gamma function, respectively, while the parameters 𝑝 and 𝑏 are functions of the Sérsic index 𝑛 (see Section 2 in Terzić& Graham 2005). The lack of spatially resolved data from the rest-frame optical or near-infraed emission prevents the fitting of twostellar components (i.e., bulge and disc), due to the strong degen-eracies between the two. The single Sérsic component employed inthe dynamical fitting should be, therefore, considered as a globaldescription of the stellar distribution. This relation is obtained under the assumption that Φ is axisymmetric • The gas component has the same distribution as the [CII] emis-sion line and it is defined by an exponential profile so that 𝑉 gas = √︄ 𝐺 𝑀 gas 𝑅 gas 𝑦 [ 𝐼 ( 𝑦 ) 𝐾 ( 𝑦 ) − 𝐼 ( 𝑦 ) 𝐾 ( 𝑦 )] (9)where 𝑦 = 𝑅 / 𝑅 gas and 𝐾 , 𝐾 , 𝐼 , 𝐼 are the modified Bessel func-tions (Binney & Tremaine 2008). Since the scale length 𝑅 gas is fixedat the value found above, the only free parameter of the fit for 𝑉 gas is the conversion factor ( 𝛼 [ CII ] ) between the total [CII] luminosityand the total gas mass, 𝑀 gas = 𝛼 [ CII ] 𝐿 [ CII ] (see Rizzo et al. 2020,for further details). • The dark matter halo is modelled as a Navarro-Frenk-White(NFW Navarro et al. 1996) spherical halo, 𝑉 DM = √︄ 𝐺 𝑀 DM 𝑅 DM ln ( + 𝑐𝑥 ) − 𝑐𝑥 /( + 𝑐𝑥 ) 𝑥 [ ln ( + 𝑐 ) − 𝑐 /( + 𝑐 )] , (10)where 𝑐 is the concentration parameters and 𝑀 DM and 𝑅 DM arethe virial mass and radius, respectively. To reduce the number offree parameters, we fixed the value of 𝑐 to the value obtained byaveraging the values of the concentration parameters for dark mat-ter halo masses in the mass range between 10 𝑀 (cid:12) and 10 𝑀 (cid:12) and assuming the mass-concentration relation estimated in N-bodycosmological simulations (Dutton & Macciò 2014, see Table C1).We notice that at these redshifts, the concentration is almost inde-pendent of the dark-matter halo mass varying by at most 8 percentfor a variation of 3 orders of magnitude in the halo mass (see Rizzoet al. 2020, for further details on this assumption). The only freeparameter of 𝑉 DM is the virial mass 𝑀 DM , since the virial radius 𝑅 DM can be expressed as a function of 𝑀 DM .To fit the circular velocities and compute the Bayesian posteriordistribution of the free parameters ( 𝑀 star , 𝑅 e , 𝑛 , 𝛼 [ CII ] , 𝑀 DM ) weuse DYNESTY, a python implementation of the Dynamic NestedSampling algorithm (Speagle 2019). We use flat prior distributionsfor 𝑅 e , 𝑛 and 𝛼 [ CII ] and log-uniform priors for the mass parameters(see Table C2 in Appendix C). The results of this lens-kinematic analysis can be visualised in threesets of figures for each lensed system. The first set shows the zeroth-,first- and second-moment maps of the data, the reconstructed sourceand the source kinematic model (Fig. 1 and Figs. A1 to A4). Thesecond set displays the data, the model, the reconstructed source andthe kinematic model for some representative channel maps (Fig. 2
MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies and Figs. A5 to A8). The third set of images shows the position-velocity (p-v) diagrams along the major and minor axis for thereconstructed sources (black contours) and their kinematic models(red contours, see Fig. 3 and Figs. A9 to A12). The most probable aposteriori lens and kinematic parameters are listed in Tables 2 and3, respectively.We note that four out of the five sources are lensed by a galaxy,while SPT0113-46 is lensed by a group of galaxies (Spilker et al.2016). When modelling SPT0113-46, we therefore also includedthe contribution to the lensing potential of the three closest galaxiesin the form of three elliptical power-law components. We found thatthis was sufficient to fit the data to the noise level (see column threein Fig. 2). The lens mass parameters listed in Table 2 are consistentwith previous analysis of the same systems (Spilker et al. 2016).To define the lowest contours in Figs. 2 and 3 and the correspondingfigures for the rest of the sample (Appendix A), we construct mapsof the SNRs in each channel of the reconstructed source and defineas reliable the pixels with SNR (cid:38)
3, allowing us to discriminatefeatures in the reconstructed source from noise artefacts. Thecharacterization of the noise in the source plane is not trivial:correlated noise features in the lens plane may be absorbed into thesource reconstruction (Stacey et al. 2020). In particular, for the datasets used in our analysis, the uncertainties in the source surfacebrightness resulting from noise artefacts are, indeed, significantlylarger than the uncertainties due to the lens model parameters. Toestimate the uncertainties in the source plane, we, therefore, assumeour maximum a posteriori source model and create mock lenseddata with 100 different Gaussian realisations of the noise at thelevel measured in the real data. We then reconstruct the source fromthe mock data sets and measure the mean and standard deviation ofthe surface brightness in each pixel and in each spectral channel. ∼ Both the figures containing the channels maps (Fig. 2 and Figs. A5to A8) and the p-v diagrams (Fig. 3 and Figs. A9 to A12) show howwell the rotating disc model is able to reproduce the emission ofthe reconstructed sources in our samples. The p-v diagrams alongthe major axis of the reconstructed sources (black contours) have,in fact, an S-shape, that is a typical signature of a rotating disc (seethe red contours). The thickness of the p-v diagrams both alongthe major- and minor-axis is the result of the velocity dispersionand the instrumental resolution. At the central channel maps, closeto the systemic velocity, the black contours of all galaxies have a"butterfly" shape, a typical pattern of a rotating disc.We note also that on top of the bulk rotation motions two galaxiesof the sample, SPT0441-46 and SPT2146-56, have anomalouskinematic components, features very common in nearby spiralgalaxies (Fraternali et al. 2001, 2002). In particular, SPT0441-46and SPT2146-56 show faint emission in the so-called forbiddenregions (i.e., forbidden for rotation) of the p-v diagrams (seethe black contours in the lower left quadrants in Fig. A10 andA11). These features are due to lag in rotation and non-circularmotions and they are usually ascribed to extraplanar gas (Fra-ternali & Binney 2006; Sancisi et al. 2008). The inspection ofthe p-v diagrams and channel maps indicates that the derivedkinematic parameters are robust also for these two galaxies. Dueto the faint emission of the anomalous kinematic components,the kinematic models are, in fact, not influenced at all by them and are able to well reproduce the bulk of the gas in regular rotation.
The rotation curves of the galaxies of our sample resemble thosefound for local spiral galaxies (Lelli et al. 2016). All rotation curvesflatten, in fact, at large radii and show a variety of shapes in theinner regions. In particular, we can distinguish three main classes: • rotation curves that steeply rise and then decline before flatten-ing out to the outermost measured value (SPT0113-46, SPT0441-46). These curves are typical of nearby massive spiral galaxies withstellar bulges (Cimatti et al. 2019); • rotation curves with an inner slow rise and then a flattening(SPT2132-58), typical of present-day disc galaxies; • rotation curves with an intermediate behaviour (SPT0345-47,SPT2146-56).The variety of the stellar distributions in these DSFGs is, therefore,imprinted in their dynamics. In particular, we confirm a previousfinding (Rizzo et al. 2020): some DSFGs at 𝑧 ≈ ± ± 𝑓 gas and the total baryonic mass 𝑀 bar . We note that the inferred values of 𝑀 gas are in agreement, within the 2- 𝜎 uncertainties, with the valuesderived from the CO luminosities and the dust masses (Aravena et al.2016). In order to derive the star formation properties of the sources wemodel their continuum emission in a narrow spectral range closeto the redshifted [CII] emission and use the strong lensing mag-nification factor (Table 6) to compute the intrinsic infrared lu-minosity from the observed values taken from the literature (Ar-avena et al. 2016, column two of Table 6). Specifically, these in-frared luminosities were obtained from spectral energy distribu-tion (SED) fitting of far-infrared and sub-millimeter observations,covering the range 250 to 3000 𝜇 m (Aravena et al. 2016). By ap-plying the Kroupa Initial-Mass-Function (IMF) conversion factorof 1 . × − M (cid:12) yr − L − (cid:12) (Kennicutt & Evans 2012), we thenderived the SFR for each source (column four of Table 6). 𝑀 star plane For most star-forming galaxies, there is a tight correlation betweentheir SFR and their stellar mass (Brinchmann et al. 2004), the so-called main sequence. Several studies (e.g., Noeske et al. 2007;
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MNRAS000 , 1–17 (2020)
F. Rizzo et al.
Table 2.
Mass model parameters of the gravitational lens galaxies. From left to right we list the mass density profile normalisation, position angle, axis ratio,slope, the external shear strength and its position angle.Name 𝜅 𝜃 𝑞 𝛾 Γ sh 𝜃 sh arcsec deg degSPT0113-46 1.20 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
15 0.95 ± ± ± ± ± ± ± ± ± ± Table 3.
Kinematic parameters of the sources. The kinematic model is a rotating disc defined by the geometrical parameters ( 𝑖 and 𝑃 𝐴 ) and the parametersdefining the rotation curve, equations (2), (3), and velocity dispersion profile, equation (4).Name 𝑖 𝑃 𝐴 𝑉 t 𝑅 t 𝛽 𝜉 𝜎 𝑅 𝜎 deg deg km s − kpc km s − kpcSPT0113-46 70 ± ± ± ± ± ± ±
12 1.5 ± ± ± ±
25 0.68 ± ± ± ±
13 2.2 ± ± ± ± ± ± ± ±
23 0.68 ± ± ± ±
18 0.34 ± ± ± ±
10 2.0 ± ± ± ±
22 0.48 ± ± ± Table 4.
Dynamical parameters of the sources. All parameters listed in columns two to six are inferred from the rotation curve decomposition assuming aSérsic profile for the stellar component, an exponential profile for the gas disc and an NFW profile for the dark matter halo. From left to right: the stellar mass,the stellar distribution effective radius and Sérsic index, the conversion factor between the [CII] luminosities and the gas mass and the dark matter mass.Name 𝑀 star 𝑅 e 𝑛 𝛼 [ CII ] 𝑀 DM M (cid:12) kpc M (cid:12) /L (cid:12) M (cid:12) SPT0113-46 6.5 + . − . + . − . + . − . + . − . + . − . SPT0345-47 2.3 . − . + . − . + . − . + . − . + . − . SPT0441-46 1.8 + . − . + . − . + . − . + . − . + . − . SPT2146-56 1.0 + . − . + . − . + . − . + . − . + . − . SPT2132-58 1.9 + . − . + . − . + . − . + . − . + . − . Table 5.
Derived physical properties of the sources. From left to right the gas mass, the fraction of total baryonic mass in gas, the total baryonic mass, thebaryonic effective radius, the gas depletion time and the disc-scale height.Name 𝑀 gas 𝑓 gas 𝑀 bar 𝑅 bar 𝑡 dep ℎ M (cid:12) M (cid:12) kpc Myr pcSPT0113-46 4.3 + . − . + . − . + . − . + . − . ±
73 100 + − SPT0345-47 1.7 + . − . + . − . + . − . + . − . ± + − SPT0441-46 1.4 + . − . + . − . + . − . + . − . ± + − SPT2146-56 1.2 + . − . + . − . + . − . + . − . ± + − SPT2132-58 1.9 + . − . + . − . + . − . + . − . ± + − MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Figure 1.
Moment maps for SPT0113-46. Panels a, b and c: the observed [CII] zeroth-, first- and second-moment maps. The beam size, shown as a whiteellipse on the lower left corner of panel a, is 0 . × .
19 arcsec at a position angle of 87.0 ◦ . Panels d, e and f: zeroth-, first- and second-moment maps ofthe reconstructed source. Panels g and h: first- and second-moment maps of the kinematic model. These maps are intended only for visualisation as the fullanalysis is performed on the data cube. Table 6.
SFR and [CII] luminosities of the sources. Column two: the ob-served infrared luminosity from Aravena et al. (2016). Column three: themagnification factor of the continuum in the infrared bands. Column four:star-formation rate derived for a Kroupa IMF. Column five: intrinsic [CII]luminosities.Name 𝐿 IR , obs 𝜇 SFR 𝐿 [ CII ] L (cid:12) M (cid:12) yr − L (cid:12) SPT0113-46 3.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Whitaker et al. 2012; Steinhardt et al. 2014; Tasca et al. 2015)showed that the main sequence holds from 𝑧 = 𝑧 ∼
6, witha redshift evolution of its normalization. Also, there are starburstsgalaxies, characterized by significantly higher SFR than normalmain-sequence galaxies. While starburst galaxies are rare in thelocal Universe (Renzini et al. 2015), they constitute a moderatepercentage ( ≈
15 per cent, Bisigello et al. 2018; Caputi et al. 2017)of all galaxies at 𝑧 (cid:38)
2, contributing up to 50 per cent of the cosmic SFR density at 𝑧 ∼ 𝑀 star diagram. We find that four out of five galaxies in our sampleare consistent with the starburst sequence, while SPT0113-46 is amain-sequence galaxy. These findings are not surprising, given thatthe selection criteria used for the identification of the SPT DSFGsare based on the observed flux rather than the intrinsic one (Vieiraet al. 2010, 2013). SPT0113-46 has, indeed, an observed infraredluminosity of 𝐿 IR , obs which is similar to the rest of the sample (seecolumn two in Table 6). However, the large magnification factor of ∼
36 leads to an intrinsic luminosity 𝐿 IR and SFR which are onaverage a factor of five below the other galaxies in the sample. In this section, we investigate the dynamical and structural proper-ties of the reconstructed sources. For the rest of this paper, we alsoinclude, if not otherwise stated, three non-lensed and one lensedDSFGs in our sample (see Table 7). These sources have 4 (cid:46) 𝑧 (cid:46) MNRAS000
36 leads to an intrinsic luminosity 𝐿 IR and SFR which are onaverage a factor of five below the other galaxies in the sample. In this section, we investigate the dynamical and structural proper-ties of the reconstructed sources. For the rest of this paper, we alsoinclude, if not otherwise stated, three non-lensed and one lensedDSFGs in our sample (see Table 7). These sources have 4 (cid:46) 𝑧 (cid:46) MNRAS000 , 1–17 (2020)
F. Rizzo et al. Figure 2.
Representative channel maps for SPT0113-46. From left to right the dirty image of the data, the dirty image of the model, the dirty image residuals,the reconstructed source and the kinematic model. MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Figure 3.
Position-velocity diagrams for SPT0113-46. The x-axes show the offset along the major (panel a) and minor axes (panel b) from the galaxy centre.The y-axis show the line-of-sight velocity centred at the systemic velocity of the galaxy. The reconstructed source is shown in black, the kinematic model inred. The black dots show the derived projected velocities.
Table 7.
Properties of the galaxies added to our sample. The kinematic properties in columns three to five are taken from the references shown in column six.The SFR value (converted to a Kroupa IMF) in column seven are taken from the references in column eight.Name 𝑧 𝑉 flat 𝜎 ext 𝜎 m Reference SFR Referencekm s − km s − km s − M (cid:12) yr − SPT0418-47 4.23 259 ± ± ± ±
65 Rizzo et al. (2020)AzTEC-1 4.34 200 ±
27 30 ± + − Sharda et al. (2019) 1261 + − Tadaki et al. (2018)J1000+0234 4.54 538 ± (cid:46) (cid:46)
60 Fraternali et al. (2020) 468 + − Gómez-Guijarro et al. (2018)AzTEC/C159 4.57 506 + − ±
13 16 ±
13 Fraternali et al. (2020) 787 + − Gómez-Guijarro et al. (2018) 𝑧 ∼ DSFGs
Recent observational and theoretical studies (e.g., Swinbank et al.2017; Wisnioski et al. 2015; Übler et al. 2019; Turner et al. 2017;Zolotov et al. 2015; Pillepich et al. 2019; Dekel et al. 2020) havesuggested that many high- 𝑧 galaxies, despite being rotationally sup-ported systems, have intrinsic velocity dispersions which are muchhigher than those of local galaxies. The level of rotational supportrelative to the amount of turbulence in a galaxy is generally quan-tified with the ratio between the rotation velocity and the velocitydispersion, 𝑉 / 𝜎 . We compute the 𝑉 / 𝜎 ratios for our sample usingtwo definitions: 𝑉 max / 𝜎 m , the ratio between the maximum rotationvelocity and the median velocity dispersion and 𝑉 flat / 𝜎 ext , the ratiobetween the flat part of the rotation velocity and the velocity disper-sion at outer radii (Table 8).In Fig. 6, the 𝑉 flat / 𝜎 ext ratios are plotted as a function of redshiftand compared with theoretical predictions (Pillepich et al. 2019;Dekel & Burkert 2014; Zolotov et al. 2015; Hayward & Hopkins2017). We find that the nine galaxies of our extended sample have 𝑉 / 𝜎 ratios which are sistematically larger than any current theoret-ical prediction. In particular, the median 𝑉 / 𝜎 ratio of the sampleis 13 + − , which is a factor of 4 larger than the highest value of ≈ ≈
10 kms − expected for a gas at temperatures 𝑇 (cid:46) K . These resultsimply a significant level of turbulence in the ISM of these galaxies,which is most plausibly (e.g. Übler et al. 2019; Krumholz et al.2018; Varidel et al. 2020) related to either star-formation feedback(e.g., momentum injection by supernova explosions, stellar windsand expansion of HII regions) or gravitational phenomena (e.g., gasaccretion, galaxy interactions and gravitational instabilities). Bothtype of mechanisms may play an important role in driving turbulentmotions also in nearby galaxies (see the discussion in Arribas et al.2014; Bacchini et al. 2020; Varidel et al. 2020). Due to the high levelof star-formation and the significant gas fraction, it is expected thatboth feedback and gravity-driven turbulence are more significantfor high- 𝑧 galaxies (Hung et al. 2019; Pillepich et al. 2019).We now compare our observations with an analytical model byKrumholz et al. (2018) that takes into account gravitational mech- The gas velocity dispersions, 𝜎 is the sum in quadrature of two contribu-tions, 𝜎 = 𝜎 + 𝜎 , where 𝜎 turb is the velocity dispersion due to theturbulence and 𝜎 th is the velocity dispersions due to the thermal motions ofthe particles within the gas (Cimatti et al. 2019). In particular, 𝜎 th dependson the temperature 𝑇 of the fluid: 𝜎 th = √︁ 𝑘 B 𝑇 /( (cid:101) 𝑚𝑚 p ) , where 𝑘 B is theBoltzmann constant, (cid:101) 𝑚 is the mean average mass of the particle in the fluidin units of the proton mass, while 𝑚 p is the mass of the proton.MNRAS000
10 kms − expected for a gas at temperatures 𝑇 (cid:46) K . These resultsimply a significant level of turbulence in the ISM of these galaxies,which is most plausibly (e.g. Übler et al. 2019; Krumholz et al.2018; Varidel et al. 2020) related to either star-formation feedback(e.g., momentum injection by supernova explosions, stellar windsand expansion of HII regions) or gravitational phenomena (e.g., gasaccretion, galaxy interactions and gravitational instabilities). Bothtype of mechanisms may play an important role in driving turbulentmotions also in nearby galaxies (see the discussion in Arribas et al.2014; Bacchini et al. 2020; Varidel et al. 2020). Due to the high levelof star-formation and the significant gas fraction, it is expected thatboth feedback and gravity-driven turbulence are more significantfor high- 𝑧 galaxies (Hung et al. 2019; Pillepich et al. 2019).We now compare our observations with an analytical model byKrumholz et al. (2018) that takes into account gravitational mech- The gas velocity dispersions, 𝜎 is the sum in quadrature of two contribu-tions, 𝜎 = 𝜎 + 𝜎 , where 𝜎 turb is the velocity dispersion due to theturbulence and 𝜎 th is the velocity dispersions due to the thermal motions ofthe particles within the gas (Cimatti et al. 2019). In particular, 𝜎 th dependson the temperature 𝑇 of the fluid: 𝜎 th = √︁ 𝑘 B 𝑇 /( (cid:101) 𝑚𝑚 p ) , where 𝑘 B is theBoltzmann constant, (cid:101) 𝑚 is the mean average mass of the particle in the fluidin units of the proton mass, while 𝑚 p is the mass of the proton.MNRAS000 , 1–17 (2020) F. Rizzo et al. V c ( k m / s ) SPT0113-46
SPT0345-47 . . . . . V c ( k m / s ) SPT0441-46 . . . . . . SPT2146-56 R (kpc) V c ( k m / s ) SPT2132-58 MeasuredStarsGas Dark MatterTotal V c ( k m / s ) SPT0113-46
SPT0345-47 . . . . . V c ( k m / s ) SPT0441-46 . . . . . . SPT2146-56 R (kpc) V c ( k m / s ) SPT2132-58 MeasuredStarsGas Dark MatterTotal
Figure 4.
Rotation curve decomposition. The green solid lines show the circular velocity profiles. The black dotted lines show the best dynamical models, andthe contribution from the different mass components as indicated by the legend and listed in Tables 4 and 5. anisms, stellar feedback or both to describe the evolution of someobserved properties of star-forming galaxies. For their models,Krumholz et al. (2018) assumed a rotating disc, made up of gas,stars and a spheroidal dark matter halo, in vertical hydrostatic andenergy equilibrium. The dissipation of turbulence is counteracted bythe injection of energy due to supernova explosions and the releaseof gravitational potential energy due to the inward flow of gas drivenby non axisymmetric torques and gas accretion. The inclusion ofeither the two turbulence-driving mechanisms or only one of thetwo results in different relations between the velocity dispersionsand the SFR. In the three panels in Fig. 7, we plot the 𝜎 -SFR rala-tions obtained by assuming the fiducial parameters for the high- 𝑧 galaxies shown in Table 3 of Krumholz et al. (2018), but with sixdifferent values of circular velocities matching those measured inour extended sample. The most important parameters defining the A parameter defining the models by Krumholz et al. (2018) is the effectivegas fraction at the mid-plane, 𝑓 gas , P = . 𝑓 gas (Übler et al. 2019). In Fig.7, we show the 𝜎 - SFR relations obtained by assuming the fiducial value 𝑓 gas , P = .
7. However, we note that our results do not change by assumingvalues of 𝑓 gas , P in the full range (0.6 - 0.8) probed by our sample. behaviour of 𝜎 in these models are the Toomre parameter 𝑄 andthe star formation efficiency per free-fall time 𝜖 ff : • in the "Gravity + feedback" model (left panel in Fig. 7), theturbulence is driven by both stellar feedback and gravitational insta-bilities. The former are able to sustain the turbulence when 𝑄 > 𝜎 (cid:46)
25 km s − ), while the latter play akey role for 𝑄 =
1. In this model, 𝜖 ff is kept constant at 0.015. • The "Gravity" model (medium panel in Fig. 7) includes only thegravitational instabilities as drivers of turbulence within the modelgalaxies. The assumptions are similar to the "Gravity + feedback"model, except that Q is always equal to 1. • The "Feedback, fixed 𝜖 ff " (dashed lines in the right panel in Fig.7) model is similar to the so-called ‘self-regulated-system’ model(Ostriker & Shetty 2011). In this case, the star formation is part ofa self-regulating cycle where the momentum injected to the ISM by The Toomre parameter considered in Krumholz et al. (2018) is definedas 𝑄 = Σ g Σ g +[ 𝜎 /( 𝜎 + 𝜎 )] Σ star 𝜅 𝜎𝜋𝐺 Σ g , where Σ g and Σ star are the gas andstellar surface densities, 𝜎 star is the the stellar velocity dispersion and 𝜅 isthe epicyclic frequency (Romeo & Falstad 2013).MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies . . . . . . ( M star / M (cid:12) ) . . . . . . l og ( SF R / M (cid:12) y r − ) MSStarbursts . . . f g a s Figure 5.
Location on the SFR - 𝑀 star of the source galaxies in our sample(circles), with markers colour-coded according to their gas fraction. Thesolid black line and the blue area show the best-fit and the 1- 𝜎 scatter formain-sequence galaxies at 𝑧 ∼ star formation balances the gravitational force confining the ISM gasin the disc, without any injection of gravitational potential energy. 𝜖 ff is kept constant at 0.015 and 𝑄 is left free to vary. • Also in the "Feedback, fixed 𝑄 " model (solid lines in the rightpanel in Fig. 7), stellar feedback is the only driving mechanisms ofturbulence. Similarly to the model developed by Faucher-Giguèreet al. (2013), the star-formation efficiency 𝜖 ff is not constant andvary as a function of different properties of the galaxy, while 𝑄 = 𝜎 and SFR for the galaxiesshown in Fig. 7 and these analytical models indicate that stellarfeedback is able to sustain the measured turbulence (see the rightpanel in Fig. 7). The two models (left and medium panels) includinggravitationally-driven mechanisms overestimate the velocity disper-sions by factors from ≈ ≈
40 (SPT2132-58) forseven out of nine galaxies at the corresponding values of SFR. Theonly exceptions are SPT0113-46 that has a position consistent withall four models shown in Fig. 7 and J1000+0234 for which only anupper limit for the measured velocity dispersion is available (Frater-nali et al. 2020). This result is at odds with recent works (Krumholzet al. 2018; Johnson et al. 2018; Übler et al. 2019) finding that grav-itational instabilities are fundamental to explain the position of theirgalaxies in the 𝜎 -SFR plane with respect to the models of Krumholzet al. (2018). The reason for this discrepancy may be ascribed to thedifferent galaxy populations and redshifts of the observed galaxies:starbursts at 𝑧 ∼ . (cid:46) 𝑧 (cid:46) As a further test, in this section we estimate whether the velocitydispersions measured for our sample can be explained by the en-ergy injected by supernova explosions using simple and reasonableassumptions. Following Rizzo et al. (2020) and Fraternali et al.(2020), the velocity dispersion due to the transferring of the super- nova energy to the ISM is 𝜎 SFR = (cid:18) 𝜖 SN . (cid:12) yr − ℎ (cid:19) / (cid:18) 𝑀 gas M (cid:12) (cid:19) − / kms − , (11)where 𝜖 SN is the efficiency of transferring kinetic energy from su-pernova feedback to the ISM and ℎ is the disc scale height. Equation(11) is obtained by assuming a supernova rate of 0.01 𝑀 − (cid:12) , validfor a Kroupa IMF (Tamburro et al. 2009). Since the calculation ofthe disc scale height is not trivial (see discussion in Bacchini et al.2019), we use an analytical approximation and leave the exact esti-mate to a future work. The scale height of the vertical distribution ofa gas disc in hydrostatic equilibrium can be approximated (Bacchiniet al. 2019) as ℎ ( 𝑅 ) = 𝜎 ( 𝑅 ) √︁ 𝜋𝐺 [ 𝜌 ( 𝑅 ) + 𝜌 rot ( 𝑅 )] (12)with 𝜌 rot ( 𝑅 ) = − 𝜋𝐺 𝑉 c ( 𝑅 ) 𝑅 𝜕𝑉 c ( 𝑅 ) 𝜕𝑅 (13)Since in this approximation, the self-gravity of the gas is not in-cluded, 𝜌 ( 𝑅 ) is the density profile of the stellar and dark mattercomponent and 𝜌 rot ( 𝑅 ) is obtained by considering equations (8)and (7).The values of the median ℎ for our sample are listed in columnseven of Table 5. In Fig. 8, we show the ratios between the ex-pected values of 𝜎 SFR and the measured 𝜎 m (column three in Table8). The 𝜎 SFR values are calculated by using three values of 𝜖 SN equal to 0.001, 0.01 and 0.1, typical of observed nearby (Bacchiniet al. 2020) and simulated galaxies (e.g., Martizzi et al. 2016; Ohlinet al. 2019). For all galaxies of our sample the ratios between 𝜎 m and one of the three values of 𝜎 SFR are ≈
1, confirming that theturbulent motions can be easily driven by supernova explosions(see Fig. 8). Larger values of 𝜖 SN , 0.8-1, are, instead, consideredan indication that other physical mechanisms, in addition to stellarfeedback, drive the observed turbulent motions (Utomo et al. 2019;Tamburro et al. 2009). We note that three galaxies in the sample(SPT0113-46, SPT0345-47, SPT0441-46) have values of velocitydispersions fully consistent with 𝜎 SFR at 𝜖 SN = . 𝜎 .For two galaxies (SPT2146-56, SPT2132-58), the 𝜎 SFR values at 𝜖 SN = . 𝜎 m by a factor of ≈
3, indicating that lowefficiency values 0 . (cid:46) 𝜖 SN (cid:46) .
001 are needed, similarly to whatfound both for high- and low- 𝑧 star-forming galaxies (Fraternaliet al. 2020; Bacchini et al. 2020). Our high-resolution 3D kinematic analysis shows that the sampleof DSFGs studied in this paper has ratios of 𝑉 / 𝜎 similar to thosemeasured for spiral galaxies in the local Universe (Lelli et al. 2016;Bacchini et al. 2019). The comparison with the intermediate- 𝑧 star-forming galaxies is, instead, challenging, due to different gas tracersused in the literature to determine the evolution of the dynamicalproperties of galaxies across cosmic time. While it is firmly es-tablished that the kinematics of the molecular gas traces the galaxykinematics, there is an open debate on the validity of this assumptionfor the ionized gas tracers at high redshift (Girard et al. 2018; Levyet al. 2018; Lelli et al. 2018). For example, by comparing the kine-matics of the ionized ([OIII]) and neutral gas ([CI]) for a starburstgalaxy at 𝑧 = .
6, Lelli et al. (2018) concluded that [OIII] traces the
MNRAS , 1–17 (2020) F. Rizzo et al. . . . Redshift V / σ Sharda et al. 2019Fraternali et al. 2020Rizzo et al. 2020This work
Figure 6. 𝑉 / 𝜎 versus redshift. The 𝑉 / 𝜎 for our sample (yellow circles), defined as 𝑉 flat / 𝜎 ext and for the galaxies from literature (Table 7), as indicated inthe legend. Note that the redshift of SPT0418-47 (orange diamond) is shifted by -0.02 for a better visualisation of all the points. The light-blue area shows theregion covered by theoretical studies (Pillepich et al. 2019; Dekel & Burkert 2014; Zolotov et al. 2015; Hayward & Hopkins 2017). SFR( M (cid:12) yr − ) σ ( k m s − ) Gravity+feedback SFR( M (cid:12) yr − )Gravity SFR( M (cid:12) yr − )Feedback V ( k m s − ) Figure 7.
The markers show the 𝜎 m (column three in Table 8) and SFR (column four in Table 6) for our sample (circles) and for the galaxies from Rizzoet al. (2020, diamond), Sharda et al. (2019, triangle) and Fraternali et al. (2020, squares). All markers are colour-coded according to 𝑉 flat (column five inTable 8 and column three in Table 7). In the four panels, the curves show the predictions of the relation between velocity dispersions and SFR in the analyticmodel developed by Krumholz et al. (2018), obtained with the parameters valid for high- 𝑧 galaxies. The left and medium panels show the models that includegravitational instabilities plus stellar feedback and only gravitational mechanisms as the main drivers of turbulence within galaxies, respectively. The rightpanel shows models with only stellar feedback-driven turbulence: "Feedback, fixed 𝑄 " (solid lines) and "Feedback, fixed 𝜖 ff " (dashed lines). outflow motions, while [CI] is a tracer of the galaxy virial motions.On the other hand, Übler et al. (2018) found that the kinematics ofthe H 𝛼 and CO(3-2) emission line are consistent in a star-forminggalaxy at 𝑧 = .
4. The comparison between the dynamic propertiesof our DSFG sample with those found for intermediate- 𝑧 galaxiesfrom ionized gas tracers, reveals that the 𝑉 / 𝜎 ratios of 𝑧 ∼ . 𝑉 / 𝜎 measured from the ion-ized gas at lower redshifts. This difference may be ascribed to acombination of the following reasons: • the samples of galaxies in the redshift ranges 1 to 2 with H 𝛼 or[OII] measurements have typical rotation velocities (cid:46)
250 km s − (e.g., Di Teodoro & Fraternali 2015; Swinbank et al. 2017; Übleret al. 2017). The DSFGs in our sample have 𝑉 flat from ≈
200 to 360km s − (see Table 8) and a median value of 280 km s − . The velocitydispersions 𝜎 ext of our sample (from 16 to 40 km s − , see Table 8)are consistent with the distribution of the H 𝛼 velocity dispersionsof 𝑧 ≈ MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Table 8.
Global kinematic parameters of the sources. Column two: the maximum rotation velocity. Column three: the median velocity dispersion. Columnfour: the ratio between 𝑉 max and 𝜎 m . Column five: the rotation velocity in the flat part of the rotation curve. Column six: the velocity dispersion in the externalregions ( 𝑅 (cid:38) 𝑅 e ). Column seven: the ratio between 𝑉 flat and 𝜎 ext .Name 𝑉 max 𝜎 m 𝑉 max / 𝜎 m 𝑉 flat 𝜎 ext 𝑉 flat / 𝜎 ext SPT0113-46 382 ± ±
22 9.2 ± ± ± ± ± ±
15 5.6 ± ±
25 40 ± ± ±
67 31 ±
21 15.8 ± ± ± ± ±
13 31 ±
11 7.0 ± ± ± ± ±
18 27 ± ± ±
14 16 ± ± SP T - SP T - SP T - SP T - SP T - σ SF R , m / σ m (cid:15) SFR = . (cid:15) SFR = . (cid:15) SFR = . Figure 8.
Ratio between the velocity dispersion expected from energy in-jection by the stellar feedback, equation (11), and the measured velocitydispersion 𝜎 m (column three in Table 8). The three markers for each galaxyshow the ratios obtained with three different values of 𝜖 SN , as indicated inthe legend. (2015) and Übler et al. (2019), but they overlap only with the lowvelocity-dispersion distribution at 𝑧 (cid:38) . • The velocity dispersions measured from the ionized tracers is,on average, higher than those measured from the molecular andneutral media tracers both at low- (Varidel et al. 2020; Girard et al.2021) and high- 𝑧 (Girard et al. 2018). • The H 𝛼 , [OIII], and [OII] emission lines are not good tracersof the galaxy dynamics. A recent study (Levy et al. 2018) on asample of local galaxies showed, for example, that the kinematicsof the molecular and ionized gas are different (Levy et al. 2018) andthat the difference may be ascribed to stellar feedback processes. Inparticular, the measured ionized gas kinematics is affected by thepresence of gas in outflows or in extraplanar layers. Unfortunately,the lack of good quality data of both the cold and warm gas for largesamples of galaxies at 1 (cid:46) 𝑧 (cid:46) • The measured 𝑉 / 𝜎 values can be biased towards low valuesdue to some residual beam-smearing effect in some studies (seediscussion in Di Teodoro & Fraternali 2015). • The galaxies in our sample are starbursts (with the exception ofSPT0113-46) while most of the galaxies with H 𝛼 measurements at1 (cid:46) 𝑧 (cid:46) 𝑧 (cid:38) 𝑧 (cid:46) 𝑉 / 𝜎 ratios measured for our DSFG sampleis similar both to that measured for a galaxy at 𝑧 = . 𝑧 ∼ . 𝑧 galaxies (e.g., Swinbanket al. 2017; Turner et al. 2017; Lelli et al. 2018; Übler et al. 2019;Wisnioski et al. 2019; Fraternali et al. 2020). We employ, indeed,a forward modelling approach similar to Di Teodoro et al. (2016),Lelli et al. (2018), Di Teodoro et al. (2018) and Fraternali et al.(2020). However, while they inferred model-independent velocitiesand velocity dispersions, we assume specific functional forms todescribe the rotation velocity and velocity dispersion profiles, sim-ilarly to Swinbank et al. (2017), Turner et al. (2017) and Wisnioskiet al. (2019).The comparison of the velocity dispersions of galaxies at differentredshifts from similar tracers (i.e., HI, [CI], [CII], see the filledmarkers in Fig. 9) indicate that there is a very weak increase of theturbulence with time. The median value of ≈
25 km s − for thesample of 𝑧 (cid:38) ≈ 𝜎 (12 ± − ) and CO 𝜎 (9 + − km s − ), measured for the sam-ple of nearby galaxies (Bacchini et al. 2019, 2020). The comparisonwith the CO measurements at intermediate- 𝑧 indicates, instead, thatthe median [CI]/[CII] 𝜎 at 𝑧 (cid:38) 𝜎 The median 𝜎 of 25 + − km s − is obtained by including the two upperlimits of 30 km s − (Lelli et al. 2018, green square in Fig. 9) and 60 km s − (Fraternali et al. 2020, pink hexagon in Fig. 9) as fiducial values. If thesetwo measurements are excluded, the resulting median 𝜎 is 24 + − km s − .MNRAS000
25 km s − for thesample of 𝑧 (cid:38) ≈ 𝜎 (12 ± − ) and CO 𝜎 (9 + − km s − ), measured for the sam-ple of nearby galaxies (Bacchini et al. 2019, 2020). The comparisonwith the CO measurements at intermediate- 𝑧 indicates, instead, thatthe median [CI]/[CII] 𝜎 at 𝑧 (cid:38) 𝜎 The median 𝜎 of 25 + − km s − is obtained by including the two upperlimits of 30 km s − (Lelli et al. 2018, green square in Fig. 9) and 60 km s − (Fraternali et al. 2020, pink hexagon in Fig. 9) as fiducial values. If thesetwo measurements are excluded, the resulting median 𝜎 is 24 + − km s − .MNRAS000 , 1–17 (2020) F. Rizzo et al. of 22 + − km s − , measured for 1 (cid:46) 𝑧 (cid:46) 𝑧 will be needed to confirm this trend andmake a robust comparison between similar tracers.In Fig. 9, we also show the relations between velocity dispersionsand redshifts derived by Übler et al. (2019) for main-sequence galax-ies. In particular, the gray dotted and solid black lines are the best-fitrelations for velocity dispersions measured from ionized (H 𝛼 , [OII],[OIII]) and atomic/molecular tracers (HI, CO) up to 𝑧 = . 𝑧 ≈
6. The comparison between the measuredvelocity dispersions at 𝑧 (cid:38) 𝜎 - 𝑧 relations are not valid at these high redshiftssince they systematically overestimate 𝜎 by a median factor of ≈ 𝜎 - 𝑧 relations were derived mainly from normalstar-forming galaxies, while both the galaxies in our sample and thefour galaxies from the literature (Lelli et al. 2018; Sharda et al. 2019;Fraternali et al. 2020) are above the main-sequence and thereforemore prone to develop high turbulent motions (e.g., Krumholz et al.2018; Hung et al. 2019). In this section, we compare the physical properties of the DSFGswith their plausible descendants, local ETGs. In particular, we focuson the position of these galaxies in the size-stellar mass plane ( 𝑅 e - 𝑀 star ). The relation between the size and stellar mass of galaxies,and how it evolves with cosmic time, provides, indeed, importantinsights onto the assembly history of galaxies (van der Wel et al.2014; Lang et al. 2014).In Fig. 10, we show both local ETGs (Cappellari et al. 2013)and high- 𝑧 massive quiescent galaxies (Belli et al. 2017; Lustiget al. 2020; Esdaile et al. 2020). Also, the gray stars are 𝑧 ∼ 𝑀 gas / SFR), respectively (see column six of Table 6). Wenote that we do not include the non-lensed galaxies listed in Table7, as the few resolution elements across their discs do not allowa rotation curve decomposition similar to that performed for thelensed sample.Interestingly, SPT0113-46, the only main-sequence galaxy inour sample (see Fig. 5), is consistent with the size-mass relationof local ETGs (panel a of Fig. 10). Its depletion time of ∼ 𝑧 (cid:38) 𝑛 ≈ 𝑛 ≈ 𝑛 ≈
6. The Sérsic indices derived from our analysis suggestthat the spheroidal components observed in 𝑧 (cid:46) 𝑧 (cid:46)
2, there is a strong increase of the galaxy sizesof the quiescent population, mainly driven by dry minor mergers(e.g., Bezanson et al. 2009; Naab et al. 2009; Oser et al. 2010;Cassata et al. 2013). Panel b of Fig. 10 shows that all galaxiesin our sample will end up with a stellar mass typical of the localETGs or cSFG already at 𝑧 ≈
4. This finding allows us to put someconstraints on the physical processes (e.g., mergers, accretion, Naabet al. 2014; Bouché et al. 2010) that will be acting on these galaxiesin the following ∼
12 Gyr: they should either preserve the stellarmass or be responsible only for a mild growth in size.We note that the depletion times for our sample range from ≈
10 to 360 Myr (see column six of Table 5), with a median valueof 20 Myr. All galaxies in the sample are at 𝑧 >
4, meaning thatpotentially they could be the progenitors of the recently discoveredquiescent systems at 3 (cid:46) 𝑧 (cid:46) 𝑧 (cid:38)
3, a comparison with our DSFGs on thesize-stellar mass plane is currently not feasible. However, we notethat in Fig. 10, we show three samples of ten, four and five qui-escent galaxies at median 𝑧 ∼ . 𝑧 ∼ . 𝑧 ∼ 𝑅 e values were derived from the rest-frame continuum emissions at ≈ 𝑧 quiescent population and our sample of DSFGs on thesize-stellar mass plane is challenging. The next generation of instru-ments, such as the James Webb Space Telescope (JWST, Gardneret al. 2009), will allow one to measure the structural (e.g., sizes) andthe stellar-population properties of these high- 𝑧 populations, facili-tating further investigations of the evolutionary connection betweenthe DSFGs and the 𝑧 ∼ In this paper, we presented ALMA observations of the [CII] emis-sion line for five gravitationally lensed dusty star-forming galaxies atredshift between 4 and 5. Using our lens and kinematic modellingtechnique, we reconstructed the background sources and inferredtheir kinematic and dynamical properties on ∼ 𝑧 ∼ MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Redshift σ ( k m s − ) Atomic + Molecular, ¨Ubler et al. 2019Ionized, ¨Ubler et al. 2019Bacchini al. 2019Bacchini al. 2020Girard al. 2021¨Ubler et al. 2019Girard et al. 2019 Swinbank et al. 2011Lelli et al. 2018Sharda et al. 2019Fraternali et al. 2020Rizzo et al. 2020This work
Figure 9.
Location on the velocity dispersion versus redshift plane of the source galaxies in our sample. The values of 𝜎 ext (column 6 in Table 8) are shownhere. The blue filled right triangles are velocity dispersions from HI data (Bacchini et al. 2019). The empty markers are CO velocity dispersions (Bacchiniet al. 2020; Girard et al. 2021; Übler et al. 2019, 2018; Girard et al. 2018; Swinbank et al. 2011) . The filled markers at 𝑧 (cid:38) 𝛼 , [OII], [OIII]) and atomic (HI)/molecular (CO) tracers (Übler et al. 2019).The dot-dashed lines show the corresponding extrapolations up to 𝑧 ≈ persion profiles allowed us to gain insights on how the extremeastrophysical processes and conditions characterising the early Uni-verse affect the properties of these young galaxies. In particular,the sample studied in this paper allowed us to confirm a previousfinding (Rizzo et al. 2020): DSFGs have 𝑉 / 𝜎 in the range 7 to 15and median velocity dispersion in the range between ≈
30 and ≈ − . To date, such dynamically cold galaxies with the measuredvalues of SFR and gas fractions are not predicted by any model(e.g., Pillepich et al. 2019; Hung et al. 2019; Dekel et al. 2020).By investigating the velocity dispersions and SFR of the studiedgalaxies, we found that stellar feedback mechanisms are able tosustain the level of observed turbulence with low efficiency. Thereis, therefore, no need of additional drivers of turbulence, such asgravitational instabilities. We then compare the velocity dispersionsin our sample with the analogue measurements from the literaturefor galaxies at lower redshifts and with the empirical 𝜎 - 𝑧 relationfound by Übler et al. (2019) for normal main-sequence galaxiesup to 𝑧 (cid:46)
3. The median values of the velocity dispersions in oursample is only a factor of ≈ (cid:46) 𝑧 (cid:46)
2. We findthat the extrapolation of the Übler et al. (2019) 𝜎 - 𝑧 relation to theredshifts studied in this paper results in a systematic overestimationof the observed velocity dispersions by a median factor of ≈ ∼ × 𝑀 (cid:12) and ∼ × 𝑀 (cid:12) . Their gas fraction ranges between 0.4 and 0.6.Furthermore, four out of six galaxies have a stellar component whichis well described by a Sérsic index of 𝑛 (cid:38) 𝑧 (cid:46)
3. Inparticular, by comparing our sample with local ETGs and 𝑧 ∼ 𝑧 samples, two galaxies are consis-tent with the low-mass end of the cSFGs, while the others coverthe low-mass, small-size region. Interestingly, SPT0113-46 has thelowest gas fraction within our sample and, is the only galaxy onthe main sequence. All these properties seem to be indicating thatSPT0113-46 is in the process of consuming its residual gas, quench-ing its star formation and transforming into a typical ETG. We alsofound that the baryonic masses in our sample are all consistent withthose of local ETGs. This result allowed us to set constraints onthe small amount of baryonic matter that can be accreted in thefollowing ∼
12 Gyr of the lifetime of these galaxies.Our results are based on just nine galaxies. While statisticallysignificant conclusions can not be drawn, these first results arepromising. In the near future, the thousands of strong gravitationally
MNRAS000
MNRAS000 , 1–17 (2020) F. Rizzo et al. . . . . . ( M star / M (cid:12) ) − . − . . . . l og ( R e / kp c ) a . . . f g a s . . . . . ( M star / M (cid:12) )bETGs, z ∼ z ∼ z ∼ . MQGs, z ∼ . MQGs, z ∼ . MQGs, z ∼ DSFG, z = . DSFGs, z ∼ . , this work . . . . . l og ( t d e p / y r) Figure 10.
Location on the size versus stellar mass plane for the source galaxies in our sample colour-coded according to their gas fraction (panel a, see columnthree of Table 5) and depletion time (panel b, see column six of Table 5). In panel b, the stellar masses and sizes are obtained under the assumption that allthe observed gas will be converted into stars, preserving the disc configuration (see the baryonic quantities in columns four and five of Table 5). Under thisassumption, the sizes should be considered as upper limits. The green squares correspond to local ETGs from the ATLAS survey (Cappellari et al. 2013),the gray stars are the cSFG at 𝑧 ∼ lensed galaxies (Oguri & Marshall 2010; Collett 2015; McKeanet al. 2015) discovered by the Euclid space telescope , the RubinObservatory and the Square Kilometer Array , combined with thecapability of ALMA and JWST will provide us with the opportunityto fully characterise the structural and dynamical properties on largesample of galaxies up to the epoch of reionization. ACKNOWLEDGEMENTS
F.R. is grateful to Cecilia Bacchini and Francesco Valentinofor useful comments and discussions. SV thanks the MaxPlanck Society for support through a Max Planck Lise MeitnerGroup, and acknowledges funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 re-search and innovation programme (LEDA: grant agreement No758853). This paper makes use of the following ALMA data:ADS/JAO.ALMA REFERENCES
Aravena M., et al., 2016, Mon. Not. R. Astron. Soc., 457, 4406Arribas S., Colina L., Bellocchi E., Maiolino R., Villar-Martín M., 2014,A&A, 568, A14Bacchini C., Fraternali F., Iorio G., Pezzulli G., 2019, A&A, 622, A64Bacchini C., Fraternali F., Iorio G., Pezzulli G., Marasco A., Nipoti C., 2020,arXiv e-prints, p. arXiv:2006.10764Barro G., et al., 2014, Astrophys. J., 791, 52Belli S., Newman A. B., Ellis R. S., 2017, ApJ, 834, 18Bezanson R., van Dokkum P. G., Tal T., Marchesini D., Kriek M., Franx M.,Coppi P., 2009, ApJ, 697, 1290Binney J., Tremaine S., 2008, Galactic Dynamics: Second EditionBisigello L., Caputi K. I., Grogin N., Koekemoer A., 2018, A&A, 609, A82Bouché N., et al., 2010, ApJ, 718, 1001Bournaud F., Elmegreen B. G., Martig M., 2009, ApJ, 707, L1Brinchmann J., Charlot S., White S. D. M., Tremonti C., Kauffmann G.,Heckman T., Brinkmann J., 2004, MNRAS, 351, 1151Brooks A., Christensen C., 2016, Bulge Formation via Mergers in Cosmo-logical Simulations. p. 317, doi:10.1007/978-3-319-19378-6_12Cappellari M., et al., 2013, Mon. Not. R. Astron. Soc., 432, 1709Caputi K. I., et al., 2017, ApJ, 849, 45Carlstrom J. E., et al., 2011, Publ. Astron. Soc. Pac., 123, 568Cassata P., et al., 2013, ApJ, 775, 106Chisholm J., Tremonti C. A., Leitherer C., Chen Y., 2017, MNRAS, 469,4831Cicone C., Maiolino R., Marconi A., 2016, A&A, 588, A41Cimatti A., Fraternali F., Nipoti C., 2019, Introduction to Galaxy Formationand Evolution: From Primordial Gas to Present-Day GalaxiesCollett T. E., 2015, ApJ, 811, 20Concas A., Popesso P., Brusa M., Mainieri V., Erfanianfar G., Morselli L.,MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies arXiv:1502.03362 )McMullin J. P., Waters B., Schiebel D., Young W., Golap K., 2007, inShaw R. A., Hill F., Bell D. J., eds, Astronomical Society of the PacificConference Series Vol. 376, Astr. Soc. P.. p. 127McQuinn K. B. W., van Zee L., Skillman E. D., 2019, ApJ, 886, 74Naab T., Ostriker J. P., 2017, ARA&A, 55, 59Naab T., Johansson P. H., Ostriker J. P., 2009, Astrophys. J. Lett., 699, L178Naab T., et al., 2014, MNRAS, 444, 3357Navarro J. F., Frenk C. S., White S. D. M., 1996, Astrophys. J., 462, 563Neeleman M., Prochaska J. X., Kanekar N., Rafelski M., 2020, Nature, 581,269Nelson D., et al., 2019a, MNRAS, 490, 3234Nelson E. J., et al., 2019b, ApJ, 870, 130Noeske K. G., et al., 2007, ApJ, 660, L43Nordon R., Sternberg A., 2016, MNRAS, 462, 2804Oesch P. A., et al., 2010, ApJ, 714, L47Oguri M., Marshall P. J., 2010, MNRAS, 405, 2579Ohlin L., Renaud F., Agertz O., 2019, MNRAS, 485, 3887Oser L., Ostriker J. P., Naab T., Johansson P. H., Burkert A., 2010, ApJ, 725,2312Ostriker E. C., Shetty R., 2011, ApJ, 731, 41Papadopoulos P. P., Thi W. F., Viti S., 2002, ApJ, 579, 270Pillepich A., et al., 2019, arXiv e-prints, p. arXiv:1902.05553Pineda J. L., Langer W. D., Velusamy T., Goldsmith P. F., 2013, A&A, 554,A103Planck Collaboration et al., 2016, A&A, 594, A13Powell D., Vegetti S., McKean J. P., Spingola C., 2020, arXiv e-prints, p.arXiv:2005.03609Renzini A., et al., 2015, MNRAS, 454, 4197Reuter C., et al., 2020, ApJ, 902, 78Rigopoulou D., et al., 2014, ApJ, 781, L15Rizzo F., Vegetti S., Fraternali F., Di Teodoro E., 2018, Mon. Not. R. Astron.Soc., 481, 5606Rizzo F., Vegetti S., Powell D., Fraternali F., McKean J. P., Stacey H. R.,White S. D. M., 2020, Nature, 584, 201Rodriguez-Gomez V., et al., 2016, MNRAS, 458, 2371Romeo A. B., Falstad N., 2013, MNRAS, 433, 1389Sancisi R., Fraternali F., Oosterloo T., van der Hulst T., 2008, A&ARv, 15,189Sargsyan L., Samsonyan A., Lebouteiller V., Weedman D., Barry D.,Bernard-Salas J., Houck J., Spoon H., 2014, ApJ, 790, 15Satyapal S., Ellison S. L., McAlpine W., Hickox R. C., Patton D. R., MendelJ. T., 2014, MNRAS, 441, 1297Schneider P., 2006, in Meylan G., Jetzer P., North P., Schneider P., KochanekC. S., Wambsganss J., eds, Saas-Fee Advanced Course 33: GravitationalLensing: Strong, Weak and Micro. pp 1–89Sharda P., et al., 2019, MNRAS, 487, 4305Silk J., 2013, ApJ, 772, 112Smit R., et al., 2018, Nature, 553, 178Speagle J. S., 2019, arXiv e-prints, p. arXiv:1904.02180Spilker J. S., et al., 2016, Astrophys. J., 826, 112Stacey G. J., Hailey-Dunsheath S., Ferkinhoff C., Nikola T., Parshley S. C.,Benford D. J., Staguhn J. G., Fiolet N., 2010a, ApJ, 724, 957Stacey G. J., Hailey-Dunsheath S., Ferkinhoff C., Nikola T., Parshley S. C.,Benford D. J., Staguhn J. G., Fiolet N., 2010b, Astrophys. J., 724, 957Stacey H. R., et al., 2020, arXiv e-prints, p. arXiv:2009.01277Stanley F., Jolly J. B., König S., Knudsen K. K., 2019, A&A, 631, A78Steinhardt C. L., et al., 2014, ApJ, 791, L25Strandet M. L., et al., 2016, ApJ, 822, 80Swinbank A. M., et al., 2011, ApJ, 742, 11Swinbank A. M., et al., 2017, MNRAS, 467, 3140Tacchella S., Dekel A., Carollo C. M., Ceverino D., DeGraf C., Lapiner S.,Mand elker N., Primack J. R., 2016, MNRAS, 458, 242Tacchella S., et al., 2018, ApJ, 859, 56Tadaki K., et al., 2018, Nature, 560, 613MNRAS , 1–17 (2020) F. Rizzo et al.
Tamburro D., Rix H. W., Leroy A. K., Mac Low M. M., Walter F., KennicuttR. C., Brinks E., de Blok W. J. G., 2009, AJ, 137, 4424Tanaka M., et al., 2019, ApJ, 885, L34Tasca L. A. M., et al., 2015, A&A, 581, A54Terzić B., Graham A. W., 2005, Mon. Not. R. Astron. Soc., 362, 197Toft S., Gallazzi A., Zirm A., Wold M., Zibetti S., Grillo C., Man A., 2012,ApJ, 754, 3Turner O. J., et al., 2017, Mon. Not. R. Astron. Soc., 471, 1280Übler H., et al., 2017, ApJ, 842, 121Übler H., et al., 2018, Astrophys. J. Lett., 854, L24Übler H., et al., 2019, Astrophys. J., 880, 48Utomo D., Blitz L., Falgarone E., 2019, ApJ, 871, 17Valentino F., et al., 2020, ApJ, 889, 93Varidel M. R., et al., 2020, MNRAS, 495, 2265Vegetti S., Koopmans L. V. E., 2009, Mon. Not. R. Astron. Soc., 392, 945Vieira J. D., et al., 2010, ApJ, 719, 763Vieira J. D., et al., 2013, Nature, 495, 344Vogelsberger M., Marinacci F., Torrey P., Puchwein E., 2020, Nature Re-views Physics, 2, 42Weiß A., et al., 2013, Astrophys. J., 767, 88Whitaker K. E., van Dokkum P. G., Brammer G., Franx M., 2012, ApJ, 754,L29Wisnioski E., et al., 2015, ApJ, 799, 209Wisnioski E., et al., 2019, ApJ, 886, 124Wolfire M. G., Hollenbach D., McKee C. F., 2010, ApJ, 716, 1191Wootten A., Thompson A. R., 2009, IEEE Proceedings, 97, 1463Zolotov A., et al., 2015, Mon. Not. R. Astron. Soc., 450, 2327van Dokkum P. G., et al., 2015, ApJ, 813, 23van der Wel A., et al., 2014, Astrophys. J., 788, 28
APPENDIX A: LENS AND KINEMATIC MODELLING
In this section, we show the outputs of the lens and kinematicfor each galaxy of the sample. As for SPT0113-46 in Section 3.2(Fig. 1 - 3), we show three sets of figures: the moment map ofthe lensed galaxies, the corresponding reconstructed source andkinematic model (Figs. A1-A4); some representative channel mapsfrom the cubes containing the data, the model, the residuals, thesource and the kinematic model (Figs. A5-A8); the position-velocitydiagrams along the minor and major axes (Figs. A9-A12).
APPENDIX B: SPATIAL RESOLUTION ON THE SOURCEPLANE
In order to estimate the spatial resolutions on the source plane wecalculate the dimensions of the triangular grid (Vegetti & Koopmans2009; Rizzo et al. 2018) in regions with SNR (cid:38)
APPENDIX C: PRIOR DISTRIBUTIONS FOR THEDYNAMICAL MODELLING
In Table C1, we show the values of the concentration parameters 𝑐 that define the distribution of the NFW dark-matter halo, equation Table B1.
Statistics of the spatial resolution on the source plane.Name Maximum resolution Median resolutionpc pcSPT0113-46 26 198SPT0345-47 47 189SPT0441-46 139 185SPT2146-56 24 173SPT2132-58 64 300
Table C1.
Concentration of the NFW dark-matter haloes, fixed parametersof the dynamical models presented in Section 3.3.Name 𝑐 SPT0113-46 3.04SPT0345-47 2.97SPT0441-46 2.77SPT2146-56 2.67SPT2132-58 2.45
Table C2.
Intervals of the prior distributions in the rotation curve decom-position (Section 3.3).Parameter Prior 𝑀 star [10 , ] 𝑀 (cid:12) 𝑅 e [0.04, 4.0] kpc 𝑛 [0.5, 10] 𝛼 [ CII ] [3.8, 238.0] 𝑀 (cid:12) / 𝐿 (cid:12) 𝑀 DM [10 , ] 𝑀 (cid:12) (7). Table C2 lists the intervals for the uniform and log-uniform(for the masses) priors employed in the dynamic fitting described inSection 3.3. This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Figure A1.
Moment maps for SPT0345-47. Same as in Fig. 1. The beam size, shown as a white ellipse on the lower left corner of panel a, is 0 . × . at a position angle of -50.6 ◦ .MNRAS000
Moment maps for SPT0345-47. Same as in Fig. 1. The beam size, shown as a white ellipse on the lower left corner of panel a, is 0 . × . at a position angle of -50.6 ◦ .MNRAS000 , 1–17 (2020) F. Rizzo et al. h m . s . s . s − ◦ RA D E C a . . . . J yb ea m − k m s − h m . s . s RAb − k m s − h m . s . s RAc k m s − − . . . kpc − kp c d . . m J ykp c − k m s − − . . . kpce − k m s − − . . . kpcf k m s − − . . . kpcg − k m s − − . . . kpch k m s − Figure A2.
Moment maps for SPT0441-46. Same as in Fig. 1. The beam size, shown as a white ellipse on the lower left corner of panel a, is 0 . × . at a position angle of -46.6 ◦ . MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies h m . s . s . s − ◦ RA D E C a m J yb ea m − k m s − h m . s . s . s RAb − k m s − h m . s . s . s RAc k m s − − kpc − kp c d . . m J ykp c − k m s − − kpce − k m s − − kpcf k m s − − kpc − kp c g − k m s − − kpch k m s − Figure A3.
Moment maps for SPT2146-56. Same as in Fig. 1. The beam size, shown as a white ellipse on the lower left corner of panel a, is 0 . × . at a position angle of -64.1 ◦ .MNRAS000
Moment maps for SPT2146-56. Same as in Fig. 1. The beam size, shown as a white ellipse on the lower left corner of panel a, is 0 . × . at a position angle of -64.1 ◦ .MNRAS000 , 1–17 (2020) F. Rizzo et al. h m . s . s − ◦ RA D E C a . . . J yb ea m − k m s − h m . s . s RAb − k m s − h m . s . s RAc k m s − − kpc − kp c d . . . m J ykp c − k m s − − kpce − k m s − − kpcf k m s − − kpcg − k m s − − kpch k m s − Figure A4.
Moment maps for SPT2132-58. Same as in Fig. 1. The beam size, shown as a white ellipse on the lower left corner of panel a, is 0 . × . at a position angle of 63.3 ◦ . MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Figure A5.
Channel maps for SPT0345-47. Same as in Fig. 2.MNRAS000
Channel maps for SPT0345-47. Same as in Fig. 2.MNRAS000 , 1–17 (2020) F. Rizzo et al.
Figure A6.
Channel maps for SPT0441-46. Same as in Fig. 2. MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies Figure A7.
Channel maps for SPT2146-56. Same as in Fig. 2.MNRAS000
Channel maps for SPT2146-56. Same as in Fig. 2.MNRAS000 , 1–17 (2020) F. Rizzo et al. − − − − − − − − − Figure A8.
Channel maps for SPT2132-58. Same as in Fig. 2. MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies − Offset (kpc) − − − − ∆ V L O S ( k m / s ) a − Offset (kpc) − − − − ∆ V L O S ( k m / s ) b Figure A9.
Position-velocity diagrams for SPT0345-47. Same as in Fig. 3. Offset (kpc) V L O S ( k m / s ) a Offset (kpc) V L O S ( k m / s ) b Figure A10.
Position-velocity diagrams for SPT0441-46. Same as in Fig. 3.MNRAS000
Position-velocity diagrams for SPT0441-46. Same as in Fig. 3.MNRAS000 , 1–17 (2020) F. Rizzo et al. − − − Offset (kpc) − − ∆ V L O S ( k m / s ) a − − Offset (kpc) − − ∆ V L O S ( k m / s ) b Figure A11.
Position-velocity diagrams for SPT2146-56. Same as in Fig. 3. Offset (kpc) V L O S ( k m / s ) a Offset (kpc) V L O S ( k m / s ) b Figure A12.
Position-velocity diagrams for SPT2132-58. Same as in Fig. 3. MNRAS , 1–17 (2020) ynamics of z ∼ . dusty star-forming galaxies x x x .
50 10 .
75 11 . x x x x x
15 30 x . . . x . . . x . . . x .
25 10 .
50 10 . x x .
50 0 .
75 1 . x .
75 1 .
00 1 . x . . . x x . . . . . .
25 10 .
50 10 . . . . . . . . . . .
90 10 .
20 10 . .
15 0 .
30 2 4 12 .
60 12 .
75 12 .
90 6 9 12 . . . . . . . . . . . . . . .
90 1 .
05 10 . . . . . . . . . . . . . . . . . . . . . log( M star/ M ⊙) R e(kpc) n α [CII]log( M DM/ M ⊙) log( M star/ M ⊙) R e(kpc) n α [CII]log( M DM/ M ⊙) R e ( kp c ) n α [ C II] l og ( M D M / M ⊙ ) R e ( kp c ) n α [ C II] l og ( M D M / M ⊙ ) R e ( kp c ) n α [ C II] l og ( M D M / M ⊙ ) SPT0113-46SPT0345-47 SPT0441-46SPT2132-58SPT2146-56
Figure C1.
Posterior distributions of the dynamical parameters for the five galaxies studied in this paper, as indicated in the legend. The dashed lines in the 1Dhistograms show the 16th, 50th and 84th percentiles (see Table 4).MNRAS000