Dust temperature in ALMA \hbox{[C $\scriptstyle\rm II $]}-detected high-z galaxies
L. Sommovigo, A. Ferrara, S. Carniani, A. Zanella, A. Pallottini, S. Gallerani, L. Vallini
MMNRAS , 1–14 (0000) Preprint 19 February 2021 Compiled using MNRAS L A TEX style file v3.0
Dust temperature in ALMA [C II ]-detected high- z galaxies L. Sommovigo (cid:63) , A. Ferrara , S. Carniani , A. Zanella , A. Pallottini , S. Gallerani ,L. Vallini Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy INAF - Osservatorio Astronomico di Padova, Vicolo Osservatorio 5, I-35122 Padova
19 February 2021
ABSTRACT
At redshift z > the far-infrared (FIR) continuum spectra of main-sequence galaxiesare sparsely sampled, often with a single data point. The dust temperature T d , SED thushas to be assumed in the FIR continuum fitting. This introduces large uncertaintiesregarding the derived dust mass ( M d ), FIR luminosity, and obscured fraction of the starformation rate. These are crucial quantities to quantify the effect of dust obscuration inhigh- z galaxies. To overcome observations limitations, we introduce a new method thatcombines dust continuum information with the overlying [C II ] µ m line emission. Bybreaking the M d − T d , SED degeneracy, with our method, we can reliably constrain thedust temperature with a single observation at µ m. This method can be applied toall ALMA and NOEMA [C II ] observations, and exploited in ALMA Large Programssuch as ALPINE and REBELS targeting [C II ] emitters at high- z . We also providea physical interpretation of the empirical relation recently found between molecular gas mass and [C II ] luminosity. We derive an analogous relation linking the total gassurface density and [C II ] surface brightness. By combining the two, we predict thecosmic evolution of the surface density ratio Σ H / Σ gas . We find that Σ H / Σ gas slowlyincreases with redshift, which is compatible with current observations at < z < . Key words: galaxies: high-redshift, infrared: ISM, ISM: dust, extinction, methods:analytical – data analysis
The Hubble Space Telescope (HST) and ground-based tele-scopes have been used to investigate the rest-frame Ultra-violet (UV) emission from early galaxies (for a recent the-oretical review see Dayal & Ferrara 2018). The advent ofhigh sensitivity millimetre interferometers such as the Ata-cama Large Millimeter Array (ALMA), allowed us for thefirst time to study also the Far-Infrared (FIR) emission fromthese sources (see e.g. Carilli & Walter 2013).ALMA can detect both the FIR continuum and thebrightest FIR lines in “normal” (i.e. main sequence) galax-ies at z ≥ (see e.g. Capak et al. 2015; Willott et al. 2015;Bouwens et al. 2016; Laporte et al. 2017; Barisic et al. 2017;Carniani et al. 2017; Bowler et al. 2018; Carniani et al.2018b; Carniani et al. 2018a; De Breuck et al. 2019; Tamuraet al. 2019; Bakx et al. 2020; Bethermin et al. 2020; Schaereret al. 2020a). The FIR continuum is emitted as thermal ra-diation by dust grains, heated through the absorption ofUV and optical light from newly born stars (see e.g. Draine (cid:63) [email protected] T d , SED and thedust mass M d , which are degenerate quantities. For the si-multaneous determination of T d , SED and M d the most com-mon approach is to fit the observed Spectral Energy Distri-bution (SED) with a single temperature grey body function.At z ≥ most of the sources observed with ALMA, whendetected in dust continuum, have only a single (or very few)data-point at FIR wavelengths (e.g. Bouwens et al. 2016;Barisic et al. 2017; Bowler et al. 2018; Hashimoto et al. 2019;Tamura et al. 2019). Consequently, T d , SED is assumed a pri-ori in the fitting to reduce the degrees of freedom. The lackof knowledge of T d , SED results in very large uncertainties onthe derived galaxy properties, such as M d , the far-infrared We underline that T d , SED does not necessarily correspond to thedust physical temperature, which is instead characterised by aProbability Distribution Function (PDF, see e.g. Behrens et al.2018; Sommovigo et al. 2020). In general, T d , SED does not neces-sarily provide a statistically sound representation of the PDF. Fora discussion see Appendix A. © a r X i v : . [ a s t r o - ph . GA ] F e b Sommovigo et al. luminosity L FIR , and obscured star formation rate (SFR; fora detailed discussion see e.g. Sommovigo et al. 2020). Fur-ther observations in a larger number of ALMA bands wouldameliorate the problem, but not necessarily solve it. Indeed,MIR wavelengths remain inaccessible to ALMA. Neverthe-less, the inclusion of ALMA band − − data would improvethe results significantly for galaxies at z > ∼ . At these red-shifts, these bands sample the SED at shorter wavelengths,closer to the FIR emission peak.Here we intend to overcome current observational lim-itations by combining dust continuum measurements withthe widely observed fine-structure transition of singly ion-ized carbon [C II ] at µ m . This line is the dominant coolantof the neutral atomic gas in the ISM (Wolfire et al. 2003),making it one of the brightest FIR lines in most galaxies(Stacey et al. 1991). Moreover, [C II ] has been proved to beconnected to the SFR of local (De Looze et al. 2014; Herrera-Camus et al. 2015) and high- z galaxies (see e.g. Capak et al.2015; Maiolino et al. 2015a; Pentericci et al. 2016; Carnianiet al. 2017; Matthee et al. 2017; Carniani et al. 2018a; Car-niani et al. 2018b; Harikane et al. 2018; Smit et al. 2018;Carniani et al. 2020).In this work we propose a novel method for the dusttemperature computation using L CII as a proxy for the totalgas mass, and therefore for M d given a dust-to-gas ratio. Ourmethod breaks the degeneracy between M d and T d , SED in theSED fitting. This allows us to constrain T d , SED with a singlecontinuum data point. As a byproduct of our method, weprovide an interpretation of the empirical relation found byZanella et al. (2018) between M H and L CII . We also derivea more general relation connecting the total gas mass M gas with L CII . Joining the two we can also study the evolution ofthe molecular gas fraction M gas / M H with redshift.The paper is organised as follows. We present ourmethod for the dust temperature derivation in Sec. 2, andwe test it on a sample of local galaxies (Sec. 3). We thenapply the method to the few high- z galaxies ( z > , Sec. 4)for which multiple FIR continuum observations are availablein the literature. In Sec. 5 we discuss additional outputs, i.e.the physical explanation for the relation by Zanella et al.(2018), and the molecular gas fraction evolution with z . InSec. 6 we summarise our results and discuss future applica-tions. Before introducing our method, we discuss two key ingredi-ents, i.e. the dust-to-gas ratio D and the conversion factor α CII = M gas / L CII . Multiplying L CII by the product D · α CII weinfer M d . We can then constrain T d , SED using a single contin-uum data point. Throughout the paper, we assume a flat Universe with the fol-lowing cosmological parameters: Ω M h = . , Ω Λ = − Ω M , and Ω B h = . , h = . , σ = . , where Ω M , Ω Λ , Ω B arethe total matter, vacuum, and baryonic densities, in units of thecritical density; h is the Hubble constant in units of
100 kms − ,and σ is the late-time fluctuation amplitude parameter (PlanckCollaboration et al. 2018). Several studies (e.g. James et al. 2002; Draine & Li 2007;Galliano et al. 2008; Leroy et al. 2011) have shown that D scales linearly with metallicity, with little scatter, down to Z < ∼ . Z (cid:12) : D = D MW (cid:32) ZZ (cid:12) (cid:33) , (1)where D MW = / is the Milky Way dust-to-gas ratio(R´emy-Ruyer et al. 2014). We adopt eq. 1 as our fiducialchoice, since almost all the galaxies to which we apply ourmethod have metallicities > ∼ . Z (cid:12) . Hence, they are mostlyunaffected by deviations from this linear scaling that mightoccur at Z < ∼ . Z (cid:12) .Moreover, the ideal targets of our method are the galax-ies observed in current high- z ALMA surveys (such as e.g.ALPINE, PI: Le F´evre, and REBELS, PI: Bouwens), whichare massive (stellar mass M (cid:63) (cid:39) M (cid:12) ), dusty, and evolvedsources. From numerical simulations galaxies at z ∼ withsimilar stellar masses ( < M (cid:63) < ) are expected to have Z > ∼ . Z (cid:12) (Ma et al. 2016; Torrey et al. 2019). This is alsoconfirmed, albeit with considerable uncertainties (relativeerrors up to ∼ ), by several studies which analyse FIRlines (such as [N II ], [N III ], [C II ], C III ] and [O
III ]) observa-tions at z > ∼ − to derive Z (see e.g. Pereira-Santaella et al.2017; Hashimoto et al. 2019; De Breuck et al. 2019; Tamuraet al. 2019; Vallini et al. 2020; Bakx et al. 2020; Jones et al.2020a,b, and references therein).Current estimates of Z at high redshift will be signifi-cantly ameliorated thanks to forthcoming ALMA observa-tions and to the James Web Space Telescope (JWST) spec-troscopy. Indeed, JWST will detect several optical nebularlines (such as H β , H α , [N II ], [O II ] and [O III ]) out to z ∼ .This will allow us to reduce the relative errors associated to Z down to ∼ even at very high- z (see e.g. Wright et al.2010; Maiolino & Mannucci 2019; Chevallard et al. 2019),improving also our knowledge of the dust-to-gas ratios. II ]-to-total gas mass conversion factor The [C II ] conversion factor, α CII , expresses the specific [C II ]emission efficiency per unit total (i.e. atomic + molecular)gas mass. To investigate the relation between total L CII and M gas , we use the following empirical relations : Σ SFR = − . Σ . (De Looze relation) (2) Σ SFR = − κ s Σ . (Kennicutt − Schmidt relation) (3) Σ gas = α CII Σ CII (conversion relation) (4)The first relation has been inferred by De Looze et al. (2014,hereafter, DL) from the Dwarf Galaxy Survey (DGS) sample At very low metallicities ( Z < ∼ . Z (cid:12) ), deviations from the lin-ear relation have been suggested (see e.g. Galliano et al. 2005;Galametz et al. 2011; R´emy-Ruyer et al. 2014; De Vis et al. 2019).For instance, R´emy-Ruyer et al. (2014) find a steeper D − Z rela-tion in their sample of local galaxies. However, the deviation isdriven especially by the fewer, widely scattered data at Z ≤ . Z (cid:12) . We adopt the standard units used for these quantities: surfacestar formation [M (cid:12) kpc − yr − ] , [C II ] luminosity [L (cid:12) kpc − ] , and gasdensity [M (cid:12) kpc − ] MNRAS000
III ]) observa-tions at z > ∼ − to derive Z (see e.g. Pereira-Santaella et al.2017; Hashimoto et al. 2019; De Breuck et al. 2019; Tamuraet al. 2019; Vallini et al. 2020; Bakx et al. 2020; Jones et al.2020a,b, and references therein).Current estimates of Z at high redshift will be signifi-cantly ameliorated thanks to forthcoming ALMA observa-tions and to the James Web Space Telescope (JWST) spec-troscopy. Indeed, JWST will detect several optical nebularlines (such as H β , H α , [N II ], [O II ] and [O III ]) out to z ∼ .This will allow us to reduce the relative errors associated to Z down to ∼ even at very high- z (see e.g. Wright et al.2010; Maiolino & Mannucci 2019; Chevallard et al. 2019),improving also our knowledge of the dust-to-gas ratios. II ]-to-total gas mass conversion factor The [C II ] conversion factor, α CII , expresses the specific [C II ]emission efficiency per unit total (i.e. atomic + molecular)gas mass. To investigate the relation between total L CII and M gas , we use the following empirical relations : Σ SFR = − . Σ . (De Looze relation) (2) Σ SFR = − κ s Σ . (Kennicutt − Schmidt relation) (3) Σ gas = α CII Σ CII (conversion relation) (4)The first relation has been inferred by De Looze et al. (2014,hereafter, DL) from the Dwarf Galaxy Survey (DGS) sample At very low metallicities ( Z < ∼ . Z (cid:12) ), deviations from the lin-ear relation have been suggested (see e.g. Galliano et al. 2005;Galametz et al. 2011; R´emy-Ruyer et al. 2014; De Vis et al. 2019).For instance, R´emy-Ruyer et al. (2014) find a steeper D − Z rela-tion in their sample of local galaxies. However, the deviation isdriven especially by the fewer, widely scattered data at Z ≤ . Z (cid:12) . We adopt the standard units used for these quantities: surfacestar formation [M (cid:12) kpc − yr − ] , [C II ] luminosity [L (cid:12) kpc − ] , and gasdensity [M (cid:12) kpc − ] MNRAS000 , 1–14 (0000) ust temperature in high- z galaxies of local galaxies . The second one is the Kennicutt–Schmidtrelation (Kennicutt et al. 1998, hereafter, KS). The “bursti-ness parameter” κ s quantifies the single sources deviations(upwards for starbursts, and downwards for quiescent galax-ies, see Heiderman et al. 2010; Ferrara et al. 2019) from theaverage relation. Finally, eq. 4 is equivalent to the definition α CII = M gas / L CII under the assumption that [C II ] is spatiallyextended as the gas.We combine eq. 2-4 into the following one, α CII = . κ / s Σ − . M (cid:12) L (cid:12) . (5)This relation shows that satisfying the DL and KS relationsat the same time implies that α CII cannot be constant. Itmust depend on the SFR and its mode (burst vs. quiescent).At a fixed SFR, galaxies with large κ s values (starbursts)have a lower α CII and therefore can produce a larger [C II ]luminosity per unit gas mass. The same is true if κ s is fixedand the SFR is larger. In high star formation regimes themore efficient [C II ] emission might depend on a more intenseradiation field or higher gas density (Ferrara et al. 2019;Pallottini et al. 2019). z As we approach the Epoch of Reionization (EoR) a preciseassessment of the KS relation becomes very difficult. H I isnot observable at z ≥ , and typical H tracers (CO anddust) suffer from severe limitations . Hence Σ gas is not reli-ably measurable. So far there is considerable evidence thatFIR-detected galaxies at z > are strong UV emitters withlarge SFRs, i.e. they are most likely starbursts ( κ s (cid:29) , seee.g. Vallini et al. 2020,Vallini in prep.).The validity of the DL relation might also be questionedat high- z . Most studies agree that this relation is still valid at z > , although its scatter is ∼ times larger than the localone (Carniani et al. 2018b; Carniani et al. 2018a; Mattheeet al. 2019; Schaerer et al. 2020a). However, in extreme cases(SFR < −
50 M (cid:12) / yr or z > ) high- z sources have beenfound to deviate more than σ from the local DL relation,being systematically below the latter (Pentericci et al. 2016;Knudsen et al. 2016; Bradaˇc et al. 2017; Matthee et al. 2019;Laporte et al. 2019).Recently, Carniani et al. (2020) showed that EoR galax-ies lay on the slightly different (w.r.t. the one in eq. 2) DL For details on the DGS sample see Sec. 3 Observing CO transitions becomes challenging due to the largercosmological distance of sources, and lower contrast against theCosmic Microwave Background (CMB, see e.g. Da Cunha et al.2013). This also makes dust emission observations more difficultat high- z . This is particularly true in the presence of cold dustnearly in equilibrium with the CMB (Da Cunha et al. 2013).Most importantly, the impossibility to simultaneously constrain M d and T d , SED due to the few available data points, results in verylarge uncertainties on M d , and therefore M H . This might, however, be due to an observational bias. Indeed,most high-z ALMA targets have been selected from UV observa-tions (i.e. by construction they are strong UV emitters). Thereare few exceptions represented by the (sub)mm-selected targets,as in the surveys ASPECS (Walter et al. 2016) and SPT (Weißet al. 2013). relation appropriate for starburst/H II -like galaxies : Σ SFR = − . y Σ CII (DeLooze relation / starbursts) . (6)once that obscured fraction of the SFR is appropriately in-cluded in Σ SFR . The factor y = r CII / r (cid:63) is introduced sincethere is growing evidence that at z > [C II ] emission ismore extended than UV emission ( . < ∼ y < ∼ at z > , seee.g. Carniani et al. 2017, 2018a; Matthee et al. 2017, 2019;Fujimoto et al. 2019, 2020; Ginolfi et al. 2020; Carniani et al.2020). The origin of a such extended [C II ] structure is stilldebated. Current explanations range from emission by a)outflow remnants in the Circum Galactic Medium (CGM,see e.g. Maiolino et al. 2015b; Vallini et al. 2015; Galleraniet al. 2017; Fujimoto et al. 2019; Pizzati et al. 2020; Gi-nolfi et al. 2020), b) CGM gas illuminated by the galaxiesstrong radiation field (Carniani et al. 2017; Carniani et al.2018b; Fujimoto et al. 2020)), to c) actively accreting satel-lites (Pallottini et al. 2017a; Carniani et al. 2018a; Mattheeet al. 2019).By combining eq. 6 with eqs. 3 and 4, we derive thehigh- z conversion factor α CII , hz = . κ / s y Σ − . M (cid:12) L (cid:12) . (7)Using the DL relation for starbusts, independently on thechosen factor y , results in a rescaling upwards of α CII at high- z with respect to z (cid:39) . The dependence on Σ SFR and κ s is almost unchanged. Additionally, at a fixed SFR and κ s ,galaxies with lower y (less extended [C II ] emission) have alower α CII , i.e. a larger [C II ] luminosity per unit gas mass. We assume an optically thin, single-temperature, grey-bodyapproximation. The dust continuum flux F ν observed againstthe CMB at rest-frame frequency ν can be written as (seee.g. Da Cunha et al. 2013; Kohandel et al. 2019) F ν = g ( z ) M d κ ν [ B ν ( T (cid:48) d , SED ) − B ν ( T CMB )] , (8)where g ( z ) = (1 + z ) / d L , d L is the luminosity distance to red-shift z , k ν is the dust opacity, B ν is the black-body spectrum,and T CMB ( z ) is the CMB temperature at redshift z .At wavelengths λ > µ m , k ν can be approximated as(Draine 2004) κ ν = κ ∗ (cid:32) νν ∗ (cid:33) β . (9)where the choice of ( κ ∗ , ν ∗ , β ) depends on the assumed dustproperties. We consider Milky Way-like dust, for which stan-dard values are ( k ∗ , ν ∗ , β ) = (52.2 cm g − , , 2), seeDayal et al. (2010). We also account for the fact that theCMB acts as a thermal bath for dust grains, setting a lowerlimit for their temperature. We correct T d , SED for this effect,following the prescription by Da Cunha et al. (2013). Which is also provided in De Looze et al. 2014 T CMB ( z ) = T CMB , (1 + z ) , with T CMB , = . (Fixsen 2009) T (cid:48) d , SED = { T + β d , SED + T + β CMB , [(1 + z ) + β − } / (4 + β ) . In the following, wedrop the apex from the dust temperature symbol for better read-ability. It is then intended we always refer to the CMB-corrected dust temperature.MNRAS , 1–14 (0000)
Sommovigo et al.
Eq. 8 has two parameters, M d and T d , SED . [C II ] observa-tions can be used to determine M d : M d = DM gas = D α CII L CII . (10)We substitute in eq. 8 and specialize to the [C II ] line fre-quency ν = . GHz. We thus introduce the [C II ] -baseddust temperature T d , CII , defined as the solution of F ν = g ( z ) D α CII L CII κ ν [ B ν ( T d , CII ) − B ν ( T CMB )] . (11)We can re-write this equation in a more compact form, yield-ing the explicit expression for T d , CII : T d , CII = T ln(1 + f − ) . (12)where T = h P ν / k B = . K is the temperature correspond-ing to the [C II ] transition energy ( k B and h P are the Boltz-mann and Planck constants). We have defined: f = B ( T CMB ) + A − ˜ F ν , (13)where B ( T CMB ) = [exp( T / T CMB ) − − . The non-dimensionalcontinuum flux ˜ F ν and the constant A are defined as ˜ F ν = λ F ν k B T = . × − (cid:32) F ν mJy (cid:33) , A = g ( z ) α CII DL CII k = . × − (cid:34) g ( z ) g (6) (cid:35) (cid:32) L CII L (cid:12) (cid:33) (cid:32) α CII M (cid:12) / L (cid:12) (cid:33) D . (14)Clearly, if ˜ F ν / A (cid:29) B ( T CMB ) the CMB effects on dust tem-perature become negligible.Eq. 12 can be used to compute T CII using a single
GHz observation (which provides both L CII and F ν ) onceone has an estimate for the two parameters D (Sec. 2.1) and α CII (Sec. 2.2).
Writing explicitly the expressions for D and α CII in eq. 14,we can show that T d , CII is ultimately a function of the fol-lowing parameters ( κ s , z , F ν , Z , Σ SFR , L CII ) . For local galaxies,all these quantities are well constrained by observations. Inpractice we solve eq. 12 performing a random sampling ofthese parameters around the measured values, within theuncertainties. Differently, at high- z κ s is largely unknown .Hence we consider a broad random uniform distribution forthis parameter.To constrain T d , CII at high- z , we add the following phys-ical conditions:(i) M d does not exceed the largest dust mass producibleby supernovae (SNe), M d , max . To quantify M d , max we take ametal yield constraint y Z < (cid:12) per SN, and assume thatall the produced metals are later included in dust grains.Then: M d , max = y Z ν SN M (cid:63) (15)where ν SN = (53 M (cid:12) ) − is the number of SNe per solar mass ofstars formed for a standard Salpeter 1-100 M (cid:12) IMF (Ferrara& Tolstoy 2000). At high- z we also introduce the parameter y . This is often wellconstrained by observations. (ii) SFR
FIR ∼ − L FIR (Kennicutt et al. 1998), does notexceed the total measured SFR. This directly relates to thedust mass and temperature as L FIR = M d ( T d , CII / . (Dayalet al. 2010). T d , CII solutions not satisfying (i) and (ii) are discarded.These conditions result in a lower (upper) cut for very cold(hot) dust temperatures corresponding to unphysically largedust masses (FIR luminosity and SFR). This allows us toeffectively constrain T d , CII at high- z despite the lack of infor-mation on κ s . We have selected local galaxies for which the needed dataare available: (a) κ s , (b) redshift, (c) metallicity, (d) totalSFR and Σ SFR , (e) total L CII , (f) at least two FIR continuumdetections, one of which at ν . These galaxies are drawn fromthe following catalogs : • Dwarf Galaxy Survey (DGS, see e.g. De Looze et al.2014; Madden et al. 2014, 2020): targeting a total of localdwarf galaxies, whose [C II ], [O I ] and [O III ] line emission aremapped with the Hershel Space Observatory; • Lyman Alpha Reference Sample (LARS, see e.g. Hayeset al. 2014; ¨Ostlin et al. 2014): consisting of 14 low-redshift( z = . − . ) mildly starbursting systems observed in mul-tiple bands with HST. This sample was intended as a locallaboratory for the study of Ly α , which is one of the dominantlines used to characterise high- z sources; • The complete database of the Hershel/PhotoconductorArray Camera and Spectrometer (PACS, see Fernandez-Ontiveros et al. 2016): a coherent database of spectro-scopic observations of FIR fine-structure lines (in the range − µ m ) collected from the Herschel/PACS spectrome-ter archive for a local sample of Active Galactic Nuclei(AGNs), starburst, and dwarf galaxies.The selected galaxies and their properties are reported inTab. 1. Hereafter we refer to these galaxies as the local sam-ple . The DL relation (eq. 2) has been derived from a por-tion of this same sample and therefore is nearly satisfied byconstruction. The galaxies in the local sample also followthe KS relation with a scatter consistent with that of localspirals and starbursts ( . ≤ κ s ≤ . , see Fig. 1, left panel).We compare the value of α CII resulting from eq. 5 withthe ratio M gas / L CII derived from observations (Fig. 1, rightpanel). We find . ≤ log α CII ≤ . . The predicted α CII areconsistent with the data at < . σ , although there are sig-nificant uncertainties.Finally, we compare T d , CII and T d , SED . For the local sam-ple galaxies we deduce T d , SED from the following equation: F ν F ν = κ ν [ B ν ( T d , SED ) − B ν ( T CMB )] κ ν [ B ν ( T d , SED ) − B ν ( T CMB )] . (16) Other local samples, such as KINGFISH (Kennicutt et al.2011) and GOALS surveys (Chu et al. 2017), lack one of the re-quired data (total [C II ] luminosity and metallicity, respectively).MNRAS000
FIR ∼ − L FIR (Kennicutt et al. 1998), does notexceed the total measured SFR. This directly relates to thedust mass and temperature as L FIR = M d ( T d , CII / . (Dayalet al. 2010). T d , CII solutions not satisfying (i) and (ii) are discarded.These conditions result in a lower (upper) cut for very cold(hot) dust temperatures corresponding to unphysically largedust masses (FIR luminosity and SFR). This allows us toeffectively constrain T d , CII at high- z despite the lack of infor-mation on κ s . We have selected local galaxies for which the needed dataare available: (a) κ s , (b) redshift, (c) metallicity, (d) totalSFR and Σ SFR , (e) total L CII , (f) at least two FIR continuumdetections, one of which at ν . These galaxies are drawn fromthe following catalogs : • Dwarf Galaxy Survey (DGS, see e.g. De Looze et al.2014; Madden et al. 2014, 2020): targeting a total of localdwarf galaxies, whose [C II ], [O I ] and [O III ] line emission aremapped with the Hershel Space Observatory; • Lyman Alpha Reference Sample (LARS, see e.g. Hayeset al. 2014; ¨Ostlin et al. 2014): consisting of 14 low-redshift( z = . − . ) mildly starbursting systems observed in mul-tiple bands with HST. This sample was intended as a locallaboratory for the study of Ly α , which is one of the dominantlines used to characterise high- z sources; • The complete database of the Hershel/PhotoconductorArray Camera and Spectrometer (PACS, see Fernandez-Ontiveros et al. 2016): a coherent database of spectro-scopic observations of FIR fine-structure lines (in the range − µ m ) collected from the Herschel/PACS spectrome-ter archive for a local sample of Active Galactic Nuclei(AGNs), starburst, and dwarf galaxies.The selected galaxies and their properties are reported inTab. 1. Hereafter we refer to these galaxies as the local sam-ple . The DL relation (eq. 2) has been derived from a por-tion of this same sample and therefore is nearly satisfied byconstruction. The galaxies in the local sample also followthe KS relation with a scatter consistent with that of localspirals and starbursts ( . ≤ κ s ≤ . , see Fig. 1, left panel).We compare the value of α CII resulting from eq. 5 withthe ratio M gas / L CII derived from observations (Fig. 1, rightpanel). We find . ≤ log α CII ≤ . . The predicted α CII areconsistent with the data at < . σ , although there are sig-nificant uncertainties.Finally, we compare T d , CII and T d , SED . For the local sam-ple galaxies we deduce T d , SED from the following equation: F ν F ν = κ ν [ B ν ( T d , SED ) − B ν ( T CMB )] κ ν [ B ν ( T d , SED ) − B ν ( T CMB )] . (16) Other local samples, such as KINGFISH (Kennicutt et al.2011) and GOALS surveys (Chu et al. 2017), lack one of the re-quired data (total [C II ] luminosity and metallicity, respectively).MNRAS000 , 1–14 (0000) ust temperature in high- z galaxies Figure 1. Left panel : Measured Σ SFR vs. Σ gas of our local sample. We associate to each galaxy an ID number which will be used in thefollowing plots. The dashed blue line represents the KS relation (eq. 3) and the blue shaded region its intrinsic scatter. Also shown forreference are a number of local spirals (black triangles, Kennicutt 1998) and starbursts (black stars, Kennicutt 1998). We distinguisheach galaxy in our local sample with a different colour and identify them in the legend with their IDs as in Tab. 1. We also differentiatethe three sub-samples with a different shape: a (star), b (square), and c (triangle, all references are the same as in Tab. 1). We notegalaxies in the local sample are consistent within errors with the KS relation. Right panel : [C II ] conversion factor computed from eq.5 vs. the observed log( M gas / L CII ) for the same galaxies as in the left panel. The solid symbols correspond to the value of α CII obtainedconsidering for each galaxy the κ s value computed from the measured Σ SFR and Σ gas . The dotted dashed black line is the bisector, i.e. itrepresents the relation log α CII = log( M gas / L CII ) . The fact that the points are lay within < ∼ . σ from the bisector shows that eq. 2.2 givesa good estimate of the observed gas mass-to-[C II ] luminosity ratio. Figure 2.
Comparison between T d , CII (from eq. 12) and T d , SED (from eq. 16) in our local template sample of galaxies (see Tab. 1 for theproperties of each galaxy corresponding to the ID in legend). The dotted dashed grey line represents the relation T d , CII = T d , SED and theshaded area a deviation from the equality of ± .MNRAS , 1–14 (0000) Sommovigo et al.
ID Galaxy ν F ν d L F ν Z log Σ SFR log L CII κ s T d , SED T d , CII [GHz] [Jy] [Mpc] [Jy] [ Z (cid:12) ] [M (cid:12) yr − kpc − ] [L (cid:12) ] [K] [K] a . ± .
007 3 . . ± .
029 0 . − .
041 4 .
119 0 . . + . − . . + . − . a . ± .
026 4 . . ± .
037 0 . − .
621 4 .
994 0 . . + . − . . + . − . a . ± .
100 3 . . ± .
800 0 . − .
721 6 .
669 3 . . + . − . . + . − . a . ± .
280 12 . . ± .
431 0 . − .
818 6 .
586 0 . . + . − . . + . − . a . ± .
020 2 . . ± .
820 0 . − .
054 5 .
977 0 . . + . − . . + . − . a . ± .
203 87 . . ± .
345 0 . − .
957 8 .
281 0 . . + . − . . + . − . a . ± .
172 20 4 . ± .
487 0 . − .
096 7 .
169 0 . . + . − . . + . − . b . ± .
011 131 . . ± .
014 0 . − .
901 7 .
230 0 . . + . − . . + . − . b . ± .
008 138 . . ± .
019 0 . − .
444 9 .
190 2 . . + . − . . + . − . b . ± .
027 169 . . ± .
034 0 . − .
850 9 .
550 1 . . + . − . . + . − .
10 LARS9 b . ± .
017 208 . . ± .
027 0 . − .
425 9 .
170 0 . . + . − . . + . − .
11 LARS12 b . ± .
004 473 . . ± .
004 0 . − .
021 8 .
580 5 . . + . − . . + . − .
12 LARS13 b . ± .
003 701 . . ± .
003 0 . − .
333 9 .
250 4 . . + . − . . + . − .
13 NGC4631 c .
11 6 . .
23 0 . − .
18 6 .
91 0 . . + . − . . + . − .
14 NGC3627 c .
68 9 . .
40 1 . − .
85 6 .
46 0 . . + . − . . + . − .
15 NGC2146 c .
93 12 . .
17 0 . − .
03 8 .
28 0 . . + . − . . + . − .
16 NGC3938 c .
41 17 . .
72 2 . − .
98 6 .
80 0 . . + . − . . + . − .
17 M83 c .
78 7 . .
93 1 . − .
87 7 .
37 0 . . + . − . . + . − .
18 M82 c .
00 2 . .
57 2 . − .
56 7 .
38 5 . . + . − . . + . − . Table 1.
Properties of galaxies included in our benchmark local sample. For the data without specified uncertainty, we consider a relative error as a conservative choice.
References : a (Nilson 1973; De Looze et al. 2014; Engelbracht et al. 2008; Cormier et al. 2019), b (Puschnig et al. 2020), and c (Fernandez-Ontiveros et al. 2016; Leroy et al. 2008; Cald´u-Primo & Cruz-Gonz´alez 2008; Jarrett et al. 2003;Groves et al. 2015). where we consider ν = GHz and ν = (4285 , GHz, corresponding to λ = (70 , µ m . This method basedon the continuum fluxes ratio is equivalent to the single tem-perature grey-body SED fitting , hence the obtained dusttemperature is indeed T d , SED .To produce T d , CII we consider a flat distribution for theburstiness parameter . < ∼ κ s < ∼ . , where the range is de-rived from gas masses and SFR measurements for the singlesources (see Tab. 1). The temperatures comparison is shownin Fig. 2. We find that T d , CII and T d , SED are consistent withina uncertainty in ∼ of the sources (precisely, allbut galaxies ID11, ID18). The two discrepant sources arethe most bursty ones in the sample, with an inferred κ s ∼ .Considering κ s ∼ in these two cases (rather than the afore-mentioned flat distribution) would allow us to correctly re-cover T d , SED within uncertainty. Nevertheless, we havepreferred to consider a flat distribution for κ s to be moreconsistent with a general application at high- z , where thisparameter is almost always unconstrained. We select these frequencies to avoid PAH contaminationpresent at wavelengths < ∼ µ m . For ID10 we take ν = ( µ m ) since observations at the above mentioned frequenciesare not available. For all the sources where multiple continuum observations areavailable we also computed the full SED fitting. We find T d , SED values fully consistent with that obtained from continuum fluxesratio.
We now apply our method to high- z galaxies. We have col-lected a small (3 galaxies) sample for which the propertiesb)-f) are measured. The high- z sample contains: • SPT 0418-47 (Weiß et al. 2013; Strandet et al. 2016):a strongly lensed Dusty Star-Forming Galaxy (DSFG) atredshift z = . ; • B1465666 (i.e. the “big three dragons”, Hashimoto et al.2019): a Lyman Break Galaxy (LBG) at z = . • MACS0416-Y1 (Tamura et al. 2019; Bakx et al. 2020):an LBG at z ∼ . The properties of these galaxies are summarized in Tab. 2.These sources are UV-selected, highly star forming (yet notextreme as SFR < ∼
500 M (cid:12) / yr ), do not host AGN, and arepresumably main sequence high- z galaxies.For all these sources the parameter y = r CII / r (cid:63) has beenestimated (Rizzo et al. 2020; Hashimoto et al. 2019; Bakxet al. 2020). Moreover, Rizzo et al. (2020) provided a con-straint on Σ gas for SPT 0418-47. Hence, we can also derivethe burstiness parameter for this galaxy finding κ s ∼ . Dueto the uncertainty in the gas mass derivation, in the compu-tation we consider a Gaussian distribution centred aroundthis value with a σ ∼ (i.e. we allow for values in the range < ∼ κ s < ∼ , consistent with previous works e.g. Vallini et al.2020).For the remaining two galaxies κ s is unknown, hence weneed to define a broader distribution of possible values forthis parameter. High redshift UV selected sources are strongUV emitters by construction, highly star forming, and con-sequently they are expected to be starburst κ s > . Bothlocally and at intermediate redshift, values up to κ s (cid:39) have been observed in such galaxies (see e.g. Daddi et al. MNRAS000
500 M (cid:12) / yr ), do not host AGN, and arepresumably main sequence high- z galaxies.For all these sources the parameter y = r CII / r (cid:63) has beenestimated (Rizzo et al. 2020; Hashimoto et al. 2019; Bakxet al. 2020). Moreover, Rizzo et al. (2020) provided a con-straint on Σ gas for SPT 0418-47. Hence, we can also derivethe burstiness parameter for this galaxy finding κ s ∼ . Dueto the uncertainty in the gas mass derivation, in the compu-tation we consider a Gaussian distribution centred aroundthis value with a σ ∼ (i.e. we allow for values in the range < ∼ κ s < ∼ , consistent with previous works e.g. Vallini et al.2020).For the remaining two galaxies κ s is unknown, hence weneed to define a broader distribution of possible values forthis parameter. High redshift UV selected sources are strongUV emitters by construction, highly star forming, and con-sequently they are expected to be starburst κ s > . Bothlocally and at intermediate redshift, values up to κ s (cid:39) have been observed in such galaxies (see e.g. Daddi et al. MNRAS000 , 1–14 (0000) ust temperature in high- z galaxies Galaxy z F ν Z log Σ SFR L CII κ s y = r CII / r ∗ M (cid:63) [mJy] [ Z (cid:12) ] [M (cid:12) yr − kpc − ] [10 L (cid:12) ] [10 M (cid:12) ] SPT0418-47 a .
23 1 . ± .
25 0 . − .
30 1 .
72 19 . ± . . . ± . B1465666 b .
15 0 . ± .
03 0 . ± .
30 1 .
32 11 . ± . − . ± . . MACS0416-Y1 c . < .
02 0 . ± .
16 1 .
16 1 . ± . − . ± . . Table 2.
Properties of our high- z template sample of galaxies. We underline that for the data where the uncertainty is not given we considera relative error which is a conservative choice given the other available data. References : a Bothwell et al. (2017); De Breuck et al.(2019); Reuter et al. (2020); Rizzo et al. (2020). Here we show the intrinsic values, which are obtained by dividing by the magnificationfactor of the source µ = . (De Breuck et al. 2019); b Hashimoto et al. (2019), and c Bakx et al. (2020). z = . found κ s ∼ , applying the[C II ]-emission model given in Ferrara et al. (2019). Applyingthe same method to B1465666 and MACS0416-Y1 Vallini inprep. finds very large values < ∼ k s < ∼ . Hence, we conser-vatively choose a random uniform distribution in the range < κ s < for these two sources.Using eq. 7 we compute the coefficient α CII , hz . We find, α CII , hz = + − for SPT 0418-47, which is consistent with therecent estimate by Rizzo et al. (2020) of M gas / L CII ∼ α CII , hz = ± . We derive α CII , hz = + − for B14-65666, and α CII , hz = + − for MACS0416-Y1. We note that these values are lower thanthe average α CII values found locally ( z ∼ , see e.g. Fig. 1).This might indicate a trend of less efficient [C II ] emissionper unit gas mass at higher redshift.We now compare the T d , CII estimated by our model withthe T d , SED from the SEDs. We summarise our findings andcompare with literature data in Tab. 3. For SPT 0418-47we derive T d , CII = + − K that is consistent, within the error,with the T d , SED from the SED fitting by Strandet et al. (2016)( T d , SED = ± ). Recently Reuter et al. (2020) derived aslightly higher dust temperature T d , SED = ±
11 K for thissource, which is still consistent with our result . Althoughthe uncertainty of our T d , CII is larger than the one of T d , SED this is somewhat expected. The SED for SPT 0418-47 is wellconstrained, featuring data points on both sides of the FIRspectrum. On the other hand, the metallicity of the sourceis very uncertain, . ≤ Z / Z (cid:12) ≤ . and this affects directlythe error on our T d , CII .For B14-65666 we find T d , CII = + − K . This value is con-sistent with T d , SED = −
61 K which is inferred considering . < β < . (Hashimoto et al. 2019). For MACS0416-Y1(Tamura et al. 2019; Bakx et al. 2020) we consider the up-per limit on F ν recently derived by Bakx et al. (2020). In-terestingly T d , CII in eq. 12 decreases with f ∝ F ν . Hence, forthis galaxy we provide an upper limit for the [C II ] deriveddust temperature: T d , CII ≤ + − K . Very hot dust tempera-tures T d , CII >
120 K are excluded thanks to the condition on Reuter et al. (2020) left λ (cid:63) = µ m as an additional freeparameter in the SED fitting (see also Spilker et al. 2016, for adetailed discussion), which resulted in a larger ( × ) uncertaintyalongside a raise in the dust temperature. The fewer FIR datacurrently available at very high- z do not allow for the applicationof a similar fitting procedure on a large scale. Hence, at the currentstage a simpler grey-body (as in Strandet et al. 2016) with littlevariation in the dust properties is uniformly applied, leading topretty consistent T d , SED derivations for different sources.
SFR
FIR (see Sec. 2.3.1) . This result is particularly relevantas Bakx et al. (2020) obtained only a lower limit for the dusttemperature T d , SED > ∼
80 K . By combining the two results wecan constrain the dust temperature of MACS0416-Y1 in therange T d , SED ∼ −
98 K .In conclusion, with our method, we can provide dusttemperature estimations comparably accurate as that ob-tained from the traditional SED fitting with multiple bandsdata out to z = . . This is very encouraging, as for thesingle high- z sources targeted by large programs, only sin-gle band measurements are generally available. Hence, com-monly used SED fitting is not applicable without some un-derlying restrictive assumption on T d , SED . Our method canbe used in these cases to improve the accuracy of the inter-pretation of FIR observations, and derive dust and galaxiesproperties.
Our method also provides reliable estimates of the dustmasses of distant galaxies. Once α CII has been determined,it is straightforward to derive M d using eq. 10. As alreadydiscussed, we discard the solutions for which M d > M d , max ,which is computed in eq. 15.For SPT0418-47 we find M d = . + . − . × M (cid:12) , avalue pretty consistent with that obtained from SED fit-ting M d = . + . − . × M (cid:12) . For B14-65666 we find M d = . + . − . × M (cid:12) . This is also consistent with the result ob-tained from SED fitting by Hashimoto et al. (2019). Theyfind . < M d / M (cid:12) < . with . ≤ β ≤ . In the caseof MACS0416-Y1 we find M d = . + . − . × M (cid:12) . Thisis consistent with the result obtained from SED fitting byBakx et al. (2020). They find M d = . − . × M (cid:12) for
70 K < T d , SED <
130 K , and β = .We also compute the dust yield per SN, y d , requiredto produce the above dust masses, which are consistentwith previous estimates in the literature. We use the for-mula in Sommovigo et al. (2020): y d = M d / M (cid:63) ν SN , where ν SN = (53 M (cid:12) ) − is the number of SNe per solar mass of starsformed (Ferrara & Tolstoy 2000). For all the three sources itis y d < ∼ (cid:12) (see Tab. 3 for the value of y d in each galaxy), i.e. Without the condition on
SFR
FIR dust temperatures as large as T d , CII > ∼
130 K would be reached. In part, this is a consequence ofthe very large uncertainty ( ∼ ) on the already low metallicityof this galaxy ( Z = . (cid:12) ). Indeed for a fixed flux F ν , T d , CII diverges as the metallicity Z → as this is equivalent to M d → ,see eq. 10.MNRAS , 1–14 (0000) Sommovigo et al.
Figure 3.
Recovered distribution of T d , CII as a function of M d for the galaxies in our high- z template sample (same order as in Tab. 2).The contours show the (16 , , percentiles of the distribution. The median value is represented by the purple square (alongside its 16and 84 percentiles marked by the error bars). We also show the temperature (right) and mass (top) PDFs. The upper limit on the dustmass, computed through eq. 15, is shown by the vertical black dot-dashed line in each panel. MNRAS000
Recovered distribution of T d , CII as a function of M d for the galaxies in our high- z template sample (same order as in Tab. 2).The contours show the (16 , , percentiles of the distribution. The median value is represented by the purple square (alongside its 16and 84 percentiles marked by the error bars). We also show the temperature (right) and mass (top) PDFs. The upper limit on the dustmass, computed through eq. 15, is shown by the vertical black dot-dashed line in each panel. MNRAS000 , 1–14 (0000) ust temperature in high- z galaxies Galaxy α CII , hz T d , SED T d , CII M d y d [K] [K] [10 M (cid:12) ] [M (cid:12) / SN]
SPT0418-47 a ± − − + − − This work + − − + − + − . ± . B1465666 b − − − . − . − This work + − − + − . + . − . . + . − . MACS0416Y1 c − ≥ − . − . − This work + − − ≤ + − . + . − . . ± . Table 3.
Comparison between predicted (“This work”) and pub-lished properties of galaxies in our high- z sample. We underlinethat the dust temperature estimates taken from the literature areobtained through SED fitting, while our predictions correspondto T d , CII . References : a Strandet et al. (2016); Bothwell et al. (2017);De Breuck et al. (2019); Reuter et al. (2020); Rizzo et al. (2020).Here we show the intrinsic values, which are obtained by divid-ing by the magnification factor of the source µ = . (De Breucket al. 2019); b Hashimoto et al. (2019), and c Bakx et al. (2020). within the allowed range given in the latest SNe dust pro-duction studies by Le´sniewska & Micha(cid:32)lowski (2019). Theyfind that up to y d ≤ . (cid:12) per SN can be produced, wherethe exact value depends on the amount of dust which isdestroyed during the explosion ( . (cid:12) corresponds to theextreme case of no dust destruction).The presence of warmer dust ( T d , SED > ∼
50 K ) in thesehigh- z sources alleviates the large dust mass requirementsset by the observed FIR luminosity. This is particularly rel-evant in the context of early galaxies. Allowing for lowerdust masses prevents from invoking super efficient dust pro-duction by stellar sources, which is difficult to reconcile withboth data and theoretical models (for a detailed discussionsee e.g. Sommovigo et al. 2020). Besides providing a reliable determination of the dust tem-perature, our method offers a physical interpretation of theZanella et al. (2018) relation. To show this, we parallel theanalysis in Sec. 2.2. Here we substitute the KS relationwith the following expression linking the star formation andmolecular gas surface density Σ H (Krumholz 2015) : Σ SFR = − Σ H t depl . (17)where t depl is the depletion time. Combining eq. 17 with theDL relation and the definition of the molecular conversionfactor, α CII , mol = Σ H / Σ CII , we find α CII , mol = t depl . × − Σ − . . (18) We adopt the standard units used for these quantities: Σ SFR [M (cid:12) kpc − yr − ] , t depl [Gyr] , and in eq. 18, α CII , mol [M (cid:12) / L (cid:12) ] . The dependence of α CII , mol on Σ SFR is extremely weak, in con-trast with α CII ∝ Σ − . S FR (total gas conversion coefficient , seeSec. 2.2). We can understand this result in physical terms asboth H and [C II ] emission trace closely ongoing star forma-tion. Since both Σ H and Σ CII scale almost linearly with Σ SFR ,their ratio is virtually independent of this quantity. Instead,the total gas reservoir is less sensitive to star formation (seeeq. 3). Therefore in the ratio Σ gas / Σ CII the dependence on Σ SFR does not cancel out.Most recent results by Walter et al. (2020) suggest that t depl is nearly constant above redshift z > , and then in-creases slightly from t depl ∼ . at z ∼ , to t depl ∼ . at z = . Substituting these values in eq. 18, we find α CII , mol = (12 − Σ − . . This result is compatible with themeasurement of M H / L CII = + − M (cid:12) / L (cid:12) derived by Zanellaet al. (2018) in a sample of galaxies at z ∼ − . Recently,Dessauges-Zavadsky et al. (2020) found this M H / L CII ratioto hold also in the [C II ]-detected galaxies at z ∼ − targetedby the ALPINE survey, albeit with some uncertainties. .On average, previous works indicated longer depletiontimes t depl ∼ [0 . ,
2] Gyr both in local and high- z galaxies( z ∼ − , see e.g. Bigiel et al. 2008; Genzel et al. 2010; Daddiet al. 2010; Leroy et al. 2013; Sargent et al. 2014; Genzelet al. 2015; B´ethermin et al. 2015; Dessauges-Zavadsky et al.2015; Schinnerer et al. 2016; Scoville et al. 2017; Saintongeet al. 2017; Dessauges-Zavadsky et al. 2020). Nevertheless,the observed scatter in t depl is within measurement errors byZanella et al. 2018. The variation of α CII , mol is significantlysmaller than that of α CII . Already within our limited sampleof galaxies, α CII varies by nearly two orders of magnitudedue to its strong dependence on Σ SFR and κ s (see Fig. 1). Armed with the expressions for α CII (eq. 5) and α CII , mol (eq.18) we intend to study the redshift evolution of the ratio α CII , mol /α CII = Σ H / Σ gas . To this aim, since α CII ∝ Σ − . , weneed to provide a qualitative prescription for the redshiftevolution of the average ¯ Σ SFR ( z ) in normal galaxies.We consider the cosmic SFR comoving density, ψ , de-rived by Madau & Dickinson (2014) in the range z = . − .We combine ψ with the evolution of the effective radius, r (cid:63) ≈ r e = . × (1 + z ) . kpc , derived by Shibuya et al. (2015)for a HST sample of ∼ , galaxies at z = − to obtain ¯ Σ SFR ( z ) = ψ/π r , and the corresponding expression for α CII ( z ) from eq. 5.For simplicity, in computing α CII ( z ) we consider κ s = as on average we expect most local and low- z galaxies to lieon the KS-relation . In parallel to the result shown in Sec. The exponent − . is an average value between the − . and − . found in eq. 5, 7. More precisely, Dessauges-Zavadsky et al. (2020) find a goodagreement between molecular gas masses derived from [C II ] lu-minosities (using the relation by Zanella et al. 2018), dynamicalmasses, and rest-frame µ m luminosities (extrapolated fromthe rest-frame µ m continuum). If κ s > the Σ H / Σ gas curve is shifted upwards, as we are re-ducing Σ gas ∝ α CII ∝ κ − / s , without affecting Σ H . Hence, at higherredshift deviations are expected to occur due to the burstiness ofgalaxies.MNRAS , 1–14 (0000) Sommovigo et al.
Figure 4.
Redshift evolution of the molecular gas fraction f H ( z ) = Σ H / Σ gas . The red line represents our fiducial estimate. The black lineshows the ρ H / ( ρ H + ρ HI ) trend observed by Walter et al. 2020; the vertical grey dashed line refers to the highest redshift considered intheir analysis. For sake of the comparison here we consider the same t depl ( z ) as in Walter et al. 2020.
5, we write α CII , mol ( z ) = (12 −
21) ¯ Σ − . . We can then computethe ratio: f H ( z ) ≡ Σ H / Σ gas = α CII , mol /α CII ∼ (cid:32) ¯ Σ SFR M (cid:12) kpc − yr − (cid:33) . (19)The redshift evolution of the molecular fraction in galax-ies has been experimentally determined by Walter et al.(2020) from observations of molecular, ρ H , and atomic, ρ HI , gas densities at z < ∼ . In Fig. 4 we show the compari-son of f H ( z ) with the observed redshift evolution of of theempirical ρ H / ( ρ HI + ρ H ) ratio.The two approaches yield a pretty consistent evolutiontrend, albeit they are both affected by large uncertainties.We find that on average f H ( z ) increases by a factor of ∼ from z = . to z = , in agreement with the trend foundby Walter et al. (2020) ( ∼ . − . ). However, at z > the two trends might be different, as we predict a possi-ble further increase in f H ( z ) . This can be due to (a) a fur-ther increase in the M H / M gas , and/or (b) an increase in theratio r gas / r H ratio. The first case seems to be disfavouredby theoretical studies (see e.g. Dav´e et al. 2017), as boththe H and H I evolution become steeper with redshift atfixed stellar mass. The second possibility is instead suggestedby recent works showing the presence of [C II ] emission at The H I density is obtained by combining measurements ofH I emission in the local universe (see e.g. Zwaan et al. 2005)with quasar absorption lines at higher z (see e.g. Prochaska &Wolfe 2009); ρ H ( z ) determination is based on CO and FIR dustcontinuum data (e.g. reviews by Carilli & Walter 2013; Tacconiet al. 2020; P´eroux & Howk 2020; Hodge & da Cunha 2020). high- z around × . − times more extended than the stel-lar (and possibly molecular) mass (see e.g. Carniani et al.2017, 2018a; Fujimoto et al. 2019, 2020; Ginolfi et al. 2020;Carniani et al. 2020). Clarifying this uncertainty is crucialas the assumption that f H ≈ at high- z is widely used toderive molecular gas masses from dynamical (Daddi et al.2010; Genzel et al. 2010; Dessauges-Zavadsky et al. 2020)and dust (Scoville et al. 2016; Dessauges-Zavadsky et al.2020) masses. We have proposed a novel method to derive the dust tem-perature in galaxies, based on the combination of continuumand [C II ] line emission measurements, which breaks the SEDfitting degeneracy between dust mass and temperature. Themethod allows constraining T d from a single band obser-vation at (rest-frame). We conveniently provideanalytic expressions in eq. 12 for a direct application.Besides, the same method offers a physical explanationfor the empirical relation found by Zanella et al. (2018) be-tween [C II ] luminosity and molecular gas. We also derive therelation between total gas surface density and [C II ] surfacebrightness, Σ gas = α CII Σ CII . By combining such relations wepredict the redshift evolution of the molecular gas fractiondefined here as Σ H / Σ gas .We summarise our main findings below: • Dust temperature from [C II ] data at high- z : using a sin-gle band observation, with our method, we can constrainthe dust temperature as well as with the commonly used MNRAS000
21) ¯ Σ − . . We can then computethe ratio: f H ( z ) ≡ Σ H / Σ gas = α CII , mol /α CII ∼ (cid:32) ¯ Σ SFR M (cid:12) kpc − yr − (cid:33) . (19)The redshift evolution of the molecular fraction in galax-ies has been experimentally determined by Walter et al.(2020) from observations of molecular, ρ H , and atomic, ρ HI , gas densities at z < ∼ . In Fig. 4 we show the compari-son of f H ( z ) with the observed redshift evolution of of theempirical ρ H / ( ρ HI + ρ H ) ratio.The two approaches yield a pretty consistent evolutiontrend, albeit they are both affected by large uncertainties.We find that on average f H ( z ) increases by a factor of ∼ from z = . to z = , in agreement with the trend foundby Walter et al. (2020) ( ∼ . − . ). However, at z > the two trends might be different, as we predict a possi-ble further increase in f H ( z ) . This can be due to (a) a fur-ther increase in the M H / M gas , and/or (b) an increase in theratio r gas / r H ratio. The first case seems to be disfavouredby theoretical studies (see e.g. Dav´e et al. 2017), as boththe H and H I evolution become steeper with redshift atfixed stellar mass. The second possibility is instead suggestedby recent works showing the presence of [C II ] emission at The H I density is obtained by combining measurements ofH I emission in the local universe (see e.g. Zwaan et al. 2005)with quasar absorption lines at higher z (see e.g. Prochaska &Wolfe 2009); ρ H ( z ) determination is based on CO and FIR dustcontinuum data (e.g. reviews by Carilli & Walter 2013; Tacconiet al. 2020; P´eroux & Howk 2020; Hodge & da Cunha 2020). high- z around × . − times more extended than the stel-lar (and possibly molecular) mass (see e.g. Carniani et al.2017, 2018a; Fujimoto et al. 2019, 2020; Ginolfi et al. 2020;Carniani et al. 2020). Clarifying this uncertainty is crucialas the assumption that f H ≈ at high- z is widely used toderive molecular gas masses from dynamical (Daddi et al.2010; Genzel et al. 2010; Dessauges-Zavadsky et al. 2020)and dust (Scoville et al. 2016; Dessauges-Zavadsky et al.2020) masses. We have proposed a novel method to derive the dust tem-perature in galaxies, based on the combination of continuumand [C II ] line emission measurements, which breaks the SEDfitting degeneracy between dust mass and temperature. Themethod allows constraining T d from a single band obser-vation at (rest-frame). We conveniently provideanalytic expressions in eq. 12 for a direct application.Besides, the same method offers a physical explanationfor the empirical relation found by Zanella et al. (2018) be-tween [C II ] luminosity and molecular gas. We also derive therelation between total gas surface density and [C II ] surfacebrightness, Σ gas = α CII Σ CII . By combining such relations wepredict the redshift evolution of the molecular gas fractiondefined here as Σ H / Σ gas .We summarise our main findings below: • Dust temperature from [C II ] data at high- z : using a sin-gle band observation, with our method, we can constrainthe dust temperature as well as with the commonly used MNRAS000 , 1–14 (0000) ust temperature in high- z galaxies SED fitting in multiple bands. We recover dust tempera-tures consistent with literature data (within σ ) out to red-shift z = . ; • Gas-to-[C II ] luminosity relation : the total gas conversioncoefficient α CII strongly depends on the SFR surface density( ∼ Σ / ) and the burstiness of galaxies (see eq. 5). Whencomputing the analogous conversion factor for the molecu-lar gas α CII , mol , we find that the dependence on Σ SFR nearlycancels out, hence α CII , mol ≈ const. (see eq. 18); • Molecular gas fraction : we find that f H ( z ) on averageincreases with z by a factor ≈ from z = . to z = . Thisis consistent with the trend observed by Walter et al. (2020).We predict a possible further increase at z > . This couldbe caused by a rise of the H content, and/or a change inthe relative extension of H and H I gas.Assuming a dust temperature, as usually done in high- z galaxy observations analysis, introduces large uncertaintieson the derived dust masses, infrared luminosities, and starformation rates (see also e.g. Sommovigo et al. (2020) for adetailed discussion). Our method can improve the reliabil-ity of the interpretation of [C II ] and continuum observationsfrom ALMA and NOEMA. This is particularly relevant inthe context of recent ALMA large programs targeting [C II ]emitters at high- z , such as ALPINE (Le F`evre et al. 2019;Schaerer et al. 2020b; Bethermin et al. 2020; Schaerer et al.2020a), REBELS (PI: Bouwens), and others. With futureinstruments such as JWST, providing more accurate metal-licity measurements, it will be possible to improve currentestimates of the dust-to-gas ratios at high- z . This will fur-ther enhance the precision of our dust temperature determi-nations. ACKNOWLEDGEMENTS
LS, AF, SC, AP, LV acknowledge support from the ERCAdvanced Grant INTERSTELLAR H2020/740120 (PI: Fer-rara). Any dissemination of results must indicate that it re-flects only the author’s view and that the Commission isnot responsible for any use that may be made of the infor-mation it contains. Partial support from the Carl Friedrichvon Siemens-Forschungspreis der Alexander von Humboldt-Stiftung Research Award is kindly acknowledged (AF). Wethank R. Kennicutt, T. D´ıaz-Santos, and C. De Breuck forthe useful discussions, and for the help in retrieving the datafrom their surveys, respectively KINGFISH, GOALS, andSPT.
Part of the data underlying this article were accessed fromthe computational resources available to the CosmologyGroup at Scuola Nor- male Superiore, Pisa (IT). The deriveddata generated in this research will be shared on reasonablerequest to the corresponding author.
APPENDIX A: HINTS FROM SIMULATIONS
The method developed in this paper can reliably determinethe dust temperature in a galaxy for which only a single simultaneous observation for the [C II ] line and underly-ing continuum is available. This value corresponds to thecanonical temperature, T d , SED , that one would normally de-fine from fitting the SED with a single temperature greybody formula. As already mentioned, T d , SED does not nec-essarily correspond to the physical dust temperature whichinstead is distributed according to a given Probability Dis-tribution Function (PDF, see e.g. Behrens et al. 2018; Som-movigo et al. 2020, Di Mascia et al. in prep.). Hence, it isinstructive to understand the relation among T d , SED and thePDF properties.In the context of theoretical studies, i.e. both analyt-ical models and simulations, the dust temperature PDF isactually available. Various weighting procedures can be ap-plied to this PDF, and then the average can be comparedto the observational results. In particular, the most com-monly adopted are the mass- (M-weighted) and luminosity-weighted (L-weighted; L ∝ M d T + β d ). The M-weighted tem-perature traces the most abundant cold temperature com-ponent; instead, the L-weighted is biased towards hotter butless massive dust component present in star forming regionswhere it is efficiently heated by the UV emission from new-born stars, see e.g. Behrens et al. 2018; Sommovigo et al.2020, Di Mascia et al. in prep.). Neither of them is traced by T d , SED . Indeed cold dust nearly in equilibrium with the CMBis not observable in emission; hot dust (if present) emitsmainly in the MIR, where it is largely responsible for dis-tortions of the single temperature grey body (see e.g. Casey2012; Casey et al. 2018).At high- z such distortion is not observable, as only thelong-wavelength part of the SED spectra is currently ac-cessible with ALMA (bands 6,7, and 8). However, locally,where the whole SED is well sampled, it has been observedand studied by e.g. Casey (2012) within z ∼ sub-millimetregalaxies. In light of these considerations, the most appropri-ate and clean choice when comparing theoretical results withobservations is to perform a single temperature grey-body fitto the simulated SEDs in order to consistently obtain T d , SED .In Fig. A1 we show the result of applying this procedureto the SED of the simulated z ∼ . galaxy Zinnia (a.k.a.serra05:s46:h0643) from the SERRA simulation suite.Full details on SERRA simulations are given in Pallot-tini in prep. and can be summarized as follows. Simulationszoom in on the evolution of M (cid:63) ∼ M (cid:12) galaxies from z = to z = with a mass (spatial) resolution of theorder of M (cid:12) ( pc at z = ) . [C II ] emission is ob-tained by post-processing using grids of CLOUDY (Ferlandet al. 2017) models accounting for the internal structure ofmolecular clouds (Vallini et al. 2017; Pallottini et al. 2019).Additionally, SKIRT (Baes & Camps 2015; Camps & Baes2015) is used to obtain UV and dust continuum emission,with a setup similar to Behrens et al. (2018). The simulation adopts a multi-group radiative transfer ver-sion of the hydrodynamical code RAMSES (Teyssier et al. 2013;Rosdahl et al. 2013) that includes thermochemical evolution viaKROME (Grassi et al. 2014, Bovino et al. see 2014; Pallottiniet al. see 2017b, for the network and included processes), whichis coupled to the evolution of radiation (Pallottini et al. 2019;Decataldo et al. 2020). Stellar feedback includes SN explosions,OB/AGB winds, and both in the thermal and turbulent form (seePallottini et al. 2017a, for details).MNRAS , 1–14 (0000) Sommovigo et al.
Figure A1.
SED for the simulated galaxy Zinnia (serra05:s46:h0643, black solid line) extracted from the SERRA simulation suite. Thedotted-dashed lines show the curves obtained through a single temperature grey body fitting of the SED, with the two following methods:(a) a canonical SED fitting performed considering the three red “data points” in ALMA bands , , (red line, T , SED ); (b) same as (a)but considering the full (i.e. MIR and FIR) galaxy SED (blue line, T d ); (c) the method presented in this work; it uses a single continuumobservation and the [C II ] emission (green line, T p d , CII ). The shaded regions mark the ALMA bands to . Subplot : comparison amongthe above dust temperatures values, the luminosity- (orange), and mass-weighted (blue) dust temperature PDFs derived for Zinnia. ThePDFs mean values ( (cid:104) T d (cid:105) L and (cid:104) T d (cid:105) M ) are indicated by dashed lines. See text for a detailed discussion.Galaxy z F ν Z log Σ SFR L CII κ s y = r CII / r (cid:63) M (cid:63) [ µ Jy] [ Z (cid:12) ] [M (cid:12) yr − kpc − ] [10 L (cid:12) ] [10 M (cid:12) ] Zinnia . .
81 0 .
07 2 .
56 2 .
05 4 .
29 1 . (cid:63) . Table A1.
Properties of our high- z simulated galaxy Zinnia (a.k.a. serra05:s46:h0643). We note that the parameter y = r CII / r (cid:63) = . isselected by definition, i.e. we only consider the emission coming from the central ∼ . region.Galaxy α CII , hz T d , SED M d y d [K] [10 M (cid:12) ] [10 − M (cid:12) / SN] serra0643 . ± . ± . - This work . ± .
07 63 . ± . . ± .
03 4 . ± . Table A2.
Comparison between the properties predicted with ourmethod (“This work”), and derived through a single temperaturegrey body fitting of the simulated flux in ALMA band 6,7,8 ofgalaxy serra05:s46:h0643. In the SED fitting procedure, we keepthe dust emissivity index fixed at β = . , as in our analyticalmethod, and consider a uncertainty on all the galaxy proper-ties derived from the simulation and listed in Tab. A1. We under-line that our predictions correspond to T d , CII . The main properties of Zinnia are summarised in Tab.A1. We proceed to compute and compare the following tem-peratures: • T d : dust temperature obtained from fitting the full (i.e.MIR and FIR) galaxy SED with a single-T grey-body; • T , SED : dust temperature obtained from fitting thegalaxy SED at the frequencies corresponding to ALMA band , , with a single-T grey-body; • T p d , CII : dust temperature obtained with our method com-bining a single continuum data at (rest-frame)with the [C II ] line emission data, as described in Sec. 4; • (cid:104) T d (cid:105) M : M-weighted dust temperature; • (cid:104) T d (cid:105) L : L-weighted dust temperature.We underline that in all these computations, as in the restof the paper, we keep the dust emissivity index fixed to β d = . . Such an assumption is reasonable as this is close to theemissivity index retrieved from the simulation ( β d = . − , MNRAS000
Comparison between the properties predicted with ourmethod (“This work”), and derived through a single temperaturegrey body fitting of the simulated flux in ALMA band 6,7,8 ofgalaxy serra05:s46:h0643. In the SED fitting procedure, we keepthe dust emissivity index fixed at β = . , as in our analyticalmethod, and consider a uncertainty on all the galaxy proper-ties derived from the simulation and listed in Tab. A1. We under-line that our predictions correspond to T d , CII . The main properties of Zinnia are summarised in Tab.A1. We proceed to compute and compare the following tem-peratures: • T d : dust temperature obtained from fitting the full (i.e.MIR and FIR) galaxy SED with a single-T grey-body; • T , SED : dust temperature obtained from fitting thegalaxy SED at the frequencies corresponding to ALMA band , , with a single-T grey-body; • T p d , CII : dust temperature obtained with our method com-bining a single continuum data at (rest-frame)with the [C II ] line emission data, as described in Sec. 4; • (cid:104) T d (cid:105) M : M-weighted dust temperature; • (cid:104) T d (cid:105) L : L-weighted dust temperature.We underline that in all these computations, as in the restof the paper, we keep the dust emissivity index fixed to β d = . . Such an assumption is reasonable as this is close to theemissivity index retrieved from the simulation ( β d = . − , MNRAS000 , 1–14 (0000) ust temperature in high- z galaxies see Behrens et al. (2018) for the Radiative Transfer details).Hence, the free parameters in the fitting procedure are thedust temperature and dust mass (
40 K ≤ T d ≤
200 K , and M (cid:12) ≤ M d ≤ M (cid:12) ).All these temperatures are compared in the subplot inthe upper left corner of Fig. A1. Our method gives a dusttemperature value T p d , CII = . ± . , consistent with theresult that one obtains with the usual SED-fitting techniqueusing three points corresponding to the available ALMAbands at this redshift, T , SED = ± . For this galaxy, thevalue of T , SED ∼ T p d , CII is also consistent with the M-weightedtemperature, (cid:104) T d (cid:105) M =
61 K . Instead both (cid:104) T d (cid:105) L = K and T d = K are larger than the previous values as they aremore sensitive to the small amount of dust with physicaltemperatures up to
150 K (see the L-weighted PDF in thesubplot of Fig. A1).This comparison shows that the single-T approximationoften used might lead to a misinterpretation of the physicalproperties of the galaxy depending directly on T d . Moreover,whenever theoretical studies and observations are compared,it is necessary to pay particular attention to the definitionof the dust temperature used and to the fitting procedure. We suggest that a uniform, meaningful comparison is bestperformed using either T , SED or, as we propose here, T p d , CII ,when only a single measurement is available.
It is very re-assuring that the two procedures yield essentially the sameresult. These quantities can be also easily derived from thesimulated spectrum, and readily compared with data.
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